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l^arbarli College libraro 

VmVGHT WITH INCOMK 
rmoM TM* flK<tpEsrT or 

HENRY LILLIE PIERCE, 

OF BOSTON. 



Under • vote of the PnaldCDt and Fellmn, 
October 24, 1898. 



TRANSFERRED 

TO 

HARVARD COLLEGE 
LIBRARY 




WORKS OF 
PROFESSOR FORRFST JONES 

PUBLISHED BY 

JOHN WILEY & SONS. 



Machine De8ls:n. Part I.— Kinematics 
of Machinery. 

Third Edition, Revised. vi+ 159 pages, 133 
figures. 8vo, cloth, $1.50. 

Machine De8ls:n. Part II. — Form, 

5tren8:th, and Proportions of Parts. 

Third Edition, Revised and Enlarged. ix -|- 
426 pages, 243 figures. 8vo, cloth, $3.00. 

I 



MACHINE DESIGN". 



Part II. 

FORM, STRENGTH, AND PROPORTIONS 
OF PARTS. 



FORREST R. JONES, 

Profeuor of Machine Design in Sibley College^ 
Cornell University. 



THIRD EDITION, REVJBED. 
PIBST THOUSAND. 



StEVf YORK : 

JOHN WILEY & SONS. 

Loitdon: chapman & HALL, Limited. 

1904. 



I ■ 



■ '' ' 'iRAjO'.> 

li . . .. K::0 I J 



OQpyHfffat, 1809, 1008, 

BT 

FORREST R. JONES. 



BOBKBT DRCTMMOND, PBIMTSR, MKW TOBK* 



PREFACE- 



The matter presented on the following pages is confined to snch 
subjects as the designer must deal with daily. Equations and 
formulas are put into such a form as to afford a ready means of 
application to problems under consideration. iNTumerical examples 
and data from practice illustrating principles are introduced wher- 
ever it seems that a clear understanding can be brought about in 
this way. The data thus presented have been gathered from 
numerous sources during the last fifteen years. That coming from 
modern practice is always given the preference, however, except 
where the older matter is undoubtedly the most valuable. When- 
ever possible, the results of practice or experiments as presented by 
some engineer or experimenter which fairly well represent the 
general experience along their lines are given in preference to 
abstract statements. This is believed to be the most satisfactory 
method, since it affords a means of studying all the facts incidental 
to the particular case, in addition to furnishing information fully as 
well as abstract statements. 

The publications of engineering societies and engineering 
journals have been freely consulted. Much courtesy has been 
shown by the representatives of engineering and manufacturing 
concerns, who have invariably been kind in answering inquiries, and 
many of them have furnished valuable data. Whenever possible, 
credit for such information is given in connection with the informa- 
tion itself, but much of it was of such a nature as to make it im- 
possible to separate it from general statements. 

The author desires to express his thanks to all those who have 
80 kindly treated his petitions for information. Of those not men- 

iii 



IV PREFACE. 

tioned in the body of the book Mr. Edw. L. Bateman of Fraser 
& Chalmers aud Mr. Peter Conner of the Oisholt Machine Co. 
have been especially kind, the former in giving information, and 
the latter in giving both information and his time in making 
sketches and drawings of many of the figures. 

The printer's proof of most of the matter closely related to the 
mechanics of engineering has been kindly read by Mr. E. R. 
Maurer, Professor of Applied Mechanics in the University of Wis- 
consin, who pointed out where several improvements might be 
made and ambiguities eliminated. By following his suggestions it 
is believed that the work has been materially improved. 

FOBBEST B. JONES. 
Madisoh, Wia.» January, 1899. 



PREFACE TO THIRD EDITION. 

Several new subjects have been included and a considerable 
number of examples introduced to illustrate the matter that appeared 
in the earlier editions. New data from leading engineering con- 
cerns have also been added. In all, the book is increased in size 
by about eighty pages of new matter. 

P. E. J. 

Ithaca, N. Y., November. 1908. 



NOTE. 

In the second impression of the Third Edition^ Tables of 
Dimensions and Capacities of Ball Bearings are added, and some 
changes made in the text. 

F. B. J. 

Ithaca, N. T., March, 1904. 



CONTENTS. 



CHAPTER I. 

Bearinos Aim Lubrication 1 

1. Introductory. 2. Planer and lathe ways or Vs. 8. Bearings 
for the ram or tool-carrier of a shaper. 4. Planer saddles, lathe tool- 
rests, and similar parts.' 5. Cross-head guides. 6. Relative length 
of sliding bearings, for rectilinear motion. 6.1. Method of lubri- 
cating cross-head. 6.2. Material for V's and other rectilinear bear- 
ings. 7. Journals, boxes, and journal-bearings. 8. Cylindrical 
Journal-bearing. 9. Self-aligning bearings. 10. Lubrication of 
journal-bearings. 11. Oil- and grease-lubrication for journal-bear- 
ings. Oil-pad lubrication. Nature of rubbing surfaces. 12. Special 
devices for journal lubrication with oil and grease. 18. Materials 
for journal-bearings. 14. Frictional resistance of journal-bearings. 
Coefficient of journal friction. 15. Effect of changing the propor- 
tions of journal-bearings. 16. Proportions of journal-bearings, and 
examples from practice. Problem tn the design of journal-bearings. 
17. Step-bearings. 18. Forms of step-bearings. Forced lubrication 
of step-bearings. 18.1. Button thrust-bearings. 19. Conical pivot- 
bearings. 20. The tractrix, curve of constant tangent, or Schiele's 
anti-friction curve. 21. Collar-bearings. 22. Cylindrical roller- 
bearings. 28. Conical roller-bearings. 28.1. Cylindrical roller 
thrust-bearings. 28.2. Ball bearings. 28.8. Ball journal-bearing 
with two-point contact. 28.4. Ball journal-bearing with three-point 
contact. 28.5. Ball journal-bearing with groove race. 28.6. Ball 
thrust-bearings with two-point contact. 28.7. Ball thrust-bearings 
with three-point contact. 28.8. Ball thrust-bearings with four-point 
contact. 28.9. Ball journal-bearings with four-point contact 28.10. 
Cup and cone two-point contact hub ball bearing. 28.11. Three- 
point contact hub ball bearing. 28.12. Four-point contact hub ball 
bearing. 24. Special forms of bearings. 

CHAPTER IL 

Bpub- aud Fbiotion-gbars 118 

26. Strength of spur-gear teeth, equations and diagrams. Numer- 
ical examples. 26. Methods of strengthening gears. Buttress and 



VI CONTENTS. 

shrouded teeth. 27. Short gear-teeth, proportions and working loads. 
28. Mortise gearing. 29. Kawhide, indurated vegetable fibre, etc., 
for gears. 30. Factor of safety for tooth gears. 81. Efficiency of 
spur gears. 32. Strength of bevel-gear teeth. Numerical examples. 
83. Efficiency of bevel gearing. 34. Frictiongears. 85. Cylindrical 
friction gears. Efficiency of friction gears. 86. Grooved frction 
gears. 37. Friction bevel gears. 88. Crown friction gears. 89. 
Double-cone, variable-speed friction gears. 

CHAPTER III. 

Belts and Ropes for Power Transmission 160 

40. Flat belts. 41. Equations for power transmission by flat belts. 
Numerical example. 41.1. Tandem belt drive. 42. Coefficient of 
friction and slip of leather belting. 48. Working strength of leather 
belting. 44. Velocity of leather belting. 45. Wear of leather belts. 
46. Weight and thickness of leather belts. 47. Rawhide, semi-raw- 
hide, rawhide with tanned leather face, and chrome-tanned belts. 48. 
Cotton belte. 49. Rubber belting. 50. Leather-link belts. 51. Effect 
of relative positions of pulleys. 52. Special system of flat -belt driv- 
ing. 58. Efficiency of flat belting. 53.1. Example showing the effect 
of varying the speed of belt. 58.2. Binder and guide pulleys. 58.3. 
Chain belts. 54. Ropee for power transmission. 55. Systems of rope 
driving. 56. Equations for ropes transmitting power. Numerical 
example. 57. Grooves for non-metal lie ropes. 58. Coefficient of 
friction of non-metallic ropes. 59. Working strength of non-metallic 
ropes. 60 Velocity of ropes for power transmission. 61. Wear and 
lubrication of non-metallic ropes. 62. Weight of hemp and cotton 
ropes. 63. Diameter of ropes for power transmission. 64. Effect of 
the relative position of pulleys upon ropes used for power transmis- 
sion. 65. Efficiency of rope belting. 66. Wire rope. Results of 
practice. Pulley grooves for wire rope. 

CHAPTER IV. 

Screws for Power Transmission Zl% 

^1» General discussion. 68. Relation between the turning moment 
and axial force in a square-thread pcrew. 69. Efficiency of a square- 
thread screw and collar. 70. Coefficient of friction for square -thread 
screws. 71. Problem in screw design. 72. Maximum stress in a 
screw. Example of application of formulas. 73. Angular- thread 
screws. 

CHAPTER V. 

Screw-gearing 280 

74. Common forms. 75. Worm and worm-wheel. 76. Equations 
for turning force and efficiency of worm and worm-wheel. 77. Tests 



CONTENTS. Vll 

PAOB 

of worm-wheel. 78. Screw gears. 79. Strength of screw-gear teeth. 
80. Equations for turning force and efficiency of screw gears. 81. 
Coefficient of friction of screw gears. 

CHAPTER VI. 

BCREW-FABTBNIKGS 847 

82. Forms of screw threads and proportions of bolt-heads and nuts. 
Set-screws. 83. Locking devices for nuts and screws. 84. Strength 
of screw-bolts. 85. Endurance of screw-bolts. 

CHAPTER vn. 

li^CHiiTB Krys, Pins, Forced and Shrinkage Fits 267 

86. Machine keys and pins. 87. Roller keys. 88. Eccentric keys 
or fastenings. 89. Shrinkage and forced fits. 90. Tension in and 
pressure against a thin ring fitted by shrinking or forcing. Numeri- 
cal example. 91. Shrinkage and forced fits for thick rings and heavy 
parts. 92. Allowance for shrinkage and forced fits. 

CHAPTER VIII. 

BSAFnNG, AND POSITIVE Shaft-couplings 282 

98. Notation for equations. 94. Torsional strength of round shafts. 
Numerical examples of solid and hollow shafts. 95. Twist of a shaft 
under torsional stress. Numerical example. 96. Bending strength 
of round shafting. 97. Lateral deflection of shafting on account of 
its own weight. 98. Shafts subjected to both torsion and bending. 
General case. 99. Solid round shaft subjected to more than one 
force. Numerical example. 100. Solid round crank-shafts and other 
shafts acted on by a single rotative force. Numerical example. 101. 
Hollow round shafts. 102. Hollow round shaft acted on by more 
than one force. 108. Hollow round shaft acted on by a single turn- 
ing force. 104. Experimental and determined value of the breaking 
tensile stress of round shafting subjected to bending and torsion. 
105. Practically determined formulas for round shafting. 106. Shafts 
of symmetrical sections other than round. 107. Rigid shaft -coup- 
lings, 108. Flexible shaft-couplings. 109. Positive clutch couplings. 

CHAPTER IX. 

FBtCnON-COTTPLINGB AND BRAKES 808 

110. General statements. 111. Cone friction couplings. 112. Mul- 
tiple-ring friction coupling. 113. 3Iaterinl and coefflcient of friction 
for friction couplings. 114. Strap brake. 115. Prony brake. 



VIU CONTENTS. 

CHAPTER X. 

Ply-whbblb and Pulleys 811 

116. Applications of fly-wheels, and equations for moment of inertia 
and kinetic energy. 117. Determination of moment of inertia and 
kinetic energy of a webbed wheel. Numerical example. 118. Prob- 
lem. To design a fly-wheel for a given moment of inertia, and ac- 
cording to a given form. 119. Problem. To design a fly-wheel which 
will furnish a given amount of energy for a given variation of speed. 
120. Moment of inertia of a fly-wheel with arms. 121. Stresses in 
fly-wheels with arms. 122. Numerical example of stress in, and en- 
largement of, a rotating ring due to centrifugal action. 123. Seo- 
tional-rim fly-wheels and pulleys. 124. Bursting tests of small cast- 
iron fly wheels by centrifugal action. 124.1. Effect of ribs on pulley 
rims. 124.2. Problem. Armature ring. 126. Hollow cast-iron arms 
with wrought'iron or steel tension rods, for fly-wheels and pulleys. 
126. Tangent arms for fly-wheels and pulleys. 127. Built-up plate 
fly-whetls. 128. Wire-wound fly-wheeL 129. Other special forms 
of pulleys. 180. Designs and proportions of fly-wheels and pulleys 
taken from practice. 

CHAPTER XI. 

OTLtNDERS, Tubing, Pipes and Pipe- couplings 847 

181. General. 132. Tension in a thin circular cylinder due to in- 
ternal pressure. Numerical example. 138. Cylinder with thick 
walls. 184. Bursting tests of cylinders and pipes. 185. Special forms 
of pipes. 186. Pipe-couplings and flanges. 187. Expansion coup- 
lings for pipes. 

CHAPTER XIL 

RiVBTED Joints 862 

188. Methods of making and forms of riveted Joints. 189. Single- 
riveted joint. 140. Rivets. 141. Pitch of rivets. 142. Efficiency 
of rivet^ joints. 148. Effects of shearing, punching, and drilling 
plates. 144. Faulty construction and grooving of riveted Joints, 
14& Examples of riveted joints taken from practice. 

CHAPTER XIIL 

FiuiuES OF Punching, Sheabtng, and Riveting Machines ... 891 

146. Punching and shearing machines. 147. Stresses in a section 

perpendicular to the motion of the pimch. 148. Numerical solution 

for a section perpendicular to the motion of the punch. 149. Section 

parallel to the motion of the punch. 150. Angular section of a 



CONTENTS. IX 

PAOX 

punch-frame. 151. (General form of a punch-frame. Iti8. Diiect- 
acting hydraulic riveter. 

CHAPTER XIV. 

fiELBonoN OF Materials 407 

153. General discussion and tables of the properties of materials 
most common to engineering. 

APPENDIX. 

A. Development of equations for an angolar-tluead soiew. B. 
Graphical determination of the moment of inertia of a plane area. 
Approximate method • 418 



FORM, STRENGTH, AND PROPOR- 
TIONS OF MACHINE PARTS. 



CHAPTER I. 
BEARINGS AND LUBRICATION. 

1. Introdnotory. — In machinery the name bearing, or bearing 
snrfEUse, is applied to a part which presses or bears against another 
and moires over it at the same time. 

If the surfaces are dry, or as left by the tool used to finish them, 
they have a strong tendency to abrade, or '^ cnt," each other when 
the materials commonly used for machine members are rubbed 
together. To prevent this catting, a lubricating substance, such 
as oil, grease, or graphite, is introduced between them. 

The necessity of thorough lubrication is often of such vital im- 
portance that the bearing most be designed with especial regard to 
securing a continuous application of the lubricant. 

BEARINGS FOR RECTILINEAR MOTION. 

2. Planer and lathe "ways" or "Vs."— The table or platen of 
a metal-working planer reciprocates on bearings which must give it 
an accurate rectilinear motion in order that the cutting tool may 
form a plane surface, or a carved surface whose elements are right 
lines. The V form of bearing, such as is shown in Fig. 1 at ^ and 
By is most coii[imonly used. In this style of bearing the weight of 
the table, together with the work upon it, is relied upon to hold 



2 FORM, STRENGTH, AND PKOPOKTIONS OF PARTS. 

the table in place. The catting tool freqaently exerts a side 
pressure tending to force the table off its bearings. The ability of 




the table to resist this pressure depends on the angle &*, Fig. 2, be- 
tween the sides of the V. It is plain that the smaller this angle the 




Fig. 2. 



greater will be the resistance of the table to being displaced. The 
smaller the angle, however, the greater the total pressure between 




Fig. 8. 



the bearing surfaces. This can be seen by the aid of Fig. 3, which 
represents the way on one side of the table when it is removed from 



BEARINGS AND LUBKICATIOX. 3 

the bed. The load which this V must support may be called P. 
The pressure against each side of the Y is normal to the bearing 
surfaces, friction being neglected. These normal pressures are 
represented by i\"and iVin the figure. The amount of iVand iV" 
is found by resolving P into two forces normal to the bearing sur- 
faces. This is done by taking AB parallel to P, and of such a 
length as will represent the magnitude of P according to any 
convenient scale of pounds per inch, or other nnits, and then draw- 
ing AC and BC parallel to iV" and N. Then AC = BC = N 
according to the scale selected for AB. In the figure it can be seen 
that 

The equation shows that reducing the angle between the sides 
of the V increases the total pressure upon the bearing surfaces. 
The angle should therefore be kept as large as possible without 
getting it so great that the table may be thrown from the bed by 
the side pressure of the tool. 

The V form of bearing is self-adjusting for wear, since it 
naturally settles down as the surfaces wear away. 

For ordinary service the V's for a small planer should be made 
with a smaller angle 6 than for a large machine. This is due to 
the fact that, as the size of the planer increases, the weight of the 
table increases more rapidly than the side pressure of the tool. 

Varying the angle 6 does not change the pressure per unit area 
on the bearing surfaces as long as the load P and the horizontal 
projection of the surfaces remain unchanged, for the bearing surface 
is increased in the same proportion as the total pressure. The 
power necessary to move the table increases as ^ is decreased, how- 
ever, and the wear on the ways is increased, so that the table will 
settle down more rapidly. There is also more liability to abrasion 
with the smaller angle, since localization of the pressure, due to 
unevenness of the bearing surfaces, causes a greater pressure on the 
high parts. 

That the pressure per unit area on the bearing surface is 
unchanged as long as the horizontal projection of the surface and 
the load remain the same, whatever the value of 0^ can be shown 



4 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

in the following manner: In Fig. 3 the horizontal width of the 
bearing surface GH is GI] therefore 

GH^GIqsq^, 

The total pressure upon GH is Nj whose value is given in the 
preceding equation. The pressure per unit area, as found by divid- 
ing the total pressure N normal to the bearing surface by the total 
area GH, is 

N P e ^, e P^2 
— ^-csc-^^GIcsc-^-^j~, 

This shows that the pressure per unit area is always equal to the 
quotient obtained by dividing the Joad supported upon the bearing 
surface by the horizontal projection of the surface, and is therefore 
not affected by the angle d. The other half of the load P is sup- 
ported by the opposite side of the V. 

The angle is generally 90*^ or slightly less in planers for general 
use up to 24 inches between housings. It grows larger as the 
machines increase in size, reaching as much as 120*^ for 72 in. to 
144 in. between housings. The width or slant height of one side 
of the V in planers built by the Niles Tool Works branch of the 
Niles-Bement-Pond-Co. is given below. The width of bearing 
surface, is given for only one side of one V, and corresponds to the 
distance GH in Fig. 3. 

WIDTH OF PLANER V'S. 



Distance between } 
housings, inches, f 


26 
2 


82 
2i 


86 
2i 


38 
2J 


42 
8 


44 
8i 


48 
8i 


54 


60 
4J 


72 
5f 


84 
6 


96 
6i 


120 
8i 


144 


Width of bearing sur- ) 

face = GH. Fig. 8. V 

Inches. ) 


9 



The Bement^Miles branch of the same company has, for a 
planer 72 inches between housings, the following values referring 
to Fig. 3: 

^=120^; (?^=4ff inches; G/=4i inches. 



BEARINGS AND LUBRICATION. 5 

The G. A. Gray Company planers have, referring to Fig. 3, 
S=30*^ and G/ = l inch in a planer 22 inches between housings; 
in one 72 inches between housings 5 = 110*^ and G/ = 4 inches. 
Between these sizes the angle 6 and width of way are proportional * 
to the distance between housings.* 

Planers for special work, where the piece to be machined is 
comparatively light, and the side pressure of the tool heavy, should 
have the angle 6 smaller than 90®, and the width of the bearing 
surfaces greater than for the larger angle more commonly used. 
The greater width of the bearing surfaces is necessary to keep 
down the pressure per unit area on the bearing. 




Fig. 4. 

Fig. 4 shows A pair of less commonly used bearings for a planer 
or other reciprocating table which is partly held in place by its own 
weight. The catting tool cannot displace a table having this form 
of ways, except when a heavy side pressare is applied against a piece 
of work at a considerable distance above the table. The pressure 
on the horizontal part of the bearings is equal to the load, instead 
of being greater than the load, as in the case of the V bearings. 
This is an advantage, since it reduces the liability to abrasion. 

The strip C is adjustable horizontally to allow for wear on the 
vertical surfaces. If attached to the bed as shown, it can be set so 
as to give the table a snug running fit from end to end of the bed, 
even though the wear near the centre of the length of the bed is 
greater than at the ends, as is generally the case. 

Fig. 5 shows a form of bearing used upon one side of the table 
of a planer taking work 120 inches square, f The flat angle of 150"* 

♦ The proportions of planer bearings given above were courteously fur- 
nished by the officials of the establishments named. 

t Used by the Wm. Sellers Co. Sketch kindly furnished by them. 



6 VOBMj STRENGTH, AND PROPORTIONS OF PARTS. 

is sufficient to hold the table in position when making the retnm 
stroke^ or when the tool is taking a light cut. The sides of th» 
way, which make an angle of 8^ with a yertical line, or 16^ with 
each other, prevent displacement of the table by a heavy side 
pressure of the cutting tool. On account of their angularity, no 
adjusting device is necessary to take up side wear, for the settling 
of the table in the ways makes this adjustment. 

V-form bearings can be continuously lubricated by placing 
oil-pockets, of the form shown in Fig. 6, along them at intervals. 
This pocket is simply a recess cast into the bed. A double-cone 





Fig. 6. 



Fig. 6. 



roller RR is placed in the pocket so as to be partly submerged in 
oil. The angles of the cones are such that they will fit the table 
Vs. The roller is supported at the cuds by bearings FB which are 
held up by some means, jxb springs or counterweights, so that the 
roller will press lightly against the V as the table passes over the 
roller. The slight frictional resistance between the surfaces turns 
the roller, so that the oil adhering to it is carried up to the V. 
Suitable means should be provided to prevent the roller from being 
raised too high when the table passes from over it. After the oil is 
thus placed on the V, some of it is earned between the bearing 
surfaces by the motion of the table. The pressure between the 



BEARINGS AND LUBRICATION. 7 

surfaces generally squeezes the oil out^ most of it going to the 
groove at the bottom of the V, and then flowing back to the oil- 
pocket; a smaller portion is forced out at the top into the grooves 
00^ and either flows along these grooves back to the pockets or 
runs down the inclined surfaces of the bearings when the table 
passes from over them. The groove at the bottom of the V not 
only serves for an oil-channel, but affords a space into which par- 
ticles of foreign substances may be scraped from the exposed bearing 
surfaces of the bed by the end of the table V as it passes over them. 
The supporting pins EE on which the cones revolve should be as 
small as is consistent with good design, and turn freely in their 
bearings: otherwise there is danger that the cones will stop turning 
and a flat place will be worn on them by the rubbing of the table. 




Fig. 6.1. 



^ig. 6.1 is a more compact form of the same device. The cone 
frusta are made short by cutting away more of the part near the 
apexes. The support is between the two parts that bear agamst 
the table, instead of outside. There is consequently less widening, 
or none, of the bed at the pocket. The cones may be pressed up 
against the bed by a coiled compression spring, as shown, or a 



8 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

counterweight may be made to act on the small pin by extending 
its ends out through the slots in the quill that supports the shank 
of the yoke bearing. As shown, this pin is a stop to prevent the 
cones from rising too high when the table clears them. 




Pig. 6.3, 



Fig. 6.2 has a feature of some importance that does not appear 
in the preceding devices. By the use of two separately supported 
mushroom-shaped oilers the material of the bed can be carried 
over the oil-pocket. This secures a continuous groove at the bottom 
of the V which may be utilized as a drain to carry the oil to welb or 
buckets at the ends of the ways. The dirt and particles of iron 
from the ways are not carried into. the oil-pocket as in the other 
designs. Clean oil can therefore be constantly supplied to the 
bearings. 

Flat bearings can be lubricated in a similar manner. In this 
case a pair of flanged rollers, as RR, Fig. 7, mounted on the 



BEARINGS AND LUBRICATION. 



9 



same spindle and forced lightly toward each other with a spring, so 
as to bear against the vertical part of the bearing, give good service. 
The faces of the flanges on the inner sides at jPand F should be 




Fig. 7. 



cormgated or indented, or the flanges perforated with small holes, 
to facilitate the carrying of the oil np to the ways. 

The ways or V's forming the bearings between the carriage and 
bed of an ordinary engine-lathe commonly have the form of an 
inverted V, as shown in Fig. 8. The pressure of the cutting tool 




Fig. 8. 

against the work often has a tendency to throw the carriage from 
the bed. Since the carriage is comparatively light, clamps are 



10 FORar, STRENGTH, AND PROPORTIONS OP PARTS. 

necessary to hold it in place. The V's can therefore have such an 
angle as will allow the carriage to rest on them without pressure 
against the clamps when the tool is taking a light cut, but the 
clamps may come into service when a heavy cut is taken. The 
tops of the ways should be sUghtly crowned to prevent marring 
of the rubbing surfaces by tools, etc., laid across them. 

The angle is commonly made 90° or less on engine-lathes for 
ordinary service. The width of the bearing, surface on each side of 
the V is about | of an inch to li inches for a lathe which will 
swing a piece 24 inches in diameter. 

While it is not common practice to provide any method for 
lubricating the ways of a lathe other than by an oil-can, a machine 
18 sometimes found in which an oil-pad or bunch of waste rubs 
against and lubricates the bearing smface. The carriage has a 
pocket over the ways, into which the pad fits. Extensions of the 
carriage are sometimes carried out over the ways to shield them 
from chips and turnings. 

These devices are found in smaller tools rather than larger 
ones. The very frequent cut and grooved appearance of the ways 
of lathes indicates very clearly that such appurtenances are a 
desirable addition to all such machines. 

3. The bearings for the ram or tool-carrier of a shaper for 
metal-working must guide the ram positively, so that it cannot be 
thrown out of position by a heavy pressure against the cutting tool 
in any direction. Fig. 9 is an end view of a common form of bear- 




Pio. 9. 

ing for this purpose. The ram reciprocates in the part 5, which 
may be either the frame of the machine or a carrier for the ram, 
according to the design of the shaper. 

In order to take up wear on the faces EF and HI^ the clamp D 
IS removed from the machine and a light cut taken off the surface 



BSABINOS AND LUBRICATION. 



11 



EJf and the clamp is then replaced. Another method is to plane 
B a little lower at JE than ^i^and place ** shuns *' or ** liners " 
of thin sheet metal or paper between D and B on ^he surface JE\ 
as wear occurs some of the shims are removed and the clamp drawn 
down tight again. It is essential that the clamp be firmly held 
against the part B. Wear along the surfaces KL aud MNi^ taken 
up in the same manner by cutting away the surface NO^ or by 
removing liners which had been placed between the clamp and 
part B. Wear along EH and LN is taken up by the set-screw 8^ 
which is used to force the clamp C against the ram^ the cap-screws 
U being loosened for this purpose. 

Another form of shaper-ram bearings that has been much used, 
but which seems to be giving way to that just described, is shown 
in Fig. 10. The bearing surfaces are ^^and HI on one side, and 




Fig. 10. 

the similar surfaces on the other. The objectionable feature of 
this bearing is that, on account of the wedge-like form of the 
bearing surfaces, which make the angle B with each other, the 
pressure between them is apt to be so great locally that cutting will 
occur. Otherwise this bearing presents the excellent feature that 
it can be adjusted for wear in all directions by simply lowering the 
clamp C slightly, or by forcing it towards the ram with a row of 
set-screws placed as the one at 8 in Fig. 9. 

The angle S, Fig. 10, is commonly from SO*' to 45** in practice. 

The width of the bearing surface EF is about 1 inch on a 
«haper having 6 = 30** and a maximum stroke of 18 inches. 

Another method of taking up the side wear in a ram having 
bearing surfaces at right angles is shown in Fig. 11, where the 
bearing strip E is held against the ram by a line of set-screws 8^ 
each provided with a lock-nut. Some method of preventing end 



12 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

motion of the strip must be supplied. This may be done by having 
the end of one or more of the screws made cylindrical, so as to fit 
into a small hole in the strip. Screws with conical points, fitting 
into conical countersinks in the strip, will hold it in place, but the 




Fig. 11. 

strip has a tendency to slip off the screws, thus causing unnecessary 
pressure between the rubbing surfaces. A cylindrical pin, driven 
into the frame on the line of the set-screws with its end projecting 
into a cylindrical hole in the bearing strip, will prevent end motion 
of the latter. Flat- or round-point adjusting screws can then be 
used against the flat surface of the bearing strip. 

On account of the thinness of the strip, the pressure is some- 
what localized at the set-screws. The consequent wearing away of 
the strip more rapidly at these places than between them, makes it 
necessary to spring the strip by setting up the screws, in order to 
hold the ram rigidly when taking a heavy cut. This causes a con- 
stant pressure of the strip against the ram, even when on the 
return stroke. The consequent wear and loss of power are dis- 
advantages. 

Another method of adjusting the bearing strip, which almost 
entirely prevents its springing so as to press constantly against the 
ram, is to tap the screws into it and allow them to pass through 
smooth holes in the frame; liners can then be placed back of the 
strip, and the screws tightened to hold it back firmly against them. 

4. For planer-saddles, lathe tool-rests, and similar parts, where 
one of the sliding parts is much longer than the other, the ^' gib " 
shown at in Fig. 12 can be conveniently used. In Fig. 12, B 



BEABINGS AND LUBRICATION. 



13 



may be taken to represent a portion of a planer cross-rail with the 
upper portion of the tool-carrying saddle resting upon it. The gib 
is tapered on two sides, from end to end ; the other sides are 







SSDOLB 












-€o^ 




CLAA4P C 


B 




UJ 


LU 


, LIJ. 


G 





.M. 


.-.M 


T 


T 


1 ' 








VlB. 1& 



parallel. One of the sides forming the taper rests against the 
corresponding sloping surface of the saddle, and the other against 
the rail. A stud S, tightly screwed into the saddle, and having 



14 POEM, STRENGTH, AND PROPORTIONS OF PARTS. 

two lock-nuts T and T, is used for adjusting the gib so as to secure 
the desired fit between the moving parts. Wear on the vertical 
faces is taken up by the clamp C at the top, and the corresponding 
clamp at the bottom. The dead weight of the saddle and the 
parts appertaining to it is carried on the horizontal surface imder 
the gib. The taper of the gib is generally between \ and f of an 
inch per foot. 

The tool-slide which moves on the carriage of a lathe may be 
made with a gib G, aa in Fig. 13. Here the gib bears against the 



SLIDE 



J\ !^^ 



BEST 



END VIEW 




SIDE VIEW 



Pig 18. 



inclined surfaces so that, by adjusting it alone, wear in all directions 
is taken up. The taper of the gib may be the same as for rectan- 
gular bearings, viz., J to | of an inch per foot. 

The angle generally lies between 45° and 60°. For a lathe 
which will swing 24 inches in diameter, EF may be 1 inch and 
FH li inches; for one swinging 12 inches in diameter, the corre- 
sponding dimensions are f inch and 1 inch. 

By chamfering off the comer of the rest and filling in the ro- 




81 DE VIEW 




Pio. 18.1. 



entrant angle at JB as shown, the slide is made stronger and the 
sharp comer of the rest is less liable to injury. 

Fig. 13.1 shows another device that is used for the same purpose 



BEABINGS AND LtTBBIGATIOK. 



16 



as the preceding. In this case the heavy beveled clamp that is 
used instead of a gib does not taper from end to end, but is pris- 
matic. It is held down rigidly against the rest by the screws as 
shown. In order to take up wear some of the material is planed 
oflf the bottom of the clamp where it is drawn against the rest by 
the screws. A somewhat less satisfactory method is to place shims 
or liners imder the gib, and remove them to compensate for wear. 




rV 



tr^ 



'f? 



=•— ^•j-~.rUMlg=u-j:t4 



y 



=3J=.»MPa; 



T-= 



REST 



SIDE VIEW 



H 



UJe 



Ie g'-'PE 



REST 



END VIEW 



Fig. 18.2. 



Fig. 13.2 is very much the same as the precedmg one. In 
this case the clamp is attached to the slide instead of the rest. 
The adoption of either of these devices must naturally depend 
upon the judgment of the designer in accordance with the conditions 
to be met. 

Another device that is often used for rectangular bearing sur- 
faces is that of Fig. 14, where the righ1>-angled piece O, resembling 




Pig. 14. 



a box-square, is adjustable with both a vertical and a horizontal row 
of set-screws, thus compensating for wear in all directions. 

The thin strip E, Fig. 11, can be used on angular bearings such 
as shown in Fig. 13. If the set-screws are kept at right angles to 



16 



FOBM, STBENGTH, AND PBOPOBTION8 OP PABT8. 



the outer vertical side of the slide, their points should be either 
conical, so as to have line contact with the side of the strip, or else 
should be fit into a hole drilled into the strip. The object in both 
these cases is to get as much contact surface between the points of 
the screws and the strip as possible, in order to prevent crushing 
the metal at the place of contact. 

Another method of adjustment is shown by the dotted lines at 
the left side of Fig. 13. This is by using an angular strip EFKL 
which bears against the surface LK of the slide, and is drawn 
upward by screws passing through smooth holes in the slide and 
into the threaded holes of the strip. Or, leaving the slide solid, as 
indicated by the full lines, the same device can be used on the rest, 
the strip being EFIJm this case. The screws must be put in from 
the opposite direction for this construction; they are not shown 
in the figure. The gib should be placed, when possible, where it 
does not receive the pressure of the cutting tool. 

5. Cross-head guides for engines, pumps, and similar machines 
having a crank and connecting-rod, are frequently made of the 





Fig. 15. Fig. 16. 

form shown in Pig. 15, which is an end view of the guides BB and 
piston-rod A. The bearing surfaces are at CDE and FOH, They 
must take the pressure due to the angularity of the connecting-rod 
with the piston-rod, this pressare being in either direction along a 
line DG^ not drawn. Since there is very little or no side pressure 
at right angles to DO^ the angles d can be made large. Adjust- 
ment for wear can be made either by moving apart the bearing 
surface of the cross-head or bringing those of the frame B together. 



'/" 



BEARINGS AND LUBRICATION. 17 

To do this, the cross-head must either be made of several pieces, or 
the gaides made of separate pieces and attached to the frame. 

Fig. 16 shows another form of cross-head guides. The bearing 
surfaces here are parts of cylindrical surfaces, which are represented 
as arcs of circles in the figure. The centres of these arcs are at 
and 0' on a line through B and B\ the radial distances OB and 
C/B' being less than half the distance BB' between the bearing sur- 
faces. It can be readily seen that this construction prevents the 
cross-head from rotating about A. 

In contrast with this construction for preventing the turning of 
the cross-head in the guides, some prominent engine-builders bore 
the guides concentric with the cylinder, in order to allow the cross- 
head to adjust itself to the crank-pin and connecting-rod. 

Flat bearing surfaces normal to the line BB'^ Fig. 16, are fre- 
quently ased for cross-head bearings. In order to prevent lateral 
motion the cross-head generally extends over the edges of the 
guides and has a bearing against their sides. 

6. The relative length of sliding bearings for rectilinear 
motion, and their positions at the ends of the stroke, should be taken 
into consideration when designing bearings which must perform 
considerable service during the life of the machine of which they 
form a part. 

If, in a machine of the iron planer type with its reciprocating 
table, held in place by gravity, the table is much shorter than the 
supporting bed, and always moves the same distance between the 
same limiting positions, then the extreme ends of the bed will not 
be worn at all, but shoulders wdll form at the Hmits of travel of 
the table. By making the table longer, or cutting off the ends of 
the bed, so that the table just reaches the end of the bed, the 
shoulders will be obviated, but still the bed will be worn more 
rapidly at the middle than at the ends. 

On the other hand, suppose that the table and bed are of the 
same length and the stroke is but slightly shorter than the table 
and bed — only enough shorter to prevent the table from tipping 
off the end of the bed as it stops moving — then the wear will be 
greatest at the middle of the table and the ends of the bed. The 
bearing parts will wear to a curved form similar to that shown 



18 FOBM, STRENGTH, AND PB0P0BTI0N8 OF PABT8. 

in Fig. 16.1. The table becomes concave and the bed convex on 
the bearing surf aces. 




Pio. 16.1. 

The remedy for the above defects is, for the kind of machine 
mentioned^ shown in Fig. 16.2. The bed in this figure is of such a 



^ 



TABLE 



1 



BED 



Fig. 16.2. 

length that the table will overhang enough at the ends of the 
stroke to make the wear due to such overhanging just counter- 
balance the tendency to wear rapidly at the middle of the bed. 

It has been demonstrated in practice that it is possible to get 
such a relative length of table and bed that the wearing parts 
will retain their straight form through many years of service. 

In a high-speed steam-engine with a crosshead of the form 
shown in Fig. 16.3 it has been found in practice that, with the 




Fie. 16.a 

length of the bearing surfaces of the slide and guide equal to each 
other and to the length of stroke, the parts retain plane bearing 



BEABINGS AND LUBRICATION. 19 

surfaces and show no appreciable wear after long service when weU 
lubricated. This is contrary to what has been found for planer- 
table and bed ways. The cause of the difference is due to the fact 
that the centre of gravity of the crosshead is not in line with, but 
lies below, the piston-rod. When the latter pulls upon the cross- 
head and rapidly accelerates it from a position of rest at the end 
of a stroke there is a tendency for the crosshead to rotate about a 
horizontal axis perpendicular to the direction of motion. This 
brings a heavy pressure against the forward end of the bearing 
surface of the sUde which causes more rapid wear at the middle 
of the guide and end of the slide than in any other part when the 
crosshead is still near one end of the stroke. When the stroke is 
nearly finished, the retarding action of the piston-rod upon the 
crosshead causes the heel of the crosshead to press hard against 
the guide, for a reason the reverse of that which causes the for- 
ward end to press hard when the speed of the crosshead is accel- 
erated. At the middle of the stroke, when the crosshead is moving 
uniformly, the wear is distributed quite equally over the bearing 
surfaces. The result of the whole is that, as already stated, the 
parts retain their form in an engine which " runs over " so that 
the crank-pin moves away from the cylinder when above the centre 
line of the piston-rod. 

6.1 The method of lubricating the crosshead and guide 
bearing surfaces that has been adopted by the Straight Line Engine 
Company for bearing surfaces of equal length is of interest. It is 
shown in Fig. 16.3. On account of the great over-reach of the 
crosshead shoe at the ends of the stroke there would be undesirable 
splashing of the oil, if any were to collect, unless some special 
device for preventing the splashing were used. As shown in the 
figure, a " sawtooth" form of corrugated surface has been adopted. 
The edges of the ribs just clear the slide. Oil is kept at a height 
just above the level of the bearing surfaces by the device shown 
at the right of the ways. It is a trap consisting of a tube standing 
as high as the bearing surfaces. About it is a ring-like hood some- 
what higher than, and reaching some distance down below, the top 
of the tube. When oil and water accumulate in the shallow recess, 
the water flows up inside the hood and down through the tube, 
leaving the oil in position to lubricate the guide. 



20 FORM, 8TBENGTII, AND PROPORTIONS OF PARTS. 

6.2. The material used for Tb and other rectilinear bearings 
in machine tools is ahnost universally cast iron. The durability 
of the rubbing surfaces is a question of some importance. There 
is a considerable difference in the amount of wear that takes place 
in the different grades of cast iron. A number of experiments 
made by the Alfred Herbert Co. of Coventry, England, upon cast- 
iron sliding surfaces submerged in an oil-bath demonstrated very 
clearly the fact that an open-grained soft cast iron wore away very 
much less rapidly than a close-grained, hard material. While the 
tool marks were not removed from the porous material, a very 
perceptible wear took place in the harder iron under the same 
duty. 

Two mating pieces were made from each grade of cast iron. 
The larger one, used as a stationary bed, was recessed at the top 
to two depths; the higher surface of the recess was planed for 
bearing surfaces, one strip along each side of the length of the 
recess. The latter was filled with oil so as to cover the bearing 
surface. The smaller piece, used as a slide, was reciprocated 
over the bed by a mechanical drive. 

The oil remained clear with the soft iron, but soon became 
cloudy with the hard. The rubbing motion was reciprocating. 
Each bearing surface rubbed through a distance of about eighty-four 
miles over its mate under a pressure of approximately 90 pounds 
per square inch. As much as one-fiftieth of an inch was worn off 
the rubbing surface of some of the harder irons.* 

Materials the same as used for journal-bearings are used for 
rectilinear smiaces for heavy duty in many cases. 

BEARINGS FOR ROTARY MOTION. 

Journals, Boxes, and Journal-bearings, 

7. Definitions. — When two parts of a machine rotate with 
regard to each other, as a wheel and its axle, the part of the one 
which is enclosed by, and whose surface rubs against, the other is 
called the journal ; the part which encloses the journal is called 
the box, or, less specifically, the " bearing." The name ' ' bear- 

*The parts tested and results obtained were shown the writer at the works 
of the Alfred Herbert Co., in September, 1902. 



BEABIKGS AND LUBBIOATION. 



21 



ing" or journal-bearing is very commonly applied to the whole 
assembly of parts embodying both the journal and its box. 

8. The cylindrical journal-bearing is the most common form. 
When the pressure upon it is always in nearly the same direction, 
the box may be made to extend only partly around the journal. 
The boxes used on the axles of railway cars furnish an example of 
this kind. These boxes encompass about one-third of the circum- 
ference of the journal. - This is sufficient to hold the axle in place, 
and furnishes almost as much efficient bearing surface as if the box 
covered half of the circumference of the journal. 

A difficulty that sometimes assumes serious proportions is liable 
to be met when a box, extending half-way around the journal, is 
used, especially if the box is well fitted to the journal before going 
into operation. The nature of this trouble can be described with 
the aid of Fig. 17, in which B is the box and A is the supporting 




Fig. 17. 

frame. Before operation the box is of the form indicated by the 
full lines, the semicircle CO just fitting the journal. When the 
journal is rotated in the box, the latter is apt to have a tendency to 
take the form shown by the broken lines, the points CG trying to 
approach each other. This action causes the sides of the box to 
press unduly hard against the journal. The result is increased 
frictional resistance and wear between the rubbing parts. 

One remedy for this is to make a very loose fit along the surfaces 
near C and C. Another and more certain one is to provide some 
means for holding the box back against the frame so that the parts 
CO cannot move away from it; this may be done with bolts passing 
through the frame and into the box, or by dovetailing the sides of 
the frame and box together. 

When the pressure acting on a bearing is reversed, as indicated 
by the arrow-heads in Fig. 18, the box must generally completely 



22 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

surround the journal. The adjustment for wear under such con- 
ditions should be in the direction of the forces. For the vertical 
pressures in Fig. 18 the separation should be on a horizontal plane 
through the centre of the journal. The short lines at D and D 
show the separation on this plane. The small shoulders at the ends 
of these lines are to hold the cap C in the proper position. Adjust* 



rrn/ ^''^"^^"c^ Nr m 




Fig. 18. 



ment is made by bringing the cap down nearer the frame as wear 
occurs. Liners placed between the cap and frame may be removed 
for this purpose, or some of the cap or frame cut away. If the 
pressure is horizontal and reversed, the separation should be made 
on a vertical plane. 

In many cases the wear is both vertical and horizontal. The 
horizontal steam-engine with its heavy fly-wheel furnishes an exam- 
ple of this. The downward pressure caused by the weight of the 
fly-wheel, as indicated by the vertical arrow in Fig. 19, wears away 




Fig. 19. 

the bottom of the box. The pressure, indicated by the horizontal 
arrows, is due to the steam-pressure on the piston ; this causes wear 
on the sides of the bearing. When close fitting of the running 
parts is not of great importance, a box divided at an angle with 
both lines of pressure, as shown in the figure, can be adjusted with 



BEARINGS AND LUBRICATION. 



23 



a fair degree of Batisfaction. For more accurate adjastment the 
box mast be divided into three or more parts. 

Fig. 20 shows a box divided into fonr parts, and adjastable 



^^^jra 




Fig. 20. 

both vertically and horizontally. The four parts of the box, C, Z>, 
E^ and F^ are enclosed by the frame A and the cap JS. The wedge 
/, at the bottom of the box, is nsed for the vertical adjustment, the 
top C being held down by the cap B. The side screws and O 
are for adjusting the side pieces E and ^horizontally; these screws 
are made with enlarged ends, for bearing against the sections of 
the box, in order to prevent crushing of the metal where they come 
in contact. With a box of this form the axis of the journal can 
always be kept in the same position, whatever the direction of the 
wear. This is an important consideration in some classes of 
machinery. 

A unique and very compact adjustable jonrnal-box is shown in 
Fig. 21.* It consists of an outer shell or casing A enclosing an 
adjustable lining B^ made in two parts. Both the shell and the 
lining have the same eccentricity, so that when the lining is placed 
in the shell with the thickest part of one next the thinnest part of 
the other, the outer surface of the shell and inner surface of the 
lining are concentric. The complete box can therefore be placed in 
a hole bored in the frame of a machine, and turned about its own 
axis without displacing the axis of a journal fitting into it. The 

♦Used by the Straight Line Engine Co., Syracuse, N. Y. 



24 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

lining does not completely encircle the jonrnaly but the circumfer- 
ence is completed by a wedge D at the thickest part, together with 




f^rafe-:-^r^-;^^!^^^ 



W 
u 



mm mm-:t''^'--:^y^%;j 



m^^^y-^^ 



v'-'''^-:^^-^ IS^S^ 



! o 




T^^ 



Ci3 





Fig. 21. 



thin strips or liners C placed at the thinnest part, when the bearing 
is new. As wear occurs these strips are removed from the thin 
side and placed alongside the wedge, thus reducing the diameter of 



BEARINGS AND LUBRICATION. 



26 



the bore of the box, and at the same time moving the centre of the 
bore slightly nearer the wedge side of the bearing. By placing the 
thin side of the bearing where the greatest wear occurs, each 




adjustment of the lining tends to bring the joarnal back to its 
original position concentric with the outer surface of the shell. 

When but a slight adjustment for wear on the journal and box 
is required, and compactness is desired, the one-piece box of Fig. 
22 is serviceable. The box B is made of a single piece, cylindrical 



26 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

ill the bore, and coned on the central part of its outer surface to 
fit the conical hole in the frame A of the machine ; it is cut through 
at Cy 2), and E^ from the large end to near the small one; the ends 
are threaded for nuts F and O. When new, these nuts are adjusted 
so that the inner surface takes the circular form and size to which 
it was bored when the box was solid. In order to take up wear on 
the rubbing surfaces, O is loosened and F tightened so as to draw 
the box into the frame, thus springing the sides together so as to 
diminish the diameter of the bore; O is then tightened to hold the 
box firmly in placiB. 

Fig. 23 represents what is probably the simplest form of adjust- 




Pio. 28. Fig. 24. 

able box that can be devised. The slot S^ cut along one side, allows 
it. to be adjusted by the cap-screw T, This design is also frequently 
used where it is desirable to have a pin pass easily through the bore 
when putting the parts together, but held tightly in place during 
the operation of the machine. 

In order to prevent excessive end motion of a journal in its box, 
collars are frequently placed at the 'end of the journal, forming a 
part of it, as in Fig. 24. It frequently occurs that if the collars 
are fitted so as to allow free running, but4o end shake, of the 
parts, there will be excessive pressure and binding between the 
collars and the box when the machine is put into operation. This 
is probably due to the fact that the heat generated by the rubbing 
of the surfaces over each other raises the temperature of the thin 
shell more rapidly than that of the journal, consequently expanding 
it more rapidly, and causing it to press against the collars as stated. 



BEARINGS AND LUBRICATION. 27 

The higher coefficient of expansion of the material of the box may 
also have a slight tendency to increase the binding with a material 
that is highly expansive. The remedy is plainly to make a loose 
fit when constructing the machine. 

A bearing having a journal with a single collar is shown in 
Fig. 25. This is a form that is used on the spindle of a lathe where it 
rests on the head-stock at the end next the face-plate or live-centre. 
A smaller adjustable collar is placed at the other end of the spindle 
to keep the spindle from moving endwise, toward the right in this case. 

9. Self-aligning bearings are desirable in many classes of ma- 
chinery. Fig. 26 represents a common and compact form. The 
outer surface of the box or sleeve B is partly made up of a portion 
of the surface of a sphere. The supporting part A has a concave 
spherical surface to conform to that of the sleeve. This device 
allows the axis of the sleeve to adjust itself in line with the journal 
and maintain such adjustment, even if the shaft of which the 
journal forms a part is temporarily sprung, or permanently bent so 
as to wabble. Movement of the support A has no effect on this 
alignment of the box. Some means of preventing the box from 
rotating in the support must be provided. This can be done by 
a pin C fitting tightly in the sleeve and loosely in the support. 

For convenience of construction and compactness of design it 
is best to place the centre of the sphere on the axis of the bore of 
the box. This is not absolutely necessary, however. When so 
placed, the centre line of the shaft will always pass through the 
same point, the centre of the sphere, in relation to the supports. 
This is not true when the centre of the sphere is not on the axis of. 
the bore. 

The requirements of construction can be met, in many cases, by 
casting the spherical recess in the frame larger than the part that 
fits into it, and filling the space between the sleeve and frame with 
Babbitt or other white metal that fuses at a low temperature. 
This obviates the usually expensive operation of machining the 
concave spherical surface. 

It is not necessary to have the spherical surface extend com- 
pletely around the sleeve; two diametrically opposite segments of 
the surface can be used, as shown in Fig. 27, where B is the sleeve 
having the spherical segments under the spherically concave points 
of the two screws C and C, which are supported by the frame A. 



28 FOBU, STRXirOTH, AND PBOPOBTIONS OF PARTS. 




I 



BEARIN03 AND LUBRICATION. 



20 



The lips D and D^ extending aroand the segments, prevent the 
sleeve from rotating, by striking against the end of the screws^ 
(See also Fig. 29.) 




Fio. 26. 



A self-aligning bearing can readily be made withont the nse of 
a spherical surface, but it is necessarily not so compact. Fig. 28 
shows such a bearing, which is also adjustable both vertically and 
horizontally. The sleeve B is supported on the points of two 



U±3 




Pig. 37. 



coaxial screws which pass through threaded holes in the extremities 
of the arms of the yoke C\ the lower part of the yoke passes through 
a smooth hole in the supporting frame A^ and has lock-nuts D and 
D for adjusting and locking it in place, still leaving it free to turn 
in the frame. 



30 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

10. Lubrication of jonmal-beaxingps. — Either of the three forms 
of labricantfi — liquid, pasty, and solid — may be applied to a journal 
bearing. The liqaid lubricants consist chiefly of oils, the pasty of 




grease, and the solid of graphite. The oils are by far the most 
commonly used. 



BEABIN08 AND LUBRICATION. 31 

11. Oil- and grease-lnbrioation for journal-bearings. — Both 
experimental iuTestigation * and practice show that when an oil- 
Inbricated jonmal is mn under a uniform load^ constantly applied 
in the same direction, a film of oil adheres to the joamal and is 
carried between the rubbing surfaces, unless the pressure between 
these surfaces is so great as to squeeze or rub it off. As long as the 
film of oil is maintained between the rubbing surfaces, the lubrica- 
tion is effectiye; but when the oil is so completely squeezed oat, or 
burned by the heat due to friction, as to break the film and allow 
the metallic surfaces to come into contact, abrasion and consequent 
seizure or destruction of the surfaces follow. Seizure can occur 
only when the box completely or nearly surrounds the journal, and 
is sufficiently rigid to grip and hold it when the metals come into 
contact. Probably the seizure is often dae to sudden heating and 
expansion of the material forming and lying jusfc beneath the rub- 
bing surfaces, when the lubricant has ceased to separate them. 

The film of oil must withstand the greatest pressure that occurs 
at any part of the bearing. If a box presses vertically downward on 
its journal, the maximum pressure will usually be at or near the top 
and centre of its length. From this point the pressure gradually 
decreases towards the sides and ends. 

If a hole is drilled in the middle of the box at the top, so as to 
communicate with the atmosphere, the oil will be forced out of it 
as the joamal turns. If a pressure-gauge is attached to the open- 
ing, it will show the pressure acting on the oil at that place, f which 
is probably about the same as woald exist if the hole had not been 
made. Any attempt to in trod ace oil at this point in the usual 
manner, as with a drip-cup, woald clearly be unsuccessful. Even 
if a groove were cut along the top of the box for some distance, but 
not reaching the ends, it would only collect the oil and force it up 
through the opening. The oil must therefore be applied at some 
other place. If an oil-cup which gradually feeds the oil on the 
bearing is used, it will give the most satisfactory results when the 
opening through the box is on the side where the surface of the 

* Tower's experiments: Proc. Inst, of Mech. Engrs , 1883, p. 682.' also 
other experiments. 

t Tower's experiments. 



32 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

journal is approaching the top or point of highest pressure of the 
ruhhing surfaces. This applies especially to boxes which only partly 
surround the journal. If the box completely encircles it, satisfactory 
lubrication can be obtained by applying the oil at any point where 
it will pass through the opening to the rubbing surfaces, provided 
suitable oil-grooves for distributing the oil over the surfaces are 
cut into either the box or journal, or both. Such grooves are 
generally cut into the box. 

When a *' half -box," as one which covers only half or less of the 
circumference of the journal is called, rests on top of the journal, 
it can be labricated very successfully by letting the lower part of 
the journal run in a reservoir of oil. This is called oil-bath lubri- 
cation. The oil adhering to the journal is carried up between the 
rubbing surfaces and thoroughly distributed over them. If the 
half-box is placed under the journal, the oil can be most success- 
fully applied at the top. 

Oil-pad Inbrioation is an excellent substitute for bath lubrica- 
tion when the latter cannot be advantageously applied on accoant 
of waste of oil or inability to provide a suitable reservoir for retain- 
ing it. The oil-pad may be made of a piece of porous woven 
material which is saturated with or dips into oil and presses 
against the journal. An excellent, and at the same time compara- 
tively inexpensive, material which is used instead of the woven pad, 
is the cotton " waste " so commonly used for cleaning machinery. 
It is almost wholly, possibly wholly, used in this country for lubri- 
cating car-axle journals. The waste and oil are placed in a closed 
box beneath the journal, filling the space so that there is good con- 
tact between the waste and journal. The capillary action of the 
waste carries the oil to the journal in sufficient quantity to lubri- 
cate it. 

Mr. E. Charbal states that woollen wicking was found better and 
more economical than cotton for railway service, the delivery of oil 
being from 50^ to 100^ better, and the renewals only 68^ of those 
for cotton.* 

While experimenting with a pad-labricated journal 4 inches in 
diameter and 8 inches long, running against a " half-box " at 266 

*Min. of Proc. Inst. Civ. Eng., 1894-95, Part II., p. 412. 



BEARINGS AND LUBRICATION. 83 

reTolations per minute ander a total load of 15132 poands = 756.6 
pounds per square inch of projected area, Mr. John Dewrance found 
a vacuum of 28.4 inches of mercury, corresponding to 13.9 pounds 
per square inch between the rubbing surfaces, at a point where 
the rotating part had passed the position of maximum pressure.* 

The nature of the rubbing surfaces of a bearing is of impor- 
tance. In general it may be stated that the smoother they are the 
more satisfactory service will they give. It has been found that by 
burnishing the surface of a journal with a roller burnisher pressed 
hard against it while rotating in a lathe, so that the burnisher 
rolled on the bearing surface, the f rictioual resistance to the rotation 
of the journal when placed in its bearing was lowered and more 
satisfactory operation secured. It is the practice of the C, M. & 
St. P. Ry. to roll the journals of their car-axlbs while turning the 
wheel-fit in the lathe. It is found decidedly beneficial. 

In contrast with the above, it is not an uncommon occurrence 
to find that a journal, turned and finished in the ordinary manner 
by polishing or grinding, which runs hot despite the best attention 
and care in both its manufacture and operation, can be pat into 
satisfactory working order by rubbing it with emery-cjoth so as to 
make the scratches of the emery parallel to its length ; a file has 
been used satisfactorily for the same parpose by draw-filing the 
journal so as to make the scratches of the file parallel to the axis of 
the journal, as with the emery-cloth, f 

There seems little reason to suppose otherwise than that the 
smoothest, truest surface that can be obtained is the best for a 
bearing, both in point of low f rictional resistance and great dura- 
bility. But if the surfaces fit very truly together over a consider- 
able area which has no oil-grooves or other indentations, the oil may 
not be carried between them in sufficient quantities for satisfactory 
lubrication. It is in such a case that the scratches of the emery- 
cloth or file ha\'e a beneficial effect. The scratches act as little 
reservoirs or pockets which carry the oil in between the surfaces and 
distribute it over them. An increased number of oil-grooves in the 
bearing, properly arranged, or a few in the journal, so as to cut the 

* Minutes of Proc. of Inst. Civ. Eng., vol. cxxv., p. 862; Engineering, 
Londonp Jan. 1, 1897, p. 29 ; 77ie Engineer, London, Jan. 8, 1897, p. 42. 
t Trans. Amer. Soc. Mech. Engrs., vol. yi., pp. 849-857. 



34 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

Tabbing sarfaces into small areas, Tvoald doubtless have the same 
effect as the numerous scratches. Oil-grooves, sufficiently large not 
to fill with dirt and gummy oil, should be provided plentifully in a 
bearing having considerable area and doing heavy service. It might 
seem that this should be applied to railway-car boxes. There is a 
reason why it could hardly be successfully done, however, without 
objectionably great expense. The reason is as follows: Owing to 
the nature of the service they perform, it is scarcely possible to 
examine the rubbing surfaces of the bearings from the time the 
boxes are put in place until they are worn out. The allowable wear 
is so great that grooves of a practicable depth would be worn out of 
the box long before it had completed its life. And even if this were 
not true, the probability is that the grooves would be filled with 
dirt in a short time, and thus become useless; this is also true of 
grooves cut in the journal, which is a less desirable place. 

An intermittent or a reversed force, causing pressure on a bear- 
ing, offers an opportunity for the lubricating oil to get between all 
parts of the rubbing surfaces. On account of this beneficial action 
greater bearing-pressure can be used than for a constant pressure in 
one direction. This is markedly shown in practice in the main 
bearing and crank-pin of the ordinary form of double-acting steam- 
engine.* 

For very heavy, continuous bearing-pressures and slow speeds 
grease is better than oil for lubricating journal-bearings. The 
greater viscosity, or " body,'* of the grease prevents it from being 
squeezed out from between the rubbing surfaces as quickly as the 
thinner oil, and slow speeds do not carry the oil in between the 
rubbing surfaces with sufficient rapidity to overcome the squeezing- 
out action. 

In journal-bearings, it may be taken as a rule that the faster 
the speed and lower the bearing-pressure, the thinner or less viscous 
the lubricant that will give the best service. It is frequently found 
that a change from a thick to a thin lubricant, or vice versa^ will 
overcome difficulties of heating and abrasion of a journal-bearing. 

A case has been cited where a journal which ran hot in its box 
when abundantly supplied with oil, cooled down and worked satis- 
factorily when the supply of oil was diminished. 

•Trans. Amer. Soc. Mech. Engrs., vol. vi., p. 856. 



BEABINGS AND LUBRICATION. 



36 



12. Special devices for jonmal-lnbrication with oil and grease. 
— On accoant of the importance of seoaring a constant supply of oil 
on a bearing when working hard, some of the typical appliances for 
this purpose are worthy of description. One of the simplest and 
most commonly used for a bearing whose journal rotates, is a light 
ring, of a greater diameter than the journal, which is hung over, 
and rests on, the top of the journal, a part of the top of the box 
being cut away for this purpose. The lower part of the ring dips 
into a chamber of oil below the journal. Since the weight of the 
ring rests on the top of the journal, the rotation of the latter causes 
the ring to turn also, so that the oil adhering to it is carried up and 
deposited on the journal. Such a ring-oiling device is shown in 
Pig. 29.* By providing a large and deep oil-reservoir, connected 




Fig. 29. 

by suitable chanDels to both ends of the box, so that the oil will be 
led back to the bottom, the particles of foreign matfcer gathered up 
by the oil while passing over the journal have an opportunity to 
settle out, leaving the oil clean for its next application. 

A short, endless chain is frequently used instead of the oil-ring. 
It has the advantage that, in certain forms of bearings having the 
box cast in one piece with the reservoir, it can be introduced 
through a smaller opening than the ring, or even without any 
external opening through the side of the box. This fact makes it 
especially applicable to line-shaft boxes and similar light bearings 
which are not attached to a heavy frame. The point of superiority 
sometimes advanced for the chain, that it will carry a more copious 

* Bearing manafactured by Rice Machinery Co. 



36 FOBM, STRENGTH, AND PROPORTIONS OF PARTS. 

supply of oil to the bearings is hardly worthy of consideration, for 
the ring will furnish an ample supply. 

A recent and apparently excellent substitute for the woven pad 
or cotton waste, mentioned in the preceding section, is a block of 
hard wood, concaved on one end to fit the journal, and having 
broad but very thin slits from this surface to the opposite end, 
which dips into oil. The concave end is held against the journal 
by a spring, and the capillary action of the slits carries i^he oil up 
to the surface of the joarnal. 

Wicking, similar to that of candles or lamps, is used for lubri- 
cating by placing it so that the lower end is in oil and the upper 
touches, or hangs over, the journal ; capillary action carries the oil 
up to the journal. 

When the box of a bearing rotates around the journal, as when 
a pulley turns on a shaft, oil can be readily applied as shown in Fig. 
30, provided the shaft does not extend far beyond the bearing on 



i 



-±Zl/^ ^ V FRAME 



----^"--1. 









Fig. 80. 



one side, at least. A hole is drilled into the end of the shaft as 
far as the centre of the pulley, and then another from the sur- 
face of the shaft to meet it. The oil-cup is attached by a piece 
of pipe which screws into the end of the shaft. This arrangement 
answers when the shaft is held rigidly in the supporting frame, so 
that it never revolves. If the shaft turns at times, a running pipe- 
coupling can be placed at F and the oil-cup supported by the 
frame. 

A comparatively short box on a long shaft, when.the conditions 



BEABINOS AND LUBRICATION; 37 

of operation are Bach that one rotates continuoasly and the other at 
intervals, as a palley on a line- or counter-shaft, presents the most 
difficult of all problems in oil-lubrication. Numerous appliances 
have been devised, but none seems to have come into general use. 
When the parts are not readily accessible, and are comparatively 
light, as line- and counter-shafts and their pulleys in most shops 
aud factories, a bearing whose box is made of a material which is 
itself a labricant, is the most satisfactory (see § 13). A grease-cup 
having a piston which is pressed against the contained grease by a 
spring, so as to gradually force the grease between the rubbing sur- 
faces, is often used when the parts are easily accessible, or heavy; it 
can be used in almost every case, but the attention required makes 
it frequently undesirable. If the hub of a pulley can be enlarged 
in diameter without inconvenience, a recess can be made in it, with 
an opening to the journal, and an absorbent pad fitted into it so as 
to press against the journal; by saturating the pad with oil at inter- 
vals lubrication sufficient for light loads can be secured. 

Grease-lubrication is often used for a bearing whose box does 
not rotate. One of the most common methods of applying it is by 
means of a grease-cup having an opening or oil-hole leading straight 
to the shaft, and a copper or brass pin which partly fills the hole. 
The pin stands vertically, with its lower end resting on the journal, 
being held in contact with it by its own weight. The grease in the 
cup completely surrounds the pin. As the journal becomes warm 
by running, the heat is carried through the pin to the grease, 
warming and melting it, so that it flows down to the journal. The 
slight heating of the pin by its own rubbing against the journal is 
hardly appreciable. While this device can be quite safely relied 
upon to keep the bearing from getting dry and cutting, it does not 
afford economy of power, since the journal must have enough f ric- 
tional resistance to generate heat for melting the grease. The end 
of the pin is apt to wear a groove in the journal, which may be 
objectionable. 

* Fig. 30.1 is a device for curing the ills of loose pulleys that 
seems most excellent. A sleeve fits over the shaft and forms the 
journal for the loose pulley. It is provided with oil holes and chan- 



* Afneriean Maehinist, April 26, 1900. 



38 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

nels for carrying the lubricant to both the shaft and pulley. An 
arm extending out from the sleeve and attached to some stationary 




Pig. 30.1. 



object, as the ceiling, wall, or floor, prevents the sleeve from rotat- 
ing. This device has the advantage that it can be applied to any 
free portion of a counter- or jack-shaft, and does not require any 



i^EAKINGS A:ND LUBRIOATIO]? 



39 



fiopponing shaft-hanger, floor-stand^ or wall-bracket. A uk>c would 
seldom occur where no stationary object could be found to which to 
attach the arm that prevents the sleeve from rotating. 

A crank-pin which revolves about a crank-shaft in the ordinary 
manner can be lubricated the most successfully by a device of the 
general nature of that shown in Fig. 31. A hole is drilled into the 
end of the pin, and another, to intersect it, from the rubbing sur- 
face at about the middle of its length; the latter hole should start 
from the part most distant from the crank-shaft. A piece of tubing 
is screwed rigidly into the end of the pin, with its free end extend- 
ing towards the centre of the crank-shaft, where a hollow ball is 
attached, the centre of the ball being coincident with the centre 
liae of the crank-shaft. A hole B in the side of the ball, drilled 
concentric with the crank-shaft, so as to remain stationary when 
the parts rotate, permits the introduction of oil by any suitable 






CRANK 
SHAFT 



rr^-^O 



Fig. 31. 

means, as an oil-cup with a drip- tube extending through the hole 
into the ball. After the oil is introduced it is carried out to the 
rubbing parts by centrifugal force. 

The pin of a double crank or pair of disks can be continuously 
oiled by means of a slight modification of the device just de- 
scribed. Fig. 32 illustrates the method. A groove (7 is turned in 
the crank or disk, or a grooved collar attached to it. An oil- 
hole leads from the bottom of the groove to the surface of the 
crank-pin at E. Oil that is put into the groove will be carried to 
the crank-pin bearing when the parts are rotating. The drip from 



40 



FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



the inner end of the crank-shaft bearing B may be thas used for 
the crank-pin, or oil can be fed into the groove by a tube leading 
from an oil-cup down to the under side of the crank-shaft. 

Reciprocating journal-bearings, such as that of a cross-head pin 
of a horizontal steam-engine or pump, are most commonly lubri- 
cated by a method represented by the following device: A piece of 
wicking is either suspended from an attachment to the frame of the 
machine, or stretched over a curved surface, and a supply of oil is 
fed to the wicking so as to keep it completely saturated and almost 
ready to allow a drop to fall from it. A metallic scraper or 
'' wiper" is fastened to the parts to be lubricated, which, as it is 
brought under the wicking at each stroke of the moving parts, 
scrapes or picks some of the oil from it; the oil is led from the 
wiper to the rubbing surfaces through suitable oil-ways. 

There are numerous modifications of this device, a notable one 
consisting chiefly of a thin strip of metal attached to an oil-cap so 




Fig. 82. 

that the oil fed from the cup will gather at the lower end, which is 
pointed. Another thin strip of metal, bent back upon itself so as to 
almost come together near the ends, is fastened to the reciprocating 
parts so that the ends will pass on each side of the point to pick off 



BEARINGS AND LUBRICATION. 41 

the drop of ofl and then let it flow down to the rabbing parks. 
These oiling devices are generally placed so as to act at one end of 
the stroke, thns preventing the throwing of the oil which woald 
occar in high-speed machinery if it were picked off near the middle 
of the stroke. 

A less common device for oiling a reciprocating journal consists 
of a pair of telescoping tubes, the end of one being fastened to the 
centre of the reciprocating pin, and one end of the other to a 
standard on the frame of the machine. The end of the standard 
to which the tube is attached is above the line of travel of the part 
to be lubricated. The fastenings at the ends of the tubes are made 
to permit rotary motion; this, together with the telescopic action, 
furnishes a means of providing a continaous passageway from an 
oil-cup on the standard to the journal. Suitable holes in the latter 
complete the path for the oil to the rubbing surfaces. 

When the parts to be lubricated reciprocate vertically, an open 
cup can be attached to one of them, and the oil dropped directly 
into it from a feeding-cup on the frame. 

As a precaution against injury to an oil-lubricated bearing, 
should it run dry from neglect or failure of the oiling device, suffi- 
cient grease to lubricate it for a reasonable period is placed in a 
suitable recess formed in the box. As long as the lubrication is 
sufficient to keep the bearing cool, the grease remains; but as soon 
as the temperature rises the grease melts and flows to the journal. 

Oil-grooves should be made with regard to the effective distribu- 
tion of the oil over the rubbing surfaces. If a horizontal journal al- 
ways rotates in the same direction in its box, it is not the best practice 
to cut the grooves longitudinally, for the oil is then aided but 
slightly, if at all, in its flow along the grooves by the action of the 
revolving surface. By cutting the grooves as right- and left-hand 
threads of very rapid pitch, both starting from the oil-hole in the 
direction that the journal turns, the oil will be drawn along the 
grooves by the journal, thus securing both a more rapid circulation 
and effective lubrication. This applies especially to ring- and 
chain-oiled bearings. 

Forced lubrication is frequently applied to journal-bearings. 
An oil pressure as high as 60 pounds per square inch or even 
higher is sometimes used. By this means the duty that a journal 



42 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

can perform is very greatly increased. It is especially useful for 
high speeds of rubbing. The pressure may be obtained by a pump 
driven by the machine itself, or by an overhead reservoir with pipe 
connection to the bearing. 

13. Materials for jonrnal-bearings. — ^As long as a suitable 
lubricant is continuously introduced between the rubbing surfaces 
of a journal and its box, almost any of the metals common to 
engineering practice can be used for these surfaces. Some will 
stand much higher bearing-pressures than others, but this depends 
generally upon the strength of the material. The serious injary 
that is almost certain to overtake many of the metals when copious 
labrication fails, even for an instant, precludes them from use 
except for slow speeds or light bearing-pressures. Others, while 
running without injury under both speeds and pressures compara- 
tively high, offer so much frictional resistance to the rubbing 
motion, that the power lost in overcoming it is so great as to be 
objectionable in some classes of machinery. A bearing that is called 
upon to perform heavy service continuously should, and often 
must, be made of a material, or materials, which will continue to 
run even though the lubrication is defective, and whose frictional 
resistance to the rubbing of the parts over each other is low. This 
last quality is generally, but not always, a necessity for continaous, 
heavy service; it is always desirable, however. If the frictional 
resistance is high, the material must either have good heat-conduct- 
ing capacity, so that it will carry away the frictional heat, or else a 
special cooling device, such as a hollow box with water circulating 
through it, may become necessary. A crude method of cooling, 
which is much used in rolling-mills on the bearings of the rolls, is 
to let a stream of water flow over the bearing. This would clearly 
be unsatisfactory where flowing water is objectionable. 

For heavy service, mild steel and wrought iron both run satis- 
factorily on brass or bronze. The journal is generally necessarily 
made of steel or iron on account of the required strength to endure 
such service. Brass and bronze exert a comparatively low frictional 
resistance to the rubbing of the steel or iron over them, and are 
both softer than iron and steel. This latter is an important prop- 
erty for the material of the box, for, should the lubrication fail, the 
iron or steel joarnal will continue to rub over it for a considerable 
time without serious injury to the bearing, this being a not infre- 



BEARINGS AND LUBRICATION. 43 

quent occurrence with the journals and half -boxes in railway prac- 
tice; or, if the box completely encircles the journal, it may seize 
and stop it without serious injury to the rubbing parts, since the 
softer metal will not cut into the surface of the steel or iron, and 
so long as the latter remains comparatively smooth, it \vi!l not cut 
and groove the softer metal to a serious extent. This cannot always 
be taken as true for the harder grades of the alloys just mentioned 

The "white metals" form a group of bearing alloys extensively 
used for lining the boxes of bearings. They have low coefficients 
of friction, a desirable property for bearing materials. On account 
of their weakness they are used only as a thin lining to fill a recess 
in the sheD of the box. They fuse at low temperatures, and readily 
flow into moulds when melted. They can be conveniently and 
economically used. For ordinary service their surfaces are left 
rough as they form when cast into place. .No machining is 
necessary. For high-grade, heavy service they are cast in place 
and then expanded to fit the shell tightly, either by peening with 
a hanuner or spinning with a blunt bumishing-tool, and then 
machined with a cutting-tool. 

These alloys do not injure the surface of the journal when 
lubrication fails, except possibly the hardest compositions. In 
case of excessive heating they melt and flow out of the box. This 
is a valuable property for some classes of high-speed machines in 
which sudden stoppage by seizure of the journal would be disas- 
trous. 

The composition of " white metal " aDoys is exceedingly varied. 
They generally contain either tin, lead, or zinc in considerable per- 
centage. Frequently much of two of these metals is present. 
Antimony is added to give hardness. 

Babbitt metal is a white alloy largely used for bearing surfaces. 
Its composition as now made covers so wide a range that the 
name cannot be taken to indicate any particular alloy. 

It is common practice among the best engine builders to line 
the principal bearings with a *' white metal " alloy. The main 
shaft and crank-pin bearings are so made. 

In railway practice the " half-box " of the axle-bearing is, often 
coated over with a thin layer of lead when new. The lead, on 
* See § 14 for compositions of bearing alloys for railway use. 



44 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

account of its softness, allows the journal to adjust itself to the 
bearing and obviates extreme localization of pressure, and at the 
same time acts as a metallic lubricant. 

Cast-iron boxes for steel and iron journals are quite extensively 
used, especially for line-shafting. The pressure between the 
Tabbing parts must be kept low, however, and even then the parts 
are in great danger of serious injury if the lubricant is not con- 
stantly applied. A continuous-oiling device, such as a ring or 
chain dipping in a reservoir of oil, becomes a necessity when the 
speed and pressure are at all high. If abrasion once begins, it is 
almost certain to injure both surfaces considerably, the cast iron 
generally suffering the most. The appearance of a bearing so 
injured seems to indicate that a particle of the cast iron is ground 
off and imbedded in the journal so as to slightly groove the box, 
and then the loose pieces of metal continue the grinding action in 
connection with the grooving, which roughens the surface of the 
journal still more and causes the destruction to go on, cutting away 
the box until the machinery is stopped. The brittle and crumbly 
nature of cast iron makes it exceedingly liable to great injury when 
abrasion once begins. The more close-grained the cast iron, the 
less liable is it to cutting. In order to keep the pressure per unit 
area as low as possible, cast-iron self-aligning boxes are made very 
long — ^as long as four times the diameter of the bore, or even 
longer in proportion. 

The comparatively low cost of cast iron, and the ease with which 
it can be cast into form and machined, make it a desirable material 
from the point of first cost; hence the reason for its somewhat 
extensive use for jonmal-boxes, as before mentioned. 

Chilled cast-iron journal-boxes are used on some classes of 
machinery not requiring accurate fitting of the rubbing parts. The 
bore of the box is cast against a chill which gives it a fairly accurate 
form. On account of the extreme hardness of the chilled surface, 
it is almost impossible to finish it with tools, but it may be ground 
smooth. In its almost exclusive application to the less accurate 
classes of machinery, the chilled cast-iron boxes are ased without 
finishing the bore other than by cleaning it carefully. The hard* 
ness of the surface eliminates the liability to the abrasion and con* 



BEARINGS AND LUBRICATION. 46 

tinned cntting common to iron which has been cast in the ordinary 
sand monid, and against sand or clay cores. 

Cast-iron joarnals are nsed on the rolls of rolling-mill machinery. 
Each roll is cast as a single piece, body and ends. The jonrnals or 
necks are cast in sand, so they have the ordinary crystalline struc- 
ture and softness of sand castings. These joarnals ran saccessfally 
on brass, bronze, or Babbijtt-llned boxes. The speeds and total 
pressnres are frequently qnite high, bat the joarnals are of lai*ge 
diameter, so that the bearing-pressnre per anit area is comparatively 
low. 

In some classes of metal-working machine-tools, where great 
rigidity is desired, a cast-iron spindle is nsed, whose joarnals ran 
on boxes of the same material. When so ased, the speed of rabbing 
is generally low, and the bearing-pressare per anit area always so. 
If well Inbricated at first, the rubbing surfaces soon take on a 
glazed, glass-like surface, which, once thoroughly fixed, will stand 
much neglect in the way of lack of proper lubrication. The same 
is true, possibly even to a greater extent, when the cast-iron journal 
runs on Babbitt metal, or on cast iron and Babbitt metal in alter-- 
nate strips. 

Hardened steel, or case-hardened iron or mild steel, journals 
and boxes, ground to form, are used frequently where it is especially 
desirable to maintain accuracy in the running parts. These hard 
parts run well together when the rubbing surfaces are smoothly 
finished. They will also run excellently on any of the materials 
that can be used on steel, wrought iron, or cast iron. Their cost 
is much greater than that of the unhardened materials. 

A self-lubricating material for journal-boxes has been invented 
in recent years. It is made of graphite held together by wood 
fibres. The process of making is to mix together the pulverized 
graphite and wood pulp in a bath of water, run the mixture into 
moulds having the form of the piece wanted, and then subject it to 
heavy pressure, forcing the water out through minute openings in 
the moulds for this purpose. The compressed ^^ fibre-graphite " is 
then treated with oil and baked to give it desirable qualities. The 
material thus manufactured is strong enough to work under 
pressures ordinarily, required for line- and counter-shaft boxes, 
pulleys, crank-shafts of small engines, etc. Ko lubricant is re- 



46 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

qaired or shonid be used, for the graphite is itself a solid lubricant. 
The application of oil or grease to a fibre-graphite box, not only 
does no good, but is harmful. The caase of their harmful action 
is as follows: The surface of a joamal is always somewhat rongh. 
When it is first revoWed in a fibre-graphite box, this roughness 
abrades the box to some extent, and the small particles of graphite 
thus loosened collect in the depressions on the journal, finally filling 
them flash with the higher parts of the metallic surface, thus form- 
ing the best possible rubbing surface. If oil is now applied to the 
bearing, it loosens these small particles from the journal and floats 
them away, leaving the surface in the original condition; the con- 
tinued application of oil will not allow particles of graphite to 
collect on the journal, and consequently the wearing away of the 
box continues much more rapidly than if it were left dry. 

The fact that the flbre-graphite bearing requires no attention 
whatever when running satisfactorily, and that oil and grease are 
entirely absent, makes it a desirable material for places difficult of 
access, and when oil and grease may be harmful. 

Some of the minerals are used in their natural condition for 
journal-boxes. Probably these are confined to the more close- 
grained ones almost exclusively. They are used without any lubri- 
cant for much the same reason as just mentioned for fibre-graphite. 
When an iron or steel journal first runs on a stone box, the latter 
grinds off the surface of the metal, which lodges in the interstices 
and irregular depressions of the stone, filling them and making a 
smooth surface which works satisfactorily. If oil or water is 
applied, the particles of metal are floated out of the concavities of 
the stone, and the grinding action is again started. 

The action of a stone box and steel journal can be readily under- 
stood by rubbing the blade of a knife over a clean oil-stone until the 
stone becomes glazed and stops cutting or grinding away the steel, 
and then applying oil, thus causing the stone to cut freely again. 

It is believed that no materials except the more precious stones 
are used in their natural state for bearing surfaces in this country. 
These are used only for light machinery, such as watches, clocks, 
electric meters, chronographs, etc. It is said that the spindles of 
large grindstones are quite commonly run on stone boxes in the 
cutlery-manufacturing establishments in England. 



BEARINGS AND LUBRICATION. 47 

Joamal-bearings that are sabmerged ia water and cannot be 
conveniently enclosed so as to be lubricated with oil or grease, fre- 
qaently have the boxes made of wood. Lignnm-vitsB, letterwood, 
and camwood are the varieties chiefly adopted. The former seems 
to be the most largely used. It is generally fixed with the end of 
the grain against the jonmal. The water itself is a good labricant 
for steel, wrought iron, or bronze, running on wood. It is therefore 
only necessary to provide a way for the water to get between the 
rubbing surfaces. This is often done by using thin strips of the 
wood set in a metal frame or casing so as to run parallel with the 
length of the journal, and extending out a short distance from the 
casing toward the journal, thus leaving small passageways between 
them for the water. 

The wood has another point in its favor, which is that there is 
little or no electrolytic corrosion of the journal or shaft passing 
through it, such as is always present when the journal runs on a 
metallic box. This corrosion is greatest in sea-water. It affects 
the journal most seriously at the end of the bearing, a groove form- 
ing around it at that locality. The steel of the shaft, the bronze 
or brass box, together with the salt water, present all the elements 
of a galvanic cell, and it can only be expected that electrolytic 
action will occur. When the shaft and box are in direct con- 
tact, the electrical resistance through the water at their exposed 
surfaces is low, and consequently the electrolysis is rapid. The 
casing holding the strips of wood can be made of the same 
material as the journal, thus reducing or eliminating the corrosion. 
Moreover, the fact that the journal and metal of the box are not 
in direct contact may be beneficial in reducing corrosion, but this 
can be of little moment if the box is supported on metallic fram- 
ing which in any way has contact with the shaft of which the 
journal is a part. 

14. Frictional resistance of journal-bearings. — If a horizontal 
journal rotates in its box under a vertical load P, and the sum of 
all the elementary frictional forces acting tangent to its rubbing 
surface, to oppose rotation, is /, then / -^ P = /i is called the 
coefficient of journal friction for the bearing. 

The quantity /may also be defined as the force which must be 
applied to a rotating piece at a distance from the axis of the journal 



48 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

equal to its radias, and normal to the axis, in order to keep it rotat- 
ing uniformly. 

In an oil-lubricated bearing, the frictional resistance partakes 
partly of the nature of the friction between two unlubricated solid 
bodies rubbing 07er each other, and partly of that of a solid moving 
through a considerable volume of liquid. The value of /i therefore 
lies between that for solid friction of the rubbing materials of the 
bearing, and that of the same surfaces passing through a volume of 
the lubricant. Any variation of the rubbing surfaces, the kind of 
oil, or the quantity that is applied, will affect the value of pi. 

Many experiments have been made to determine the value of /4 
and its method of variation with change of speed and pressure. 
These experiments, made by different investigators, are unquestion- 
ably nearly all, probably all, thoroughly reliable. The values 
obtained for /i, and the method of its variation, are so greatly 
different in nearly all cases that it seems impossible to arrive at any 
definite conclusion regarding them. The following facts seem to 
be fairly well established for mild steel running on brass, bronze, 
or white-metal alloys, operating under a steadily applied load: 

1st. In an oil-lubricated journal-bearing, the coefficient of fric- 
tion fJi at first decreases as the velocity of rubbing increases from 
zero, then increases with further increase of speed. The speed at 
which this change in the variation of fx occurs becomes higher as 
the bearing-pressure is increased. 

2d. In an oil-lubricated journal-bearing, the coefficient of fric- 
tion /I decreases, rapidly at first, then more slowly, and finally begins 
to increase, as the load is increased from zero up to the limit of 
working. 

3d. In an oil-lubricated journal-bearing, for the higher speeds, 
the coefficient of friction /i decreases as the temperature rises, until 
the lubricant either becomes so thin as not to sufficiently separate 
the rubbing surfaces, or is disintegrated; but for low speeds /i 
increases as the temperature rises. 

A series of tests for the coefficients of friction of the alloys given 
in Table I were made by Messrs. Joseph Kuhn and Bobert T. 
Mickle.* One journal each of wrought iron, steel, and cast iron 

* *' Variation of the Coefficient of Friction with Different Loads and Bearing 
Metals." Bngineering New$, May 18 and 25, 1898, pp. 468 and 404. 



BEARINGS AND LUBRICATION. 

Table I.* 

COMPOSITION OF BEARING-METAL ALLOYS. PERCENTAGES. 



49 





1 




i 


^ 


1 


BemarkB. 


A 

B 
C 

D 
E 
H 

I 


79.70 

74.90 

78.45 

74.90 
87.60 
85.71 
88.34 


10.00 

9.40 
9.80 

9.30 
12.50 
14.29 
16.66 


9.50 

9.45 

11.76 

11.15 


6.45 
6.36 


.80 
.80 


Peon. Ry. standard phosphor- 

bronze. 
Same as A with 6% ziuc added. 
Baltiinore & Ohio Ry. standard 

bearing alloy. 
Same us C with 6f^ zinc added. 
Copper 7, tin 1. 
Copper 6. tin 1. 
Copper 6, tin 1. 



* Sngineering Netoa, May 25, 1898, p. 496. 

Table II. f 

TESTS OF FBICTIONAL RESISTANCE OF ALLOY A, 





Wrought-iron Journal. 


Steel Journal. 


Cast-iron Journal. 


Pressure, 

pounds per 

square 

inch. 


Velocity 

of 
Bubbing, 
feet per 
minute. 


Temper- 
Fahr.deg. 


Coef.of 
Friction, 


Temper- 
ature, 
Fahr.deg. 


Coef.of 
Friction, 


Temper- 
ature, 
Fahr. deg. 


Coef.of 
Friction, 


125 

250 ] 
400 


246 

608 

1066 

280 

488 
968 

222 
451 
866 

247 

600 

234 


76.0 

89.0 

127.6 

86.3 
102.1 
136.0 

89.3 
100.9 
137.3 

86.6 
105.4 

91.5 


.0176 
.0199 
.0204 

.0086 
.0112 
.0122 

.0044 
.0074 
.0092 

.0038 
.0025 

.0018 


76.5 

88.0 

112.8 

83.8 

96.9 

128.6 

87.3 

99.1 

133.8 

87.0 
107.9 

94.0 


.0161 
.0181 
.0256 

.0095 
.0076 
.0107 

.0054 
.0065 
.0048 

.0050 
.0042 

.0035 


77.8 

91.6 

115.8 

84.5 
108.0 
129.8 

92.6 
100.8 
133.6 

89.8 
104.8 

92.1 


.0169 
.0179 
.0178 

.0092 
.0108 
.0137 

.0120 
.0078 
.0082 

.0057 
.0040 

.0038 



t Engineering Netoa, May 25, 1898, p. 495. 



50 



FORM, STRENGTH, AND PROPORTIONS OP PARTS. 



was used. The dimensions of all were the same, riz., 2 feet in cir- 
cumference and 1| inches long. The boxes were each 1.42 inches 
long, parallel to the axis of the journal, and 2 jf inches wide circum- 
ferentially. The projected area was 4 square inches. The journal 
was moved endwise ^^ of an inch 26 times a minute. The lubri- 
cant was " Extra ^' lard oil, as used by the Pennsylvania Railway 
Company. The results of the tests are given in Tables II and III. 



Table III.* 

COMPARATIVE TESTS OP BEARING ALLOYS UNDER PRESSURE OF 
250 POUNDS PER SQUARE INCH, AT A RUBBING SPEED OF 
500 FEET PER MINUTE. 





1 

s 

1. 


Wrought- iron Journal. 


Steel Journal. 








M-S 


c 




OB 

is 


c 




i 


o 




1 


It 


SI 


hi 


& 


II. 


1 




.P. 


Z 

1*4 




►» 




iss 


^4 






•i 




Ess 


*- 


1 

< 


1 

> 


P 




o 

8 ^ 


|i 







$^ 






A 


500 


105.4 


83.4 


.0025 


107.9 


41.2 


.0042 


104.8 


22.5 


.0040 


B 


475 


117.8 


51.7 


.0040 


125.8 


47.1 


.0061 


121.0 


35.2 


.0<»61) 


C 


478 


110.5 


29.8 


.0055 


113.5 


29.5 


.0040 


122.4 


54.9 


.0091 


D 


458 


113.0 


43.5 


.0056 


117.8 


68.7 


.0049 


127.8 


35.0 


.0070 


E 


473 


113.1 


49.5 


.0055 


113.1 


31.4 


.0052 


116.1 


9.9 


.0074 


H 


468 


111.1 


7.7 


.0043 


112.6 


25.9 


.0045 


116.1 


16.6 


.0066 


I 


500 


115.0 


25.7 


.0043 


117.3 


32.1 


.0088 


120.1 


13.9 


.0046 



* Et\giiieering News, May 25, 1893, p. 495. 

In Table II it can be seen that, for the wrought-iron journal, 
when working under 250 pounds per square inch, /j, falls from .0038 
to .0025 when the velocity of nibbing increases from 247 to 500 
feet per minute; evidently the speed at which ja begins to increase 
has not been reached. The same is true for the steel and cast-iron 
journals for the same pressure and speeds. At a pressure of 125 
pounds per square inch, the wroiight-iron journal seems to have 
passed the speed for the change in the variation of /i from decrease 
to increase; the cast-iron journal seems to have nearly reached the 
turning-i)oint, for /i increases from .0078 to only .0082 when the 



BEARINGS AND LUBRICATION. 



51 



speed changes from 451 to 865 feet per minute. In nearly all the 
other cases the turning-point has been passed. There is, as shown 
by the table, a considerable increase of temperature in all the cases 
of increase of speed, but the rise of temperature runs fairly imifonn 
for similar increases of speed. 

It was found, in a test of commercial-power-transmission 
machinery, that the coefficient of friction increased rapidly after a 
rubbing speed of 2.9 feet per second = 174 feet per minute was 
reached.* 

Table II also shows a continuous decrease in /z as the pressure 
increases for an approximately constant speed. The pressure does 
not reach a value high enough to cause [i to begin to increase. 

Fig. 33 t shows, for a rubbing speed of 314 feet per minute and 




aOOO 4000 tfOOO sooo 

LOAD ON AN AREA OF 8 
SQUARE INCHES. POUNDS. 

Fig. 33. 



10000 




Fig. 83.1. 



a temperature of 120° Fahrenheit, a decrease of /£ up to a bearing- 
pressure of about 800 pounds per square inch, and a slight increase 
of fi for increasing pressure above 1000 pounds per square inch, of 
projected area of the journal. 

Fig. 34 shows how // decreases as the temperature rises, for a 
speed of 314 feet per minute and a load of 1000 pounds per square 
inch of the projected area of the journal. 

X Exceedingly low values of the coefficient of journal friction 



* •'Experiments upon Friction in Electric Motors and Transmission Shaft- 
ing," by 8. Hanappe. Minutes of Proc. Inst. Civ. Eng., vol. cxxir., p. 496. 

t Fig. 33 may be found in Digest of PhyHcal Tests, July, 1897, p. 175; Fig. 
34 on p. 174. 

X Private communication from Professor Kingsbury, March, 1908. 



62 FORM, STRENGtH, AND PROPORTIONS OF PARTS. 

have been obtained recently by Prof. Albert Kingsbury. The 
tests were made on a hardened, ground, and polished steel journal 
If inches in dianxeter and 2 inches long. Each of the two " brasses" 
embraced 120*^ of the joumaFs circumference, as shown in Fig. 33.1. 
Great care was taken to obtain good rubbing surfaces and an accu- 
rate fit. The parts were submerged in a bath of the lubricant, 
with the axis of the journal vertical. The pressure between the 
bearing surfaces was secured by a spring that tended to force 
the two " brasses" together when they were in position on opposite 
sides of the journal. Table IIIa shows the minimum values 
obtained for the coefficient of friction at a bearing-pressure of 340 
pounds per square inch of projected area of the journal and differ- 
ent speeds. The lubricant was " velocite," a mineral spindle-oil. 
Kerosene or other oils of slight viscosity gave practically the same 
results. The bearing-pressure was never put on until the journal 
was at full speed, and relieved before stopping. Otherwise the rub- 
bing surfaces would be liable to Injury at slow speeds. 



Table IIIa. 

friction tests of an accurately fitted journal-bearing. 

Journal 1} inches diameter by 2 inches long, hardened, ground and 
polished. Two brasses each embracing 120** of circumference of the journal 
on opposite sides. Bearing submerged in oil. Axis vertical. No oil grooves 
or channels. 



Rev. per minute. 


Velocity of rubbing. 
Ft. per min. 


Temperature of oil - 
bath. Degrees Fahr. 


Minimum coeffldeDt 
of friction, fi. 


186 

118 

75 


67.1 

42.4 
27.1 


175 
186 
120 


.000458 
.000465 
.000458 



A striking example of the effect of the condition of the rubbing 
surfaces upon the coefficient of friction, fi, has been stated by 
Mr. A. H. Emery.* The materials rubbing together were hardened 



♦Trans. Amer. Soc. Mech. Engrs., vol. vi., p. 852 



BEARINGS AND LUBRICATION. 



63 



steel and non-hardened steel. By changing the hardened-steel 
rubbing sarface, which had been ground on a fine, solid emery-wheel, 
to the fii:^e finish produced by a polishing-wheel, /i was changed 
from 30^ to 3^; the pressure per square inch and the lubricant 
remained the same, and the same pieces of steel were used in both 
cases. The pressures were exceedingly high, reaching 50000 
pounds per square inch in some cases.* 

When no liquid lubricant is used, but the solid surfaces of the 
journal and box rub directly against each other, as in the fibre- 




4 8 12 16 IS 

TIME OF RUNNINQ. MINUTES. 

Fio. 84. 



graphite bearing, it is probable that there is little, if any, change 
in the value of ^ with change of load and speed, except such as 
may be caused by deformation of the box, unless there is excessiye 
heating. 

The power which is expended in overcoming the frictional 
resistance of a journal is transformed into an equivalent amount of 
heat at the rubbing surfaces. This heat must be conducted away, 
first by the lubricant to the rubbing surfaces, when a lubricant is 
used, and then through the materials of the journal and box, so 
that it may radiate, or be carried away by some cooling substance, 
as water, when special means of cooling are provided. The oil, 
therefore, must have, as one of the qualities of a good lubricant, 
considerable heat-conducting capacity when used for heavy service. 
It is always desirable, however, that its unctious properties shall be 

* Mr. Emery states in a private communication that this pressure was used 
on tbe wedge-sbaped jaws oi' a it-sting machine holder. 



54 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

such as to reduce the generation of heat to a minimnm, thna 
reducing the power expended in overcoming friction. 

15. Effect of changing the proportions of jonrnal-bearings. — ^In 

the majority of cases where a bearing is to perform heavy service, 
the diameter is made as small as the requisite strength will permit. 
The length is then made such that the bearing-pressure per unit ' 
area will be low enough for satisfactory operation. 

In order to examine the effect of changing the diameter of a 
journal, let it be assumed that a bsaring, running with a solid 
lubricant, and designed to operate under a given load and speed, 
heats to excess. It therefore becomes necessary to change its size 
in order to secure cool running. First, assume that the diameter 
is increased, the length remaining unchanged. For convenience 
in pointing out the effect of this change it will be assumed that the 
diameter is doubled. The frictional resistance to rotation =/, 
acting tangent to the surface of the journal, is unchanged ; for the 
coefficient of friction fi is the same in both bearings before excess- 
ive heating occurs, and the load is the same for both. The dis- 
tance through which / acts, during one revolution of the larger 
journal, is twice that for the smaller. If /> = the diameter of the 
smaller, and 2*Z> that of the larger, journal, the energy converted 
into heat by the frictional resistance during one revolution will be 
nDf for the smaller, and ^nDf for the larger; that is, twice aa 
much mechanical energy is wasted, and twice as much heat pro- 
duced, in the larger as in the smaller journal. The larger bearing 
has twice as much area of material at the rubbing surface for 
carrying away the heat, as the smaller; therefore the quantity of 
heat to be conducted away per unit area of the material is the same 
for both. This shows that the liability to excessive heating is just 
the same in both bearings, and that, while increasing the diameter 
brings no improvement in the way of cooler running, it doubles the 
waste of energy in the bearing. 

Again, it may be assumed that the length of the journal is 
doubled, the diameter remaining D = that of the original journal. 
The frictional resisting force /is the same as before; the diameter 
is also the same. The mechanical energy transformed into heat is 
therefore the same as before, and equals nDf. The area of the 
material through which the heat is conducted away from the 



BEARINGS AND LUBRICATION. 55 

Tabbing anrfaces is twice as great in the longer as in the shorter 
bearing; hence the heat that mast pass throagh a nnit area of the 
material is bat one half as great in the elongated bearing as in the 
original. The longer joamal is conseqaently less liable to excessive 
heating ; the mechanical energy lost in it is the same as in the shorter. 
In a general way, it may be said that doabling the length, without 
changing the diameter, of a bearing ranning with a solid lubricant, 
redaces its liability to excessive heating by one half, without chang- 
ing the power necessary to overcome its f notional resistance. 

When excessive heating of an oil-labricated bearing cannot be 
prevented by the best labrication, or by putting the robbing sur- 
faces in their best condition, together with proper precautions for 
cooling, etc., it is evident that there is need of change in the 
dimensions. It is assumed that the journal is to run at a given 
rotative speed and under a given total load. If the bearing-pressure 
per square inch is very high, and the velocity of rabbing compara- 
tively low, it may be remedied by increasing the diameter; or, if 
the bearing-pressure is low, and the velocity of rubbing high, 
decreasing the diameter may prove effective. The strength of the 
journal must be considered, of coarse. Lengthening the journal 
will reduce the liability to heating; but if, on account of too great 
length, the pressure becomes localized, as at one end, the trouble 
may be increased. On account of the numerous causes affecting 
the successful running of an oil-lubricated bearing, it does not seem 
possible to arrive at any more definite conclasions than these. 
Each problem must be considered with regard to the conditions 
affecting it, and dealt with accordingly. 

16. Proportions of journal-bearings, and examples from prac- 
tice. — The most recent and valuable formulas for the proportions 
of the bearings of engines have been given by Professor John H. 
Barr of Cornell University.* The formulas, or at least the con- 
stants in them, are derived from American practice in engine-con- 
struction. The engines are divided into two classes: '^ low-speed " 
for Corliss and other long-stroke engines usually making not more 
than 100 to 125 revolutions per minute, and "high-speed" for 
those having a stroke from one to one and a half times the piston 

♦Trans. Amer. Soc. Mech. Eng., vol. xviii., 1897, p. 756. 



66 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

diameter, and a rotative speed of 200 to 300 reyoIationB per minute. 
The data for the formulas were obtained from about eighty separate 
•engines classed as high-speiBd, and about eighty-five classed as low- 
fipeed. The high-speed engines ranged from 20 to 240 H.P«, and 
the low-speed engines from 46 to 740 H.P. The practice of 
thirteen builders is represented in the high-speed, and of twelve 
builders in the low-speed, engines. The equations are given for the 
average maximum, average minimum, and mean sizes of the parts. 
The notation for journal-bearings is: 

A = area of piston, square inches; 

£ = a constant; 

67= a constant; 

D = diameter of piston, inches; 
H.P. = rated horse-power; 

ir= a constant; 

L = length of stroke, inches; 

if = a constant; 

JV"= revolutions per minute; 

8 = steam-pressure, taken at 100 pounds per square inch 
above exhaust, as a standard pressure; 

a = dl=i projected area of journal, square inches; 

d = diameter of journal, inches; 
f = length of journal, inches. 

Maik joubkals: 

''=<^{/¥' (•) 

z = jrrf (2) 

The projection of the journal equals 

dl = MA (3) 

The values of the constants for the main journals are: 

Hi^h-speed Centre-crank Engine. Low<«peed Stde-cmnk Engine. 

For each of two journals. One journal only. 

Mean. Maximum. Minimum. Mean. Maximum. Minimum. 

(7=7.8 as 6.5 6.8 8.0 6.0 

iT =2.2 8.0 2.0 1.9 2.1 1.7 

Jf= .46 .70 .87 .66 .64 .46 



BEARINGS AND LUBRICATION. 57 

Cbank-pin: 

l=C^ + B; , (4) 

JTA 
d= f^ ox dl^KA (5) 

The yalaes of the constants for the crank-pin are: 

High-speed Engines. Low-speed Engines. 

Mean. Maximum. Minimum. Mean. Maximum. Minimum. 

-B=:2.5" 2.6" 2.6" 2.0" 2.0" 2.0" 

C^ .80 .46 .18 .6 .8 .4 

^= .24 .44 .17 .09 .115 .065 

Gboss-hbad pin: 

a = rf? = (7-4 ; (6) 

l:=Kd. (7) 

The yalaes of the constants for the cross-head pin are: 

High-speed Engines. Low-speed Engines. 

Mean. Maximum. Minimum. Mean. Maximum. Minimum. 

(7= .08 .11 .06 .07 .10 .054 ' 

K = 1.25 2.0 1.0 1.8 1.5 1.0 

Problem in the design of journal-bearings. — The application 
of the above formulas may be illustrated in the following problem: 

It is required to determine the dimensions of the main bearings, 
crank-pin, and cross-head pin of a centre-crank engine haying a 
stroke £ = 14 inches, piston diameter 2> = 12 inches (correspond- 
ing to ^ = 113.1 square inches), and rated at 100 H.P. for a speed 
JV'= 250 revolutions per minute. 

This comes under the high-speed class. 

Main journals: 

d = Cy/^ = 7.3//12? = 7.3 X .7368 = 6.38"; 
l = Kd=2.%x 6.38 = 11.84". 



58 FORM, STRENGTH, AND PBOPOBTION8 OP PARTS. ^^ 

This corresponds to a Talae of M = .66 in the equation 
dl = MA. 
Crank-pin : 

I = c^^ + B = .3^ + 2.5" = 2.14 + 2.6 = 4.64"; 
KA_ .24 X 113.1 _ „ 

Cross-head pin : 

dl= GA= .08 X 113.1 = 9.048 sq. in. ; 
l = Kd=1.25d. 

Therefore; 

d X l.25d = 1.25rf' = 9.048; 
d* = 9.048 -4- 1.25 = 7.238; 
tf = 2.69"; 
I = 1.25t? = 1.25 X 2.69 = 3.36". 

The nearest convenient working dimensions would naturally be 
used in practice. 

With regard to the difference in the allowable working pressures 
for constant, as compared with intermittent, loads, Mr. Babcock 
made the following statement:* '* I found that in crank-pins with 
good fitting I could allow as high as 1200 pounds maximum to the 
square inch ; pins, perhaps 4 to 6 inches diameter, running up to 
60 or 70 revolutions, would stand that continuously without getting 
warm. The main journal of the same engine would not stand over 
300 pounds to the square inch without getting warm.'' 

Regarding the locomotive Lady of the Lake, Druitt Halpin 
states f that, at the beginning of the stroke, the total pressure on 
the crank-pin 3^ X 3 inches is 28140 pounds, which gives 28140 -r- 
(3 X 3^) = 2680 pounds per square inch of projected area. The 
modifying effect of the reciprocating parts is not considered. In 

* Trans. Amer. Soc. Mech. Engrs., vol. vi., 1885, p. 856. 

f Minutes of Proceedings of the Inst. Mech. Engrs., 1883, p. 657. 



BEARINGS AXD LUBRICATION. 59 

the same engine the main bearings are 6 X 6 = 36 sqnare inches, 
and carry 8 tons each, or 498 pounds per sqaare inch. 

Mr. Beaachamp Tower quotes Mr. Tomlinson as saying that 
300 pounds per square inch is undesirable in locomotiye-azle bear- 
ings, while 1000 pounds per square inch, and considerably more, 
can be used on the crank-pins.* 

Mr. Henry Davy states that in pump-bearings, for speeds of 
rubbing up to 12 feet per minute, 600 pounds per square inch for 
continuous pressure in the same direction, and 1000 pounds per 
square inch for intermittent pressures, can be satisfactorily used.f 

In a freight locomotive built by the Schenectady Loco- 
mofcive Works for the Boston & Maine Eailroad Company, the 
bearing-pressures, due to the weight of the parts supported, 
calculated from data given in EngiJieering Review of March 26, 
1898, are: 215 pounds per square inch of projected area for the 
driving-wheel axles, and about 250 pounds per square inch for the 
tender-bearings. The driver-axle bearings are 8 inches diameter 
by 10 inches long; the tender- journals 4^ inches diameter by 8 
inches long. 

Dr. C. B. Dudley states that in railway practice, bearing-pres- 
sures as high as 350 to 400 pounds per square inch are used.^ 

President Joseph Tomlinson of the Institute of Mechanical 
Engineers is recorded as saying: '* The practical limit at which he 
had arrived watf 2^ cwt. per square inch; and if more than this 
pressure were allowed to the bearings of a locomotive engine, not- 
withstanding their freedom to wabble from side to side, they would 
not run cold. . . . Whenever he had an engine which would not 
run cold, and in which the weights upon the bearings could not be 
changed, he had put a bigger axle in and thereby cured the heating 
directly." § 

Several of the leading builders of large engines, such as are used 
for direct driving of electrical generators, state that the highest 
pressure they consider safe for crank-shaft bearing is 150 pounds 
per square inch of projected area of the journal. The same value 

*Pioc. Inst. Mecb. Engrs., 1884, p. 80. 

t Ihid., 1883, p. 654. 

I Journal of the Franklin Institute, 1882, p. 88. 

§Proc. Inst. Mech. Engrs., 1891, p. 181. 



60 FORM, STRENGTH, ANb PROPORTIONS OF PARTS. 

is given as the safe limit for small high-speed engines by the buildera 
of this class of machinery. 

Mr. Edwin Reynolds, of the Allis-Chalmers Co., is qnoted as 
follows regarding journal-bearings :* *^ The sqaare root of the speed 
in feet per second multiplied by the pressure per square inch of 
projected area, should not exceed 500." In reply to an inquiry, 
he is further quoted in the same place as saying: *^ It is true I have 
used the rule you mention for a limit and never go up to 500, ex- 
cept in Yertical engines where the steam pressure in the cylinder is 
sufficient to lift the shaft against the cap. 350 to 375 should be 
the limit for horizontal engines. This method for determining size 
and proportions has proved satisfactory in a very large number of 
machines." 

Step-hearings and Button Thrust-bearings. 

17. When the weight of a vertical shaft and the parts attached 
to it, together with whatever end thrust may come on it, are sup- 
ported by a box which bears against its end, and at the same time 
prevents it from moving sidewise, the whole combination of the 
rubbing parts and others in the immediate neighborhood is called 
a step-bearing. 

18. Forms of step-bearings. — The simplest form of step-bearing 
is shown in Fig. 35, where the step B is bored out to fit the end of 



Fig. 85. 

the vertical shaft A, In this case the bearing is made up of only 
two parts. The sloping part at the top of the box holds the oil that 
is used for lubrication. Badial grooves are generally out across the 

^American Machinist, Oct. 16, 1898, p. 879. 



BEARINGS AND LUBRICATION. 



61 



bottom of the shaft to allow the oil to gain access to all parts of the 
horizontal bearing sarface. Bearings of this class are commonly 
nsed where either the speed of rotation is low, or the pressure is 
light. Pillar-cranes are ordinarily sapplied with such a bearing for 
stepping the mast. In this application the pressure may be high^ 
for the speed of rotation is very low. 

When the shaft in Fig. 35 makes one revolution, the parts on 
the end farthest from the centre rub over the box through a dis- 
tance equal to the circumference of the shaft, while those near the 
centre rub over a much smaller distance, and the geometrically 
central point does not have any motion over the box. On account 
of this inequality of rubbing there will be a corresponding uneven- 
ness of wear, so that if the parts are fitted together accurately when 
new, so as to make the pressure uniform over the end of the shaft, 
the outer portion will wear away most rapidly in seryice, thus caus- 
ing the pressure to become heaviest at the centre, and lightest at 
the outer part, of the rubbitig surfaces. The pressure at the centre 
may become so intense as to crush the material at that point. Even 
if this does not occur, abrasion and cutting are likely to take place. 
The inequality of weiir and pressure may be partly obviated by 
removing some of the material at the centre of the rubbing parts, 
leaving a pair of annular rings for the rubbing surfaces. This is 
advisable in nearly all cases. 




Fio. 86. 



If the speed of the shaft is high, difficulty will be experienced 
in lubricating it on account of the centrif tigal action of the rotating 



62 rORM, STRENGTH, AND PROPORTIONS OF PARTS. 

part throwing the oil from the centre and not allowing it to retam 
again, unless some special provision is made for its doing so. Sach 
provision can be readily made, however, as shown in Fig. 36, by 
making an oil-i>a88age from the top of the step to the centre of the 
bottom of the bearing. This arrangement forms a small centrifugal 
pump, which draws the oil in at the bottom of the bearing throngh 
the oil-passage, throws it to the oater part of the bore, and forces 
it to the groove aroand the top of the step, so that it will again be 
ready to start on the same circuit. Complete and free circulation 
may be thus secured. 

For heavier duty, either on account of increased speed or pressure, 
this form of bearing may be made more durable by placing a number 
of disk-shaped washers between the end of the shaft and the box, 
as shown in Fig. 37. One set of these washers is generally made 




Fig. 87. 



of some hard material, such as steel, and the other set of a softer 
material, as brass or bronze. The two kinds are then placed alter- 
nately, so that each washer rubs against a material different from 
itself. If the shaft is of mild steel and the box of cast iron, which 
is a common constraction, the top washer is often fastened to the 
shaft, and the bottom one to the box, thus making all of the wear 
come upon the washers. The number of pairs of bearing surfaces 
over which the wear is distributed is one more than the number of 
free washers. The series of washers permits a slower speed of rubbi ng 
between each pair of the surfaces, and, in case abrasion should begin 
between any pair, the rubbing motion will cease there until the oil 



BEARINGS AND LUBRICATION. 



63 



has an opportanity to get between them, or until repair can be 
made, withont serions injary to the machine or the necessity of 
stopping it. The washers generally have a hole bored through the 
centre, and are grooved radially for oil. The same device for 
securing circulation of the oil as is shown in Fig. 36 can be used, 
of course. Hardened and ground tool-steel, or case-hardened and 
ground mild-steel, washers running on brass or bronze give most 
excellent service. 

In machinery where the shaft and box cannot be accurately 
aligned, or where they may get out of line from some cause, such 
as settling of the supporting parts or springing of the shaft, 
lenticular washers with spherical faces of the form shown in Fig. 
38 may be used. By making them smaller in diameter than the 



p 


'■■ 


1 


31 


1 ' 


L- iin 












_..^--- 




„. 


^_^ 


- — ■ : 


■■^— ., 




^— ' ■"'■ 


■■ -•.■''^<Hs^L->aiiiiJ 




■■# 


M 


li 


w ■■ 



^^ 



JMii^^ilMi 




mm ^^^^^^^^^ 



Fie. 88. 



Pig. 88.:. 



bore of the box, they will adjust themselves to a perfect bearing for 
any relative position of the shaft and box, within the limits for 
which they are designed. As with the flat washers, however, the 
wear will be more rapid at the parts more remote from the centre. 

Fig. 38.1 shows another method of allowing for self-adjustment 
of a step-bearing. It is less expensive than the preceding design. 
The step proper is spherical at the bottom and the base-plate is con« 
caved to fit it. The step may be machined to form on the bottom and 
A " white-metal " alloy cast in the floor plate to fit it. The two pins 
are to prevent rotation of the step. If they press against the wings of 
the step at points on a horizontal diameter of the sphere, there will 



64 



FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



be no tendency to throw the step from its place when a motion of 
adjustment takes place. The Bottom of the step may be cut away 
as indicated by the broken line and only a corresponding portion of 
the socket used. It is not necessary to have both surfaces spherical, 
especially for light service. One, the support, may be conical or 
any other form that will hold the spherical surface and allow it to 
adjast itself. 

The step-bearings for turbine water-wheels running on vertical 
shafts often have a lignum-vitsB step. Fig. 39, which supports 
the metallic shaft. The step is made crowning on top, and the end 
of the shaft cupped to fit it, the rubbing surfaces are spherical in 
form. Water lubrication is easily obtained by cutting radial grooves 
in the rubbing surfaces, for the bearing is surrounded by water, 
which will flow into the grooves. If the 
speed is high, the water can be made to circu- 
late freely through the grooves by boring a 
hole through the centrp of the wooden step 
from the bottom upward to the rubbing sur- 





Fio. 89. 



Pig. 89.1. 



faces, as shown. The centrifugal action will throw the water out 
through the grooves to the circumference of the bearing, and at the 
same time draw it in through the hole in the centre of the step. 

Exceedingly heavy service, both as to speed and pressure, is 
sometimes required of a step-bearing. It is not advisable, or even 
practicable in many cases, to meet this requirement by increasing 
the diameter of the bearing; for not only does the wear increase 
with the diameter, as has already been pointed out, but there is also 



BEARINGS AND LUBRICATION. 65 

a correBponding increase of f rictional resistance with its accompany- 
ing increase of power loss in the bearing. Moreover, the liability 
to abrasion and catting is also increased. Some other means mast 
therefore be adopted for secaring the desired qaalifcies. 

Forced lubrication affords what seems to be the most satisfactory 
solution for secaring the saccessf al operation of step-bearings for 
heavy daty. This can be applied most easily and economically in 
the water-labricated bearing with the wooden step, Fig. 39, when 
nsed in connection with a water-wheel. In such a case it is only 
necessary to connect a pipe to the bottom of the hole in the bearing, 
and lead it to the water in the fore-bay. Practically the whole 
head of water is thas made available for forcing the labricant, 
which is the Water itself, in between the robbing surfaces. This 
assumes that the bearing is below the wheel, so that the lubricating 
water can flow freely from the bearing to the tail-race. The oil- 
grooves, if used, should extend from the centre only part way to 
the circumference. If extended clear across the rubbing surfaces, 
the lubricant would be forced through them without performing its 
function. 

To secure a higher pressure, for forcing the lubricant into the 
bearing, than the head of water will give, a force-pump may be 
attached to the pipe connected with the bearing. 

Any of the step-bearings shown in the preceding figures can be 
lubricated with oil in a manner similar to that just described for 
using water, it being necessary to provide a reservoir for catching 
the oil which overflows from the top of the bearing, and a pump 
for taking up this oil and forcing it into the bearing again. 

18.1. Button thrust -bearings are a suitable form for light 
woik. They are even used for comparatively heavy pressures 
and high speeds in some cases.* 

Usually a crowned surface presses and spins against a flat 
plate. Both are hardened for durability. Since steel balls have 
come into such common use that they can be obtained at a low 
cost, they have been utilized to some extent for this form of bearing. 

Fig. 39.1 shows an easily constructed thrust-bearing of this 

type. The end of the shaft is bored and a steel ball fitted snugly 

into it. The step is recessed for a hardened steel disk. When 

used in a vertical position as shown, the rubbing surfaces may 

♦See § 77, Table XVI, for worm-gears with button thrust- bearing. 



66 



FORM, STRENGTH, AND PROPORTIONS OP PARTS. 



be submerged in oil. A high speed would throw it away from the 
rubbing parts. 

Fig. 39.2 has a loose ball between two plane surfaces. The 
ball is kept in place by a quill that has a loose fit over the two 



;.QLflLL 




Fig. 89.2. 

end plugs of the rotating parts. This design has the advantage 
of two pairs of wearing surfaces. The ball may rotate so as to 
present different parts of surface to the plugs. One plug is tapered 
so as to be easily removable from the shaft, but is more expensive 
than the other, which is cylindrical. 

Pivot-hearings. 

19. Conical pivot-bearings of the form shown in Fig. 40 are 
extensively used in light machinery. The ease with which they can 
be adjusted for wear is the chief factor in bringing them into nse 
for light work. The wedge-like action of the point, and the 
unequal wear on the rubbing surfaces, prevent their use for heavy 
machinery to any considerable extent. 

The pressure acting over the conical bearing surfaces may be 
assumed as acting at two diametrically opposite points for the pur- 
pose of finding the amount of the total normal pressare between 
them. Thus, in the figure, for the thrust P the normal pressure 
iV = (P -T- 2) CSC (^ -7- 2), and the total normal pressure is 



2N= Pcsc 



e 



The intensity of pressure per unit area is the same as if the load 
were supported on a flat-ended shaft of the same area as the projec- 
tion, on a plane normal to its axis, of the bearing surface of the 
pivot. In other words, the angularity of the pivot-point does not 
affect the pressure per unit area on the bearing surfaces. This is 
showQ by dividing the total normal pressure 2iVby the area of one 
of the rubbing surfaces. The area A of the conical rubbing sur- 
face of the pivot, Fig. 40, is 



BEAKIKOS AND LUBRICATION. 



67 



A = %itR X i( VB) - %nr X i( VC) 

6 6 

= nR X -B CSC - — ;rr X r C8C r- 



e 



= ;r(i2' — r')c8C^. 




Fig. 40. 
Hence for the pressare p per unit area 



2N ^^4 

^ = ■3- = 



;r(ir — r') csc - ^ ' 

The angle 6 does not enter the last member of this eqnation; 
hence p is not affeoted by the angle of the cone. 

The wear will be more rapid on the parts having the greatest 
radial distances from the axis, and conseqaently the greatest rabbing 
action. The result is that after use the pressare between the 
rabbing sarfaces will be greater near the point of the cone than 
near the base. For this reason a portion of the box is generally cut 
away at the centre, as indicated in the figure. 

It is common practice, in light machinery, to use a conical 
bearing at each end of a shaft or spindle. It will then withstand 
side pressure as well as thrust. 



68 FOBM, STRENGTH, AND PROPORTIONS OF PARTS. 

20. The <<tractrix" or "curve of constant tangent" is the only 
theoretically correct outline for a step- or pivot-bearing, since it is 
the only one that will wear nniformly over the rubbing surfaces? 
and thus maintain a uniform pressure per unit area between the 
surfaces. It is also called "Schiele's anti-friotion curve/' after 
its discoverer. The nature of the curve can best be described 
by the method of drawing it. This can be done on a piece of 
smooth horizontal paper in the following manner: In Fig. 41 
the line AB is taken for the directrix of the curve. A beam- 




compass is placed so that one point, (7, is on the directrix, and 
the other, />, lies on a line drawn normal to the directrix at (7. 
C is then moved along AB toward B so that D trails freely after 
it, care being taken to hold the beam-compass by the point C so 
that D is not thrown out of the path it will naturally follow when 
there is nothing to press it aside. If /> is a smooth, round pencil- 
point, it will trace the tractrix DE. A wedge-pointed pencil may 
also be used at D by placing it so that its edge takes the direc- 
tion CD. When using this kind of a pencil-point, it is best to draw 



BEARINGS AND LUBRICATION. 



another curve, FG^ ou the opposite side of the directrix, starting 
with G at the same point as before. They can then be tested for 
accuracy by folding the paper along AB^ so that they will coincide 
if correctly drawn. 

From the method of drawing the tractrix, it is evident that, when 
the compass is in any position cd^ the line joining the points c and 
e/must be tangent to the curve at d\ the distance cd^ being that 
between the points of the beam-compass, remains constant; there- 
fore, on any line drawn tangent to the tractrix, the distance from 
the point of tangency to its intersection with the directrix is a con- 
stant. Hence the name " curve of constant tangent." 

The tractrix can be continued to an indefinite length, but in 
practice only a portion of it is used. This is generally taken from 
D toward ^ as far as desired. Fig. 42 represents the end of a shaft 




turned down to this form of profile, and resting upon the support- 
ing part of the bearing. The directrix coincides with the centre- 
line of the shaft, and the largest radius is equal to the distance 
between the compass-points that were used for describing the curve. 
The proof that the tractrix will wear away uniformly over the 
entire rubbing surface is as follows: In Fig. 43 a narrow band or 
ring of the bearing shown in Fig. 42 is represented. The width of 
the band, measured along the rubbing surface, is 8\ this width is 
assumed to be so small that the curved surface may be treated as if 
it were conical in form, but it is necessarily magnified in the draw- 



70 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

ing so that the parts may be seen. The mean radias of the ring is 
r, and its horizontal thickness is t. The line R has the direction 
of the carve at its mean radias, and the length of iZ, between the 
point of tangency and the directrix ABj is the same as the greatest 
radias of the bearing, as shown in Fig. 42. 

It has been shown in the preceding section that the pressure on 
a conical pivot-bearing is not affected in its intensity per unit area 
by the angle of tho cone; therefore, in a correctly fitted tractrix- 




bearing, the pressure per unit area is the same over the entire 
rubbing surfaces. The wear upon each unit area of one surface 
may therefore be taken as proportional to the distance through 
which it rubs against the other while making one revolution. For 
convenience it may be assumed that 

V = volume of material worn from a unit area of one rubbing sur- 
face when the sarf aces rub over each other through a unit 
distance under any given pressure. 

The area of the rubbing surface of the ring is ^Ttrs; the distance 
through which a point on the ring rubs during one revolution is 
2nr; therefore the volume of the material worn from the ring 
during one revolution of the shaft equals 

F=27rr« X 2w X,v. 



BEARINGS AND LUBRICATION. 71 

In order to see more clearly how the ahaft will settle Tertically 
on its sapport when this amount of material is removed from the 
small band which has been selected, it may be assomed that this 
band is the end of a thin tube, of a thickness ty cat entirely free 
from the rest of the bearing. The sectional area of this tube, on -a 
plane perpendicular to its axis ABj is 27rrt. In the fignre it can 
be seen that, in the similar right triangles, one having the hypothe- 
nuse 8 and side /, and the other the hypothennse R and side r, 

8T 

t : 8 =: r : Rf OT t = si therefore the area of the cross-section of 
the tabe equals 

A = 27trt = 27rr%. 

The amount JTby which the tube is shortened during a revolu- 
tion is the quotient found by dividing the quantity of material 
removed by the area of the cross-section of the tube, which gives 

A ^ 8r 



This equation shows that the shortening of the tube is independ- 
ent of its radius r, and is represented 6y the continued product of 
the constant 2^, and the quantities v and 72, which are also con- 
stants for a given bearing, load, and solid lubricant. The shorten- 
ing of this tube is therefore the same as that of any other that may 
be cut from the bearing, which shows that the complete bearing 
wears away so that a uniform pressure is maintained between the 
rubbing surfaces, and the bearing will retain its original form. 

The equation of the tractrix, which may be used for obtaining 
points through which to draw the curve, can be developed as 
follows: In Fig. 44, AB is the directrix of the curveri? any point 
on the tractrix, and R = CD the tangent at p; the coordinates of 
p are x and y. Taking the differential portion d8 of the curve at 
j9, and the corresponding dy and dx^ it can be seen that, in the 
similar right triangles, one having the hypothenuse d8 and sides 



72 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

ix and dy^ the other the hypothenose R = CD and aides x and 



dy ^ VR-7? 
dx ^ X * 




Fig. 44. 

Integrating this expression, and taking into account only the 
positive signs, gives 



= Elog,^+^^^^^^-VW^ 



The natural system of logarithms must be used in this equation. 
This is indicated by the subscript €. (e = 2.7182818.) The 
natural logarithm of a number may be found either in a table of 
natural logarithms, or by multipljring the common logarithm of the 
number (i.e., the logarithm to the base 10, which is written log^J 
by log. 10 = 2.3025851. 

In the equation it is convenient to put iZ = 1 when solving for 
corresponding values of x and y. Any system of units or scsaJe of 



BEARINGS AND LUBRICATION. 



73 



drawing can be adopted for plotting the curve so as to give it the 
required size. By patting R = 1 the equation becomes 



y = natural log —^ i^l — a?% 



or 



y = 2.3025851 common log '^ + ^^ ^' _ l/T^T?, 



For a point having a; = .6, the solution for y is, with natnral 
logarithms, 



y = nat. log ^ "^ ^^" ^'^^' - Vr^^^Q 

.0 

= nat.log^+f^^-.8 
.6 

= nat. log 3 — .8 

= 1.0986 - .8 = .2986. 

The solution by common logarithms differs from this only in 
taking the logarithm of 3. Thus, by common logarithms, 

y = 2.3025851 com. log 3 — .8 
= 2.3025851 X .4771213 —.8 
= 1.0986 - .8 = .2986. 

Other values of the coordinates, calculated as above, are given 
in Table IV. 

Table IV. 

COORDINATES OF TRAOTEIX. 



X 


V 


X 


V 


1.0 





A 


.650 


.9 


.031 


.3 


.920 


.8 


.093 


.2 


1.818 


.7 


.182 


.1 


1.997 


.6 


.298 





00 


.5 


.451 







74 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



The tractrix-bearing can be labricafced, when rnnning at high 
speed, in the same manner as the step-bearing, Fig. 36. It is 
possible that, when oil-labrication is used, the wear is not aniform 
on acconnt of the variation of the coefficient of friction with speed 
of rubbing at the different radial distances. 

This form of bearing is but little used, the only apparent reason 
being that it is difficult to make. 

Collar-hearings. 

21. Collars extending out from the surface of a shaft are gen- 
erally provided for taking the end thrust of a shaft that cannot be 
supported by a step-bearing. Fig. 45 shows a shaft with such 
collars resting in a box of suitable form to receive it. 

If, instead of a number of collars, a single one were used to 
resist the thrust, the area of its side pressing against the box would 
have to be equal to or greater than the combined areas on one side 




. ■• ■: ■ ■ ; ■ I ' :'~y>}l ■'-■'-' -<^ ' !■"'■':■- ' - ■ ' v ; ' 



1 1.1- ^ i 



Fig. 45. 

of all the collars shown, in order to keep the pressure between the 
rubbing surfaces the same per unit area. This would necessitate 
making the collar of a comparatively large diameter. This increase 
of diameter would bring into action the uneven wearing so marked 
in flat step-bearings, and also increase the frictional resistance to 
turning, in about the same proportion that the mean diameter of 
the rubbing surface of the single collar is greater than that of each 
collar when several are used. This is one reason, probably the 
most important one in most cases, for using the multiple-collar 
thrust-bearing instead of a single collar; another reason is that a 
greater economy of space can be secured with the smaller diameter 
of the multiple-collar bearing. 



BEABINOS AND LUBRICATION. 



75 



When the thrnst is always in the same direction, as indicated by 
the arrow in Fig. 46, the bearing may be shortened, while still 



m 



u 



ly 



uru 



B- 



Fig. 46. 



retaining eqnal strength and wearing surface, by making the collars 
of the form shown in this figure, where the flat faces are of the same 
area as in the preceding figure, but are brought nearer together. 

In order to obtain the best service from a collar-bearing working 
under heavy pressure and at high rubbing speed, each collar should 
have its own individual bearing-ring, separately adjustable. By 
this means, which is the common practice for large machinery, each 
ring can be adjusted for equal bearing-pressure, even though the wear 
on the different ones may be unequal; and in case heating should 
occur at any collar, its ring can be adjusted to partly relieve the 
pressure between them until they run cool again, or it may even be 
removed for repair while the machinery is operating, the load being 
carried by the remaining collars in the meantime. Elaborate collar- 
bearings are used on the propeller-shafts of screw-propelled vessels. 

When used for heavy service, collar-bearings are lubricated by 
an oil-bath, or by pipes leading to holes in the bearing-rings which 
open on the rubbing surface of the ring; four pipes and openings 
are often used on a single ring, placed at equal distances apart 
around the ring. Suitable oil-grooves are cut in the rubbing sur- 
face, starting from the oil-opening and running zigzag a short 
distance from it in a direction corresponding to that in which the 
collar rubs over it. 

Table V represents the practice in thrust-bearings on merchant 
and naval vessels built in the yards of the Newport News Ship- 
building & Dry Dock Co. The horseshoe bearing has each part, 
which comes into coatact with the collars of the shaft, made in the 
shape of the letter TJ, or a horseshoe. This form allows the bearing 
parts to be removed and replaced readily. 



76 FORM, 8TBENGTH, AND PR0P0BTI0N8 OP PARTS. 

Table V.* 

COLLAR THRUST-BEARINGS. 

Steel collars running against white metal of bearing-rings; oil-bath lubri- 
cation. 



Horse- power of one 
engine 



Gunboats. 



Revolutions per min- 
ute — 



Diam. of shaft, inches 

Inner diam. of rub 
bing surface of col- 
lar, inches 



Outer diam. of rub- 
bing surface of col- 
lar, inches 



Number of collars. . . 

Estimated total thrust 
of one engine ex 
erted on one collar- 
bearing, pounds .. . 

Method of cooling... 



Type of bearing. 



875 
800 



8J 
9 



18750 

Water 
circu- 
lation 

Collar 



800 



5J 



8J 
9 



18100 

Water 
circu- 
lation 

Collar 



Batlle- 
ships, 



5000 

120 
14 

14i 

21i 
11 

79800 

Water 
circu- 
lation 

Horse- 
shoe 



Merchant Ships. 



600 

115 
8 

8i 

12i 
6 

12300 
Collar 



250 

110 
6 

H 
6 



Collar 



8500 
115 

m 

181 

19 
15 

55900 

Water 
circu- 
lation 

Collar 



8800 

80 
15J 

16 

24 
11 

71200 

Water 
circu- 
lation 

Collar 



400 

iia 

7 
7i 

11 
6 

960a 



* Data kindly furnished by Mr. C. B. Orcutt, President of the Newport News Shipbuild- 
ing & Dry Dock Co., builders of the battlestiips Kearsarge, Kentucky, Illinois, etc. 

The practice of the Marine Iron Works of Chicago, for their 
gmaller work, is shown in Table VI and Figs. 47, 48, and 49. Fig. 
47 is a steel shaft with seven collars integral with the shaft. The 
box has a Babbitt lining cast around the collars so as to fit them. 
An oil-trougli at the top of the bearing, from which oil-holes lead 
to the top of each collar, famishes a means of labricating. The 
data relative to this bearing are given in Table VI. This bearing 
is designed for a 50-H.P. engine at 300 revolutions. 



BEARINGS AN1> LUBRICATION. 



77 



Table VI.* 

COLLAR THRUST-BEARINGS. 



Type of Bearing 


Fig. 47. 


Fig. 48. 


Fig. 49. 




Horse-power of engine .... 


60 


25 


76 


Revolutions per minute. . . . 


800 


400 


275 


Diam. of shaft, inches. 


2f 


2i 


4 


Inner dlam. of nibbing sur- 
face of collar, inches 


^ 


2i 


4 


Outer diam. of rubbing sur- 
face of collar, inches .... 


H 


H 


9 


Number of collars 


7 
1680 


1 
1040 


1 


Estimated total thrust on 
collar-beariDg, pounds.... 


2100 


Material of collar 


Steel 
Babbitt metal 


Cast iron 
Bronze washer 


Steel 


Material of bearing-rings . . 


Bronze washer 


Method of lubricating 


Oil-trough 
above journal f 


Grease-cups. Solid 
oil in cups placed 
on oil-holes 


Grease-cups { 



* Data kindly furnished by the Marine Iron Works, Chicago, HI., except the estimated 
total thrust on collar- bearing, which was calculated by the writer, 
t Oil-holes lead from trough to edges of collars. 
t Chamber in bottom part of box can be filled to make an oil-bath if desired. 






UAijyfTy^jiunu^ 



Fio. 47. 



78 FORM, 8TEENGTH, AND PROPORTIONS OF PARTS. 



Fig. 48 is a thrust-bearing for a 25-H.P. engine making 400 
reyolntions per minute. It consists of two cast-iron collars fastened 
to the shaft on each side of a cast-iron box-bearing. Loose collars 
of bronze are interposed between each end of the box and the collar 
adjacent to it. It is lubricated by means of grease-cups attached * 




$ S V- f g 

— e o 0- - 

q XZ J 9 



^ 



-g 




Fig. 4a 

to oil-holes leading to the journals and the faces of the collars. 
The remaining data are given in Table. VI. 

Fig. 49 is for a 75-H.P. engine making 275 revolutions per 
minute. It has a steel collar, fastened to the shaft, which bears 





Fig. 49. 

against loose bronze collars, which in turn bear against cast-iron 
boxes. Lubrication is secured by grease-cups, and, if desired, by 
filling the reservoir at the bottom of the box with oil, thus obtain- 
ing bath lubrication. Other data are given in Table VI. 

Roller Journal-hearings, 

22. Cylindrical roller-bearings for working conditions similar 
to those which an ordinary journal-bearing is designed to meet have 



BEARINGS AND LUBRICATION. 



79 



come into extensive use. Such a bearing is illustrated in Fig. 49.1. 
The shaft or journal is surrounded by several cylindrical rollers, 
which roU inside of an accurately bored /jasing or shell. The bore 
of the casing is slightly greater than the diameter of the journal 
plus twice that of the rollers, so that the latter will roll freely 
when not on the side of the journal where the load forces it against 
them. Under some conditions a steel sleeve is used over the 
shaft. 

The journal, rollers, and casings must all be accurately cylin- 
drical, and have their axes parallel, in order to work correctly. It 
is not possible to keep the rollers parallel to the journal without 
some additional device for that purpose. 

A very effective method is to use a slotted tube somewhat 
longer than the rollers and bored to have a free running fit on the 
journal; the slots are cut longitudinally as long as the rollers, 
and each is wide enough to allow a roller to drop into and turn 
freely in it; a complete ring of metal is left at the ends of the slots, 
so that the whole slotted tube forms a rigid *'cage'' for holding 
the rollers in position. Such a cage is shown in Fig. 49.1. The 
cage rotates less than half as fast as the journal. 




CAQE AND ROLLERS 

Fio. 49.1.* 

The rollers rub against the sides of the slots with a pressure 
that depends on the accuracy of construction and, after the bear- 
ing has been in use for some time, the durability of the materials 
used. If any of the bearing surfaces wear out of their true cylin- 
drical form the roller will have a tendency to twist about out of 
line with the shaft and to move lengthwise. This will cause an 
• Made by Mossberg & Granville Mfg. Co , of Proviileuce, K. T 



80 FORM, STRENGTH, AND PROPORTIONS OP PARTS. 

end pressure of the roller against the cage at the end of the slot. 
Rapid wear and the early destruction of the bearing are apt to 
follow. 

Various devices other than the plain slotted cage are used for 
retaining the rollers in position. One method is to drill a hole in 
each end of the roller and insert a ball after the manner of Fig. 39.1. 
The cage used in connection with this device consists of a pair of 
rings each with as many countersinks as there are rollers in the 
bearing. One of the rings is placed at each end of the rollers so 
that the balls are in the countersinks. The two rings are then 
fastened together with rods in such a way as to hold the rollers in 
place by the balls fitting into the countersinks. The baU is some- 
times let into the ring and the rolls countersunk at the ends to bear 
against the balls. Another method is to groove the ends of each 
roller circumferentially and place two balls between each adjacent 
pair of rollers, one ball at each end of the rolls, so that the balls 
run in grooves. Twice as many balls as rollers are used. The 
balls are in turn held in place by retaining rings at each end of 
the bearing. These rings are rigidly fastened together. Still 
another method is to place small rollers between the ones that 
carry the load. These small separating rollers are kept in place 
by a retaining cage or other device. 

There is another trouble that is liable to occur if a roller gets 
out of alignment, especially if it is long and the material is brittle. 
As soon as the roller gets out of position the line of its contact with 
the journal is curved instead of straight. The roller must therefore 
bend and become liable to fracture by the bending. A broken 
roller is almost certain to cause rapid destruction of the bearing. 
Several short rollers, lying end to end in the same slot of the cage, 
are sometimes used for long bearings; also a special form of 
flexible roller (see Fig. 50). 

Solid rollers of tool steel hardened and ground to form and 
running upon bearing surfaces of the same material are successfully 
used for heavy pressures and high speeds ; mild steel case-hardened 
and ground has also proved satisfactory. Mild steel and tool steel 
unhardened are used to a large extent. Wrought iron (puddle 
iron) and cast iron are unsuitable for any service but the lightest 
pressures and slow speeds, and are uncertain for that. 



BEARINGS AND LUBRICATION. 8l 

The frictional resistance of a properly constructed roller-bear- 
ing is exceedingly low, especially at slow speeds. 

A journal supported on rollers starts from rest with a very 
small proportion of the effort required for a journal-bearing, even 
of the best construction. 

The wear is very slight in a well-constructed roller-bearing 
of durable materials, hence its friction remains low for a very 
long time — as long as the wear is inappreciable. In poor bearings 
with consequent rapid wear the friction increases rapidly and 
reaches a high value before they are worn out. 

Roller-bearings should always be lubricated with a liquid 
lubricant. A single application lasts a long time, as long as two 
years or more on shafting at 400 revolutions per minute or slower 
speeds, when an oil pocket is provided. 

Great improvement has been made in the construction of roller- 
bearings since their extensive use began in recent years. This 
has been accomplished by more careful construction and the use 
of suitable materials. Roller-bearings that are thoroughly reli- 
able are now obtainable. 

The avoidance of rapid wear in some classes of machinery 
where accurate adjustment is necessary under high pre^ure, as in 
calenders and cold-metal rolling-machines, is often of equal or 
greater importance than saving of power. Both have been suc- 
cessfully obviated in practice by the use of roller-bearings. 

The load which a roller-bearing will carry is greater for high 
speeds than can be put on the common journal-bearing with sliding 
surfaces, practically the same for medium speeds, and less for very 
low speeds. 

The safe load in pounds for a well-made solid roller journal- 
bearing with six or more rollers is 

P= 100.000 D^ ^ i^' 7^"^ °^ ^ °'^* 17 
3iS than 50 feet per minute) ; 

in which D = diameter of rollers, inches; 
L= length of one roller, inches; 
JV= number of rollers; 

P= total safe load or pressure on bearing, pounds; 
<S= linear velocity of convex bearing surface relatively to 
concave bearing surface, feet per minute. 



82 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

For values of S less than 50 feet per minute the pressure may 
be taken the same as for S=50. For exceedmgly slow speeds the 
pressure may be taken somewhat higher, however. 

This equation is based upon the assumption that the safe load 
per inch of total effective length of rollers is 2,000 D* pounds, 
considering one-third of the roUers under pressure. 

Fig. 49.2 and Table VIa show the form and give the dimensions 
and loads for solid roller journal-bearings that have been success- 





PiG. 49.2. 



ful in practice. They refer to the bearing iUustrated in Fig. 49.1. 
The effective length of the rollers is taken as one-third of the 
aggregate length of all the rollers in the bearing. 

A roller journal-bearing for a 6-inch shaft has the following 
values given in Table VIa: D= 11/16 inch; NL/S=72; and, in 
column B, diameter of convex bearing surface = 7 inches, corre- 
sponding to a circiunference of 22 inches =1.83 feet. 

At 200 revolutions per minute the linear velocity of the convex 
bearing surface is 5=200X1.83=366 feet per minute. 

The safe load for this bearing is, by the preceding equation, 

P = 100,000^jiyX^ = 9300 pounds. 

This value comes within less than one per cent, of that given 
in the table. 



BEARINGS AND LUBRICATION. 



83 



Table VIa. 

SAFE LOADS FOR ROLLER JOURNAL-BEARINGS OF THE TYPE FIG. 49.2. 





SI 

P 


1 ^ 

ill 




1^ 


II 


1 
1 


1^ 




Safe Pressure on Journal, 
Pounds. Total -P. 


Rev. 
iSfn'. 


Rev. 
iSfn. 


Rev. 
lS?n. 


Rev. 
iSfn. 


A 


B 


c 


if 


E 


D 


JV 


L 


NL 
3 


0-60 


200 


400 


800 


H 


^, 


u^ 


11^ 


1*4' 


"He 


8 


1.25 


3.33 


233 


233 


160 


75 


%• 


»%• 


l«h« 


1%« 


1»4 


'He 


10 


1.25 


4.16 


290 


290 


180 


90 


P 




IH 


2 


24 




10 


IHie 


5.63 


700 


700 


360 


175 


\h 


m 


m 


24 


24 


Vi 


10 


UVie 


5.63 


700 


700 


360 


176 


4 


iv^ 


j;^ 


24 


24 


4 


10 


2.00 


6.66 


830 


700 


360 


175 


1'-^ 


2-4 


24 


4 


12 


2.00 


8.00 


1.000 


800 


400 


200 


1 


1^ 


2 


24 


2*4^ 


M 


14 


2.5 


11.66 


1,460 


900 


460 


225 


\Y^ 


24 


2H 


2?4 


Vi 


14 


2.5 


11.66 


1.460 


900 


i-o 


225 


IH 


IM 


24 


2K 


3 


/^ 


16 


2.5 


13.33 


1.660 


830 


-ll/j 


210 


i?5 


1^ 


2='8 


2H 


3 


4 


16 


2.6 


13.33 


1.660 


830 


415 


210 


J^ 


2 


2H 


3 


34 


4 


18 


2H 


16.5 


2.100 


1.000 


.5(.^J 


2S0 


m 


24 


2H 


34 


34 


4 


18 


2.76 


16.5 


2.100 


1,000 


500 


250 


]r* 


i^l 


2»4 


34 


34 


4 


20 


3.00 


20.0 


2,500 


1,060 


534) 


265 


IH 


2Ji 


3-^^8 


34 


4 


20 


3.00 


20.0 


2.600 


1,060 


530 


265 


2 


24 


3 


S'A 


3»4 


4 


22 


3.25 


23.63 


2.960 


1,110 


555 


280 


2H 


2'i 


3»^ 


4 


44 


■he 


20 


3.50 


23.33 


4.630 


1,630 


815 


410 


2^* 


?^ 


ill 


44 


44 


•he 


20 


3.50 


23.33 


4.630 


1,630 


815 


410 


2;'' 


3 


44 
44 


44 


<He 


20 


3.75 


25.0 


4.900 


1,600 


KiH) 


400 


L^ 


2!^ 


3=»4 


4,4 


yi« 


22 


3.75 


27.6 


5,390 


1,650 


«-;'> 


415 


2H 


S 


3J^ 


44 


5 


^ie 


22 


4.25 


31.16 


6.100 


1,860 


m:y 


460 


2^ 


4 


4H 


5 


•^ie 


22 


4.25 


31.16 


6.100 


1350 


S2,=> 


460 


2>i 


3»/^ 


4H 


54 


64 


4 


22 


4.6 


33 


9.240 


2.540 


1^70 


m& 


3 


24 


44 


64 


^A 


•H 


22 


4«%2 


34.6 


9.690 


2,540 


1.270 


635 


SH 


3J^ 


4H 


6»^ 


H 


22 


42%2 


34.6 


9.690 


2,540 


j.l»70 


aas 


^^ 


4 


434 


5'^ 


6 


»^ 


24 


5.0 


40 


11,200 


2300 


1 JIKi 


700 


3H 


4,4 


4Ji 


55^ 


6 


?^ 


24 


5.0 


40 


11.200 


2.800 


1 Am 


700 


34 


4}-i 


5 


5:U' 


64 


•^^8 


24 


5.5 


44 


11,200 


2.800 


\ ,1(10 


700 


35^ 


44 


hH 


6^ 


6'4' 


Tie 


24 


5.5 


44 


11,200 


2.800 


l,4m> 


700 


354 


4H 


54 


64 


6>4 


Tie 


24 


5.5 


44 


11.200 


2.800 


IJI'Kt 


700 


33^ 


4H 


6»^ 


l«1 


7 


Vie 


24 


5.75 


46 


14.000 


3.500 


l,75i> 


875 


4 


4H 


5?4^ 


7 


Tie 


24 


5.75 


46 


14.000 


3.500 


! 7r^\ 


875 


4' 8 


5 


6 


6>8 


74 




24 


6.25 


50 


18.750 


4,690 


2,3i.> 


1.170 


4H 


54 


64 


7 


74 


^ 


24 


6.25 


50 


18.750 


4,690 


'2Mh 


1470 


4-H 


6.4 


64 


74 


8 


4 


24 


6.75 


54 


19.600 


4.900 


2,4J)0 


1*225 


44 


64 


74 


8 


4 


24 


6.75 


54 


19.600 


4.900 


^.450 


1,225 


4«/^ 


64 


64 


7H 


84 


1^ 


24 


7.0 


56 


19,600 


4.900 


2.45U 


1.225 


4?4 


5»i 


6^ 


7H 


84 


4 


24 


7.0 


56 


19,600 


4.900 


2A^y 


14225 


4'^ 


5Ji 


7 


8 


84 


«>ie 


24 


7.5 


60 


24.600 


6 150 


:iMyo 


l.fiW 


5 


6 


74 


8'-^ 


8'2 


^ie 


24 


7.5 


60 


24.600 


6.150 


3J(¥t 


1550 


5H 


64 


7->^ 


8-^8 


94 


*/^ 


22 


8^i« 


60 


30 400 


7 600 


:i.M^M^ 


lOGO 


5H 


64 


74 


84 


94 


'^'^ 


22 


8V,« 


60 


30.400 


7.600 


ri will 


1.&00 


5H 


6H 


7*^ 


8^8 


94 


^ 


22 


S^ie 


60 


30.400 


7600 


:* sCM \ 


1900 


54 


64 


75i 


8^4 


9^' 


5-^ 


24 


8.5 


68 


31 .800 


7.950 


:j 075 


1.990 


5H 


W^ 


7J'8 


84 


94 


H 


24 


8 5 


68 


31.800 


7 9.50 


A U75 


19m 


5H 


6'^ 


84 


94 


10 


Hie 


24 


8.5 


68 


35 200 


8.800 


4 liHi 


2200 


5H 


6K 


8'4 


9'4 


10 


^Vie 


24 


8.5 


68 


35.200 


8.800 


4 M><\ 


2200 


6 


7 


8^8 


9-^^ 


104 


iVie 


24 


9.0 


72 


37.400 


9 350 


4 tlT5i 


2 340 


6K8 


7Vg 


84 


94 


104 


^Vie 


24 


9 


72 


37 400 


9,350 


4 ,^75 


2,340 


6k^ 


?5| 


8H 


9H 


lOH 


iVie 


24 


9.0 


72 


37 400 


9.,350 


4,(1 7 5 


2340 


6^8 


8J/8 


94 


114 


=^^ 


22 


9.25 


72 


37 400 


9 350 


4i^7r. 


2340 


64 


?.1 


9 


10 


11,4 


5i 


22 


9.25 


72 


37.400 


9,350 


4 iV7'i 


2 340 


6^ 


94 


104 


114 


■4' 


24 


10 


80 


45.000 


11.250 


fy tvj:> 


2,810 


6^4 


7»^ 


94 


104 


114 


'1 


24 


10.0 


80 


45.000 


11.250 


flA'l'2^~i 


2.810 


63^8 


71-^ 


9-^8 


1»4 


114 


24 


10 


80 


45.000 


11 2bO 


5,(1 '^5 


2.810 


7 


8 


W^ 


10»/^ 


124 


I'He 


22 


10 75 


80 


52.800 


13.200 


fl nnc* 


3300 


7^ 


84 


9H 


103^ 


124 


"^ie 


22 


10.75 


80 


52,800 


13 200 


6.01 Ml 


3,300 


7»^ 


84 


9Jg 


104 


124 


iMe 


24 


10 75 


86 


56.800 


12.400 


7JIMI 


3550 


7S 


8«s 


10 


11 


124 


i^e 


24 


10.75 


86 


56300 


14.200 


7 iml 


3,550 


74 


84 


104 


114 


124 


''He 


24 


10.75 


86 


56.800 


14.200 


7.HK> 


3.5i50 



84 



FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



Table VIa. — Continued, 

SAFE LOADS FOR ROLLER JOURNAL-BEARINGS OF THE TYPE FIG. 49.2. 



li 


'o _ 




!l 


•c 


i 


i 






Safe 


E^ressure 


on Journal. 


II 




5S 


1 


« 


& 


P 


Pounds, Total -P. 


""5 


II 

o 






II 


•s 

£ 


"35 

f 


•2^ 








III 
1" 


R«v. 
Alfn. 


Rev. 


Rev. 
per 


Rev. 

:6!n. 


A 


B 


c 


H 


£ 


D 


A' 


L 


NL 
3 


0-50 


200 


400 


800 


7% 


SVs 


10^ 


\\% 


134 


n 


22 


11.75 


86 


59 400 


14.850 


7 ■l^J.-, 


3,710 


7H 


SH 


104 


114 


13^, 


H 


22 


11.75 


86 


69,400 


14350 


7 A2?x 


3 710 


7H 


SH 


Ws 


US 


134 


u 


22 


11.75 


86 


59.400 


14.850 


7.4-.-I 


3.710 


8 


9 


io:«4 


11!^ 


134 


'A 


22 


11.75 


86 


59 400 


14.860 


7.l-.-» 


3,710 


syn 


9^^ 


1U4 


124 


144 


'^<ia 


22 


12.25 


90 


63,200 


^«ooo 


7 .1^( H ] 


3 950 


»H 


94 


IPs 


12H 


144 


'«^16 


22 


12.25 


90 


63,200 


i:..>00 


7 ,'.■>£ H 1 


3,950 


8^1 


m 


114 


12H 


144 


1%6 


22 


12.25 


90 


63,200 


1,-. -00 


7 IK HI 


3.950 


8>4 


9»X 


iiH 


12K 


144 


I'^ie 


24 


12.5 


100 


63 200 


\ :. :-oo 


7 .yiKi 


3.950 


SH 


9j| 


ii's 


13,ks 


15 


1 


24 


12.5 


100 


75.000 


!^.:50 


%:viT. 


4,660 


8H 


10 


12 


13M 


15 


1 


24 


12.5 


100 


75 000 


1^.750 


9.r:^o 


4.660 


SVi 


lOH 


12«s 


13^8 


15 


1 


24 


12.5 


100 


75.000 


IH750 


9^25 


4 660 


9 


lOU' 


12'i 


134 


154 


1 


24 


12 75 


102 


75 000 


IN.T50 


9 325 


4.660 


9H 


104 


124 


\^H 


16 


1 


24 


13.75 


110 


82.500 


-:^',25 


io.aio 


5,155 


P. 


10J4 


12''4 


14 


16 


1 


24 


13.75 


110 


82.500 


■.-..25 


10 310 


5.155 


11 


13 


14^4 


16 


1 


26 


15.0 


130 


91000 


50 


11.325 


5,660 


10 


UK 
1P4 


13H 


144 


16 


1 


26 


15.0 


130 


91000 


22.750 


11.325 


6660 


lO^i 


14 


154 


16 


14 


26 


15.0 


130 


91000 


22.750 


11 :t25 


6 660 


104 


12 


14'4 


mi 


18 


14 


26 


15^ 


136 


103 200 


25.800 


] 'J * n ?i > 


6.450 


lOH 


\2M 


144 


16 


18 


VH 


26 


15^^ 


136 


103.200 


25300 


IL'.IMHJ 


6,450 


' 11 


124 


14?:^ 


16M 


18 


14 


26 


15-4 


136 


103 200 


26.800 


Il'AHllf 


6,450 


im 


13 


15!^ 


164 


18 


14 


26 


154 


136 


103.200 


25.800 


vi.Sim 


6,450 


12 


134 


16 


174 


18 


1*4 


26 


15^ 


136 


110 000 


27,500 


n-:^\ 


6375 


12^^ 


14 


164 


18 


18 


14 


26 


154 


136 


110000 


27,500 


i:v7,vi 


^^li 


13 


144 


17 


184 


18 


lU 


28 


15.0 


140 


119 900 


29 975 


14 WA} 


7.495 


134 


15 


174 


19 


20 


14 


28 


17.0 


160 


125 000 


31 250 


] t' CiJ.i 


7310 


14 


154 


18 


194 


20 


1'4 


28 


17.0 


160 


125 000 


31 .250 


i;>.f;i^.-) 


7310 


14H 


16 


184 


20 


20 


14 


30 


17.5 


175 


136 500 


34 125 


IT 1IH> 


8.550 


15 


164 


19' 4 


20^4' 


22 


14 


30 


19.0 


190 


160 000 


40 000 


:!i^ 1 II JO 


10 000 


164 


17 


19»i 


21M 


22 


14 


30 


19.0 


190 


160.000 


40.000 


3. iintf 


10, (KX) 


16 


17»4 


204 


22»i 


22 


14 


30 


19.0 


190 


160 000 


40.000 


2*' liiHk 


lO.tKXJ 


16>r2 


11^ 


21 


2234" 


22 


14 


30 


19.0 


190 


160 000 


40 000 


1\^ 1M(I 


10 000 


17 


214 


23 » 4' 


22 


14 


32 


18.75 


200 


160 000 


40,000 


2^1.1 iiKl 


10000 


174 


}^il- 


22 


22>H 


22 


1-^:^ 


32 


18.75 


200 


160 000 


40,000 


211 lion 


10.000 


18 


224 


24H 


22 


1-4 


32 


18.75 


200 


160.000 


40 000 


2U iiiK^ 


10.000 


19 


20>4 




254 


24 


V4 


34 


21.0 


238 


175 0(X) 


43 750 


2) >2n 


10.900 


20 


22 


25 * 


27 


24 


1'4 


36 


21.0 


252 


175.000 


43.750 


'l\ ^J.-V 


1 10.900 


22 


24 


27 


29 


26 


40 


22.5 


300 


210 000 


62,500 


Jf. iJ lO 


' 13.125 


24 


26 


29 


31 


30 


m 


42 


27.0 


380 


256.500 


64,125 


\{\1 <HHI 


16.000 



BEARINGS AND LUBRICATION. 



85 



A special roller, of which a group is shown in Fig. 50, has been 
designed to obviate the danger arising from the fracture of a roller, 



^ V 



'ffr^rffrrYr^^^ 



Fig. 5u.* 

either from getting out of line or unevenness of the surfaces of the 
bodies in contact, and also to secure contact along the entire 
length of the roller even if the surfaces on which it rolls are not 
exactly true. It is made by winding a steel ribbon about a man- 
drel, in the same manner that a strip of paper may be wound 
about a round pencil, so that the edges of successive convolutions 
just clear each other. The cross-section of the ribbon is rectan- 
gular, and is varied according to the speed and load under which a 
bearing is to operate. 

A box made up of these flexible steel ribbon rollers, as applied 
to a line shaft, is shown in Fig. 50.1. The cage, frame, or yoke is 
shown both in place among the rollers and removed from the box. 
In this particular box the cage is made up of slightly more than 
half an annular ring at each end, connected by three bars that are 
cast together with the part rings. In some designs steel stamp- 
ings are used iastead of castings. Some of the rollers lie in the 
spaces between the bars of the cage, and the others complete the 
circumference of the journal outside of the cage. In a box for a 
shaft about two inches in diameter, having fourteen rollers, four 

♦Made b}' the Hj-att Roller Bearing Co. 



86 



FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



of them lie in each pocket of the cage and the remaining six outside 
of it. These six rollers are longer than the eight in the pockets. 




Fig. 50.1. 



In other styles of cages the rollers are all of the same length, and 
the cage is made up of two halves resembling the cage illustrated. 
The box is fitted with a pad at each end to prevent the oil from 
flowing out along the shaft. 

Figs. 50.2 and 50.3 show a form of the Hyatt roller-bearing 
adapted to parts that have a hole bored in them to receive the 
box. The outer broken ring of sheet metal fits into the bore and 
forms a shell or casing inside of which the rollers run. A similar 
inside ring of metal, shown in Fig. 50.3, is used as a sleeve for 
iron shafts, or when the shaft is not suitable for the rollers to bear 
directly upon. 

On account of the flexibility of the roller, hardened surfaces 
are not necessary for either the rollers or the bearing surfaces 
upon which they run. The shell and sleeve are made of sheet or 



BEARINGS AND LUBRICATION. 87 

plate steel bent to fomi and used without finish by grinding or other 
operation. 




Fig. 50.2. 

As to lubrication and the rubbing or grinding action of the 
rollers against each other and the cage, the following statements 
are made: 

"The rollers do not grind against each other when under load 
for the reason that they are separated, or, in other words, when 
one reaches the zone of pressure it has a tendency to stop an instant 
and separate from the preceding one. The yokes or cages are usu- 
ally made of very soft brass. In bearings that have been in use 
for years doing severe work, the file marks in the yokes are just as 
plain apparently as the day the yoke was finished. Many cases 
are on record where these bearings have operated for two years on 
shafting nmning at 400 revolutions per minute without attention 
beyond the original lubrication, and without perceptible wear." * 

Lubrication is especially effective on account of the reservoir 
capacity of the rollers and the alternation of right- and left-hand 

♦Communication from Mr. Alfred P. Sloan, Jr., General Manager Hyatt 
Roller Bearing Co., June, 1908. 



88 



FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



Table VIb. 

approximate dimensions of hyatt roller journal-bearings. 

For speeds less than 50 rev. per minute. The diameter of roller and 
hence that of the casing vary for different speeds. The higher the speed the 
larger the roller. The ratio of length of bearing and diameter of shaft is 
practically the same as for other forms of journal bearings. 

The outside diameter of casing is slightly smaller than the diameter of 
hole into which the bearing fits to give suitable clearance. On account of 
variation in the thickness of the steel the outside diameter of the casing is only 
approximate. The sleeve fits directly upon the shaft. It is intended for 
shafts of materials other than steel. 



1: 

li 

li 

2 

2i 

2" 

2j 

3 

3i 

3i 

3i 

4 

4i 

4i 

4J 

5 



Load, 200 Lbs. per Sq. In. of Projected 
Area of Journal. 



Diam. 

of 
Roller. 



1 
1 

1 

u 
1* 

H 
U 



Thick- 
ness 
of 
Sleeve. 






Thick- 
ness 
of 
SheU. 



Hi 
% 

% 



Outside 

Diam. 

with 

Sleeve. 



2 

3i 

3i 

4 

41 

4} 

5 

51 

5i 

6 

61 
6i 
6f 

7i 
71 
8 
8i 
81 
.91 
9i 
91 



Outside 
Diam. 
with 
Sleeve 
and 
SheU. 



2] 

I 

31 

*t 
4i 

5 

51 

5J 

51 

61 

61 

6S 

71 

7' 

8 
8l 
8^ 
94 
98 
9i 
101 



Ix)a<i, 400 Lbs. per Sq. In. of Projecte I 
Area of Journal. 



I Diam. 
I of 
I Roller. 



1 
1 
1 
1 

H 
ij 
H 

I H 

I U 

H 

I ^» 

n 



Thick- 



of 
Sleeve. 



'A 
'A 
}i 

i: 
r 
': 
:i 
:r 
i: 
i: 
:■ 
•: 
:■ 
■r 
:r 
:r 
:: 
:; 
'.: 
:■ 
■r 
:r 



Thick- 
ness 

of 
SheU. 



i 



Outside 

Diam. 

with 

SleevB. 



2f 
31 

3i 

4t 

4i 

5 

5J 

5i 

5i 

6t 

6i 

6i 

7 

7i 

7i 

8 

8i 

8i 

9 

9i 

9i 

10 



Out- 
side 

Diam. 
with 

Sleeve 
and 

SheU. 



2J 

^1 

31 

4 

4J 

4i 

5i 

51 

5i 

6 

6i 

61 

7i 

71 

7* 

8t 



9. 

lOi 
lOj 



B£L\BINQS AND LUBRICATION. 



Table VIb. — Continued. 



U 
H 
li 

2 

2i 

2i 

2f 

3 

It 

3i 

4 

4i 

4J 

4} 

6 

5i 

6i 

5i 

6 



Load, 600 Lbs. per Sq. In. of Projected 
Area of Journal. 



Diam. 

of 
RoUcr. 



Thick- 
ness of 
Sleeve. 






_. . . Outaide 
Thick- : Diam. 

"o''ifi?'' *'»•> 
Shell. Sleeve. 



2* 
3J 
3» 

at 

4 

4i 

4i 

51 

5i 

5i 

6 

6i 

6i 

7 

71 

7} 

8 

81 

8i 

9 

91 

91 
10 
101 



Outside 
Diam. 

with 
Sleeve 

and 
SheU. 



3i 

3 

4t 

4f 

5 

5i 

5i 

6 

6i 

6} 

74 

7f 

7f 

8i 

8f 

81 
81 

n 

10} 
lOf 



Load, 800 Lb«. per Sq. In. of Projected 
Area of Journal. 



Diam. 

of 
Roller. 



Thick- 
neneof 
Sleeve. 



Thick- 
ness of 
Shell. 



Outside 
Diam. 

with 
Sleeve. 



3f 

3} 

4i 

4i 

5 

5i 

5i 

6 

6i 

6} 

7 

7i 

7i 

8 

8i 

8i 

8i 

9i 
10 
lOi 
lOJ 



Out- 
side 

Diam. 

with 

Sleeve 
and 

SheU. 



«0 



FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



spirals, the latter serving to distribute the oil from end to end of 
the bearing. 

The dimensions that are used for roller-bearings of the form 
of Fig. 50.2 are given in Table VIb for convenience of reference 
and to show what loads may ! e used for speeds not greater than 
50 revolutions per minute. 

Table VII gives the results of tests made by Wm. Sellers & Co. 
on several Hyatt roller-bearings. 



Table VII. 

TP:STS of HYATT ROLLER-BEARINGS.* 
Coefficient of friction /i given in the body of table. 



Dimensions of Bearings. 


Total 

Load. 

Pounds. 


Revolutions i)er Minute. 




5 


25 


48 


128 


214 


Diameter of journal. U^W- 
Length of bearing. 3". 
Bearing bored .009'' smaller than sum of 
two liners, two rollers, and shaft. 


1.000 
2,000 
3,000 


/» 


M 


.16964 


/« 


/» 






. 08974 








.0r>730 












Diameter of journal, l^'He". 
Length of bearing.3". 

Bearing bored .004'' larger than sum of two 
liners, two rollers, and shaft. 


1,000 
2.000 
3,000 


.02958 
.01874 
.01249 


.01578 
.00986 
.00789 


.00789 
.01080 
.01020 


.00786 
.00789 
.00789 


.00789 
.00592 
.00526 


Diameter of journal, 8". 
Length of journal, 12". 
Bore of bearing .023" larger than sum of 
two liners, two rollers, and shaft. 


10.000 
20.000 
30,000 
40.000 
50.000 

500 
1,000 
1..500 
2.000 


.02399 
.01923 
.OlS.'if) 
.01805 
.01708 




.03826 


.02540 
.01560 
.01280 
.01240 






.02386 






.01959 
01795 








.01692 














.03156 


031.56 




.02367 


.01972 




.02762 .01973 
.01841 .01710 
.01282 .01775 




.01578 


.01381 


Hyatt coDunercial, 2" shafting-box. 




01578 


01315 




.01578 


.01282 















♦ Age of Steel, April 10, 1897, p. 17 ; American Machinut. June 24. 1897, p. 20. 

Two street-railway ears, one fitted with Hyatt roller-bearings 
and the other with common journal-bearings, were tested at Provi- 
dence, R. I. The roller-bearings showed a saving of 13% of the 
total power used to drive the car with common bearings.* 

Fig. 50.3 is a flexible roller journal-bearing for heavy duty. 
It is 24 inches in diameter inside and 12 inches long over all. There 
are 54 rollers li inches in diameter and 11^ inches long. The 
rollers are held in position by a cage with rods passing through 
every sixth roller. 

* American Machinist, Oct. 21, 1897, p. 794. 



BEARINGS AND LUBRICATION. 



91 




24X 12 in. special bushing for heavy duty. 
Safe working load 50,000 lbs. at 60 revolutions per m'nute. 

Fig. 50.3. 



KI' 



[5Fi,h..."H.V C4.ATE 
i 


ftOLLEfl 


r-| 




nousf?. 


CJi^t 




BEfcWN', or P^T!; 



:j 




Fig. 51. 



92 FORM, STRENGTH, AND PROPORTIONS OP PARTS. 

RoUer Thrust-hearings. 

23. Conical roller-bearings are used to resist the end thrust of 
a shaft or other part, performing the same function as the more 
common forms of collar-bearings and of step-bearings already 
described. Fig. 51 shows the typical fonn of such a bearing. 
The rollers are truncated cones of such a taper and so placed that 
their vertices, as well as those of the conical surfaces on which they 
roll, all coincide on the axis AB oi the shaft. When so constructed 
there is true rolling action between the parts; slipping is entirely 
absent. It is not necessary that the rollers shall have their axes 
at right angles to that of the shaft, but they may be incHned at 
any angle within practical limits. It is often convenient to make 
either the step or the end of the shaft flat on the surface where the 
cones roll and the other bearing surface coned to suit the rollers. 

On account of the taper form of the rollers, there is a tendency 
to force them out radially from between bearing plates. 

The rollers should be small in comparison with the diameter of 
the bearing in order to keep their tendency to move out radially 
as small as possible. It would hardly be advisable to make the apex 
angle, embraced between two diametrically opposite elements of the 
conical surface, greater than 15° in any case; 10° or less is a more 
suitable value. It should not be forgotten, however, that the 
rolling resistance of the cones increases as the diameter decreases. 

In order to prevent radial movement of the rollers, and also 
to hold them from twisting around so that their axes do not 
intersect that of the shaft, a retaining "cage" of some form is used. 
Such a cage is shown in Fig. 51. It is a very simple design, but 
not the best form for heavy service. As shown it consists of a flat 
ring perforated to form pockets to receive the rollers. The conical 
surface of each roller has a snug-running fit in its pocket. By 
making the crowned outer end of the roller more convex than the 
corresponding end of the pocket, the frictional resistance is kept 
lower than if these rubbing surfaces have the same radius of curva- 
ture. The cage has a free-running fit in the thrust-block or step 
of the bearing. 

The angular speed of rotation of the cage is half that of the 
moving bearing plate when the bearing plates have equal angles. 



BEARINGS AND LUBRICATION. 



93 



Figs. 51.1 and 51.2 show a conical roller thrust-bearing whose 
dimensions, loads, and speeds are given in Table VIIb. This bear- 
ing is guaranteed to work under pressures as high as 5000 pounds 
per square inch on the bearing surface of one bearing ring at slow 




^. Si- 



Fig. 51.1. 



speeds. At high speeds the bearing pressures must, of course, 
be lower. Each roller is coned on the part that comes into contact 




Fig. 51.8. 



with the bearing rings. The larger end is cylindrical beyond 
the bearing surface, and of a diameter equal to that of the larger 
end of the cone frustum. The cage is one soUd piece of metal 
bored radially with combined conical and cylindrical holes to fit 



94 



FORM, STRENGTH, AND PROPORTIONS OP PARTS. 



Table VIIb. 
safe loads for conical roller thrust-bearings of the 

TYPE FIG. 51.2. 

If the thrust-box or thrust-block is large enough to fit over plates P, use 
construction 1 ; if not, use construction 2. 



















gfC 


Safe Pressure 


Diameter 












Num- 
ber of 


A^le 


IH 


on Bearing. 
> Pounds. 


of Shaft. 












Roll- 
era. 


RoU- 
ere. 


11 & 






Inches. 




















Deg. 


Rev. 


Rev. 


















r 


>??n. 


:8fn'. 


A 


B 


c 


H 


J? 


F 


N 




Sq. In. 


0-50 


100 


1 tolM 




2H 


i;i 


M 


H 


28 


""64" 


3.338 


420 


420 


1 1 


3 


I'Hh 


Jie 


28 


64 


4.307 


600 


525 


-;h« 


3%« 


4^6 


I^ 




30 


64 


6.21 


1,113 


835 


vviii ;• 2 




4 


47^ 


>» 


^ 


30 


64 


7.66 


1,530 


960 


2M6 ;; 2H 


^'='i« 


4%« 


6^0 


'•He 


iV,e 


30 


6 


10.137 


1.940 


1,000 




:■! I 


6 


6*8 


1 


?;^ 


30 


6 


11.966 


2.520 


1,260 


Vf/\% *' 2*'4 




^, 


6?^ 


1'^ 


?^8 


30 


6 


15.23 


3.340 


1,670 


21^6 •• 3 


:^-A6 


7'8 


1'8 


Ji 


30 


6 


19.83 


4.650 


2,325 


3Me •* ^H 


j'.fl 


6»/i6 


8 


l\i 


1 


30 


6 


20.862 


6.120 


2.560 


7lh\. " 34 


1 . 


7«^ 


81 Wo 


14 


IV^ 

14 


30 


6 


24.84 


5.600 


2.800 


1 .. 


7H 


8"yie 


m 


30 


6 


28.86 


6.900 


3.450 


r . 


7J^ 


94 


IH 


1»8 


30 


6 


30.5 


7.500 


3.750 


4Me " 4|i 


"i' ^ 


ill 


10«Vie 


IH 


1« 


30 


6 


35.097 


• 8.000 


4.000 


m^ "48 


^' , 


101 «He 


m 


30 


6 


38.104 


9,100 


4.550 


4^1*1 " 4Si 




^H 


11«H6 


m 


\% 


30 


6 


41.2,34 


10.000 


6.000 


4i«H« " 6 


ri 


10 


iiv, 

12^8 


VA 


\H 


30 


6 


50.266 


11.000 


5.5.50 


«i« •• 5K 


\, - 


lO'/i. 


2Vio 


1^4 


30 


6 


64.2 


11,860 


5.925 


i: , 


10^^ 


121^ 


2Me 


1?^ 


30 


6 


57.101 


12,600 


6.250 


5»..« " 5^^ 


■7 


iiji^d 


13^i« 


2«H6 


2 


30 


6 


62.52 


14,900 


7.450 


SJ'f^e " 6 


7 ' 




14 


2*^6 


2 


30 


6 


67.148 


16.900 


8.450 


m^ " 64 


7 1 


14T^ 


27^6 


24 


30 


6 


78.017 


18,900 


ft,450 


0",, •• 7 


V, : ^ 


134 


15^ 


2».^« 


2^^ 


30 


6 


89.684 


22.400 


1 1 .200 


'.^1 "74 


'=' 


14H 


17 


2H 


27.^6 


32 


6 


104.373 


25.600 


12.800 


7 ... •' 8 


«>a 


154 


17T^ 


2'^ 


2»/^e 


32 


6 


117 808 


31.600 


15,800 


8V,i " 84 


10 


18»<r 


3 




32 


6 


132.96 


.36.600 


18,300 


8',« " 9 


104 


17I4 


19H 


34 


2H 


32 


6 


147.12 


42.000 


21.000 


9Vie '* 94 


11 


r 


204 


3"^ 


24 


32 


6 


162.98 


48.000 


24.000 


9»,i« •* 10 


IIH 


214 


35^ 


3 


32 


6 


179.6 


54.000 


28.000 



the rollers. The radial thrust of the rollers is taken by the external 
ring. The apex angle of the rollers is generally 6°; never more 
than 6i° in tWs design. 

Fig. 51.3 illustrates a thrust-bearing for comparatively light 
duty not exceeding a bearing pressure of 250 pounds per square 
inch of bearing surface when the speed is as high as 250 revolu- 
tions per minute. 



BEARINGS AND LUBRICATION. 



95 



Another form of cone-roller thrust-bearing is shown in Fig. 51.4. 
The radial pressure of the roller is resisted by a ball between two 
conically cupped surfaces. The apex angle of the rollewrdoes not 





Fio. 51.4. 



Pig. 51.5. 



exi'eed 15° in this design as nianufactured. The stock forms of 
rollers are shown in Fig. 51.4a. 

\ case is cited where a step-bearing with conical rollers work- 
ing under a pressure of 104.5 pounds per square inch had a coef- 
ficient of friction /£ = .0025 when running on steel.* 

23.1. Cylindrical roller thrust - bearings are used to a con- 
siderable extent, and for very heavy duty in some cases. Such 
a bearing is sho^\Tl in Fig. 51.5. The bearing plates have flat 
surfaces. The rollers are, of course, all of the same diameter. 
Several rollers are placed end to end in the same pocket. The 
pockets are at different distances from the centre of the bearing, so 
that the rollers will travel in different paths and have less tendency 
to wear grooves in the bearing plates than if several rollers fol- 
lowed the same path. The axes of the rollers are radial to the 
bearing. The ends of the rollers are crowned. 

A bearing 22 inches outside diameter of the type Fig. 51.5 
has been successfully tested under 80,000 pounds pressure at 300 
revolutions per minute. f Another 18 inches outside and 8 inches 
inside diameter has been operating 800 days of 16 hours under a 



♦ Ccusier's Magagi'ne, May, 1897, p. 66. Several forms of roller-bearings 
are illustrated and described in this article. 
\ Machinery, April, 1903, page 898. 



96 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 




BEARINGS AND LUBRICATION. 97 

constant pressure of 100,000 pounds at 125 revolutions per minute. 
The parts were of case-hardened steel in both bearings.* 

In thrust - bearings with cylindrical rollers running on flat 
plates there is a combined rolling and slipping or spinning action 
between the rollers and plates. The natural path of the roller is 
straight. It is made to follow a circular path by the cage. Each 
time a roller passes over its complete circular path it spins around 
once, in relation to the plate, in addition to rolling over the latter 
a distance equal to the mean circumference of its path. With the 
very short rollers used in practice the rubbing of the rolling sur- 
faces over each other is of small consequence. 

The ends of the rollers should be well rounded at the edges 
to prevent small pieces from breaking off. 

The wear of the bearing surfaces of cylindrical roller thi-ust- 
bearings operating under heavy pressure is first shown by a rough 
appearance of the surfaces. The roughness is caused by the 
flaking off of a thin layer or skin of the metal. It is readily per- 
ceptible to both the eye and touch. Once started, rapid destruc- 
tion is almost certain to follow. There is also a wear of the outer 
end of the roller against the cage due to the pressure of the latter 
necessary for guiding the roller in its path against its tendency 
to move in a straight line, and, at high speeds, some appreciable 
centrifugal action. 

A bearing with 184 cylindrical rollers each ^ inch in diameter 
by i inch long running on a path lOf inches outside and 4| inches 
inside diameter, on a pair of bearing plates 1 inch thick and 11 
inches diameter, lasted well. It was kept in service till the plates 
were worn away unevenly to an average depth of about ^^ of 
an inch. 

The cage pockets at the outer radius were worn most on the 
side next the centre of the bearing. This was caused by the tip- 
ping of the roller as the depression was worn in one plate while 
the other remained flat, and its tendency to crowd toward the 
centre on accoimt of the sloping edge of the depression. Some 
of the pockets were worn through into the next one radially. Many 
of the rollers were fractured. The bearing was used for passenger- 

*Maehinery, May, 1908, page 489. 



98 FORM, STRENGTH, AND PROPORTIONS OP PARTS. 

elevator service. The thrust on it when the elevator was loaded 
and at rest or moving uniformly was about 4500 pounds. The 
entire weight to be accelerated, including the counterweight to 
the passenger cage, etc., was about 9500 poimds. This was 
brought up to a speed of from 350 to 400 feet per minute within 
a distance of 20 feet in regular service. The full speed of rotation 
was more than 350 revolutions per minute. The case is cited to 
show how much such a bearing can be worn and still operate. 

By supporting a step- or thrust-bearing on a spherical base, the 
pressure may be kept imiformly distributed among the rollers. 

The combined speeds and pressures that roller thrust- and step- 
bearings will carry are from four or more times as great as for ball 
bearings occupying the same space, except for very low speeds. 

BaU Bearings. 

23.2. Hardened steel balls rolling on hardened steel sur- 
faces are used for machine bearings to a considerable extent. 
Their chief field is for light service, especially for Ught pressures. 
They have also been used successfully under both high speed* and 
pressure with certain forms of bearing surfaces. For such duty 
the material must be extremely and uniformly hard, yet strong 
and free from microscopic flaws and cracks, and the surfaces 
must be highly polished without scratches. Accuracy of form is 
also essential. The same is desirable for light service, but is not 
so absolutely necessary. 

Wear is not to be taken into consideration in the design of 
ball bearings for heavy service, and no adjustments for it are 
required. When wear becomes perceptible, the life is practically 
ended. On account of the great number of failures that have 
occurred in improperly made ball bearings, there is often undue 
prejudice against them, even for light service. In a great many 
of the forms there is not pure rolling motion between the ball and 
race, but a combined rolling and spinning motion instead. On 
account of the localization of pressure upon a small area at the 
place of contact, the sliding or rubbing action due to spinning is 
apt to produce abrasion even under moderate pressures; it is 
almost certain to do so under heavy loads on the bearing. In 



BKA&IXGS AND LUBEICJkTION. 



99 



bdD beuin^ hr heavr soirice pure rolUng niotioii is especially 
desiraUe. 

SS.S. Ball j««raal-beari]i|^ with two-poiiit cotttact. — Fig. 51.6 
shows a ban bearing in which the balls have a puie tolling action 





¥m. SLfk 



on C3dindrical bearing surfaces. The ball cage is a hollow circular 
cylinder with radial perforations which serve as ball pockets. 
The latter are so distributed that no two balls travel in the same 
path. A simple method of preventing the balls from falling out 
of the cage when the latter is removed from between the bearing 
surfaces is shown. The pockets are made by drilling radially 
toward the axis of the cage with an ordinary cone-pointed drill. 
By stopping the drill feed before the lips have passed completely 
through the shell, a flange of metal is left that prevents the ball 
from falling inside the cage when put into the pocket. After 
the ball is in the pocket the outer edge of the hole is pressed down 
by a hollow pimch or set so as to flange in the outer end of the pocket 
and thus retain the ball. 

This form of bearing offers no resistance to thrust and end 
motion. It is easily made without special appliances, and is very 
useful. 

23.4. Ball journal-bearing with three-point contact. — Fig. 51.7 
is a bearing in which all the balls travel over the same path on 
the races. Each race is a solid ring. One is cylindrical and the 
other grooved. Both sides of the groove should make the same 
angle with the centre line of the bearing, so that the two points of 
contact of a ball against its sides shall be equidistant from the 
axis of the bearing. A ball then has pure rolling on the cylindrical 
surface. In the groove it has a combined rolling and spinning 



100 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



motion, the latter about the axis passing through the points of 
contact a and h. This form of bearing does not resist end motion 
except to the extent of the frictional resistance to sliding between 
the balls and the cylindrical bearing surface. 




wm 










Fig. 61.7. 



Fig. 51.8. 



If the angle between the sides of the groove and a parallel 
to the axis of the bearing is too small, the ball is apt to bind or 
cramp between the cylindrical bearing surface and one side of the 
groove, especially if there is any end motion of the shaft. Such 
cramping generally results in the fracture of one of the parts, 
usually the ball. The more highly finished the surfaces, the smaller 
the angle that can be safely used. Probably 25° is as small as is 
safe, and 30° is advisable. 

The normal pressure of a ball against each side of the groove 
(Fig. 51.7) is found by the expression 
Pressure against each side of groove = (i Radial pressure on ball) sec d 

Fig. 51.8 is similar to the preceding one, except that the inside 
ring is grooved instead of the outside one. 

23.5. Ball journal-bearing with grooved race, two-point contact. 
— Fig. 51.9 is a form of bearing which has proved satisfactory 
both experimentally and in commercial and miUtary service. 
The balls of a single unit all roll in the same grooves of a pair of 
races. The profile of the groove is an arc of a circle with a radius 



BEARINGS AND LUBRICATION. 



101 



somewhat greater than that of the ball. One of the races or 
bearing rings is a single piece. The other is made up of three 
parts: the ring itself, a small removable piece, and the screw for 





Fig. 51.9. 

holding the latter in place. The removable piece on one ring is 
necessary for introducing the balls and taking them from the 
bearing. It should be placed either on the inner or outer race 
where no pressure due to the load comes upon it. This form of 
bearing will resist end thrust such as comes on the hubs of auto- 
mobiles, on engine shafts, etc., as well as carrying the load normal 
to the axis of the bearing. 

Professor Stribeck * gives for this form of bearing, when the 
radius of the grooves is 9/8 times that of the ball. 



Safe load in kilograms 
for one ball 



> = 1 50 X (diameter of ball in centimetres)* 

and, if 2 = the number of balls when there are not less than ten, 

' , . r = Z-150X (diam. of balls in centimetres)*, 

grams on beanng ) 5 

By expressing these values in pounds and inches, and using the 
following notation for a ball journal-bearing of Ihe form Fig. 51.9, 

* Glasers Annalen fftr Gewerbe und Bauwesen, 1901. 
t ^66 tables for safe loads, pages 420-420«. 



102 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

P= total safe load on ball journal-bearing, pounds; 
i\r= number of balls when not less than ten; 
€?= diameter of balls, inches, 
the following approximate expressions are deduced: 

Safe load in pounds for one ball=2100d^ pounds {d in inches) 
and 

P==420N(P pounds. 

The loads for bearings with 14 balls are given in Table VII a 
in accordance with the last formula. 

Table VIIa. 
safe loads for ball bearings w'lth fourteen balls of the. 

TYPE FIG. 51.9. 

Diameter of balls, inches J } 1 1} 2 

Load, pounds 1,470 3,800 5.880 13.200 23,500 

23.6. Ball thrust - bearings, two - point contact. — The . end 
pressure or thrust of a shaft may be taken by balls placed between 
two flat circular plates or disks whose planes are perpendicular 
to the axis of rotation. The plates must be hardened and ground 
to a smooth, accurate surface. A retaining cage is generally 
used for keeping the balls in position. If no cage is used, and 
there are enough balls to cover the bearing surfaces of the plates,, 
when the radius of the latter is several times the diameter of the 
balls the latter crowd hard together and offer much frictional 
resistance to the motion of the parts. The crowding is due to the 
tendency of each ball to roll on the bearing surfaces in a straight, 
path nonnal to a line joining the centre of the ball and axis of the 
bearing. This tendency is the same as with a short roller. Since 
the balls all try to move away from the centre of the disks, they 
crowd together when stopped by the casing that necessarily sur- 
rounds the bearing. 

By the use of a retaining cage with individual pockets for 
the balls, the frictional resistance of the bearing and the wear of 
the balls are both reduced. Such a cage may be made by perforaii>- 
ing a disk with circular holes as in Pig. 51.10. After a ball is in 
its pocket, the ends of the hole may be flanged in so as to retain it* 



BEABINGS AND LUBaiCATlON. 103 

This may be done by the use of a cup-end punch or die, as already 
described for the cyUndrical journal-bearing. No two balls should 
travel in the same path. The cage should be of some compara- 
tively soft material such as brass or bronze. The motion of a ball 
relatively to a bearing disk is combined rolling and spinning about 
an axis through the two points of contact with the disks. The 
pressure of the balls against the cage in a direction radial to the 
bearing plates is apt to be so great as to wear the pockets unduly 
in heavy service. By the use of conical bearing surfaces similar 
to those for a cone-roller thrust-bearing, for the balls to roll upon, 
the radial pressure and spinning of the balls are both reduced, 
always provided that the bearing plates do not differ so much 




Fig. 51 10. 

from flat surfaces as to cause the balls to slip, not roll, out radially 
on account of the wedge-like action of the divergent surfaces. 
By coning the plates so that the two apexes are at the san:ie ]ioint 
on the axis of the bearing (the condition for cone-roller bearings) 
the tendency of the ball is to roll in a circle about the axis of the 
bearing. The ball will itself seek and remain in the position which 
brings the cone apexes into coincidence. If placed too far from 
the axis, it will creep toward the centre of the bearing; if placed 
too near, it will creep from the centre. When the bearing plates 
are coned, and there is no retaining ring or other device to prevent 
the balls from moving radially, the pressures between the ball 
and plates are not normal to the conical surface, or radial to the 
sphere. The direction of the pressure must be along the line 
passing through the two places of contact between the ball and 



104 



FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



plates. On account of its deformation under pressure, a ball 
acts very much like a short cone roller between coned plates. 

If the balls are all of the same diameter, they must all be at 
the same distance from the centre of the bearing, thus forming a 
ring of baUs. They should be held in place by a retaining ring. 

23.7. Ball thruBt-bearings, three-point contact. — The single 
groove-ball thrust-bearing (Fig. 51.11) makes use of one ring of balls 
without the necessity of a retaining ring. The use of the latter 
may be advisable, however, to keep the balls from rubbing against 
each other. Pure rolling of the ball on the flat bearing plate can 
be secured by making the sides of the groove at such angles that 
the two points of contact of a ball placed in the groove will lie 



»A0IU8 OF BALL CIRCLE - 




BEARING RINQ 






K A 1 J JJJ 



PLATE 



Fio. 51.11. 

on a line whose intersection with the axis of the bearing lies in the 
plane of the flat bearing surface, as shown in the figure. 

In a bearing with 1-inch diameter balls with their centres in a 
circle 9 inches in diameter, or 4^ inches in radius, if the outer side 
of the groove makes an angle ^" = 30° with the surface of the flat 
bearing ring, then the corresponding angle of the inner side of the 
groove is fl'=52.2°, in order to cause the line passing through the 
points of contact a and b to meet the flat bearing surface of the 
plate at its centre. The angle of 30° for the outer side of the 
groove is probably as small as should ever be used, on account of 
the tendency of the balls to wedge or jam between the plate and 
side of the groove. On the other hand, larger values of 5" give 
a greater spinning action of the ball against the sides of the 
groove, as well as higher pressures at the same places. 

The relation between the pressures against the bearing surfaces 
in a bearing of the form of Fig. 52.11 are expressed by the equations 



BEARINGS AND LUBRICATION. 



105 



N' ^ 



T sin 0'' 



N"= 



sin[180°-(/?' + ^'0]' 

T sin d' 
sin[180°-(l!?' + ^'0r 



By substituting in these equations the values 5'' = 30° and 
^'=52.2°, the pressures against the sides of the groove, Fig. 5U1> 
for r=100 pounds, are: • 



iV' = 



100 sin 30 



iV"- 



sin [180 -(52.2 + 30)] 
100 sin 52.2 



100X5 ^ 50 
sin 97.8 .99075 

100X79012 



= 60.4 lbs. 



sin [180 - (52.2 -f 30)] 99075 



- = 79.81 



Fig. 51.12 has the same size'of balls and ball circle as the pre- 
ceding design, viz., 1" diameter of ball for a 9" diameter ball 
circle. The groove has the same bottom angle as before, 97.8°, 




RADIUS OF BALL CIRCLE 



BEARINQ RiN3 



BEARIN 



\^KJUJJl 



Fig. 51.12. 

but the sides make equal angles with the flat face of the ring. 
The mating surface is coned to secure pure roUing of the ball upon 
it. The cone elements differ from a flat plate by the angle a = 11° 8'. 
A force r=100 pounds, acting parallel to the axis of the bearing, 
will cause a pressure iV=102 pounds between the ball and bearing 
cone. The angle between iV, normal to the sphere, and the two 
normals at the places of contact in the groove, differ but very 
slightly from those in the preceding figure, only 0° 2' in each case. 
The pressures AT' and N'' also differ very slightly from those of the 
flat plate bearing. 

In the matter of construction, there is little ground for choosing 
either of the two bearings, Fig. 51.11 or 51.12, in preference to the 



106 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

other. The former, on account of having a flat plate instead of a 
cone, will work with very fair satisfaction when the axis of the plate 
does not coincide with, but is parallel to, that of the ring. The lat- 
ter, on account of the conical surface, must have the ring and 
cone co-axial. 

23.8. Ball thrust-bearing, four-point contact. — ^Fig. 51.13 is a 
form of bearing that is much used, and which frequently gives 
trouble. When the two bearing rings are grooved so that two 
lines passing through the points of contact of the ball wdth the 
grooves meet on the axis of the bearing as shown, then the best 
kinematic conditions for four-point contact are secured. The 
diflSculty of making the grooves so that the pressure will be prop- 
erly distributed is great. Thus, suppose that the two rings are to 



/ o / ^0=*4O°2O' "<- RADIUS 



0'^iOW 




OF BALL CIRCLE 



BEARIt^Q RINQ 






ET 



) ) ) )) 



BEARIhQ RINQ 



Fig. 51.18. 



be exactly alike, as in the figure, and that the outer sides of the 
grooves are made correctly. If the inner sides are not then so 
made that the intersection of the sides of the groove lies at the 
same distance from the centre of the plate in both cases, a con- 
struction difficult to make exact, then the pressure ^vill not be 
properly distributed, for the side of the groove that is "low" 
will not carry its due load. In fact, it will carry none at all until 
the deformation of the parts under pressure is sufficient to bring 
the ball against this side of the groove. The unduly heavj'^ pressure 
on the other side of the groove will cause destruction of the bear- 
ing much sooner under heavy loads than it should occur. The 
bearing may be designed with unlike rings, but the difficulties 
of construction still remain. 

The axes of the two bearing rings must coincide in the four- 
point contact thrust-bearing. If they are not coincident, although 



BEARINGS AND LUBRICATION. 



107 



still parallel, the pressures will be thrown unduly on one side of 
a groove. 

The figure is drawn to scale and shows the proper distribution 
of pressure in a bearing with balls 1 inch in diameter with their 
centres in a circle 9 inches in diameter, as for the two preceding 
bearings. 

23.9. Ball journal-bearing, four-point contact. — Fig. 51.14 
shows a ball bearing whose chief function is to resist pressure 
normal to its axis of rotation. It will also resist thrust and end 
motion. Each groove should have the same angle on both sides. 



wmi 



S^ 





Fio. 51.14. 



The two grooves do not have to be of the same angle, however. 
Two rigidly supported bearings of this form should not be used on 
the same shaft, for any change of length of the shaft between the 
bearings would force one side of the groove hard against the balls 
in each bearing. It can, of course, be used satisfactorily in con- 
nection with other bearings that do not resist end motion. The 
grooves can be made flatter than when only one part is grooved. 
An angle less than 30° between any two adjacent sides of the 
grooves is not advisable. 

23.10. Cup-and-cone two-point contact hub ball bearing. — Fig. 
51.15. This is the very famiUar form of ball bearing so exten- 
tensively used for the wheels of road vehicles. Each ball runs 



108 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

in a pair of races whose surface profiles in a cross-section are con- . 
cave conoids with a radius of curvature somewhat greater than 
the radius of the ball. The actual locations of the places of con- 
tact on the ball are indeterminate. For the purpose of deter- 
mining the approximate values of the bearing pressures under a 
given load, the centres of the bearing areas have been assumed 
to be at the ends of a diameter of the ball in the figure. This is 
not a stable condition, however, for as the parts move, the ball, 
when in this initial position, will try to roll farther out from the 
axis of the bearing. This tendency will be resisted by the increas- 
ing slope of the sides of the ^races, at the places of contact, and 




Fig. 61.16. 



the ball kept nearly in the position shown. On the other hand, 
if the ball happens to get in closer to the axis than shown, so that 
a pair of lines tangent to the two areas of contact meet before 
intersecting the axis of the bearing, then the rolling of the ball 
will have a tendency to crowd it still further in between the sur- 
faces. This tendency of the ball to force itself farther in either 
direction out of its true position when it has once started, and 
the fact that it is always tr3ang to leave the correct position when 
under pressure, make it necessary to lubricate the bearing to pre- 
vent the ball from wedging in the race. The radius of curvature 
of the races must, on the same account, be made sufficiently small 
to give the ball but Uttle room for side motion. The curvature 
shotild be to a radius between IJ and IJ times the radius of the 
baU. 



BEARINGS AND LUBRICATION. 



10» 



With the proportions shown in Fig. 51.15, in which the diameter 
of the ball circle is taken as 2, and the distance between the ball 
circles as 3, units, a force P= 100 pounds normal to the axis of the 
bearing and at a distance of 1 unit outside of the right-hand ball 
circle produces the pressures indicated between the balls and 
races. These forces are obtained by the force diagram to the right 
of the bearing. It is assimied for convenience of illustration 
that the pressure is all carried by one ball when forces are shown 
acting upon a ball. The pressure is actually distributed among 
several balls in an ordinary commercial bearing. 

If the cones were reversed in position and placed with their 
bases next to each other, and the cups also reversed to correspond 
with the new position of the cones, leaving the diameter of and 
distance between the ball circles unchanged, the pressures against 
the rightrhand lower and left-hand upper balls would be nearly 
doubled with the force P of the same value and acting the same as 
before. 

23.11. Three-point contact hub ball bearing. — Fig. 51.16 is 
of the same general form as the preceding one. The difference is 





Fig. 51.16. 



that true conical surfaces are used for the inner bearing surfaces 
and grooves for the outer ones. The ratio of the diameer of the 
ball circles to the distance between them must be made greater in 



110 FORM, STRENGTH, AND PROPORTIOx\8 OF PARTS. 

this form than in the cup-and-cone design in order to secure a 
sufficiently large cone apex angle to prevent the balls from wedging 
between the cone and outer side of a groove. The half -apex angle 
of 30® is about as small as should be used for bicycle wheels where 
there is no great side pressure against the tire. In three- or four- 
w^heeled stably balanced vehicles, or elsewhere when two wheels 
are placed on the same axle, it is better to make the apex angle 
-60° or larger, 30° or more for the half angle. 

The force diagram and pressures against the balls are shown 
for a pressure P= 100 pounds normal to axis of the bearing and at 
s, distance of one unit outside of the right-hand ball circle. Re- 
versing the cones in this case would have the same effect as for 
the cone-and-cup bearing. 

23.12. Four-point contact hub ball bearing. — Fig. 51.17 is 
intended for the same use as the preceding two. It has the same 



i 




I * ■■■■■ ■? X P .1 I 




V 



B^ DIlTAItOE ICrWEEM ^ * 



Fig. 51.17. 



ratio between the diameter of and distance between ball circles 
as the two-point bearing. These two have the advantage of com- 
pactness over the three-point bearing. 

The adjacent sides of the grooves should not have an angle 
much smaller than 30° between them. The force diagram and pres- 
sures upon the balls are shown in the figure for a force P=100 



BEARINGS AND LUBRICATION. 



Ill 



pounds acting at a distance of 1 unit outside of the right-hand 
ball circle. 

The objections to the four-point contact are not so serious in 
this design as in the thrust-bearing. 

Special Forms of Bearings. 

24. The bearing shown in Fig. 52 is often used on light 
machinery, such as circular saws, wood-planing machines, etc. It 






n 



■f»*iii-? 



no 



Fig. 62. 









V^i 



m r^j 



n 



Fig. 52.1. 

is something of a combination between a journal- and thrust-bear- 
ing, and serves in a measure the purpose of both. 

The collars are integral parts of the shaft. The box is divided 



112 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

lengthwise and lined with a soft bearing metal, which is cast in» 
while the journal is in place. 

A bearing (Fig. 52.1) of a similar nature is made by cutting 
grooves into the shaft to serve the same purpose as the collars. It. 
answers very well when only light service is required, but the shaft 
is weakened so as to be unfit for heavy service. 



aJT^ 




Fig. 53. 



Rg. 53 is a taper bearing provided with lock-nuts a and 6 for 
adjusting it and taking end thrust. The box surrounding it is 
solid. The taper is commonly about IJ inches in diameter per 
foot of length. This form of bearing is chiefly used on light ma- 
chinery running under Ught journal-pressure. 



CHAPTER II. 
SPUR- AND FRICTION-GEARS. 



SPUR-GEARS, 

25. strength of spur-gear teeth. — When a rotating spnr-gear 
transmits power to its mate, there must be pressure between the 
pairs of teeth that are in contact. If the teeth were coiTectly 
formed and accurately spaced, the pressure necessary to turn the 
driven gear would be nearly uniformly distributed between the 
pairs of teeth in contact. The elasticity of the teeth might 
theoretically affect the uniformity of the distribution, but not to an 
extent to be worthy of consideration in practice. 

On account of the inaccuracy of spacing that always exists to 
fiome extent, and the loss of the correct form of the teeth by wear, 
it is customary to assume, for purposes of designing, that the entire 
force that is required to rotate the driven gear is applied by a single 




Fig. 54. 

tooth of the driver to its mate on the driven. If there is a prob- 
ability that the teeth may be thrown out of alignment by the 
springing of the shafts or movement of the supports, or that foreign 
substances may come between the teeth, the pressure may be local- 
ized at the top of a tooth at one end, as at P in Fig. 54, and break 
it off as shown by the irregular line of fracture. To allow for such 

113 



114 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

a coQdition, it is customary to assume that, however wide the face 
of the gear, the strength of the tooth is no greater than if the gear- 
face had only a width x measured across the face of the gear. The 
value of X is different for teeth of the same pitch and height, but 
of different profile, and cannot be accurately determined in any 
case. For the more common forms of gears having teeth whose 
height is roughly .7 of the circular pitch, it is generally safe to take 
X as 1.5 to 2 times the circular pitch, according to whether the 
teeth are narrow or broad at the base. The wider the face of the 
gear, the less rapidly will it wear; and if accurately made and 
rigidly supported, the strength will be about proportional to its 
width for widths as great as six or more times the circular pitch. 

When a pair of teeth first come together, there is a shock caused 
by the blow as they strike each other. The intensity of the shock 
increases with the speed at which the teeth travel; hence gears 
running at high speed will not stand as much pressure between their 
teeth as those running slower. There may also be shocks caused by 
loads suddenly applied to the driven gear, as in rock- and ore- 
crushers, and rolling-mill machinery. The gear should be propor- 
tioned to resist such shocks. 

The pressure between a pair of teeth would be normal to their 
surfaces along the line of contact, if there were no frictional resist- 
ance acting to prevent the combined motion of rolling and slipping 
of the one over the other. This frictional resistance always does 
exist, however, to an extent depending on the material of the teeth, 
the finish of their working surfaces, and whether they are lubricated 
or dry, clean or covered with dirt and grit. 

Fig. 55 shows the profiles of a pair of teeth A and B just as they 
come into contact when A is driving B, the rotation about the 
centres of the gears being in the direction indicated by the arrows. 
The point of contact is at (7, and the common normal to the tooth 
curves passes through the pitch-point P. The pressure between 
the teeth would be in the direction CP if there were no friction 
between them. The friction causes the line of pressure to be 
inclined to CP by some angle PC//, whose value depends on the 
coefficient of friction of the tooth surfaces. The lever-arm, about 
the centre of the driven gear, of the force acting along C'/T, is shorter 
than that of a force acting along OP ; hence the force that acts along 



8PUK- AND FRICTION-GEARS. 



115 



CH to turn the driven gear against a given resistance must be pro- 
portionally greater than that which would act along CP. 

E and D are the positions of the same pair of teeth just as they 
are on the point of separating. As before, the common normal to 
the curves at the point of contact F passes through P, and the 
frictional resistance causes the pressure between them to make some 




angle PFHinih this normal. In gears that are not lubricated, or 
are lubricated and covered with dirt and grit, it is possible that the 
coefficient of friction may be large enough to increase the angle 
PFH to such an extent that the direction of pressure FH will 
become tangent to the gear- face at the top of the tooth ; or, what 
is practically the same, normal to the radial line which passes 
through the middle of the tooth profile. This radial line will be 
hereafter referred to as the median line of the tooth. 

A tooth of the driven gear is subjected to the greatest stress 
when in the position B^ just as it comes in contact with its mate, 
for the pressure is then applied at the greatest distance from ita 
base; and a tooth of the driving gear is working under the greatest 
stress when in the position />, just as the pair are separating. 

For the tooth B^ on the driven gear, it can be seen that the 
pressure against it, acting along a line inclined to the median line 
of the tooth, produces both radial compression and bending stresses 
in the material. This is true whether friction is considered or not. 
Friction increases the proportion of compressive to bending stress. 

The tooth i>, on the driver, also has both compressive and bend- 
ing stresses caused by the pressure against it when friction is not 
considered. Friction reduces the ratio of compressive to bending 



116 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

stress, and if the friction becomes great enough to bring the line of 
pressure normal to the median line of the tooth, the compressive 
stress is eliminated, and the action is a purely bending one. 

If such a material as cast iron, which is much weaker in tension 
than compression, is used for gears having equal diameters and the 
aame form of teeth, D will be more apt to break than B\ if the 
material is one that is equally strong in both tension and compres- 
sion, B would probably fail first. 

The force P that will break a gear-tooth when exerted normal to 
its median plane at the top of the tooth, and uniformly distributed 
across the face of the gear, as in Fig. 56, may be found by the 




Fig 56. 

formula for a cantilever. The quantities entering into this formula 
when applied to a gear-tooth are : 

P = force applied tangent to top of tooth ; 
S = maximum fibre-stress per unit area in the material ; 
V = shearing stress per unit area in the material; 
b = breadth of gear-face; 
h = thickness of tooth at breaking section ; 
/ = distance from top of tooth to breaking section. 

The formula is 



P = 



Sbh' 
6/ ' 



On account of the filleting which is commonly used at the 
bottom of the tooth, the distance I of the breaking section from the 
top of the tooth is less than the total height of the tooth. The 
location of the breaking section can be found for any tooth, 



SPUR- AND FRICTION-GEAES, 117 

whether the profiles on both sides of the median line are similar or 
not, by taking sections normal to the median plane and at different 
disfcanoes from the top of the tooth, introdnoing the I and h of each 
in the formula, until the one that gives a minimum yalae of F is 
found. When the profile of the tooth is symmetrical about the 
median line, the value of P can be obtained more directly by draw- 
ing a parabola whose vertex is at the centre of the top of the tooth, 
and whose sides are tangent to the tooth curves. The value of F 
for any section parallel to the direction of F is the same as for all 
other sections, this being the property of a cantilever of parabolic 
profile and uniform breadth. 

In any system of interchangeable gears, such as the cycloidal 
with constant size of generating circle for a given pitch, or the 
involute with constant angle of obliquity, the thickness of the teeth 
at the base is less for gears of small diameter than those of large ; 
consequently the teeth of the smaller gears are weaker. 

The curve, Fig. 57, was obtained by calculating, according to 
the formula for the cantilever just given, the strength of cycloidal 
teeth generated by a describing circle having a diameter half that 
of a 15-tooth gear of the pitch adopted, and a fillet at the bottom 
of the tooth of a radius equal to one sixth of the width of the 
space at the addendum circle.* This is practically the form of 
tooth adopted by the Brown and Sharpe Mfg. Go. for their inter- 
changeable gears. The length of the tooth from top to bottom is 
0.6866 times the circular pitch, and the thickness on the pitch- 
circle equals half the pitch. It can be seen by the diagram that the 
teeth on a 120-tooth gear are twice as strong as those on one having 
12 teeth. By multiplying or dividing the reading on the scale of 
'^ pounds pressure at tooth-point " by a suitable constant, the curve 
can be made to apply to any pitch, width of gear-face, and fibre- 
stress. 

With regard to the effect of the position of the line of contact 
between a pair of engaging gears, it is shoum by construction that, 
in a pair of perfectly made and aligned gears having 13 teeth each, 
at the instant contact changes from one to two pairs of teeth, or 

• *• Diagrams for the Relative Strength of Gear Teeth," by Forrest R. Jones. 
Trans. Amer. Soc. Mech. Engrs., vol. xviii., p. 766. 



118 



FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



Table VIIc. 

GEAR wheels: TABLE OF TOOTH PARTS. 
DIAMETRAL FITCH IN FIK8T COLUMN. 



Diametral 
Pitch. 


Circular 
Pitch. 


Thickness 
of Tooth on 
Pitch Line. 


Addendum 
1" 
or-^-. 


Working 

IVpth 
of Tooib 


Depth of 
Sitace below 
Pitch Uue. 


Whole 

Depth 

of Tooth. 


P 


P' 


t 


• 


D" 


•■f/ 


!>"+/ 


i 


6.2882 


8.1416 


2.0000 


4.0000 


2.3142 


4.8142 


ft 


4.1888 


2.0944 


1.3833 


2.6666 


1.5428 


2.8761 


1 


8.1416 


1.5708 


1.0000 


2.0000 


1.1571 


2.1671 


1ft 


2.5133 


1.2566 


.8000 


1.6000 


.9257 


1.7257 


1* 


2.0944 


1.0472 


.6666 


1.8838 


.7714 


1.4381 


1ft 


1.7952 


.8976 


.5714 


1.1429 


.6612 


1.2326 


2 


1.5708 


.7854 


.5000 


1.0000 


.5785 


1.0785 


2ft 


1.8963 


.6981 


.4444 


.oooo 


.6148 


.9587 


2J 


1.2666 


.6288 


.4000 • 


.8000 


.4628 


.8628 


2ft 


1.1424 


.5712 


.8C36 


.7278 


.4208 


.7844 


8 


1.0472 


.6236 


.8333 


.6666 


.8857 


.7190 


^ 


.8976 


.4488 


.2857 


.5714 


.3806 


.6168 


4 


.7854 


.3927 


.2500 


.5000 


.2898 


.5898 


6 


.6283 


.8142 


.2000 


.4000 


.2314 


.4314 


6 


.6236 


.2618 


.1666 


.8833 


.1928 


.8595 


7 


.4488 


.2244 


.1429 


.2857 


.1658 


.8081 


8 


.8927 


.1968 


.1250 


.2500 


.1446 


.2696 


9 


.8491 


.1745 


.1111 


.2222 


.1286 


.2897 


10 


•8142 


.1671 


.1000 


.2000 


.1157 


.2157 


11 


.2856 


.1428 


.0909 


.1818 


.1052 


.1961 


12 


.2618 


.1309 


.0833 


.1666 


.0964 


.1798 


18 


.2417 


.1208 


.0769 


.1588 


.0890 


.1659 


14 


.2244 


.1122 


.0714 


.1429 


.0826 


.1541 



SPUR- AND FRICTION-GEAKS. 



lift 



Table VIId. 

GEAB wheels: TABLE OF TOOTH PABTS — Continued. 

DiAHETBAL FITCH IN FIBBT COLUMN. 



Diametral 
Pitch. 


Circular 
Pitch. 


Thickness 
of Tooth on 
Mtch Line. 


Addendum 
or- 


Workinfc 

Depth 
of Tooth. 


Depth of 
Space below 
ntchLine. 


Whole 

Depth 

of Tooth. 


P 


P' 


t 


• 


D" 


•+/ 


lyi+f 


15 


.2094 


.1047 


.0666 


.1833 


.0771 


.1438 


16 


.1968 


.0962 


.0625 


.1250 


.0728 


.1848 


17 


.1848 


.0924 


.0588 


.1176 


.0681 


.1269 


18 


.1745 


.0873 


.0555 


.1111 


.0648 


.1198 


19 


.1653 


.0827 


.0526 


.1053 


.0609 


.1185 


20 


.1571 


.0785 


.0500 


.1000 


.0579 


.1079 


22 


.1428 


.0714 


.0465 


.0909 


.0626 


.0980 


24 


.1809 


.0654 


.0417 


.0883 


.0482 


.0898 


26 


.1208 


.0604 


.0385 


.0769 


.0445 


.0829 


28 


.1122 


.0561 


.0357 


.0714 


.0418 


.0770 


80 


.1047 


.0524 


.0383 


.0666 


.0886 


.0719 


82 


.0982 


.0491 


.0312 


.0625 


.0362 


.0674 


84 


.0924 


.0462 


.0294 


.0588 


.0340 


.0634 


86 


.0873 


.0486 


.0278 


.0555 


.0321 


.0599 


88 


.0827 


.0418 


.0263 


.0526 


.0804 


.0568 


40 


.0785 


.0898 


.0250 


.0500 


.0289 


.0539 


42 


.0748 


.0374 


.0288 


.0476 


.0275 


.0514 


44 


.0714 


.0357 


.0227 


.0455 


.0263 


.0490 , 


46 


.0688 


.0841 


.0217 


.0435 


.0252 


.0469 


48 


.0654 


.0327 


.0208 


.0417 


.0241 


.0449 


50 


.0628 


.0314 


.0200 


.0400 


.0231 


.0431 


66 


,0561 


.0280 


.0178 


.0357 


.0207 


.0385 


60 


.0524 


.0262 


.0166 


.0338 


.0198 


.0860 



120 



FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



Tablb VIIe. 

GEAR wheels: TABLE OF TOOTH PARTS. 
CIRCULAR FITCH IN FIRST COLUMN. 



j 


t 


j 


^3 


^[a, 
§ 


1^ 










P' 


1" 

p7 


F 


t 


s 


D" 


•+/ 


2)"+/ 


P'X .81 


P'X.885 


2 


i 


1.5708 


1.0000 


.6366 


1.2782 


.7866 


1.8782 


.6200 


.6700 


u 


A 


1.6755 


.9375 


.5968 


1.1987 


.6906 


1.2874 


.5818 


.6281 


u 


♦ 


1.7963 


.8750 


.5570 


1.1141 


.6445 


1.2016 


.5425 


.6868 


H 


A 


1.9338 


.8126 


.5178 


1.0845 


.5985 


1.1158 


.6088 


.5444 


1* 


f 


2.0944 


.7500 


.4775 


.9549 


.6525 


1.0299 


.4660 


.5025 


lA 


if 


2.1866 


.7187 


.4676 


.9151 


.6294 


.9870 


.4456 


.4816 


H 


A 


2.2848 


.6875 


.4877 


.8764 


.5064 


.9441 


.4262 


.4606 


lA 


M 


2.8986 


.6562 


.4178 


.8856 


.4884 


.9012 


.4069 


.4897 


n 


t 


2.6183 


.6250 


.8979 


.7958 


.4604 


.8588 


.8876 


.4188 


lA 


i* 


2.6466 


.5987 


.8780 


.7560 


.4874 


.8166 


.8681 


.8978 


n 


ft 


2.7926 


.6626 


.8581 


.7162 


.4143 


.7724 


.8488 


.8769 


lA 


H 


2.9668 


.6812 


.8882 


.6764 


.8918 


.7295 


.8294 


.8569 


1 


1 


8.1416 


.5000 


.8188 


.6866 


.8688 


.6866 


.8100 


.8860 


it 


lA 


8.8610 


.4687 


.2984 


.6968 


.8458 


.6437 


.2906 


.8141 


i 


H 


8.5904 


.4876 


.2785 


.6670 


.8228 


.6007 


.2718 


.2931 


it 


lA 


8.8666 


.4062 


.2586 


.5178 


.2998 


.6579 


.2519 


.2722 


i 


U 


4.1888 


.8760 


.2887 


.4775 


.2762 


.5150 


.2825 


.2618 


u 


lA 


4.5696 


.3487 


.2189 


.4377 


.2582 


.4720 


.2181 


.2808 


f 


U 


4.7124 


.8888 


.2122 


.4244 


.2455 


.4677 


.2066 


.2238 



SPUR- AND FMCTION-GEAB8. 



121 



Tablb VIIf. 

GBAB WHEELS: TABLE 07 TOOTH PABTS — GtmHmud. 
dBcniAB nTOB nr fibot columji. 



E 






1" 



1| 

IJ 

2 

^ 

2* 
31 
8 

8J 
8i 
4 

4 
6 

6 

7 

8 

9 

10 
16 



5 0265 
5.5851 
6.2882 
7.1808 
7.8540 
8.8776 
9.4248 
10.0531 
10.9956 
12.6664 
14.1872 
15.7080 
16.7552 
18.8496 
21.9911 
25.1827 
28.2748 
81.4159 
60.2655 



U 

I 






.8125 
.2812 
.2500 
.2187 
.2000 
.1875 
.1666 
.1562 
.1429 
.1250 
.1111 
.1000 
.0987 
.0883 
.0714 
.0635 
.0555 
.0500 
.0312 



S 






.1989 
.1790 
.1392 
.1893 
.1278 
.1194 
.1061 
.0995 
.0900 
.0796 
.0707 
.0637 
.0697 
.0581 
.0455 
.0898 
.0854 
.0818 
.0199 



I 






.8979 
.8581 
.8188 
.2785 



.2122 
.1989 
.1819 
.1591 
.1415 
.1278 
.1194 
.1061 
.0910 
.0796 
.0707 
.0687 



)S 






.2801 
.2071 
.1842 
.1611 
.1478 
.1881 
.1228 
.1151 
.1052 
.0921 
.0818 
.0787 
.0690 
.0614 
.0526 
.0460 
.0409 
.0868 





ll 




D"+f 


P' X .81 


P'X.886 


.4291 


.1988 


.2094 


.8862 


.1744 


.1884 


.8488 


.1550 


.1675 


.8008 


.1356 


.1466 


.2746 


.1240 


.1840 


.2575 


.1168 


.1256 


.2289 


.1088 


.1117 


.2146 


.0969 


.1047 


.1962 


.0886 


.0957 


.1716 


.0775 


.0888 


.1626 


.0689 


.0744 


.1873 


.0620 


.0670 


.1287 


.0581 


.0628 


.1144 


.0517 


.0558 


.0981 


.0448 


.0479 


.0858 


.0388 


.0419 


.0768 


.0344 


.0872 


.0687 


.0810 


.0885 


.0429 


.0194 


.0209 



122 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

vice versa^ which is the time when the greatest stress is brought 
upon the material of the tooth, the pressure is applied at such a 
distance outside of the pitch-line of the tooth subject to the greater 
bending moment, as to allow an increase of pressure of more than 
70^ of that which could be applied at the top of the tooth. With 



4fm 






^ 




] 


r —r 






. 


-F- 




^ 


r^ 






__-^ 














I [ 








: i 






















1-- 
























;' 






" 


f r_ 


-^ 




^ 


!_ ^ 




















ma 














_>^ 


' 






































r^ 










































^ 


„i*^ 












































,^ 














































^ 




































t 








_4^ 












































T 










































_^ 












































^ 








































^ awt 






^ 






















































































_ 














1 






























r^ 1 y: 


1 










































^iT 












































^8 


A\ 


■! 












































ganxf 


y\ L 


I 












































■ ^r 












































' 1 1 




































































- 






















' 


J 


1 F 






































fiMO 


iC-i 


f 




















































































I 
















1 




















1 
























t 










































- - 




























i , l^ton 
















^ 




















Tl"' 








P«lODD 






















* 








































































































1 


































1 












f 
























'\ I 


























SOl> 


















■ ] 




























J 


































1 


: f 






1 




-i 
































t : I 










t-i' 


■ "f4- 






























1 \ 1 




"^ I f 


' 


_i I T . 
















i 














^l:-*-h^ 






a 


Z % 


^ % 1 


\ \ 


g 1 


Nu 




er 


of 


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In 


8 


Mur. 


: 


1 


3 


a 


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\% 


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g 



DiaKram showiiiK fhe pressure at the point of a single tooth In a gear of M diametral 
(-8.28 inches olrcular) pitch and 1 inoh face, which will produce a fibre stress of 0000 pounds 
per square inch in the material of the gear. 

Fig. 67. 

a greater number of teeth in either or both gears, a greater increase 
18 allowable. 

In calculating the strength of a tooth by the cantilever formula, 
no account is taken of the shearing stress on the section under con- 
sideration. The value of this shearing stress per unit area of the 
section normal to the median plane is 

v = P -f- area of section. 

Combining this with the fibre-stress due to bending gives for 
the maximum tension or compression, both being the same in value. 



Maximum stress = ~ -f y v* -f — . 



8PUB- AND FRICnON-OEARS. 123 

The yalne of the *' maximnm stress "for a given preasare P is 
bnt A^ greater than that of S obtained by the formola for the can- 
tilever. This 4^ excess is for a rack-tooth, which is the strongest 
of all, internal gears excepted; it grows smaller as the number of 
teeth decreases. On account of its comparatirely small yalue, when 
compared with the uncertain elements which enter into any attempt 
to calculate the sjbrength of gear-teeth, it may be safely neglected. 

The maximum shearing stress, according to the formula for com- 
bined shear and flexure, 



/ 9^ 

Maximum shear = y v* -\- — , 

is about 54^ of the maximum tension. 

The curves of Figs. 58 and 59 were laid out from Pig. 67 by 
making use of the fact that, when the number of teeth and width 
of a gear remain constant, the strength of the teeth varies as the 
circular pitch ; and by taking account of the fact that, when the 
number of teeth remains constant, and the width of the gear-face 
bears a constant ratio to the circular pitch, the strength of the teeth 
varies as the square of the pitch, the curves of Figs. 60 and 61 were 
plotted. 

In the diagrams, Figs. 58, 59, 60, and 61, each of the diagonal 
lines, some of which are straight and some broken, represents a 
different fibre-stress in the material of the tooth. The break in 
some of the lines has no significance, being made only as a means 
of shortening the diagrams to a convenient length. It should be 
noted, however, that a change of pounds per inch is thus made 
necessary on the pressure-scale, the change occurring at the pressare- 
line where the angle is made in the diagonals. Figs. 58 and 59 are 
for gears having a face 1 inch wide. They are of exactly the same 
nature, one for small and the other for large pitches, and might 
have been combined in a single diagram. Such a combination 
would make it difficult to read values for the smaller pitches unless 
the whole diagram were excessively large. The same is true of 
Figs. 60 and 61. 

The diagrams, Figs. 58, 59, 60, and 61, are for use in finding 
any one of the four factors, circular pitch, pitch diameter, pressure 



124 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 





POR QCAR FACK 1 INCH WIDC. 








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«ijg - — - - - 


1:": 3? 


^ V XX?^' 


S^~\C^^ 






a \> \v. X^ 


,3 




i- \ %^ 


x^ ^ 






"^ 3. i wft 


1.73- '■^'-I-- 




fc " T" \^ 


\ N. \^ 






-t -uaL^^X- 


sL^ ^sT 






r L N 


^-N X^ 






1 S X 


--.^:..-:s 






tt t:il _ 


: :^": : 


LSJ * 




^ s 


^^ \: :: 






A 


ts^ X I 


1 






k^-U4 ..S^. . 



I Bhowlng relatian between pressing at tooth point, stress in ooter fibie, and 
pitch ol a gear too(h. Flgoree in diacT&m ahoye "RACK" huUoate fibre stroM-^hoaa 
Mow, diaiTwf4ir of gear. 

Fig. 58. 



SPUBr AKB FRIOIION-OSABS. 

FOR gcau rAcc t tnCH wioc 



196 




Diagram showing relation between pressure at tooth point, stress In outer fibre, and! 
pitch of a gear tooth. Figures In diagram above "RACK** Indicate fibre stress thoi» 
below, diameter of gear. 

Fig. 59. 



126 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



WrOTH OF GEAR FACE^CTRdrkAR PITCR. 




Diagram showing celation between pressure at tooth point, stress in outer fibre, and 
pitoh of a gear tooth. Flgoree in diagram above "RACK** indicate fibre Btrosa-^thow 
lelow, diameter of gear. 

Fio. 60. 



SPUR- AND FBIOTION-GEARS. 

WIDTH or GEAR FACE - CI RCULAR PITCH, 



127 




Diagram showing relation between pressure at tooth point, stress in outer fibre, and 
J/KUAi of a gear tooth. Elgusea in diagram above "RACK** indicate fibre a tr o w t faoaa 
Mow, diameter of gear. 

Fig. 61. 



128 FORM, STRENGTH, AND PROPORTIOJ^S OP PARTS. 

at tooth-point J or fibre-stress, when the other three are known or 
assumed. 

The following formula has been developed from Fig. 57 by 
Prof. John H. Barr.* 

The notation, slightly changed from his, is: 

C = circular pitch, inches; 
D = pitch diameter of gear, inches; 
F^ width of gear-face, inches; 
N '=' number of teeth in the gear; 
P = pressure on teeth, i)ounds; 

S = tensile or compressive fibre-stress in the material of the gear, 
pounds per square inch. 

P=(7^.S(0.106-^) (8) 

From this the following equations may be deduced: 
For a gear-face 1 inch wide. 



P=<74.106-?:|2),] 



PN 

a = 



For use when 
' N is known. 



(9) 
(10) 
(11) 



For use when 
D is known. 

(12) 



• • • • 



SiQ.lO&N- 0.678)* J 
P= (75(0.106 -0.215^); 

C = 2)[_0.246 - Y .0606 - *-65^]. 
For a width of gear-face = circular pitch, 

P=C-s(0.1(«-°f?); 1 . . (IS) 

^ -^^ ' I For use when 
I p^ I iV' is known. 

,^'^\/>S'(0.106iV^- 0.678)' J '^^^ 

?=<?-5(o.io»-o..4f^'iXS°- • • • • <») 



.•Trans. Amer. Boc. of Mech. Engrs., vol. xvni., 1897, p. 776. 



SPUR- AND FRIdTlON-GfiAM. 129 

When the width of the gear-face is equal to the circniar pitch, 
or bears a given ratio to it, the equation involving the diameter of 
the gear contains both the cube and sqoare of C; hence no equation 
having a general solution for obtaining C when Dy P, and S are 
known can be obtained from the original form given above. The 
last equation given can be used tentatively, of course, by substitut- 
ing assumed values of C in it and solving, continuing until a satis- 
factory value is found. 

In designing gears it is customary to assume that the pressure 
against the teeth is the same in amount as would have to be applied 
tangent to the pitch-circle of the gear in order to drive it. The 
difference between the pitch radius and the radial distance to the 
point where the pressure is applied to the tooth surface is not great 
enough, in any ordinary system of gears, to need consideration. 

When, for a gear, the pitch radius R in inches, and the greatest 
torsional moment M in inch-pounds to be exerted upon it, are 
known, the force P that must act tangent to the pitch-circle can 
be found by the equation 

P = -g- pounds (16) 

If the highest rate of working, expressed in horse-power (H.P.), 
or in inch-pounds of energy B transmitted per minute, together 
with the number of revolutions per minute F, and pitch radius R 
inches, are known, P can be found by the equation 

-, 33000X12XH.P. , 

^"^ 2i^EV pounds,. . . . (17) 

or 

^ = 2ilTP«'^°^« <^«> 

Table VIII, representing experiments made by the Brown & 
Sharpe Mfg. Go. upon cut gears of their own manufacture, shows 
che observed breaking pressure of the teeth and the calculated fibre- 
stress which would exist if the pressure were all applied at the top 
of one tooth, normal to its median plane, and uniformly distributed 
across the face of the gear. 



130 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

Example. — Let it be required to find the pitch of a gear 18 
inches in diameter^ 4 inches face, working at 4000 pounds maxi- 
mum stress in the material, to withstand 2000 poimds pressure at 
its point. This gives a pressure of 2000 4-4= 500 pounds per 
inch of width of face. In Fig. 58 on the right-hand scale, marked 
"pounds pressure at tooth-point," find the 500-pound line and 
follow it to its intersection with the 4000-pound fibre-stress Une; 
from this point drop down to the curve marked "18 inches 
diameter," and thence to the scale on the left of the diagram, thus 
obtaining a reading of 2.25 diametrical pitch, or about 1.4 inches 
circular pitch. 

Example. — ^A pressure of 4550 pounds is to act on a gear of 
8 inches pitch diameter and 7 inches face, the limit of fibre-stress 
being 6000 pounds per square inch. The pressure per inch of width 
in this case is 4550 -h 7 =650 pounds. By the same method as 
before, following the 650-pound pressure-line to its intersection 
with the 6000-poimd fibre-stress line, and thence toward the 
bottom of the diagram, it is found that the vertical line does not 
cut the 8-inch diameter-line, but falls to the right of it, the latter 
terminating at 1.5 dianr.etrical pitch, this being the greatest pitch 
that can be used when the number of teeth is not less than twelve, 
which is the lower limit in the diagrams. The fact that the vertical 
and diameter lines do not intersect shows that no gear having 
twelve teeth or more can be designed to fulfil the conditions given. 
With a fibre-stress slightly greater than 6000 pounds per square 
inch, however, a gear of 1.5 diametrical pitch will answer. 

Example. — For a width of face equal to the circular pitch, 
Figs. 60 and 61 are used. The method is the same as above, except 
that the pressure for a width of face equal to the circular pitch is 
read on the scale of pressures at tooth-point, instead of the pressure 
per inch of width, as was done before. Thus, for a gear 30 inches in 
diameter, whose face width is to be three times the circular pitch, 
to work at a fibre-stress of 14,000 pounds per square inch, under a 
pressure of 90,000 pounds at the tooth-point, we have 90,001 -r- 
3=^30,000 pounds pressure on a width of face equal to the circular 
pitch. Following the 30,000-pound line from the right-hand side 
of Fig. 61 to its intersection with the 14,000-pound diagonal line. 



SrUK- AXD FBK1IQ»-€XAB&. 131 

and thence to the SO-inch cfiaineter curre, giTos o^To inches^ rir- 
ralu- pttdu viikh is something ksss than .5 dibnaetrical pitch. 

PteUia. — fiTtd the hoise-poveis that can be transmitted 
by a pair of ^eais eooskting of a sted (Hnion and biunae gwr- 
The gear is nm at 25 revolutions per minute. The pinion ki 6 
inches pitch diameter* has 12 teeth of 2 dian^trical ]»tch. ainl iis^ 
limiting fibie^ress is 12,000 pounds per square inch. The ^e^ecjur 
is 30 inches {Mtch diameter, has 60 teeth, and its limiting fibre- 
stress is GOOO pounds per square inch. The gear faces are 4 
inches wide. 

By Fig. 58 the pressure at the tooth-point that toII pnxlucf 
12,000 pounds per square inch stress in the funion is 960 povmds 
per inch of width of gear face. And, by the same figure* 900 povmils 
per inch of width of gear face will cause a fibre-stress of 6000 
poim^ds per square inch in the gear. 

Since the two gears work together, the lower of the two tooth 
pressures, i.e., 900 pounds, is one that must be useit for calculating 
the capacity of the gears for transmitting power. The total pres- 
sure for the gear 4 inches wide is 

P=4X900=3600 pounds. 

The horse-powers that can be transmitted may be found by 
equation (17). The pitch diameter and revolutions per minute of 
the gear will be used. By substituting in equation (17) 

„ ^ 2^15X25X3600 ^, , 
^•^'- 33000X12 ^^^•^• 

Problem. — ^A gear whose pitch diameter is 6 inches — .6 of a 
foot is required to transmit 5 horse-powers at 120 revolutions per 
minute. It is to be assumed that, on accoimt of inaccuracy of 
form and poor supports, the teeth may bear on one corner only. 
The working stress is to be 6000 pounds per square inch. It is 
required to find the greatest allowable number of teeth for this 
gear which will fulfil the above conditions. 

The pressure exerted against a tooth according to the assumed 
conditions is, by equation (17), 

^ 33000 X5 o„ - 



132 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

The diagram, Fig. 60, for a width of gear face = to the circular 
pitch may be used for the solution. Since the pressure is at a 
comer of the tooth, a width of gear face equal to only 1.5 times 
the circular pitch will be taken as effective. The pressure for a 
width of face = circular pitch will therefore be 875-5-1.5=583 
pounds. Find 583 on the "pounds pressure at tooth-point" 
scale. Follow this horizontally toward the left to its intersection 
with the 6000-pound oblique line, then vertically downward to 
the 6-inch diameter line and then horizontally to the left. The 
reading thus found on the diametral pitch scale is about 2.5. The 
corresponding circular pitch is 1.28 inches. The number of teeth 
for 2.5 diametral pitch is 15. 

Problem. — Design a pinion and gear to transmit 50 horse- 
powers when the pinion runs 100 revolutions per minute and the 
gear 25. The gear faces are to have a width three times as great 
as the circular pitch, and the gears to be of the smallest diameter 
possible, using nothing less than 12 teeth, that will transmit the 
power with a fibre-stress not greater than 3000 pounds per square 
inch in the material of the teeth, according to the assumptions 
made for the diagrams, Figs. 57 to 61, and equations 8 to 15. 
The gears are to be accurately cut and rigidly supported. The 
Brown & Sharpe system, having a describing circle of a diameter 
half as great as that of the pitch circle of a 15-tooth gear of the 
pitch under consideration, to be used. 

The total turning moment that must be exerted by each gear 
is, by equation (17), 

rr ^ w • * 33000X12XH.P. -,-, 
Total turning moment = ^-y = PR 

33000X12X50 
27rlOO 

= 31600 inch-pounds. 

The turning moment for a width of gear face equal to the cir- 
cular pitch is, since the face is to be three times as wide as the 
circular pitch, 

^^^31600^ 10530 inch-pounds. 



SPUR- AND FRICTION-GEAES. 133 

Equation (14) can be used for determining the circular pitch. 
When both terms of the fraction imder the radical are multiplied 
by R the equation becomes 



-v; 



PRN 



iSR{0AO&N-0,67Sy 

In this equation N must be taken equal to 12^ for, of all gears 

of the same diameter and not less than 12 teeth, the 12-tooth one 

is the strongest. This can be seen by inspection of the diagrams 

Figs. 5S-61. R may be substituted in the last equation in terms 

of C, in accordance with its value, 

6C 
R= — for a 12-tooth gear. 

By making these substitutions, and squaring both sides, 
^^ 10530Xl2X7r 

3000X6C(0.106X 12-0.678) ' 

_1053qxl27r_^ 

3000X6X0.594 ' 

C =3.336 inches. 

Pitch diameter of the pinion = — — = 12.75". 

Pitch diameter of the gear is 4X12.75=51 inches. 

Width of gear face is 3X3.336=10", about. 

31600x2 
Total pressure at tooth-point= -^-^ t^r— =4970 pounds. 

1^.7 o 

Problem. — Same as preceding one, except that the greatest 
fibre-stress shall not exceed 10,000 pounds per square inch, instead 
of 3000. 

This solution could be made, of course, in the same manner as 
before. It can be done much more readily, however, by making 
use of the fact that, in similar gears for transmitting equal amounts 
of energy at the same rotative speed, the fibre-stresses have a 
ratio to each other which is the inverse of the ratio of the cubes 
of the diameters of the gears, or of their other similar linear 
dimensions. (The proof of this statement is given after the solu- 
tion of the problem.) Accordingly, putting: 

D= diameter of gear working at 3000 pounds per square inch 
fibre-stress; 



134 FORM, STRENGTH, AND PROPORTIONS OP PARTS. 

D'= diameter of gear working at 10,000 pounds per square inch 
fibre^stress; 
the value of D' is expressed by the equation 



-"i 



"'-"■ S--^ 0^-.6694C. 



The value of D for the pinion in the preceding example is 12.75 
inches. 

Therefore the diameter of the pinion is 

2)'= .6694X 12.75= 8.53 inches. 
Diameter of gear =4x8.53=34.12"; 
Pitch= .6694X3.336=2.233"; 
Width of face=3x2.233=6.7"; 

Total pressure at tooth-point = — ^-r^ — ^ = 7410 pounds. 

The proof of the property of gears just made use of can be 
given by the aid of the equation on page 78, which, by multiply- 
ing both its terms by R, becomes 

Since the turning moment must remain the same, whatever 
the size of a gear, when transmitting a fixed amount of power at 
a given speed of rotation, the following is true for two similar 
gears, represented by the two terms of the equation respectively: 
Sbhm S'b'jh'yR' 
6Z 61' • 

Corresponding linear dimensions of the two gears must all 
have the same ratio since they are similar. Calling this ratio », 
the last equation may be written 



whence 
Therefore 



6Z 6lx « *' 






R' As S (RT 

i2 ^ NlS" """^ S' i2» ' 



Which may be read, for this particular problem, 

Stress in larger pinion (Radius of smaller)' 



Stress in smaller pinion (Radius of larger) 



s • 



SPUR- AND FRICTION-GEARS. 

Table VIII. 



135 



BREAKING LOADS FOB CUT CAST-IRON GEARS, EXPERIMENTALLY 

DETERMINED. • 



Diametral 
Pitch. 


Circular 
Pitch, 
inches. 


Width 

of 
Face, 
inches. 


Num- 
ber of 
Teeth. 


Pitch 

Diaxn., 

iuches. 


Revo- 
lutions 


Velocity 
at Pitch- 
circle, 
feet per 
min. 


Observed 
Breaking 
Pressure 
at Pitch- 
circle, 
pounds. 


Stress in 
Outer Fibre, 
pounds per 

sq. inch. 


10 
8 
6 
5 


.8142 
.8927 
.5286 
.6288 


ii- 
It 


110 
72 
72 
90 


11 
9 

12 
18 


27 
40 
27 
18 


78 
94 
85 
85 


1060 
1460 
2220 
2470 


38000 
29000 
24000 
20600 



The following formnlas, 19 to 32, based npon the assnmption 
that the pressure is always normal to the tooth surface, have been 
developed from investigations made by Mr. Wilfred Lewis.* In 
them the force is assumed as being effective at the point where the 
normal to the top of the tooth curve intersects the median line of 
the tooth. The pressure between the teeth is resolved into two 
components at this point — one radial, and the other perpendicular 
to the median line of the tooth. The radial component is neg- 
lected, only the one normal to the radius and exerting a purely 
bending action on the tooth, being considered. 

Formulas 19 to 32 are applicable to cycloidal gears developed by 
a generating circle whose diameter is half that of a 12-tooth gear of 
the same pitch as that under consideration, and to involute gears in 
which the normal to the tooth curve at the pitch point makes an 
angle of 75° with the line drawn from the pitch point to the centre 
of the gear. 

The proportions of the teeth to which the following formulas 
apply, taking the circular pitch C as the unit of measurement, 
are: thickness on pitch-line = .47(7; addendum = .3(7; clearance 
= .06((7+l). 

Two methods of filleting the bottom of the teeth are represented. 
In one the fillets are as large as will just clear the tops of the teeth 

* Proceedings Engineers' Club of Philadelphia, vol. x., 1898, p. 16 ; Ameri- 
can Machinist, May 4, 1898, p. 8, and June 8, 1898, p. 7 ; Trans. Amer. Soc 
Mech. Engrs., vol. xviii., 1897, pp. 776, 781. 



136 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

of an intenneshing rack, and in the other the radios of the fillets 
is equal to the clearance between the top of a tooth and the bottom 
of a space. 

For convenience of application, the formnlas are given for both 
a gear-face one inch wide, and for a width of face equal to the 
circular pitch. It is assumed that the pressure is uniformly dis- 
tributed across the face of the gear. The notation is the same as 
in the preceding equations. 

The formulas deduced from Mr. Lewis's investigation of 
strength of gear-teeth are: 



For a gbab-facb one inch wide: 

Iiarse Fttlets. 



c = 



PIT 



For use when 
JVis known. 



/S(0.124J\^- 0.684)' 
P = C» [0.124 - 0.218 -^1 ; 
O = i)[_0.284 +*/.0807 - 4.6 -^1. 



(19) 
(20) 



.... (21) 
For nse when 
D is known. 

.... (22) 



SmaU XtUeto. 



P=C^[0.124-2f!]; 



= 



PJV 



For nse when 
Jf is known. 



i8'(0.124JV- 0.888)" 
P = (75r0.124 - 0.282 ^1; 
O = D\0.n — "v/ 0.049 - 3.57 -^J. 



(23) 
(24) 



.... (25) 

For use when 
D is known. 
.... (26) 



SPUR- AND FBICTION-GEABS. 



137 



For a width of gear-face = circular pitch: 

lA^se FiUete. 



P=(7«^[0.m-^]; ' 

C- ! PN — 

V '5(0.124iV^- 0.684)* J 



For use when 
JV is known. 



(37) 
(28) 



P= (7'>Sr0.124 - 0.218^1 ^y 



use when ^„o\ 

known. • • • • V-^^J 



SmaU FiUete. 



- / PN 

~ V 'Sf(0.124iV"- 0.888)* J 

P= C»<Sr0.124 - 0.282^1. ^^ 



P 

C 



For nse when 
N is known. 



use when 
known. * 



(30) 
(.31) 

(32) 



Gases that require a large amonnt of power to be transmitted, 
when the diameter and speed of a gear are so limited by the condi- 
tions to be fulfilled as to require an excessive and objectionable 
breadth of face, can sometimes be met by using two or more gears 
of the required diameter, on the same shafts with faces of such a 
breadth that the sum of all their breadths is equal to that which 
would be required for a single gear rigidly supported and perfectly 
aligned. When several gears are used on the same shaft in this 
manner, they should not be placed so that their teeth are in line, 
but ^^ stepped" by placing the teeth of each successive gear in 
advance of (or behind) those of its neighbor by a distance eqaal to 
the circular pitch divided by the number of gears on the shaft; 
such an arrangement gives smoother running and less liability to 
breakage. 

26. Metl^ods of strengthening gears. — For cases where very 
strong teeth are required, and where the pressure against the teeth 



138 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

is UBually or always in the same direction, the form of tooth shown 
in Pig. 62 can be used, in which A is the working side. "When 




Fig. 62. 

nsed for sach a purpose as hoisting, where the machinery is reversed 
and driven backward to lower the hook or other device for suspend- 
ing the load, the side opposite A^ as well as A^ should be made of 
some correct tooth outline; but when A is the only side that works, 
the opposite side may be made of any outline that will give strength 
and, at the same time, clear the tops of the teeth of the intermesh- 
ing gear. 

For cycloidal gears this buttressed tooth can be obtained by 
using a large describing circle for the face and flank of A^ and a 
small one for the corresponding parts of the opposite side, when it 
is desired to make the latter of a form suitable for backward driv- 
ing. Involute gears of this form require a large base-circle for the 
working side A^ and a small base-circle for the opposite side. 

The strength of such a tooth, as compared with that of the 
ordinary form having the same pitch, height, and breadth, can be 
determined quite approximately by developing a tooth of each form, 
and comparing their thicknesses at or near the bottom ; when a large 
flUet is used at the bottom of the tooth, the weakest plane lies a 
short distance above the bottom. The strength of the teeth is 
approximately proportional to the squares of their thickness at the 
sections lying the same distance from the top. Having once 
obtained the ratio of strengths, it can be used for all teeth of the 
same form, as long as the pitch and breadth of gear face of those 
compared are equal to each other. 

"Shrouding" is another method of adding strength to teeth. 
It consists of adding an annular ring or disk to one or both ends of 
a gear. This shroud may extend either partly or entirely to the 



SPUR- AND FRICTION-GEARS. 139 

tops of the teeth. It fonns an integral part of the gear-casting or 
forging. Since the shrond forms a rigid support for the ends of the 
teeth, shearing of the metal between the ends of the tooth and the 
shroud must occur at this point in order to allow the tooth to break 
near the bottom as a cantilever. A full shroud at each end of a 
gear evidently increases its strength more than when part of the 
tooth stands above the shroud. The thickness of a shroad, 
measured parallel to the axis of the gear, is generally at least as 
great as that of a tooth at the pitch-line. 

The strengthening effect of a shroud depends upon the breadth 
of the gear, a narrow one being more strengthened than a wide one 
of the same pitch and diameter; for, while in both cases the shear- 
ing resistance added by the shrouds is the same, it forms a greater 
ratio to the cantilever resistance as the gear grows narrower. 

When a pinion engages with a large gear, the former wears more 
rapidly on account of having its teeth come into mesh more fre- 
quently; and, if they are of the same material, the pinion conse- 
quently grows weaker more rapidly than the spur-gear, in addition 
to having been the weaker of the two at first, on account of the form 
of its teeth; hence the pinion is the one to be shrouded if this 
device is used at all. Two meshing spur-gears of the same size and 
material should both have equal shrouds extending about half-way 
to the top of the teeth, provided it is necessary to strengthen them. 

27. Short gear-teeth are much stronger than those having the 
proportions commonly used, in which the height of the tooth is 
roughly 0.7 of the circular pitch. The shorter teeth also run more 
quietly. Modern practice is adopting them to a considerable extent 
when the gears are not intended to be interchangeable. A tooth 
height of about 0.4 of the circular pitch is commonly used. Such 
teeth are especially satisfactory for cast gears used without 
machine-finishing on the working surface of the teeth. 

Mr. C. W. Hunt gives the following proportions and, in Table 
IX, working loads for the involute gears adopted by the company 
bearing his name.* They are used for coal-hoisting engines and 
similar machinery, which generally do not have solid foundations. 
The teeth are cast to form and used without machine-finishing. 

* Trans. Amer. Soc. of Mech. Engrs., vol. xviii., 1897, p. 787. 



140 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



The notation is changed, to agree with that of the formnlas given 
above. 

Table IX. 

WORKING LOADS FOR SHORT-TOOTH UNCUT CAST-IRON GEARS. 



c 


P' 


p„ 


C 


P' 


P" 


Circular 


Working 


Maximum 


Circular 


Worlcing 


Maximum 


Pitch, 


T.oa(1, 
pounds. 


Load, 


Pitch, 


Load, 


Load, 


inches. 


pounds. 


inches. 


pounds. 


pounds. 


1 


1820 


1650 


^ 
H 


6700 


8800 


1 


2800 


2600 


8800 


10500 


8000 


8700 


2f 


10000 


12500 


ij 


4100 


6000 


8 


12000 


14800 


2 


5800 


6600 









The addendum and dedendum are each equal to 0.2 of the cir- 
cular pitch (7; the clearance equals .05(C7+ 1) inches; the width 
of the gear-face is 2(7+ 1 inches. 

Non-metallic Spur-gears. 

28. Mortise-gearing is made by keying or pinning wooden teeth 
into cast-iron rims designed to receive them. Fig. 63 shows end and 








T 




fc;- 


' ^^ 




^^- 


'^ 


^> ' 


\N 






^ 


R- 


r^ 


^ 




1^ 


1 










^ 


1 




s 


^ 






:^ 




V • 


-^ 




^ 


,V,^^^ 








* 







Fig. 68. 

side Tiews of a mortise-gear, the rim R being in section, so as to 
show full end and side views of the wooden teeth T and the wooden 
keys jK' which hold the teeth in the tapered openings through the 



SPUR- AND FRICTION-GEARS. 



141 



rims. Fig. 64 is another form, the tooth-shank here being smaller 
than the bottom of the tooth, thus making a shoulder which rests 
against the rim and holds the tooth from dropping down into it 
when the wood shrinks or is compressed by the pressure due to trans- 
mitting power. A pin P is used to hold the tooth in place, instead 




Fig. 64. 

of the key of the preceding form; the key could be used in both 
forms equally well, however. 

The working side of a tooth of the form of Fig. 64 is generally 
made with a deeper shoulder than on the opposite side, in order to 
allow for the wear due to service, the tooth retaining nearly its 
original strength till worn past the shoulder. Bevel- and spur-gears 
can both be made with wooden teeth when it is thought that the 
existing conditions to be met can be more satisfactorily fulfilled by 
them than with metal gears. 

The more common practice is to mate a cast-iron and mortise 
gear, the smaller being of iron when they are of different diameters, 
the iron teeth having a thickness on the pitch-line of about 0.4 of 
the circular pitch, that of the wooden teeth being the remaining 
0.6 of the pitch, due allowance being made for backlash and rough 
work by reducing these values slightly. The smaller gear is made 
of iron, because the work performed by any one tooth upon it is 



142 FOBM, STRENGTH, AND PROPORTIONS OP PARTS. 

greater than for a tooth of the larger one; consequently greater 
darability is obtained by thns making the smaller gear of a material 
better able to withstand wear; also, the gears are stronger, since the 
teeth of the smaller, whose form makes them weaker than those of 
the same thickness on the pitch-line woald be on the larger, are 
made of the stronger material. Sometimes, bat very infrequently, 
both gears are made with wooden teeth. 

Comparatively noiseless running at high speeds is one of the 
good qualities of mortise-gearing, this being especially marked when 
comparison is made with a pair of cast gears; the elastic quality of 
wood makes them able to resist sudden shocks that might break 
cast-iron gears of the same size and running at the same speed ; they 
are very durable when run under moderate pressure with proper 
care in lubricating the teeth. 

In practice, the wooden tooth-blanks are first keyed in the rim 
and then machined to proper form in the same manner as metal 
gears, the only difference in the two processes being that different 
cutting edges are required for shaping wood and metal, as, for in- 
stance, a saw, running at high speed, is used in a bevel-gear planing- 
machine when cutting wooden teeth, instead of the sharp-cornered 
planer-tool required for metal. 

The woods more commonly used for teeth are maple, hickory, 
and locust. In order to prevent swelling and shrinking as the at- 
mosphere changes its humidity, the blanks are thoroughly saturated 
with paraffin or some other oily substance before putting in place. 

29. Bawhide, indurated vegetable fibre, etc., are frequently 
used for small gears where quiet running is desired. The gears are 
usually made of a number of thin disks placed side by side and held 
together by a pair of metallic disks at the ends. Most of them are 
durable under light service, but are not strong. Under heavy 
service they may wear rapidly, if they do not break. Rawhide and 
some of the other materials of a similar nature are liable to shrink 
considerably when used in a dry place. This may be a serious 
objection on account of reducing the diameter of the gear and caus- 
ing the disks to become loose. 

30. Factor of safety for tooth-gears. — While, as in most cases 
in machine-designing, it is impossible to fix a factor of safety, since 
its value must depend upon the conditions of each individual case. 



8PUB- AND FRIOTION-GEARS. 143 

an ezamination of a few of the points to be considered may be of 
aid in selecting its yalae. 

Thus, in a hand-driven. crane, where the speed is slow and any 
nnnsaal strain is not liable to oocnr, the factor nsed needs only to 
indade an allowance for flaws, heterogeneous material, and internal 
stress, it not being necessary to include shocks and stresses that 
cannot be estimated, since neither of these last two really exist when 
proper safety-locking devices are nsed. If the change is made to 
power-driving, however, and high speeds are nsed, then an allow- 
ance must also be made for the shock due to the striking of the 
teeth together. The greatest stress that can come on the crane at 
any time is calculable within practical limits, being limited to the 
breaking strength of the chain, plus frictional resistances; hence 
no allowance for unknown stress need be made. Repeated stress 
certainly does occnr in the teeth of a gear in service, but the allow- 
ance for flaws, heterogeneous material, and internal stresses is 
generally so large that the material does not regularly work near 
enough to its elastic strength to make necessary any allowance for 
this cause of fracture. 

Rolling-mill machinery, stone- and ore-crushers, and other 
machines applied to similar purposes, are subjected to shocks and 
unknown stresses. In such cases the necessary factor of safety can 
be determined only by experience and some knowledge of the nature 
of the material to be operated upon. 

31. The efficiency of spur-gearing depends very largely upon 
the frictional loss in the supporting bearings. The greater the loss 
of power in the bearings the lower the efficiency. The pressure 
between the teeth causes an equal amount of pressure upon the 
bearings supporting each gear if the bearings are on each side of 
the gear. The weight of the gear, as well as other weights and 
forces, must be taken into account when calculating the total 
pressure on the bearings. 

The experiments made by Wm. Sellers & Co. upon a spur-gear 
18.62 inches pitch diameter, having 39 teeth of H inches circular 
pitch, running on journals 2\^ inches diameter, and driven by a 
spur-pinion 5.73 inches pitch diameter, with 12 teeth of the same 
pitch, and running on one journal 2^^ inches and the other 1|| 
inches diameter, both placed close to the hub of the pinion, show 



144 FOKM, STRENGTH, AND PROPORTIONS OF PARTS. 

the average efficiencies given in Table X for pressareB of 430, 700, 
1100, 1600, and 2600 pounds pressure between the teeth.* 

Table X. 

EFFICIENOIES OF SPUR-OEABS. 

Beyolutions of pinion per minute 8 5 10 20 50 100 200 

Efficiency, per cent 90 93 04 96.6 97.8 9B.2 96.6 

If the same amount of power were transmitted with the larger 
gear driving the smaller, the efficiency would be lower. This may 
be more readily seen by supposing that the journals of both gears 
are of the same size. The pressures on the journals of the two 
^ears are equal with or without friction. When the resisting 
moment is applied to the pinion-shaf t,the increase of pressure between 
the teeth, due to friction, over that which would be required were 
there no frictional resistance, is greater than that when the larger 
gear is the driven. The ratio of the increase in the two cases is 
inversely as the radii of the gears. The power required to drive a 
gear is proportional to the pressure against its teeth if the coefficient 
of friction remains constant. The change of pressure on the bear- 
ings would probably cause a slight change in this coefficient, but 
even if this should occar, the amount of driving power to be applied 
to the large gear when driving the small one, in order to deliver a 
given amount of power to an operating-machine, would be greater 
than that necessary if the driving gear were smaller than the driven. 

BEVEL-GEARS. 

32. strength of bevel-gear teeth. — In order to investigate the 
nature of the pressure between the teeth of an accurately con- 
structed and adjusted pair of bevel-gears, and of the fibre-stress in 
the material) let it first be assumed that they are not rotating, and 
that there is no pressure between their teeth. Then assume that 
one is locked so as to prevent its turning, and that a turning force 
is applied to the other, thus producing pressure between the 
engaging teeth. 

On account of the elasticity of the material, a slight deflection 

* " Experiments on the Transmission of Power by Gearing," by Wilfred 
Lewig. Trans. Am. Soc. Mech. Engrs., vol. vii., 1886, p. 278. 



SPUB- AND FRICTION-GEARS. 145 

of the teeth will be caused by the pressare on them. The corre- 
epoDdingly slight rotation thus allowed in the other parts of the 
gear to which the taming force is applied, will cause a point at the 
top of the large end of a free tooth to moTe throngh a linear dis- 
tance whose ratio to the corresponding linear motion of a point 
similarly situated at the small end of the same tooth is equal to 
the ratio of the radial distances of the two points from the axis of 
the gear. 

If the line of contact between a pair of engaging teeth is at the 
top of the working surface of a tooth on the locked gear, the top of 
the tooth will be deflected more at the large end of the gear than 
at the small, the deflection of any point along the top of the tooth 
being proportional to its radial distance from the axis of the 
gear. The linear dimensions of the tooth profile at the large 
and small ends of the tooth are proportional to the radii of the 
addendum-circles at the ends of the gear. The profiles are, of 
course, similar. Hence, in accordance with the property of canti- 
levers of similar profile that (referring to Fig. 56), when the 
breadth b remains constant, as well as the ratio of the length I to 
the height A, the linear deflection at the end i^ proportional to the 
load P, it can be seen that the distribution of pressure along the 
line of contact is in proportion to the radial distances from the axis 
of the gear, and also in proportion to the linear dimensions of the 
tooth sections. And, again in accordance with the property of 
cantilevers of similar profile, that, when the breadth b remains un- 
changed, the end deflection that wUl produce the same maximum 
fibre-stress S in each is proportional to the linear dimensions, it is 
evident that the maximum fibre-stress is the same in the tooth 
from end to end of the gear. 

If ff is the large, and h the small, addendum radius of the gear, 
then the resultant pressure against a tooth acts at a radial distance 
equal to 2(H* — h*) -f- ^H* — A") from the axis of the gear. 

The mean value of the pressure per unit length of the line of 
tooth contact equals the total or resultant pressure divided by the 
width of gear-face. In a tooth under pressure, this mean value is 
exerted at the middle of the gear-face. Therefore, a spur-gear 
having the same face width, and teeth of the same form and size as 
those of the bevel-gear at the middle of its face, is of the same 



146 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



strength as the hevel-gear. If m is taken as the mean addendum 
radins of the bevel-gear^ and a is the angle between the axis of 
the bevel-gear and the elements of the addendum-cone, the radius 
of the spur-gear must be m sec or, in order to agree with the system 
of gear-teeth used for the bevel-gear. 

As in the case of spur-gears, the pitch-surface dimensions may 
be used without serious error for practical forms of gears. The 
dimensions of the pitch-cone will therefore be used hereafter 
instead of those of the addendum-cone. 

Fig. 65 is a section of a bevel-gear. The dimensions given are: 




Fig. 65. 

A = largest pitch radius of bevel-gear; 
a = smallest pitch radius of bevel-gear; 



C = 



_ A'\-a ^, _ 



2A 



C = circular pitch at middle of gear-face; 



£7' = circular pitch at large end of gear; 
^= width of gear-face, measured on pitch-cone; 
R = length of a normal to pitch-cone at middle of gear-face, 
measured between pitch surface and intersection with axis 



SPUR- AND FRICTION-GEARS. 147 

of gear; this is the radius of the pitch-circle upon which 
the teeth are generated for the middle of the gear-face; 
a = angle between gear-axis and pitch-cone elements. 

The notation for the pressure is: 

-P(mean) = mean value of pressure on bevel-gear tooth per inch of 

length of contact-line; 

P(feB) = resultant of all elementary pressures on bevel-gear tooth; 

P^ = pressure which, if applied at the greatest pitch radius of 

the gear, would produce the same torsional moment on 

the gear-shaft as the actual acting pressure. 

In accordance with the assumption just made, the resultant 
pressure P(re8) acts at a radial distance 2(J[' — a') -7- 3(-4" — a") 
from the axis of the gear. This distance is inconvenient to use in 
designing, it being more satisfactory to use the pitch radius A of 
the large end of the gear. The relation between P' and P(ree) is 
expressed by the formula 

Whence 

The relation between P(mean) &nd P(re8) ^ given by the formula 



•^(reB) = FPimeaxiy 



Therefore 



^ ^3 A{A' - a«)^^<"^«»>- • • • (^^) 

The value of P(mean) corresponds to those given on the scale of 
^^ pounds pressure at tooth-point *' in Figs. 58 and 59. The value 
of /'(moan) therefore can be found on one of these diagrams by using: 
gear diameter ^R\ circular pitch C, taken at the middle of the 
bevel-gear face; and any given or assumed working fibre-stress S. 



,148 FORM, 8TBENGTH, AND PROPOBTIONS OF PABTS. 

If the force P' is giyen, and the pitch C required for a given 
Tslae of 8 and a specified gear-blank, the yalae of P(mean) is first 
fonnd by the equation 

-f^Cmfian) — 2" A* — a* "F^ .... \0±} 

and then^ by the diagram^ the value of Gy from which O' can be 
determined. 

Example. — ^Let -4 = 10 inches, jP= 4 inches, a = 50**, 0' = 
1.0472 inches (corresponding to 3 diametral pitch), and S= 3000 
ponnds per square inch. Then : 

o = 10 — 4 sin 50° = 10 — 4 X .766 = 6.936 inches; 
O^ ^^O' = (16.94 X 1.047) -^ 20 = 0.887 inch; 
^ ^ (A+a) ^ ^ ^ 1^1,556 ^ 13,18 inches; 
2i2= 26.36 inches. 

The value of P(mean) for 0.887 of an inch pitch, a pitch diam- 
eter of 26.36 inches, and 3000 pounds per square inch fibre-stress 
is found on the diagram Fig. 58 to be 260 pounds. Therefore, by 
equation (33), 

^' = 3- loff)"- (04)'] ^ ^ ''' = '"^ P^'^"'^- 

The horse-power that would be transmitted by the gear at 100 
revolutions per minute is 

„ -. 900 X 27r X 10 X 100 _ « .^ 
^•^- - 33000 X 12 ^•^^• 

Example.— Take P' = 1000 pounds, S = 2000 pounds per 
square inch, and the dimensions of the gear-blank the same as those 
in the preceding example. The pitch C is required- 



SPUR- AND FBIGTION-OEAB8. 149 

By equation (34) 

^ 3 10( 10)' - (6.94)' 1000 ^^^ . 

P(mean) = ^ (10)' - (6,94)' ^4- = ^^^ PO^nds^ 

In the diagram Fig. 58, (7= 1.65 inches* Therefore 

^, 2AC 2 X 10 X 1.65 , ^^ . , ra ;^ * ^ \ 

C = , = — = 1.95 inches. (End of example.) 

The Talae of P(niean) in equation (33) can also be determined by 
formula (11), P for spur-gears and P(mean) for bevel-gears being 
identical. Substituting, in equation (33), the .value of P(inean) as 
given in equation (11) changes (33) to the form 

The value of P(meaD) as given by formulas (21) and (25) can also 
be used for the system of gears that they represent. 

The solution of the next to the last example by equation (35) 
gives 

^ = I ^^f^'-^^ X . X soo«[o..o. - 0,«,»-^] 

= 900 pounds. 



Problem. — Find the size of a pair of bevel-gears to transmit 
40 horse-powers, and another pair to transmit 100 horse-powers, 
with a velocity ratio of 5 between a pair of shafts at right angles 
to each other; the pinion to have 16 teeth with a widfli of gear 
face equal to 2.5 times the circular pitch at the large end of the 
gear, to run at 100 revolutions per minute, and work at a fibre- 
stress of 10,000 pounds per square inch; the supports to be rigid 
and the teeth accurately formed; gears to be of the smallest possible 
dimensions to fulfil these conditions. 

The direct solution for obtaining the required sizes of the 
gears is not as convenient as to assume some dimensions of the 
gears and find their capacity for transmitting power. Then, 



150 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

by use of the data thus obtained, the size of the required pair 
of gears can be determined. 

Accordingly, it will be assumed that the circular pitch (at 
the large end of the pinion) is 2 inches, then the width of the 

face is 

F=2.5X2 = 5". 
The pitch radius at the large end of the pinion with 16 teeth is 

2?: 
For a velocity ratio of 5 with shafts at 90®, the tangent of 
the half pitch-cone angle of the small gear is 

tan a = .2 ; whence a = 1 1*^ 19'. 
The pitch radius at the small end of the pinion is, for the face 
width F=5'', 

a=5.093-5 sin a = 5.093-5x.l962=5.093-.981=4.112''. 
The mean circular pitch is 

A+g 5.093+4.112 9.205 
^''^2A^ 2X5.093 "^ ~ 5:093 "^'^"^ ' 
The equivalent diameter of a spur-gear having the same form 
of tooth as that at the middle of the face of the bevel pinion is 
2/2=(il+a) sec a = 9.205 sec 11^ 19' -9.205X1.019 = 9.38''. 
The mean value of the pressure against the point of one tooth, 
as found by equation (11), is 

P(».an) =C5(.106- .215g) =CS(.106- -^l^g:^) 

= 1.807X 10,000 (.106- .0414) = 18070X .065= 1170, 
The equivalent pressure at the largest pitch diameter of the 
pinion is, by equation (33), 

p,_2^ ( 5.09)«-(4.11)» 

3 6.09[(5.09)»-(4.11)»]^ ^"""^ 
_2 131.87-69.43 
3 ^ 5.09 (25.91 - 16.89) ^ <'°~°' 
2 fi2 44 

■=3^5-:o9x9:^x^x^i^<^=^«^ P°"^^ 

The power that the gears will transmit is 

M P _ 5300X 100 X2;r 5.093 ^^ 
^•■^•" 33000x12 ~^^'^' 



SPUR- AND FRICnON-GEARS. 151 

The dimensions of the pinion for transmitting a different 
amount of power can now be detenrined by making use of the 
fact that, for similar gears working at the same fibre-stress and 
speed of rotation, 

H.P. of smaller _ (Radius of smaller )' 
H.P of larger ~" (Radius of larger)' ' 
Accordingly, for this case, the pitch radius of the pinion for 
transmitting 40 H.P. is 



40 



il'=A»^=Ai/.935=5.093X. 978=4,98", 

which corresponds to a pitch diameter of 9.96 inches at the large 
end of the pinion. 

The circular pitch to agree with 9.96 inches diameter and 16 
teeth is 

Width of gear-face=5 »1^=5X.978=4.89". 
To transmit 100 H.P. the dimensions of the pinion will be 
^'=i4»l^=AX'^2:336=5.093Xl.32=6.72", 
which corresponds to a diameter of 13.44 inches, and 

^'=2i^3X2=2-6*"- 
Width of gear-face = F = 1 .32 X 5 = 6.6". 
The side and end wear of the supporting bearings of a pair of 
bevel-gears, caused by the pressure between their teeth, both tend 
to localize the pressure between the teeth at the large ends. In 
allowing for such wear, when calculating the strength of the teeth, 
it is therefore correct to assume that the load is carried by a part of 
the larger end of the tooth. Probably a width of gear-face equal to 
the circular pitch at the large end is as great as can be safely taken 
when the apex angle a of the pitch-cone approaches near to 90®; 
when this angle is very small the gear becomes more like a spur- 
gear, and the width of gear-face maybe increased to 1.5 times the 
circular pitch. Allowance for localization of pressure at the small end 
of the tooth may be made in the same manner as for the large end. 



152 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

Such localization may be caused by foreign matter between the 
teeth, settling of supports, etc. 

Mortise bevel-gears are very commonly used. Shrouding the 
teeth is also practised to a considerable extent. Teeth correspond- 
ing to the double helical teeth of spur-gears are less frequently 
adopted ; these do not, of course, have an end sliding motion like 
that of screw-bevel and screw gears. A machine for cutting bevel- 
gears whose elements do not intersect at the apex of the pitch-cone 
has recently been put into practical use. 

33. The efficiency of bevel-gearing is less than that of spur- 
gearing, for the reason that the pressure between the teeth causes 
an end thrust on the bearings, in addition to the side pressure 
corresponding to that of spar-gears. The frictional resistance due 
to this end thrust causes a reduction of efficiency. 

FRICTION-GEARS. 

34. Friotioa-gears, having smooth surfaces held in contact under 
pressure, and transmitting power by means of the frictional resist- 
ance between their surfaces, find a very considerable application in 
certain classes of machinery, notably that which is frequently 
stopped and started, and whose source of power is a constantly 
rotating shaft or pulley; they also afford a convenient method of 
obtaining different speeds when the primary shaft of a machine 
rotates uniformly. Less frequently they are used for transmitting 
power between two uniformly rotating shafts. 

When a considerable amount of power is transmitted, and the 
pressure between the gears is heavy, one of a pair of gears which are 
in contact, is generally made of iron, and the surface of the other of 
some material such as wood, paper, leather, hard rubber, etc. 

35. Cylindrical friction-gears. — The turning force P which 
can be exerted by one smooth cylindrical friction-gear upon another 
when they are pressed together with a force F^ the coefficient of 
friction of the material being fJL^ is given by the formula 

P = /aR 

The coefficient of friction ^ for paper friction-wheels, as deter- 
mined by a series of experiments made by Prof. W. F. M. Gobs,* 

♦ Trans. Am. Soc. Mecli. Engrs., vol. xviir., 1897, p. 102. 



SPUR- AND FRICTION-GEARS. 153 

is given below. In these experiments one of the pair of wheels in 
contact was iron, and the other of compressed straw-board in '' thin 
disks cemented together under heavy pressure and strenghtened by 
iron side-plates, or fitted over iron centres.^' The edges of the 
disks were pressed against the iron wheel when power was trans- 
mitted. 

The experiments were made upon paper friction-wheels approxi- 
mately 5iy 8, 12, and 16 inches in diameter, all in contact with a 
16-inch cast-iron wheel. " The contact pressure was varied from 
75 pounds per inch of width to more than 400 pounds, and the 
speed limits gave a peripheral velocity varying from 450 to 2700 
feet per minute." 

It was found that: 

The coefficient of friction jx increases as the rate of slipping 
between the gears increases; 

When the slip is as great as 3^, there is apt to be a sudden 
increase in its value to 100^; i.e., motion ceases to be transmitted 
to the driven wheel ; 

'' The coefficient of friction is apparently constant for all pres- 
sures of contact up to a limit which lies between 150 and 200 
pounds per inch of width of wheel-face, beyond which limit its value 
apparently decreases " ; 

** Friction-wheels of 8, 12, and 16 inches diameter give nearly 
the same value for the coefficient, while results from a 6-inch wheel 
are lower by about 10^ " ; ■ '" 

*' Variations in peripheral speed between 400 and 2800 feet per 
minute do not affect the coefficient of friction." • 

Fig. 66,* taken from Professor Goss's paper, shows the relation 
between the slip and the coefficient of friction which he found could' 
be easily maintained with paper friction-wheels 8 inches or more in 
diameter. 

It is probable that the coefficients of friction for the other 
materials that are most commonly used in contact with metal foi^ 
friction-gears operating under comparatively high pressures, are- 
lower than those generally found by the laboratory experiments 
where plane surfaces are moved over each other at unit pressure 

* Trans, Am. Soo. Mecli. Engrs., vol. xvm., 1897, p. 108. .1 



164 FORM, 8TBENGTH, AND PE0P0BTI0N8 OF PARTS. 

yery much lower in comparison. It is believed that the following' 
yalnes of }x are as high as can be safely used for pressures of 100 
pounds or more per inch of width of gear-face: 

Metal on metal 0.2 

Leather on metal 0.3 

Wood on metal 0.3 

A system of cylindrical friction-gears that is quite commonly 
used where it can be applied, and which has many advantages in 



JBO 



.16 



.u 



j«^ 















^^ 


z 
o 






/ 








E 
11. 
o 




/ 










i 




/ 










8 


/ 
















SUP 


N PER 


CENT 







12 8 

Fig. 66. 

the way of economy of construction and ease of operation, is shown 
in Fig. 67. The larger gears, which are of iron, are placed on the 
driving and driven shafts, whose centres remain at a fixed distance 
apart. The small gear, intermediate between the other two, is so 
supported that it can be pressed against the others, or withdrawn 
from contact with them, at will. It is generally made of some of 
ihe so-called *' friction materials," and thus affords a means of 
securing a high coefficient of friction, and a somewhat elastic sur- 
face in contact with the more rigid iron. 

In a pair of friction-gears having different materials on their 
working faces, it is advisable to use the softer material on the 
driver; then, in case of excessive slipping, there is not so much 
danger that a flat place will be worn in the»driven one. 

Another system of cylindrical friction-pulleys that has been 
found satisfactory in some cases, where the distance between the 
centres of the shafts can be varied^ consists of two pulleys, one on 



SPUE- AND FRICTION-GKARS. 



165 



the driving shaft and the other on the driven, and an endless belt 
of leather fitting loosely around one, which is flanged to hold it in 
place. The face of the second pulley is slightly narrower than the 
space between the flanges of the other, so that when the two are 
drawn together the belt is pressed between the pulley faces and 
forms a cushion against which they work. The coefficient of fric- 




FiG. 67. 

tion in snoh a system is, of course, that of leather on iron. This 
device has advantages in its simplicity and comparatively low cost 
of construction. It is hardly applicable for the transmission of 
large amounts of power. 

The power that can be transmitted by a pair of cylindrical 
friction-gears in practical operation is about the same per inch of 
face width as that per inch of width of the kind of flat leather belt 
that would be used on the same machinery. It depends, of course, 



156 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

upon the pressure and coefficient of friction. When a given amount 
of horse-power H.P. is transmitted, the value of P can be found 
by equation (17), § 25. 

The effloienoy of friotion-gears depends very largely upon the 
frictional resistance of their supporting journals. The heavy 
pressures between them, necessary to produce the required turning 
force, brings a correspondingly high pressure upon the journals, 
with its attendant frictional loss. While no efficiency tests upon 
this class of transmission machinery seem to have been made public, 
it would appear probable that the friction loss in it is much greater 
than in spur-gears or belt-connected pulleys constructed with the 
same care. 

36. Orooved Motion-gears, having their surfaces grooved cir- 
cumferentially, are used when it is desired to have a tight grip 
between them without excessive pressure upon the bearings. Pig. 
68 shows a section through the rims of such a pair of gears in con- 




FiG. 68. 



tact, cut by a plane passing through the axes of both gears. The 
wedge-like action gives them increased holding power. 

There is a sort of rubbing or grinding action between a pair of 
grooved friction-gears in action which is detrimental to both their 
life and efficiency of service. The grinding action increases with 
the radial depth of the engaging surfaces; hence, if such gears are 
used, it is better to make them with a large number of shallow 
grooves than a few deep ones. 

Both of a pair of grooved friction- wheels are frequently made of 
the same metal, cast iron being most used. Better service can be 
obtained by making them of such metals or alloys as work well 
together in journal- and step-bearings under heavy pressure. 



SPUR- AND FRICTION-GKABS. 



157 



The angle between the sides of a groove should seldom be 
smaller than 30^, but the increase of grip over that of a pair of 
smooth cylinders is not great if the angle exceeds 40°. 

The total pressure acting between the sides of the grooves of a 
pair of gears with parallel axes, when the gears are pressed together 
with a force F normal to their axes, the angle of the groove being 

0, is jP CSC ~ ; and the tangential or turning force acting at their 

n 

pitch-point is approximately ^Fcac —, in which ja = coefficient of 
friction between the materials of the grooves. 




Fig. 69. 



37. Friction bevel-gears are often used for connecting intersect- 
ing shafts where the same conditions of operation as have been 
mentioned for cylindrical friction-gears exist. In addition to tbe 



158 FOBM, STBENGTH, AKD PROPOETIONS OF PABTS. 

side pressure on the bearingB, there is an end thrust which maj be 
80 great as to require special provisions, in the way of a step- or 
collar-bearing, to withstand it when the apex angle a of the gear- 
cone is large and the service is heavy. If F is the normal pressure 
between the gears, and a the angle between the axis of the gear 
and its face, then the value of the end thrust E may be expressed 
by the equation 

E^ jPsin a. 

38. Crown Motion-gears are used on light machinery where it 
is desired to vary the speed of the driven shaft while the driver runs 




uniformly. Pig. 69 represents such a device. A ring Z, of leather 
or other suitable material, is held between a pair of disks on the 
shaft F, which is in the same plane, and at right angles to the axis 
of the metal disk JT, against which L presses. 

IC £ is the driver, moving it across the face of K will change 
the speed of the latter, and the direction of JT's rotation will be 
reveiised by moving L across its centre. When possible L shoald 
be the driver, since it is faced with the softer material. When this 
method of driving is used on a drill-press, where the force required 



SPUE- AND FBIOTIOK-OEARS. 159 

to tarn the driU-spindle increases as the 8X>eed of rotation decreases, 
as is the case when changing from a small to a large drill-bit, 
L should always be the driver, since its lever-arm about the axis of 
JT increases in the same proportion that the speed of ^decreases, 
the speed of L remaining constant. 

39. Double-oone, variable-speed friction-gears. — ^A special form 
of friction-gears for variable speeds is shown conventionally in Fig. 
70. The mechanism consists of a pair of similar cones, A and By 
with smooth surfaces, placed side by side on parallel axes, and 
separated a slight distance to allow a short endless belt O to pass 
between them. Their pressure against the belt serves to transmit 
power between them as they rotate. The belt is held in position 
by a shifter, not shown; by moving it along from end to end of 
the cones when they are running, their speed ratio can be varied. 



CHAPTER m. 

BELTS AND ROPES FOR POWER TRANSMISSION. 

FLAT BELTS. 

40. When a belt is transinitting power by its frictional resist- 
ance against the face of a palley over which it passes, there most 
necessarily exist a difference of tension in it, at the points of tan- 
gency with the pulley, equal to the torsional force exerted upon the 
pulley. The tensile stress gradually increases from the point where 
the belt first comes in contact with a driven pulley to where it leayes 
it. The torsional force that can be transmitted to the pulley 
depends jointly upon the belt tensions, coefficient of friction 
between the belt and pulley face, radius of pulley, velocity at which 
the belt travels, and the weight of the belt per unit of length. 
The last three items must be included on account of the centrifugal 
force exerted on the belt as it passes around the pulley, which force 
reduces the pressure against its face, and is the principal factor 
which limits belt speed. The centrifugal force becomes greater as 
the speed increases, until, at high velocities, the belt is partly lifted 
from the pulley, and consequently but little turning force can be 
transmitted by it. 

41. Equations for power transmission by flat belts. — By the use 
of the equations developed below, the torsional force that can be 
exerted by a belt, and the relations between all quantities involved, 
can be determined. The following notation is used: 

A = sectional area of belt, square inches; 
H.P. = horse-power transmitted; 

P = total turning force exerted by the entire width of belt 

against the pulley = T^— Toy pounds; 
R = radius of pulley or sheave, feet; 

i6o 



BELTS AND ROPES FOB POWER TRANSMISSION. 161 

T^ = total tension in belt on tight side^ pounds; 
To = total tension in belt on slack side^ pounds; 
^V = total tension in belt when at rest^ pounds; 

V = velocity of belt, feet per minute; 

c = centrifugal force for 1 cu. in. of belt, pounds; 

g = acceleration due to gravity, = 32.2 ft. per sec. per sec. ; 

p = turning force transmitted by a belt having a cross-sec- 
tional area of 1 square inch, or by a rope 1 inch in 
diameter, pounds; 

q = pressure of pulley against 1 linear inch of belt, pounds; 

r = radius of pulley or sheave, inches: 

t = tension, in pounds per square inch, at any section of the 
belt; 

t^ = tension, in pounds per square inch, on tight side of belt 
= working tension of belt; 

f^ = tension, in pounds per square inch, on slack side of belt; 

tr = tension, in pounds per square inch, on both sides of belt 
when at rest or running without load; 

V = velocity of belt, feet per second; 

w = weight of belt per cubic inch, pounds; 

a = arc of contact between belt and pulley, degrees; 

8 = .01746« = arc of contact between belt and pulley, cir- 
cular measure (radians); 

fji = coefficient of friction between belt and pulley; 

e = 2.71828 = 100*»*»« is the base of the hyperbolic, Naperian, 
or natural logarithms. 

The general equation for the centrifugal force acting on a body 
moving in a curved path is 

, ^ .. , . . (mass) X (square of velocity) 

(centrifugal force) = ji- m 1 • 

° ' radius of curvature 

The specific form for the present case is 

'-JE-' g(r-^ 12) " 9660 "T~ - •^^^^^^^^^- ' (^^^ 

Fig. 71 shows a pulley with a belt passing around it, the angle 
of contact being 6. The nature of the forces that act along the 



162 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

arc of contact^ and the relation between the belt teneionB t^ and to, 
when the belt is jaet beginning to slip, may be determined by tak- 





Pig. 71. 

ing an elementary length of the belt = ds, embracing an angle dff 
on any part of the arc of contact^ and considering it as a free body, 
as shown in Fig. 72. 

The forces which hold this elementary body in equilibrium are, 
first, t and {t + dt), normal to the end sections and making an 
angle dO with each other, whose resultant force acts to press the 
belt against the pulley; second, o(2«, the centrifugal force; third, 
qds, the pressure of the pulley against it; and fourth, dp = piqds, 
the frictional resistance along its surface where it is in contact with 
the pulley. 

The resultant of t and {t + dt) can be taken as normal to the 
surface of the belt, since dt is small in comparison with t. This 
resultant is counterbalanced by qds and cdSy dp not having any com- 
ponent normal to the belt surface, since it acts along the surface. 
The following equation for the value of qds can now be written 

qds + cds = t sin -^ + (t + dt) sin — , 

4 A 



in which dH is so small that its sine can be taken as equal to the 



BELTS AND HOPES FOR POWER TRANSMISSION. 163 

sin -^j, being the product of 

two small quantities^ can be neglected; the equation thus becomes 

qds + cds = tdd, 
or 

qds = tde -- cds, (37) 

in which ds = rd0, and c = -^ as given in equation (36). By 

gJi 

substituting these values in equation (37) it becomes 

qds = tde - ^^rde 
gR 

z:ztdd^\2~de (38) 

9 

By equality of moments about (Fig. 72), 

t + di = t + fJiqdSy 
whence 

dt = fjiqds. 

Substituting the value of qds as given in equation (38) in this 
equation, it takes the form 

or 

t - 12!^ 



By further solution 



^o 9 



whence 



't.- 



Hyperbolic log. ( ^ ) = /ift 



164 FOBM, STRENGTH, AND PB0P0BTI0N8 OF PABT8. 

By substituting for v its equiyalent value F-r- 60, and reducing. 

The taming force per sqnare inch of area is 

i> = (<» - .0001036wF') - (to - .OOOlOSetoF') 

= <»-<»= (<.-.0001036wF')(^!^l^) ... (40) 

= (f,-.0001036«>F') (^"^Q.„„^/ ) (41) 

From equation (39), 

<, = <.6^»-.0001036wF*(e^«-l) (42) 

_ <jo.«^»8^ - .O001036toF'(10-«»«^ - 1). . (43) 
And from equations (40) and (41), 

tn = p-^^^^ + .(mm^v,V* (44) 

1Q.00758Ma 

= i^ lO.OQ758^ - 1 + >0001036tgr', ... (45) 



Tablb XI. 

ANGLE OF CONTACT a ON THE SMALLBB OF A PAIB OF PTTLLBY8 
DIBECTLY CONNECTED TOGBTHEB BY AN OPEN BELT. NO 
ALLOWANCE FOB SAG OF BELT. 

2) = diam. of large pulley ; d = diam. of small pulley ; (7= distance be 
tween pulley-centres. 



D-d 


ADgle of Contact. 


D-d 


Angle of Contact. 


D-d 


Angle of Contact. 


C 


Degrees. 


C 


Degrees. 


C 


Degrees. 


.05 


177.13 


.80 


162.74 


.56 


148.07 


.10 


174.27 


.85 


159.84 


.60 


146.07 


.15 


171.37 


.40 


156.90 


.65 


142.07 


.20 


168.50 


.45 


153 67 


.70 


189.00 


.25 


165.64 


.60 


161.04 


.75 


135.84 



BELTS AND ROPES FOR POWER TRANSMISSION. 165 

Example. — Find the secfcioniil area of a belt to transmit 200 
H.P. when running ab 4500 feet per minate, and working at 300 
pounds tension per square inch on the tight side; the arc of contact 
on the working pulley having the smallest portion of its circumfer- 
ence embraced by the belt, being 160°. 

The value of the turning force P which must be exerted by the 
belt is found by the equation 

^^ 33000 XH.P. . ^^^^ 

which becomes, by the substitution of numerical values, 

o 33000X200 ._^ , 

^ = 4500— = ^^^^P^^^^'- 

Assuming that good fulled leather is to be used, the weight 
per cubic inch may be taken as tt; = .035 of a pound, and the coeffi- 
cient of friction may be taken as ;/ = 0.3. 

By equation (41) the turning force which a belt having 1 square 
inch of sectional area will transmit is 

p = [300 - .0001036 X .035(4500)"] — jp^gj =.J 

= [300 « 73.4][?:|^] = 226.6 X -gjj = 128 pounds. 

The total area of belt required is therefore 

^ = P -^i? = 1467 -5- 128 = 11.45 sq. in. 
The total tensions are 

p 

Tn = Atn=-tn = 11.45 X 300 = 3435 pounds; 

To= Tn- P — 3435 - 146?.= 1968 pounds. 

If a belt .5 of an inch thick is used, it would have to be about 
23 inches wide. 

The turning force per inch of belt-width for a belt 23 inches 
wide would be 1467 -t- 23 = 63.8 pounds. 

A tension of 400 pounds per square inch is sometimes used, but 
18 higher than is advisable for a belt that works nearly or quite up 



166 FOBH, STRENGTH, AND PROPORTIONS OF PARTS. 

to its rated capacity continuously; from 200 to 300 pounds per 
square inch is more advisable. 

When the belt is at rest, the tensions are practically equal in 
both stretches between pulleys. The sum of the tensions, T^ + T^y 
is greater when a belt is transmitting power than when running idle 
or standing still. For a horizontal belt working at 300 pounds 
tension per square inch on the tight side, and having 2^ slip (i.e., 
the surface of the driven pulley moving 2^ slower than that of the 
driver) on cast-iron pulleys, the increase of the sum of the tensions 
may be taken, for speeds up to 1000 feet per minute, as an average 
of about \ of the value when the belt is idle. At high speeds the 
centrifugal action tends to reduce this increase, until, at the speed 
giving a centrifugal tension equal to the initial tension, they become 
equal on the two sides, and of the same value as when the belt is 
idle. It is on the safe side, so far as the stress in the belt is con- 
cerned, to allow for the full increase of tension, however, hence it 
will be so taken. Galling the tension in each side, when the belt 
is at rest, TV, the relation between T^ and the sum of the working 
tensions is expressed by the equation 

t(2?;) = ?;+?;; 

whence 

7;. = f(yn + 7;). 

The value of T^ has just been found above; that of T^ may be 
obtained by the equation 

Substituting this value of T^ in the last equation gives 

Tr = f (27; — P) = f [(2 X 3435) - 1467] = 2026 pounds. 

This is the tension that must be maintained in the belt, when 
at rest or running idle, in order that it shall not have more than 2^ 
'^if slip when working under the full load for which it is designed. 
It corresponds to a tension tr per unit area of section, when the belt 
is at rest, of 

t^^iTr-^A^ 2026 -$- 11.45 = 176 pounds per square inch. 



BELTS AND ROPES FOR POWER TRANSMISSION. 



167 



The required tension may be fairly well maintained by having 
the piiUeys so that the distance between their centres can be 
adjusted, or by using an adjustable idle pulley as a tightener. 
When no means of pulley adjustment is used, the belt must be 
shortened at intervals in order to allow for the permanent stretch 
that continually occurs throughout its life of service. To allow for 
this stretch, the belt must, at each adjustment, be made shorter 
than is necessary to produce the required tension of rest. The 
tension, when shortening, may be weighed by a pair of belt-clamps, 
fitted with springs and a graduated scale, used to draw the ends 
together while the belt is in its working position. 

If, in the example just given, fi is taken as 0.4 instead of 0.3, 
the following values are found: 

r«=2880; r^=1610; and f^=168. 

41.1. Tandem Belt Drive. — ^When a belt drives more than 
one pulley, conditions additional to those for a two-pulley drive 

1, ifli ^ 






Pig. 73.1. 



enter into the determination of the tensions in it. The tension 
that determines the size of belt may not be the highest found by 



168 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

calculation for each pulley separately. It is sometimes higher 
than necessary for any one pulley of the group. This can be best 
illustrated by an example. 

Fig. 72.1 shows four pulleys, three of which, A, B, and C, are 
to receive power from the driver D. In the upper part of the 
figure the arrangement, dimensions, direction of rotation, and 
horse-power to be delivered are given. In the lower part each 
pulley is shown separately, with the angle of belt ^vrap, turning 
force P, and belt tensions necessary to drive it considered apart 
from the others. 

These tensions correspond to the assumed values 

i„=400; //=.3; w;=.035. 

Inspection of the values found shows that B requires the 
highest tension, 1642 pounds. Since the turning force for B is 
467 pounds, the tension of the slack side of -B is 

1642-467= 1175 pounds. 

The tension on the slack side of D must therefore also be 1175 
pounds since B and D are joined directly by the belt. And since 
the turning force for D is 600.4 pounds the tension on the tight 
side of D must be 

1175+600.4= 17^5.4 pounds. 

This must therefore be the tension in the stretch of belt joining 
D and C. .This stretch of belt is also on the tight side of C. The 
turning force for C is 66.7 pounds. The tension on the slack side of 

C is therefore 

1775.4 - 66.7 = 1708.7 pounds. 

This is also the tension on the tight side of A, The turning 

force for A is 66.7 pounds. Therefore the tension on the slack 

side of A is 

1708.7-66.7 = 1642 pounds, 

which agrees with that found for the tight side of B, as it should. 

The tensions found for the pulleys collectively are either equal 
to or greater than those calculated for each one taken individually. 
They are therefore satisfactory. 



BELTS AND ROPES FOR POWER TRANSMISSION. 



169 



it may be noted that the sum of the turning forces for the 
Jriven pulleys A, B, and C miist be equal to that for the driver D. 
AJso that these turning forces are divided among the driven pulleys 
in proportion to the power to be deUvered to them. This would 
still remain true if the driven pulleys were of different diameters. 

In Fig. 72.2 the only change of conditions from the above is in 
the way the belt passes around the pulleys. The necessary belt 




Pig. 72.2. 



tensions for each pulley, considered individually, are shown in the 
same manner as before. The 1642 pounds for D is the highest 
tension found for any one pulley. The tension on the slack side 
of 5 is 1175, which must also be the tension on the tight side of C, 
Reducing this by the turning force 66.7 pounds gives 1108.3 pounds 
for the slack side of C and tight side of A, Reducing this again 
by 66.7 pounds gives 1041.6 for the slack side of A and also of D. 
Adding to this 600.4 gives 1642 pounds for the tight side of D, 
which agrees with that for B, 

The tension in each stretch as just found and of the values 
shown in the upper part of the figure are either equal to, or greater 



170 FORM, STRENGTH, AND PROPORTIONS OP PARTS. 



than, that calculated for any individual pulley; therefore the 1642 
pounds determine the area of belt. 

Fig. 72.3. The only change in the drive is in the method of 
wrapping the belt around the pulleys. All quantities are shown 
in the same manner as before. Inspection shows that the highest 
tension calculated for any individual pulley is 1263.5 pounds for 
driver D. The tension on the slack side of C is therefore 1263.5 — 
66.7 = 1196.8 pounds. 

This is also the tension of the tight side of B, which, reduced 
by the turning force of 467 pounds for B, gives 729.8 pounHs for 




Fig. 72.8. 

the slack side of B and tight side of A, Reducing this by the 
turning force of 66.7 pounds for A, gives 663.1 pounds for the slack 
side of A J which Is the same as calculated for the slack side of D. 
The tension of 1263.5 pounds as calculated for the tight side of the 
driver D therefore determines the area of the belt section for this 
method of wrapping it around the pulleys. 

The belt is more than 40% larger in cross-section in the first 
case than in the last. It is about 16% longer in the last ca.se, how- 
ever. 



LYICIJSflZ 




1.6 



20 40 60 do 



--^lI: 



1.5 



i 



t-rrt 



H^ 



±1 



"HT 



100 



120 140 



ANGLE OF "^^{1 
160 180 



33 






M 



Wr 



1.4 



4=W- 



'^-~ 



REUTIVE VALUES OF TURNING FORCE 

FOR DIFFERENT ANGLES OF WRAP OF A BELT OR ROPE BASED 
ON THE TURNING FORCE FOR 180°WRAP AS UNITY. 

THE POWER TRANSMITTED IS PROPORTIONAL TO 
THE TURNING FORCE FOR A GIVEN SPEED. 



1.3 




> 
D 

00 ^-i 



< 

X 
H 
Z 
O 

o 
liJ 

OQ 
UJ 

o 
a: 
O 
u. 

O 

Z .7 

z 

3 

U. .6 
O 

(O 
liJ 

D 

> 
liJ 

> 



liJ 



140° 100° 180" 

Pig. 72.0 



P-DEGREES. 

X) 220 240 



340 SCO 




at page 171). 



i 1800S 



WTf 



m- 



DIAGRAM SHOWING THE RELATION BETWEEN THE PC 
TRANSMITTED AND TENSION IN BELTS AND ROPE 

p — FRICTIONAL TURNING FORCE OF BELT OR ROPE, POUNDS; 
tm — TENSION ON TIGHT SIDE OF BELT OR ROPE, POUNDS; 
ia Bs WEIGHT OF ONE INCH OF LENGTH OF BELT OR ROPE, POUNDS; 
V —VELOCITY OF BELT OR ROPE, FEET PER SECOND, 
V= " " " " " " " MINUTE; 

a -* ANGLE OF WRAP AROUND PULLEY OF BELT OR ROPE, DEGREES 
/i = COEFFICIENT OF FRICTION BETWEEN BELT AND PULLEY FACE; 
= " " " " ROPE AND SHEAVE GROO\ 

TAKING ACCOUNT OF THE WEDGING OF THE ROPE INTO THE G 
HORSE POWER = pv-r- 660 




1100: 



1000: 



fiOO- 



800: 



O ^ 

> o 

Q CO 600- 



600- 



400- 



800: 



500- 



100: 



400 860 300 250 300 150 lOO" 60 

CENTRIFUGAL TENSION— .3727 «v«— .0001036 «V* 

ONE SMALL DIVISION— 5 



■14-LM4f 




100 150 200 250 800 8M 400 450 

EFFECTIVE TENSION -^- .3727 «v« — f,—. 000 1036 «V* 

ONE SMALL DIVISION — 5 



600 



BELTS AND ROPES FOR POWER TRANSIOSSION. 171 

41.2. The diagrams, Figs. 72.4 and 72.5, may be used con- 
jointly to find the sectional area of the belt in the example, § 41. 

Fig. 72.4 is for belts having 180*" angle of wrap, and Fig. 72.5 
is for modifying the results to agree with the actual angle of wrap. 
For the example imder consideration find, on the bottom, "veloc- 
ity" scale of Fig. 72.4, a reading of 4500 feet per minute, and deter- 
mine the intersection of the corresponding vertical line with the 
curve for T=300 and //=.3; then go horizontally to the straight 
diagonal and, finally, vertically upward to the top scale. The 
reading thus found on this scale is 18.81 H.P., which is the amount 
of power 1 square inch of belt will transmit when making 180° of 
wrap, instead of 160*^, as given. 

By Fig. 72.5 it can be seen that a belt making 160*^ of wrap 
will transmit 7% less power, or .93 as much as one wrapping half 
way round the pulley. 

The amount of power that 1 square inch of belt will transmit 
under the condition given is, therefore, 

H.P. for 1 square inch section= 18.81 X. 93 =17.49. 

The sectional area of belt is 

4 = 200^17.49=11.45 square inches. 

If the thickness of the belt is predetermined as .5 inch, then 
the H.P. for 1 inch of width can be found on the scale for a "belt 
1 inch wide," as follows: Find as before the intersection of the 
4500 feet per sectional line with the curve for ju=.3 and T=300. 
This intersection corresponds to a reading of 9.4 H.P. on the scale 
for a belt 1 inch wide and .5 inch thick, making 180® wrap. 

Correcting for 160° wrap, 

H.P. for section 1'' by .5"= 9.4 X. 93 =8.74. 

And the width of belt is 

Belt width =200 ^8.74 = 22.9 inches. 

The diagram, Fig. 72.6, has a far wider range of usefulness than 
the pair just used. It can be applied to both belts and ropes of 
any kind of material. It is good for very much larger angles of 
wraps than the preceding diagram. While hardly so convenient 



172 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

to use for the example just solved, it may be applied to the solution 
by way of illustration. 

First obtain by calculation the product nyo^, in which v=75 feet 
per second to correspond with the required 4500 feet per minute. 

From this 

T^y'=.035X(75)2=197. 

Find on the left-hand scale of values of mv^ the reading 197, 
go horizontally to the right to the intersection of the 197 line 
with the single oblique line to the left of the zero of the bottom 
scale and from the intersection drop down to the scale of "cen- 
trifugal tension,'' where the reading is 73 pounds. From the point 
just found go to the right a distance corresponding to 300 pounds. 
This gives a reading of 227 pounds on the scale of " effective tension.'' 

The value of [la Is .3X160=48. From the poiiit just found 
on the scale of effective tension go vertically upward to the oblique 
line (not drawn on diagram) for the value //<f=. 3X160 =48, and 
thence to the right to obtain a reading of the scale of "turning 
force exerted by belt on pulley face." The reading thus obtained 
gives the value of p as 128 pounds for 1 square inch of belt section. 
The completion of the problem is as in the solution given just after 
the statement of the example. 

42. The coefficient of friction /i, and slip of leather belting. — 
The coefficient of friction is an exceedingly variable quantity, 
generally lying between 0.25 and 1.0 for leather in good working 
order running on smooth iron pulleys ; its value is even greater than 
this in some cases. The coefficient is somewhat higher for the same 
belt on. wooden pulleys with unvarnished faces than on iron, and 
leather-covered pulleys give still higher values. The varnished 
faces of new wooden pulleys give very high coefficients of belt fric- 
tion. A special varnish is sometimes used on iron pulleys to 
increase [i, A change in the intensity of pressure between the belt 
and pulley affects the value to a slight extent, decreasing it as the 
pressure increases for belts that are sufficiently oiled or "dressed '' 
to secure durability; for ordinary working conditions /i may be 
considered as constant between the limits of pressure that can 
be used without rapid wear of belting. 

Corks inserted in the faces of iron or wood pulleys so that they 
project a small distance (.03 to .05 of an inch) above the face 
Increase fi to some extent and reduce the slip greatly. 



BELTS AND ROPES FOR POWER TRANSMISSION. 173 

It has been found by several experimenters that the coefficient 
of friction increases as the slip of the belt over the pulley increases. 
The higher values of [i therefore appertain to a high rate of 
slipping. If too much slipping occurs, there is danger that the 
heat generated will dry, or even bum, the surface of the belt, and 
thus not only weaken it, but at the same time reduce the coefficient 
of friction. 

A slippage of 3% (i.e., a velocity of the driving-pulley face 3% 
faster than that of the driven) is probably as much as should be 
allowed in general practice, 2% being a good value. The rate of 
slipping, at which the higher values of the coefficient are obtained, 
varies from 10% to as much as 20% in some cases. 

Belt-driven machinery that works against a high resistance for 
a short interval, as is customary with punching and shearing ma- 
chines, should not have a maximum reduction of speed, and con- 
sequent rate of belt-slip, greater than 20% during the working 
period. Machines expending energy during a greater portion of 
each cycle should have a smaller variation of speed on account of 
the greater liability of the driving belt to slip from the pulleys. 

With a belt in fair working condition, the coefficient of friction 
/i can safely be taken as 0.3 when the slippage is as much as 2% on 
iron and unvarnished wood pulleys; probably 0.4 is allowable in a 
majority of cases. Fair working condition means that the surface 
next the pulley is not hard, dry, or cracked, or the belt stiff for the 
lack of belt dressing, but soft, pliable, clean, and running without 
excessive vibration. 

When a belt becomes hard and dry, it can be softened, and the 
coefficient of friction increased, by the application of a suitable belt 
dressing, provided the belt has not been too long neglected. 

43. Working strength of leather belting. — ^Tension causes a 
leather belt to elongate continuously throughout its life; upon 
removal of the tensile stress it will partly return to its original 
length, but a permanent elongation always remains. Both the total 
and pertnanent elongations increase much more rapidly during the 
earlier part of its life than later, if the belt is always used under 
the same conditions. During the time of its efficient service the 
elongation is very nearly uniformly distributed throughout the belt; 
but when it has elongated a certain percentage of its length, the 



174 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

stretching becomes uneven, and consequently the belt soon gets 
crooked and unfit for service. 

It may be stated almost as an axiom that the elongation is more 
rapid the greater the stress. The life of a belt, therefore, grows 
shorter as the working stress increases. The permanent stretch 
that a belt will endure before becoming useless is probably a nearly 
constant percentage of its length, which, for good belts, may safely 
be taken at least as great as 6% for leather.* 

Mr. F. W. Taylor, in a carefully kept record of the performance 
of belting in continuous use for nine years in the machine-shop of 
the Midvale Steel Co., finds that double leather belting lasts well 
under a total load, t, of 240 pounds per square inch of sectional 
area, the annual cost of repairs, maintenance, and renewals amount- 
ing to not more than 14% of the first cost. When the stress is kept 
at 400 poimds per square inch of section, he finds that the annual 
expense for the same items becomes about 37% of the first cost. It 
should be kept in mind that these are the continuous working 
values and not, as is often the case, the maximum load under which 
a belt may work for a short time, the stress being much less most 
of the time, as, for instance, a belt driving a dynamo which carries 
its full load of Ughts or motors during only a small part of its run. 

When a belt is laced together at the ends, the strength of the 
joint varies from 50 to 95% of that of the belt ; the average efficiency 
of the laced joint is about 70%. t The average strength of joints 
made with metal fastenings is less than that for lacing. A carefully 
made cemented joint gives a strength about the same as that of the 
belt. This method of splicing, making what is commonly called 
an endless belt, is unquestionably the best for all cases of ordinary 
application, and becomes almost a necessity for high speeds, since 
any heavy place, such as a laced or metal-fastener joint, causes 
vibration. Almost any well-made leather- or wire-laced or metal- 
fastener belt joint is strong enough to give working tensions of 400 
or even 600 pounds per square inch in the belt, and last until there 
is need of shortening the belt when there is no tighting device. 



♦ Notes on Belting, by F. W. Taylor. Trans. Am. Soc. of Mech. Engrs., 
vol. XV., p. 204 

Digest of Physical Tests, January, 1896, p. 46. 



BELTS AND ROPES. FOR POWER TRANSMISSION. 176 

The ultimate strength of leather under a static load is several times 
the values used for transmitting power. 

By inspection of equation (44) it can be seen that, for a given 
belt working under uniform conditions as to velocity and turning 
force transmitted, the total tension, Tn, decreases as the coefficient 
of friction, jn, increases; hence if, in designing a belt, the value of 
jn is taken as small as it will probably ever be for that belt, it is 
reasonably safe to say that Tn will never exceed its calculated value 
as long as the belt works against a constant load. 

A long belt is more durable than a short one because it makes 
fewer bends about a pulley for a given number of rotations. In 
certain classes of work there is also another reason: If there are 
sudden resistances and consequent sUght reductions of speed 
with the following recovery, a long belt, on account of its ability 
to stretch more in its entire free lengths, allows the slight check 
with less strain in it; or, if a body is to be started from rest, as 
when a cam strikes and lifts its follower in a stamp-mill, a long 
belt gives a greater length of time for acceleration and thus 
receives less strain than a short one. 

In practice it is seldom possible to measure the tension in a 
belt when it is working. There is no reason, however, that the 
tension, practically equal in both stretches between pulleys, should 
not be known at the time of putting it in place or of tightening 
after it has become loose by service. This can be done by drawing 
the ends together with spring belt-clamps made to weigh the 
tension, as has been mentioned before. The belts used in Mr. 
Taylor's experiments, already cited, were adjusted in this manner. 
He finds that double leather belts, tightened to 240 pounds per 
square inch of section, or 71 pounds per inch of width, when at 
rest, and when made to exert a turning force on the pulley of 65 
pounds per inch of width, will stretch so that the tension falls to 
106 pounds per square inch, or 33 pounds per inch of width, in an 
average time of two and one-half months of service, their average 
tension during this time being 150 pounds per square inch, or 46 
pounds per inch of width. 

Mr. Taylor concludes that, for continuous working, a double 
oak-tanned and fuUed-leather belt will give an eflFective pull of 
35 pounds per inch of width on the face of the pulley when the 



176 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

arc of contact is 180°; and that other types of leather belts, and 
6- to 7-ply rubber belts, will give an effective pull of 30 pounds per 
inch of width for the same arc of contact. 

In iron-working machinery, such as lathes, planers, and drill- 
presses, 50 pounds per inch of width is quite commonly taken as 
the effective turning force or pull per inch of width for 180** of 
contact. 

44. Velocity of leather belting.— Below 2500 to 3000 feet per 
minute, the velocity of a belt has but slight effect upon the length 
of its life. The most economical speed is probably from 4000 to 
5000 feet per minute. Higher velocities, up to about 6000 feet per 
minute, will generally increase the driving power if the belt is 
worked at high tension, but the life of the belt is shortened, and 
vibration, commonly called " flapping," is liable to occur. It may 
also run from side to side of the pulley, an action known as 
'^ chasing " ; this is more apt to occur in a thin, pliable belt than a 
thick, stiff one of the same width. At higher velocities the cen- 
trifugal force becomes an objectionable feature in addition to these, 
and, even if the belt runs smoothly, less power can be transmitted 
than at slower speeds. 

When V* = 1^ in the equations for belting, the turning 

force P = 0^ the tension in the belt due to centrifugal action being 
equal to the working stress. For a belt weighing .035 pounds per 
cubic inch, and adjusted to work at 400 pounds per square inch 
tension, P = when the velocity is 10509 feet per minute; and for 
a working tension of 200 pounds per square inch P = for a veloc- 
ity of 7430 feet per minute. These are the velocities at which the 
turning force and the power transmitted become zero. 

Belt speeds of 5000 feet per minute, and even more, are common 
in wood-working machinery having pulleys as small as 4 inches in 
diameter, or less. Thin leather belts are successfully used for such 
work. • 

When speeds as high as 5000 feet or more per minute are used 
in connection with pulleys as small as one inch in diameter, a woven 
linen web or tape has been found better than leather.* 

*Mr. John T. Hawkins, in Trans. Am. Soc. of Mech. Engrs., vol. vii., 
3886, p. 582 Belt used for stereo typer's routing-machine. 



BELTS AND ROPES FOR POWER TRANSMISSION. 177 

45. Wear of leather belts. — Each time a belt is bent by passing 
over a pulley, the fibres farthest from the palley are elongated, and 
those next to it compressed. The continuons repetition of the 
bending when the belt is in service has a tendency to crack the 
leather, particularly on the side away from the pulley. The smooth 
or hair side of a single belt is more easily cracked by this action 
than the flesh side. It is therefore advisable to run the hair side 
next the pulleys. Again, there is a slight wear on the surface of 
the belt that comes in contact with the pulleys. The flesh side is 
much stronger than the hair side. This is, therefore, another 
reason for running the hair side next the pulleys. 

Double leather belts are made by cementing together the flesh 
«ides of two thicknesses of leather, thus leaving the hair sides 
exposed to surface wear. Good double belts, properly cared for, 
are not subject to cracking on the side away from the pulley when 
working under proper conditions. Triple and quadruple thick- 
nesses of leather are used for making very thick, heavy belts. It 
is possible, although there is no very definite proof, that a very 
thick belt, as the quadrnplCj when working up to the same stress 
per square inch that would ordinarily be used for lighter belts, may 
have so high a stress per inch of width, and consequent pressure 
against the pulley, that the heat generated by the slipping of the 
belt over the pulley face will dry and burn the surface of the 
leather so that it will become hard and crack on the surface. Such 
an action would, at the same time, reduce the coefficient of friction, 
and thus induce more slipping and injury to the belt. 

The diameter of the pulley has a very considerable effect on the 
life of the belt; for, if a pulley is very small, the belt will be bent 
80 short that there will be excessive wear between its particles, and 
the greater strain due to the short bend will have a greater tendency 
to form cracks. A common practice among engineers is to fix a 
minimum diameter of pulley for single, double, triple, and quad- 
ruple belts. This limitation will answer only for the ordinary 
thicknesses of such belts, however, and should be treated accord- 
ingly. To say that a belt is double does not fix its thickness by any 
means, on account of its great variation in common use. (See 
§46.) 



178 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

Mr. Taylor concludes, as a result of his experience, "that it is 
safe and advisable to use: 

" A double belt on a pulley 12 inches diameter or larger; 

"A triple belt on a pulley 20 inches diameter or larger; 

"A quadruple belt on a pulley 30 inches diameter or larger; 
and that it is inadvisable to use double, triple, and quadruple belts 
on pulleys respectively as small as 9 inches, 15 inches, and 24 inches 
diameter." 

He also considers it safe to say " that the life of belting is 
doubled by splicing and cementing the belt, instead of lacing, 
wiring, or using hooks of any kind." 

A "quarter-turn" belt (i.e., one connecting a pair of pulleys 
whose axes are at right angles to each other, no intermediate or 
idle pulleys being used) is subjected to heavy stresses at its edges 
if the pulleys it connects are large in comparison with the distance 
between them. Belts used in this manner generally wear out 
rapidly, frequently becoming stretched and uneven on the edges. 
By giving the belt a half twist at the splice, so that in a single 
belt the hair side and flesh side come together so as to form a 
continuous surface, the edge-stretching effect is reduced to a min- 
imum. The hair side and flesh side will alternately run next the 
pulley face when so spliced. It is advisable to avoid a quarter- 
turn. (See latter part of § 53.2.) 

46. Weight and thickneBs of leather belting. — ^The weight of 
good leather belting varies from .03 to .04 of a pound per cubic 
inch when new. The various processes of tanning, and the dif- 
ference between fulled and unfulled leather, are the principal causes 
of this very considerable variation of weight. Unfulled belts that 
are light when new, generally increase in weight when not allowed 
to become hard and dry in service; the application of belt dressing 
and, in many kinds of ser\'ice, the accumulation of oil that gets upon 
them in various ways cause increased weight. 

The thickness of belting cannot be given with any considerable 
degree of accuracy. The average thickness of single belts is about 
.22 of an inch, and often reaches .25 of an inch; it can readily be 
seen that such leather can be trimmed to any desired thickness. 
Double belts, such as are commonly used, vary in thickness at least 
from .22 to .35 of an inch, the majority lying between .30 and .36 



BELTS AND ROPES FOR POWER TRANSMISSION. 179 

of an inch. Unless specified as light or heavy, the thickness may 
ordinarily be taken as about .33 of an inch. 

47. Bawhide, •emi-rawhide, rawhide with tanned-leather face, 
and chrome-tanned belts. — ^All of these classes of belts have a higher 
coefficient of friction than ordinary tanned leather belts, but do not 
seem to be so durable or satisfactory for service in dry places. 

Rawhide seems to give better service in damp places than tanned 
leather, the moisture apparently not affecting its capacity to trans- 
mit power, or causing it to crack and distort as the tanned leather 
does, especially when alternately dry and wet. The weight of 
rawhide is about the same as that of tanned and fulled leather. 

Semi-rawhide belting is made of hide that has only a thin sur- 
face layer tanned, the inner portion retaining the qualities of 
rawhide. 

Rawhide belts with tanned-leather faces have the tanned leather 
pasted on the rawhide with sizing or glue. Under heavy service 
this facing is apt to come loose after a considerable time. 

Chrome-tanned belts are especially adapted to damp places, 
as in dye-houses, etc. They are extremely elastic, and elongate 
greatly during life. They have a high coefficient of friction. 
The color is green. 

48. Cotton belte are generally made by folding together several 
thicknesses of cotton canvas or ducking and fastening them by 
stitching or otherwise. They are sometimes woven solid so that no 
folding is necessary. Their strength is probably greater than that 
of any other form of belts commonly used. The coefficient of 
friction is rather low when the belt is used dry without any filling, 
sizing, or dressing. When properly sized or dressed, however, the 
coefficient of friction is equal to that of good leather belts. A 
weather-proof brand which is placed on the market seems to give 
excellent service for outdoor work of the most tr3dng kind. In it 
the interstices seem to be completely filled with the sizing, which, 
on the outside at least, is water-proof. It seems to be as durable as 
rubber belting, and has the advantage that there is no thin layer of 
rubber to rub or roll oflf, as is frequently the case with the latter 
when excessive slipping occurs. Some kinds of cotton belts have 
excessive elasticity. 



180 FORM, STRENGTH, AND PROPORTIONS OP PARTS. 



The weight of cotton belts depends largely upon the kind and 
amount of sizing that is used. Belts showing the following weights 
have been found in sendee: 0.026, 0.033, 0.037, 0.044, and 0.050 
of a pound per cubic inch. The latter is the weight of the weather- 
proof belt mentioned above. 

Cotton-leather belting is made by stitching a piece of thin 
leather to a cotton belt so as to make a leather facing on one side, 
which is used next the pulley. An unsized belt can be given a 
high coefficient of friction in this way. The facing is apt to tear off, 
especially in service where the belt must be shifted from step to 
step of a cone pulley. 

49. Bnbber belting is composed of a cotton web with a compo- 
sition of rubber filling all its interstices and completely covering it 
generally. When of good material it is not injured by moisture, and 




Fig. 78. 



Fio. 74. 



is therefore excellent for damp places and out-of-door service. The 
coefficient of friction of good rubber belting is high. If the rubber 
compound is poor, the coefficient of friction may be low and the 
compound will crack. When overloaded and caused to slip on the 
pulleys, the rubber is in danger of peeling loose and rolling up so 
as to tear off considerable areas on a belt that has a complete cover- 
ing of rubber over the cotton, thus destroying the belt. 

In some kinds of rubber belting the cotton web is not com- 
pletely hidden by the rubber, but stands out distinctly on the 
surface. 

The weight of rubber belting is abont 0.045 pounds per cubic 
inch. The joints of rubber belting can be cemented by coating 
the surfaces with uncured rubber in solution and vulcanizing the 
splice with steam-heated clamps placed over it. 

50. Leather-link belts. — Figs. 73 and 74. These are made of 
small pieces of leather, generally from one to two inches long and 



BELTS AND ROPES FOR POWER TRANSMISSION. 181 

^YO-eighths to seyen-eighths of an inch broad, perforated at each 
■end at right angles to the natural surfaces of the leather. These are 
fastened together by pins through the holes so as to form a chain or 
belt of any desired width and length. The pins are generally of iron 
or steel, and of such a size as will fit the holes in the links. The edges 
of the links are exposed and form the broken surface which comes in 
contact with the pulley. In the better makes of this kind of belts 
each pin extends only half-way across its width, and the two half- 
belts thus formed are fastened side by side, to form the complete 
width of belt. By this means the belt is allowed to take the form 
of a ^^ crowned " pulley whose face is made of two cone frusta 
placed base to base. A round piece of leather is sometimes used for 
uniting the links, instead of the steel pins. A belt thus made will 
adjust itself to a pulley whose crowning consists of a smooth curve, 
such as an arc of a circle. 

The coefficient of friction of link belting seems to be about the 
same as that of good leather belts, possibly somewhat lower. The 
weight per cubic inch generally runs, for metal pins, from 0.035 to 
0.050 pounds. On account of its great thickness it is much 
heavier per unit area of working surface than solid leather belt- 
ing of the same capacity for transmission. 

Leather-link belting is especially applicable to connecting a large 
imd small pulley that are placed near together in the same horizon- 
tal plane, their axes being horizontal. This is because, on account 
of its weight, it can be very slack, and, the slack side being above, 
it sags down so as to make a large arc of contact on the pulleys, 
thus increasing the turning force without making the belt exces- 
sively tight, as would be necessary with any of the ordinary solid 
belts. 

When a link belt breaks, it is generally without warning, and, 
on account of its weight, it is capable of doing serious damage, 
when running at high speed, by striking whatever may be in its 
path. 

51. EfTeot of relative positions of pulleys. — ^When the axes of 
the driving and driven pulleys are in the same horizontal plane, and 
the loose side of the belt is uppermost, the arc of contact on each 
pulley is increased, both by the sagging of the loose side as the load 
is applied, and by the tightening of the lower side to a more nearly 



182 FORM, STRENGTH, AND PROPORTIONS OP PARTS. 

straight line on account of the increased tension in it. On account 
of this increase of the arcs of contact, the tension is smaller on the 
tight side of the belt than it would be if the rotation of the pulleys 
were reversed, thus bringing the slack side below and decreasing 
the arc of contact; hence it is always advisable, when possible, to 
run the pulleys so that the slack side of the belt will be uppermost. 

Again, suppose that a large pulley is driving a comparatiyely 
small one placed yertically below it; the arc of contact on the small 
pulley will be much less than on the large one, and, in addition to 
this, the tension in either stretch of the belt at its point of tangency 
with the small pulley will be less than at the similar point on the 
large one. The difference of tension at the top and bottom of a 
stretch of the belt is the same as the weight of a portion of the belt, 
equal in length to the vertical distance between the points of 
tangency. Now suppose the small pulley is placed above the large 
one; the greater tension in the belt, still remaining at the upper 
pulley, will be applied so as to counteract, in a measure, the effect 
of the reduced arc of contact, while before, with the small pulley 
below, it was cumulative with it. No further investigation is 
necessary to show that it is advisable to place a small pulley above 
a large one, especially when the belt has a considerable length. 

For inclined positions of a belt connecting pulleys running on 
horizontal axes, the two facts brought out above should be taken 
into consideration. They indicate that the slack belt should always 
be kept above, and the small pulley higher than the large one, when 
they are not at the same level, and when other conditions will ad- 
mit of such an arrangement. 

Whatever the relative positions of a pair of pulleys, the tension, 
in a clear stretch of belt, at the point of tangency with the upper 
pulley, is greater than that at the similar point on the lower one, 
by an amount the same as the weight of a length of the belt equal 
to the vertical distance between the points of tangency. 

52. A special system of flat-belt driving is illustrated in Fig. 
75. It is especially applicable to close grouping of machinery. 

The belt is tightened by the mechanism shown near the engine- 
cylinder. 

The driving capacity of this device, when used with ordinary 
double belts, and with a contact on the large pulley at least as great 



BELTS AND ROPES FOR POWER TRANSMISSION, 



183 



in linear measure as that on the small or driven pulley, can be 
safely taken as 1 11. P. per inch of belt width for a belt speed of 
750 feet per minute.* 




Pig. 75. 

53. Efficiency of flat belting. — The power losses in flat leather 
belts are chiefly due to journal friction and belt slip. There is also 
a small loss caused by bending and straightening the belt as it runs 
on and leaves the pulley, but it does not seem to be great enough 
to need consideration. The slip is sometimes considered as made 
up of two elements, the creeping of the belt over the pulley, due to 
its elongation as it passes from the slack to the tight side, and the 
actual slip, which allows all parts of the belt to move at the same 
rate, faster than the driven pulley face or slower than the driving 
pulley. The effects of both are the same upon the efficiency, hence 
it does not seem necessary to separate them when dealing with this 
property. 

If the journals are larger than necessary to withstand the pull 
of the belt, or the latter is working above or below its normal 
capacity, the efficiency is lowered. 

♦ Communication from A. L. Ide & Sons. 



184 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

The efficiency of flat leather belting is probably never greater 
than 97^ in the grades of machinery that are generally used in 
practice. This refers to well<made machines with joarnal-bearings. 
If bearings having a very low frictional resistance, such as ball- 
bearings, are used, the efficiency might be increased to 98^, or 
possibly more. Probably 95^ is a fair average for good practice 
when a belt is working at or near its most economical rate. 

Crossed belts require less tension to transmit a given amouut of 
power, on account of having greater arcs of contact on the pulleys, 
than open ones; hence, if there is no rubbing together of the two 
stretches between the pulleys, the efficiency should be higher. 
Bubbing is almost certain to occur, however, thus adding an 
additional power loss. The rubbing may be of small consequence 
if the pulleys are of the same size; but if their diameters are greatly 
different, and especially if they are near together, a very consider- 
able rubbing pressure is almost certain to be caused. The power 
loss under such a condition is proportionately large. It is seldom, 
possibly never, advisable to use a crossed belt when considered with 
regard to power transmission and belt economy. 

The efficiency of the other kinds of flat belting mentioned in 
the preceding paragraphs does not differ enough from that of 
leather to be practically appreciable. 

53.1. Example showing the effect of varying the speed of 
belt. — ^Two shafts, 30 feet apart between centre lines, in the 
same horizontal plane, are to have 25 horse-powers transmitted 
between them, both running at 200 revolutions per minute. Leather 
belting is to be used at a tension of 300 pounds per square inch 
on the tight side. 

The coefficient of belt friction may be taken as .3 and the 
weight of the leather as .035 pound per cubic inch. 

Since the belt wrap is 180°, the diagram, Fig. 72.4, may be used 
directly for finding the sectional area of the ]>elt. 

The calculations will be made for three pairs of pulleys of 3-, 
6-, and 9-foot diameters. (See Figs. 75.4, 75.5, and 75.6.) 

For the 3-foot pulley 

r = 200 X 3r -^ 60 = 3 1 .4 feet per second. 



BELTS AND ROPES FOR POWER TRANSMISSION. 



185 



432 LBS. TOTAL TENSION 
350 " EFFECT. " 



750 LBS. TOTAL TENSION 





Z' OIAM. 
180° ARC 
998 


718 " EFFECT. " 

\ 2.B OQ. IN. BELT SECTION 
< 8 300 LBS. PER 6Q. IN. 


\ 


200 R.P.M. 
25 H.P. 
P-438 LBS 


(i)».036 " •• CU. IN. 

J'- 

^12 LBS. TOTAL TENSION 



EFFECT, 



Pig. 75.1. 




6' DIAM. 
180" ARC 
200 R.P.M. 
26 H.P. 
P»219 LBS. 




1.44 SQ. IN. 
BELT SECTION 



213 LBS. TOTAL TENSION 
140 " EFFECT. " 

Pig. 76.3. 



391 LBS. TOTAL TENSIOW 




1.8+ SQ. IN. 
BELT SECTION 



9 DIAM. 
180** ARC 
200 R.P.M. 
26 H.P. 
P — 146 LBS. 

^832 ^ 




245 LBS. TOTAL TENSION 



Pig. 75.8. 

By the diagram, Fig. 72.1, a belt of 1 square inch cross-section 
running 31.4 feet per second will transmit 10 horse-powers. The 
sectional area of the belt is therefore 



A = 25 -^ 10 = 2.5 square inches. 



186 FORM, STRENGTH, AND PROPORTIONS OP PARTS. 

Since the pulleys are 30 feet apart, 

Length of belt=2x30+37r=69.42 feet; 

Volume of the belt =69.42X12x2.5=2083 cubic inches; 

Superficial area of belt 5/16 inch thick =46.3 square feet. 

For the 6-foot pulley: 

^;=200x6;r-^ 60=62.8 feet per second. 

By the diagram the power transmitted by one square inch of 
cross-section of belt at a speed of 62.8 feet per second is 17.3 horse- 
powers. The sectional area of the belt is therefore 

il= 25 4- 17.3= 1.44 square inches. 

Length of belt =2x30+6;r= 78.85 feet; 

Volume of belt = 78.85 X 12 X 1 .44 = 1360 cubic inches ; 

Superficial area of belt 5/16 inch thick =30.1 square feet 

For the 9-foot pulley: 

v=200x97rH- 60=94.2 feet per second. 

The power transmitted by one square inch of section at 94.2 
feet per second is 19.2 horse-powers. The sectional area of belt is 
therefore 

-4=254-19.2=1.3 square inches. 

Length of belt = 2 X30+9;r= 88.27 feet; 

Volume of belt =88.27X12X1.3 =1375 cubic inches; 

Superficial area of 5/16 inch belt =30.6 square feet. 

The 6-foot pulleys require the least volume of belt. The 9-foot 
ones need about the same amount; but 53% more is necessary for 
the 3-foot pulleys. The drive with 6-foot pulleys is the cheapest 
for the distance of 30 feet between pulley centres. This might 
not be true for shafts very near together, for then the cost of pulley 
may have greater importance than that of the belt. 

While there is a very considerable difference in the belt sec- 
tions for the 3-foot and 6-foot pulleys, there is, as has already 
been pointed out, but little further reduction by using a 9-foot 
pulley. The reason that the reduction of the 9-foot pulley is so 



BELTS AND ROPES FOR POWER TRANSMISSION. 187 

slight is because, at the comparatively high belt speed with this 
diameter, the centrifugal tension begins to have an appreciable 
value. If the size of the pulley were increased to 12 feet, a larger 
belt than for the 6-foot pulley would be required on account of 
the high centrifugal tension. For a 12-foot pulley the belt would 
have nearly 2 inches of cross-section when working at 300 pounds 
per square inch tension on the tight side. 

The centrifugal tension in a belt has no effect on the pulley. 
It causes no pressure against the pulley, and hence exerts no 
turning effort upon it. 

The tension in the belt which is effective in turning the pulley 
is what remains of the total tension after the centrifugal tension 
has been deducted. 

The required effective tension decreases, under the conditions 
for this example, in the same proportion as the diameter of pulley 
and belt speed increase. Therefore the pressures on the bearings 
due to the belt tensions also decrease in the same ratio as the 
pulley diameters and belt speeds increase. This will be shown in 
numerical values. 

For any belt: 
Centrifugal tension = .0001036i£?F' = .3727tm;' pounds per square in. 

For the 3-foot pulley: 
Unit centrifugal tension =. 3727 X. 035 X (31. 4) »= 12.8 pounds per 

square inch. 
Total centrifugal tension=2.5Xl2.8=32 pounds. 
Total tension, tight side,= 7'i= 300X2.5 =750 pounds. 
Effective tension, tight side, = 750 — 32 = 718 pounds. 

For the 6-foot pulley: 
Unit centrifugal tension =. 3727 X. 035 X (62.8)' = 51.1 pounds per 

square inch. 
Total centrifugal tension =1.44X51.1 = 73 pounds. 
Total tension, tight side, = ^1= 300x1.44=432 pounds. 
Effective tension, tight side, = 432 — 73 = 359 pounds. 

For the 9-foot pulley: 
Unit centrifugal tension =. 3727 X. 035 X (94.2)' =116 pounds per 
square inch. 



188 FORM, STRENGTH, AND PROPORTIONS OP PARTS. 

Total centrifugal tension= 1.3X116 = 152 pounds. 
Total tension, tight side, = 7^1= 300X1.3+ =391 pounds. 
Effective tension, tight side, = 391 — 152=239 pounds. 

The centrifugal tensions have the same ratio as the squares of 
the velocities. The latter have the ratios 1, 2, and 3. Therefore 
the centrifugal tensions per square inch have the ratios 1, 4, and 9. 

To find the pressure on the bearings it is necessary to know the 
effective tension on the slack side of the belt. This is obtained by 
deducting the turning force P from the effective tension on the 
tight side. 

For the 3-foot pulley: 

^ 550 H.P. 550X25 ._ , 

P= -3j^ =438 pounds. 

Effective tension, slack side, = 718—438=280 pounds; 
Bearing pressure =718 +280 =998 pounds. 

The corresponding values obtained in a similar manner for the 
6-foot and 9-foot pulleys are: 

For the 6-foot pulley, 

P =219 pounds; 

Effective tension, slack side, = 140 pounds; 

Bearing pressure =499 pounds; 

and for the 9-foot pulley, 

P=146 pounds; 

Effective tension, slack side, = 93 pounds; 
and 

Bearing pressure =332 pounds. 

It will be seen that, by using the 6-foot pulley, the friction 
loss at the journals and the belt section are both reduced, even for a 
shaft of fixed diameter, as a line shaft. For a head shaft whose 
diameter may depend largely upon the effective belt tensions, 
the shaft itself may be made smaller for the 6-foot pulley than for 
the 3-foot, which is an additional advantage for the larger pulley. 

53.2. Binder and gnide pnlleys, — ^Tlie use of idle pulleys for 
tightening an endless (cemented-joint) belt is very frequently 
desirable. They can be made adjustable to take up the elongation 



BELTS AND ROPES FOR POWER TRANSMISSION. 189 

of the belt, to relieve its stress when not in service, and to adjast 
its tension according to the work it must do, thus increasing its life, 
and, if properly applied, the efficiency of power transmission. 

To be efficient, a binder pulley should be large in diameter, 
placed on a slack side of the belt in a position to materially increase 
the angle of belt wrap, and supported on journals only as large 
as are necessary for the pressures on them. 

When only one binder is used on a belt connecting a large pulley 
with a small one on parallel shafts comparatively near together, 
it should be placed near the small pulley so as to make its angle 
of belt wrap as great as on the large one. 

The distance between a binder and working pulley should not 
be less than 18 inches for ordinary double belts. If the pulleys 
are closer together than this, there is apt to be noisy running on 
account of the stiffness of the belt joints. A stiff joint, passing 
from one pulley to another very near it, strikes the latter a hanmier- 
like blow accompanied by noise, especially on iron pulleys with 
uncovered faces. 

A binder pulley may sometimes be placed so as to press against 
the slack side of a belt at its point of tangency on the driving or 
driven pulley. This is not advisable except when it cannot be 
otherwise located. 

In belt drives for shafts that are not parallel, one of the guide 
pulleys can be made adjustable as a belt tightener. 

The guide pulleys may be placed so as to give large angles of 
belt wrap. This is always desirable. As binders only, the puUeys 
should be large in diameter and supported on bearings not unduly 
large. 

Guide pulleys should be located so that the direction of motion 
can be reversed without displacing the belt. 

The stretching of the edges of the belt is reduced to a minimum 
by placing the guide pulleys in this manner, and, by giving suf- 
ficient crawn to the pulley faces, the edge stretch is made practically 
zero. 

There are many excellent belt drives operating in the New 
England States with binder and binder-guide pulleys for trans- 
mitting amounts of power ranging from a few horse-powers to 
two hundred or more. Many of them are for shafts not parallel, 



190 FORM; STRENGTH, AND PROPORTIONS OP PARTS. 

as for connecting a vertical-shaft water-wheel to a horizontal line 
shaft in a mill or factory. The belts on these drives run many 
years without repair or resplicing. 

On the other hand, there are in operation throughout the country 
a far larger number of poor examples of the use of binder and guide 
pulleys. Small pulleys improperly placed and supported by ex- 
cessively large bearings are numerous. They are inefficient, short- 
lived, and usually noisy. The belt is often injured by a short 
bend over a small pulley. 

Example. — ^The requirements of the example, § 53.1, may be 
met by belt drives with binder pulleys. The results for 3-, 6-, and 




608 LBS. TOTAL; 678 EFFECTIVE 

-J 



165 LBS. TOTAL 
140 EFFECTIVE 

2.01 Sa LN. BELT SECTION 

57 8_ 

Pig. 75.5. Fio. 75.6. 




FiQ. 76.4. 



9-foot pulleys with binders placed so as to give 270® of belt wrap 
are given below. The arrangement of the pulleys at one end of 
each drive are shown in Figs. 75.4, 75.5, and 75.6. 

The velocities, centrifugal tensions per square inch, and the 
required turning force P are the same as before in each case. Other 
values are: 

For the 3-foot pulleys: ^1 = 603; ^2=165; effective total ten- 
sions =578 and 140; A =2.01; length of belt -73.4 feet; volume of 
belt =1770 cubic inches; superficial area of belt 5/16 inch thick = 
39.3 square feet. 

For the 6-foot pulleys: 7',=350; 7^3=131; effective total ten- 
sions=289. and 70; A = 1.17; length of belt=87 feet; volume of 
belt =1240 cubic inches; superficial area of belt 5/16 inch thick = 
27,1 square feet. 

For the 9-foot pulleys: ^1=315; 72=176; eflfective total vol- 
ume of belt =1275 cubic inches; superficial area of belt 5/16 inch 
thick =28.3 square feet. 



BELTS AND ROPES FOR POWER TRANSMISSION. , 191 

The amount of belt required without binders is greater than 
the above by about 19% for the 3-foot, 9% for the 6-foot, and 7% 
for the 9-foot pullejra. 

There are two additional pulleys with their supports to counter- 
act the saving of belt by the use of binders. The pulley faces are 
not so wide, however, and the supports may be lighter. 

The bearing pressure on one of the 3-foot pulle}^ due to the 
belt tensions is the resultant of the effective tensions of 578 and 
140 pounds acting at right angles to each other. This resultant, 
as shown in Fig. 75.5, is 595 pounds. The corresponding bearing 
pressure on the 1-foot-diameter binder is the resultant of the two 
practically equal tensions in the stretches of belt leading from it 
at right angles. These tensions are each 140 pounds. Their 
resultant. Fig. 75.6, is 198 pounds. 

The bearing pressures for the 6- and 9-foot pulleys are obtained 
in the same way, as indicated in Figs. 75.7, 75.8, 75.9, 75.10, 
75.11, and 75.12. 

The simi of the bearing pressures on a working pulley and its 
binder is smaller than the bearing pressure for the same size of 
working pulley without a binder, and the latter has a smaller 
journal than the working pulley. The frictional loss is therefore' 
less when the binders are used. 

53.3 Chain belts, running upon sprocket-wheels which re- 
semble in a general way the teeth of spur-gears both as to form, 
thickness, and distance between them, are coming into extensive 
use for the transmission of power. These chains have projections 
upon one side to fit into the spaces between the teeth of the 
sprocket-wheel. 

In the Reynolds chain the construction is such that the chain 
has practically no rubbing action over the teeth of the sprocket- 
wheel while coming into contact with and disengaging from 
them. It is quiet-running. The limit of speed is that at which 
the lubricant is completely thrown off by centrifugal action. This 
limit is found to be about 1300 feet per minute. As the pitch of 
the chain increases with wear, it adjusts itself to a larger radius 
on the sprocket, thus maintaining a correct fit. • 

In the Morse chain the necessity of lubrication is obviated by 
a blunt knife-edge bearing surface between the pins joining the 



192 FORM, STRENGTH, AND PR0POR;riONS OF PARTS. 



850 LBS. TOTAL; 289 EFFECTIVE 




Fig. 76.9. 



Fia 76.7. 




Fig. 7S.ia 



Fig.75.1Sl 



BELTS AND ROPES FOR POWER TRANSMISSION. 193 

links. This chain mns succeesfnlly at higher speeds than one 
requiring lubrication. 

BOPES. 

54. BopeSy running on grooved pulleys or sheaves, are largely 
used for the transmission of power. The name ^^sheaye" is applied 
to a pulley with but one groove; when there are two or more 
grooves the name "grooved pulley" is generally used. 

While ropes are very frequently used for transmitting power 
between shafts that are parallel and at snch a distance apart as is 
common for leather belts, their especial application seems to be that 
of connecting shafts that are not parallel, or are at a great distance 
apart. On account of their approximately circular form of section, 
they bend with equal ease in all directions, hence " quarter turns '' 
and bends in various directions are not nearly so severe upon them 
as upon flat belts. Hemp, cotton, and wire ropes are the varieties 
almost exclusively used for power transmission, although leather, 
rawhide, and other materials are used to some extent, generally for 
light work. Manila hemp is generally called " manila," and the 
special make, which is largely used for power transmission, is called 
"Stevedore." 

Fihrmia Ropes. 

55. Two systems of driving with fibrous ropes are in general 
use. They are commonly known as the Gontinuons or American 
system, and the Multiple or English system. 

In the multiple system there are as many separate, endless ropes 
as there are grooves in each of the system of pulleys over which all 
the ropes run. Each rope always runs in the same groove of each 
pulley. The ropes are, of course, parallel to each other, practically 
speaking. The maltiple system is generally used for transmitting 
very large amounts of power. 

In the continuous system, shown in Fig. 76, there is a .single 
endless rope, wound continuously over the pulleys. The winding 
is such that a point in the rope, starting at the groove at one end 
of any pulley, passes from this groove to the first groove of each of 
the other pulleys, then to the second groove of the first palley and 



194 FORM, BTBENGTH, AND PROPORTIONS OP PARTS. 

of the other palleys in snccession, thence to the third groove, and 
80 on, nntil it has passed through all the grooves; it then passes 
oyer a single-groove gnide-sheave, which leads it back to the first 
groove of the first poUej. The gnide-sheave is often made to serve 
the doable purpose of both guide and tightener. When used as 
a tightener it is supported on a carriage which is free to travel back 
and forth on a track or guide; weights are attached to the carriage 
to keep the tension of the rope uniform. The tension carriage must 
be free to have a considerable range of travel, in order that it may 
adjust itself for variation in the length of the rope, which may be 
caused by change of load, condition of the atmosphere, the gradual 
lengthening of the rope with service, and other influences. The 
tension-carriage should always be placed on the slack side of the 
rope. In Fig. 76 it can be seen in the upper middle portion of the 
figure. The tension-weight is shown alongside the column. 

There are numerous modifications of the arrangement of the 
tension-carriage and guide-pulleys, to conform with local conditions, 
one or more idler sheaves frequently being added to gaide the rope, 
but the principle is the same in all. 

The. method of transmitting power by ropes to the different 
floors of a building is shown in Pig. 77. 

A varying load on a continuous-system rope-drive causes un- 
equal tension in the stretches of rope between the pulleys. This 
can be seen by assuming that a drive which is running for some 
time without any load other than the journal-friction of the 
machinery immediately appertaining to the drive, has a load equal 
to the full capacity of the drive suddenly thrown upon it. While 
running light the tension in each strefcch of rope, on both sides of 
the pulleys, is practically equal to that caused by the tension- 
carriage, i.e., equal to one half the effective weight on the tension- 
carriage. As the full load is suddenly applied, all the stretches of 
rope on the slack side are slackened somewhat more by the stretch 
of those on the tight side; the tension-carriage, however, maintains 
the same tension in the stretch between it and the driven pulley. 
Consequently when a length of rope approximately equal to one 
half of what would be required for connecting both pulleys with a 
single band has passed from the tension-carriage to the pulley, the 
tension on the tight side in the first stretch after leaving the 



es 

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ay 



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lie 
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lal 
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iD- 

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196 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

the same pulley. As the rope continaes to ran over the pulley, the 
first stretch on the slack side becomes tighter than the others on the 
slack side, and the second stretch on the tight side increases its 
tension to a higher value than that of the stretches coming later on 
the tight side. After a time the tension becomes uniformly dis- 
tributed among the stretches on the tight side. 

On account of the inequality of tension in the stretches of the 
rope on the tight side, when working under varying loads, which 
may be so marked as to very appreciably shorten its life when there 
are many turns of a continuous rope around a pair of pulleys, it is 
advisable to limit the turns to a comparatively small number; the 
greater the variation of load, the smaller should be the number of 
turns of a continuous rope. 

When, in order to fulfil the requirements of power transmission, 
a large number of turns of rope are required, and the continuous 
system is adopted, it is often advisable to use two or more contin- 
uous drives, operating side by side on the same pulleys if desired; 
the main pulleys in such a case would be the same as for a single 
continuous drive; an additional tension-carriage and guide sheave 
or sheaves, if the latter are used, must be supplied for each con- 
tinuous rope. The additional cost of the latter may be more than 
counterbalanced by the increased life of the rope, however. 

56. The equations for ropes transmitting power are similar to 
those for flat belts. On account of their approximately circular 
sectional form, it is more convenient to take a rope of unit diameter 
as the basis of calculations, instead of a unit area, as is done for flat 
belts. The coefficient of groove friction 0, which is generally used, 
is that of the rope in the groove, and is, of course, greater than 
that of the same rope on a flat surface of the same material as the 
pulley, on account of the wedge-like action of the rope in the 
groove; the sharper the angle between the two sides of the groove, 
the higher this coefficient of friction. 

The following notation is applicable to ropes only; the symbols 
nsed in the equations, but not given in this notation, are the same 
as for flat belts: 

Tr= weight of a rope 1 inch in diameter and 1 foot long, pounds; 
w = weight of a rope 1 inch in diameter and 1 inch long, pounds; 



BELTS AND ROPES FOB POWEB TBANSHI88ION. 197 

r = working strength of a rope 1 inch in diameter, pounds; 
/? = angle between sides of groove, degrees; 

^ = /i CSC -^ = coefficient of groove friction. 

The equations for ropes, deduced in the same manner as those 
for belts, are 

j?= (r - .0001036tt; F') (^^^^ ^ ) 

= (r-. .00010361. F-)(-^p5^3^), (47) 

and 

^ = P^^rzn + -OOOlOSett; F« = P ^J^^L^, _ ^ +.0001036te^ r\ (48) 

Bzample. — ^It is required to design a rope-drive to transmit 200 
H.P. when running at 4500 feet per minuto, and working at a 
tension on tho tight side equivalent to 200 pounds for a rope of 
1 inch di&meter; the arc of contact on the working pulley having 
the smallest portion of its circumference embraced by the rope 
being 160°. Manila hemp rope to be used. 

This problem is essentially the same as the example given under 
flat belting; the only difference being that ropes are used instead 
of a belt. As in that problem, the turning force 

P = 1467 pounds. 

By substituting in equation (47), taking = 0.3 and ia = .024, 

[10.00?68X.8X180_ 1-1 
— — io:s^4 ^J 

= (200 - 50) ';^,, = 85 pounds. 

Assuming that rope 1^ inches in diameter will be used, and 
that it will work under a tension equivalent to 200 pounds for a 
rope 1 inch in diameter, the working strengths being taken as 
proportional to the squares of the diameters, gives, for the turning 
force of the 1 J-inch rope. 



198 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

26 

— X 85 = 133 pounds. 

The number of IJ-inch ropes required is, therefore, 
1467 -^ 133 = 11 (about). 

The total tension T^ bears the same relation to the total turning 
force P as r does to p, therefore 

T; = ~P = ^1467 = 3460 pounds. 
p 85 

The tension that must be put on a rope when at rest may be 
determined in the same manner as for a flat belt. It is not known, 
however, that any experiments have been made to determine 
whether there is the same increase of the sum of the tensions in 
the two sides of the rope when working, over that when at rest, as 
there is in flat belts, but it seems reasonable to assume that such is 
the condition. Upon the assumption that the increase is one third 
of the sum of the tensions of rest, the tension in each stretch of 
the rope when at rest is, as for the flat belt, taking into account 
the fact that there are 11 stretches of rope between the pulleys on 
each side, 

Tr 3 ^ 27; - P 3 (2 X 3460) - 1467 ,^^ , 

n^sX^n-^S*^ n = 185 pounds. 

If a tension-carriage is used, as in the continuous system, the 
effective weight for producing the total tension T^ on the slack 
side of the rope must be 27^-^- 11, since there are 11 ropes in this 
particular drive. Therefore the effective weight of the tension- 
carriage must be 

27; _ 2(7; - P) _ 2(3460 - 1467) _ _. ^^,^ 
— = jj '- = jj^ = 360 pounds. 

57. The grooves for non-metallic ropes found in practice are of 
numerous forms. The most common angle between the sides of 
the groove is about 45% however. It varies from 30° to 60° in 
extreme cases. It is clear that the smaller this angle, the tighter 
the rope will wedge into it, and the less liable will it be to slip. 



BELTS AND ROPES FOR POWER TRANSMISSION. 199 

More power will be required to pall it oat of a sharp-angled groove, 
however, thus causing more power loss; it is also possible that there 
will be more wear of the rope. In some forms the sides are 
straight, as in Fig. 78, while in others they are curved, as in Fig. 




Fig. 78. 



79,* where is the centre of curvature for the left side of the 
groove; the right side has a similar curvature. 




In a groove with curved sides, the angle between the sides 
where the rope comes in contact with them, when resting lightly in 
the groove, is smaller than would be used for straight sides; at the 



200 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

bottom it is larger. This arrangement secares a good grip of the 
rope when it is at or near the top of the working part of the groove, 
but allows it to slip more readily when, on account of wear, it is 
brought nearer to the bottom. Such a provision for more ease of 
slipping when near the bottom of the groove is of considerable im- 
portance in the working of a rope-drive, especially of the multiple 
system. 

In order to show the action of a multiple-rope drive, let it be 
assumed that one has been in use until some of the ropes have 
become so worn that they must be replaced by new ones, and that 
the pulleys are of greatly different diameters, as is frequently the 
case when a line-shaft or some high-speed machine is driven directly 
from a pulley on the main shaft of an engine. The old ropes, 
being smaller than the new ones of the same nominal diameter, on 




Fig. 80. 

account of wear and stretching, will lie nearer the bottoms of the 
grooves than the new ones, and their effective radii will accord- 
ingly be less on each pulley, the whole system appearing as in 
Fig. 80. For convenience it may be assumed that the grooves are 
of the same form on both pulleys, this being very frequently the 
condition in practice. The reduction of the effective diameter will 
therefore be the same on both pulleys as the rope wears. By 
putting 

D = effective diameter of the large pulley with new rope, 

a ■= redaction of effective diameter of each pulley, 

d = effective diameter of the small pulley with new rope, 

the conditions of working can readily be expressed mathematically; 
it will be assumed that D = dd. 



BELTS AND BOPES FOR POWEE TBAN8HIS8ION. 201 

If there were no slipping of the new ropes in the grooves, the 
small pulley would make three revolutions for one revolution of 
the engine; and if there were no slipping of the old ropes, the por- 
tion of length of one of them which would pass over the large 
pulley during one of its revolutions is 7r{D — a). This length of 
rope, by passing over the small pulley at an effective diameter of 
d^-Uy would turn it through 

-f3 ^ = -^TT / = 3 + 3 revolutions, 

n(d — a) n{d — a) ' d — a 

This shows clearly that there is a tendency for the smaller, or old, 
ropes to drive the small pulley faster than the new ones. As a 
result, the entire load will be carried by the old ropes unless they 
slip in their grooves. A form of groove which will let a rope slip 
more easily as it becomes smaller with wear is better than one 
which does not allow such slippage. 

If, instead of a large pulley driving a small one, the small one 
drives the large, there will be a tendency for the load to be thrown 
upon the larger or new ropes. The increased tendency to draw 
these ropes down into the grooves acts as a corrective, however, and 
thus prevents the excessive loading of the larger ropes. 

In order to show the action of the ropes in a drive having curved 
grooves similar to that of Fig. 79, let it be assumed that an engine 
is driving a line-shaft through a rope-drive, the pulley on the engine 
being larger in diameter than the one on the line-shaft. It may be 
further assumed that the drive has been in service for some time, and 
that half the ropes have become so worn that they must be replaced 
by new ones. The old ropes left in place will, on account of wear 
and stretch, be smaller than the new ones of nominally the same 
diameter, and will therefore lie nearer the bottom of the groove. 
The effective diameter at which the new ropes work on each pulley 
will therefore be greater than that of the old ones. The difference 
of the effective pulley diameters, for the new and old ropes, is less 
than for straight-side grooves, provided the groove angle where the 
new rope comes in contact with the groove is the same as used for 
straight sides; at the bottom the angle is larger. The grip on the 
rope is therefore not as great when it approaches the bottom of the 
groove as it would be if the sides of the latter were straight. The 



FOBM, STRENGTH, AND PE0P0RTI0N8 OF PARTS. 

curved sides of the grooves thus afFord a means of preventing the 
excessive tension which would be thrown on the rope if it were run- 
ning in y grooves. The life of the ropes is therefore lengthened. 

A groove whose angle grows more flat at the bottom strains 
the rope less when an excessive load is thrown on the system than 
does one with straight sides making an angle which is the mean of 
that of a curved-side groove. 

Another method of preventing unequal loading of the ropes in 
a multiple drive, when pulleys of different diameters are used, is to 
make the groove of the larger one with a smaller angle than those 
of the smaller. The decrease of the effective diameter of the larger 
pulley is thus made more rapid than for the smaller, as the rope 
becomes smaller. A case is cited where the engine driving pulley 
was about three times the diameter of the driven pulley. By 
making the angle of the groove 30^ on the larger pulley, and 45^ 
on the smaller, unequal pulling was obviated.* 

There seems to be a difference of opinion among engineers as to 
what form of groove should be used for idler- or guide-pulleys. 
On the one hand it is maintained that a groove of circular section 
at the bottom, and large euough to let the rope run in it without 
wedging against the sides, is the best on account of causing no 
frictional loss as the rope enters and leaves the groove; the contrary 
argument is that the rope slips in the round-bottom groove, and 
thus causes as much loss as the V groove, together with more rapid 
wear of the rope. It woald seem that the round-bottom groove 
would be at its best when the rope is in contact with a considerable 
portion of the circumference of the pulley, and the V groove when 
the arc of contact is small, the round-bottom groove being the more 
apt to permit slipping with a small arc of contact; but whether the 
round bottom is better, even for large arcs of contact, does not seem 
to be positively settled. 

Cast iron and hard wood, such as maple, are materials which are 
almost universally used for the rope to run on; the grooves are 
generally turned in the rim of the pulley, no lining being used. 
Cast-iron and wooden grooves therefore correspond to pnl ley-rims 
of these materials. 

* Kent's " Mechanical Engineers' Pocket-book." 



BELTS AND K0PE8 FOR POWER TRANSMISSION. 203 

The sides of the grooVe should be perfectly smooth in order to 
prevent rapid external wear of the rope. 

68. Coefficient of friction of non-metallic ropes. — There seem 
to be no experiments showing the coefficient of friction of ropes 
running in grooved pulleys. The experiments made on ropes 
running over flat pulleys are scarcely applicable to modern practice 
in rope-driving, for there is little probability that the ropes used in 
the two cases were in even approximately the same condition with 
regard to lubrication. The coefficients given below are not experi- 
mentally determined, but based upon modern practice. It is 
believed, however, that they are sufficiently correct for all practical 
applications. 

The value of pi for well-lubricated manila ropes running in 
polished grooves generally lies between 0.1 and 0.3; 0.12 to 0.15 
are safe values to use in designing. These latter two values give 
for 45** groove angles the corresponding values of the coefficient of 
groove friction 

= 0.12 CSC -^ = 0.32; and = 0.15 esc =^ = 0.40, 
and for 30° grooves, 

= 0.12 CSC -^ = 0.46; and = 0.15 esc ^ = 0.58. 

When the rope is dry, or lubricated with a somewhat sticky sub- 
stance, the coefficient of friction is higher. 

Other qualities of hemp rope have practically the same coefficient 
of friction as manila, when lubricated in the same manner. 

Cotton rope has a higher coefficient of friction than hemp. 
This may be due to the fact that it does not require so much lubri- 
cation, on account of its containing a natural lubricant, as well as 
its softer texture. It is doubtless safe to take /i as high as 0.2 in 
designing; this gives for 45° grooves = 0.62, and for 30° angles 
= 0.77. 

Rawhide rope does not seem to have been used extensively 
enough in large sizes to give a very definite knowledge of its fric- 
tional qualities. It is much used in small sizes cut flat from the 
hide and twisted into a round rope. Its length can be varied and 



204 PORM, STRENGTH, AND PROPORTIONS OF PARTS. 

tension adjusted by slightly twisting or untwisting it. Several 
small rawhide ropes used in parallel are more satisfactory for oily 
places, as automatic screw machines, than a single flat belt. It is 
doubtless safe to take M as high* as for cotton ropes when the raw- 
hide is kept in good condition and not allowed to get moist; if 
there is any liability to moisture, /i may be taken the same as for 
well-lubricated hemp ropes. 

69. Working strength of non-metallio ropes. — When manila 
ropes are used for power transmission they have been found to be 
durable when working under a stress of 200d' pounds on the taut 
side, where d = diameter of rope in inches. Cotton ropes have 
given good service under the same tension. Rawhide ropes can be 
successfully operated at a tension at least one quarter higher than 
that of hemp and cotton, i.e., 250d* pounds or more. 

The above values are for economical working and reasonable 
durability. As with flat belts, ropes can be operated at much 
higher stresses, for a short time, than those just given; double the 
above values, or even more, may be used for a short time. 

60. The velocity of ropes for power transmission is limited by 
the action of centrifugal force in the same manner as for flat belts. 
Non-metallic ropes run much more steadily at high speeds, how- 
ever, the flapping and chasing common to flat belts being practically 
absent. On account of this latter advantageous quality they are 
commonly run at higher speeds than are found satisfactory for belts; 
a speed of 5000 feet per minute, or more, is frequently used. The 
tension due to centrifugal force in a rope weighing 0.32 of a pound per 
foot, and running at 8493 feet per minute, is equal to 200 pounds; 
this is practically the weight and working strength of a 1-inch 
manila rope. Boughly speaking, therefore, no power can be trans- 
mitted by a well-lubricated hemp rope working at 200d* pounds 
tension when the velocity reaches 8500 feet per minute. The 
speed at which maximum power can be transmitted with such a 
rope, working under 200<{' pounds tension, is about 5500 feet per 
minute for an arc of contact (^ = 160° to 180"*. Cotton rope, 
being somewhat lighter, has its speeds of maximum power trans- 
mission and of theoretically no power transmission both somewhat 
higher. 



BELTS AND ROPES FOR POWER TRANSMISSION. 205 

The most economical speed for manila rope does not vary much 
from 4500 feet per minute, although there is no great change in 
economy for variations of 1000 feet per minute on either side of 
this value. The speed that is most economical, for the rope alone, 
is that which gives the minimum cost of rope per horse-power 
transmitted, which cost includes first cost and maintenance of the 
rope only. 

61. Wear and lubrication of non-metallio ropes. — The wear of 
vegetable-fibre rope belts is of two kinds, external and internal. 
External wear is caused by the slipping of the rope in the groove, 
and the rubbing against the side of the groove as it winds on and 
is unwound from the pulley. The outer strands are gradually worn 
away by external wear, and the rope weakened accordingly. In 
order to prevent any considerable weakening of the rope by external 
wear, a covering, of some material weaker and cheaper than that of 
the body of the rope, is sometimes placed over it, thus forming a 
rope which maintains a nearly uniform strength until the covering 
is worn through, provided the rope is well lubricated internally. 

Internal wear requires serious consideration, for, unless some 
suitable lubricant is used to reduce the friction between the 
strands and fibres, the life of the rope is apt to be short. When 
an unlubricated or improperly lubricated rope has been in service 
for some time, a fine dust is formed in it by the particles worn off 
by internal friction. This dust can be easily seen by opening the 
strands. A reverse bend in a rope is a cause of greatly increased 
internal wear. For this reason all sheaves and pulleys should be 
placed, as far as practicable, so that the belt will bend in the same 
direction in passing over them. 

A large number of the lubricants used for rope are made of 
graphite mixed with some substance such as molasses, beeswax, or 
tallow. The best way of applying a lubricant is to saturate the 
strands as they are being laid up to form the rope. Hemp rope for 
power transmission is generally so treated during its manufacture. 
A lubricant that is applied externally must be of such a nature that 
it will penetrate the rope and act upon all the strands to reduce 
their frictional resistance to rubbing against each other. 

Cotton fibres are covered with an oleaginous wax in their natural 



206 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

condition; this coating serves as a lubricant for the rope, and 
eliminates the necessity of artificial lubrication. 

When a vegetable-fibre rope is exposed to the weather while in 
service, a dressing which forms a water-proof coating should be 
Applied to it, care being taken first to have the interior lubricated. 
A mixture of beeswax and graphite is frequently used for water- 
proofing. Substances of an adhesive nature, such as tar, while 
answering excellently as a preservative for a rope that does not run 
over pulleys or sheaves, is not suitable for those used for power 
transmission. Such a substance cements the fibres of the rope 
together so that they cannot slip over each other freely when the 
rope is bent around the pulley, thus causing them to tear or break 
each other. It may also cause the rope to adhere to the pulleys so 
that the fibres will be picked off by the pulley, which, of course, 
causes rapid deterioration. 

Houghness of the grooves is certain to cause rapid wear, and, as 
has already been stated in § 57, care should be taken to have them 
perfectly smooth and without fiaws. 

The diameter of the pulleys over which a rope runs has much 
to do with the wear upon it; the smaller the pulley, the more rapid 
the wear on account of the sharper bend. A diameter of the pulley 
equal to about 40 times the diameter of the rope represents a fair 
average of the size found in practice. A somewhat smaller size 
than is represented by this may be used for the smaller diameters 
of rope up to | inch ; but for ropes as large as 2 inches diameter 
the pulley may be a few inches larger than this indicates. At high 
velocities it is belieyed to be advantageous to use somewhat larger 
pulleys than are suitable for low speeds. 

62. Weight of hemp and cotton ropes for power transmission. — ' 
The weight of new hemp transmission rope as made by two manu- 
facturers is given in Table XIa. 



BELTS AND ROPES FOR POWER TRANSMISSION. 

Table XIa . 
weight op hemp transmission rope. 



207 





Founds per 100 Feet. 


Diameter 
of Rope, 
Inches. 


Pounds per 100 Feet. 


Diameter 


Plymouth 

CJordage Co., 

a- and 4- 

Btrand. 


C. W. 

Hunt Co., 

"Stevedore" 

4-8trand. 


Plymouth 

Cordage Co., 

3- and 4- 

strand. 


C. W. 

Hunt Co., 

"Stevedore" 

4-8trand. 


fe 


9i 
12J 
16 
20 
30 
34 
42 
46 


12 


If 

2 
2i 


50 
65 
70 


54 
69 


15 
21 
31 
34 

48 


79 
94 


hi 


112 
130 


114 
135 
175 







The average weight of cotton transmission rope that has been in 
service for some time is about 0.26^* pounds per foot.* 

The splice for power transmission must be much longer in a 
rope than for most other purposes. Table XIb gives lengths for 
good practice. 

Table 'XIb. 
length of splice in hemp transmission rope. 



Diam. of rope, inches.. . 


H 


^ 


H % 


1 


IK 


IK 


IH 


IH 


IH 


\H 


2 


2H 


Length of splice, feet. . 


10 


10 


10 


10 


12 


12 


12 


12 


14 


16 


18 


20 


22 



63. The diameter of ropes used for power transmission depends 
very largely upon the size of the sheave that can be conveniently 
used. A rope 2 inches in diameter is the largest that is found 
in practice to any considerable extent. From 1^ to 2 inches in 
diameter are the limits of the sizes generally used where there is a 
large amount of power to be transmitted , and the pulleys can be 
made large without inconvenience, as in rope-drives connecting the 
main pulley of a large engine with a line-shaft, electric generator, 
or cable-drums of a cable railway. Smaller sizes are used for 
general transmission in buildings using power for manufacturing 
purposes. 

64. The effect of the relative position of pulleys upon ropes 
used for power transmisjuon is of the same nature as for flat belts. 
In practice, however, it is demonstrated that hemp and cotton ropes 

* Rope Transmission of Power, by John J. Flather. 



208 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

work more satisfacfcorily than flat leather belts when one palley id 
at a considerable distance above the other. This is doabtless largely 
dae to the fact that a tension-carriage is used to take up the slack 
in the rope, while the belt, ordinarily operated without any device 
to make adjustment for stretch, soon becomes loose on the lower 
pulley. The weight of a leather belt for such a drive is generally 
greater than that of a hemp or cotton rope, which is disadvan- 
tageous for the belt. 

Hemp and cotton ropes are often used successf ally, in factories, 
for directly connecting sheaves or pulleys three or four stories apart, 
one being nearly or quite vertically above the other. 

65. The efficiency of rope belting seems to be about the same 
as that of flat belts when the distance of transmission is short, so 
that only a single stretch is necessary between the driving and 
the driven pulleys. For long distances of transmission the rope is 
more efficient than flat leather belting; this is partly owing to the 
fact that much longer stretches of rope can be used than of flat 
belting. 

WIRE ROPES. 

66. Wire ropes made of iron or steel, and generally with a 
hemp core, have been extensively used for transmitting power 
through long distances. They are being replaced by electrical trans- 
mission machinery for the greater distances, however, and by non- 
metallic ropes for the lesser. The large diameter of the sheaves, 
generally from 100 to 140 times that of the rope, is an objection- 
able featare on account of the space required for them, their 
great weight and comparatively high cost. The rope itself is much 
more expensive than non-metallic ones for transmitting the same 
amount of power. 

The sheaves for wire ropes are generally made of cast iron. 
The grooves are made much wider than the diameter of the rope, 
which runs against the bottom only; the flanges act only as gaides 
to prevent it from leaving the sheave in case of excessive swaying. 
The bottom of the groove is lined with wood, leather, gutta percha, 
or some similar material, soft as compared with the iron. 

Wire rope sometimes has a strong tendency to sway when 
running. One of the chief causes of this is lack of roundness in the 



BELTS AND ROPES FOR POWER TRANSMISSION. 209 

sheaye. After the lining is placed in the groove, it should be 
accurately turned to run true; a small depression, sufficient to fit 
possibly one quarter of the circumference of the rope, will hold it 
in the centre of a sheave rotating on a horizontal axis, when the 
rope is running without swaying. 

The Richmond Manufacturing Company of Lockport, New 
York, have doubtless had the most extensive experience with wire- 
rope power transmission of any concern in the United States. As 
a resalt of an experience of thirty years, they have arrived at the 
following conclusions: 

'* First. The best motion for the pulleys is not less than 100 or 
more than 140 revolutions per minute. The power will often do 
well at a less or greater motion, but generally the result is not so 
satisfactory. 

" Second. If a cable is run over level ground for a long distance 
a support will be needed every 400 or 500 feet, but where both ends 
of the cable are high enough so as to allow plenty of room for the 
sag in the middle, a support is not required for less xlistance than 
1000 feet. Where supports are required, a stout post set firmly in 
the ground and extending about 20 feet above it, with two small 
pulleys upon the top, answers the purpose. 

** Third. Where the cable is short, some method of tightening 
it should be provided, either by arranging the driving pulleys so 
that they can be pushed farther apart, or by using an extra pulley 
as an idler; where the cable is long, this is not required, as the 
weight of the cable itself will then prevent any slipping on the 
pulleys. 

'^ Fourth. Large cables must never be used upon small pulleys, 
as the continual bending of coarse wires aroand too small pulleys 
would soon break them." 

Table XII has been adopted by this company for its own prac- 
tice, and is recommended for. general work. 



210 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



Table XII. 

SHOWING THE NUMBER OF HORSE-POWER EACH SIZE OF CABLE WILL 
SAFELY TRANSMIT ; THE DIAMETER OF DRIVING AND DRIVEN 
LEATHER- PACKED GROOVED PULLETS NECESSARY TO BE USED 
WITH EACH SIZE ; THE NUMBER OF REVOLUTIONS THEY 
SHOULD MAKE, ETC. 



Diameter of Cable. 
Inche*. 


Diameter of Pulley. 
Feet. 


Revolutions per 
Minute. 


Horse-power 
Transmitted. 


J' 
1 

A- 

A 

1- 






i:::::::: 


8 
3 
4 
4 
6 
5 
6 
6 
7 
7 
8 
8 


100 
140 
100 
140 
100 
140 
100 
140 
100 
140 
100 
140 


2 
8 

4 
6 
9 
18 
14 
20 
28 
82 
88 
42 



Their experience has also shown thafc 100 feet is the shortest 
distance between the driving and driven pulleys that will give satis- 
factory operation. They find that a cable working according to 
the conditions given above will last from five to six years when 
working ten hours a day.* 

In the use of wire rope on cable railways, and where power is 
transmitted over a long distance by a single endless rope, it is 
desirable to have the tension on the tight side of the rope several 
times that on the slack. This is accomplished by using a pair of 
winding drums at the power plant of a railway, and at both ends of 
the line when power is transmitted between two points. The two 
drums of a pair are placed near together, with their axes parallel; 
the tight side of the rope winds in the groove at one end of one 
drum, passing half-way around it, and then goes to the correspond- 
ing groove on the other, then back to the second groove of the first 
drum, and so on, winding consecutively in all the grooves of both 
drums. This device is most effective when the drums are geared 

* Data and information kindly furnished by the Hon. William Richmond, 
president of the Richmond Manufacturing Co. 



BELTS AND ROPES FOR POWER TRANSMISSION. 



211 



together so that both act as drivers, or, if they are at the driyen 
end of the line, power can be taken from both. 

The wear on such a pair of winding drums, used as drivers, is 
most rapid in the groove where the rope first winds on, and 
gradually decreases in each successive groove. When the first 
groove has become worn to a smaller effective diameter than the last, 
there is a tendency toward unequal winding, and, as a consequence, 
heavy strains are produced in the rope on and between the two 
drums, which can only be relieved, in a measure, by the rope's 
slipping in the groove. 

In order to prevent the rapid destruction of the rope and drums 
by this action, Mr. John Walker * has designed a differential pulley. 
A section of the rim of this pulley is shown in Fig. 81. Each 
groove is cut in a ring^ separate 
from the rest of the pulley, and 
a number of the rings are placed 
side by side on what corresponds 
to a flanged pulley. The rings 
have a free-running fit on the 
cylindrical part of thu pulley, 
but their resistance to turning 
is regulated by one of the 
flanges, which is separate from 
the pulley and held in place by 
bolts that can be adjusted to 
give the desired pressure against the sides of the rings. A cushion 
of some elastic material is placed between the loose flange and the 
main part of the pulley. The loose flange should be adjusted so 
that the rings will "turn on the pulley a little more easily than the 
rope will slip in the grooves. 

The differential pulley can be equally well applied to any system 
of rope-driving, using . any kind of rope, where there are two or 
more grooves in a pulley, f 

* Formerly of the Walker Co. of Cleveland, Ohio; now Consulting Engi- 
neer, located in Chicago. 

t A large amount of data on power transmission by wire ropes is given in 
Kent's ' * Mechanical Engineers' Pocket-book." 




Fig. 81. 



CHAPTER IV. 
SCREWS FOR POWER TRANSMISSION. 

67. In some classes of machines, screws are used for applying a 
great force acting through a small distance. Examples of sach 
application may be seen in testing-machines for determining the 
strength of materials, and in presses for copying, baling cotton, etc. 
Again, in sach machines as those for planing the edges of boiler- 
plates, a proportionately smaller force, exerted by the screw against 
the tool-carriage, acts through a greater distance. 

The thread used on such screws is generally square, and in all 
cases, unless some special requirements make it necessary to have 
some other form, the surface of the driving side of the thread should 
be such as is generated by a line always remaining perpendicular to 
the axis of the thread. In other words, the working side should be 
the same as that of a square thread. 

The following notation for screws will be used: 

A = sectional area of screw at bottom of thread, square inches; 
D = mean diameter of collar- or step-bearing, inches; 
E = efficiency of screw and collar for forward motion ; 
E' = efficiency of screw and collar for backward motion or over- 
hauling; 
jP= turning force applied to the screw for raising the load, 

pounds; 
F' = turning force applied to the screw for lowering the load, 

pounds; 
F" = turning force exerted by screw when overhauling, pounds; 
Ip = polar moment of inertia of the section of the screw, biquad- 
ratic inches; 
y = tension or compression in screw, pounds; 
d = mean diameter of thread, inches; 

d. = diameter of top of thread, inches; 

212 



80EEWS FOR POWER TRANSMISSION. 213 

e = efficiency of screw-thread alone for forward motion; 
e' = efficiency of screw-thread alone for backward motion or 

overhauling; 
I = length of lever-arm of the turning force, inches; 
p = pitch of thread, inches; 
r, = radius of screw at bottom of thread, inches; 
s = shearing stress per unit area caused by the screw-thread 

resistance, pounds per square inch ; 
t =: tensile or compressive stress per unit area caused by the axial 

force, pounds per square inch ; 
/5 = angle between surface of thread and a normal to the axis of 

the screw, degrees; 
= angular pitch of thread, which is the angle between the 

mean helix and a plane perpendicular to axis of screw, 

degrees; 
// = tan = coefficient of friction between threads; 
M' = tan 0' = coefficient of friction between collar and supporting 

surface ; 
= angle of friction between threads, degrees; 
0' = angle of friction between collar and supporting surface, 

degrees. 

The pitch p is the distance, parallel to the screw axis, between 
similar points of adjacent turns of the thread on a single-thread 
screw; in a screw having more than one thread, jo is the axial dis- 
tance between similar points on successive turns of the same thread. 
The mean thread-diameter d is taken as that of a helix lying 
midway between the top and bottom of the thread, and, for 
convenience, all the pressure of the nut against the thread is con- 
sidered as acting along this mean helix. Theoretically the diameter 
of this helix is slightly greater than this mean value, but the differ- 
ence is so small as to be far within the necessary limits of accuracy 
for any practical requirements. The same is true of D. The mean 
angular pitch 6 is found by laying out a right triangle, making one 
side equal |7, and the other equal ndy being the angle between the 
latter and the hypothenuse. This gives 

^^^i^ (4^-> 



214 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

Square-thread Screws, 

68. Relation between the turning moment and axial force in & 
square-thread screw. — Whon a screw is used for power transmission, 
as described above, it is often desirable to know the turning moment 
which must be applied to produce a given axial force, or vice versa; 
the axial force may be either tension or compression. 

Fig. 82 may be taken to represent a portion of such a screw. 
As a convenient method of dealing with the problem, it may be 
assumed that a load is suspended from it by means of a nut (not 
shown) which fits on the thread ; also, that the screw is supported 
by a bearing on which the collar near the top rests. 

It is evident that friction must be considered. Since the nut 
and supporting surface under the collar may be of different 
materials, or differently lubricated, the coefficient of friction 
between the screw and nut, and that between the collar and the 
material against which it bears, may have different values; to make 
the case general, they will be considered as different. 

If there were no friction between the threads of the nut and 
screw, the pressure between them would be normal to their working 
surfaces and make an angle with the axis of the screw. On 
account of friction, however, the direction of the force acting 
between them at any point makes the angle of friction with a 
normal to the thread at that point. When the screw is turning to 
raise the load, the direction of pressure between the threads is at an 
angle {6 -{- <p) with the axis of the screw. 

An elementary portion ab (see figure), of the tension T, is 
therefore held in equilibrium by the forces ac and cb; cb = 
ab tan {0 + 0). The latter is the external force, normal to the 
axis of the screw, which must be applied to turn the screw against 
the resistance due to the elementary axial force ab. The nut re- 
sistance to the turning of the screw, acting as it does at a distance 

^ from the screw's axis, is of a value {cb) x t- = aJ tan {0+</>) X t. 

Tbe total nut resistance, for the total load T, is the sum of the 
resistances for all the elementary forces, each equal to abj and is 
expressed by the equation 

Nut resistance = T tan (^ + 0) X ^r. 



U0HUW8 FOB POWEK TKANSMISSION. 



216 




216 FOBM, STRENGTH, AND PROPORTIONS OF PARTS. 

This represents the valne of the taming moment which mnst be 
applied to o7ercome the resistance in the nut. 

By the same method, the tarning moment necessary to overcome 
the friction of the pressure T against the collar, is shown by the 
equation 

Collar friction = rtan 0'^ = Tii'^. 

In order to prevent side pressure of the body of the screw 
against the supporting frame, the torsional moment acting to turn 
the screw may be applied as a couple, each force having a lever-arm 

equal to — about the axis of the screw, thus making the arm of the 

couple eqaal to 2. GalliDg each of the external turning forces F^ 
the driving torsional moment becomes Fh 




Fig. 82.U 



The following equation of turning and resisting moments can 
now be written for a square-thread screw with thrust-collar when 
lifting a load (Fig. 8?.t): 



SCREWS FOB POWEK TRANSMISSION. 



217 



Screw Resistance, 
^=iPtaii(6»+0)f 



Collar Friction. 



_ tan_^_-Man_0 rf Z) 

~l- tan # tan 0^2"^'^ "2" 



-p+jiTtd d 
7td-MP 2 



+ Tm'4' 



(50) 



(61) 



In equations (50) and (51), as well as those between them, the 
screw resistance is expressed by the first part of the right-hand 
members, and the collar friction by the last part of the right-hand 
members. 




Fto. 82.2. 



For lowering the load by turning the screw backward, the 
angle of friction <f> must be taken on the opposite side of the 



218 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

normal ad to the thread from that shown in the figure, because 
the frictional resistance acts in a direction opposite that when the 
load is lifted. The equation of turning moment and tension 
then becomes, for lowering the load, when is greater than B 
(Fig. 82.2), 



Thread Resistance. Ck>llar Friction. 



^'/=3'taii(0-<?)| + r/i'^. 



(62) 



or 






(63) 




And when 6 is greater than <t> (Fig. 82.3), 



or 



SCREWS FOR POWER TRANSMISSION. 219 

^Al^m^ v^Mi^r. I>rivlng Moment of 
Collar Friction, Screw-thread. 

J'7=J>'-:|-7'tan(e- 0)|, . . . . (54) 

ri^TM^^^T^-P:i;-^xl. .... (55) 

The yalae of ff that will jast hold the load at rest, or allow it to 
descend aniformlj, when no external turning force is applied, is 
given by the equation 

tan (^-0) =//'-§- (56) 

Any larger value of than obtained by this equation will require 
a resisting moment applied to the screw to hold the load. 

A screw that will allow a load to descend by its own weight is 
said to OYerhauL 

The turning moment F''l that a screw will exert when over- 
hauling is 



^&'1K.'" collar Friction. ^ 

F"l=^Ti^{d-<p)l-TM'^, . . . . (57) 



or 



F'*l = TP^:^xi^TM'^.. . . . (68) 

If there were no collar friction, overhauling would occur for any 
value of greater than 0. 

The principle of overhauling is applied to advantage in some 
mechanisms. Probably the most familiar example is the screw- 
driver having a handle which contains a nut that engages with a 
screw-thread of rapid pitch on the stock for holding the bit or 
blade: by forcing the handle down over the screw, the blade is 
rotated. 



220 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

69. Effioienoy of a square-thread screw and collar. — ^For for- 
ward motion, when the end thrast of the screw is resisted by a 
collar, the efBiciency may be foand by dividing the usefnl work 
accomplished during one rcYolution of the screw, = ?)?, by the 
work, F{%frl)^ applied to turn the screw through one revolution; 
the effioienoy for forward motion is therefore expressed by the 
equation 

^=^ (««> 

By substituting in this equation the value of p, as expressed in 
equation (40), and that of Fl as given in equation (50), it becomes 

TTtd tan ^ tan ^ 



If the coefficients of collar friction and thread friction are equal, 
which corresponds to /i' = >u = tan 0, the maximum value of ^ is 
obtained when 



to» = 7=^-- • • • <"> 

sec + tan 0y 1 + -7 

The efficiency e of the screw-thread alone is obtained by dropping 
from the denominator of equation (60) the expression TuD^i' for 
the work done in overcoming collar friction, which gives 

tan e .^ . 

•"toEn(9 + 0) f«^) 

The maximum value of is obtained when 



tan6^ = sec0-tan0 = VT+/? - /i. . • (63) 

The value of the thread angle d is less for maximnm efficiency 
when there is no collar friction than when there is. 



SCREWS FOR POWER TRANSMISSION. 221 

When a screw is oyerhaaling, the points of application of the 
driving force and resistance are exchanged; the axial force or load 
becomes the driving force, and the external resistance is applied at 
the same point as the driving force when lifting the load. 

The efficiency E' for overhaaling is therefore expressed by the 
eq nation 

^' = ^^- (64) 

The value of F^% in this expression, is given in equations (67) 
and (58) ; that in (57) will be sabstituted. Therefore, 



„, Tnd tan( g - 0) - TnDpf _ c? tan(g - 0) -- Z>/i^ . . 

^' = kdteh = Jtsrs ' <^^> 



Dropping the collar friction TnByi' from this equation gives 
for the efficiency e' of the screw-thread alone, when overhauling, 

•'=^tj^ w 

When a driving force F* must be applied to a screw when 
lowering its load, there is no efficiency. The efficiency is zero when 
the load runs down uniformly of its own accord. 

70. Coefficient of friction }i for square-thread screws. — This 
quantity has been very carefully determined by Prof. Albert Kings- 
bury for different materials and lubricants.* The dimensions of 
all the bolts and nuts tested were as follows: 

Outside diameter of screw 1.426 inches 

Inside diameter of nut 1.273 " 

Mean diameter of thread 1.352 ** 

Pitch of thread i ** 

Depth or height of nut (effective) ItV ** 

Area of rubbing surface of thread 1 sq. in* (abooi) 



• Trans. Amer. Soc. Mech. Engrs., vol. xvn., 1896, p. 96. 



222 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



The nuts fit the screws loosely in order that there might be no 
friction other than that between the working sarfaces. ^' The 
threads were cat carefully in the lathe and had been worn to good 
condition by trials previoos to those here recorded. Screw No. 5 
was not qaite so smooth as the others.'' The lubricant was applied 
when the nut and screw were put together, and no more was added 
during the taking of the series of cards for them. The speed was 
slow; about one revolution in two minutes. 

The results of the tests are gi^en in Table XIII. 

Table XIII. 

COEFFICIENTS OF FRICTION FOR SQUARE-THREAD SCREWS. 

Working surface of thread approximately 1 square inch in area ; pressure 
per square inch of thread approximately the same as the total tension in screw- 
bolts. 





Ck>efficient8 of Friction ft.. 

Average Values of Four 

Readings. 


Value of ft for a 
Single Card. 


Pressure 


Screws. 


i 


1*- 


i 


9 a 


Highest. 


Lowest. 


and 
Lubricant. 


1. Mild steel .... 

2. Wrought iron. 
8. Cast iron 

4. Cast bronze. . . 

5. Mild steel, 
case-hardened. 


.141 
.189 
.125 
.124 

.138 


.10 
.14 
.189 
.135 

.143 


.136 
.188 
.119 
.172 

.13 


.136 
.147 
.171 
.132 

.198 


Screws 
Nut 9 
/i = .20 


Screw 8 

Nuts 


Pressure 10000 

lbs. per sq. inch. 

Machinery-oil. 


1 


.12 


.105 

.1075 

.10 

.10 

.0975 


.10 

.10 

.095 

.11 

.105 


.11 

.12 

.11 

.1325 

.1375 


Screw 4 
Nut 9 
// = .26 


Screw 8 
Nuts 
;u = .09 




2 


.1125 

.10 

,1150 


Pressure 10000 


8 

4 


lbs. per sq. inch. 
Lard oil. 


5 


.1175 










1 


.111 


.0675 

.07 

.071 

.045 

.055 


.065 
.075 
.105 
.044 


.04 

.065 

.059 


Screw 5 
Nut 6 
/i = .15 


Screw 5 
Nut 9 
// = .03 




2 


.089 


Pressure 10000 


8 


.1075 


lbs. per sq. inch. 


4 


.071 


Machinery- oil 


6....:.... 


.1275 


.07 -035 


and graphite. 












1 


.147 

.15 

.15 


.156 
.16 
.157 
.13 
1775 


.132 

.15 

.14 

.18 

.1675 


.127 

.117 

.12 

.14 

.1325 


Screw 5 
Nut 7 
/i = .19 


Screw 2 
Nut 9 
M = .11 




2 


Pressure 8000 


8 


lbs. per sq. inch. 
Machinery-oil. 


4 


.127 
.155 


6 













SCREWS FOR POWJCR TRANSMISSION. 223 

The coefficient of friction was low enough, in some cases, to 
allow the screw to " overhaul "; i.e., a pressure against the screw, 
parallel to its axis, caused it to rotate and pass through the nut. 

Following out the fairly well established fact that, in oil-lubri- 
cated journals, the coefficient of friction decreases rapidly as the 
velocity of rubbing increases from slightly above zero to 10 or 100 
feet per minute, according to whether the pressure is high or low, 
it would seem that lower values of the coefficients than given in the 
table might be found in screws running at the higher speeds that 
are frequently used for power transmissioQ. 

71. Problem. — Design a screw to lift 7 tons = 14000 pounds; 
end thrust of screw to be taken by collar-bearing; screw must not 
overhaul; driving gear attached to screw to be 18 inches in 
diameter. Screw-thread to be "square." 

For finding the pitch angle to prevent overhauling, the lowest 
values of the screw and collar friction that may occur with the 
materials used should be adopted. It will be assumed that the 
screw is to be of mild steel; the nut and collar-bearing may be of 
cast brass. The lowest values of the coefficients of friction may be 
taken as 

/i = .03 = ten 0, whence = 1° 43'; and /i' = .025. 

The ratio of the mean diameter of the collar to that of the' 
thread will be taken bs1.6 = D -r- d. 

The value of the pitch angle is found by equation (56), by 
which 

ten (^ - 0) = m'^ = .025 X 1.6 == .04, 

whence 

(9-0 = 2° 18'; 

0- r43' = 2° 18'; 

^ = 4° 1'. 

Since the pitch must generally be such that the screw can be 
cut in an ordinary lathe, it will be assumed that^ = .5 inch, and 
the mean diameter d of the screw, to give the required angle 0y 
calculated. 



224 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 
By equation (49), 



whence 



•""=4 



d - ^ = -^ -. -g - 2 a7 

w-tanfl «'tan4*'^ .06993;r "" 



Taking the outside diameter of the screw as 0.45p greater than 
the mean diameter d for a single-thread screw gives 

Outside diameter of screw = 2.27 + .45 X .5 = 2.5". 

The mean diameter of the collar is 

B z= 1.6d = 1.6 X 2.27 = 3.63 inches. 

The turning moment Fl which is necessary to raise the load is 
found by equation (51). The greatest values of the coefficients of 
friction, /j, and pi\ that are probable to occur at any time when the 
lubrication is poor, should be used in this equation in order that 
the maximum value of the turning moment may be found. 

The values jw = .15 and /i' = 0.1 are probably the highest 
that occur in well-made machines operated with reasonable care 
and cleanliness. 

Applying equation (51), 

- 11000 -^ + '^^^^'^^ :c ^-^^ I 11000 X 1 ^'^^ 

- ^^^ ;r2.27 - .15 X .5 ^ 2 + 1*«^^ X ^- 1 -3- 

Screw Resistance. Collar Friction. 

= 3532 + 2541 = 6073 inch-lbs. 

And the force F acting tangent to the pitch circle of a gear 
whose pitch radius is 9 inches is 

F = 6073 -5- 9 = 675 pounds. 



80BBW8 FOB POWEB TBAIfSMISSIOlT. 

No allowaiioe has been made for journal friction in the aboye 
ealcnlationB. This makes the Yalne of F greater than 675 pounds. 
The pressure between the teeth of the drinng gears is greater than 
F on account of their obliquity of action. Assuming that the actual 
pressure between the teeth, taking into account both journal fric- 
tion and obliquity of action, is 800 pounds, and that the coeflScient 
of journal friction is .06, then the corresponding yalue of the tan- 
gential force is 

^= 675 + (.06 X 800)^ = 682. 

•/ 

The efficiency of the screw when working with the highest 
assumed coefficients of friction, and including journal friction, is, 
by equation (59), 

Tp _ 14000X0.5 __ .Q^.QV 
^ " Fi^ "" 682 X 2^ X 9 " "^^ " ^^^• 

78. Maximum stress in a screw. — ^A screw that is turning and 
lifting a load is subjected to an axial tensile stress equal to the 
weight of the load, together with a torsional moment equal to the 
turning moment that must be applied to overcome the resistance of 
the screw-thread. The combination of these two stresses produces 
what may be called a maximum tensile stress and a maximum shear* 
ing stress. The values of these maximum stresses should not ex- 
ceed the working strengths, tensile and shearing, of the material. 

If the axial force acting on the screw is compression instead of 
tension, and the distance between the nut and collar is small 
enough to allow the part in compression to be considered as a short 
column, not liable to bend or buckle, the maximum compressive 
stress will be equal to the maximum tensile stress for the same load, 
and the maximum shear will be the same in both cases. 

The formulas for maximum stress in a rod of circular section, 
as given in works on the ^' mechanics of materials," are: 



t /? 

Maximum tension or compression = « "^ \/ T + **> • (^'') 

Maximum shear ~\/t"^''* * * ^^^^ 



226 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

In dealing with the strength of a screw of ordinary construction^ 
the. thread may be neglected, since it adds but little to the strength. 
The screw may therefore be considered as a cylindrical bar of a 
diameter the same as that of the bottom of the thread. Under 
this assumption the value of t is 

and that of «, for square-thread screws, is 

' /p-^r,"" 1 , "" Ttr* 

By substituting these values of t and 8 in equations (67) and 
(68), the following equations are obtained for square-thread screws: 

Max. tension or 1 y /~T^ T* tan'. (^+ 0)d' 

compression P^^ f = ^2 "*" V O^ "* ^rV* "' 

sq. in. ) V • 

which reduces to 

Max. tension or ) ^ tT / f d\*'\ 

compression per V = i j[^l + V 1 + 4 tan* (^ + 0)(^j J. (69) 
BU» in. J 

In the same manner, 

Example of application of formulas for maximum stress in a 
gcrew.— In the problem solved in § 71 the following data were 
assumed, or else determined for the highest coefficients of friction: 
T= 14000 pounds; tan = .15; = 8° 31' 10"; ^ = 4^ 1'; 
^ _|. _ ;i^2° 32' 10"; and d = 2.27 inches. In addition to these 
data, the radius at the bottom of the thread may be taken as 
r, = 1 inch; whence A = 3.1416 square inches. 



SCREWS FOB POWER TRANSMISSION. 
By the subetitation of these quantities in equation (69), 



227 



Maximum tension = | —fl + ^l + 4 tan' (12^32' lO")^?:??^ 



= i- -jLl + Vl + 4(0.22236)-(2.27)*] 

= I ^^q{^ + 1-4211) = S390 lbs. per sq. in. ; 



and, by equation (70), 
Maximum shear = - X 1.4211 = 3180 lbs. per sq. in. 

Angular-thread Screws. 

73. Selation between turning moment and axial force in an 
angular-thread screw. — When the working surface of a screw thread 
makes an angle /3 with a normal to the axis of the screw, as in, Fig. 
83, the pressure between the threads of the nut and screw is 







Fig. 88. 

increased by the wedge-like action of the thread. 

The following equations apply to an angular-thread screw when 
lifting a load.* (Notation given in § 67.) 

* Tbe development of the equation for thread resistance is given in § A of 
the Appendix. 



22B VOBM, STBBNOTH, AJSTO PBOPOBTIOirS OF PARTS. 

Thread Reristanoe. Sf^ 

Friction. 

pn ^ y tang + taD08ec/? d M . . 

'' ■'l-taii6'tan0oo8/?^2 + -'^2 * ' ^^^> 

_ ( j> -j- >rd) + // Bee /? ^d M 

~ 1 - (j> -T- a:d);< cos /? ^ 2 "^ '^ 2 

^y p+;*>rrf8ec^ I ^^^ ^ 

H-d — >wp COB /? 2 2' ^ ' 






(73) 



__ tan ^ (1 --■ tan g tan 0co8 >g) , 

~ tan ^ + tan sec /? ' ' ' • ' • v / 

For the backward motion of an angalar-thread screw when a driv- 
ing force is applied to lower the load: 

Thread Resistance. Collar Friction. 

F'l = T ^^f^J-J^^. X i + 7>4. . (75) 
1 + tan 6/ tan COS p 2 ' ^2 ^ ' 

If tan is greater than tan sec /? in equation (75), then the 
thread resistance is negative, and most be subtracted from the 
collar friction. 

The thread resistance for backward motion becomes zero when 

tan = tan sec >5, 
or 

tan = tan cos /3. 

The last two equations indicate the values of and which 
would just allow the screw to overhaul if there were no collar fric- 
tion. 

When overhauling, the driving moment, exerted by the nut, is 

Driving Moment of Screw.thread. Collar Friction. 

-.^-, ^ tan — tan sec d ^ . D 



SCREWS FOR POWER TRANSMISSION. 229 

Experimentally determined efficiency of a V-thread screw-bolt. 
— A series of experiments upon a screw-bolt were made by James 
McBride to determine its efficiency.* An ordinary Y-thread screw* 
bolt was purchased on the market, and was not specially prepared for 
the experiments; the nnt fitted the thread freely, so that it could 
be run along the thread easily with the hand; the flat end of the 
nnt rested on a washer which formed the bearing-surface; both nut 
and washer were of malleable iron with bearing-surfaces left rough 
or unfaced. Lard-oil was used as a lubricant. The bolt was sus- 
pended vertically by the nut, which rested on a support 4 feet 6 
inches high; a load was hung on the lower end of the bolt. The 
torsional moment was applied by difiFerent men pulling on a hori- 
zontal, single-ended wrench which spanned the nut. The following 
are the more important data: 

Diameter of bolt = 2 inches; 

Pitch of screw = 0.22 of an inch = 4.5 threads per inch; 

Standard V thread; 

Load suspended = 7500 pounds. 

The highest efficiency found was 12.29^, the lowest 9.71^, the 
ayerage being 10.19^. 

At the ayerage efficiency of 10.19^ a torsional moment of 
1 inch-pound, applied to the wrench, would produce an axial stress 
r = (2;r -i- 0.22)0.1019 = 2.91 pounds in the bolt. 

* Trans. Amer. Soc Mecli. Engra., yoL xii., 1891, p. 781. 



CHAPTER V. 

SCREW.GEARINa. 

74. The most common form of screw-gearing that is nsed, when 
there is to be any considerable amount of power transmitted, is the 
worm and worm-wheel; and, of this mechanism, that having the 
axes of the worm and wheel at right angles is most generally 
adopted, donbtless because it is only when the shafts are at a right 
angle that the face of the wheel can be concaved to embrace a por- 
tion of the worm. In the worm and worm-wheel there is generally 
a great difference of angular velocities, the worm making many 
revolutions to give one turn to the wheel. The diameter of the 
wheel is ordinarily much greater than that of the worm. 

When the axes are at any other angle than 90°, the face of the 
wheel cannot be concaved, and its teeth, instead of being curved so 
as to secure line contact with the worm, must be portions of a 
many-threaded screw, or else similar to those of a spur-gear, both 
of which give only point contact theoretically. 

When the worm and wheel do not have greatly different angular 
velocities, and the teeth on both are short lengths of the threads 
of a many- threaded screw, the name *' screw-gears" is commonly 
applied. This term will be used to designate all such mechanisms 
in which the wheel does not have a concave face. The axes may 
be at any angle. 

WORM AND WORM-WHEELS.* 

76. The strength of a worm and worm-wheel seldom needs con- 
sideration, for, under ordinary conditions, the allowable pressure 

* The coefficient of friction for wotm-gearing is probably about the same 
as that of screw-gears (Table XXI) or square-thread screws (Table XIII), ac- 
cording to the Telocity of rubbing. 

230 



SCREW-GEARING. 231 

between them, that will not prodace abrasion, is mach less than 
would be required to break the teeth when made of any of the 
materials common to engineering. 

Since the teeth are cut on a wheel concaved to fit the worm, 
they are much stronger than those of the same proportions that are 
straight. Their strength cannot be satisfactorily calculated, and it 
is scarcely desirable to make such a calculation. 

The strength of the worm-thread cannot be calculated, but when 
of the same material and thickness at the pitch surface as the teeth 
on the wheel, it is the stronger of the two. 

76. Equations for taming force and efficiency of worm and 
worm-wheel. — The equations of the relation between the turning 
moment and pressure against the teeth in a direction parallel to the 
axis of the worm are essentially the same for a worm-wheel as those 
of an angular-thread screw, given in § 73. This assumes that the 
thread is angnlar, not square, for it is not known that a square- 
thread worm is ever used when it must perform a service that 
demands even moderately high speeds and pressures. 

If a worm works in engagement with two wheels that are tangent 
to its pitch surface at diametrically opposite points, and whose axes 
are parallel to each other and at right angles to that of the worm, 
the equations for an angular-thread screw apply to it without modi- 
fication. For the more common mechanism, in which the worm 
engages with a single wheel, there is a side pressure on the support- 
ing journal-bearings which may materially increase the turning 
moment required to rotate the worm against a given pressure of the 
wheel against it. This is especially true if the journal is large or 
has a high coefficient of friction. 

The thrust-bearing of a worm can be obviated by using two 
worms, one right-hand and the other left-hand, on the same shaft, 
each engaging with a suitable worm-wheel; the shafts of the worm- 
wheels must be geared together so as to give them both the same 
rate of rotation in opposite directions, provided each worm and 
wheel forms a part of the mechanism differing from the other only 
in the direction of the thread. It can readily be seen that the 
thrust of one worm will be annulled by that of the other, thus 
eliminating the need of a thrust-bearing. 

For convenience of reference the notation used in the equations 



233 FOBM, STRENGTH, AND PROPORTIONS OF PARTS. 

for worm-gearing will be given in fnll, although most of the 
symbols are the same as for screws: 

D' = mean diameter of thrust-bearing, inches; 
D" =■ mean diameter of joarnal, inches; 

B = efficiency of mechanism for forward motion; 
JS' = efficiency of mechanism for overhaaling; 

F = force acting to rotate the worm, pounds; 
F^' = turning force exerted by the worm when overhaaling, 
pounds; 

P = turning force acting on driven wheel, pounds; 

y = thrust of worm or screw-gear i)arallel to its axis, pounds; 

d = mean diameter of worm- thread, inches; 

e = efficiency of worm-thread alone, journal and thrust-bearing 
friction not included ; 

e' = efficiency of worm-thread alone when overhauling; 
I = length of lever-arm of turning force, inches; 

p = pitch of worm-thread, inches; 

fi = angle between an element of the thread surface and a piano 
normal to the axis of the screw; 

6 = angular pitch of screw-thread; 

/i = tan = coefficient of friction between worm and wheel ; 
/i' = tan 0' = coefficient of thrust-bearing friction; 
;i" z= coefficient of journal-friction ; 

<p = angle of friction between worm and wheel ; 
0' = angle of friction for thrust-bearing. 

In the following discussion it is assumed that the axes of the 
worm and worm-wheel are at right angles. 

The pressure, due to the forces acting between the teeth, against 
the journals supporting a worm that engages with but one worm- 
wheel, assuming that there is a journal at each end of the worm, is 
equal to the resultant of two forces, one approximately equal to the 
force resisting the rotation of the worm when driving the wheel, 
acting parallel to the axis of the wheel, and at a distance d/2 from the 
axis of the worm, and the other equal to the component, normal to 
and intersecting the axis of the worm, of the pressure 7 tangent to the 
mean diameter of the wheel and parallel to the axis of the worm.* 

* Strictly the force T causes some additional pressure on the journal since 
it acts with a lever arm d-i-2. This pressure is so small as to be negligible 
in any ordinary design. 



SCREW-GEARING. 233 

« 

The first of these forces, acting parallel to the axis of the wheel, 

, -- tan + tan sec /J * , , 

is approximately T= 7 — ttz — ^n — ^*; the second, normal to 

^^ "^ 1 — tan 6^ tan cos /? ' 

and intersecting the axis of the worm, is T tan /?. Since these two 

forces act at right angles, their resaltant, which is the 



Pressure on \ __ rp // tan g + tan sec /Sf V , x^ « ^ 
worm.journals J "*" ^ y Vi - tan » tan cos /?y + tan p. 

The journal friction, assuming that both journals are of the 
same diameter D'^ and have the same coefficient of friction ^\ is 

Joumid fric- ) n" //tan6^ + tan08ec/?\\^ .^ ,^,, 

tjonofworm. [ =/^"^l-y^( i,tangtan0cosg ) +^^ ^- ^^'^ 

If the journals supporting the worm are of different diameters 
or have different coefficients of friction, the pressure on each 
journal may be found and the friction of each determined. The 
total pressure is divided between the two journals in amounts 
inversely proportional to their distances from the pitch point of the 
worm and wheel. Such a refinement as this would seldom, if ever, 
be worth applying in practice. 

There is a|i end thrust on the worm-wheel shaft approxi- 
., ,,^tan^4-tan0sec>S , , 

mately equal to ^ 1 _ t^^ ^ tan cos /? ^ *^ * ^^'^"^^ pressure 

having a value TV\-\' tan' ft. 

The following equations may now be written by referring to 
those for an angular-thread screw. In them the frictional losses in 
the bearings supporting the worm-wheel are not taken into account 
as producing part of the thrusting force T. The equations of 
efficiency, therefore, do not exactly represent the efficiency of a 
complete mechanism, on account of the friction of the worm-wheel 
bearings being neglected, but the speed of the worm-wheel is so 
slow that the friction losses in its bearings are comparatively small 
ordinarily. The efficiencies given by the following equations are 

* See Appendix, § A. 



234 FORM, STRENGTH, AND PROPORTIONS OP PARTS. 

far more accnrate than the assamptions that can be made for the 
coefficients of friction. 

The Talne of the ^' journal friction " in the following equations 
;nay be obtained by equation (77). 

When the worm drives the wheel: 

Th««d Resistance. "^^Jj^ 

1 — tan 6/ tan0co8>5 2 ' ^ 2 ' friction, ^ ' 

or 

_ p+M7td^ec/3 d ^ D^ Journal ,7o^ 

^^^nd--^pco^P^2 + ^^ 2 + friction, ^^^^ 

and 

Tp __ 7atang . 

^"■;f(^-~^^7"' ^^^> 

__ tan ^(1 — tan ^ tan cos ft) 

"" tan ^ + tan sec >5 ' • • • v ; 

If it is desired to neglect the effect of the angle >5, then, for the 
approximate turning moment, 

Approximate Thread Thrust- bearing 
Resistance. Friction. 

^/=Ttan(^ + 0)|+7>4' + J-™J . . (82) 

or 

Fl = yP+_il!L^ X f + 2>'-^' + J?''™"! . . (83) 
nd — MP 2 2 ' fnction, ^ ^ 

And, for the approximate worm-thread efficiency, neglecting /5, 

^ = tanT^+0) ^^PP''°^'"'*^^y)- • • • <®*) 

When the wheel drives the worm, which action corresponds to 
overhauling in a screw : 

Taming "Force due to Pres- Thrust-bear- 

sure of Wheel against Worm. ing Friction. 

J.7 = t'^^-JTV ^ X ^ - ^^'f - ? -"r^ (85) 
1 + tan ^ tan cos >5 2 2 fnction, ^ ' 



SGREW-GEARINGk 285 

or 

;rd + /<p COS /S^ 2 ^ friction, ^ ' 

Mid 

^^"^--fateO' (^^^ 

. _ tan g - tan sec /? 

tan «(1 + tan Stan COB >5) ^^ 

As is indicated by the above equations, the efficiency of worm- 
gearing increases with the angular pitch up to a certain limiting 
value, which depends upon the coefficient of friction of the worm 
and the frictional resistance of the thrust-bearing. This limiting 
angle is greater than has apparently ever been found satisfactory in 
practice. There seems to be a general tendency, however, to use a 
greater angular pitch than formerly, when the primary function of 
the worm-gearing is to transmit power in considerable amounts. 
A pitch angle of 20°, or even more, is quite commonly used for 
worm-driven machine-tools. 

77. Tests of worm-gearing. — In 1885 Wilfred Lewis published 
the results of an extensive series of tests made by Wm. Sellers & 
Go. on worm and spiral gearing. The result of each individual 
test on a particular mechanism, at different speeds and pressures, 
was plotted on a diagram, which also had a curve of the mean 
efficiency of the mechanism for the entire range of speed covered. 
From these curves of mean efficiencies the readings given in Table 
XrV were taken. The teeth were approximately of the involute 
system. 

In the experiments it was found that abrasion, or cutting, be- 
gan between the surfaces of the worm-thread and wheel-teeth at 
certain limiting pressures and speeds. The mechanism could still 
be run after abrading, but the efficiency was materially reduced. 
This is shown very glearly in the third and fourth columns of 
Table XIV, which give the efficiency both before and after 
cutting ; the drop in efficiency in the last column, at 100 revolu- 
tions, also shows the effect of cutting. 



236 FOBM, STRENGTH, AND PROPORTIONS OF PARTS, 



Table XIV. 

EFFICIENCY OP CAST-IEON WORM-GEARING.* 

Efficiency = ratio of power delivered by worm-wbeel shaft to that applied 
to worm-shaft. 

All worms 4" pitch diameter ; all worm-wheels 18.62'' pitch diameter, 89 
teeth, l^inch pitch. Worm ran in oil-bath. When two values are given for 
the efficiency in the same column, the lower one was obtained after abrasion 
began. The drop from 69 to 65 in the last column is due to abrasion. 





Efficiency, per cent. 


BeTolutlona per 
minute, approxi- 
mately equals 
the velocity of 
■lidlng in feet 
per minute. 


9 threads, 8 In. pitch. AnRular 

pitch 180 61'. Oast thread 

and teeth. 


1 thread. U in. pitch. Angular pitch 

6*49'. Step-bearing 1.6 in. 

mean dlam. 


Collar thrust- 
bearinfir \'* mean 
dlam. Pressure 

aoo to 0000 lbs. 


Step-bearing \,V' 

mean diam. 

Pressure 1200 to 

5600 lbs. 


Cast thread and 

teeth. Pressure 

450 to 6600 lbs. 


Machine^sut 

thread and teeth. 

Pressure 1200 to 

6600 lbs. 


8 


65 
59 
62 
65 
68 
70 
72 
74 
75 
76 


49 

52 a 

57 8 
59 ^ 

64 5 

68 ^ 

70 and 59 

71 and 60 

72 and 61 . 
78 and 62 
74 and 68 

65 
66 
67 




42 


5 




45 


7 
10 
15 
20 
80 
40 
60 
80 
100 


46 

57 "^1 
59 

64 and 46 

67 and 48 

69 and 50 

71 and 52 

78 and 58 

54 

56 

57 

57.5 
58 
59 


48 
51 
55 
57 
61 
68 
67 
69 

67 f 1 


120 




150 




68 it 


200 




70 \ < 


800 




•^ / ^ 


400 






500 






600 




900 

















* Readings of efficiency taken from diagrams by Wilfred Lewis in Trans. Amer. Soc 
Mech. Eng., yol. yii., p. 878. 

The second colnmn gives the efiSciency of a worm whose thrust 
was taken by a collar-bearing formed by turning down the end of 
the thread to a flat surface ; this did not form a complete ring of 
metal to bear against the bearing, about half the material being re- 
moved on account of the space between the threads. In all the 
other tests the thrust was carried by a step-bearing made of two 
hardened-steel disks, carefully ground, with a hard-brass washer 



8CBEW-GBARING. 



237 



interposed. The step-bearing was 2{( inches outside diameter. 
The shaft at one end of the worm was 2{^ inches diameter, and 
lf|- inches at the other. 

An examination of the table shows that the efficiency increased 
with the speed, in all cases, until abrasion began. The greater 
efficiency of the donble-thread worm may be seen clearly for the 
lower speeds, but is not so decided as the speed increases. 

A series of experiments were also made on the single-thread 
worm with cast teeth, whose efficiency is given in the fourth 
column of Table XIV, to determine the speeds and prepsures liable 
to produce cutting during a ten-minute run. The results of these 
experiments are given in Table XV, taken from Mr. Lewis's paper. 

Table XV. 

SPEEDS AND PBESSURB8 LIABLE TO PRODUCE CUTTING IN 
CA8T-IB0N WORM-GEAKING. 



Velocity of 
sliding. 


Pressure 
on teeth, 
pounds. 

1786 
1780 
1205 
448 
2822 
8481 
4887 
6658 


Temperature. 


Efficiency. 


Duration 
of run, 
minutes. 


Ft. Ib8. per 
min. con- 
sumed in 


feet per 
minute. 


Initial. 


Final. 


Initial. 


Final. 


friction 
before cut- 
ting began. 


800 
880 
880 
800 
480 
400 
8(K) 
•806 1 


106* 

118 

187 

118 

144 

170 

188 

168 


140* 

132 

150 

188 

167 

180 

166 

186 


.600 
.607 
.575 
.694 
.601 
.689 
.641 
.677 


.887 
.462 
.860 
.445 
.450 
.415 
.478 
.677 


6 
8 
8 

10 
7 
8 
6 

10 


117,600 

129.300 

97.000 

29.400 

117,800 

98.800 

122.400 

102.000 



* No cutting at 306 feet per minute. 

In the last of these experiments, at a speed of nibbing of 306 feet 
per minute, and a thrust of 5558 pounds on the worm, no abi-asion 
occuiTed. The temperatures given are those of the oil-bath in 
which the worm ran ; the initial efficiency was obtained before 
abrasion, and the final after cutting began. 

In 1883-84: Dr. E. H. Thurston made a series of experiments on 
worm-gearing for the Yale & Towne Mfg. Co. The results are 
given by Henry E. Towne in the Transactions of the American 
Society of Mechanical Engineers, f The data show that the worm 

fVol. VII., 1886. p. 800. 



238 FORM, STRENGTH, AND PROPORTIONS OF PARTS, 

and wheel were both of cast iron^ with machine-cat threads and 
teeth ; the worm was 6^ inches pitch diameter^ double-threaded, 
and 4 inches long on the thread ; the worm-wheel was 15|f inches 
pitch diameter, with 50 teeth and 2-inch face. The pitch, calcu- 
lated from these data, is 2 inches linear, or 5^ 59' angular. 

During the experiments three forms of thrust-bearings for tak- 
ing the thrust of the worm were used. They were : First, a 
collar thrust-bearing having a collar 1 inch wide and 2^ inches 
mean diameter. The rubbing surfaces were the faced ends of the 
worm-hub and of the cast-iron supporting frame of the mechanism. 
Second, a button thrust-bearing, or step, made by capping the end 
of the worm-shaft with a piece of hardened steel, haying its ex- 
posed face slightly convex, and letting it run against the hardened 
end of an adjusting set-screw. Third, a roller thrust-bearing, con- 
sisting ^'of 12 chilled cast-iron coned rollers of ^ inch mean di- 
ameter, contained within a brass cage having a separate pocket for 
each cone, the cones travelling at a mean radius of If inches from 
the axis of the shaft, between two steel collars or rings, one bearing 
against the hub of the worm, and the other against the face of the 
frame-bearing, the faces of these collars being coned to the shape 
of the rollers. The centrifugal thrust of the cones was resisted by 
a wrought-iron ring surrounding the cage, the ends of the cones 
being convex.'* 

Table XVI shows a comparison between the button-bearing and 
Table XVI. 

COMPABATIVB EFFICIENCIES OF THE SAME WORM AND WHEEL 
WITH BUTTON THRUST-BEARING AND ROLLER STEP-BEARING. 



Horse-power 
per 100 reyolu- 
tions of worm 

per mlDute. 


Efficiency, per cent. 


Horse-power 
per 100 revolu- 
tions of worm 

per minute. 


Efficiency, per oent. 


Button thmst- 
bearlng:. 


Roller thrust- 
bearing. 


Button thrust- 
bearing. 


Roller ttmiBt- 
bearing. 


.25 
.50 
.75 

1.00 

1.5 

2.0 

2.5 


6 
10 
18 
16 
21 
25 
29 
82 


9 
12 
18 
22 
27 
81 
85 
88 


4.0 
5.0 
6.0 
7.0 
8.0 
8.28* 
9.00 
10.5t 

• 


88 
44 
49 
54 
59 
60 


48 

47 
50 
52 
54 

56 


8 




57 









* Highest power given for button-bearing. t Highest power given for roller-bearing. 



SOBEW-OEABING. 



239 



roller-bearing for different rates of working, and Table XVII be- 
tween the collar-bearing and roller-bearing. The readings in the 
tables are taken from diagrams showing the results of the experi- 
ments. 

Table XVIL 

compabativb efficiencies of the same worm akd wheel 
with collar thrust-bearing and with roller step-bearing. 



Revolutions 

of driver 
per minute. 


Efficiency, per cent. 


Revolutions 

of driver 
per minute. 


Efficiency, per cent. 


Collar thrust- 
bearinfc. 


Roller thrust- 
bearing. 


Collar thrust- 
bearing. 


Roller thrust- 
bearing. 


60 
100 
150 


8S 
42 
48 
42 


48 
62 
56 

68 


250 
800* 
850 
400 1 


40 
88 


60 
68 
66 


200 




67 









* Highest speed for collar-bearing. 



t Highest speed given for roller-bearing. 



The difference in the efficiencies obtained with these three forms 
of thrust-bearings shows clearly how much power may be lost in 
this part of the mechanism. It should be noted that while the ef- 
ficiency of the roller-bearing is higher than that of the button-bear- 
ing for rates of working up to 6 horse-power, it becomes lower at 7 
horse-power. The original diagram shows equal efficiencies at about 
6.5 horse-power. 

The results of a number of experiments on three tool-steel 
worms, hardened, running against cast-iron worm-wheels, are given 
by Bertram P. Flint.* Two of the worms were of the same diameter, 
but one was of practically twice as great lead or pitch as the other. 
The results of the experiments are given in Table XVIII. That 
the limit of pressure is lower at the highest speeds of rubbing is 
clearly shown. 

The double worm-and-wheel meohanlim shown in Fig. 83.1 
requires no thrust-bearing for the worms. Two worms, one rigbt- 
and the other left-hand, are placed on the sanie shaft. The worm- 
wheels are geared together by a pair of spur-gears. lu this par- 
ticular design, used for elevator-service, a worm-wheel and spur- 



Sngineering News, April 9, 1892, p. 848. 



240 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 




SCREW-GEARING. 241 

gear are made together as a unit. The spur-gear has helical or 
spiral teeth cut at an angle with its axis equal to the pitch angle 
of the womi thread. The middle of the gear is hobbed out to 
form a true worm-wheel. In some designs where a spur-gear 
and worm-wheel are a unit, the spur-gear has teeth running straight 
across its face parallel to its axis. This necessitates cutting a 
pair of circumferential grooves around the wheel so as to separate 
the worm-wheel teeth from those of the spur-gear on each side 
of it. The worm-wheel and spur-gear can, of course, be made en- 
tirely separate and fastened to the same shaft. In the design 
shown, a winding drum for the elevator cables is placed upon one 
worm-wheel shaft so that all the power is finally transmitted to 
the drum shaft by one worm-wheel. • 

The worm-and-wheel drive, Fig. 83.1, with bronze worm- 
wheels and steel worms, submerged in castor-oil or high-grade 
cylinder oil, will operate satisfactorily at 500 revolutions per 
minute of the worm when exerting a turning force of 6000 pounds 
on each worm-wheel tangent to its pitch-circle. This corresponds 
to 6000 pounds tension or compression in the worm-shaft between 
the worms. The rubbing velocity is 793 feet per minute at the 
pitch-circle of the worm. 

The tangential pressure between the teeth of the spur-gears 
is slightly less than 6000 pounds on account of the somewhat larger 
pitch diameter of the spur-gears. This pressure is 

9Q fin97 

^rZ X 6000 = 5780 pounds. 
o0.7o 

A smaller, similar mechanism, with worms 3* inches outside 
diameter, 3.0226 inches pitch diameter and i inch pitch, engaging 
with worm-wheels having 80 teeth each on a pitch radius of 9.5493 
inches operates successfully at 750 revolutions i)er minute of the 
worms and 4400 pounds tension or compression in the part of the 
worm shaft between the worms. This corresponds to a rubbing 
speed of 710 feet i)er minute and a tangential pressure 4400 
pounds on each worm-wheel. The spiral spur-gears are 20 inches 
pitch diameter with 80 teeth of 4 diametral pitch. 



242 FOBM, STB£NGTH, AKD PROPOBTIOfTS OF PARTS. 

Table XVIIL 
efpicibncy op w0bm-6eabing.* 

Efficiency = ratio of power delivered by worm-wheel sb&ft to that recelTed 
by the worm-shaft. 

Worm of tool-steel, hardened ; wheel of cast-iron ; axes of worm and wheel 
both horizontal ; worm on top of wheel ; bottom of wheel dipped in oil-bath. 
Thrust of worm taken by phosphor-bronze plate against which the end of the 
shaft turned. The limit of pressure indicates the highest pressure that did not 
cause abrasion. 



Thnut of 
Worm. 
Founds. 



Efficiency, per cent. 



Single-thread Worm. 
Angular Pitch 6» SO'. 



If 



m 

is3 






Double-thread Worm. 
Angular Pitch IS* W. 



lit 

Sag 

9SM 



p. 

Its 






Double-thread 

Worm. 

Angular Pitch lO*. 



m 



SSOD 






200 

800 

400 

600 

600 

700 

800 

1000 

1200 

1400 

1600 



89.6 
uni- 
formly 

in- 
creases 
to 
41 



87 

89 

41 

48 

44 

44.6 

44 

43 

42 

89 



80 

82 

88 

84.6 

86 

87 

87 

86.8 

86 



48 
47 
61 
68 
64 
64 
68 
61 
60 
48.6 



49 

68 

66 

67 

64 

62 

61.6 

49.6 

48 



47.6 

49.6 

61 

61.6 

61 

49 

48 

47 



60 
68 
64 
64 
68 
62 
61 



68 
64 
64 
63 
61 



Limit of pres- 
sure, pounds. 



1400 



1226 



1470 



1260 



1060 



1276 



710 



* Readings of efficiency taken from diagram by Bertram P. Flint, Engineering Newa, 
April 9, 180S, p. 848. 

t The efficienciea shown in Table XIX were obtained with a 
pair of Hindley wonns in an elevator drive arranged in general 
as shown in Fig. 83.1. Two ^vinding drums, one on each wonn- 
wheel shaft, were used. The experiments were conducted by driv- 
ing the worms with an electric motor whose armature was direct- 
connected to the worm-shaft, and hoisting a weight correspond- 



1[ American MachtnUt, Jan. 21, 1897, p. 46. 



BCBEW-GEABINO. 



243 



Table XIX. 

ZFFIOIEKCT OF HIKDLET WOBK AKD WOBM-WHBBL 
DBIYEK BLBVATOB.* 



EflSciencj = 



Load lifted X distance traTened. 



' Electrical energy received by motor. 
Two Hindley wormB, right- and left-hand, on same shaft. No thrust-bear- 
ing or step. Pitch diam. of worm, 5.47 inches ; pitch diam« of gear, 26.89 
inches ; doable- thread worm ; lead or pitch of each thread, 2.86 inches nearly; 
angular pitch at pitch line 9* 89^; Telocity ratio 29^ to 1; speed of worm, 500 
rev. per min.; Telocity of rubbing at pitch line, 718 ft. per min.; speed of 
travel of worm-wheel at pitch line, 112 ft. per min. 



Prensure in Fdanda on 




Pressure in Pounds on 






Effldenoj, 




Effldenej, 






percent. 






percent. 


Both Womu. 


One Worm. 




Both Worms. 


One Worm. 




1000 


500 


44 


8000 


1500 


68 


1500 


750 


54 


4000 


2000 


72 


2000 


1000 


60 


6000 


8000 


76 


2500 


1250 


65 


8500 




78 



* Readings taken from diagram in AmericanMachiniMt of Jan. 81, 1897, p. 45. 

ing to an eleyator-cage, which was lifted by means of a rope running 
from the winding-drams on the worm-wheel shafts. Measurements 
of the electrical energy received by the motor, and of the work done 
upon the weight, were taken. The ratio of the latter to the former 
corresponds to the efficiency given in the table. This, of course, 
does not represent the efficiency of the worm-gearing alone, as has 
been given in the preceding tables, since the motor losses, as well as 
^ those of the machinery between the worm-wheels and weight lifted, 
are included. The experimenters estimate that the efficiency of the 
worm-gearing alone '' can scarcely be less than 90^/' Attention 
is also called to the fact that the percentage motor loss increases as the 
load becomes lighter; hence the efficiency-curve drops more rapidly 
than the real efficiency of worm-gearing. 

A parallel series of tests on a pair of worms of the ordinary 
form, and working under the same conditions as the Hindley, gave 
an efficiency 2^ lower at 2000 pounds pressure on both worms, 
which gradually decreased to 10^ lower at 6000 pounds. 

The requirement of the Hindley worm for accurate adjustment 
should be kept in mind when applying it where the service is heavy. 



244 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



SCRBW-GBARS. 

78. Screw-gears seem to be rapidly increasing in favor for use 
where the service is light. Numerous examples are found in the 
feed mechanism of both light and heavy machine tools. In a great 
many cases they are now used where heretofore bevel-gears were 
almost universally applied. The objection against them, that the 
teeth cannot be accurately cut in a milling-machine, but must be 
finished by hand, is not sufficiently strong to prohibit their use. 

79. Strength of screw-gear teeth. — When the velocity ratio is 
great, and the teeth of the gear corresponding to a worm-wheel run 
nearly or quite straight across its face, they may be dealt with 
for strength as those of a spur-gear having pressure applied at one 
point. While the point of application may be at the top of the 
tooth, it should never come at the end of its length across the gear 
face ; therefore it is doubtless safe to consider the effective width 
of thd gear face for resisting fracture at least as great as 1.5 times 
the circular pitch. When the teeth run at a considerable angle 
across the face of the gear, they are somewhat stronger than those 
of a spur-gear of the same diameter and thickness. When the 
angle between the axes of the gears is not so greatly different Irom 
90° as to prevent cutting the teeth of only such a length, in the 
middle of the gear face, as is necessary to allow the engagement of 
other gears, the teeth are much stronger on account of the support- 
ing material at their ends; such a gear corresponds in a way to a 
worm-wheel or one that is shrouded. 

80. Equations for turning force and efficiency of screw-gears. — 
When the axes of the gears are at right angles, the equations for 
worm-gearing apply equally well to screw-gears. If the angle 
between the axes does not differ greatly from 90°, the same equa- 
tions will give results more accurate than our knowledge of the co- 
efficients of friction entering into them. 

The slight practical use to which equations for smaller angles 
between the gear-axes can be put does not seem to warrant their 
presentation. 

It is worth noting, however, that, as the angle between the 
axes decreases, the amount of sliding between the teeth of the 
gears also decreases, and in consequence of this, together with the 



SGBEW-OEABING. 



245 



reduction of thrust on the spiral pinion, the efficiency * increases 
uniformly, reaching that of spur-gears when the axes become 
parallel. 

In the tests already referred to as being made by Mr. Lewis, he 
found the screw-gear efficiencies given in Table XX. The larger 

Table XX. 

EFFICIENCY OF CA8T-IE0K 8CEBW-GEAR8.* 

Efflciency = ratio of power delivered by spur-gear shaft to that applied to 
spiral-pinion shaft ; or, ratio of output to input. 

All spiral pinions 4" pitch diameter ; all spur-wheels 18.63" pitch diameter, 
89 teeth, 1^ inch pitch. Spiral pinion ran in oil-bath. 

Oycloidal teeth, accurately cut. 







Efflciency, 


per cent. 




BeTolutioiis 










of 


1 thread. 


2 threads. 


4 threads. 


6 threads. 


Screw-pinion 
per minute. 


hSll" pitch. 


3.069'' pitch. 


6.828" pitch. 


12.894" pitch. 


«• 61" angular 


18« AV angular 


28^ 31' angular 


45« 44' angular 




pitch. 


pitch. 


pitch. 


pitch. 


8 




60 
68 
66 


'70 
78 
76 


81 


5 




83 


7 


45* 


84.5 


10 


46 


68.5 


78 


86 


16 


48 


72 


80 


87.5 


20 


49 


74 


82 


89 


80 


61 


77.5 


84.6 


91 


40 


58 


80 


86 


92 


60 


65.6 and 55.5 


88 


89 


94 


80 


68 and 57.5 


85 


90 


94.8 


100 


70 and 59 


86 


91 


95.8 


120 


71.5 and 60.5 


87 


92 


96 


150 


78.5 and 62 


88.5 


92.8 


96.5 


200 


75.5 


89 


92.8 


96.6 


275 


77.5 




92 


96.4 









* Readings of efflciency taken from diagrams by Wilfred Lewis in Trans. Amer. Soc. 
Eng., vol. vii., p. 873. 

wheel in each test was a spur-gear with accurately cut cycloidal 
teeth; the screw-pinion was also accurately cut to fit the gear. 
The angle between the gear-axes was 90® — 6^ the pinion-shaft 
being set at the pitch angle 6 with a normal to the axis of the spur- 
gear, in order to obtain accurate intermeshing of the gears. The 
two efficiencies of the single-thread pinion, at speeds from 60 to 
150 revolutions, are due to different conditions of the rubbing sur- 



246 FOiOi, 8TBENGTH, AND PROPORTIONS OF PARTS. 

faces; the higher efficiencies correspond to an improToment of 
these surfaces, secured by running the mechanism for some time 
under a light load. 

81. Coefficient of friction of sorew-gears. — By determining the 
journal friction as accurately as possible, and allowing for it, Mr. 
Lewis found, by calculation, the coefficient of friction ^ for the 
two-, four-, and six-thread pinions given in Table XX. These 
coefficients are giyen in Table XXI, together with their average 

Table XXI. 

COEFPIGIEKT OF PRIGTIOK OF SGBEW-GEABS. 
For Three of the Qears given in Table XX. 







Coefficient of FrieUon, il. 




ReTolations of 










Screw-pinion 
per minute. 










2 threads W AV, 


4 threads 28<' 81', 


6 threads 45« 44^ 


ATerage 




ancrular pitch. 


angular pitch. 


angular pitch. 


Values. ' ": 


8 


.066 


.105 


.004 


.095 


6 


.078. 


.097 


.089 


.088 


10 


.064 


.081 


.076 


.074 


20 


.060 


.065 


.061 


.059 


60 


.086 


.042 


.088 


.088 


100 


.026 


.080 


.024 


.026 


200 


.018 


.026 


.016 


.020 



values for various speeds. In view of the low values of pi for 
square-thread screws, given in Table XIII, when lubricated with 
graphite and oil, it would seem that such a lubricant might be 
excellent for worm- and screw-gears. In fact it is recommended 
by a leading concern building large worm-driven metal-working 
planers. 

Valuable data could doubtless be obtained by a series of experi- 
ments on worm- and screw-gears of different materiab and with 
different lubricants. 



CHAPTEE VI. 

SCREW-FASTENINGS. 

82. In machine oonstraction it is frequently desirable to fasten 
parts together in sach a manner that they can readily be separated 
and put together again without injuring the fastening or destroy- 
ing its usefulness. In order to accomplish this, fastenings haying 
screw-threads cut upon them are used* 




SELLERS UNITED STATES 

STANDARD THREAD 

PlO. 84. 



The Sellers United States Standard thread, shown in sectional 
outline in Fig. 84, has the sides inclined at an angle = 60^ = 2>9. 



247 



248 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

The proportions are obtained by extending the lines of the sides 
nntil they intersect, then catting off one eighth of the distance b 
from the top, and filling in to the same extent at the bottom, of the 
. angles thus formed; this makes the height of the thread = A = 0.65 
of the pitch. 

Table XXII gives the proportions of TJ. S. Standard threads. 

Table XXII. 

IT. S, OB SELLERS SYSTEM OF SCREW-THREADS. 



Diameter of 

Bolt, 

IncheR. 


Tlireodsper 

iBCh. 


Diameter of 
Boot of 
Thread. 
Inches. 


Width of Flat, 
Inches. 


Area of BolU 

body in 
Square Inches. 


Area at Boot 

of Thread in 

Square Inches. 


1/4 


20 


.186 


.0062 


.049 


.027 


6/16 


18 


.240 


.0074 


.077 


.045 


8/8 


16 


.294 


.0078 


.110 


.068 


7/16 


14 


.844 


.0089 


.150 


.098 


1/3 


18 


.400 


.0096 


.196 


.126 


9/16 


12 


.454 


.0104 


.249 


.162 


5/8 


11 


.507 


.0113 


.807 


.202 


8/4 


10 


.620 


.0125 


.442 


.802 


7/8 


9 


.781 


.0188 


.601 


.420 


1 


8 


.887 


.0156 - 


.785 


.550 


1* 
l| 


7 


.940 


.0178 


.994 


.694 


7 


1.065 


.0178 


1.227 


.898 


If 


6 


1.160 


.0208 


1.485 


1.057 


u 


6 


1.284 


.0208 


1.767 


1.295 


If 


6i 


1.889 


.0227 


2.074 


1.515 


If 


5 


1.491 


.0250 


2.405 


1.746 


n 


5 


1.616 . 


.0250 


2.761 


2.051 


2 


4i 


1.712 


.0277 


8.142 


2.802 


^ 


4i 


1.962 


.0277 


8.976 


8.028 


2i 


4 


2.176 . 


.0812 


4.909 


8.719 


2f 


4 


2.426 


.0312 


5.940 


4.620 


8 


H 
8 

8i 


2.629 


.0857 


7.069 


5.428 


8f 


2.879 


.0857 


8.297 


6.510 


8i 


8.100 


.0884 


9.621 


7.548 


8f 


8 


8.317 


.0418 


11.045 


8.641 


4 


8 


8.567 


.0413 


12.566 


9.968 


H 


^ 


8.798 


.0485 


14.186 


11.329 


4i 


U 


4.028 


.0454 


15.904 


12.758 


4f 


2f 


4.256 


.0476 


• 17.721 


14.226 


5 




4.480 


.0500 


19.685 


15.768 


5f 


<u 


4.780 


.0500 


21.648 


17.572 


6i 


2f 


4.958 


.0526 


28.758 


19.267 


5f 


2% 


5.203 


.0526 


25.967 


21.262 


6 


2i 


5.428 


.0555 


28.274 


28.098 



SCRBW-FASTKNINOS. 



848 



The propcnrtMHiB of heads mnd nats, as made by the leading 
manii&ctiiTerB, do not genenlly conform with the XJ. S. Standard, 
and not always with each other. It is advisable to consult the 
catalogae of the mannfactorer whose bolts are to be used. 

Table XXTIT gives the proportions of bolt-heads and nuts 
adopted by some of the leading makers. 

Table XXm.* 



PBOPOBTIONS OF BOLT-HEADS ADOPTED BY DIFFSBEIJ^ 
MANUFACTURERS. 

The dimensions are Uie same whether finished or rough. 





Dimensions of Square and Hexagon Bolt^eads.t 


DSam. 
of 


Hoopes^ 
Townsend. 


Rhode Island 
Tool Co. 


Wm. H. Haskell 
&Co. 


J. H. Sternbergh 
ft Son. 




Width. 


Thick- 
ness. 


Width. 


Thick- 
ness. 


Width. 


Thick- 
ness. 


Width. 


Thlok. 
neis. 


1/4 
5/16 
8/8 
7/16 
1/2 
9/16 
5/8 
3/4 
7/8 
1 

n 
u 


7/16 
1/2 
19/32 
11/16 
8/4 
27/82 
15/16 
IJ 
lA 
1* 
1| 
If 


8/16 
1/4 
9/32 
3/8 
7/16 
1/2 
17/82 
5/8 
8/4 
7/8 

1 

li 


3/8 
15/32 
9/16 
21/32 
8/4 
27/82 
15/16 
li 
lA 
l.i 


8/16 

1/4 
5/16 
8/8 
7/16 
1/2 
17/82 
5/8 
8/4 
7/8 


7/16 

1/2 

5/8 

23/82 

13/16 

15/16 

1 

lA 
If 
lA 


8/16 

1/4 

9/32 

11/82 
8/8 
7/16 
1/2 
6/8 
8/4 

18/16 


7/16 
17/82 

5/8 

28/82 

18/16 

29/82 

1 

lA 

If 

lA 

If 

i« 


8/16 

1/4 

6/16 

8/8 

18/82 

15/82 

1/2 

9/16 

11/16 

8/4 

7/8 

1 





















• Maefiinery, April 1807, paice 248. 

t The width of the nut is the same as that of the head ; its thickness Is oommonlj e<ittal 
to the diameter of the thread. 

Fig. 85.1 shows the thread section recommended by an inter- 
national committee for use on the European continent. It differs 
from the U. S. standard thread only in having clearance between 
the top and bottom of the threads. The clearance space is rounded 
at the bottom and has a depth equal to 1/16 of what would be tlio 



260 FORM, STRENGTH, AXD PROPORTIONS OF PARTS. 

total height of the thread if it ran to sharp angles. A bolt made 
with this thread will fit correctly into a nnt of the IT. S. standard, 
or a nnt of the international standard thread will fit oyer a U. S. 
standard bolt thread, provided of course that each has the same 
number of threads per inch. 




-k. 



WHITWORTH ENGLISH 
STANDARD THREAD 

Fig. 85. 



Fig. 85.1. 



The Whitworth English Standard thread, Fig. 85, has an angle 
of 55^ between the sides, and the top and bottom are rounded for 
a distance equal to one-sixth of the depth of the corresponding 
V thread; this leaves the height of the thread = A = 0.64 of the 
pitch. 

By cutting off the sharp points of the older forms of V threads, 



BCREW-PASTBNINGS. 



261 



80 afi to form the standard threads just mentioned, they become 
less subject to bruising bj accidental blows, and the taps, dies, and 
turning tools used to form them are more durable than those with 
sharp threads. 

The square thread, Fig. 86, is commonly used when there 
is considerable wear due to relative motion of the threaded 
parts engaging together; it has the advantage of presenting 





SQUARE THREAD 

Fig. 86. 



BUTTRESS THREAD 

Fig. 87. 



a surface almost at right angles to the line of pressure, which 
is ordinarily parallel to the axis of the thread. For fastenings 
the pitch is usually twice as great as for the IT. S. standard 
thread. 

The buttress thread, Fig. 87, is sometimes useful when the 
pressure against the thread is all, or nearly all, in one direction. 



263 FORM, STRENGTH, AND PROPORTIONS OP PARTS. 

The surface taking the thrust is made perpendicular to the axis of 
the thread, the other having any convenient angle with it. By 
this means a strong thread is obtained. 

The forms of screw-fastenings are almost infinite in num- 
ber; a few of the more common ones, together with their 
ordinary functions, are given below, chiefly in order that 
they may be referred to with a clear understanding of what 
each is. 

A through bolt consists of a bar of metal, forming the body of 
the bolt, having a thread and nut at one end, and a head, forming 
an integral part with the body, at the other. Ordinarily it is used 
to clamp together parts of machinery by passing through them bo 
that the inner surfaces of the head and nut press against the pieces 
held together by the bolt, as shown in Fig. 88. A washer is fre- 
quently placed between the nut and the clamped piece, to prevent 
marring the latter, and to give a better bearing surface for the nut. 

A cap-screw, Fig. 89, is of the same form as a bolt with the nut 
removed. It is commonly used to fasten parts together by screwing 
into a threaded hole in one of them. The depth of the threaded 
hole depends largely upon the material; in all cases where the parts 
are drawn tightly together, its depth should be as great as the 
diameter of the screw. When the material tapped into is much 
weaker than that of the screw, as when a machine-steel cap-screw 
is screwed into cast iron, the threaded depth of the hole should be 
made considerably greater; twice the diameter of the screw, or 
more, is advisable under such conditions. 

A stud. Fig. 90, consists of a bar threaded at both ends. One 
end is screwed into a tapped hole in one of the pieces to be held 
together, and the other receives a nut which presses against the 
clamped piece. A stud performs the same service as a oap-sorew. 
It is useful where the parts are to be separated frequently. If a 
cap-screw is drawn down hard and removed several times from cast 
iron, there is danger that the threads of the hole will break and 
crumble away, unless the material is close-grained and of good 
quality. By using a stud this is obviated, for it can be screwed 
tightly into the hole and left there. The nut and stud can both be 



SC RE W-FASTENINO& 



263 



made of a strong, durable material, and in case of injarj to the 
threads, can be replaced readily and at small expense. A stud does 




p^ipjaMM^Y:;^..: 





Pig. 88. 

not need as long a thread in the tapped hole as a oapHScrew, for 
the reason that it does not turn when under stress. 

Fig. 91 shows a stad that can be nsed, when only a shallow hole 
can be made in a weak material, by screwing the larger end into the 
threaded hole and putting the nut on the smaller. Fig. 92 shows 
another form that can be used in the same way, or it can be used 



254 FORM, 8TBENOTH, AND PROPORTIONS OF PARTS. 

when the stud is screwed into a thin piece, through which the 
threaded hole passes. For such a purpose the small end is screwed 
in until the shoulder prevents it from entering farther; the nut is, 
of course, placed on the large end in such a case. 




Fig. 89. 

A set-screw is used to. prevent sliding or rbtation between parts 
that fit together; an ordinary application is that of fastening pulleys 
in place upon shafting. A common form of set^screw is that of 
Fig. 93 ; the point shown is round ; other points, largely used, are 
the cup, cone, and pivot, shown in Figs. 94, 95, 96, and 97. 

The method of using a set-screw is shown in Fig. 98, which 
represents a collar or pulley-hub held in place on a shaft by the 
set-screw. When the piece through which the set-screw passes 
must be held very firmly in position, so as to resist forces of con- 



SCREW-FASTENINGS. 



256 



siderable magnitade, tending to displace it, a recess, generally 
conical, is drilled in the shaft to receive the end of the screw; for 
light work, the screw is tightened against the smooth shaft so that 
the point is set into it far enough to hold. 




Fie. 00. 

The eonical and cnp points hold more firmly than the round 
point wben set lightly against the smooth shaft, bat at the same 
time they mar it to a greater extent, thas making it more difficult 
to separate the parts. 

Prof. Gaetano Lanza gives the resalts of a series of tests on the 
holding power of set-screws made on screws having different kinds 
of points.* The experiments were made on a pulley held in place 

♦ Trans. Amer. Soc. Mecli. Eng., vol. x., p. 280. 



256 FORM, STRENGTH, AND PROPORTIONS OP PARTS. 



on a )^^-iiich shaft by two f -inch set-screws having ten threads to 
the inch. The screws were tightened with a force of 75 poands 
applied to the end of a 10-inch wrench. All screws were of wrought 







Pig. 91. 



Pio. 9i, 



Pig. 93. 







Fig. 95. Fig. 96. Fig. 97. 

iron, and only one was case-hardened at the point. The shaft was 
of rather hard steel, and the set-screws made bat little impression 
npon it. Six tests were made on each of the points A, B, and D; 
four on C. A summary of the results is given in Table XXIV. 



SOSEW-FASTSNINOS. 



267 



Table XXIV. 

HOLDIMrG POWER OF SET-SCREWS. 






A 

Flat 

ends 


B 

End 
rounded 




End 
rounded 

toi" 
radius. 


D 
End 

•ndoAse- 
hATdened. 


Lowest holding power, pounda.. 

Highest " 

Meaa " •• «• 


1413 
2294 
2064 


2747 
8079 
2912 


1902 
8079 
2578 


1962 
2958 
2470 




>< 


f^ 










Fio. 98. 
The following remarks are made by Professor Lanza regarding 
the screws: 

'^A. The set-screws were not entirely normal to the shaft; 



258 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

hence they bore less in the earlier trials before they had become 
flattened by wear. 

*'B. The ends of these set-screws, after the first two trials, were 
found to be flattened, the flattened area having a diameter of about 
i of an inch. 

^^G. The ends were found, after the first two trials, to be 
flattened as in B. 

''D. The first test held well because the edges were sharp, then 
the holding power fell off till they had become flattened in a 
manner similar to B, when the holding power increased again.'' 

A pulley that does not fit its shaft is difficult to hold in place by 
a single set-screw, or even by two or. more placed on the same side 
of the shaft. In either case, the screw-points, pressing against the 
shaft on one side, cause contact between the hub and shaft on the 
opposite side, as at a. Fig. 98, while the remaining parts of the 
bore and shaft do not touch. If a torsional moment acts, tending 
to rotate the pulley clockwise around the shaft, the set-screw, hold- 
ing firmly to the shaft, will cause the hub to slide toward the left 
over the shaft at a; a reversal of the turning force will allow the 
surfaces at a to slide back over each other. Numerous repetitions 
of this action will cause the point of the set-screw to enlarge the 
indentation in the shaft, and, finally, the point may wear away, or 
the screw work backward in the hub, so that the parts can rotate 
relatively to each other. Such a repeated application and removal 
of the turning force or resistance is of common occurrence in 
practice. 

There are two methods of obtaining a more reliable set-screw 
fastening than the above, when there is not a close fit between the 
hub and shaft. One is to place two set-screws at an angle of about 
90° with each other, in the same cross-sectional plane of the shaft 
and hub, both screws being radial; the other is to cut away a por- 
tion of the hub, as shown by the dotted line below a, Fig. 98, still 
retaining the single set-screw, or more, as the case may be, on one 
side of the shaft. Both these methods give three-point contact in 
a plane normal to the axis of the shaft, and passing through the axis 
of the screw. This prevents slipping between the hub and shaf t, 
thus giving a firmer fastening. 

In all the above fastenings the form of the head or nut, whether 



80REW-FA8TENINOS. 



designed for a wrencli or a Bcrew-driyer, does not change the name 
of the fastening. Very small screws, with heads slotted for a 
screw-driver, are, however, commonly called machine screws. 

83. Locking devices for nuts and screws. — Since the constant 
jarring to which many kinds of machinery are sabjected, frequently 
causes nuts and screws to work loose, and even to fall from their 
places, the necessity of securing them by some safety locking 
appliance has brought forward numerous devices for this purpose. 
A few will be described. 

One of the most common is the lock- or jam-nut shown in Fig. 
99. The ordinary proportions are given. They are locked together 





Fio. 90. 



Fio. 99.1. 



by screwing them tightly against each other. When they are also 
tightened against the piece held in place, a greater stress is brought 
upon the nut nearest the end of the bolt, which therefore should 
be thicker than is necessary for the one next the clamped piece. 
In practice the thin nut is frequently placed on top, but this is 
wrong. It is probably due to the fact that ordinary wrenches are 
not thin enough to turn the thin nut, when it is under, without 
catching against the top one. 

Fig. 99. 1, in which the small sorew draws the split part together, 
is a useful device. 

Coiled-spring nut-locks, Figs. 100 and 101, are used by placing 
one just under the nut in the place ordinarily occupied by a washer; 
those in the figures are intended for bolts with right-hand threads. 
When the nut is tightened by taming it clockwise, its bearing sur- 



260 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



face slips freely over the nnt-lock, and presses it down against the 
clamped piece; as soon as the nnt is turned backward, however, the 
sharp comers of the lock cut into the surfaces of both the nnt and 
the piece under it, thus preventing the loosening of the former. It 
can easily be seen that such a nut-lock can be used only where the 
nut is not to be removed, or where the cutting of the material by 
the lock is not objectionable. 





Fio. 100. Fig. 101. 

Another device, in which the thread is made so as to lock the 
nut, is shown in Fig. 102. The thread of the bolt is undercut so 





Fig. 102. 



Fig. 108. 



that its working side makes about five degrees less than a right angle 
with the axis of the bolt, and the apex of the thread is cut to a 
knife-edge. The not has a thread cat so that its working side 
makes about five degrees more than a right angle with the axis of 



80 EEW-FA8TENING8. 261 

the nnt. This leaves a cayitj of abont ten degrees between the 
bolt- and nnt-threads, so that when the nnt is tightened the thin 
edges of the bolt-thread are forced ont into the nnt, thns locking 
them together. It is claimed that this deyice is serviceable even 
where the nnt is to be removed as often as a dozen times daring its 
nse. 

Fig. 103 illustrates a bolt with a cold-rolled or '' undercut " 
thread as made by J. H. Sternbergh & Son, of Beading, Penn.* 

84. Strength of screw-bolts. — ^In many cases a bolt must have 
sufScient tensile strength to resist a known or estimated stress, as, 
for example, when used to attach cylinder-heads to engines, pumps, 
etc. ; to fasten together flanged pipes carrying some substance under 
pressure; or to support a suspended load. 

If, as is ordinarily the case, the nut is to be screwed up tightly 
enough to prevent even the slightest separation of the clamped 
parts when under stress, the initial tension in the bolt, due to the 
tightening of the nut, must be at least as great as the service ten- 
sion, in order to prevent lengthening of the bolt under the service 
tension, and the consequent separation of the parts clamped 
together by it. This assumes that the parts are rigid and large in 
comparison with the bolt. 

For example, sappose that, in Fig. 104, the part ^ is to be 
clamped against A by the bolt, so that they will not separate when 
a given load T is suspended from B^ the line of action of T being 
coincident with the axis of the bolt. The initial tension in the bolt 
must therefore be at least as great as T. In order to allow for a 
factor of safety, assume the working strength of the material of the 
bolt as 6000 pounds per square inch. The load 7^ may be taken as 
5000 pounds. The required area of the bolt to resist tension only, 
in a section across the roots of the thread, is therefore 
5000 
-6000 = '-^'^ ^^- ^^- 

The sectional area at the roots of a U. S. Standard thread, on 
a bolt having a body 1^ inches in diameter, is 0.89 of a square inch, 

* Other designs of nut-locks are illastrated in Rose's " Modem Machine-shop 
Practice," vol. i., pp. 118-121; Unwin's "Elements of Machine Design," Part 
I., pp. 150-168; Reuleaux's "Constructor," p. 66; Klein's "Elements of Ma- 
chine Design," p. 9 and Plate II. 



262 FORM, STRENGTH, AND PROPORTIONS OP PARTS. 



the root diameter being 1.065 inchee. Since this is the nearest 
standard size having an area as large as that required, it would be 
Qsed in practice. 

The taming moment necessary to give the tension of 5000 



3} 



B 



^i. 



J 




pounds can be found by equation (71)* or those following it, in 

. . 1 ^ 1.25 + 1.065 ^ -^. , 
which, for this case: p = \ inch; d = '-^ = 1.16 mches; 

D = 1.7 inches, about; >5 = SO"* ; and, since the coefllcient of friction 



* Page ltt4, notation p. 150. 



SCREW-FASTENINGS. 263 

is uncertain and should be taken large enough to cover poor lubri« 
cation and rough bearing-surfaces, it is admissible to take pi = 0.15, 
and/i' = 0.12. 

Substituting in equation (72), 

^ + .15 XTTX 1.16X1.15 

Fl = 5000^ i^ + 5000 x .12 X il^ 

;r 1.16 -,15 X^X .866 

= 650 + 510 = 1160 inch-pounds. 

Assuming I = 16rf = 16 x 1.25 = 20 inches, this being about 
the value used in practice for a solid wrench, gives 

• 1160 ^^ , 

F = -^-p = 58 pounds, 

which is 1 pound of torsional force for every 5000 -5- 58 = 86 
pounds tension in the bolt. 

The value of Fl can also be obtained by assuming an efficiency 
E for the bolt, and substituting in equation (59), which can be 
applied to angular- as well as square-thread screws. Taking 
^ = 0.1, and substituting (see § 79), 

^^ 5000x1/7 ^, 10X5000 _.^. , , 

^•^ = -:pp-^7r> "^ ^? = -^^^^ = 1140 inch-pounds. 

The maximum tensile and shearing stresses can be found by 
equations (67) and (68); the values of t and « to be used are 
/= T-^ A= T -^ nr^^ and 8 = Fl-r- i7tr^\ These maximum 
stresses do not need consideration, however, unless the nut is 
tightened while under the stress of the load. 

If the material through which the bolt passes in Fig. 104 were 
totally without elasticity, and the bolt tightened to produce a ten- 
sion 7 in it, then, for any suspended load not exceeding T^ the 
tension in the bolt will always be equal to T; and when the load 
equals T there would be no pressure between the surfaces clamped 
together, for the bolt would elongate enough to relieve the pressure. 

If a spring were interposed between the surfaces before tighten- 
ing the bolt, and put in compression, without clamping it solid, 
by screwing down the nut till a tension T is produced, then a 
suspended load would make the tension in the bolt equal to 7 plus 



264 FOBM, 8TBENGTH, AND PEOPOETIONS OF PARTS. 

the load. This assumes that the elongation in the bolt is so slight 
as to be inappreciable compared with the capacity of the spring for 
elongation. 

Since all material is elastic, there is always something of the 
spring action on the bolt. This, of course, is exceedingly slight 
when heavy parts of rigid material are bolted together; when short 
bolts are used to clamp together pieces separated by a springy sub- 
stance, such as rubber packing, the spring action may be great 
enough to require consideration. 

Experiments show that grooves used for standard bolt-threada 
very materially increase the tensile strength per unit area, although 
the total strength may be made less than that of the original bar. 
This is due to the fact that fracture is made to occur at the groove, 
which, on account of having a large amount of resisting material on 
each side, requires a greater force to fracture it than a bar of the 
same diameter as a section at the bottom of the groove. 

The effect of cutting a V thread upon a bolt is nearly the same 
as for a single circumferential groove of the same form as the thread- 
groove. It has been found by experimental investigation that, on 
account of this strengthening effect of the thread, with fairly good 
rubbing surfaces reasonably well lubricated, the axial tensile stress 
per square inch which will cause rupture in a U. S. Standard screw- 
bolt, while tightening the nut, is practically the same as the break- 
ing strength per square inch of the body of the bolt.* This means 
that the twisting effect of the nut may be neglected. 

The sectional area of U. S. Standard screw-bolts at the bottom of 
the thread is roughly 0.7 of the area of the bar on which the thread 
is cut, for diameters up to 2 inches. It therefore seems safe to say 
that the axial tensile stress which will rupture such a screw-bolt, 
when tightening the nut, is 0.7 as great as will fracture the body 
of the bolt, it being assumed that the body and the top of the 
thread are of the same diameter. 

Major Wm. K. King found that by doubling the number of 
threads per inch on a screw-bolt, the total tensile strength was 
mcreased about 20^, and the resilience or shock -resisting power to 
a much greater extent. The gain was somewhat greater when the 

* Zeitscbrift des Vereines deutscber Ingenieare, April 27, 1896, p. 005. 



la 


18 


1.21 


1.23 


.06 


.08 


.0726 


.0984 



SCBEW-FA8TENIN08. 265 

nnmber of threads per inch was tripled.* The average resnlts of 
several tests were as follows: 

Threads per inch 6 

Belative tensile strength. ... 1. 

Elongation 025 

Belative work or resilience. . .025 

The stress was applied at the nnt and head. Stripping of the 
thread did not occur in any of the experiments, but the reduction 
of the diameter of the screw by elongation was so great as to let a 
portion of the threads of the nut and bolt slip past each other. 

It has been shown by numerous tests upon screw-bolts having 
the thread made by the cold-rolling process, which forces the metal 
up so that the top of the thread is somewhat larger in diameter than 
the body of the bolt, that, when subjected to tensile stress applied 
to the working faces of the head and nut, the bolt invariably frac- 
tures in the body.f These tests show very clearly the strengthening 
effects of the thread, for the sectional diameter across the bottom 
of the thread is, of course, considerably less than that of the body 
of the bolt. 

85. Endurance of screw-bolts. — Repeated stresses in a screw- 
bolt, such as occur in the fastenings of a trip-hammer, steam or 
pneumatic rock-drill, connecting-rod ends of pumps, engines, etc., 
frequently cause bolts to fracture across the threads between the 
nut and body. This is caused by the slight temporary elongation 
of the bolt every time the shock or stress occurs. Such an elonga- 
tion may occur without allowing the surface of the parts held 
together to separate even to the slightest extent; for the elasticity 
of any two parts clamped together with a bolt will, when an addi- 
tional stress is applied to separate them, especially if it is a shock, 
cause them to increase their dimensions in the direction of the 
length of the bolt, and thus elongate it. 

By reducing the sectional area of the body of the bolt so that it 
is not greater than that across the bottom of the thread, the bolt is 
made more durable. This is due to the fact that, by reducing the 

* Trans. Amer. Inst. Mining. Engrs., 1885, p. 90. 
t Catalogae of J. H. Sterabergh & Co. 



266 FORM, STEENGTH, AND PROPORTIONS OF PARTS. 



r 



H 



Bectional area of the body, the bolt elongates more readily, the 
slight elongation that occurs with each shock distributing itself 
nearly uniformly throughout the length of the bolt, thus reducing 
the actual stress which occurs in it with each shock. When the 
body of the bolt is of the same diameter as the top of the thread, 
the slight elongation which occurs with each repetition of stress is 
largely localized at the bottom of the thread between the nut and 




Fig. 105. 



L_ 




Fig. 106. 

body of the bolt, and causes the metal to gradually give way by 
fatigue. 

The reduction of the sectional area of the body of a bolt may be 
accomplished either by reducing its diameter, as in Fig. 105, or by 
drilling a hole in the centre, as in Fig. 106. The hollow bolt is the 
stronger torsionally, and will therefore withstand a greater tensile 
stress while the nut is being screwed on. The increased endurance 
of bolts so made has been demonstrated in practice. 



CHAPTER VII. 

MACHINE KEYS. PINS, FORCED AND SHRINKAGE PITS. 

MACHINE KEYS AND PINS. 

86. The principal f anction of machine keys is to prevent rotary 
motion of one part aboat another, as of a pnlley about a shaft on 
which it fits; less frequently they are used to prevent lateral motion 
also, when the tendency to such motion is comparatively small. 
They are in a general way divided into three classes, commonly 
oalled fiat, square, and feather or sliding keys. 

There are no accepted standards of keys arid key-way propor- 
tions. Tables XXV, XXVI, and XXVII, by John Eichards,* give 
the average proportions of general practice, however. 

The flat key, Pig. 107, is most suitable for heavy machinery, 
such as is used for mill-work, when both the relative rotation of the 
parts and the lateral motion incidental to the vibration of the machi- 
nery, and, in some cases, the weight of the parts, are to be resisted. 
A fiat key should completely fill the key- way; the pressure against 
the sides should be greaterthan at the top and bottom, where a light 
pressure is all that is necessary. A heavy pressure against the top 





Fig. 107. Fig. 108. 

and bottom of the grooves has a tendency to spring the parts out at 
true and fracture the hub. The top of a key is the side remaining 

* Richards' "Manual of Machine Constraction." CcuH&r'i Magazine, April, 
1898, p. 416. 

267 



268 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



completely exposed after the key has been put in positioD in one of 
the parts to be keyed. 

"Square" keys, Fig. 108, which are generally only approxi- 
mately square in section, are generally used for the lighter classes 
of work, and especially for machine-tool construction. They should 
fit tightly on the sides, only merely touching at the top and bottom 
so as to prevent their tipping over under a heavy load, or not 
touching at all at the top. 

Table XXV.* 

DIMENSIONS OF FLAT MACHINE KEYS, INCHES. 



8 

1* 



Diam. ofsUaft. 


1 


n 


u 


11 


2 


34 


8 


1 ■ 


4 5 


6 


7 


Breadth of keys 


i 


A 


i 


t 


i 


« 


f 


1 H 


i! 


1 


Depth of keys.. 


A 


tV 


i 


h 


f 


tV 


i 



Table XXVI. * 

DIUENSIOKS OF APPROXIUATELY SQUARE MACHIITE KEYS, IKCHES. 



Diameter of shaft 
Breadth of keys. . 
Depth of keys . . . 



A 
A 



A 

i 



H If 



Table XXVIL* 
dimensions of feather or sliding machine keys, inches. 



Diameter of shaft 
Breadth of keys. . 
Depth of keys. . . 



u 


u 


If 


2 


3i 


2i 


8 


Zi i 


i 

f 


i 

f 


s 


t 


i 
i 


1 


f 


*; 1 



1 



* Taken from Casaier'a Magaxine^ April, 1808, p. 416. 

If only a small portion of the power that the shaft is able to 
transmit is taken off through the key, the latter may be made 
smaller than given in the tables. The side pressure on the key is 
practically inversely as the diameter of the shaft. 

A feather or sliding key is adapted to a service requiring lateral 
motion of the parts over each other, but not relative rotation. On 
account of the wear on the sides, due to the lateral motion, and in 



MACHINE KEYS AND PINS. 



269 



order to give the key a firm hold upon the part to which it is 
attached) its radial height is greater than for '^square" keys. 
Such a key is fastened to one of the parts, either tightly or loosely, 
as is most suitable for other requirements, and moves laterally with 
regard to the other. Probably the simplest method of fastening 
such a key to a shaft is to dovetail it slightly on the sides and ends 
near the bottom, then force it into place and rivet the edge of the 
key-way down against it with a key-set. Another method of 
attaching a short feather to a shaft is shown in Fig. 109. The key 




Pig. 109. 



is made with a couple of round lugs or pins which pass through 
holes drilled in the shaft and are riveted on the side opposite the 
key. The same device is applied to a feather attached to a hub in 



Y^^ 



y^ 



Ji l , -,;:.;:.; 







Pig. 110. 

Fig. 110. Another method is shown in Fig. Ill; the feather may 
be made a loose fit in this form, and can be removed by slipping the 
hub off the end of the shaft. 

A method of using a square key which is not very generally 
applied in practice, but which appears to possess an advantage in 
the ease with which it can be fitted, is shown in Pig. 112; the key 



270 FORM, STRENGTH, AND PROPORTIONS OF FARTS. 



is square, and is placed so that one diagonal is radial. A heavy 
load would exert a strong bursting pressure on the hub. For this 
reason the key is hardly suitable for heavy service. 



r-\: m^oMkmS^^^^^^^y 



n 







Fio. lit 






Fig. 112. 



Fig. 118. 



Fig. 114. 



A oylindrical pin can be used, as in Fig. 113, when the shaft 
and hub are of the same material, or near enough alike to make it 
practicable to drill a hole for the pin as shown. Such a key or pin 
is hardly suitable for heavy machinery. 

A taper pin, Fig. 114, can be used to advantage when there is 
considerable end pressure to be resisted, as well as turning force. In 
practice the smallest size of such a pin is ordinarily determined by 
the smallest taper reamer that is long enough to ream out the hole. 
Morse standard taper pins are generally used. The torsional 
moment, which a pin that is not large enough to weaken the shaft 
unduly will resist, is much less than for a key of the proportions 
given in the tables, if the length of the key is as great ^ the 
diameter of the shaft. 



MACHINE KEYS AND PINS. 271 

87. Boiler keys are used to some extent for fastening small 
pulleys, etc., to shafting. The hub of the pulley is bored a short 
distance at each end to fit the shaft; the centre is bored eccentric 
with the ends, and larger in diameter by something more than the 
diameter of the roller key to be used. A hardened cylindrical key 
of about the same length as the larger part of the bore is placed in 
it, and the shaft slipped through the hub. A slight rotation of the 
latter causes the key to bind between it and the shaft so as to pre- 
vent further rotation in that direction; a slight rotation in the 
opposite direction loosens the connection. It is self-evident that 
this device can be used only when the tendency to turn the parts 
over each other is always in the same direction. 

88. Eccentric keys or fastenings of the form shown in Fig. 115, 




Fig. 115. 
known as Blanton Patent Fastenings, are used to ^ood advantage in 
certain classes of machinery. The surface of the shaft is turned into 
a series of corrugations as shown, and the hub is bored to the same 
form, being made large enough to slip along the shaft freely when the 
parts are put in the loose position. A slight angular rotation of one 
causes it to lock with the other and drive it. By turning them in 
the opposite direction relatively to each other they are loosened. 

This fastening has been found especially applicable to the lift- 
ing-cams of ore stamp-mills, largely on account of the ease with 
which the cams can be removed and new ones substituted. When 



272 tOKM, STKENGTH, AND PROPORTIONS OF PARTS. 

there is danger that the fastening may become loose on account 
of the momentum of the parts, slight variations of speed, reversals 
of motion, etc., a key is applied, as in Fig. 116, to prevent the 



1 


w 


¥ 


T 


1 


\ 


l- 


V 


^ 



/ 



r" 



Fig. 116. 




Fig. 117. 



loosening. The fastening, although thas modified, is still adapted 
to driving in only one direction. 

The Blanton fastening is adapted to round shafts as shown in 
Fig. 117. The turning moment is largely resisted by the friction 



SHKINKAGE AND FORCED FITS. 273 

between the shaft and corrugated sleeve, thns relieving the rectan- 
gular key of much of the pressure that would come upon it if pre- 
venting rotation between two round parts.* 

SHRINKAGE AND FORCED FITS. 

89. Shrinkage and forced fits are adopted frequently when it is 
desired to have parts fit together very tightly; the parts are almost 
invariably either cylindrical or slightly conical on the surfaces in 
contact. For a shrinkage fit, the outer member, of tvro parts that 
are to be fastened together, is finished to a diameter slightly smaller 
than the inner. It is then heated so as to expand it enough to 
pass over the inner member, put into place and cooled. The con- 
traction of cooling causes it to grip the inner part firmly. 

For a forced fit the parts are prepared in the same manner as 
for shrinkage fits, but are forced together cold instead of the outer 
one being expanded by heat and then shrunk into place. 

90. Tension in and pressure against a thin ring fitted by 
flhrinking or forcing. — The tension in a ring that is thin radially 
in comparison with its diameter, as well as the pressure betvreen the 
ring and body it encircles, can be calculated if the modulus of 
elasticity E of the material is known. This assumes that the ring 
is either forced into place, or is heated to a uniform temperature 
throughout and all parts cooled at the same rate. While sueh heat- 
ing and cooling can probably never be attained in practice, they can 
be nearly enough approximated to warrant the above assumption. 
If cooled unevenly, the greatest tension will be at the place cooled 
last. 

The following notation will be used for shrinkage and forced 
fits: 

A = sectional area of ring, square inches; 
D = diameter of ring before heating, inches; 
2), = diameter of ring after putting into place, inches; 
E = tensile modulus ol elasticity of material of ring, pounds per 
square inch ; 

* The Blanton Patent Fastening is used in the United States by Fraser & 
Chalmers of Chicago, who control its use on stamp-mills. (Statement from 
pamphlet issued by the Blanton Patent Syndicate, Ltd., London.) 



274 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



i^= force required to dip the ring from place, pounds; 

P = total pressure betweea surfaces of ring and inner part| 

pounds; 
7'= total tension in ring, pounds; 
h = radial thickness of ring, inches; 
p = unit pressure between surfaces of ring and inner member, 

pounds per square inch; 
r = radius of ring, inches; 
t = unit tension in ring, pounds per square inch; 
to = width of bearing surface of ring, measured parallel to axis of 

bore, inches; 
/u = coefficient of friction between surfaces bearing together. 

Remembering that E = (stress in pounds per square inch) -r- 
(elongation per inch of length), the two following equations may be 
written: 



t = B 



D-D 



T=At = AE 



D ' 
D • 



(89) 
(90) 



The equation for the unit pressure p between the surfaces may 
be obtained by assuming that the ring is cut in two on a diametral 

c 




Fig. 118. 



plane, and one half considered as a free body with the forces neces* 
sary to hold it in equilibrium acting upon it, as shown in Fig. 118. 



SHRINKAGE AND FORCED FITS. 275 

These forces consist, first, of the two tensions 7 and T which the 
other half of the ring exerted when in place, and, second, of the 
elementary forces normal to the inner surface of the ring and 
uniformly distributed over it. T and T are. both normal to the 
plane of the section of the ring. The forces acting normal to the 
inner cylindrical surface of the ring may be taken as having a unit 
value of p pounds per square inch. 

Take an elementary portion ds of the circumference of the ring. 
The corresponding inner surface of the ring is wds. The total 
pressure acting on this elementary area is pwds. This pressure may 
be resolved into two components, parallel and perpendicular to T 
and T. The one which is normal to T and T is annulled by the 
equal and opposite component of the pressure acting on an equal 
area symmetrically situated with regard to the median line 0(7. 
This component may therefore be neglected. The other component, 
parallel to T'and 7, and acting in the opposite direction from them, 
must be resisted by a portion of the tensile forces ^Tand T acting at 
the ends of the half-ring. The value of this component is 
pw{d8) cos 6^, in which 6 is the angle between the median line 00 
and a radial line drawn to the elementary area w{d8). The sum of 
the components parallel to T and T for all the elementary areas of 
the half -ring must equal 27^, since T and T are the forces which 
hold these components in equilibrium. Therefore 



fpwids) coaO = 2T 



Since r is the radius of the ring, and dO is the angle between 
the two radial lines drawn to the ends of the elementary length ds 
of the circumference, ds = rdO. The last equation may therefore 
be written 



'/: 



pwr I cos edd = 2T, 

which gives 

2pwr = 2T. 



276 FORM, STRENGTH, AND PROPORTIONS OF PA.RTS. 

Transposing and sabstitnting the valae of T^ as given in equa- 
tion (90), gives 

p = l-=.^=AE^^.. . . . (91) 

^ wr wr Dwr ^ ' 

The total pressure P between the surfaces bearing together is 

TV/) 
P = pwnD = ^^-^ = 2T7r = ^Atn. . (92) 

The force F necessary to slip the ring laterally along the surface 
against which it presses is 

F ^ piP =^ ^}xTn = ^jxAtn (93) 

Example. — If a steel ring 4 inches wide, 0.7 of an inch thick 
radially, and 60 inches in diameter before putting in place, is shrunk 
or forced on another part which increases its diameter .06 of an 
inch, the tension per square inch is, by equation (89), putting 
^=30,000,000, 

t = 30,000,000'— = 30000 lbs. per sq. in. 

The total tension is, by equation (90), 

7= ^^ = (4 X 0.7)30000 = 84000 lbs. 

The pressure per square inch between the cylindrical surfaces 
is, by equation (91), 

T 84000 ^^^,, 
^ = — = . ^^ ^^ = 700 lbs. per sq. m. 
-^ wr 4 X 30 r n 

The total pressure between the bearing surfaces, determined by 
equation (92), is 

P = 27V = 2 X 84000;r = 527780 pounds. 

The force required to slip the ring laterally, taking /i = .15, is, 
by equation (93), 

J' = .15 X 527780 = 79170 pounds 
= 19.6 tons (about). 

91. Shrinkage and forced fits for thiok rings and heavy parts. 
—When the ring or outer member is very thick or heavy, the 



SHRINKAGE AND FORCED FITS. 277 

material in it is not sabjected to even an approximately uniform 
stress when two parts are shrank or forced together. The layers 
near the smaller diameter of the outer member are put into greater 
tension than those more remote, on account of the elasticity of the 
material and the supporting action of each successive annular layer 
against the one inside of it. On account of this nneven distribu- 
tion of stress, it is not probable that any mathematical formula can 
be deduced that is of suflBcient practical value for very thick pieces 
to warrant its adoption. This applies to such a member as a crank 
shrunk or forced on its shaft, or having a pin forced or shrunk 
into it. 

Again, when a heavy part is shrunk or forced over another, the 
great pressure exerted on the inner piece decreases the diameter of 
the latter on account of its elasticity. This has been perceptibly 
shown when the steel tires are shrunk on locomotive drive-wheel 
centres. The increase of diameter of the tire, due to putting it in 
place, is less than the difference between the bore of the tire and 
the outer diameter of the wheel centre before putting the tire on. 
The extent of the reduction of the inner part depends upon the 
form and strength of the parts. It is therefore a quantity that 
cannot well be introduced into a formula, except empirically. 

92. Allowance for shrinkage and forced fits. — For locomotive 
drive-wheels and steel-tired car-wheels, the bore of the tire is quite 
commonly made .001 of the diameter smaller than the wheel centre. 
The Midvale Steel Co. makes a greater difference of diameter than 
this, their rnle being to bore the tire 7^ of the nominal diameter 
smaller than the wheel centre. 

The collection of data. Tables XXVIII, XXIX, XXX, and 
XXXI,* shows the practice of several machine-builders in allowing 
for shrinkage and forced fits. The force necessary to press the 
parts together is given in nearly all cases. 

It is the practice of some leading concerns to slightly taper the 
parts that are forced together. This obviates the necessity of slowly 
forcing the parts together throughout the entire length of the bore, 
and reduces the liability to abrasion and cutting between the sur- 
faces. The parts are readily separated, since they become loose 

* Taken from Machinery, May, 1897. 



'J78 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

wbeu given a small relative lateral movement. The difficnUy of 
securing tapers acoarately the same on both parts is, of coarse, 
detrimental to their general application. 

Table XXXII indicates the practice of the Chicago, Milwaakee, 
and St. Paul Eailway in the Milwaukee shops, for both taper and 
cylindrical fits. 

Table XXVIII.* 

EXTRACT FROM LAKE & BODLEY CO.'S RECORD BOOK OF 
FORCED FITS. 



No. 


1" 


f 


ll 

i' 

1.8767 


^i 


I 
ll 

< 


^ 1 

§11 

< 


111 
ill 


1 
g . 


i 

10 


I' 


1 


1.8798 


6.125 


.0031 


.oori 


86 


16.7 


2 


20 


2 


1.8819 


6.125 


1.877 


.0042 


.0022 


86 


16.7 


2 


15 


28 


8 


1.8774 


4.375 


1.8764 


.001 


.00062 


24.4 


13.7 


.5 


1 


1 


4 


2.7455 


4.6 


2.7387 


.0068 


.00247 


88.7 


26.5 


8 


12 


25 


5 


2.7465 


4.5 


2.7487 


.0028 


.001 


38.7 


26.5 


5 


12 


23 


6 


8.261 


5 


3.2542 


.0068 


.0021 


51 


41.5 


5 


20 


45 


7 


3.2625 


5 


3.2555 


.007 


.002 


51 


41.5 


6 


15 


30 


8 


3.267 


5 


3.261 


.006 1.0018 


51 


41.5 


5 


15 


20 


9 


4.2505 


6. 


4.2402 


.0103 .0024 


79.8 


85.1 


5 


22 


44 


10 


4.2388 


6.625 


4.2478 


.0091 .0021 


78.1 


98.4 


12 


30 


60 


til 


4.2303 


6.5 


4. 2224 


.0079 


.0019 


95.8 


91. 


10 


60 


125 


12 


5.9343 


4.0625 


5.9216 


.0127 


.0022 


75.7 


112.2 


6 


16 


25 


13 


5.9381 


4. 


5.9252 


.0129 


.0022 


74.4 


110.4 


3 


18 


85 


14 


5.9294 


4.125 


5.9194 


.01 


.0017 


76.7 


113.8 


5 


15 


25 


15 


6.8829 


5.125 


6.8697 


,0132 


.002 


110.7 


190.1 


8 


20 


42 


16 


6.889 


5. 


6.8785 


.0105 


.0015 


108. 


18 .9 


5 


22 


45 


17 


6.8692 


4.875 


6.855 


.0142 


.0021 


104.8 


180.4 


5 


85 


65 


18 


7.8884 


5.5 


7.873 


.0154 


.002 


185.9 


267.8 


5 


32 


64 


19 


7.8715 


6.5 


7.8575 


.014 


.0018 


160.5 


815.9 


5 


25 


50 


20 


7.862 


5.625 


7.846 


.016 


.002 


138.2 


272.8 


8 


40 


80 


21 


8.924 


6.125 


8.905 


.019 


.0021 


170.8 


378.9 


20 


45 


68 


22 


8.9 


6.75 


8.8848 


.0152 


.0017 


188.4 


419.9 


5 


47 


96 


23 


8.878 


6.5 


8.8669 


.0112 


.0013 


180.7 


401. 


10 


45 


92 



• Taken from Machinery, vol. iii., No. 9, May, 1897. t No. 11 was a caat^teel crank-dtelu 



SHRINKAGE AND FORCED FITS. 



279 



Table XXIX.* 

DATA FURNISHED BY C. O. HEGGEM, 8UPT. RUSSELL A CO., 
MASSILLON, OHIO. 

Allowance is for both press and shrinkage fits for steel cranks. 

FOR PREBB FITB— CABT-IRON CRAITXB. 

Diameter of hole 4 to 6 inches. Allow .0090 inch 





6 ** 7J 


If 


ft 


.0060 ** 




7i " 9 


n 


It 


.0055 " 




10 •' 12 


« 


«i 


.0050 " 




12 •• 16 


'* 


«< 


.0040 *• 




16 - 18 


(f 


«f 


.0080 '* 


FOR SHRINK FITB— -CAST-IRON 


CRANKS 




meter of hole 


4 to 5 


inches. 


Allow 


.0045 inch 




5 " n 




*t 


.0030 *' 




7i '* 9 




t€ 


.0027 •* 




10 '* 12 




<< 


.0026 •• 




12 *' 16 




f< 


.0020 " 




16 " 18 




<« 


.0015 - 



Table XXX.* 

FROM DATA FURNISHED BY H. BOLLINOKX, BRUSSELS, BELGIUM. 

CRANK-FINS. 



Diameter, 
inches. 


Y^^^: 


Total 

Allowance, 

inches. 


Allowance 

per inch, 

inches. 

.0018 
.0018 
.0018 


Total 
Pressure, 
pounds. 


2.2 

2.79 

8.145 


2.76 
3.78 
4.82 


.0089 
.0049 
.0058 


43800 
49500 
55500 



LEYBK-PINS. 



.865 


1.57 


.0028 


.0027 


6800 


1.1 


1.77 


.0025 


.0028 


6800 


1.18 


1.965 


.0027 


.0028 


6800 


1.875 


2.86 


.0081 


.0023 


9060 



CYLINDER CASES. 



11.4 
17.7 



.0098 
.0098 



.00086 
.00055 



87100 
87100 



1.49 


4.82 


.0059 


.004 


37100 


1.46 


4.82 


.0059 


.004 


49500 


1.77 


4.76 


.0070 


.004 


55500 


2.24 


11.18 


.0039 


.0017 


61800 


2.68 


11.18 


.0039 


.0015 


74000 


2.95 


18,12 


.0049 


.0017 


113200 



* Taken from Machinery^ vol. in., No. 9, May, 1807. 



280 FORM, STRENGTH, AND PROPORTIONS OB' PARTS. 






S5 



! 



0^ a 




i illil 



S 00' > BO > C^" > 

Isgsgss 







B 
O. , 

JM2 



I 



£ 

9 



,0 A 



^ ■So && «, 






OOaO Mi-i A 






SiS Si^3 



22' 



""S^g 



5? 
2| 

i 

J3® . 



li 



S « i 



8§8 

8§iiiii222! 






si §§l 



8§§§§8iii 8i88^ 

fiiii iiii 



• o 

2S 



8' 



II 

« e 



Issf 

Ma 

si 



«i« 



ssss 



i§ 



So© 









83_ 

oco 



oo22w^^^2522 



S5 



^ o»«e^to« 



Ba22i222222s2o 






c8 






6 



J 



£ 
d 



JS 






a 

i 

5 _ 



o o 

2X 






2 



3 



I 

I 
i 

I 

§ 



SHRINKAGE AND FORCED FITS. 281 

Table XXXIL 

forced fits. practice of chicago, milwaukee, and 
st. paul railway. 



Parallel-rod 
pin * 



crank- 



Maiu crank-pins*. . 

Engine driving-axles 

Engine iruck-axle. 
Car-axles 



Diameter 

at larfi^e eod. 

Inches. 



Approximate 
leoKth, inches. 



} 4H 

111 

I ?! 

HI 

5t 



6 
6 
6 

81 
7 
7 to 9 J 



Taper s= 

variation of 

diameter per 

fool of length. 



1/32 

1/32 
1/82 
1/32 

none 

none 
1/16 



Pressure to force 

parts together, 

tons. 



20 

25 

80 

85 

60to80;t 

80 preferred 

I 30 to 40; 

25 will bold. 

40t 



4' End riveted over after forcing into place. 

t If only 60 tons is used, more care is taken in fitting key. 

X Only slight variations from this amount allowed. 

In order to bring the parts into alignment for forcing them 
together, two diameters are sometimes used, one but very slightly 
smaller than the other. Each size extends half the length of the 
surfaces that bear together. 



CHAPTER Vni. 

SHAFTING, AND POSITIVE SHAFT-COUPLINGS. 

93. Notation. — ^The following notation is used in the formulas 
for the strength and deflection of shafting: 

D = diameter of round shaft, or width of side of square shaft, 

inches; 
E^ = shearing modulus of elasticity of material of shaft, used 

for torsion, pounds per square inch ; 
Et = tensile modulus of elasticity of material of shaft, pounds 
per square inch ; 
/ = moment of inertia of cross-section of shaft about a gravity 

axis in the plane of the section, biquadratic inches; 
J = polar moment of inertia of cross-section of shaft, biquadratic 

inches; 
L = length of shaft, inches; 
Mj, = bending moment, inch-pounds; 
Mf = twisting moment, inch-pounds; 
{Mi,)i = ideal bending moment which would induce the same fibre- 
stress as the combined bending and twisting moments ; 
{Mt)i = ideal twisting moment which would induce the same fibre- 
stress as the combined bending and twisting moments; 
P = tiiming force, pounds; 

St = section modulus of shaft for bending = / -t- c; 
St = section modulus of shaft for torsion = J -z. c; 
W= bending force., pounds; 

c = distance from centre of shaft to most remote part of sec- 
tion, inches; 
d = diameter of bore in hollow shaft, inches; 
/ = fibre-stress in material of shaft due to bending, pounds per 

square inch ; 
I = length of leyer-arm of taming force, inches; 

282 



JlXLESj 8HAFTIKG9 AND POSITIVE SHAFT OOUPUNOS. 283 

n = the ratio d -h D; 

8 = shearing-stress in shaft dne to taming force, pounds per 

square inch; 
w = weight of shaft per inch of length, pounds; 
= angular deflection or twist of shaft in degrees = 57.29578 

X (angular deflection in radians). 



BOUKD SHAPTIKG. 

94. Torsional strength of round shafts. — ^When a shaft is used 
simply for transmitting power from one point to another, there 
being no intermediate devices, such as gears or pulleys and belts, 
for receiving or delivering power, the torsional strength and angular 
deflection of the shaft are all that ordinarily need consideration. 
This assumes that the bearings of a horizontal shaft are near enough 
together to prevent excessive sagging or bending between them, 
and that a vertical shaft is supported by thrust-bearings so as to 
prevent excessive local end thrust or buckling. The angular de- 
flection or stiffness needs especial attention when steady running 
under a variable load is important. 

The relation between the turning moment Jiff and shearing- 
stress « in a shaft is expressed by the equation 

Mt=:Pl = 8^. (94) 

For a solid round shaft J = -^ and = —. 

Substituting these values in the last equation, it becomes: 
For solid round shafts 



if, = P/=.1963«Z>*, (95) 



whence 



l^=.^^jp. = 7^21^1 



(96) 



284 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

For a hollow round shaft J = — ^^ — -^ -^ and c = -5 as be- 

fore. Equation (94) becomes, by substituting these values in it 
and putting d = 71D, for hollow round shafts 

Mt = Pl = .19638^-^ = .1963sD\l - n*). . (97) 

Transposing puts the equation into a convenient form to solve 
for D when the ratio n of the bore to the diameter is assumed. The 
equation thus becomes: For hollow round shafts 



The slight weakening effect of removing the centre of a shaft 
is worthy of note. A shaft having an axial hole of a diameter 
d = .3Z) through its centre has 1 — (.3)* = .9919 of the torsional 
strength of a solid shaft of the same material; if d = AD, the 
strengths are as .9744 to 1; for d = .62>, the ratio is .9375 to 1; 
and for d = .6D the ratio is .87 to I. 

The torsional strength of a hollow shaft having a bore half as 
large in diameter as the shaft is 1.443 times that of a solid shaft of 
an equal sectional area (i.e., the same weight per linear foot) and 
of the same material. 

The above statements regarding the relative strength of solid 
and hollow shafting are based upon the assumption that the mate- 
rial in each is of the same strength. In shafts of the same sec- 
tional area, however, the material of the hollow-forged shaft can be 
made stronger than that of the solid one, because it can be worked 
more thoroughly under the hammer or forging-press. The ordi- 
nary method of making a hollow forging is to cast the ingot solid 
and. bore out the centre. This takes out the poorest material, 
which is always at the centre. Practically the same result is ob- 
tained when the piece is forged solid and the centre bored out 
afterwards. For forgings of very large diameter, the method of 
boring before forging is undoubtedly preferable. 

In modem practice a working shearing-strength as high as 



AXLES, SHAFTING, AND POSITIVE SHAFT COUPLINGS. 286 

12000 pounds per square inoh is used for hollow-forged steel shafts 
for the screw-propellers of large vessels, etc. 

Example. — The turning force F^ applied at a distance / = 40 
inches from the centre of a shaft 15 inches in diameter when work- 
ing at a maximum shearing fibre-stress 8 = 9000 pounds per square 
inch, may be found by equation (95). By substituting the given 
values in this equation, 

P X 40 = .1963 X 9000 X(15)", 
whence 

P = 149000 pounds. 

Sectional area of shaft = 176.71 square inches. 

Example. — The diameter of a hollow shaft having a hole half 
as large as the shaft, to resist a turning force P = 149000 pounds, 
acting on a lever-arm of a length Z = 40 inches, and working at a 
maximum shearing fibre-stress of 9000 pounds per square inch, can 
be found by substituting in equation (98), thus obtaining 



-v^^ 



X 149000 X 40 ,.oo. , 

= 15.33 mches. 



(.6)*] 9000 



Sectional area of shaft = 138.4 square inches. 

The area of a solid shaft for the same strength, as shown in the 
preceding example, is 27 per cent greater than this. 

96. Twist of a shaft under torsional stress. — The relation be- 
tween the angle of twist and the turning moment acting on a solid 
round shaft is expressed by the following equation, which gives 9 
in degrees: 

. = 583.6|§.=583.6g. («^) 

For hollow shafts the same relation is expressed by the follow- 
ing equation, giving in degrees: 



FORM, STRENGTH, AND PROPORTIONS OP PARTS. 

The following equation of the relation between the shearing- 
stress per unit area and the twist is applicable to both solid and 
hollow round shafts: 

e = 114. ej^ degrees (101) 

Example. — The angular twist of a solid machine-steel shaft 15 
inches in diameter and 100 feet = 1200 inches long^ when sabjected 
to a turning force ^= 149000 pounds, applied at a distance 2 := 40 
inches from the centre, assuming that the modulus of elasticity for 
shearing E, = 12000000,* is by equation (99) 

^ 149000 X 40 X 1200 _ o 
6^=583.6 12000000 X (16)* ^ ^'^^ ' 

The data in this example, with the exception of the length of 
the shaft, are the same as used in the examples of the preceding 
section. The fibre-stress used in these examples is 9000 pounds 
per square inch. Substituting this value of 8 in equation (101), and 
solving, gives 

_ 9000 X 1200 __ o 

^ = ^^^•^2000000 X 15 -^-^^^ 

which is, of course, the same as obtained by equation (99). 

The shaft, if working at a shearing-stress of 12000 pounds per 
square inch, would have a twist of 9.17^ per 100 feet of length ; 
the value of the turning force for this stress would be P = 198700 
pounds. 

96. Bending-strength of round shafting. — A round shaft that 
is subjected to bending only, as when a horizontal rotating shaft 
supports a load that does not exert a twisting moment upon it, can 
be calculated for strength as if it were a beam, the working fibre- 
stress /being selected with due regard to the fact that the stress is 
repeated at every revolution. Each fibre is alternately put into 
tension and compression during every revolution of the shaft. The 
safe working stress cannot be taken so high, on this account, as is 

* ^« = 10000000 is more nearly correct for the materials generally used for 
line shafting in factories, etc. 



AXLES, SHAFTING, AND POSITIVE SHAFT COUPLINGS. 287 

permissible for a stationary beam supporting a static load. In the 
majority of cases the fibre-stress, as* determined by the required 
rigidity of the shaft, is lower than the safe working strength for 
repeated stress. 

The values of the maximum bending moment M^, acting on a 
round shaft when resisting a bending force W, applied according 
to the more common methods of loading, and for the weight of the 
shaft itself, are as follows : 

Load applied to projecting end at a distance L from supporting 
bearing (cantilever), 

M^^WL (102) 

Load applied midway between two end supports, 

M,= ^. (103) 

Load applied between two supports at distances a and b respect- 
ively from them (a + * = ^)* 

^^ Wdb /,^^v 

M^ = -g-- iX^^) 

Two equal loads applied, each at the same distance a from the 
end nearest it, 

M^=^Wa (105) 

Bending moment due to weight of horizontal projecting shaft 
(cantilever), length of projection = X, 

M.^'^ (106) 

Bending moment due to weight of shaft, end supports, 

J^. = -f-. (107) 

The relation between the bending moment J/^, tensile or com- 
pressive fibre-stress/, and the diameter of the shaft is given by ilie 
following four equations: 



288 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 
For solid round shafts, 

Jf, = .0982/D'; (108) 

D = 2.168 VM^ -5-/. (109) 

For hollow round shafts, 

Jf,= .0982/(Z)'-rf'); .... (110) 






2> = 2.168//-.---'-^ (Ill) 



The actual point of support of a shaft is frequently difficult to 
decide upon. If the bearings are self-aligning, the point of sup- 
port may be taken in the plane, perpendicular to the shaft, con- 
taining the point about which the bearing-sleeve swivels; this is 
generally at the middle of the sleeve. When a bearing is rigidly 
supported, however, the point of pressure against the shaft may be 
near one end of the box at the time the shaft is subjected to the 
greatest bending moment. This applies to a shaft working under 
a vai7ing bending force, as the crank-shaft of a steam-engine. 

A long, tight-fitting hub, as of a fly-wheel supported on a 
horizontal shaft, prevents the shaft from bending where it is 
encircled by the hub. If the shaft is of the same cross-section 
throughout, the maximum fibre-stress in it will be at the end of 
the hub. While this shortens the lever-arm of the bending force 
and reduces the maximum fibre-stress to a lower value than if the 
load were (theoretically) applied at the middle of the hub, it also 
localizes the strain at the ends of the hub, and thus increases the 
liability to rupture, on account of fatigue of the metal, by the 
repetition of stress due to the rotation of the parts. Probably the 
best way of reducing the liability to fracture at the end of the hub 
is to enlarge the shaft where the hub is placed upon it, joining the 
enlargement to the smaller part by a fillet of as large a radius as 
can be conveniently used. 

It is assumed in the above discussion that the centre of the hub 
contains the centre of gravity of the fly-wheel. 

In practice it has been found that shafts and axles which are 



AXLES, SHAFTUfQ, AND POSITIVE SHAFT COUPLINGS. 289 

subjected to repeated stress, and especially to shocks and vibra- 
tions, are more durable when proportioned so that the stress and 
strain are distributed over them with at least an approximate 
degree of uniformity, and all re-entrant angles well filleted, than 
when made of uniform section throughout, or with sharp re- 
entrant angles. 

It is common practice in car-axles to make them small in the 
middle in order that they may spring more readily, and thus 
reduce the strain upon the parts near the wheels and journals. 
This is an application of the same principle that makes a hammer- 
handle more durable when reduced in section between the head 
and the part that is held in the hand. 

97. The lateral deflection of shafting on account of its own 
weight seldom needs consideration, except, possibly, for the 
smaller sizes. The remedy for the excessive deflections is to de- 
crease the distance between bearings, or to increase the diameter of 
the shaft. The same is generally true of deflections caused by a 
load or belt pull. 

The amount of the deflection can be calculated by the formulas 
ordinarily given for beams, but such a calculation is seldom needed. 

It should be remembered, however, that a small shaft having 
supports so far apart as to allow it to bend considerably may, when 
rotated at a high speed, run out of true to a dangerous extent on 
account of the centrifugal action. This is due, not to the deflec- 
tion of the shaft, which may be considered only as a measure of 
the liability to excessive bending, but to the fact that such a shaft 
when rotating will not keep its axis in exactly the same position. 
As soon as the shaft moves laterally to the least extent, the centrif- 
ugal action increases its deflection. 

98. Shaft subjected to both torsion and bending, general case. 
— While this is the most common case that is found in practice 
with shafting, the experiments that have been made with a view to 
establishing the formulas that have been developed to meet it are 
very few and meagre. Of all the formulas that have been pre- 
sented by different writers the following two, (112) and (113), 
seem to be the most convenient and correct. 

The method adopted in these formulas is to find a bending or 
torsional moment equivalent to the actual moments acting simul- 



290 FOBM, STRENGTH, AND PROPORTIONS OP PARTS, 

taneously^ and design the shaft accordingly. The shaft must, of 
course, be designed to resist the moment which has the greatest 
tendency to break it. 

The value of the ideal or equivalent bending moment (J4)i 
which will produce the same tensile or compressive fibre-stress in 
the material as the combined bending and twisting moments, M^ 
and Jlfff is expressed by the equation 



{M,), = iM, + lVM,^ + M,^ (112) 

And the value of the ideal torsional moment which will cause 
the same shearing-stress in the material as the combined bending 
and torsional moments is shown in the formula 

(M,), = I Jf, + VM^TW. .... (113) 



Solid Round Shafts. 

99. Solid round shafts subjected to more than one force. — ^The 
value of the resisting moment required to withstand the ideal bend- 
ing moment {M^)i of equation (112) is expressed by the formula^ 
for a solid round shaft, 

w)«=^ (^i*> 

By substituting this value of {3Qi in equation (112) it reduces 
to equation (115). 
For a solid round shaft: 



whence 



and 



/!>• = 6.366[.6Jf6 -f i^ifft* -f- J/;]. . . (116) 



!>• = ^[.6Jf, + VWT^'] . . . (116) 



f^^[.6M, + VWT^l . . (U7) 



AXLES, SHAFTING, AND POSITIVE SHAFT COUPLINGS. 291 

The value of the resisting moment required to withstand the 
ideal twisting moment (Jff)^ of equation (113) is, for a solid round 
shafts 

W), = ^ (118) 

Substituting this value of (Jf|)| in equation. (113), it becomes, 
by a slight reduction, 

For a solid round shaft: 



«2>' = 5.093[.6J/, + ^M^ + Mi\ . . . (119) 
whence 

/)• = ^[.6J4 + VWl^'] . . . (120) 
and 



6.093 



s = ^^[.6Jf, + VM^' + M,']. . . . (121) 

In equations (115) to (121) the bracketed quantities are iden- 
tical. In (119) the exponent 5.093 is 0.8 that of (115); therefore, 
if the shearing-strength of the material is taken as 0.8 of the ten- 
sile strength, which is very commonly done for iron and steel, the 
same value will be obtained for D* by both equation (116) and (120.) 
Hence either one of these equations may be used for determining 
D when 8 = .8/. 

If the shearing-strength is taken greater than .8 of the tensile 
(i.e., 8 > .8/), then equation (116) should be used for finding D; 
but if the shearing-strength is taken as less than .8 of the tensile, 
(i.e., 8 < .8/), equation (120) should be used. 

Example. — A belt whose turning force and total effective ten- 
sions are predetermined is to run on a pulley 5 feet in diameter, 
whose shaft is supported by bearings on both sides of the pulley, one 
at a distance of 5 feet and the other 3 feet from the middle of the 
pulley-hub; also that all the power is to be transmitted in one direc- 
tion through the shafting leading from the pulley. What should be 
the diameter of the shaft for working strengths of 12000 pounds 
per square inch tensile, and 10000 pounds per square inch shearing 
for a belt wrapping 160** around the pulley ? 



292 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

The values of the turning force P and the belt-tensions T^ and 
To9 as determined^ may be taken as 

P = 1467 pounds; 
T^ = 2960 pounds; 
To = 1493 pounds. 

The torsional moment M^ that the shaft must resist is 

Mt = Pl= 1467 X (2.6 X 12) = 44010 inch-pounds; 

and the bending force W that the shaft must resist is the resultant 
of 2; and To, which act at an angle of (180° — 160°) = 20^ with 
each other. This resultant is 4360 pounds. (The difference between 
this resultant and the sum of the two belt- tensions (^n + ^o) = 
4453 is so small comparatively as not to need consideration in gen- 
eral practice. The sum of the belt-y:)nsions would ordinarily be 
used for so small an angle between the two stretches of belt. The 
actual resultant will be used, however, in the following calcula- 
tions.) 

The bending moment J/5 which the shaft must resist is, ac- 
cording to equation (104), 

^^ ^ 4360(3 X 12)(5 X 12) ^ ^3^^^ i.eh-pounds. 

O X l/w 

Since the ratio (10000 : 12000) is greater than .8, equation (116), 
involving the tensile strength of the material, must be used, whence 

D* = ^^-^ X 98100 + f (98100)' + (44010)']; 
Z) = 4.46 inches. 

Problem. — Fig. 118.1. A solid shaft 7 feet long between self- 
aligning bearing centres has a pulley 6 feet in diameter placed 3 
feet from one bearing centre and 4 feet from the other. The pulley 
is to receive 200 horse-powers from a belt wrapping 150** arouud it. 
The shaft is to run at 250 revolutions per minute. Of the 200 
horse-powers, 120 are to be delivered on the 3-foot shaft side of the 
pulley and 80 horse-powers through the 4-foot length of head shaft. 



AXLES, SHAPHNG, AND POSITIVE SHAFT-COUPLINGS. 293 



Find the diameter of the shaft for limiting fibre stresses of 
10,000 poimds per square inch tensile and 8000 pounds per square 
inch shearing. 



02 



120 Vl.P. 



80 H.P. 



mj 




Fig. 118.1. 



The linear velocity of the belt is 



v= 250X^7: = 78.54 feet per second. 



The turning force required is 



P= 



550XH.P . 550X20 
V 78.54 



= 1400 pounds. 



Since the power is transmitted in both directions from the 
pulley; neither part of the shaft is subjected to the entire torsional 
effort necessary to transmit 200 horse-powers. 

The torsions in the two parts of the shaft on e£ch side of the 
pulley are proportional to the power transmitted through them. 
The greatest torsional moment wiU, of course, be on the 120 horse- 
power side. Its value is 

120 
Mt^i^fZ^X 1400 X 3 X 12 = 30,240 inch-pounds. 

Since the centrifugal tension in a belt has no effect in produc- 
ing either torsion or bending in a shaft supporting the pulley upon 
which it runs, only the effective tensions in the belt need to be 
considered for designing the shaft. These effective tensions may 



294 FOKM, STRENGTH, AND PROPORTIONS OF PARTS. 

most readily be obtained by assuming that the belt has no weight 
and that therefore all the tension in the belt is effective. The 
turning force per square inch of belt having 150** wrap aroimd a 
pulley with a coefficient of friction of .3 and working at a tension 
of 300 pounds per square inch is 163 poimds. The total effective 
tension on the tight side of the belt is 

1400 
Effective Tn^-j^ 300=2670 pounds, 

and on the slack side 

Effective 7^0= rn-f'=2670- 1400= 1170 pounds. 

The resultant of 2570 and 1170 pounds, acting at an angle SOP 
with each other, is 3650 pounds. 

The maximum bending moment in the shaft produced by the 
resultant of the two belt tensions is 

3/6= 3650X ^^^^^~^^y^^ = 75,000 inch-pounds. 

The diameter of the shaft may now be found by either equation 
116 or 120 for combined bending and torsion, since the limiting 
shearing stress is given as .8 of the tensile stress. Equation (ISff) 
will be used. By substitution in it ^ (- 

i)'=^^ [.6x75000+i/(75000)H(30200)T 

whence D=4.31 inches. 

100. Oyerhanging solid round orank-shafts and other over- 
hanging shafts acted on by a single rotative force. — ^When a 
single rotative force P is applied to an overhanging shaft, the 
equations of turning and resisting moments take a simpler form 
than those in § 99. This is due to the fact that both the bending 
moment Mi, and the twisting moment Mf are due to the same tor- 
sional force P instead of different forces. In this ca4>e M^^PL 
and Mt^PL 



AZLESp 8HAETIKG, AND POSITIVE 8HAFT-00UPLIN0S. 295 

Substitating these valuee of Mf, and M^ in equation (116)^ it 
reduces to— 

For a solid round shaft: 



ly = 6.366j[.6i + Vr + n ; . . . (122) 



.P 



/=6.366=[.6i + 4/i* + P]; . . . (123) 



and similarly for the shearing-stress equation (120) reduces to^ 
For a solid round shaft : 



D* = 6.093f:[.6i + VL* + P]; . . . (125) 



$ = 6.093^[.6i + Vr + r]i . . . (126) 

^ ~ 6.093[.6i+ VZM=T*] ^^^^^ 

Example. — An overhanging crank-shaft is to be designed to 
drive a piston 35 inches in diameter against a constant pressure of 
60 lbs. per sq. in. The following dimensions are also fixed : length 
of stroke, 40 inches; length of connecting-rod 6 feet; ** overhang^' 
of crank 15 inches, measured from middle of crank-pin to a point 
3 inches back from the front of the crank-shaft bearing; tensile 
stress in material of shaft not to exceed 12000 lbs. per sq. in. ; 
ghearing-stress not to exceed 10000 lbs. per sq. in. The speed is to 
be so slow that the inertia of the moving parts need not be considered. 

Since the limiting shearing-stress is greater than .8 of the ten- 
sile stress, the diameter of the shaft can be determined by equation 



296 FORM, 8TBENGTH, AND PROPORTIONS OF PARTS. 

(122). The values of the quantities entering into this equation 
for this particular case are: 

F = ^^^'eO = 57726.6 pounds; 

i = 16 inches; 
Z = 20 inches; 
/ = 12000 lbs. per sq. in. 
Therefore 



D' = 6.366f}^[.6 X 15 + 4^(15)' + (20)*] 
= 1041; 

whence 

D = 10.13 inches. 

Hollow Round Shafts. 

101. The equations for hollow round shafts are similar to 
those for solid ones and may be written directly from them. The 
more important ones are given below for convenience of reference. 

102. Hollow round shafts acted on by more than ond force. 
For tensile fibre-stress: 



6.366 



J, = ^^^''^^..^ l^M, + VM,^ + jy,']; . • (128) 



6.366i>. 



/ = F=T* t-^"^* + ^^^" + ^'^' ' • • ^^^^^ 



For shearing fibre-stress: 



5.093 



/>' = s{i-n') ^'^^' + ^^^' + ^''^> • • ^^^^^ 



AXLES, SHAFTING, AND POSITIVE SHAFT^OUPLINGS, 297 
6.093Z) 



^rzra^l6M, + VM,^ + M,^]. . . . (131) 

103. Overhanging hollow round shafts acted on by a single 
tnming force. Crank-shafts^ etc. 
For tensile fibre-stress: 

^\= /(f- n-) !^-^^ + ^^^^T^ ■> ' • • • (132) 
6.36Q PD 



/= ^.J^ IQL + VLi + 7] ; .... (133) 



P = /^^' - '^^ (134) 

6.3&GDI.QL + VL' + f] ^ 



For shearing fibre-stress : 



5.093P 



^ = 8(1 -n') ^-^^ + i^X' + H; » • . . (135) 



'^ = ^^^l-GL + Vr-+f]i .... (136) 

104. Experimentally determined values of the breaking tensile 
stress of round shafting subjected to combined bending and torsion. 

— Professor Gaetano Lanza made a series of experiments on shaft- 
ing subjected to both bending and torsion while rotating.* Some 
of the values obtained by him are given in Table XXXIII. They 
represent the breaking stress by fatigue of the metal, but not the 
static strength. 



• Trans. Amer. Soc. Mech. Eng., .ol. vni., 1887, p. 180. 



298 FORM, STRENGTH, AND PROPORTIONS OP PARTS. 

Table XXXIIL* 

bbbaking-stebngth of shafting subjected to combined 
bending and twisting. 



DUmeter 

of Shaft, 

Inches. 


Total 


Hone-power 


^6 
Max. Bending 


Max. Twisting 


Fibre-stress 
Caasing 


BevolutioiiB. 


Transmitted. 


Moment. 


Moment. 


Fracture. 






Inch-lbs. 


Inch-lbs. 


Lbs. per sq. in. 


1.26 


7,040 


11.717 


11,514.1 


8.926.4 


62.168 


1.35 


88.889 


8.181 


10,507.8 


2.656.8 


55,876 


1.25 


81,641 


5.291 


9,891.0 


1,714.6 


52,062 


1.25 


108,002 


4.881 


9,241.7 


1.899.2 


48.589 


1.25 


80,694 


6.276 


9,241.7 


2,027.6 


48,911 


1.25 


19,883 


6.842 


8.917.1 


2.028.2 


47,245 


1.25 


82,741 


6.283 


8.917.1 


2,029.7 


47,246 


1.25 


108,739 


6.192 


8.592.5 


2,081.6 


45.582 


1.26 


88,208 


6.888 


8.267.8 


2.026.8 


48,914 


1 


185.288 


6.288 


8.781.5 


2,029.7 


41.788 



* Taken from Transactions Am. Society Mechanical Engineers, 1887, toL yiii.. p. IM. 

Samples of the shafting tested for comhined stress were also 
tested for static tensile strength. The results are given in Table 
XXXIV. 

Table XXXIV. 

STATIC TENSILE STRENGTH OF SHAFTING. 

Refers to sliaftiog represented in Table XXXIII. 

Breaking Tensile Strength, 
lbs. per sq. in. 

1 26"diam 5^0.1 46,800 

1.^ ^»°^-^No.2 49,865 

Average 48.888 

l"diam i^o.l 58,687 

Average 60,250 

106. Practically determined formulas for round shafting. — 

There are many cases in practice where the relations between the 
jorsional and bending moments^ or even the value of either, cannot 
be estimated with much accuracy. To meet such conditions, for- 
mulas have been devised to accord with transmission-machinery that 
has given satisfactory service. These formulas are given in many 



AXLES, SHAFTING, AND POSITIVE SHAFT-COITPLINGS. 299 

hand-books and treatises on power transmission. They may be 
found in Kent's Mechanical Engineers' Pocket-book, together with 
tables for different speeds, etc., calculated from the formulas. 



SHAFTS OF STMMETBICAL SECTIONS OTHER THAN BOUND. 

106. Shafts of any section other than round are seldom used in 
machine construction. Since they are sometimes used, however, 
the method of determining their strength for combined stress will 
be pointed out. 

In order to obtain the equation of strength of any form of sec- 
tion, it is only necessary to substitute the resisting moment of the 
shaft against bending for the ideal bending moment {Mj,)i in equa- 
tion (112), or its resisting moment against torsion for the ideal tor- 
sional moment (Mt)i in equation (113), and solve as for round 
shafts after making the substitution. 

Galling / the working strength of the material for tensile or 
compression stress, and Sf, the section modulus of the shaft for re- 
sisting bending, the following equation may be written: 

(!£,),= fS^; (138) 

and, similarly, calling s the working shearing-strength of the ma- 
terial, and 8t the section modulus of the shaft for resisting torsion, 

(M,),=^8St (139) 

The values givec in these two equations are the ones to be sub- 
stituted in equation (112) or (113) in order to obtain equations of 
the same general nature as (115), (116), and (117). 



POSITIVE COUPLINGS FOE SHAFTS. 

107. Bigid shaft-oouplings. — When no allowance is to be made 
for lack of alignment of the connected shafts, a rigid coupling is 
commonly used. 

The most common form of rigid coupling for large shafts is a 



300 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

pair of flanged hubs or collars. The hub of each is solid and bored 
to a diameter to fit the shaft, to i^hich it is generally keyed. The 
flanges are bolted together after putting in place. The shafts are 
generally brought into line either by allowing one shaft to enter 
into the bore of the half coupling on the other, by a shoulder on one 
flange fitting into a corresponding recess in the other, or by making 
an accurate fit of the bolts in the flanges. Other less common 
methods are also used. 

For the medium and smaller sizes of shafting used for power 
transmission, couplings bored to a slightly smaller diameter than 
that of the shaft, and split longitudinally, either upon one side or 
both, are very commonly used. 

A very simple form of split-sleeve coupling is shown in Fig. 119. 
It consists of a sleeve bored to the same diameter from end to end. 



Fig. 119. 

and given a slight conical taper toward each end on the outside. 
It is either split longitudinally down one side, or cut completely in 
two longitudinally, so as to form two halves. When used, it is 
placed upon the two sections of shafting, which are brought to- 
gether, and then a ring, bored to the same taper as the outside of 
the sleeve, is either driven or shrunk upon each end of the coup- 
ling, thus causing it to grip the shafts tightly. When the torsional 
force to be transmitted is small, the coupling may be used without 
a key, but for heavy work a key is generally used after the manner 
common to solid couplings. 

It can readily be seen that a split-sleeve coupling of much the 
same nature as the above one can be made by using bolts, instead of 
the rings, to clamp the two halves together. 

A largely used form of coupling, conimonly known as Seller's 



AXLES, SHAFTING, AND POSITIVE SHAFT-OOUPLINGS. 301 

coupling, consists of an outer sleeve, bored at each end with a coni* 
cal hole to receive a bushing with a corresponding taper on the out- 
side. The bushing has a bore slightly smaller than the shaft. 
Bolts, extending from end to end through the sleeve, serve to draw 
the two conical bushings towards each other, and thus cause them 
to grip the shaft. These bolts also serve as keys to prevent the 
rotation of the bushings in the outer sleeve. Keys may be used or 
not between the shaft and bushing, according to the nature of the 
work. 

108. Flexible Bhaft-conplingB. — When there is a probability 
that there will be a slight relative movement of the shafts coupled 
together, as in the case of the shafts of a dynamo and engine resting 
on separate foundations and direct-connected together, it is necessary 
to have a coupling which will adjust itself to the slight throwing 
out of alignment of the shafts that occurs under such conditions. 

The coupling Fig. 120 has been successfully applied for mod- 




^ 



r-.,^^^-£ 



V 



Pig. 130. 

erate amounts of power transmitted. It consists of a pair of disks 
very similar to those of the rigid coupling, but instead of being held 
firmly together by bolts they are separated as shown, and each has 
a number of strong pins projecting from it so as to almost touch 
the other. The other disk has the same number of pins. The pins 
are spaced uniformly around both flanges near their peripheries. 
Each pin of one flange is connected to one of the other by a short 
link of some elastic material, as rubber or leather. As one flange 
rotates it draws the other after it by means of the connecting-links, 
whose elasticity allows them to adjust themselves to the lack of 



302 FORM, STRENGTH, AND PROPORTIONS OK PARTS. 

alignment of the shafts. The links can be made longer or shorter 
than shown in the figure if desired. 

Another flexible coupling, which allows the shafts to have bend- 
ing motion relatively to each other, but does not allow lateral motion, 
is shown in Fig. 121.* It consists of a large flange upon one of the 
shaft-ends to be coupled, and a small one upon the other. The two 
flanges are connected together by a washer-shaped ring, which may 




Fig. 121. 

be made by cutting it from sheet metal, the outer diameter being 
made much larger than the inner. This flat ring, being of an elas- 
tic material, allows a slight bending motion of the shaft as stated 
without injury to the parts. f 

109. PoBitive clutch-couplings are often used for connecting 
parts of machinery which are to be disengaged at times. The most 
common form consists of two parts resembling, in a manner, the 
halves of the coupling shown in Fig. 120, but having projecting 
teeth or jaws instead of pins. These jaws are made so that they 
will engage with each other for transmitting rotative motion. The 
coupling is generally disengaged by slipping one of its halves along 
the shaft so as to separate the jaws. 



* Trans. Amer. Soc. Mecb. Eng., vol. vii., p. 526. 

f For Oldham's and Hooke's universal couplings see Part 1. of Machine De> 



sign. 



CHAPTER IX. 

FRICTION-COUPLINGS AND BRAKES. 

110. In the operation of machinery it is of ten desirahle to start 
and stop some part which receives its motion from a constantly 
running driver, or to bring two parts into engagement without 
changing their relative positions. For this purpose **fi:iction- 
conplingB^' are used almost universally. Such a coupling trans- 
mits power by means of two smooth surfaces, held together with 
sufBcient pressure to make their frictional resistance to slipping 
great enough to produce the required transmission of power. 

The Motion-brake, for retarding the motion of a part^ is 
practically the reverse of the friction-coupling, so far as its general 
principle is concerned, but differs from it in that there is generally 
much more rubbing between the surfaces, and consequently the 
materials must be selected with more attention to their qualities of 
not abrading and cutting. This applies to brakes such as are 
used on hoisting-machinery, but hardly to those for railway-wheels, 
etc., where the conditions are such that it is impossible to prevent 
abrasion on account of foreign matter getting between the rubbing 
surfaces. 

FRICTION-COUPLINGS 

111. Cone friction-conpIingB, consisting of two parts having 
conical surfaces, one external and the other internal, fitting 
together as in Fig. 122, are often used in machine construction, 
and especially in machine tools, for connecting the ends of two 
shafts, or for transmitting power from a shaft by means of a gear 
or other device attached to the part of the coupling which is free to 
rotate upon the shaft when the clutch is open, but is driven by the 
shaft when the two parts of the clutch are forced together. 



804 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

The notation in the equations for a cone friction-clutch is: 
M = torsional moment transmitted through clutch ; 
N = total normal pressure between the conical surfaces of the 

clutch ; 
P = turning force acting at a distance R from the centre of the 
shaft; 




Fig. 123. 

li = mean radius of conical surfaces; 
T = axial force acting to close the clutch ; 
fi = angle between the conical surfaces and the axis of shaft; the 

apex angle of the conical surfaces is therefore 2ff; 
ji = tan = coefficient of friction between the conical surfaces; 
^ = angle of friction between the conical engaging surfaces of thj 
clutch. 
If the two parts of the coupling are forced together when 
they have no relative rotation, the pressure between the conical 
surfaces would make an angle with a normal to the surfaces, 
as indicated by the dotted lines in Fig. 122. This frictional 
resistance would have to be taken into consideration if it were 
desired to set the clutch so that there would be no slip whatever 
between the parts at the time of closing or afterwards. But if 
the clutch is closed while one part is rotating relatively to the other, 
or if a slight slip at the time of throwing on the load is not objec- 
tionable, it is not necessary to consider the angle of friction when 
determining the axial force T required for closing it. A slight sli]) 
at the time of closing is almost invariably, probably always, allow- 
able. Of the following equations, the ones not including <f> are 
therefore applicable to cone friction-clutclies in nearly all cases. 



FRICTION-COUPLINGS AND BRAKES. 305 

For cone friction-clutches closed while having a rotative slip 
between the engaging surfaces: 

N=T cosecp; (140) 

P=fiN = fiT cosecP; (141) 

M^PR^fiTR cosec^ (142) 

For a cone friction-clutch closed without rotative slipping 
between the engaging surface^: 

N=T cosec(P +<!>); (143) 

P^fiN^fiT cosec(p + (l>)] (144) 

M=PR^fiTRcosec{^-^<l>) (145) 

When P=^<1> the parts will separate of their own accord with 
no load on the clutch and when r=0. 

Problem. — Design a cone friction-clutch of the type shown 
in Fig. 122 for transmitting 25 horse-powers at 250 revolutions per 
minute. 

The mean radius of the cone surface may be assumed as 6 
inches. The turning force necessary on this radius is, by equa- 
tion (17), 

^ 33000X12XH.P. 33000X12X25 ,^^^ 

^ 2;^ 2;rX6x250 -^^^Vounds. 

The coefficient of friction /£ may be taken as .15. Then 

M=P^/£=1054-f- .15-7027 pounds. 
The clutch will just open if tan /?=/£. Since /£ may at times 
become greater than the assiuned value of .15, its maximum value 
may be taken as .2; then 

tan ^= tan <f>=pL=.2, 
i^=11.33^ 
The axial closing force must be, from equation (140), 

T=N-rC8cp=^N sin /?=7027X. 196= 1380 pounds. 
For a pressure of 50 pounds per square inch between the cone 
surfaces the area of each surface will be 

7027-^50=140.5 square inches. 
The mean diameter of the cone surface is 12 inches. The 
width will therefore be 

140.5^12;: = 3.75 inches. 
Note. The resultant of the turning forces acting tangent to 
the conical surface is actually at a slightly greater distance from 



306 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



the axis of the clutch than the melan radius of the cone surface. 
The error introduced by using the mean radius is small and on the 
safe side. 

112. Multiple-ring firiction-conpling. — ^When it is desirable to 
keep the size of the coupling small, especially in diameter, the 
form shown in Fig. 123 is appUcable.* As illustrated, the outer 



*Jv- 



HPiiii-iKi 1 








Fig. 128. 
cylindrical sleeve A is fitted on the shaft with a feather key, so- 
that it may slip along it. The hub B has a running fit on the 
shaft, but i3 prevented from moving along it by a shoulder. A 
number of rings are placed between the sleeve and hub, each alter- 
nate one engaging with the sleeve by means of several projections 
on the latter corresponding to feather keys, so as to be free to slip 
along the sleeve and hub. The remaining rings engage with the 
hub in a similar manner. By applying an axial force to press the 
sleeve toward the hub, a pressure, equal to the axial force, is in- 
duced upon each side of every ring. 

Putting, for multiple- ring friction -couplings: JIf = torsional 
moment transmitted through coupling; P = turning force exerted; 
i2 = mean radius of rubbing surfaces of rings; 7^= axial force 
exerted to press the sleeve toward the ring; n = number of pairs 
of rubbing surfaces in contact; // = coefficient of friction of the 
rubbing surfaces; then 

F = MnT; (146) 

M=PR=M7iTR (147) 

As represented in the figure, there are as many pairs of rubbing 

* Weston friction-coupliDg used by Yale & Towne Mfg. Co. on hoisting- 
ma -^liinery. 



FRICTION-COUPLINGS AND BRAKES. 307 

Burfaces as there are rings. This is not always true of snch coup- 
lings, however. 

113. Materials and ooefficient of friction for friotion-conplings. 
— In machine tools and the connter-shafts for driving them, unless 
they are exceptionally large, the material of a cone friction-coup- 
ling is generally cast iron for both rubbing surfaces. When accu- 
rately ground together with a fine abrasive, as flour-emery, the 
cast iron gives excellent satisfaction for this use. The^ coefficient 
of friction pi must be taken with regard to whether the surfaces are 
oily or dry. Since there is nearly always a certainty of oil getting 
on them at some time, the value )u = 0.15 is probably as high as it 
is safe to assume for cast iron on cast iron. 

When a friction-coupling is to perform heavy service with con- 
siderable slipping at the time of setting the machinery into motion, 
cast iron on wood or leather gives good service. The wood may be 
either set on end or with the grain parallel to the direction of 
' rubbing. Any of the metals that run well together for journal- 
bearings may be used if the coupling is kept clean and lubricated. 
Otherwise wood or leather rubbing against metal is safer. For 
dry wood on cast iron pi may be taken from 0.16 to 0.18; for oily 
wood fi falls as low as 0. 10. The value of /i for leather on cast 
iron does not seem to be well determined for such service as is re- 
quired for friction-clutches, but it can probably be taken as high 
as 0.20 for oily leather; it is much higher for dry leather. Leather 
should not be used where there is sufficient slipping to bum it. 

The best service for heavy work and hard usage seems to be 
obtained with wood rubbing on metal. In practice the latter is 
generally cast iron. 

FRICTIOK-BRAKBS. 

114. strap brake. — A form of strap brake much used for 
hoisting-machinery is shown in Fig. 124. The brake-drum, 
whose centre is at 0, is partly encircled by a strap whose ends are 
attached to a lever for tightening it upon the drum. The strap 
may be made of any material, according to the requirements to be 
met. For heavy service on mine-hoists, a strap of Swedish wrought 
iron, lined with blocks of basswood placed with the grain parallel to 
the length of the strap, is very commonly used. Experience has 



308 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

shown that a steel strap is very liable to fracture under such service, 
apparently on account of fatigue of the metal, caused by the re- 
peated application of stress due to tightening and loosening the 




Pig. 124. 



drum, and to the almost incessant yibrations common to such 
service. Basswood is selected because in it are combined both 
durability and uniform gripping power. For lighter service, a 
wrought-iron strap is often used directly against the cast-iron brake- 
drum. This can be done saccessfully only where the rubbing sur- 
faces can be kept lubricated. 

The tensions T, and T^ in the two free portions of the brake- 
strap between the drum and brake-lever, for a given force P re- 
sisting the rotation of the drum, and applied at the surface of. the 
latter, may be foijnd by the equations for belting. . The value of V 
in these equations is zero for a brake-strap, and no centrifugal force 
due to its own weight acts upon it. 

Equations (39) to (45) are applicable to the solution. In these 
equations the coefficient of friction yu may be taken as 0.1 for wet or 
oily surfaces of wood ou cast iron, and from 0.15 to 0.18 for dry 
rubbing surfaces of the same materials. 

The pressure that can be safely put on wood-lined strap or crab 
brakes for the heavy duty of mine hoists is 25 lbs. per sq. in. or less. 

115. The Prony brake, Fig. 125, is a special form of strap 
brake. Its almost universal application is as an absorbent dyna- 
mometer for tests of the power developed by motors. As shown in 



FRICTION-COUPLINGS AND BRAKES. 



309 



the figure, it consists of an iron strap A encircling a brake-drum 
whose centre is at 0. The strap is lined with wooden blocks which 
bear against the surface of the drum. The ends of the straps are 
held together by a bolt which affords a means of adjustment of 




FiQ. 125. 

tension in the strap. The two members B and O each have one 
end attached to the brake-strap, and are joined together so as to 
form a rigid arm. As the brake-drum rotates in the direction of 
the arrow, the tendency of this arm to rotate with it is resisted by 
a force Q acting against the end of the arm. 

If the two members of the brake-arm are attached to the strap 
at points diametrically opposite, so that each portion of the strap 
covers 180° of the drum, and if Q acts vertically upward and is of a 
value equal to the portion of the weight of the brake that is sup- 
ported by the drum, then the following equatioi^ are applicable 
for determining the relation between the maximum tension T^ in 
the brake-strap and the force Q. It can be seen that these equa- 
tions are of the same nature as those for belting. The brake-strap 
corresponds to two portions of a belt having an arc of contact of 
180** upon the pulley. 

The notation for the Prony brake is: 
L = distance from centre of drum to line of action of Q (not nec- 
essarily the distance from the centre of the drum to the 
point of application of Q) ; 



310 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

P = total frictional resistance at surface of brake-drum; 
Q = force applied at end of brake-arm to resist its rotation; 
R = radius of brake-drum; 
T-=- greatest tension in brake-band; 
6=2.71828; 
n = 3.1416. 

By taking moments about 

PR^QL ox P^^Q (148) 

In accordance with the equations for belting, taking into ac- 
count the fact that there is no centrifugal force acting on the 
brake-strap, 

P €^» P io.«m«iii80 

^ = 2 e^» — 1 ~ 2" lO-««^i«> — 1* • • • ^"^ *^^ 

Whence, by substituting the value of P given in equation (148), 
QL 6^' _QL 10.ooTBqm9o 

2^ €^» - 1 " 25 10.00768^180 _ 1- • • V^^"^ 

If the points of attachment of the two members of the brake- 
arm to the brake-strap are not diametrically opposite, it is hardly 
possible to get equations for the maximum tension in the strap that 
are of any practical value. If the material of the strap were totally 
inelastic, such equations might be deduced, but on account of its 
elasticity they cannot be. The equations just given may be applied 
with a fair degree of accuracy, however, when the arc between 
points of attachment does not differ greatly from 180^, and if Q 
does not differ greatly from the portion of the weight of the brake 
supported by the drum. 

The coefficient of friction /i, and the materials for the Prony 
brake, may be taken the same as for the strap brake of the pre- 
ceding section. Since the service required of a Prony brake is 
seldom so trying as that of the strap brake for hoisting-machinery, 
it is generally not necessary to exercise so much care in selecting 
the materials. 



CHAPTEE X. 

FLY-WHEELS AND PULLETS. 

116. A fly-wheel, as commonly used in machine constrnction, 
is a rotating wheel which serves as a storehouse for energy. At 
certain periods during the operation of the machine the rotative 
speed of the fly-wheel is increased, thus increasing its store of 
kinetic energy, and at other periods the speed is decreased, the 
wheel thus returning or supplying energy to the machine. 

Examples of the application of fly-wheels are very common. 
Three cases may be cited, however, to point out different classes of 
service. 

In the ordinary double-acting steam-engine with a single 
cylinder the fly-wheel has its speed slightly increased twice during 
each revolution, and also slightly decreased twice during the same 
revolution, provided the average speed of rotation does not change 
during a series of successive revolutions. The function of the fly- 
wheel in this case is to prevent great fluctuation of speed during a 
single revolution, as well as to maintain an approximately uniform 
speed when the load against which the engine works varies 
suddenly. 

In a punching and shearing machine the fly-wheel receives a 
store of energy while being brought up to its normal speed. As 
the punch or shears are driven through the metal operated upon, 
the speed of the fly-wheel is decreased and a portion of its energy 
given up to perform the work upon the metal and to overcome 
frictional resistances. The time interval of working on the mate- 
rial punched or sheared, which corresponds to the period of reduc- 
tion of speed, is small in comparison with the time the machine is 
running idly. After punching the material the fly-wheel again 
receives energy while being brought back to its normal speed.. 

In the Howell torpedo a fly-wheel is brought up to an 
exceedingly high speed. When the torpedo is launched into the 

811 



312 FORM, STRENGTH, AND PROPORTIONS OF PAISTS. 

water^ the kinetic energy stored in the wheel is utilized to drive a 
screw propeller for forcing the torpedo through the water. The 
fly-wheel gradually stops as it gives up its energy. 

117. Determination of moment of inertia and kinetic energy of 
a webbed wheel. — The total kinetic energy, K.E., stored in a 
rotating fly-wheel is expressed by the equation 

K.E. = ^(mass) x (angular velocity)' X (radius of gyration)', or 

K.E. = ^(moment of inertia) X (angular velocity)'. 

The form of fly-wheel that is most generally adopted has a rim 
of rectangular cross-section with slightly rounded corners, a hub 
that is approximately cylindrical, and arms tapering uniformly 
from the hub to the rim, growing smaller toward the rim; for 
small sizes of wheels a solid web is often used, the web generally 
being thinner at the rim than at the hub. 

The moment of inertia of a wheel with uniformly tapering arms 
can be found by reducing the arms to an equivalent web. This 
reduction can be made with an accuracy far within the limits 
required for ordinary practice, and the results obtained by this 
method of substituting a web for the arms are ordinarily more 
accurate than those obtained by dealing with the arms directly. 
(See § 120.) The method of finding the moment of inertia of a 
webbed wheel ol the form shown in Fig. 126 will therefore be given 




Pio. 136. 

as being more generally applicable than that for any other form of 

wheel. 

* The notation for the equations applying to a webbed wheel is : 
E= energy given out by wheel for a given change of speed of 
rotation, foot-pounds; 

* See § 120 for notation of lly- wheel with arms. 



FLY-WHEELS AND PULLEXa 3^3 

/ = moment of inertia of entire wheel about its axis, foot- 
pound-seconds; 
hxibf ^ynbf Mid /rt„ = moment of inertia of the hub, web, and rim, 

respectively, foot-pound-seconds; 
E.E. = total kinetic energy stored in the rotating wheel, foot- 
pounds; 
JV= a? -!- 2w = reyolutions per second; 
N^ = initial speed of rotation, revolutions per second; 
iV, = speed of rotation after slowing down, revolutions per 
second; 

^ ^ y, f 

B = -^ -2-£ = radial distance at which sides of web would 

s "■ *• 

intersect if extended, used for convenience only, feet; 
T = •^-~- — = -n^ — = length of axis intercepted between 

sides of web if extended to axis of wheel, used for con- 
venience only, feet; 

g = acceleration due to force of gravity = 32.2 feet per second 
per second; 

p = tensile stress in rim dae to rotation, pounds per square inoh; 

r = mean radius of rim of wheel, feet; 

r, = radius of bore of hub, feet; 

r, = outer radius of hub = inner radius of web, feet; 

r, = outer radius of web = inner radius of rim, feet; 
• r^ = outer radius of rim, feet; 

t^ = thickness (or length) of hub, feet; 

t^ = thickness of web at inner edge, feet; 

<, = thickness of web at outer edge, feet; 

t^ = thickness of rim measured parallel to axis of wheel, feet; 

V = velocity, feet per second; 

io = weight of material, pounds per cubic foot; 

A. = increase in radius of ring due to centrifugal action, feet; 

TTs 3.1416; 

CO = 2;rJV= velocity of rotation, radians per second; 
a?, = initial speed of rotation, radians per second; 
09, =r speed of rotation after slowing down, radians per second. 



314 FOBM, STBENGTHy AND PR0P0BTI0N8 OF PABT8. 

The moments of inertia of the three parts, the hub, web, and 
rim, are: 

^u. = *-^(r/-r/); (161) 



/>-.= 



--J-l—l 6^J' • • ^^^^^ 

/..- = ^*(r;-r/) (153) 

The moment of inertia of the entire wheel is the sum of the Ps 
for all the parts; whence 

/=/hub+/w.b+/*n (154) 

The total kinetic energy K.E. stored in the wheel when rotat- 
ing at an angular velocity of go radians, or If revolutions, per 
second, is 

K.E.= !(«•/= 2;r'JV^V. .... (155) 

The energy B transformed from kinetic energy into mechanical 
energy or heat, or both, when the speed of rotation of the fly-wheel 
drops from a?, to a?, radians, or from iV, to 2f^ revolutions, per sec- 
ond, is shown by the expression 

^•= !((»,•- a7/)/= 2;r«(JVr« - JV,«)r. . . (166) 

The same amount of mechanical energy E must, of course, be 
applied to the wheel to bring it back from iV, to N^ revolutions per 
second, friction neglected. 

Example. — It is required to find the kinetic energy K.E. at a 
speed of 300 revolutions per minute of a cast-iron fly-wheel of the 
form shown in Fig. 126 and having the following dimensions: 

= 6 in. = .6 ft. ; r, = IJ in. = ^ ft. ; 

= li in. = .125 ft. ; r, = 3 in. = .25 ft; 

= 1 in. = iV ft.; r, = 18 in. = 1.6 ft; 

= 6 in. = A ft.; r, = 24 in. = 2 ft 



FLY-WHEELS AND PULLEYS. 316 

The weight of cast iron may be taken as 450 pounds per 
cubic foot. 

If the acceleration due to force of gravity is taken as 32.2 feet 
per second per second, then the units of measurement must be the 
foot, pound, and second. 

The moment of inertia of the three parts may first be de- 
termined. 

The moment of inertia of the hub is, by equation (151), 



J __ 4607r X 

-*hub — 



!#[<•-)•-©■] 



2 X 32.: 
= 10. 97(. 0039062 - .0001174) 

= .0416. 

For the web equation (152) can be applied. In this equation 
the quantities T and R enter. T may be measured on the drawing 
with considerable accuracy, but the intersection of the lines deter- 
mining R is generally difficult to determine. The values of both 
will be calculated for this problem according to the equations given 
in the' notation; whence 

jj^l8xl^5^_pa, ^ 43 inches = 4feet, 
and 

T = ^^^f = 1.6 inches = ^ foot. 

By substituting in equation (152) 
2;r450 X 2 



/we.= 



r (1.5)--(.25)* _ (1.5)*-(.25)n 
L 4 5 X 4 J 



32.2 X 15 L 
= 11.7 X .885 
= 10.4. 

And for the rim, by equation (153), 
= 100.3. 



316 FOKM, STRENGTH, AND PROPORTIONS OF PARTS. 

For the entire wheel the moment of inertia is therefore 

/=. 0416 + 10.4 + 100.3 
= 110.74. 

It can be seen that the effect of the hub on the total moment of 
inertia is practically inappreciable, and may therefore be neglected 
in a wheel whose rim diameter and weight of rim are as great in 
proportion to the similar quantities for the hub as for the wheel 
just considered. 

The kinetic energy of the entire wheel when rotating at 300 
revolutions per minute is, according to equation (155), 

K.E. = ^n^NU 

= 27r'(3/gft)Ml0.74 
= 64648 foot-pounds 
= 27.32 foot-tons. 

118. Problem. To design a fly-wheel for a given moment of 
inertia and according to a given form. — Let the required moment 
of inertia /= 300, the wheel to be similar to Fig. 126. 

Since the required wheel is to be similar to the one considered 
in the preceding problem, it is only necessary to change the dimen- 
sions of the wheel, all in the same proportion, to such an extent as 
will give the required /—in other words, to apply such a scale to 
the drawing as will give the required /. 

The proportionate change of the dimensions can readily be de- 
termined by making use of the fact that the / of similar wheels is 
proportional to the fifth power of .their linear dimensions. There- 
fore the linear dimensions of the wheel in the preceding example^ 
which has an /= 110.74, must each be multiplied by 



\/ 



110.74 



in order to obtain the required / = 300. 

This scale gives the outer radius of the rim 

r, = 1.22 X 24 = 29.28 inches. 



FLY-WIIEEi.8 AND PULI.EY8. 317 

and the other dimensions must be increased in the same propor- 
tion. 

The above solution shows that when it is desired to design a 
wheel for a required / it can be done by making a drawing accord- 
ing to the form desired^ considering it full size, and finding its 1 
accordingly. The scale which must be applied to the drawing to 
obtain the required / can then be determined by dividing the re- 
quired / by the / of the drawing, considered full size, and extract- 
ing the fifth root of the quotient. 

119. Problem. To design a fly-wheel which will famish a given 
amount of energy for a given variation of speed. — Let it be re- 
quired that the wheel shall furnish 40000 foot-pounds of energy for 
a 55^ speed reduction, and that the full speed shall give the outer 
circumference of the rim a velocity of 4800 feet per minute. 

This problem can most readily be solved by making use of the 
fact that in similar wheels having the same circumferential linear 
velocity the kinetic energy is proportional to the cubes of similar 
linear dimensions. 

For convenience it may be assumed that the form of wheel se- 
lected is that of Fig. 126, and that the proportions selected for the 
first trial are those of § 117. 

For the wheel considered in § 117, / = 110.74. When this wheel 
runs at a circumferential velocity of 4800 feet per minute, its an- 
gular velocity is 

4800 4800 ,^ ,. ^ 

^* ^ "60r" "^ 60x^ = ^^ radians per second. 

When the speed has dropped 6^, the angular velocity is 

fl7, = 40 — .05 X 40 = 38 radians per second. 

The energy given out by the wheel while slowing down m is^ 
by equation (166), 

^=.i[(40)'-(38V]110,74 
= i X 156 X 110.74 
= 8638 foot-pounds. 



818 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

In order to secure a fly- wheels similar to the one just considered, 
which, when running at 4800 feet per minute circumferential 
speed, will furnish 40000 foot-pounds of energy for a 6^ reduction 
of speed, the proportions given in § 117 must all be multiplied by 
the same factor, whose value is 



V 



8/40000 T fi^^Q -,- , . 
—-—-. = 1.6668 = If about. 
8638 



The outer radius of the rim must therefore be 1.6668 X 2 = 
3.3336 feet, and the other dimensions must be increased in the same 
proportion. 

In the fly-wheel considered above, part of the energy given out by 
it during its reduction of speed is converted into heat on account of 
journal friction, etc. If the frictional resisting forces remain con- 
stant, the mechanical energy converted into heat while the speed is 
being checked is proportional to the number of revolutions made 
during the change of speed. If the speed of the wheel is checked 
by frictional resistance only, all of the energy given up by the 
change of speed is converted into heat. 

120. Moment of inertia of a fly-wheel with arms. — The / of the 
hub and rim can be found by equations (151) and (153), the numer- 
ical solution being similar to that in § 117. Only the method of 
dealing with the arms will therefore be considered. It is assumed 
that the hub and rim have the form shown in Fig. 126. 

Suppose that the arms are sheared off from the hub and rim so 
as to leave the surfaces of the latter two parts smooth, and that the 
arms are flattened out to form a web which will just fit in between 
the hub and rim. The moment of inertia of the web will be prac- 
tically the same as that of the arms from which it was made, and the 
area of its outer edge will be the same as the total area of all the 
sheared outer ends of the arms ; the area of the inner edge will be equal 
to the total of all the sheared surfaces at the inner ends of the arms. 

If, in a pulley with arms : 
A^ = total sheared area of the inner ends of the arms; 
A, = total sheared area of the outer ends of the arms; 
i^ = thickness of inner edge of equivalent web having the same 

area as the sheared area of all the inner ends of the arms; 
/, = thickness of outer edge of equivalent web; 



FLY-WHEELS AND PULLEYS. 319 

and the remainder of the notation is the same as for the webbed 
wheel, Fig. 126; then 

t,=^ and t, = ^ (157) 

' 27rr, • 2Tr, ^ ' 

Ab an example, let it be required to find the moment of inertia 
of six pulley-arms according to the following data: 

^4, = 80 square inches; 
A, = 48 square inches; 

r, = 11 inches (see Fig. 126); 

r, = 40 inches (see Fig. 126). 



By equation (157) 



/ = = 1.157 inches; 

' 2;rll ' 



t, = -r^ = .1909 inch. 



These are the values of t^ and ^„ according to Fig. 126, for the 
web whose moment of inertia is the same as that of all the arms. 
The corresponding values of B and T are : 

^ 40 X 1.157 - 11 X .1909 ,^ ^^ . , 
^ = 1.157 .-.190 -9 = ^''^^ ^^^^^«' 

-, 1.157 X 45.73 , „^ . . , 
^= 45.73-11 =l-g^^^^^^es. 

The moment of inertia of all the arms is, therefore, by equation 
(152), using the values for the equivalent web, 

_ __ 27r450 X 1.524 r(40)* - (U)J _ (40)' - (11)* H 

^•rm. - Aeb - 33 g x 12 L 4(12)* 5 X 46.73(12)*J 



-n .g. p545359 102238949 "] 
- 11.151 [^ g^^^^ 4741300 J 

= 11.151(30.688 - 21.566) 
= 11.151 X 9.122 
= 101.7. 



320 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

It is not generally convenient to obtain the areas of the sheared 
ends of the arms, since these ends are curved surfaces which are por- 
tions of cylindrical surfaces of radii equal to the distances from the 
centre of the pulley to the outside of the hub and inside of the rim. 
In any ordinary design the error is inappreciable when the area 
taken is that of a plane section of the arm at right angles to its 
length. 

Even if the arms taper uniformly from end to end, and the 
areas of the curved sheared surfaces are taken, there is a slight error 
in this method ; for if the arms were flattened out so that each 
particle remains at its original distance from the centre of the 
wheel, the web thus formed will not have straight lines for its sides, 
as shown in Fig. 126, but the sides will be slightly concave. 

The greater the width of the arm in the direction of the wheel's 
rotation, the greater the curvature of the side of the web. In or- 
dinary wheel designs, however, this curvature is so slight as to be 
negligible. 

The two errors, one due to taking plane cross-sections of the 
arm, and the other to assuming that the equivalent web has straight 
lines bounding its axial section, are compensative, instead of ac- 
cumulative. 

121. Stresses in fly-wheels with arms. — The stresses which 
occur in a fly-wheel of the ordinary design when in operation are 
so complicated that it is not believed they can be computed with 
even a practical degree of accuracy. The general nature of the 
stresses may be shown in such a manner, however, as to be a guide 
to the designer when considering the methods of reducing them 
with a view to decreasing the liability of the wheel to rupture on 
account of excessive speed or sudden variation of speed. 

Fig. 127 is a portion of a built-up pulley. The hub is complete^ 
but the ends of sections of the rim are not fastened. The pulley is 
represented as being under stress applied by a weight TT hanging 
from the end of an arm attached to the hub or shaft which sup- 
ports the wheel. The wheel is prevented from rotating by cords or 
tension-bars attached to pins at ^, B, and (7, extending out from 
both sides of the rim and having a cord attached to each end, the 
cords all having the same tension and pulling at right angles to the 
arms. 



FLY-WHEELS AND PULLEYS. 



821 



Each arm acts as a cantilever to resist the bending action of the 
cords pulling on the pin near its end, and is accordingly bent or 
deflected as a cantilever. This bending makes the centre line of 




Pig. 127 

each arm convex on the side opposite that toward which the cords 
are pulling, assummg that the centre lines were straight before 
stress was applied, and the sections of the rims are consequently 
tipped forward so that their ends are slipped over each other as 
shown. 

Now suppose that, while the arms are still under stress due to 
the hanging weight W, the ends of the rim-sections aje brought into 
line by radial forces Q applied near the ends of each section as 
shown. This will cause stress in the rim and arms. In the neigh- 
borhood of the angles E where the arms join the rim, the material 
will be in tension, whose maximum value will occur near B. The 
maximum value of this stress may occur in the arm or rim, accord- 
ing to their relative strengths. 

The material of each arm is in tension near the hub on the side 
opposite JEJ on account of the pull of the cords. 

The bending of the arm shortens the radial distance from the 



322 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

centre of the pulley to the outer end of the arm ; hence, when all 
the sections of the entire pulley have their ends brought into line, 
the rim will be drawn in somewhat at the end of each arm, and, 
instead of being circular, will be slightly wavy around its circum- 
ference. The effort of the rim to assume its circular form, assum- 
ing that this is the form when not under stress, will cause tension 
in the arm ; there will also be bending tensile stress in the rim near 
E on account of its being bent inward. 

If, instead of having cords attached to the pins in the rim of 
the stationary fly-wheel, it were started or stopped from rotating by 
power applied through its shaft, the inertia of the parts would in- 
duce stresses similar to those caused by the pull of the cords. A 
belt running on the rim and transmitting power to or from the 
shaft has much the same effect. 

In addition to the stresses corresponding to those caused by the 
pull of the cords on a stationary wheel, there are in a rotating 
wheel others due to the centrifugal tendency of the rotating parts. 
These stresses may be considered by dealing with the rim and arms 
separately. 

When a ring, thin radially, of a mean radius r, rotates about its 
geometrical axis with a linear velocity of V feet per second, circum- 
ferential tension is induced in the material. The value of this 
stress p in pounds per square inch is expressed by the following 
equation, in which {w -^ 144)*equals the weight of a piece of the 
material 1 inch square and 1 foot long: 

Equation (158) shows that the tensile stress in a thin rotating 
ring is proportional to the square of the linear velocity v, and that, 
if V remains constant, the tensile stress is constant whatever the 
radius of the ring. 

The radius of the ring is increased on account of this stress by 
an amount A. in feet given by equation (159), in which Et = ten- 
sile modulus of elasticity of the material : 

A = ^. • . . . (159) 



FLY-WHEELS AND PULLEYS. 323 

If the ring is thick radially, say 1 foot thick for 20 feet of diam- 
eter, the mean radius (r, + rj -^ 2 can be useil without an error 
great enough to require practical consideration. 

For the arms equations of the same nature as (158) and (159) 
can be written. They are probably of no practical value, however, 
so they will be omitted. The general nature of the effect of rota- 
tion upon the arms may be expressed, however, by the statement 
that, if a thin, prismatic bar is rotated about an axis at one end and 
normal to the length of the bar, the maximum tensile stress in the 
bar is one half as great as in a ring having a radius equal to the 
length of the bar and rotating at the same angular velocity; the 
increase of the length of the'bar is one third that of the increase of 
the radius of the ring. 

Suppose that a ring of metal, corresponding to the rim of a fly- 
wheel, is placed on a horizontal table, and a "spider," correspond- 
ing to the hub and radial arms, is placed in the ring, the arms Le- 
ing of such a length as to just fit in the ring without pressure 
against it. If the table is rotated about a vertical axis coincident 
with that of the ring and spider, carrying these parts with it, the 
ring, on account of the centrifugal action, will increase in diameter 
more rapidly than the arms will increase in length, and will there- 
fore separate from them, leaving a space between the end of each 
arm and the ring. If, on the contrary, the arms are attached to 
the ring so that no separation can occur, the arms will be somewhat 
elongated on account of the pulling-out action of the ring, and the 
ring will be bent inward where the arms are attached. Tensile 
stress will therefore be induced in the arms and in the inner side of 
the ring at and near the arms. The relative intensity of the stress 
in the arms and ring will depend on their relative strength. 

The pulling-in action of the arms, when there are several, 
reduces the tension in the rim caused by centrifugal action. 

As a summary of the above, it may be stated that, if a fly-wheel 
which has no stress in its parts when at rest is rotated uniformly 
about its axis, tensile stress will be induced in the arms and rim, 
that in the rim having the greatest intensity midway between the 
arms and at or near the arms; also, if the speed of rotation is varied 
by a driving or resisting force applied through the shafJ; or hub, 
there will be additional tensile stress induced about one of the angles 



324 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

between the end of each arm and the rim. A pulley carrying a belt 
that is transmitting power has similar stresses. 

Only tensile stresses have been considered in the discussion of 
the fly-wheel, because they are greater than those of compression, 
and also because most fly-wheels, and a considerable proportion of 
large pulleys, are made of cast iron, which has a much lower tensile 
than compressive strength. 

When a fly-wheel is supported on a horizontal shaft, there is an 
additional tension in the lower arms. The tension in an arm 
extending vertically downward from the hub is much greater than 
that due to the portion of the rim belonging to the arm, i.e., greater 
than that due to a weight equal to that of the whole rim divided by 
the number of arms. 

122. Numerical example of stresB in, and enlargement of, a 
rotatii^^ ring due to centrifugal action. — The tensile stress in a 
thin cast-iron ring of any diameter, rotating about its geometrical 
axis at a circumferential velocity of 6000 feet per minute, is, by 
equation (158), taking w = 450, 

w ,^, 460 /6000\' ^„^,, 

^ = 144^ ^ = 144X32.;? X 1-60-) = ^^^ ^^'' ^'' ^^^ ^^- 

The increase of the radius caused by the rotation is for a rin<r 
20 ft. in diameter, by equation (159), taking E^ = 14000000 lbs. 
per sq. in. , 

The corresponding increase in the circumference is 
2/T X .0083 = .052 in. 

The arms in a fly-wheel of ordinary design are considerably 
smaller near the rim than at the hub. "The elongation is therefore 
not uniformly distributed along an arm, but is greatest at or near its 
smallest cross-section. For a rotating wheel having a heavy, stiff 
rim, it seems fair to assume that the greatest elongation per unit 
length in the arm is at least as great as would be obtained per unit 
length in a prismatic bar of a total length equal to the diameter of 
the rim, by subjecting it to a tensile stress sufficient to elongate it 



FLY-WHEELS AND PULLEYS. 325 

to the same extent as the diameter of the free rim would be increased 
by the rotation. 

Upon this assumption the tension in each arm supporting the 
ring just considered, taking the arm as 83 inches long and allow- 
ing for no distortion of the hub, would be 

0083 
Tension in arm = ' ,, X 14000000 = 1400 lbs. per sq. in. 

The rim-tension of 970 lbs. per sq. in., and the tension of 1400 
lbs. per sq. in. in the arms are of comparatively small value when 
considered separate from stresses due to other causes. But, since 
the rim- and arm-tension increase as the square of the velocity, 
they may attain considerable values for very high speeds, such as 
sometimes occur when an engine "races." 

123. Sectional-rim fly-wheels and pulleys. — When the rim of a 
fly-wheel or pulley is divided into sections by being cut through mid- 
way between each pair of arms, as in Fig. 127, the ends of the sec- 
tions must be held together, of course, by some kind of fastening. 
As has already been stated, centrifugal action tends to bend or bow 
the rim out between the arms. If the wheel is rotating uniformly 
and has no belt upon it, each part of the rim between two arms 
acts much like a beam supported at the ends and uniformly 
loaded from end to end. The centrifugal action in the wheel cor- 
responds to a uniformly distributed load acting radially outward. 
This bending outward of the rim has a tendency to open up the 
joint more at the outer than the inner side of the rim. For this 
reason the fastenings, which are tension-members, should be placed 
as near as possible to the face or outer circumference of the rim, 
and should have a considerable thickness of metal, measured radi- 
ally, between them and the inner circumference of the rim. 

If the rim is thin, as for band-wheels, lugs or flanges must 
generally be added for the fastenings. These lugs should be of suf- 
ficient radial height to make the distance from the fastenings to the 
inner side of the lug large in comparison with the distance from the 
centre of the fastenings to the face of the pulley, and they should 
be braced well with circumferential ribs running back towards the 
arms. In order to give the rim rigidity to resist bending, these 
ribs should also have a considerable radial depth. 



3:26 FORM, STRENGTH, AND PROPORTIONS OF PARTS, 

In recent years a number of large pulleys have been designed 
with sectional rims divided at the end of each arm, there being no 
joint in the rim between the arms. 

124. Bursting tests of small oast-iron fly-wheels by centrifugal 
action. — A number of tests were made upon small cast-iron fly- 
wheels by Prof. C. H. Benjamin in order to determine the sx)eed 
at which they would burst when rotating with practical uniform- 
ity.* The wheels tested were all of cast iron, and most of them 
were miniatures of large fly-wheels designed by leading manufac- 
turers of such machinery. Some of the miniatures were 15 inches, 
and the others 24 inches, in diameter. Part had solid rims, others 
had rims made in two sections and fastened together with bolts, while 
still others had rims in two sections fastened together with links. 
The wheels were direct-connected to a steam-turbine, which drove 
them at a gradually increasing speed until they flew to pieces. No 
brake or pulley was applied to cause tangential resistance to rota- 
tion, and no belt was run on the wheel. 

The tests showed that, as would naturally be expected, the 
wheels having sectional rims fastened together midway between two 
arms were much weaker than those having solid rims ; and that, 
although the tension-members of the fastenings at the ends of the 
sections were of less tensile strength than the rim itself, they gave 
way in but one instance, in which case the bolts broke; the rupture 
occurred in the jointed rims of all the other wheels. 

It was also shown that a wheel cast from the same pattern as 
another, but having its rim turned down so as to be much thinner 
radially, broke at a much lower speed than the one having the rim 
left thick, as it had come from the mould. 

Tests upon wheels similar in every respect except the number 
of arms, some having three and some six, showed that the three- 
arm pulleys were weaker than those having six. The pattern for 
the three-arm wheels was made by removing the alternate arms 
from the patterns for the six-arm wheels. 

Tables XXXV to XLI show the dimensions of the fly-wheels 
tested and the results of the tests. When bolts were used to fasten 
the joints, flanges, extending across the inner surface of the rim, 

* The Bursting of Small Cast-iron Fly-wheels. Trans. Amer. Soc. Mech. 
En(r.. vol. XX.. 1899. 



FLY-WHEELS AND PULLEYS. 



327 



furnished holes for the bolts. When two links were used, they 
were imbedded in the side of the rim, after the method of Fig. 128. 
The third link, when used, gripped a pair of lugs extending out 
from the inner surface of the rim. 



Table XXXV. 

FIFTEBN-INOH WHEELS. 





Rim. 


Anns. 


Weight 


No. 


Style. 


Diameter, 
inches. 


Breadth, 
Inches. 


Depth. 
incLes. 


Area, 
sq. In. 


No. 


Area, 
sq. in. 


Wheel, 
pounds. 


1 
2 
8 
4 
5 
6 
7 
8 
9 
10 


Solid 
«< 

«i 

<« 

Sectional 
Solid 

« 

«( 

n 


13 

15 

IS 

141 
14i 
14i 


2 

2 
2 
2 

'2* 
2 
2 

ii 


.70 
.65 
.615 
.52 

.69* 
.615 
.475 
.400 
.847 


1.4 
1.8 
1.28 
1.04 

i!88 

1.28 

.95 

.75 

.65 


6 
6 
6 
6 
6 
8 
8 
8 
6 
6 


.46 
.46 
.46 
.46 
.46 
.46 
.46 
.46 
.46 
.46 


20.87 
20.44 
19.12 
16.62 
20.87 
19.25 
16.56 
18.68 
12.68 
18.00 



Table XXXVI. 

FIFTEBK-INCH WHEELS. 





Bursting Speed. 


Centrifugal 




No. 






V* 


Remarks. 




Revolutions 


Feet per 


= To- 






per Minute. 


Second = V. 






1 


6,525 


480 


18,600 




2 


6.525 


430 


18,500 




8 


6.035 


395 


15.600 


Tbln rim. 


4 


5.872 


880 


14,400 


•< 1* 


5 


2,925 


192 


8.700 


Joint. 


6 


5,600* 


868 


18.600 


Three arms. 


7 


6,198 


406 


16,500 


ti t< 


8 


5,709 


868 


18,600 


<i «< 


9 


5,709 


865 


18.800 


Thin rim. 


10 


5,709 


861 


13,000 


<4 << , 



• Doubtful. 



328 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

Table XXXVIl. 

TWENTY-FOUR-INCH WHEELS. 









Shape and Size of Rim. 


Wheel, 


No. 














Diameter, 
inches. 


Breadth, 
inches. 


Depth, 
incnes. 


Area, 
sq. in. 


Style of Joint. 


pounds. 


11 


24 


^ 


1.5 


8.18 


Solid rim. 


75.26 


12 


24 


4 


.75 


8.85 


lulerual flanges, bolted. 


98 


18 


24 


4 


.75 


:5.85 


H »• l< 


91.75 


14 


24 


4 


.75 


8.86 


l< <l *t 


95 


15 


24 


ft 


.75 


3.85 


tt €i l< 


94.75 


16 


24 


2.1 


2.45 


Three lugs and links. 


65.1 


17 


24 


1.2 


2.1 


2.45 


IVo * 


65 



Table XXXVIII. 

FLANGES AKD BOLTS. 





- Flanges. 


Bolls. 


No. 


Thickness, 
inches. 


Effective 

Breadth, 

inches. 


Effective 
Area, 
inches. 


No. to 
Each 
Joint. 


Diameter, 
inches. 


Total Tensile 
Strength, 
pounds. 


12 
18 
14 
15 


11/16 
11/16 
15/16 
15/16 


2.8 
2.75 
2.75 
2.5 


1.92 
1.89 
2.58 
2.84 


4 
4 
4 
4 


5/16 
5/16 
5/16 

8/8 


16,000' 
16,000 
16,000 
20,000 



BT TESTtNO-MACHINB. 

Tensile strength of cast irpn = 19,600 pounds per square inch. 
Transverse strength of cns^t iron = 46,600 pounds per square inch. 
Tensile strength of 5/16 bolls = 4.000 pounds. 
TensilesU'engtU of. 3/8 bolts = 5,000 pounds. 



FLY-WHEELS AND PULLETS. 



329 



Table XXXIX. 

FAILURE OF FLANGED JOINTS. 





Area 

of 

Rim. 

sq. in. 


Effeotlve 
Area 

Flanees, 
sq. in. 


Total 
Btrensrth 

Bolts, 
pounds. 


Bursting Speed. 


Centrifugal Tension 




No. 


Rev. 

per 

Minute. 


Ft. per 

Second 

= V. 


Founds 
per So. In. 

~ io' 


Total 
Pounds. 


Remarlcs. 


11 
12 
13 
14 
15 


8.18 
3.85 
8.85 
8.85 
8.85 


i!92 
1.89 
2.58 
2.84 


16,666 
16,000 
16,000 
20,000 


8,672 

V,766 
1.875 
1,810 


885 

184 
196 
190 


14,800 

* 8.466 
8.850 
8,610 


47,000 

18,'lbb 
14.800 
13,900 


Solid rim. 
FlHiige broke. 

Bolts broke. 
Flange broke. 



Table XL. 

LINKED JOINTS. 





Lugs. 


Links. 


Rim. 


No. 


Breadthi 
in. 


Length, 
in. 


Area, 
sq. in. 


Number 
Used. 


Effective 

Breadth. 

in. 


Thick- 
ness, 
in. 


Effective 
Area, 
sq. in. 


Max. 
Area, 
hq. in. 


Net 
Area, 
sq. in. 


16 
37 


.45 
.44 


1.0 
.98 


.45 
.48 


3 
2 


.57 
.54 


.827 
.880 


.186 
.205 


2.45 
2.45 


1.98 
1.98 



BT TESTING-MACHINE. 

Tensile strength of cast iron = 19,600. 
Transveree strength of cast iron = 40,400. 
Av. tensile strength of each link = 10,180. 

Table XLI. 
failure of linked joints. 





Strength 

of 

Links. 

pounds. 


Strength 
of 
Rim, 

pounds. 


Bursting Speed. 


Centrifugal Tension. 




No. 


Rev. 

per 

Minute. 


Ft. per 

Second 

= V. 


Poundj« 
persq.^ln. 

~ To- 


Total. 
Pounds 


Remarks. 


16 
17 


80.540 
20,360 


88,800 
38,800 


8,060 
2,750 


320 
290 


10.240 
8,410 


25,100 
20,600 


Rim broke. 
Lugsand rim broke. 



330 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

124.1. Effect of ribB on pulley rims. — In the use of circum- 
ferential ribs on the rims of pullejrs care should be taken to .see 
that they do not weaken the rim instead of strengthening it. They 
will generally stiffen the rim whether it is made stronger or weaker. 
In Tabic XLIa three cross-sections of equal area are shown. The 
section modulus or resistance factor of each is given for both steel 
with its tensile and compressive strengths equal, and for cast iron 
four times as strong in compression as in tension. 

With the steel the section modulus of a ^ven form of section 
is the same with the ribs in tension a*s in compression. This is 
not true of the cast iron for a section ribbed on one side only. 

Deep ribs should be used in steel and cast-iron sections whose 
ribs are in tension. In cast-iron sections whose ribs are in com- 
pression the ribbed section is always stronger than the rectangular. 

124.2. Problem. — Find the speed of rotation of a st«el arma- 
ture ring with 24 lugs of 100 pounds each, attached as shown in 
the Fig. 127.1, which will cause a tensile stress of 14,500 pounds 
per square inch in the ring, and 12,000 pounds per square inch in 
the lug-bolts. 

The following notation will be used in addition to that in § 117 : 
C = centrifugal force of one lug, pounds; 
P = P+P'= total tension in rotating armature ring due to the speed 

and weight of both itself and the attached lugs, pounds; 
P = total tension in the rotating armature ring due to its own speed 

and weight, pounds; 
P'= total tension, due to the speed and weight of the lugs, in the 

armature ring, pounds; 
a =eflFective area of bolts to resist tension, square inches; 
d = diameter of body of bolts for attaching the lugs to the armature 

ring, inches; 
p = tension, due to its own speed and weight, in a rotating ring of 

one square inch section, pounds. 
If V is taken as the linear velocity of the ring at its mean radius 
of 52.5 inches, then the linear velocity of the middle of the lugs, 
which lie in a circle of 57 inches radius, will be 

^X.= 1.0861,. 



FLY-WHEELS AND PXJLLBTS. 



331 



i 


i 1 1 




1 




i 


' ^ 


1 


* 




1 








T 


• ^ 


Y 


- 




r TT 


^Y 


f- 




-> 


f 


e 










^ Z 








f 2 








Z-' 




































5 




























ii 


























S-- 




-* 


3 


t— 


— > 




t— 






fe- 




c » • 










J 


r 


1 


1 


T 




< 


1 


ECT 
IN 1 
TRA 


















1 






^ o o 






1 






1 














5S-" 












2 


A t 














jp » 










J 


kj > 
















1 








1 


















i-« 






-t5- 




1 


1 


«r 






r X 


1 






1 




' 




t 


T ' 


1 




o o o 


AREA OF SECTION 




Ot Ot GTf 


SO. INCHES. 




S S[ a 






S CO CO 


MOMENT OF INERTIA. 




•co lO 00 






•-»• 






» CO •♦^ 


•< 




-^ 






(0 








N 




CD 








» 2 








s»| 








5S^ 


& 




g ^ ^ 


8TE 

TREN 
9TRE 
TENS 


§ 




^ CO i-k 


» H X l^ 


;i 






o F 1 


2 o 






g S 


3g 






y ? 








^ 






^ CO Hi> 


is 

2 Z 


CAST IRON 

COMP. STRENGTH as 
4 X TENS. STRENOTH 


ii 






S a 




.8 ? S 

h9 00 U 


11 


m 






O 










z 










332 



FOKM, STRENGTH, AND PROPORTIONS OF PARTS. 



The force necessary to prevent a lug weighing 100 pounds 
from flying off the armature ring, when the ring rotates at a linear 
velocity of v feet per second, is expressed by the equation 



C= 



100(1.086v)' lOOX 1.1794t^X 12 



<7X57^12 



32.2X67 



' .771t;* pounds. 




O 
O 



• 




Fig. 127.1. 



The effective sectional areia of the bolt that holds the lugs in 
place is that at the bottom of the threads. Since the diameter of 
the bolt is not yet determined, the exact ratio of the diameters of 



FLY-WHEELS AND PULLEYS. 33:> 

the body and of the root of the thread is not known. It is on the 
safe side to assume that the body is 1.18 times as large in diameter 
as the root of the thread. This is a somewhat larger ratio than 
really exists for bolts from 1.75" to 3" diameter of body. 

Under this assumption a bolt in tension, and of a body diam- 
eter d, has an effective area of bolt 

ncP 



4(1.18)»' 

Two bolts are to be used for each lug. Their combined tensile 
strength must equal the centrifugal force of one lug, whose valuo 
has just been determined as .771v'. 

The equation of the bolt area and the centrifugal force of (! > 
lug is therefore, for two bolts in each lug, at a tensile str^s 
12,000 .pounds per square inch, 

12000x27rrf^ 771.^. 
4X(1.18)» ==^^^^' 

whence 



|2v 771 

d=1.18i;X J-j|^^=.007547v inches. 

The tension per square inch in the ring due to its rotation, 
taking the weight of steel as 490 pounds per cubic foot, is, for ita 
own mass, 

P°^'° 144X32.2 ^'^^^^^' pounds per square inch. 

The given cross-section of the ring is 

3// X 12'' =36 square inches. 

The total tension in the ring due to its own weight and speed 
is therefore 

P- 36X .1057t?*= 3.804v^ pounds. 



334 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

The tension in the ring due to the weight and speed of the 
attached lugs may be found by a method analogous to that com- 
monly used for determining the circumferential tension in cylin- 
drical boiler shells, tubing, pipes, etc., due to the internal pressure 
of an enclosed fluid or gas. The relation between the circum- 
ferential tension and the internal pressure is 

Total circumferential ) press, per unit area Xdiam.X length 
tension in tube ) "" 2 

_ total ra dial pressure 
" 2;r 

Adapting this to the armature, the centrifugal force of the 
24 lugs corresponds to the total internal pressure of the tube. The 
centrifugal force for one lug of 100 pounds has been found to be 
.771v' pounds. The tension in the ring due to the speed and 
weight of the 24 lugs is therefore 

P'=?i><^=2.95.^ 
2;r 

The total tension in the armature ring is the sum of the ten- 
sions P and P'. 

The tension per unit area in the ring is found by dividing this 
amount by the effective area. The least effective area of the 
armature ring is on a section through the centre Unes of a pair of 
bolt holes. The solid area of the ring is here reduced by an amoimt 
«qual to the thickness of the ring times twice the diameter of the 
bolt. Therefore, for a fibre stress of 14,500 pounds per square 
inch, the following equation can be written: 

P+P' 



14,500= 



effective area of armature ring 
3.804v»+2.95t;^ 



and 



36 -(3X 2 X. 007547V " 
i;=233.9 feet per second 



y= 14030 feet per minute, 
which corresponds to a 

Speed of rotation =i^^= 496.2 R.P.M. 



FLY-WHEELS AND PULLEYS. 335 

The outer diameter of the armature ring is here used in order 
to roughly counterbalance the error introduced by using the mean 
diameter of the ring and the diameter of a circle through the 
middles of the lugs, instead of slightly larger diameters, in the 
calculations for tension due to rotation. 

The diameter of the bolts may now be obtained by introducing 
the value of v just determined into the equation for d developed 
earlier in the problem; thus 

d= .007547^= .007547X233.9- 1.766 inches. 

A diameter d of 1.75 inches would naturally be used in practice. 
The values of the tensions in the ring and bolts may now be 
detennined as a check, using 1.75-inch bolts. 

Net sectional ) ^3x12-3X2X1.75=36-10.5=25.5 square ins. 
area of ring ) ^ 

Tension m\ P+P' 6.750i;^ 



} - 



ring ) net section 25.5 

6.750X ( 233.9)' ...^^ , . , 

*' 9^"^ '^ 14480 ix)unds per square inch. 

A bolt of 1.75" body has, according to tables of standard 
bolts, an 

Area at root of thread = 1.746 square inches. 

Therefore 

Tension ) .771v' .771 X (233.9)* ,^^^^ , 

in bolts I =2-^0746= 2x1.746 ^^^^^P^^^^^ Ver sq. m. 

There are, of course, bending stresses in the ring, due to the 
localization of centrifugal force at the points where the lugs are 
attached. The arms of the spider supporting the ring are also a 
cause of bending stresses. 

At any other speed of rotation the tensions in the ring and 
bolts are proportional to the square of the velocity. 



336 FOBMj STRENGTH, AND PROPORTIONS OF PARTS. 

Special Forms of Fly-wheels and Pulleys. 

125. Hollow cast-iron arm8 with wrought-iron or steel tension* 
rods for fiy-wlieels and pnlleys. — On account of the comparatively 
low tensile strength of cast iron and the predominance of tensile 
stress in the arms of pulleys and fly-wheels, it is obvious that the 
liability of the arms to fracture will be reduced if, by some means, 
the cast iron can be relieved partly or wholly of tensile stress, al- 
though by doing so it may be subjected to high compressive streso. 
A simple and comparatively inexpensive way of doing this is to cast 
the arms hollow and put a steel or wrought-iron tension-rod through 
the centre of each from rim to hub. The rod is put in tension by 
means of its end fastenings when the pulley is constructed. The 
surrounding cast-iron walls act as a column to resist this tension 
and are thereby subjected to compression. A pulley so constructed 
is unquestionably much safer under heavy service than one having 
only cast-iron arms. This device has been adopted by at least one 
concern which has placed many large pulleys in commercial service. 

126. ''Tangent" arms for pulleys and fly-wheels. — Pulleys of 
this description have arms which, instead of being radial, are tan- 
gent to a circle of fixed diameter, as in the construction common 
to bicycle-wheels. Tangent-arms have been adopted for wheels of 
considerable size performing service where the demands upon the 
fly-wheel for energy are sudden and severe. Such fly-wheels have 
been applied to mine hoi sting- machinery. The arms, instead of 
being round, are generally of rectangular cross-section. Being 
purely tension-members, they are of course made of some such ma- 
terial as wrought iron or steel, cast iron being totally unfit for this 
purpose. The tangential direction of the arms prevents any neces- 
sity ol their resisting the torsional moment of the shaft or rim by a 
bending moment in the arm. It is not probable that even a very 
approximate calculation of the stresses in the arms can be arrived at. 

127. "Built-up" plate fly-wheel.— A fly-wheel made of rolled 
structural-steel plates is in use in the power-station of the Union 
Eailroad Co. at Providence, K. I. The web of the wheel is made of 
a number of segmental plates placed so as to break joints, and held 
together by rivets passing through from side to side. The rim is 
built tip of several pieces of plate metal cut so as to form a rin^: 



FLY-WHEELS AND PULLETS. 337 

when placed together. These sections are placed together so as to 
break joints and are riveted through from side to side. . Steel 
plate 1 inch thick is used for the web, and If and 1^ inches thick 
for the rim. The hub is of cast iron 72 inches in diameter, made 
of two disks. It can be seen that the pulley has what might be 
called a built-up web. The pulley is 18 feet in diameter and has a 
rim 15|X16 inches. The web is made of two thicknesses of one- 
inch plate. The factor of safety is about forty. 

128. Wire-wound fly-wheel. — A fly-wheel having a rim com- 
posed of wire wound circumferentially under tension around a 
web was constructed for use in a rolling-mill at Ladore, Wales, 
using the Mannesmann process of rolling tubing from the solid bar. 
In this class of rolling a very great amount of power must be sup- 
plied to the machinery in a short time. The strain upon the fly- 
wheel is accordingly very great on account of the sudden reduction 
of speed which it must undergo in order to deliver its energy, as 
well as on account of the high speed at which it runs in order to 
have sufficient energy stored in it. 

The construction adopted is the strongest and safest that can 
be devised. Two steel disks, 20 feet in diameter, are bolted to a 
cast-iron hub. The outer edges of the disks form a groove into 
which wire is wound to form the heavy rim. The groove is filled 
with 70 tons of No. 5 steel wire wound on under a tension of 50 
pounds. The wheel is run at 240 revolutions per minute, which 
corresponds to a circmnferential velocity of 15080 feet =2.85 
miles per minute. 

129. Other special forms of pulleys. — ^Many large pulleys are 
now constructed with built-up rims and cast-iron arms. The rims 
are frequently made heavy enough to serve as fly-wheels. 

Smaller pulleys are often made completely of wood built up in 
sections and having but a few arms, generally two or four. 

Medium-size pulleys are often made with wrought-iron or sheet- 
steel rims and cast-iron arms. Small-size pulleys are also frequently 
constructed in this way. 

An "all-steer' pulley has recently been placed on the market. 
It is made up completely of sheet metal, the arms and hub being 
pressed into form. 

Two sets of arms are very conmionly used for wide pulleys, and 
in some cases as many as three or more sets are \ sed. 



FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



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FLY-WHEELS AKD PULLEYS, 



339 



Designs and Proportions of Fly-wheels and Pulleys Taken from 

Practice.* 

ISO. Since it is evidently impossible to make calculations for 
fly-wheels and pulleys with regard to their strength, it seems advis- 
able to present some of the larger sizes, as made by leading builders, 
which have shown themselves to be satisfactory in use; also to 
give some of the proportions of the smaller sizes of pulleys. 

Table XLII gives the sizes of arms for pulleys from 10 to 96 
inches in diameter and having rims with faces up to 26 inches 
wide. These are the proportions adopted by Struthers, Wells & 
Co. for the pulleys used upon their engines. Table XLIII gives 
the thicknesses of rims adopted by the same company. 

Table XLIII. 

ATEBAGB THICKNESS OF ENGINE-PULLEY EIMS. 



Wide Pulleys. 



Diameter, 
inches. 



13 
18 
20 
28 
86 
44 
44 
48 
54 
60 
60 
72 
72 
72 
78 
78 
78 
78 
84 
84 
96 
96 
108 
108 
120 
120 



Face Width, 
inches. 



Thickneai 
of Rim, 
inches. 



5/16 

3/8 

7/16 

1/2 

9/16 

1/2 

9/16 

9/16 

11/16 

8/4 

15/16 

7/8 

15/16 

it 

11 

11 

u 
n 

2 
2i 

2* 
8 



Narrow Pulleys. 



Diameter, 
inches. 



6 

8 

10 

14 

16 



86 
86 
42 
48 
48 
54 
54 
54 
60 
72 
72 
78 
78 



Face Width, 
inches. 



Thickness 
of Rim, 
inches. 



1/4 

5/16 

8/8 

7/16 

7/16 

7/16 

1/2 

5/8 

5/8 

7/8 

7/8 



* The following illustrations of fly-wheels and pulleys are taken from 
drawings and blue-jmnts kindly supplied by the establishments whose names 
are mentioned in connection with them. 



340 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



Fig. 128 is a 20-foot band-wheel with a 62-inch face and having 
a rim divided into eight sections as made by the Edward P. AUis 
Co. The rim-sections meet at the ends of the arm. The weight 
is 65000 pounds. 




Fig. 129 is the working drawing of the hub, one arm, and one 



( 

i ' ■ 

I 

! \ 



4 • \ 



. > , .' l .i . ' All- 




Pie. 128. 
2(Kfoot band-wheel, 82-inch face Cast iron. 8 rim sections. Weight 




Tofactpag* \ 



)00 ])ound8. Edward P. Allis Co., Milwaukee, Wis. 



FLT-WHEELS AND PULLSTS. 



341 




I 
I 

CO 

I 

c 

I 

ti 



I 



'-' o o 

111 

si 

d 



I 



o 

.13 












842 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



section of the rim of a 10-foot band-wheel having a 62-inch face 
and weighing 60000 pounds as made by the Hooven, Owens & 




Eentschler Co. The rim is divided into ten sections, each section 
having one arm. The arm in this design is different from those 



,. ■ / 




Fig. 
35-foot flj-wheeL Cast iion. 8 aectiona. Weight 161 




To face pag€ 9iZ^ 



pounds. Fraser & Chalmers, Chicago, 111. 



FLY-WHEELS AND PULLETS. 



343 




844 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

Qsnally adopted. It is made in the fonn of a cast*iron I beam* 
This form presents advantages in the way of getting a sound cast- 
ing and in making an arm which is strong enough to resist a turn- 
:€ig force acting upon the rim or hub. A wheel of this design ia 
operating successfully at 76 levolationB per minute. 




Figs. 130 and 131 are from the working prints of a fly-wheel 23 
leet 6 inches in diameter as made by the same company. This 
#heel is also divided into ten sections. The net weight is 128421 
pounds. One of the wheels made after this design is direct-con- 



FLT- WHEELS AND PULLEYS. 



345 



nected to an 800-kilowatt generator speeded at 80 revolutions per 
minute. 

Fig. 132 is a 25-foot fly-wheel weighing 160000 pounds and 
divided into eight sections. Four of these wheels have been made 




by Fraser & Ohalmers, and put into operation in the power-station 
of the- West Chicago Street Eailway Co. They are on 34 and 
54 X 60 inches, compound condensing Corliss engine^ which are 
direct-connected to electric generators. 



346 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

Fig. 133 represents another fly-wheel as built by the same 
company. It is 12 feet in diameter and made in two parts* The 
weight is about 12000 pounds. 

Fig. 134 is another 12-foot fly-wheel made by Fraser & Chal- 
mers. The design is peculiar in that the rim is overhung in order 
to bring its weight over the bearing which supports it. The 
weight of this pulley is 12500 pounds. Two of these wheels are in 
use on each of the pumping-engines for delivering water at high 
pressure to the hydraulic accumulators in the works of the Fope 
Tube Co. The water pumped is utilized for tube-drawing benches, 
etc. 

Fig. 135 represents a 20-foot fly-wheel weighing 80000 pounds 
as constructed by the Southwark Foundry & Machine Co. The 
wheel is made in two parts, held together by links at the rim and 
bolts at the hub. 

Fig. 136 is the wheel uded on the direct-connected electric gen- 
erating units of the Metropolitan Railway, N"ew York City. 

Fig. 136.1 is a band-saw wheel with staggered arms which give 
rigidity and allow at the same time for shrinkage of the casting. 



CHAPTER XL 
CYLINDERS, TUBING, PIPES, AND PIPE-COUPLINGS. 

131. In the following discnssion of cylinders, tubing, and pipes, 
they are separated into two classes, according to whether the walls 
are thin or thick, -on account of the difference in the nature of the 
stress occurring in thin and thick walls. It is assumed that all the 
cross-sections discussed are circular. 

The notation for cylinders, tubing, and pipes is as follows: 

D = external diameter, inches; 

V = cross-sectional area of wall of cylinder, square inches; 
T = total circumferential tension per inch of length in the wall of 

the cylinder, pounds; 
T = total longitudinal tension in the wall of the cylinder, pounds; 
a = thickness of wall, inches; 
d = internal diameter, inches; 
f = internal pressure, pounds per square inch ; 
t = circumferential tension, pounds per square inch; 
V = longitudinal tension, pounds per square inch ; 
n = 3.1416. 

132. Tension in a thin oircular cylinder due to internal pres- 
iure. — ^When a pipe or tube is closed at the end and contains a 
liquid or gas under pressure, there is a tendency to burst the pipe 
which produces tensile stress in its walls. 

If the pipe is long and the walls thin in comparison with its 
diameter, the relation between the internal bursting-pressure and 
the total circumferential tension in an inch of length, taken at some 
distance from the end so that there will be no strengthening effect 
of the cap or head, is expressed by the equation 

T^^ipd. (160) 

847 



348 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

This expression is obtained in the same manner as that for the 
tension in a shrunk-on ring (see § 90 and Fig. 118). 

The circumferential tension per square inch in the walls when 
they are thin in comparison with the diameter is 

rp -I ^j 

^ "^ "^ "^ 2 a (*PP^^^™*^^^^' • ^^^^^ 

The total longitudinal force tending to separate the ends of the 
pipe is 

r=^ (162) 

The wall area resisting this force is that of the cross-section cut 
by a plane normal to the length of the pipe^ and is 

F' = n^L-^ (163) 



The walls being very thin in comparison with the diameter of 
the cylinder, their cross-sectional area is 

F' = Tida (approximately) . . (164) 

The longitudinal tensile stress per square inch, as obtained by 
combining equations (162) and (163), is 

'' = 5^. W 

And the approximate longitudinal stress per square inch, as 
obtained by combining equations (162) and (164), is 

^ = J — (approximately) . . (166) 

Equations (161) and (166) show that for pipes with thin walls 
the circumferential tensile stress per unit area is twice as great as 



CYLINDEK8, TUBING, PIPES, AND PIPE-COUPLINGS. 349 

the longitudinal. In other words, the tendency to burst the pipe 
longitudinally is twice as great as circumferentially. 

The cylinder-head, flange, or cap upon the end of the cylinder 
or pipe aids the material near the end of the pipe in resisting the 
circumferential tensile stress due to internal pressure. If, there- 
fore, the cylinder is short in comparison with its diameter, its 
capacity to resist pressure will be greater than that of a long one. 

Example. — The circumferential tensile stress in a pipe 7 inches 
inside diameter and 7f inches outside, when withstanding an 
internal pressure of 2500 lbs. per square inch, is, for thin walls, by 
equation (161), 

t = ^(2500 X 7) ^ T^ = 2736 lbs. per square inch. 

The total end pressure on the pipe may be obtained by equation 
(162), whence 

r' = ^«-«^ = 96212 lbs. 
4 

And the longitudinal tension per unit area is, by equation (166), 

/' = i(2600 X 7) -4- T^V = 1367 lbs. per square inch. 

133. Cylinder with thick weUb. Stress in material due to 
internal pressure. — When the walls of the cylinder are thick in 
comparison with its diameter, the circumferential tensile stress in 
them due to the internal pressure is not uniformly distributed 
throughout the material, but, on account of the elasticity of the 
walls, is greatest at the inner layer of metal and least at the outer, 
gradually diminishing from the inside toward the outside. 

Equations (167) and (168) agree with those given in several 
books on the mechanice of materials, etc. They are based upon 
the assumption that the volume of the material forming the cyl- 
inder does not change when put under stress (i.e., the material is 
assamed to be incompressible). Under this assumption the cir* 
cumferential stress in any fibre of the material is inversely propor- 
tional to the square of its distance from the axis of the cylinder. 



360 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 
The equations for cylinders with thick walls are: 



(167) 



' = \^, <»«> 



The influence of flanges and closed ends is doubtless greater in 
thick cylinders than in thin ones. For such short cylinders as are 
commonly used for hydraulic presses, etc., the strengthening effect 
of the closed end is very great. 

134. Bursting tests of cylinders and pipes. Lap - welded 
wrought-iron pipe.* — The results of a series of tests made by 
R. W. Hildreth & Co. of New York upon a large number of pieces 
of lap- welded wrought-iron pipe ilhisfcrate so well the manner in 
which piping and couplings may give way under pressure that a 
very brief abstract of the log of tho tests will be given. 

The pipes tested were all 7 inches inside diameter and 7f inches 
outside. The sections were held together by cast-iron flanges of 
the form shown in Fig. 137. Three 1-inch bolts were used in each 
pair of flanges. The pipes and flanges were threaded with 8 
threads per inch, having a taper of -^ of an inch in 2^ inches. 
When screwed together, the pipe protruded beyond the flange -^ of 
an inch. 

The gaskets were of gutta-percha 8.7 inches external diameter, 
so as to fit into the recess of the flange. 

In the first series of experiments six lengths of pipe, each about 
20 feet long, selected at random, were coupled together and tested 
under high pressure. The longitudinal tension due to the water- 
pressure against the heads of the pipe was resisted by the pipe and 
flanges, no auxiliary devices being used to resist this end thrust. One 
flange broke at a pressure of 1700 pounds per square inch, and 
after this was repaired another broke at an internal pressure of 2000 
pounds per square inch. 

* Engineering New§, March 21, 1895. 



CYLINDERS, TUBING, PIPES, AND PIPE-COUPLINGS. 361 

In the second series of tests the ends of the pipes were tied to- 
gether from end to end with two 3-inch rods having sleeve-nuts for 
tightening; the pipe was also secured against buckling. At the 




Pig. 187. 



first trial one pipe burst six feet from the end at an internal pres- 
sure of 2400 pounds per square inch. The fracture showed a blis- 
ter about 12 inches long. At the second trial a flange cracked at 
800 pounds pressure. It was supposed that the bolts had not been 
tightened properly. At the third trial another flange broke with 
violence at 2300 pounds pressure. And at the fourth trial still 
another flange broke at 2500 pounds pressure per square inch. 

All the flanges that broke were of the form Ay Fig. 137. Each 
broke about midway between the lugs through which the bolts 



As a result of the tests it was recommended that a round flange 
with five or six |-inch bolts be specified instead of that with three 
lugs and bolts. It was stated that ^' with such flanges perfect pipe 
of the same quality and dimensions as those ordered should 
withstand a pressure of from 2500 to 3500 pounds per square 
inch." 

Tests of drawn tubing. — A number of so-called ^' seamless'^ 



364 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

gasket of strawboard saturated with oil, this difficulty was largely 
obviated. This packing held except when the projection on the 
head was smaller in diameter than the counterbore in the cylinder. 
One of the serious difficulties met during the test was leakage 
through the cylinder walls on account of minute blowholes. Some 
of them were' almost inyisible to the naked eye, but at high pressure 
the water would spurt out several feet in a slender stream. Water 
oozed through the walls at all points. 
The log of the tests is as follows: * 

^* Cylinder a. — Wire-gauze packing; leaked at 400 pounds. 
Substituted copper wire No. 22, A. W. 6.; this leaked at 600 
pounds. Substituted soft-rubber gasket; pressure carried to 800 
pounds several times. Leak at blowhole stopped by peening. On 
raising pressure to 775 pounds cylinder failed on a circumference 
just below the upper flange, the crack starting at blowhole and 
running each way about 90 degrees. 

" Cylinder b. — Gasket of lead fuse wire ^-inch diameter with 
ends fused together. Leakage at pressure of 450 pounds, and the 
flange cracked. Substituted rubber and graphite packing; leak at 
crack with pressure of 600 pounds; no further rupture. 

" Cylinder c. — Rubber and graphite packing inserted, heated to 
260 degrees Fahr. by live steam ; bolts screwed down and packing 
left one day to harden. Leaked badly at 600 pounds; renewed 
packing, but it leaked again at 550 pounds. Flanges showed signs 
of failure and experiment was abandoned. 

** Cylinder d. — Counterbored joint, with gasket of straw- 
board soaked in linseed-oil. Leakage at blowholes with 70O 
pounds pressure. Blowholes peened and coated with paraffin, 
when pressure was raised to 800 pounds several times. One blow- 
hole calked on outside; on applying pressure of 700 pounda 
rupture occurred on longitudinal line through blowhole. Several 
small blowholes found in line of fracture. 

'* Cylinder e. — (On this and all subsequent cylinders the 
counterbore and straw-board gasket were used.) Pressure raised 
gradually to 1325 pounds, when rupture occurred on circumference 
under flange. The crack began at several small blowholes. 

* Ulastrations and references to tliem are omitted. 



CYLINDERS, TUBING, PIPES, AND PIPE-COUPLINGS. 3B6 

" Cylinder /. — Pressure raised gradually to about 2500 pounds 
(above graduation of gauge)^ when cylinder failed in same manner 
as preceding one; cylinder leaking badly at time of rupture. 

'* Cylinder No. 1. — Broke at 600 pounds on a longitudinal line 
along a row of blowholes. 

" Cylinder No. 2. — Broke at 1060 pounds around a circumfer- 
ence just under flange. Fracture very clean. 

'' Cylinder No. S. — Broke at 975 pounds in the same manner as 
No. 2, the crack beginning where there was a slight flaw. Fracture 
clean. 

" Cylinder No. 4. — A number of small blowholes near the centre 
of shell caused considerable trouble by leakage, and had to be 
calked inside and out. Rupture finally occurred at 700 pounds 
pressure along a longitudinal line. 

** Cylinder No. 5. — Bnpture occurred ai 875 pounds, a crack 
starting under the flange running part way around and then up 
through flange and head. 

" Cylinder No. 6. — At 476 pounds pressure the bottom head 
broke. On renewing this and raising pressure to 900 pounds the 
top head failed in the same manner. These heads had been used 
for several cylinders, and were probably weakened. The test was 
abandoned at this point for lack of time. 

Qreat pains were taken in casting these cylinders, and they may 
be considered good examples of cast-iron cylinders as made for 
engine- or pump-work. The blowholes mentioned were most of 
them very minute, and under ordinary circumstances would have 
remained unnoticed." 

The percolation of liquids througn the walls of cast-iron cylin- 
ders at high pressure is of common occurrence. Such cylinders 
may be lined with copper, brass, etc., to prevent this leakage. 

135. Special forms of pipes. — ^The high steam-pressures com- 
mon to modem practice, as well as the high pressures used for 
hydraulic and other purposes, have necessitated the use of piping 
made of a stronger material than cast iron. For such purposes 
pipe made of steel plates riveted together, as is ordinarily done 
in the construction of shell boilers, has been adopted to a consid- 
erable extent. The bursting-strength of such a pipe is measured 
by the strength of the riveted joints. 



856 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

** Spiral-riveted" pipo is used for neariy all the sizes of pipes 
common to practice, and especially for those which are thin in com- 
parison with their diameters. It is made by wrapping a long strip 
of flat material, as a thin strip of paper would be wound around a 
lead-pencil, so that the edges overlap. In the pipe the overlap is 
made sufficient for a row of rivets used to fasten the successive 
convolutions together. The edges of the plate must be prepared 
before wrapping, by thinning or offsetting them, so that the diame- 
ter of the pipe will not increase with each turn of the strip from 
which it is made. On account of the necessity of such offsetting 
it is difficult to make this kind of pipe of any great thickness in 
comparison with its diameter. The helical joint is sometimes 
welded instead of riveted. 

136. Pipe-conplingB and -flanges.* — The most essential features 
of a pipe-coupling are that it shall be strong enough to prevent 
fracture when under pressure, and that it shall not leak either 
between the faces of the two flanges or between the pipe and flange. 

The stresses which a flange mast withstand are those due to 
internal pressure in the pipe, together with those caused by bolting 
the flanges together, expansion and contraction of the pipe, and its 
tendency to bend under its weight or to buckle on account of the 
elongation of the pipe by expansion when heated by steam, etc. 
Steel flanges are used for the high steam-pressures of modern prac- 
tice, as well as for pipes for high hydraulic pressures. 

The packing must be so held in position that it will not blow 
out under pressure. If it is held between two perfectly plane 
flange-faces, the friction between it and the flanges, together with 
its own strength, is all that holds it in position. The end thrust 
due to internal pressure has a tendency to separate the flanges, thus 
reducing their clamping force on the packing. This is apt to cause 
leakage and possibly to tear the packing to pieces and blow it out. 
This was plainly shown in the experiments on cast-iron cylinders 
cited in § 134^ By recessing or counterboring one flange and 
forming its mate to fit into the counterbore, the packing, if of the 



* The dimensions of the standard flanges adopted by the Amer. Soc. Mech. 
Engrs. are given in Kent's *' Mechanical Engineers' Pocket-book," together 
with numerous other designs of flanges. 



CYLINDERS, TUBING, PIPES, AND PIPE-COUPLINGS. 



357 



same external diameter as the counterbore and fitted into it^ will 
be held in place much more effectively. 

Fig. 139 represents a method of attaching pipe-flanges that is 
suitable for the high steam pressures of modern engineering. The 




Fig. 189. 

proportions shown in the figure are those for roUed-steOx flanges. 
In order to attach the flange it is expanded by heat and then 
shrank on the pipe so that the end of the latter protrudes beyond 
the face of the flange. The pipe is then peened or spun out so as to 
expand it over the rounded corner of the flange, and is then turned 
off so that it is just flush with the face of the flange. The packing 
P fits in the counterbore and covers the joint between the flange 
and pipe, thus serving the double purpose of preventing leakage 
between the flange-faces and also between the pipe and bore of the 
flange. As an extra precantion a dovetail counterbore is some- 
times made as shown by the dotted lines at A^ into which a piece 
of some malleable material, such as copper, can be calked to pre- 
vent leakage between the pipe and flange. 

For exceedingly high pressures and correspondingly thick pipe, 
the pipe and flange are both threaded and screwed together, the 
rest of the details remaining the same. 



358 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

Fig. 1 and Table XLV show the form and proportions of 
weldless rolled-steel flanges for pipes from 8 to 72 inches outside 
diameter and for pressures up to 150 pounds per square inch. 

->i N 




Fto. 140. 
Fig. 140.1 shows two pipe couplings of nearly similar design. 
The proportions are given in Table XLVa. The bolt holes are 
1/48 of the diameter of the bolt larger than the bolt. Pattern A 




PATTERN A 



PATTERN B 



Fig. 140.1. 



is for pipes from ^ inch to 7 inches diameter, inclusive. Pattern B 
is for sizes larger than 7 inches up to about 17 inches. The flanges 
are of rolled steel. When used on the thicker pipes these joints 
have been satisfactory under an internal pressure of 7000 pounds 
per square inch. It is suggested that the flanges could be reduced 
in weight without injuring their efficiency by beveling the comer 
on the inner surface as shown by the dotted lines of Pattern A.* 



CYLINDERS, TUBING, PIPES, AND PIPE-COUPLINGS. 



359 



A pipe is sometimes riveted to the flange by a circumferential 
row of rivets extending radially through a lip running back over 
the pipe on the side of the flange opposite its face. This lip cor- 
responds to the ones extending out toward A and B in Fig. 139. In 
such a joint the pipe extends only partly through the flange and 
is calked at the end to prevent leakage. 

Flanges forged or cast as an integral part of the pipe are iLsed 
to some extent, chiefly for thick pipes and high pressures. 

Fig. 140.2 shows a form of pipe coupling designed by Mr. A. 
H. Emery and used on small sizes of pipe for high hydrauUc pres- 
sure. The illustration is full size. The small pipe, shown is steel 
and the other parts bronze. The joint is made tight by giving 
slightly different angles of bevel to the ends of the pipes so that 




Fig. 140.3. 
the parts coming together are in contact only in a narrow ring of 
the smallest diameter posssible. A fine thread is generally used, 
12 or 20 per inch, preferably the latter, except for cast iron. 

The particular joint shown was subjected to an internal pres- 
sure of 6000 pounds per square inch and showed no leak. Joints 
similarly constructed have held without leak under 10,000 pounds 
per square inch internal pressure.* 

137. Expansion-couplings for pipes. — In order to allow for 
the change of length which always occurs in pipes with change of 
temperature, some form of expansion-coupling must be introduced 
at intervals in order to prevent excessive stresses in the material. 
Numerous devices have been adopted for this purpose. Those most 
* Communication from Mr. A. H. Emery, Jr. 



360 form, strength, and proportions of parts 

Table XLV. 

dimensions of weldless rolled-steel flanges for pipe 
carrying pressure up to 150 pounds per square inch. 

(Dimensions in Inches.) (Refers lo Fig. 140.) 



Diameter, inside. 

A 
O 

G 
H 
L 
M 

N 

^ Rough Wt, 
Fin. Wt. I 
Male f 
Finished Wt. Female 



% 



12 

mi 



!ft 



18 



186 
115 
ISO 



15 

iP 

149 
156 



16 



:ft 



18 

5 

235 
904 

218 



Diameter, inside. 

A 

G 
H 
L 
M 
N 

i Rough Wt, 
Approx.-JFin^Vt. I 

Finished Wt. Female 



19 

19) 
1 
265 

232 

242 



20 



i^ 



295 
257 



28^ 

m 

871 
887 



80 






482 



86 

45H 
89 
2H 



666 

595 
612 



42 

45 



705 



used may be divided into two general classes, however: Ist. Thosa 
which operate by telescopic action, generally having an accnrately 
turned tube which slides back and forth through a stuffing-box 
packed in a manner suitable to allow such motion after the method 
common for piston-rod glands of steam-engines. 2d. Those in 
which the elasticity of one of the parts, called the expansion-piece, 
is depended upon to allow a slight end motion of the pipes. One 
form of this coupling consists of a short length of pipe made of 
rather thin copper, corrugated circumferentially. The corrugations 
allow the piece to change its length endwise under moderate end 
pressure. Another form consists of a pair of ilanges, one much 
larger in diameter than the other, fastened together in a manner 
somewhat similar to that shown for the shaft-coupling in Fig. 121. 
Instead of having a plane disk, however, as shown in this figure, a 
corrugated copper disk is generally used, the corrugations running 
circumferentially around the disk. The corrugations allow a com- 
paratively free end motion of the parts relatively to each other. 



CYLINDERS, TUBING, PIPES, AND PIPE-COUPLINGS. 



361 



Pattern "B.' 



Pattern *'A.' 



O- «0- OO 



M OJ 0« *. CO MM IO^»-^^ ►-^ 

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CHAPTER Xn. 



RIVETED JOINTS. 



138. It is common practice in engineering to fasten together the 
edges of plates, angles, beams, etc., with rows of rivets. If the 
plates riveted together are to be used for the retaining- walls of 
a vessel to contain a flnid under pressure, the joint must be 
designed for tightness as well as strength. This means that the 
rivets must be put close enough together to prevent the plates from 
springing apart between the rivets to any considerable extent. In 
addition to placing the rivets near together it is generallj necessary 
to calk the joints with a calking-tool, whose typical form is that 
shown in Fig. 141. This tool resembles a cold-chisel whose 
chipping edge has been ground off blunt. It is held in the posi. 







im 



Fig. 141. 



tion indicated in the figure and struck with a hammer to calk the 
edges of one plate down against the other. 

If the joint is for vessels which are to hold solid material, or for 

862 



KIVIUTED JOINTS. 363 

the members of bridges and buildings, it is necessary to design it 
for strength only, there being no necessity for tightness. 

In hand-riveting the rivet is heated to a fall red heat and then 
passed through holes previously punched or drilled in the plates and 
brought into proper position to receive the rivet. The rivet is then 
held in place with a *' dolly-bar '* while the opposite end is first 
upset with hammers and then finished with a ** set '* or " snap." 
The '' set " resembles a sledge-hammer with a depression cut in its 
face to conform with the kind of head that is desired, as button- 
head, cone-head, pan-head, etc. It is held against the rivet and 
struck with a sledge. Very small rivets are often put in place cold. 

Machine-riveting is used much more in modern practice than 
hand-riveting. A machine-riveter has two dies, cupped to the form 
the rivet-head is to have, which press on the opposite ends of the 
rivet-blank after it has been put in place. The body of the rivet is 
swelled out in the hole and the head formed by the heavy pressure 
that is exerted. The dies of riveting-machines are commonly 
operated by either steam, air, or hydraulic pressure. In hydraulic 
riveting one method is to press the rivet down so as to form the 
heads while it is still very hot, then relieve it from the pressure of 
the die for a short time until it has cooled somewhat, and then put 
it under pressure again until it has become cold enough to prevent 
its stretching materially under the tendency which the plates may 
have to spring apart. In steam and pneumatic riveting the rivet- 
head is first formed by a steady pressure of the die, then allowed to 
cool somewhat, after which it is struck a few sharp blows with the 
die driven out by the action of the steam or air against it in order 
to bring the plates together so that they will be gripped tightly 
when the rivet has cooled completely. For very heavy riveting 
the riveter sometimes has a pair of closing- or gripping-dies, called a 
*' plate-closer," which are used for pressing the plates together and 
holding them until the rivet has cooled. 

If the work is well done, the rivets should fill the holes com- 
pletely. Since the rivet blank is always smaller in diameter than the 
hole it is to fill, this necessitates its being upset from end to end so 
that it may swell to the size of the hole. It has been found that 
this can be accomplished better by having the blank hottest at the 
head end, so that it will swell first under the head, gredually fill* 



364 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

ing the hole from the head toward the point, and finally forming 
the second head. 

When the holes are made by punching, they are larger at one 
end than the other, being roughly conical on acconnt of the punch 
being smaller than the die on which the plate rests during the 
operation of punching. The plates should be placed so that the 
small ends of the holes come together. When so placed, the end 
contraction of the rivet tends to draw it more tightly against the 
conical walls, thus eliminating, in a measure at least, the loosening 
effect of the decrease in diameter due to cooling, and giving the 
rivet a better fit in the hole; the rivet also grips the sides of the 
holes and draws the plates together, thus relieving the heads of a 
portion of the strain. If, on the contrary, the large ends of the 
holes are placed together, the swelling of the rivet when under the 
dies tends to force the plates apart, and its contraction to loosen it 
in the hole; also the end contraction is resisted by the heads only. 

Numerous styles of riveted joints are in common use. Some of 
the simpler ones will be shown in order to explain the nature of the 
stresses that act upon the plate and rivets. The seams most com- 




Fig. 142. 



monly used are of two general classes, namely, lap-joint and butt- 
joint. The names are derived from the manner in which the edges 
of the plates are placed relatively to each other. In the lap-joint 



RIVKTED JOINTS. 



365 



the plates overlap each other, examples of this form of seam heing 
shown ia Figs. 142, 143, and 144; in the batt-joint the edges of 






C) 



m 




Fig. 143. 



the plates are butted against each other, and one or two cover- 
plates, straps, or welts placed over their junction, the rivets passing 



C) 




Fig. 144. 



through one plate and one side of the strap or straps, as shown in 
Figs. 146 and 147. Fig. 145 is a lap-joint with cover-plate. 



366 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

The seams are farther classified according to the namber of row9 
of riyets that are nsed, and the positions of the rivets of one row 




Fig. 145. 



relatiyely to those of the other rows. The rows of rivets ran parallel 
to the length of the seams. Single-riveted joints are shown in Figs* 




oo 
oo 



BUTT JOINT, SINGLE COVER-PIATE OR STRAP, SINOLE MVFrCD. 



Fig. 146. 



RIVET KD JOINTS. 



367 



142, 146, and 147; in each of these jointB the edges of the plates 
are pierced by bat a single row of rivets, although two rows are 





Fio. 147. 

embodied in the seam in two of these three cases. Donble riveting 
ia shown in Figs. 143 and 144, the edges of the plates being pierced 




Fro. 148. 
with two rows of riyets. The riyets are said to be staggered when 
those of one row are opposite the spaces of the adjacent row, as in 
Figs. 143 and 145 ; when they are opposite each other, as in Figs. 
144, 146, and 147, the seam is chain-riveted. 

The pitch of rivets is the distance between the centres of 
adjacent rivets in the same row. 

The margin is the distance from the edge of the plate to the 
edge of the rivet-hole. (See wi, Fig. 142.) 

The overlap is the discance one plate laps over the other. 

All the rows of a seam do not need to have the same pitch, as 



368 FOKM, STKEiNGTH, AND PROPORTIONS OF PARTS. 

can be seea by reference to Fig. 14d, where the pitch of the enter 
rows is double that of the middle one. 



PROPORTIONS OP RIVETED JOINTS. 

139. A single-riveted joint may yield in one of the five follow* 
ing ways when forces are applied as indicated by the arrows ia 
Fig. 142: 

1st. Shearing the rivets in the plane of the plate surfaces that 
are in contact. 

2d. Crushing the rivets, or the plate in front of them, by the 
'* bearing-pressure " of the rivets against the plate. 

3d. Tearing the plate between the rivets. 

4'th. Splitting or tearing the plate between the rivet and the 
edge of the plate. 

5th. Shearing out the plate in front of the rivet, the piece 
sheared out having a width approximately equal to the diameter of 
the rivet. 

Double-riveted joints may fail by either the first or third 
method, or by crushing the rivets. Any other manner of failure 
would be a combination of two or more of the five methods given 
above. The failure of joints with several rows of rivets may be still 
more complicated. 

The frictional resistance to the slipping of the plates over each 
other, due to their being clamped together by the end contraction 
of the rivets, is, in a carefully made hydraulic machine-riveted 
joint, generally from one third to one half of the total strength of 
the joint. A hand-riveted seam generally offers less frictional 
resistance to the slipping of the plates over each other than one 
which is machine-riveted. In designing riveted seams it is not 
customary to take this frictional resistance into consideration. 

The notation for riveted joints is as follows: 
A = sectional area of rivet on a plane perpendicular to its axis, 
square inches. (It is assumed that the rivet fills the hole, 
and therefore its diameter is equal to that of the hole where 
the plates touch each other); 



RIVETED JOINTS. 369 

c = ultimate bearing- or crushing-strength of plate or rivet for 

single shear, pounds per square inch; 
c' = ultimate crushing-strength of plate or rivets for double shear, 

pounds per square inch ; 
d = diameter of rivet-hole, inches; 
/ = ultimate tensile strength of plate between rivet-holes, pounds 

per square inch ; 
n = number of rivets passing through one plate along a length of 

edge equal top; 
p = pitch of rivets, inches; 
8 = ultimate shearing-strength of rivet for single shear, pounds 

per square inch; 
s' = ultimate shearing-strength of rivet for double shear, pounds 

per square inch; 
t = thickness of plate, inches; 
w = distance between rows of rivets. (See Fig. 143.) 

A rivet is in single shear when the shearing force acts only in 
one plane, as in a lap-joint, or a butt-joint with one cover-plate. 
It is in double shear when the tendency is to shear it off in two 
planes, as in the double-welt butt-joint, Fig. 147. 

Table XLVI gives ultimate values that can be safely used in 
* practice with good material. A suitable factor of safety must be 
introduced. The common practice in this country is to use steel 
plates with iron rivets for boiler construction. Steel plates with 
mild steel rivets are used to a considerable extent for other classes 
of work. 

140. Rivets. — Since a rivet may fail either by crushing or 
shearing, it is first necessary, for a given thickness of plate, to find 
the diameter of rivet having equal crushing and shearing strengths 
by equating the strengths as follows: 

For single shear. 



whence 



cat = -j-s; 



'=^L-''''t ■ ' ■ ' ' <"«> 



370 FOHM, STRENGTH, AND PROPORTIONS OF PARTS. 



Table XLVL 
itltihatb strength of rivets and plates. 





Founds perSquare Inch. 




Iron. 


Steel. 


Rivets, shearing-strength in single shear, # 


40,000 
85,500 
65,000 
80,000 
40,000 


50 000 


Rivets, shearinir-stren firth in double shear, tt 


45 000 


Bearing-pressure in single shear,* t 


90 000 


BearinflT-Dressure in double shear, c' .' < . < 


110,000 
56,000 


Plates, tensile strength between rivet-holes, % 







* Bearing area taken as the projected aroa of the bearing surface. 

For donble shear, 



whence 






^- 07087 --^^^y 



(170) 



Sabstitnting in equation (169) the valaes given in Table XLVI 
gives for single shear: 

Iron rivets, 

rf=1.27H*M^ = 2.0M (171) 

Steel rivets, 

rf = l.«7|HM^ = 2.28^. .... (172) 
Sabstitnting in equation (170) gives for donble shear: 
Iron rivets, 

ii«.635||JH/ = 1.43< (173) 

Steel rivets, 

rf=.635i,VWi^ = 1.65^ (174) 



RIVETED JOINTS. 371 

The diameters given by equation (171) hold good in practice for 
plates from -j^ to f of an inch thick, bat beyond this point the diam- 
eter of the hole becomes smaller in proportion to the plate thickness. 
The difficalty of driving the larger sizes of rivets is largely accoant- 
able for this fact. Since the sectional area of a rivet varies as the 
sqaare of its diameter, it is clear that any rivet smaller than the size 
giving eqaality of bearing and shearing strengths will fail by shear- 
ing; therefore in designing a joint whose rivet is not larger than 
the size having equal bearing and shearing strengths, it is not 
necessary to use the bearing-strength of the rivet and plate, the 
shearing-strength of the rivet being all that needs consideration; 
but if the rivet is larger than the size for equality of these two 
values the bearing-strength must be used. 

Table XLVII gives average diameters of holes for plates as 
commonly used in practice. 

Table XLVII. 

DIAMETERS OF RIVBT-H0LE8 COMMONLY FOUND IN PRACTICE FOR 
LAP-JOINTS AND SINGLE-STRAP BUTT-JOINTS. 

Thickness of plate, inches 3/18 M 5/l« « H « « % 1 IH 

Diameter of rivet, Inches « H % « « % 1 1 IJi IJi 

The form and proportions of some of the rivet-heads most com- 
monly used are shown in Figs. 149, 150, 151, and Table XLVIII. 

141. Pitch of rivets. — For a given thickness of plate and 
diameter of rivet, the pitch required for equal strength of rivets 
and plates may be found by the following equations, in which it is 
assumed that the margin is large enough to prevent the rivets from 
breaking out toward the edge of the plate, and that rupture must 
occur either by shearing the rivets, tearing the plate along the line 
of rivet-holes, or crushing the rivets or plate. 

When the diameter of the rivet equals or is smaller than the 
value for equality of shearing- and crushing-strength, then — 

For single-riveted joints having the rivets in single shear, 

'^s = {p-d)tf; 



372 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

whence 

Table XLVIIL 
diheksions of countersunk rivet-heads.* 

(See Fig. 151.) 

Dimensions in Inches. 

Diam. of rivet, D H 5/l« H 7/l« H »/!« H 11/1« H 18/1« fi t IH ^H 
Diam.ofhead,^ H 19/SS 11/16 35/88 H 81/88 1^^ lA IM lU lA IH lU 2 



!♦- 



ZfXiZIA^ 



Fig. 149. 



For doable shear, 



.— ^. 



-1.75D— f— *i }*- 

h — -0— H 



-^.750- -*! 

Fio. 150. 



\ 

— o[- — -I 

wetaAupm 

PROPOttTIOIW 

Fig. 151. 






whence 



y= (p ^ ef)^/; 






. (176) 



And iu donble-riveted joints, both rows of rivets being of the 
same pitch : 

For single shear, 



2;rrf' 



8^{p^d)tf, 



* Taken from drawings kindly famished by J. H. Stembergh & Co. 



BIVETED JOINTS. 378 

whence 

For double shear, 

-^«'= (^ - d)tf'. 



P='^f.+ d = 2A'+d.. . . (177) 



whence 



p = ^d'^+ d = ^A^.+ d. . . . (178) 



If the rivet is larger than the theoretical valae for eqaality of 
shearing and crashing strengths, failure will occur by crashing. In 
accordance with this — 

For single-riveted joints having the rivets in single shear: 

cdt = {p^ d)tfi 
whence 



For doable shear, 



P = '-^^- (179) 



P = ^d. (180) 



For doable-riveted joints whose rivets have the same pitch in 
both rows and are in single shear: 

2cdt=^{P'^d)tfi 
whence 

P = ?^ (181) 

For doable shear, 

P = ^d. (182) 

In general, in any joint of ordinary form having two or more 
rows of rivets passing through the edge of one plate, the pitch of 



874 FORM, STUENGTH, AND PROPORTIONS OF PARTS. 

the rivets can be determined, when jt? is taken equal to the pitch of 
the row farthest from the edge of the plate, by equation (183) or 

(184) when the rivets are small enoagh to fail by shearing, or by 

(185) or (186) when large enoagh for the joint to fail at the 
bearing surfaces. 

For single shear, 

p^^I.^d = nAtj^d. . . . (183) 

For doable shear, 

2n7rd* «' 



+ e; = 2«^~ + rf. . . . (184> 



^ 4 ^/ • tf 

For failare by crashing when the rivets are in single shear. 



P^'^d. (185> 



For double shear, 

P = '^^d. (186) 

142. The efficiency of a riveted joint is the ratio of its strength 
to that of the solid plate. The theoretical efficiency can readily be 
calcalated if it is assumed that the portion of the plate between the 
holes is of the same strength per unit area as at other places. 
Thas, for a single-riveted lap-joint of |-inch steel plates and f-inch 
iron rivets, the pitch, by equation (175), is 

Taking js? = 1\^ inches. 

Efficiency = ?^ = {l\} -1)^-1^1 = .55 -f . 
P 

The actual efficiency of a joint may differ considerably from the 
theoretical on account of various influencing causes, the principal 



RIVETED JOINTS. 375 

ones being the effect of punching upon the sfcrength, hardness, and 
ductility of the plate, and the pressure of the plates against each 
other due to the contraction of the rivets. It is probable that, with 
a well-made joint, working with a factor of safety of four, no 
slipping of the plates occurs so as to bring the rivets hard against 
the holes on the bearing side. 

143. Shearing, punching, and drilling plates preparatory to 
riveting. — The plate metal, coming from the rolling-mill in irreg- 
ular shapes, must be sheared to form and size suitable for its pur- 
pose. After this the rivet-holes are made in it, there being three 
methods of making them, as follows: 1st. Punching to the required 
size with a power-punch by a single stroke of the punch for each 
hole; 2d. Punching a hole from tV ^^ i ^^ ^^ ^^^^ smaller than 
required, and then enlarging it to full size with a reamer or 
similar cutting tool; 3d. Drilling. Of these three methods punch- 
ing is by far the quickest and cheapest; the punching and reaming 
process is considerably more expensive; and drilling is the most 
costly of all. 

When a hole is punched in a plate of metal, it is found that the 
** wad " or *' plug " forced oat is much smaller in volume than the 
hole formed by its removal (unless the plate is hard and brittle, and 
hence unsuitable for riveting up to withstand pressure). It is 
about the same diameter as the hole, but sometimes not more than 
half as Ipng as the thickness of the plate. Since the metal in the 
wad is found to be little, if any, more dense than the plate, it is 
evident that there must be a flow of the metal as the punch presses 
upon it, of such a nature that part of the metal, instead of being 
punched out of the hole with the wad, is forced into the walls of 
the hole. The thickening of the plate around the hole is proof of 
this. 

This flow of metal into the walls of the hole, which is really a 
cold working of the material, changes its physical qualities to a 
depth varying from ^ to i of an inch, according to the size of the 
hole and the quality of the metal, the result being that a thin ring 
or sleeve, harder and less ductile than the original plate, is formed 
around the hole. As to the effect of this hardened sleeve upon the 
strength of the plate along the line of holes, and consequently upon 
the efficiency of the joint, there has been much discussion with 



376 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

regard to whether it increases, decreases, or has any effect apon 
these qualities. The nuneroas experiments made upon riveted 
joints seem to indicate, however, that punching is injurious to hard 
plates and those made from a poor quality of iron or steel, hut that 
soft plates may actually he henefited hy the same treatment. 
Whether any benefit is ever derived or not, it is certainly true that 
when the operation is injurious its deleterious effects are more 
marked the poorer the quality and the greater the hardness of the 



By reaming out the holes after punching, the effects of punch- 
ing are nearly or quite removed, and the joint shows practically the 
same efficiency as when the holes are drilled. The effects of punch- 
ing can also be partly, sometimes completely, removed by annealing 
the plate after punching; this process is rather inconvenient and 
expensive for large plates, howev^. 

The effects of shearing the plate to approximate size are removed 
by planing off a strip of. metal along the edge, which at the same 
time reduces it to size and makes a smooth edge. The sheared 
edge always presents a rough surface when it comes from the 
shears, which in itself would not be acceptable for good work on 
heavy plates, especially where tight joints are to be made. 

144. Faulty construction and grooving of riveted joints. — 
When the holes in the two plates do not coincide, it is evident that 
the rivet will be hard to drive into place, and an offset will be 
formed on it where the plates press together when it is headed; and 
if the heads are directly opposite each other they cannot both be 
concentric with the body of the rivet. The whole result is that 
the rivet is improperly formed and is not as efficient as a perfect 
one. By careful work very good seams with punched holes can be 
made. 

By punching the holes small, clamping the plates together in 
the relative positions they will occupy when riveted, and reaming 
through both plates at the same time, a practically perfect rivet- 
hole can be obtained ; . the reaming can be done by hand or 
power. 

Drilled holes can be more accurately located than punched ones, 
consequently better coincidence of the holes can be obtained when 
the plates are drilled separately than when punched ; and drilling 



RIVETED JOINTS. 377 

has the further advantage that the plates can first be clamped 
together and then the holes drilled perfectly concentric. 

In machine-riveting it is necessary for the dies to be held rigidly 
so that no side motion can occur, for if it does the head will be 
formed eccentrically with the body and the rivet weakened in con- 
sequence. In practice this fault is sometimes so great that the edge 
of the head is tangent to the body of the rivet. Fortunately such 
great faults are not found in boiler- work turned out by makers of 
any reliability. 

The stress upon a lap-riveted seam, causing a tendency of the 
plates to bend, as in Fig. 148, localizes the tension to some extent 
at A and B. The localization does not generally need serious con- 
sideration, however, unless the plates are grooved by calking or 
corrosion along the edges at A and B. If deeply grooved, the 
tendency to bend becomes greater, and in badly grooved plates 
rupture may occur along the groove at the edge of the seam. 

The slight bending of the plates at a seam also tends to 
localize their pressure against the rivets. This localization is at or 
near the plane of contact of the plates. It has the effect of causing 
the rivets to shear more readily than if there were no bending of 
the plates. 

When lap-joints are tested to destruction, the head of the rivet 
is sometimes pulled off on account of the bending of the plate, as 
indicated in Fig. 148. 

145. Examples of riveted joints taken from practice. — Figs. 
152 to 163 represent the practice of the Baldwin Locomotive Works 
with regard to riveted joints.* 

Figs. 164 to 171 illustrate the riveted seams adopted by the 
Continental Iron Works for cylindrical-shell marine-boilers, f 

* Taken from blue-prints kindlj furnished by Burnliam, Williams & Co., 
Baldwin Locomotive Works. 

f Taken from printed designs kindly furnished by the Continental Iron 
Works. 



378 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 




Fig. 152. 

Double-riveted lap-joint. 1/2" plate, 13/16' rivets. 7/8" holes. 

Baldwin Locomotive Works. 





-w-w-^^-m 



-%^ 



-«e 



1. 



J 



Fig. 153. 

Double-riveted lap-joint. 5/8" plate, 15/16" rivets, 1" holes. 

Baldwin Locomotive Works. 



BIYKTED JOINTS, 



379 





Fig. 154. 

Donble-riveted lap-joint. 5/8" plate, 15/ir rivets, 1" holes. 

Baldwin Locomotive Works. 




Fig. 155. 

Double-riveted lap-joint. 11/16" plate, 1^" rivets, If'' holes. 

Baldwin LooomotiTe Works. 



380 FORM, STRENGTH, AND PROPORTIONS OF PARTS.. 







^/^ 



m 



Ht 



^^ 






'C^Ky 



^^ 



r^V 



r 



*M~ 



^•H%?i' 



^W 



r 



+f!4t- 



//'^\ 



■^ 



# 






^\ 




^ 



o 

I 






I- 

i 



I'M 



bo 

a 

•c 



d 



I 

d 



RIVETED JOINTS. 



381 




FORM, 8TRENOTH, AND PROPORTIONS OF PARIS. 




Fig. 158. 

Butt-joint with doable covering-strips. 9/16" steel plate, 1" riyeta. 

Baldwin Looomotive Works. 




Fio. 169. 

Butt-joint with doable covering-strips. 6/8" steel plate, I'' riveta 

Baldwin Locomotive Works. 



BIVETED JOINTS. 



383 




384 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 




BIVETED JOINTS. 



385 



13- 








vX^' 



lOBM, STKENQTH, AND PBOPOBTIONS OF PABTS* 




BIV£I£I> JOINTS. 



387 






A^ 








Fig. 164. 

Bteam-pressare, lbs. per sq. in., 100. Inside diameter of shell 6' 0". 

7/8"' rivets, 15/16" lioles. The Continental Iron Worka. 






^ ^LJ rl U^!^ ^♦--l 

I e 9 •* SV *• • • • • • • • • •_• K^i. 



m 



**«Hf 




L*.i 



Fig. 165. 

Steam-pressare, lbs. per sq. in., 180. Inside diameter of .shell ff 6^« 

ly* rivet^ 1^^ holeB. The Continental Iron Works. 



388 FOBM, STRENGTH, AND PROPORTIONS OP PARTS. 




Fig. 166. 

Steam-preBsnre, lbs. per sq. in., 160. Inside diameter of shell C' 6'. 

7/8" rivets, 15/16" holee. The Continental Iron Works. 




Fig. 167. 

Stetm-pTessure, lbs. per sq. in., 200. Inside diameter of shell (f fTm 

1" rivets, 1^" holes. The Continental Iron Works. 



RIVETED JOINTS. 



889 



iH'-i^'i^j^- 



i 



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A i,e e ® A^\s o o ® « aoR 
\© <l e e .e^9\e » 9 ^ e qj 



5 2! 





Pig. 168. 

iSteam-pressare, lbs. per sq. in., 100. Inside diameter of shell 11' 6". 

7/8" rivets, 16/16" holes. The Continental Iron Works. 




T 



\e j» ,©^ *>\tt\ © • • © © 1«5 • 



j» ,©^*»\tt\© • © © © Jeg » • 
** \ ©X \ e • © 

]:itk:^ir~" — — 



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Pig. 169. 

Steam-pressare, lbs. per sq. in., 180. Inside diameter of shell 11' 0^'. 

If rivets, 1^" holes. The Continental Iron Works. 



390 FORM, STBBNQTH, AND PROPORTIONS OF PARTS. 




Fig. 170. 

Steam-pressare, lbs. per sq. in., 160. Inside diameter of shell 11' 6''. 

If rivets, l^j" holes. The Continental Iron Works. 



2-s'i^*T~S8Ji" 



2i]^8^«ll 








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® ® ® ® ^ 
e ® ® ® 1 


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FiGf. 171. 

Steam-pressure, lbs. per sq. in., 200. Inside diameter of shell 11' ^, 

H" rivets, l^y" holes. The Continental Iron Works. 



CHAPTER Xm. 

FRAMES OF PUNCHING, SHEARING, AND RIVETING MACHINES. 

PUNCHING AND SHEARING MACHINES.* 

146. A frame of the general form shown in Fig. 172 is very 
commonly osed for punching and shearing machines. The punch 




Fig. 172. 

is attached to a slide which works in the guides O and operates 
against a die supported at H, The material to be punched or 
sheared is placed between the punch and die. 

If the frame is for a stationary machine which is to set in the 
position shown in the figure, it is made with a flat base as shown ^ 
When the jaws are to stand vertically instead of horizontally, the 
base illustrated is not necessary, of course, but some suitable means 

* It is believed that Professor Albert W. Smith was the first to present the 
method of dealing with the stresses in C frames that is given In this chapter 

891 



392 FORM, STRENGTH, AND PROPORTIONS OF PARTS, 

muab be sapplied for supporting the frame. Portable punches are 
generally suspended by a chain or rope, hence no base is necessary. 

The frames for heavy stationary punching and shearing machines 
are generally made of cast iron. Forgings and steel castings are 
generally used for portable machines. 

In designing a punch-frame for strength it is necessary to con^ 
sider the external forces acting npon it, together with .the internal 
stresses due to the externally applied forces. Sections must be 
selected at different places along the frame, and each designed so as 
to give the required, working stress in the material, due allowance 
being made for the localization of shrinkage stresses at re-entrant 
angles, and for the general appearance of the machine. 

In general it will be found that for cast iron the sections near 
the punch will have to be made heavier than is necessary for the 
requisite strength in order to obtain walls thick enongh to cast 
well in connection with the remainder of the frame. 

The external forces acting on the frame when punching a piece 
of metal have practically the same effect on the material near the 
back part of the throat of the frame as if two opposite forces, both 
coincident with the axis of the punch, were applied to separate the 
jaws. In ordinary construction the centre line of the punch lies 
in a plane dividing the frame into two symmetrical halves. This 
median plane of the frame is parallel to the paper in Fig. 172. 
When the punch is so located, its pressure against the piece to be 
punched has no other tendency than to separate the jaws. 

When used as a shearing-machine, the cut begins near one end 
of the shear-blades and travels across them to the other end as they 
are brought together. Assuming that the blades are set so that 
they are perpendicular to the median plane, it can be seen that at 
the beginning of the cut the pressure against the blades will be to 
one side of the median plane, although approximately parallel to it. 
On account of being to one side of the median plane, the pressure 
against the shears at the beginning of the cut has a tendency to open 
the frame more on one side than the other, and thus cause a tor- 
sional or twisting action upon it in the two parts extending forward 
to form the throat and jaws. When the cut has passed to the 
opposite end of the blades, a similar twisting action occurs, but in 
the opposite direction. 



FRAMES OF PUNCHING AND SIMILAR MACHINES. 393 

The frame should be of each a sectional form as will offer the 
greatest resistance to the twisting action just mentioned, as well as 
to the greatest (in ordinary cases) tendency to separate the jaws. 
This can be done, when the frame is cast, by making it of the hollow 
or box form. 

If the shear-blades are placed parallel to the median plane with 
their cutting edges in it, and are sharp, tliere would probably be 
no great force acting on the frame when shearing other than one 
tending to separate the jaws in the same manner as when punching. 
As the blades become dull, however, there is a tendency for them to 
move sidewise relatively to each other, thus inducing a side bending 
action on the parts of the frame forming the jaws, and a twisting 
action on sections back of the throat. The box section is the best 
to resist this action, and is generally used for cast-iron frames. 

The following notation may be used for punch-frames : 

A = total area of section of frame; 
/ = moment of inertia of a section about its gravity axis which is 

normal to the median plane of the frame; 
Ig = moment of inertia, about its gravity axis normal to the median 

plane, of any part of the section ; 
/« = moment of inertia, about the gravity axis of the entire section, 

of any part of the section ; 
P = force necessary to drive punch through the material to be 

perforated ; 
8c = I-r- e^, = section modulus of the section with regard to com- 

pressive fibre-stress; 
iS'i = / -^ ^1 = section modulus of the section with regard to tensile 

fibre-stress; 
a = area of any selected part of the section ; 
b = greatest dimension of any partial section, measured parallel to 

the gravity axis about which the moment of inertia is 

required ; 
d = depth of throat of frame; 
I = distance from line of action of F to gravity axis of section 

under consideration ; 
00 =s distance from gravity axis of section to outermost fibre in 

compression; 



394 FOKM, STEENGTH, AND PEOPOETION8 OP PARTS. 

fi = distance from gravity axis of section to ontermost fibre in 

tension; 
f^ = greatest compressiye fibre^strees in section; 
/, = greatest tensile fibre-stress in section; 
h = greatest dimension of any partial area measured perpendicular 

to the gravity axis aboat which the moment of inertia is 

required; 
3 = shearing-stress per unit area uniformly distributed over entire 

section; 
t = tensile stress per unit area uniformly distributed over the 

entire section; 
z =distance between gravity axis of any selected part of the area 

and the gravity axis of entire section. 

147. Stresses in a section perpendicular to the motion of the 
punch. — The stress in the material on the section YZy Fig. 172, 
taken normal to the pressure P against the punch and die, can be 
determined by considering the portion of the frame above ¥Z as a 
free body, as shown in Fig. 173. 




Fig. 173. 



The forces acting on this part of the frame may be taken as 
those due to punching a plate. They are: 

ist. The pressure P against the punch as it is forced through 



FRAMES OF PUNCHING AND SIMILAR MACHINES. 395 

the piece to be perforated. The line of action of P coincides with 
the centre line of the panch. 

2d. The fibre-stress in the material due to the bending action 
of P. P acts with a lever-arm whose length is the distance d + «< 
from its line of action to the centre of gravity of the section. The 
tensile fibre-stress /^ at F, and the compressive fibre-stress /« at Z, 
are: 

/. = :^^^; (187) 

■f' = ^^^ <i««) 

3d. A uniformly distribnted tensile stress of a value 

over the entire section. 

By combining the stresses acting on the section TZ it can be 
seen that the maximum total tensile stress per unit area is at F, 
and is eqaal toft + t* 

The eqaation for obtaining the total stress is 

Maximum tensile stress )_/.,/_ P(d + e^ P .- ^ . 
on section YZ | - /* + ^ - 3^ + -j' ^l^^) 

Similarly, the maximum total compressive stress per unit area 
on the section YZ is at Z, and is equal to the numerical difference 
of /o and t. Therefore 

Maximum compressive ) _ ^ _ ^ _ P(d+6y) P . 
stress on section YZ) ■~^«""'~ s^ "^ A' ' ^ ' 

If the centre of gravity of the section lies midway between Y 
and Zy then/i =/,. Equations (189) and (190) show that if the 
centre of gravity is thus located the maximum tensile stress at Y 
will be greater than the maximum compressive stress at Z by an 
amount equal to 2^ = 2P ~- ^. This assumes that the modulus of 
elasticity of the material is the same for tension as for compression, 



396 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

which is practically trae for the materials common to snch oon- 
straction. 

If a material having eqnal strength in both tension and com- 
pression is used for the frame, the cross-section should be made 
heavier on the side next the punch or shear-blades in order that 
{ft + shall equal (/, — /), this being the condition for equal 
maximum tensile and compressive stresses. 

For snch a material as cast iron, which is much stronger ir com- 
pression than tension, it becomes necessary for economy of material 
to make the section much heavier on Ihe side next the punch than 
at the back. This is done to make (/, + t) smaller than (/. — /). 

148. Numerical solution for a section perpendicular to the 
motion of the punch. — This corresponds to the section on YZ in 
Fig. 172. 

Let the work required of the machine be to punch a 1-inch hole 
19 inches from the edge of a plate | of an inch thick, having a 
shearing-strength of 56000 pounds per square inch; also to be 
operated with shear-blades for shearing the same plate. The frame 
to be cast iron. The throat to be 20 inches deep, so as to allow the 
edge of the plate to clear it 1 inch, and to be stressed to practically 
2000 pounds per square inch tension when punching the hole. 

The force P necessary to drive the punch through the plate is 
found by multiplying the shearing-strength of the material by the 
area to be sheared by the punch. The sheared area is that of the 
wall of the hole made by the punch. Whence 

P = 550007r X IX I = 129600 lbs. 

It can safely be assumed that the force exerted to separate the 
jaws of the frame will not be so great when shearing as when punch- 
ing the size of hole specified. If the frame is made strong enough 
to operate as a punch, it is therefore only necessary to give it such 
a form of cross-section as will secure sufficient rigidity against 
twisting sidewise when shearing. 

A drawing of any convenient size, and of a form considered 
suitable for its purpose, may be made for the required section. 
Fig. 174 may be taken as the sectional form selected for this 
problem. The moment of inertia /of this section, and thence its 



FRAMES OF PUNCHING AND SIMILAR MACHINES. 397 

section modulus, may now be found, the scale of the drawing being 
taken as unity (i.e., scale 1 inch = 1 inch). The section thus 
represented in Fig. 174 is not by any means large enough for the 




i.tesaiN. 

.561 



«V I KIAHULeV B AND B ■■ . M II »| 

11 RECTANQIxaCANDC- 1.04ii .. 
RECTANGLE " .8611 1 1 

TOTAL AREA - 8.71 6Q.IN: 



Fig. 174. 



required strength but it is not necessary that it shall be, for the 
scale that must be applied to it to obtain the dimensions for the 
requisite strength can be determined by trial, as shown later. 

In Fig. 174 the section has been taken of a simple form in order 
that the calculations for / might be readily made. The actual 
working form should have rounded comers and filleted re-entrant 
angles. 

For convenience in determining /, the section is divided into six 
parts, namely: 

1st. A rectangle R\ 

2d. Two similar and equal triangles B and B'\ 

3d. Two similar and equal rectangles (7 and C'\ 

4th. A rectangle D. 



b98 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

The divisions tliroagfa the body of the section are indicated by 
broken lines. The centre of gravity of each partial area of the 
section is indicated by X. 

In order to determine /, the gravity axis XX of the section 
mast first be located. This can be done conveniently by taking 
moments aboat the line QH^ considering the areas as forces propor- 
tional to the areas, whence 

^ = 1.306 in. and «« = 2.195 in. 
For the rectangle Rx 

I„ = ^hV = tV X 2.5(.7)' = .0714; 
a2;' = L75 x (.955)' = 1.696; 
/„ = 7^ + az^ = .0714 + 1.596 = 1.6674. 
For the triangles B and B* : 

^g = 2 X TrV*A* = 1^ X .2(2.8y = .8439; 
a^ = .56(.328)' = .0602; 
7, = .2439 + .0602 = .3041. 
For the rectangles G and G ' : 

7, = 2 X ^hV = :^i|^' = .6866; 

a««=1.04(.696)' = .5023; 
7. = .6866 + .6023 = 1.0879. 
For the rectangle D\ 

Ig^^iV = tVX 1.8(.2)" = .0012; 
7»= a^' = .36(2.095)* = 1.6800; 
/«=. 0012 + 1.58 = 1.5812. 



FRAMES OF PUNCHING AND SIMILAR MACHINES. 399 
For the entire section: 

/ = 1.6674 + .3041 + 1.0879 + 1.5812 = 4.6406. 
The section modulns for tension is 
7 4.6406 _ 

This is the section mod alas for the dimensions given in Fig. 
174. It is clearly tdo small for the required service. A scale of 
drawing mnst therefore be assumed which will give a larger section. 

Assume that the drawing Fig. 174 is one ninth size. The 
yalne of ^ for the enlarged section will therefore be 9 X 1.305, and 
8t for the same will be (9)* x 3.556, since the section moduli of 
similar sections are proportional to the cubes of their linear dimen- 
sions. The area will be (9)' X 3.71. 

The maximum tensile stress in the enlarged section will there- 
fore be, by equation (189), 

129600(20 + 9 X 1.305) 129600 

•^ (9)* X 3.556 "^ (9)« X 3.71 

= 1590 + 430 = 2020 lbs. per sq. in. 

This value is practically satisfactory. Therefore by multiply- 
ing the linear dimensions of Fig. 174 by 9 a section of suitable size 
may be obtained. The dimensions obtained in this manner should 
be modified to agree with shop methods of measurement, of course. 
The corners should be rounded and re-entrant angles filleted. This 
need not be done to such an extent as to seriously affect the section 
modulus, however. It is often necessary to apply a fractional scale 
to the drawing, such, for instance, as 9.1, 8.7, etc. 

If the section adopted is of such a form as to make it impossible 
to divide it into geometrical portions, the moment of inertia may be 
found graphically, the gravity axis being located during the 
process.* The section modulus can then be readily determined. 

* For finding moment of inertia graphically see appendix, g B. 



400 FOKM, STRENGTH, AND PROPORTIONS OF PARTS. 

149. Section parallel to the motion of the pnnch. — This corre- 
spondB to a section on WXy Fig. 172. The portion of the frame to 
the left of WX may be considered a free body, as in Fig. 175. The 





, 


./• 


c 






G 




^ 


\ 




■p 


T' 



Fic. 175. 
forces and stresses acting on it are indicated in the figure. They 



are: 



1st. The force P, coincident with the centre line of the punch. 

2d. The fibre-stress in the material due to the bending action 
of P. P acts on a lever-arm of length Z, measured from the line 
of action of P to the plane of the section. The tensile fibreHstress 
is at PF, and equals 






The compressive fibre-stress is at X, and equals 

PI 



/o = 



& 



(191) 



(192) 



3d. A shearing-stress uniformly distributed over the section. 
The value of this shear per unit area is 



8 = 



(193) 



The maximum tension, compression, and shear are found by 
combining the stresses obtained by the last three equations. The 
formulas for maximum tension and compression, as given in works 
on the strength of materials, are: 



FRAMES OF PUNCHING AND SIMILAR MACHINES. 401 



Maximum tension ~ 2 "^ \/ *" "I" (2) ' 



Maximum compression == 2 "*" \/*' "^ \i) ' 

Since the value of a is the same for all parts of the section, the 
maximum shear will occur where the fibre-stress due to bending is 
greatest. This will be at X if the gravity axis of the section is 
farther from X than from TT. The equation is 



V-'+i)' 



Maximum shear = a / «• + r^j .... (194) 

The section must, of course, be designed strong enough to resist 
both tension and shear. In general, it is not possible to make it 
equally strong to resist both when such a material as cast iron is 
used. At a section near the punch the shearing-stress may be the 
one to fix the proportions, but when the section is taken near the 
back of the throat the tensile stress is more apt to be the one 
requiring consideration. 

It should be noted that for the section WX the lever-arm I 
remains the same length whatever scale is applied to the drawing 
of the section. 

The section WX must have such dimensions normal to the 
median plane of the frame as will allow it and the section on YZ to 
be joined together to form the frame. 

In order to give a symmetrical appearance to the frame, and to 
obtain a form that will cast well, the sections near the guides at the 
working end of the frame are generally made much stronger than 
calculations show to be necessary. 

150. Angular section of a punch-frame. — A section taken at an 
angle, as the one on TU^ Fig. 172, has the following forces and 
stresses acting upon it. They may be seen by the aid of Fig. 176. 

1st. The force P due to the pressure of the punch against the 
plate. 

2d. The fibre-stress due to the bending action of P. The lever- 



402 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

arm of P is Z, measured from the line of action of P to the gravity 
axis of the section which is projected as a point at 0. The tensile 
fibre-stress at T^ dae to the bending action of P, is 






(195) 



, ^ 

I 
_ QJ 



Fio. 176. 
and the compresBive fibre-stress at U, due to the same canse, is 

PI 



/. = • 



(196) 



3d. A shearing-stress /acting as shown in Fig. 176. The ralne 
of / is found by resolving P into two components, / and T, one 
parallel and the other perpendicular to the plane of the section. 
The shear per unit area due to / is 

8 = 7-^- A. 
4th. A tensile stress T acting normal to the plane of the section. 
The tension per unit area due to T is 

the formulas for maximum stresses are: 



Maximum tension ^-^^ + a A' + {^^^Jy 



FRAMES OF PUNCHING AND SIMILAR MACHINES. 403 



Max. compression -="^^-^ 1" \/*' "f" V9 — ) ' 



Maximum shear = ^^« + (.^l±-^y or = W^^« + (..^_jy. 

For the maximam shear the greater of the two quantities under 
the radicals is to be used. In many cases a numerical solution is 
necessary to determine which of these two quantities is the greater. 

This section roust have dimensions, normal to the median plane 
of the frame, intermediate between those of the sections on WJl 
and rZ. 

151. General form of a punch-frame. — In designing the frame 
enough sections should be tested to secure a nearly uniform strength 
throughout the frame, due allowance being made for the fact that 
the sections near the punch-gnides may have to be heavier than is 
actually necessary for strength in order that the frame may have a 
symmetrial appearance. The walls of the frame should not be made 
so thin at any part that they will not cast well in connection with 
the heavier parts. 

On account of the tendency of the shrinkage stresses in a casting 
to be greatest about re-entrant angles, it is generally advisable to 
make the sections bordering on the curve at the back of the throat 
somewhat stronger than those along the straight sides of the jaws. 
When the distance between the jaws is not greater than shown in 
Fig. 172, the back of the throat can be made an arc of a circle, or 
approximately so, as shown in the figure. When the jaws are far 
apart, however, this cannot generally be done well, but the re- 
entrant angles should be well filleted in order to prevent excessive 
shrinkage stresses in the material about them. 

If the frame is made with an open flat bottom, suitable for rest- 
ing on a foundation, the sections along the lower horizontal part, 
referring to Fig. 172, should be tested in the same way as those 
above, unless it is certain that each is at least as strong as the one 
immediately above it. 

In most designs the frame tapers, somewhat in the manner of 
a pyramid, from the guides toward the back of the throat. The 



404 FOKM, STRENGTH, AND PKOPORTIONS OF PARTS. 

thickness HM^ Fig. 174, of the rib generally decreases from the 
throat toward the panch, as indicated in Fig. 172; less freqaentij 
it is kept of the same thickness all the way around the throat and 
jaws. 

Having obtained the general form of the frame with regard to 
strength and symmetry, the auxiliary parts, such as bearings, etc. , 
can be added to complete the frame. 



DIEECT-ACTING HYDBAULIC RIVETER. 

152. A hydraulic riveter is required for driving rivets in boiler- 
plate up to 60 inches wide and ^ i^t an inch thick, the largest rivet 
to be f of an inch in diameter, corresponding to 0.307 of a square 
inch area. 

Experiments show that a pressure of 160000 pounds per square 
inch on a hot rivet of this diameter will set the rivet and form the 
head satisfactorily. The pressure required for this case will there- 
fore be 

0.307 X 160000 = 49120 = 50000 pounds, in round numbers. 

By calling this value 50000 there is no greater deviation from 
the correct value than is shown in the experiments on riveting. 

Fig. 177 is a side elevation of the outlines of the most common 
form of such machines as they are found in practice. The cylinder 
Q contains a piston attached to a piston-rod, one end of which pro- 
jects at Ry where one of the dies for compressing the rivet and 
forming the head is attached. The piston-rod serves as a ram for 
driving the die. The stationary die is at S, 

In order to rivet cylindrical shells of small diameters, such as 
flues, the stake should be made as slender as is possible for the 
required strength, so that the shell can be passed over it for driving 
the rivets in the longitudinal seams when riveting up a single sec- 
tion, and for the circular seams when riveting two sections together. 
It can be made of either a steel forging of rectangular cross-section 
or a steel casting having an I section. 

The frame may be cast iron or a steel casting. Since the oyer- 



FRAMES OF PUNCHING AND SIMILAR MACHINES. 405 

reach of the riveter^ which is the distance ib will reach over a plate 
from its edge to drive a rivet, is small, the frame can be so made of 
cast iron as to have snfficient strength without excessive weight. 




Pig. 177. 

The stake and frame are held together by two bolts, one passing 
on each side of the stake and through a bar passing over the top. 
This bar, section-lined in the figure, clamps against the top or back 
of the stake. 

The lugs T and F are a part of the frame, their office being to 
hold the sfcake in position. 

In proportioning for strength the stake and frame can be dealt 
with separately, each as a beam supported at the ends and loaded 
eccentrically with a single load. The moment and shear diagrams, 
identical for both, are shown in the figure. 

It will be noticed that the distance from the centre of the dies 
to the plane of the bolt centres is 64 inches, which is 4 inches in 
excess of the distance the centre of the rivet is to be from the edge 
of the widest plate. An examination of the figure shows that about 
this much must be allowed for the lag and bolts. 

By taking moments about a point 64 inches from the centre line 
of the punch the pressure on the lug V is found. Its value is 

F = (64 -h 32)50000 = 100000 pounds. 

That on ^is 



T = 50000 + 100000 = 160000 pounds. 



406 

The bending moment is a maximum at the plane of the c^entre 
lines of the bolts, and equals 

M = 60000 X 64 = 3200000 inch-pounds. 

If the stake has a cross-section fairly thick, measured perpen* 
dicular to the plane of the paper, in proportion to its width, and is 
made of good steel, it will, on account of the length being large in 
proportion to its thickness and width, have fibre-ti tresses due to 
bending which are large in proportion to the shearing-stress 
uniformly distributed over the section. Under such conditions the 
shear can be neglected, and the stake designed to resist the stresses 
due to bending only. This is in accordance with the common 
method of dealing with beams whose height is small in proportion 
to their length. In general, the stake should taper from its maxi- 
mum sectional dimensions at the bolts, growing smaller in both 
dimensions toward both ends. The thickness is generally reduced 
uniformly, but the height changes more rapidly as the distance from 
the largest part increases. 

There should always be sufficient thickness to preyent the upper 
or stake die from springing sidewise to such an extent as to make 
the head of the riret objectionably eccentric with the body. 

The frame may be of either a solid I section or hollow. In 
either case the bending moment and shear can both be dealt with 
in the same manner as for the section WX oi the punching and 
shearing machine. 

The bolts must be tightened with a combined initial tension 
somewhat greater than that which is thrown upon them when a 
rivet is driven, in order to prevent the springing apart of the stake 
and lag. The lug T^ therefore, must resist in this case a compres- 
sion somewhat greater than 150000 pounds. The lug V must, of 
course, resist a compression of 100000 pounds. 

The frame may either stand as in the figure, suitable flanges 
being provided for resting upon a foundation, or it may be placed 
with the axis of the cylinder vertical. The nature of the service 
required of the machine is the determining factor for the position 
in which it shall be placed. 



CHAPTER XIV. 



SELECTION OF MATERIALS. 



153. In selecting materials for machine parts, the properties 
that must most commonly be considered with regard to their adapt- 
ability are as follows: strength, resilience, stiffness, coefficient of 
friction, durability, convenience and cost of working into the 
reqaired form, and cost of the material as found on the market in 
the merchantable form. It is seldom that all of these properties 
need to be taken into account when deciding upon the material for 
any given machine-member. 

A few examples may serve to show more clearly than any other 
means how the qualities have to be taken into account. 

In an ordinary engine-lathe the bed is of a somewhat compli- 
cated form, having ribs, sometimes hollow, extending from side to 
side, bosses and other raised places for attaching parts, and the 
sides are ordinarily flanged or ribbed at both top and bottom in 
order to give rigidity. This complication at once sets forth the 
necessity of making the bed of some material that can be cast in 
moulds. There are only two materials, whose cost is such as to allow 
them to be used for such purposes, that can be formed by casting, 
namely, cast iron and steel. Cast iron is much the cheaper of the 
two, and can be planed and otherwise machined much more readily 
and, consequently, more cheaply than steel. It is a much weaker 
material than steel and is not so stiff. These two qualities 
apparently make it objectionable, but this is really not the case on 
account of the following reasons: In order for a lathe tool to take 
a broad cut when removing only a thin turning from the surface of 
the piece worked upon, it is necessary for the lathe-bed to be both 
rigid and heavy, so as to prevent the tool from springing away from 
its work and '* chattering " against it; the mass of metal in the bed 
has much to do in the prevention of the small vibrations that 

407 



408 FORM, STRENGTH, AND PROPORTIONS OF PARIS. 

cause tlie chattering. Therefore the mass of metal necessary to 
hold the tool steady for wide cats will make the lathe strong and 
stiff enough for the heaviest cuts that the driving mechanism of the 
lathe can carry. Cast iron, therefore, in comparison with steel, has 
for the lathe-bed the advantage in cost and ease of machining, 
while its lower strength and moduli of elasticity are of small con- 
sideration, on account of the mass of metal to be used; clearly, cast 
iron is the best material for the purpose. 

Even if the vibration of th« tool had not been an item in the 
conditions above, the cast iron would have still been the best 
material on account of its softness making it easy to machine as 
well as the low cost of the raw material, both together making the 
finished product much less costly in cast iron than in steel. This 
is also true of numerous other machine-tool parts and machines. 

The spindle or arbor of an ordinary engine-lathe rests in two 
bearings supported by the head-stock, and is driven by a gear 
attached to it near the bearing next to the face-plate or live-centre, 
as the case may be. A stepped pulley occupies all or most of the 
remaining length of the part of the spindle which lies between the 
bearings. When the lathe is working, all the power transmitted 
to overcome the resistance of the cutting edge of the tool must 
pass through the part of the spindle lying in the bearing just 
back of the face-plate, thus causing a torsional moment in the 
spindle. In order to withstand this twisting moment, and at the 
same time the weight of the face-plate and the work upon it 
or the centres, the spindle, unless large in diameter, must be made 
of some strong material, which should also be resilient in order to 
withstand any of the shocks that are apt to occur accidentally. By 
keeping the spindle small the frictional loss in the head -stock bear- 
ings is kept down, and there is probably less liability to cutting and 
seizing of the journal by the bearings when running at high speeds. 
There is also less frictional loss between the bearing surfaces of the 
stepped pulley and spindle when the latter is not larger than is 
necessary to give ample bearing surface. The best material for the 
spindle is, therefore, a moderately hard machine steel, used without 
hardening. In case great accuracy and very little wear of the bear- 
ing surfaces are desired, wrought iron or very low carbon steel 
can be used by case-hardening and then grinding to form in an 



SELECTION OF MATERIALS. 4C9 

emery grinding machine. This is fche method frequently ased in 
making milling-machine arhors. 

In very heavy lathes, sach as are used for turning locomotive 
drive-wheels, shafts for ocean vessels, or heavy ordnance, many 
designs have the driving-cone and back-gears placed on shafts serv- 
ing to support them only, and the spindle is driven by a pinion 
meshing in a gear attached directly to the face-plate. By this 
means the spindle is relieved of all duty except supporting the face- 
plate and the load upon it or the live-centre. Since there is no 
torsional force to be resisted by the spindle, and furthermore 
because such lathes run at moderate speeds, the spindle can be 
enlarged and made of a weaker material than steel. Cast iron is 
found to give good service in such cases when made much larger 
than would be necessary for steel, lightness and rigidity being 
secured by making the spindle hollow. The large diameter gives a 
low pressure per unit area npon the bearing surfaces and, with 
proper care during the early life of the machine, they become 
glazed and wear well, even when the bearing as well as the spindle 
is of cast iron. The cost is much less for such a spindle of cast iron 
than for steel, even when the diameter is kept down with the latter. 

In general, a steel lathe-spindle does not run satisfactorily npon 
cast iron, and since the head-stock of a lathe is almost invariably 
made of this material, it is necessary to provide some other material 
for the bearing surface npon which the journal of a steel spindle 
runs. In this case the material of the boxes is selected with regard 
to its coefficient of friction, durability in resisting f rictional abrasion 
and wear, and possibly cost. The last item is generally not a very 
important one, however, since there is a comparatively small amount 
of material in them ; brass, bronze. Babbitt metal, or some of the 
alloys resembling some one of them are the materials commonly 
chosen. Brass and bronze are more expensive, but they give a 
better and more finished appearance to the machine than Babbitt's 
or similar alloys which are used as linings for a cast-iron shell. 
The cost is far less for these metal linings with cast-iron casings 
than for brass or bronze. 

In many cases the conditions to be fulfilled indicate clearly the 
necessary material; thus a spring that must have considerable 
strength and resilience mnst be made of steel containing enough 



410 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 

carbon to caase it to harden and temper under proper heat treat- 
ment, or it must be made of brass or some similar alloy; in the 
general case, therefore, there are narrow limits to choose between. 
As soon as the purpose of the spring is defined, as for car- or 
wagon-springs, the cost immediately limits the selection to a single 
choice, steel being the only material admissible. 

Hydraulic presses when made of cast iron allow the water to 
percolate through the walls of the cylinder when subjected to high 
pressures. This was formerly a source of great trouble, and various 
devices were invented to overcome it, one being to line the cylinder 
with copper, brass, or bronze. When steel castings came to be 
successfully made, it was found that they were water-tight even 
under pressures much higher than had been attempted with the cast 
iron. Naturally the material to be selected for a high-pressure 
hydraulic press is steel, the selection in this case being based mainly 
upon the density and consequent water-tightness of the metal, 
although the greater strength is a considerable factor, these two 
more than overbalancing the increased cost due to the higher price 
of steel castings and the more expensive work of machining. 



SBLSCTION Oy MATERIALS. 



411 






S 

M 



O 

o 
(J 

O 












1 




* 


•5 








1 


1 






t9 

1 


1 


1 


1 














^ 


t« 


to 






VI 


5 






a 


a 


•a 






bobo bobo 1 


5 


U 


U 






5- 


a ^. 


a 


? 


^^ 


-'^ 


^^ 




1 


1 s 


^ 
s 


1 
1 


3^ 


If 


II 


a a 


^1 's't . 


■8s ai 


31 


•a 

n 


a 

1 


n n 


1 


a 

1 


i 

• 

• 

: 




2 . 


''S ^ 




^ 














11 


2 .2 


i 


ja 


•a •§) 










ll 


ii 


a 


!2i 


£ 


s 










•^ 


.: S 




!> 




a 


S 


If a>' S 




^ 3 

^ ^ 

S 


•a 




« 1 


,1 




'^t 








P 


1 




1: 






■■■ 1 










8 - 


f*^ 




,2 






a 








a 

2 




• 1 




1 














; 1 


|1 


i 1 

: "2 




i 

• s • 








11 


i 


1 




1 
■ 1 


Is 


i 1 




•3 






: 11 






■ s 


s^- 


i i 




.S 






: -o-^ 






"S 


o| 


08 




s 






SbO 




1 


a 
■ 2 

i 
f 

1 




1 


: S 

E 

; s 
' 1 




9 

1 

PC 


' 1 


1 f 
' 1 


a 

i 



412 FOBM, STBENOTH, AND PJOOPOBTIOITS OV PABTS. 






M 

o 



o 

o 
» 

3 

M 

•a 

H 
O 



;3 

O 

3 

Q 
O 

- Q 

GO 

W 

H 



1 



§- 



las 



I 



s 

i 



II 



1 



§ § § 
fit 

oT »' of 



I 




§■ 



§ § 



§§§§§§§§ 

^ :^ & s' ^ sf 9 sf 



g 



§§§§§§§§§ 

S' 9 « ^ S^ i: if gf S^ 



l§S§l§ 



IstN 



8f 



!§§§§§§§§§ 

S S S' S* 8 S S 9 !f 8' 



1^ 



§o§|§ 




APPENDIX 

A. Deyelopment of eqnations for an angnilar-tliread screw. — 

Fig. 178 shows a portion of an angular-thread screw of which HH^ 




Fig 178. 



is the mean helix, and KK' a tangent to it at T. Fis a point on 
HH\ and VY is parallel to the axis of the screw. VY may be 
taken as representing an elementary force acting on the thread 
at Y. O'Y is the position of the generating-line when passing 
through the point Y\ the angle of O'Fwith the radial line R'Y 
is ft. R' Y is, of course, normal to the axis XX of the screw. 

The elementary force VY may be resolTed into two forces, one 
radial along YR'y and the other along the line WY^ normal to O'Y 
and lying in the axial plane containing (?'Fand R'Y. The radial 

418 



414 APPENDIX. 

tsomponent is annnlled, so far as forces external to the screw are 
concerned, by the radial component of the elementary force corre- 
sponding to VY^ and lying diametrically opposite it; therefore the 
radial component does not need fnrther consideration. 

The component acting along WY is held in eqnilibrinm by two 
forces, one i^NY) normal to the sarface of the thread, and the other 
(not shown in the figare) perpendicalar to the axial plane through 
Y (i.e., perpendicular to the plane containing ff'J'and R'Y). 
This latter force is the one which, acting with a lever-arm equal to 
the radius of the mean helix, tends to produce rotation of the screw. 

In order to determine the relative values of this rotative force 
4ind that along WY it is necessary to know the value of the angle 
a = angle NYW in terms of the thread-angle /S and the helix- 
angle 0. (d is the angle between KK' and a normal to the axial 
plane through Y.) 

To facilitate the determination of the value of a^ the spherical 
ttriangle GRTmB,yhe formed by extending YG' and YE\ In this 
triangle- the side g = 90° and /3 are known, as well as the solid 
.angle E along RYy which equals 90° + 6. The angles G and K 
«re each less than 90°. 

The line NY is normal to the plane KYGj and since GYis the 
intersection of this plane with the axial plane RYGj a is the com- 
j)lement of the solid angle ff, or or = (90° — G). 

The determination of the value of a in terms of and y5 is as 
follows: 

cos r = cosg cos /? + sin ^ sm /3 cos R 

= cos 90° cos /? + sin 90° sin /5 cos (90° + 6) 
= + sin /? sin ^ (197) 



and 



. ^ sini? . . -o sin (90° + (?) 

sm G = smg — = sm 90 — —> - 1 . 

sm r |/i _ cos' r 

Substituting the value of cos r given in equation (197) gives 

.cos ff 



sin G = 



Vl - sin* /S sin' 



APPENDIX. 415 

By trigonometrical relations 



cot G = 



sin 



, , COS* 6/ i^l - sin" 6 sin^ 



V' 



1 - sin* /S sin" 6^ ^ cos 6^ 



_ Vl- sin" /y sin' g - cos" g 
~ cos ' 

This reduces to 

cot (? = tan ^ cos /5 = tan a (198)^ 

The relation between the force VY and the turning force- 
through Y and normal to the axial plane RYG is as follows: 

The force WY= FFsec /?, and the turning force = IFF tan. 
(a + 0), in which <f> is the friction-angle. Therefore 

Turning force = VY sec /5 tan {a + <f}) 

1 — tan a tan <p 
And by substituting the value of a given in equation (198)- 

r« . , Tr-r^ ^ Ttan 6^ cos i5 4- tan 

Tummg force = FFsec /? r — ^ x. , . 

^ '^ 1 — tan & COB p tan 

__ — „ tan ^ + sec /^ tan <p 



1 — tan 6^ cos /S tan 0' 



By adding together all the elementary forces VY and calling 
their sum T= tension in bolt, the total turning force ^becomes 

^^ y tan g + sec /? tan 
1 — tan (^ cos /3 tan 0. 



416 APPKNDIX. 

B. GrapMcal determination of the moment of inertia of a plane 
area. Approximate method. — ^In the numerical solution giyen in 
§ 148 it can be seen that the I„ of the rectangle R (referring to 
Fig. 174) is made up of the sum of two quantities, one of which is 
the moment of inertia Ig of the rectangle about its own gravity axis, 
and the other the product a^ of the area of the rectangle R by the 
square of the distance of its centre of gravity from the gravity axis 
XX of the entire plane area. If the rectangle R were very narrow, 
.as measured perpendicular to the gravity axis XX, its Ig would 
become so small as to be negligible for practical purposes. Its I^ 
would therefore practically be equal to az^. The graphical solution 
which follows is a method of finding the sum of all the a«*'s for the 
«mall areas into which a given plane area may be divided. 

In Fig. 179 ^ is the plane section whose approximate moment 
of inertia about its gravity axis normal to the centre line of the 
area is required. This approximate moment of inertia will be given 
the same symbol !„. as the accurate moment of inertia about the 
gravity axis. 

Divide the section A into a number of thin strips, 1, 2, 3, 4, ... , 
13, 14, parallel to the gravity axis about which the moment of 
inertia is to be determined. The dotted lines indicate the divisions. 
In Fig. 179 these strips are made somewhat wide in order to secun 
sufficient space for lettering and to make the drawing clear. For 
very accurate work they should be made narrower. These strips 
may be of the same or different widths. The position of the centre 
of gravity of each strip may now be estimated, and a line drawn 
through it parallel to the gravity axis about which /„ is to be found. 
The same can also be done for all the other strips. The area of 
strip 1 may now be represented by a line AB parallel to the given 
direction of the gravity axis. The line through the centre of 
gravity of strip 1, whose area is represented by the vector AB^ is 
indicated by the letters ah written on opposite sides of the line. In 
the same manner he is the line through the centre of gravity of 
strip 2, and BO the vector representing the area of 2. BG is a 
continuation of AB. By proceeding in this manner for all the 
strips, the line AOy which is the vector representing the entire area 
of the section A^ is obtained. 

Select a point P at a distance \A from the line A Oi and from 



APPENDIX. 



417 



it draw rays to the points A^ By 0^ . . . ^ 2fy 0. One position of 
P may be readily located by drawing lines from A and 0, each 




Fig. 179. 



making an angle of 45^ with AO] their intersection determines cm 
position of P. 



418 APPENDIX. 

Prom any point a' on ab draw a'X parallel to AP^ and «f 
indefinite length; from a' also draw a'V parallel to BP^ intersect- 
ing he at V ; through V draw Vc' parallel to CP, intersecting cd 
at c' ; through c' draw c'd' parallel to DP^ intersecting de at d\ 
Continuing this operation, n'X is finally drawn through n' parallel 
to OP, intersecting a'X at X. The .point X lies on the gravity 
axis of the entire section which is parallel to the strips into which 
the area was first divided. The gravity axis is therefore determined 
by drawing XX' perpendicular to the centre line of the area. (The 
proof that X determines the grayity axis is similar to that for find- 
ing a point through which the resultant of a number of parallel 
forces passes. It does not seem necessary to give it here.) 

Extend a'V to intersect XX' at q. The triangles a'Xq and 
PAB are similar, since their sides are parallel by construction. The 
altitude of a'Xq is z^\ that of PAB is \A0. 

Therefore 

^AB)^{Xq)^{\AO)^z,^ 

whence 

(AB)z, = (i^O)(Xy). 

By multiplying both sides of this equation by z^ it becomes 

{AB)z^ = {AO)i\Xq X «,). 

In this last equation the first member (AB)z^ corresponds ta 
the required product represented by the general expression a^ for 
the strip under consideration; AO represents the entire area of the 
given section; and ^Xq X z^ equals the area of tlu9 triangle a'Xq. 
Therefore for strip 1 

Approximate /« = (area of given section) x (area of triangle a'Xq). 

In the sa^e manner it may be shown for the strip % that its 
4(pproximate 7a, = (area of total section) x (area of triangle Vqr)^ 

i^ud so on for all the strips into which the given section is divided. 
The sum of the areas of all the triangles of a nature similar to 



APPENDIX.. 419 

a'Xq and Vqr eqaals the area of the f anicnlar polygon By shaded 
in the figure. Therefore for the entire section A^ 

Approx. / = (area of given section) X (area of f anicnlar polygon) 
= (area A) X (area E). 

Both of these areas can be conveniently measured with a pla^ 
nimeter. 

In a manner similar to the above it may be shown that the 
moment of inertia of the section A about any axis YY' parallel to 
XX' \a 

Approximate I^ = (area A) X (area a'SV . . • m'n'vsa'). 

The points «, /, and u are obtained by prolonging a'%^ a'q^ and 
aV to intersect YY'. 



420 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 




Fig. ISa 




Fia. 181. 



BEABINGS AND LUBRICATION, 



420flJ 



Table LI. 

Ball Bearings. Dimensions and Loads. 

Refers to Figs. 180 and 181. 

(Prom Catalogue of the Deutsche Waffen- und Munitions-Fabriken, Berlin, 
with slight modifications.) 

25.4 millimetres = 1 inch; .4536 kilograms «1 pound. 



Number Correepond- 


Diameter of Rings, 


Width of 




Safe Load 




ing 


to 


MUlimetres. 


'"T- 


Revolu- 
tions per 


Normal to 
Axis, 


Weight of 
Bearing, 
in Kilo- 










Milli- 


Minute. 


in Kilo- 








Inside. 


Outside, 


metres. 




grams. 


grams. 


Fig. 180. 


Fig. 181. 


/>!. 


Da. 










1 




12 


37 


9 


1500 


140 


0.050 


2 


"2 


15 


40 


9 


^ 


160 


0.060 


3 


3 


20 


52 


10 


260 


0.110 


4 


4 


25 


62 


12 


s 


380 


0.175 


5 


5 


30 


72 


13 


1 


500 


0.260 


6 


6 


35 


80 


14 


s» 


670 


0.330 


7 


7 


40 


90 


16 


850 


0.475 


8 


8 


45 


100 


17 


1 


1100 


0.625 


9 


9 


50 


110 


19 


1250 


0.825 


10 




55 


117 


19 


v* 


1400 


0.935 


11 




60 


127 


20 





1600 


1.160 


12 




65 


137 


22 




1900 


1.46f 


13 


..... 


70 


147 


24 




2200 


1.83i 


14 




75 


157 


25 




2500 


2.200 


15 




80 


168 


27 


500 


2900' 


2.715 


52 


52 


20 


65 


14 


500 


480 


0.240 


53 


53 


22 


72 


la 


■ %% 


600 


0.350 


54 


54 


25 


80 


17 


750 


0.450 


55 


55 


27 


88 


19 


900 


0.600 


56 


56 


30 


95 


20 


•^•i 


1100 


0.738 


57 


57 


35 


103 


22 


300 


1300 


0.935 



Thrust bearings for 1500 to 4000 R.P.M. are made in numbers 5 to 10 
similar to Rg. 180. except that one less ball is used. The loads for thrust 
bearings of this type are one-third of the values in the ''safe load'' colunm 
of the above table. 



4206 FORM, STRENGTH, AND PROPORTIONS OF PARTS. 



-sr*\ 




FiQ. 182. 



}•— «p** 




FiQ. 183. 



BEARINGS AND LUBRICATION. 



420c 



Table LIL 
Ball Bearikos. Dimbnsioks axd Loads. 

Refers to Figs. 182 and 183. 

(From Catalogue of the Deutsche Waff en- und Munitions-Fabriken, Berlin, 
with slight modifications.) 





25.4 millimetres - 


-1 inch; . 


4536 kUograma- 1 


pound. 




Number Corraspond- 
inc to 


Diameter of Rinse, 
Millimetree. 


*Width of 
Rings. 


Revolu- 


Safe Load 
Normal to 


Weight of 










S. 
MiUi. 
metiee. 


tione per 
Minute. 


Axis, 
in Kilo- 
grams. 


Bearing. 


Fig. 182. 


Fig. 183. 


Ineide. 
Di. 


Outeide, 
Da. 


in Kilo- 
grame. 


102 




10 


32 


9 


3500 


95 


0.035 


103 


ios' 


15 


37 


9 


^ 


115 


0.050 


104 


104 


20 


42 


9 


135 


0.060 


105 


105 


25 


52 


9 


s 


250 


0.087 


106 


106 


30 


62 


10 


1 


340 


0.137 


107 


107 


35 


70 


10 


1 


375 


0.175 


108 


108 


40 


80 


11 


420 


0.250 


109 


109 


45 


85 


11 


1 


450 


0.275 


110 


110 


50 


90 


11 


500 


0.320 


111 


111 


55 


100 


12 




650 


0.410 


112 


112 


60 


105 


12 




o 


700 


0.450 


113 


113 


65 


115 


14 






900 


0.620 


114 


114 


70 


120 


14 






950 


0.650 


116 


115 


75 


130 


16 






1200 


0.765 


116 


116 


80 


135 


16 






1250 


0.950 


117 


117 


85 


145 


18 






1520 


1.160 


118 


118 


90 


150 


18 






15S0 


1.235 


119 


119 


95 


160 


20 






1875 


1.530 


120 


120 


100 


165 


20 






1950 


1.585 


121 


121 


105 


180 


22 






2300 


2.235 


122 


122 


110 


185 


22 


1500 


2350 


2.275 



Thrust bearings for 1600 to 4000 R.P.M. are made in numbers 102 to 122 
similar to Fig. 182, except that one less ball is used. The. loads for thrust 
'bearing9 of this type are one-third of the values in the "safe load" column 
•of the above table. 



420d FOBM^ STRENGTH, AND PROPOBTIONB OF PART& 




TOP RINQ HAS n^T BEARING SURFACBr 
FiC. 184. 




Pig. 185.' 




BEARINGS AND LUBRICATION- 



420e 



Table LIII. 
Ball Thrust Bearings. Dimensions and Loads. 

Refers to Figs. 184, 185, and 186. 

(From Catalogue of the Deutsche Waffen- und Munition»-Fabriken, Berlin ) 

The bearings are made either with both rings grooved, as in Fig. 185. or 
with one ring flat and the other grooved, as in Fig. 184 Numbers 21. 23. 
25, and 28 have a separate spherical-surface plate as shown in Fig. 186. 

25.4 millimetres^ 1 inch; .4536 kilograms — 1 pound 





DimensioDB in IfiUi metres. 


Safe Tx>ad8 in Kilograms 




1 
1 




u 






il 


Fig. Ig4. 


Fit ise 


3 


1 


S 


i ' 




!l 


.If 


ill 


II 




U 

^5E 


5.2 


h 


1 




-oc 


«J 


«60 


--0^ 


--cS 


s-^s 


R-HS 


:^-;s 


"-K 


^-S 


4 
3 


1 


Q 


1 






= £ *< 




^t^ 
1^1 
D 










D 


J>i 


2?:t 


J^ 


B 
















6 


30 


32 


53 


18 


40 


im 


200 


250 


300 


350 


1100 


0.15 


7 


35 


37 


62 


21 


50 


200 


2m 


300 


400 


450 


1500 


0.21 


8 


40 


42 


64 


21 


50 


250 


300 


350 


450 


550 


1600 


0.24 


9 


45 


47 


73 


25 


60 


300 


350 


400 


550 


700! 


2100 


0.36 


10 


50 


52 


Ts 


25 


65 


360 


400 


600 


650 


^00 


2300 


0.40 


U 


m 


57 


HS 


^ 


70 


400 


500 


600 


750 


1000 


2900 


0,57 


12 


60 


62 


m 


2S 


75 


450 


550 


700 


850 


um 


31000.80 


13 


65 


67 


100 


32 


SO 


550 


650 


800 


1000 


1200 


3^O0!0.88 


14 


70 


72 


103 


32 


^5 


600 


700 


900 


1100 


14m} 


4rx>o 


0.91 


16 


HO 


82 


115 


35 


95 


700 


800 


1100 


1200 


1700 


5000 


1,10 


17 


P5 


gg 


125 


38 


105 


850 


950 


1300 


1500 


2000 


6000 


1.44 


19 


05 


9S 


140 


41 


115 


1000 


1150 


1600 


19(X» 


2400 


7000 


1 90 


2] 


106 


lOS 


155 


40 


130 


1200 


1400 


ISOO 


2200 


2700 


KOOO 


2.42 


23 


115 


118 


1*^5 


43 


140 


Vim 


1 1 .00 


2200 


2500 


3200 


10000 


2 91 


25 


125 


128 


175 


46 


150 


1400 


IIIOO 


2400 


2900 


3700 


lUHX) 


3.57 


2S 


140 


143 


200 


52 


170 


1700 


2200 


3000 


3700 


4SO0 


13000 


4.99 



INDEX. 



A]k>3rB, eo m pcwtion of, 49. 

tests of, 49. 
Antifriction curve, 67. 
Armature ring, deeagn of, 320. 

Babbitt metal, 43. 
BaU bearings, 98, 42a 

cup and cone, 107. 

for hub. 107. 

journal type, 99. 
Ball joumal4>earing, 99, 420. 

safe load, 101. 
Ball thrust-bearing, 102. 420(2. 
Bearings, Babbitt metal, 43. 

ball thrust, 102, 420t2. 

button thrust, 65, 238. 

cast iron, 44. 

chilled cast iron, 44. 

collar, 26, 74. 

conical, 112. 

cooling of, 42. 

fibre graphite, 45. 

formiuas for, 56. 

friction of, 47. 

lathe spindle, 28. 

length of rectilinear, 17. 

line shaft, 29. 

materials for, 42. 

metallic alloys, 49, 50l 

mineral, 46. 

nature of surfaces, 33. 

pad lubrication, 37. 

pivot, 66. 

planer saddle, 13. 

pressure, 89. 

constant load, 34. 

intermittent force, 34 

reversed force, 34. 

pulley, 38. 

rectiTmear, 1. 

roller step, 238. 



Bearings, thnist, 95, 238, 420d. 

rota^ motion, 20. 

Schide's, 68. 

self-aligning, 27, 28. 

shaper ram, 10. 

special forms. 111. 

step, 60. 

submerged, 47. 

surface, 1. 

lubrication of, 1. 

test of, 52. 

tractrix, 68. 

white metal, 43. 

wood, 47. 
Bearing alloys, 43. 

tests of, 49, 50. 
Belts, 160. 

angle of contact table, 164. 

binder pulleys, 188. 

chain, 191. 

chrome-tanned, 179. 

coefficient of friction, 172, 17^ 
180. 

cotton, 179. 

crossed, 184. 

diagrams for power of, 7. 

diameter of pulleys, 177. 

efficiency, 1^. 

elongation of, 167, 174. 

equations for power of, 160. 

fnction of leather, 172. 

euide pulleys, 188. 

high speed, 176. 

leather link, 180. 

linen web, 176. 

portion of pulleys, 181. 

power transmitted by, 160. 

pull of leather, 175. 

pulley diameters, 177. 

quarter turn, 178. 

rawhide, 179. 

481 



422 



IND£X. 



Belts, rubber, 180. 

semi-rawhide, 179. 

sUp of, 172. 

speed of, 176. 

strength of, 172. 

tandem drive, 167. 

tension at rest, 166. 

thickness of, 178. 

tightener piiUeys, 188. 

varying speed of, 184. 

velocity of, 176. 

wear of, 177. 

weight of, 178. 

wofting strength, 173. 
Belt drives, speci^d forms, 1S2, 18& 
Bevel gears, 144. 

efficiency, 143. 

examples, 148. 

friction, 167. 

strength, 144. 

wear, 151. 
Blanton patent fastening, 271. 
Bolts, 252. 

endurance of, 265. 

special forms, 266. 

strength of, 261. 
Bolt heads, proportions of, 249. 
Brake, bearing pressure, 308. 

friction, 307. 

Prony, 309. 

strap, 307. 
Button thrust-bearing, 65, 238. 

C-frames of machines^ 391. 
Calking riveted joints, 362, 377. 
Cap screw, 252. 
Chain belts, 191. 
Coefficinet of friction, 152. 

journal friction, 47. 
Collar bearings, 26, 74. 

proportions from practice, 76, 77. 
Collar friction of screws, 216. 
Conical pivot-bearings, 66. 
Conical roller-bearings, 92. 
Connecting rod end, 26. 
Couplings, cone friction, 304. 

expansion for pipe, 360, 

flexible, for shaft, 301. 

friction, 302. 

materials for friction, 307. 

multiple ring friction, 306. 

packing for pipe, 356. 

pipe, 356. 



itive clutch, 302L 

shaft. 
Crank shaft, 294. 

lubrication, 39. 
Crank pin, lubrication, 39. 
Cross-head bearing, length of, IS. 

lubrication of, 19. 
Cros&-head guide, 16. 
Crown friction gears, 158. 
Cylinder, bursting tests of, 360. 

packing for, 352. 

stress in, 347. 

tension in, 347. 

tests of cast iron, 352. 

Engine cross-head guides, 16, 18. 
Expansion couplings, 360. 

Fibre gears, 142. 
Fibre graphite bearing, 26b 
Flanges for pipe, 356. 
Fly-wheels, 311. 

for band saw, 346. 

built-up plate, 336. 

bursting tests, 326. 

composite arms, 336. 

design of, 360. 

designs from practice, 830. 

hollow arm, 336. 

inertia of, 318. 

kinetic energy, 314. 

moment of mertia, 312. 

numerical solution, 316. 

overhung, 344. 

sectional rim, 325. 

stresses in, 320. 

tangent arms, 336. 

tests of, 326. 

wire- wound, 337. 
Forced fits, 273. 

allowance for, 277. 

stress in parts, 273. 

tables from practice, 278. 
Forced lubrication, 65. 
Friction brakes, 303, 307. 

Prony, 30vS. 

strap, 307. 
Friction couplings, 303. 

conical, 304. 

material for, 307. 

multiple ring, 306. 
Friction gears, 152. 

capacity of, 155. 



INDEX. 



423 



Friction gears, coeffident of friction, 
163. 
double cone, 159. 
efficiency of, 156. 
Friction of Journal bearinfiss. 49, 50, 

52. 
Friction wheels, 152. 

Gafikets, 350, 356. 
-Gears, bevel, 144. 

bevel, wear of, 151. 

capacity of friction, 166. 

crown friction, 15S. 

double cone friction, 150. 

efficiency of bevel-, 152. 

efficiency of screw-, 245. 

efficiency of spur-, 143. 

examples solved for bevd-, 148. 

friction, 152. 

friction bevel, 157. 

friction of screw-, 246. 

grooved friction, 166. 

interchangeable, 170. 

mortise, 140. 

problems in bevel-, 149. 

problems in spur-, 130. 

screw, 244. 

spur, 113. 

stepped, 137. 

tests of, 135. 

variable speed friction,' 168, 150. 

wear of bevel, 151. 
Clear teeth, breaking load, 135. 

buttressed, 138. 

(liagrams for strength of, 122. 

factor of safety for, 142. 

formulas for, 136. 

formulas for strength, 128. 

mdonted fibre, 142. 

loads for short, 140. 

non-metallic, 140. 

pressure on, 114. 

rawhide, 142. 

short, 139. 

shrouded, 138. 

stepped, 137. 

strength of, 113. 

strength of bevel, 144. 

strengthening of, 137. 

table of proportions, 118. 

tests of, 135. 

woridng loads for short, table, 140. 
Ohaphite bearing, 45. 



Horse power, equations for, 122, 

165. 
HydauUc riveter, 406. 

Journal bearings, 21. 

adjustable, ^. 

ball, 98. 

capacity of, 60. 

changing proportions, 64. 

collar, 26. 

cylindrical, 20. 

eccenCnc sleeve, 2^. 

formulas for, 56. 

friction of, 47. 

loose pulley, 38. 

lubricating appliances, 86, 88. 

lubrication of, 30, 41. 

materials, 42. 

pressures, 69. 

problem in design of, 67. 

proportions of, 66. 

changing, 54. 

rod end, 26. 

roller, 78. 

rubbing surfaces, 38. 

self-aligning, 27. 

submerged, 47. 

test of, 52. 
Journal boxes. Babbitt metal, 43. 

brass or bronze, 42. 

cast iron, 42. 

chilled cast iron, 44. 

conical, 25. 

fibre graphite, 45. 

hardened steel, 46. 

mineral, 46. 

white metal, 43. 
Journals, abrasion, 44. 

burnishing, 33. 

cast iron, 45. 

hardened steel, 46i 

rolling, 33. 

seizure of, 44. 

Keys, 267. 

Blanton patent fastening, TTL 

dimensions of, 268. 

eccentric, 271. 

feather, 268. 

roUer, 271. 

diding, 268. 

square, ^. 



424 



INDEX. 



Lathe bed, material, 408. 
Lathe carriage, 12. 
Lathe slide, angle of, 14. 
Lathe spindle bearing, 28. 
Lathe ways, 1. 

angle of, 10. 

lubrication of, 10. 

pTX>portions of, 10. 
Lubrication of crank |nn» 80. 

crank shaft, 99. 

cross-head bearing, 10. 

forced, 41, 65. 

journal bearings, 30. 

oil bath, 32. 

oil pad, 32. 

oil ring, 24, 35. 

ropes, non-metalfic, 20& 

rotary pullej^, 38. 

special appliances of» 32. 

step bearings, 62. 

Machine keys, 267. 
Materials, for machines, 407. 

general properties, 411. 

methods of working, 411. 

modulus of elasticity, 412. 

table of strength and moduli, 
412. 
Moment of inertia, 416. 
Mortise gears, 140. 

Nut locks. 260. 

Oil bath lubrication, 32. 
Oil pad lubrication, 32. 
Open frames, 391. 

Packing, 350. 

Pin fasteninjs, 270. 

Pipes, bursting tests of, 360. 

special forms, 355. 

spiral riveted, 356. 
Hpe couplings, 356. 

dimension tables, 360, 361. 

expansion, 360. 
Rpe flanges, 358. 

dimension tables, 360, 861. 
Rvot bearings, 66. 
Planer saddle, 12. 

angle of, 14. 
Planer ways, 1. 

angle of, 3. 



Planer ways, double angle, d. 

flat, 5. 

lubrication of, 6. 

pressure on, 3. 

proportions from pncAkm, 4.. 

V-snape, 2. 
Ptony brake, 302. 
Pulley for band saw, 340. 

bearing for, 38. 

binder, 188. 

cork face, 172. 

designs from practice, 330. 

dimensions of, 338, ^0. 

effect of rim ribs, 330. 

friction, 155. 

grooves for ropes, 199. 

guide, 188. 

hollow arm, 336. 

idle, 188. 

leather face, 172. 

lubrication of, 38. 

material of, 172. 

ribbed rims, 330. 

sectional rim, 325. 

special form, 337. 

table of dimensions, 338, 8301 

tangent arm, 336. 
Punching, 375. 
Punching machine, 391. 

frame, 402. 

Rawhide gears, 142. 
Rivets, 369. 

dimensions, 372. 

pitch, 371. 
Rivet holes, diameters, 371. 
Riveted joints, 362. 

efficiency, 374. 

equations, 369. 

examples from practice, 377.. 

faulty construction, 376. 

grooving, 376. 

pitch of rivets, 371. 

porportions of, 368. 

punching effect, 375- 

shearing effects, 375. 
Roller bearines, conical, 92L 

angle of rollers, 92. 

flexible roller. 85. 

lubrication of, 81, 87. 

material for, 80. 

proportions from practio8| 88y S 

test of, 90. 



INDEX. 



425 



Roller journal-bearings, 78. 

safe load, 81. 
Roller thrust-bearings, 92. 

angle of roller, 92. 

proportions from practice, 94. 

tests of, 95. 
Ropes, capacity of wire, 210. 

coefficient of friction, 203. 

dressing for, 206. 

equations for power, 196. 

grooves for, 198, 208. 

lubrication of, 205. 

position of piilleys, 207. 

pow^r transmission, 160, 108. 

sheaves for, 198. 

sheaves for, size of, 206. 

sheaves for wire, 208. 

size for power tranamiarion, 207. 

speed of, 204. 

splice length, 207. 

strength of, 204. 

velocity of, 204. 

wear of, 205. 

weight of non-metaDic, 207. 

wire, capacity of, 210. 

wire, for power transmissdon, 208. 
Rope drive, differential driving, 200. 

efficiency of, 208. 

idle sheave, 202. 

Schiele's bearing, 68. 
Screw-bolt, endurance of, 266. 

fine thread, 264. 

reducing diameter of body, 266. 

strength of, 261. 
Screw fastenings, 247. 

cap-screw, 252. 

endurance of bolts, 265. 

fineness of thread, 264. 

locking devices, 259. 

set screw, 254. 

strength of bolts, 261. 

stud, 252. 

through bolt, 252. 
Screw gears, 230, 244. 

coefficient of friction, 246. 

efficiency of, 245. 
Screw threads, 247. 

buttress, 251. 

international standard, 25QL 

Sellers, 247. 

square, 251. 

(J. S. Standard, 247. 



Screw threads, Whitworth, 250. 
Screws, angular thread, 227. 

coefficient of friction for squaie- 
thread, 221. 

efficiency of square thread, 220. 

efficiency of V-thread, 229. 

equations for turning force, 213^ 

equations for V-thr«ad, 413. 

overhauling, 291. 

pitch of, 213. 

power transmisdon, 212. 

strength of, 225. 

stress in, 225. 

tests of, 222. 
Set screws, 254. 

holding power of, 257. 
Shaft couplings, 299. 

flexible, 301. 
Shafting, bending strength, 286k 

deflection of, 289. 

head shaft problem, 292. 

hollow, 296. 

not round, 299. 

over-hanging crank, 299. 

round, 282. 

tests of, 297. 

torsional strength, 283. 

torsion and bending, 2SSk 

twist of, 285. 
Shaper bearing, 10. 

adjustment for wear, IL 
Shearing, 375. 
Shearing machine, 391. 

form of frame, 403. 

stresses in frame, 394. 
Sheaves, diameter of, 206, 208b 

differential for wire rope, 211, 

material for, 202. 

ropes, 198, 206. 

wire rope, 211. 
Shrinkage fits, 273. 

allowance for, 277. 

stress in parts, 272L 
Spur gears, 113. 
Step bearing, 60. 

baU, 64. 

self-adjusting, 38. 

submerged, 64. 
Strap brake, 307. 
Stud bolt, 252. 

Thrust bearings, 74, 
antifriction, 68. 



426 



WBSSL 



Thnist-bearinga, ball, 102. 


Worm gearing, 23a 


button, 65. 


double, 239. 


conical roller, 92. 


double drive, 221. 


cylindrical rollera, 95. 


efficiency of, 239. 


properties from pnictioeb 76^ 


efficiency of, 242. 
equations for. 231. 
TTindly, 243 


rollers, 96. 


Schiele's, 68. 


speeds and pressuioi, 237. 


Tractrix, 68. 


tables of efi&iency. 236, 288. 


bearing, wear of, 66L 


tests of, 235. 




test of Hindly., 243 


White metal. 43. 





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Text-book of Organic Chemistry. (Walker and Mott) 8vo» a so 

^ Liboimtory Manoal of Organic (Hiemistry. (Walker.) xamo, x oo> 

Hopkins^ Oil-chemists' Handbook 8to» 3 00 

Jackaoa^ Directions for Laboratory Work In Physiological Chemistry. .8to. i 25 

Kesf^B Cast Iron 8vo, a 50 

Laddie Manual of Qnantltati?e Chemical Analyris xamo. x 00 

I^adansff'k Spectrum Analysis. (Thigle.) 8vo, 300 

Lassar-Cohn's Practical Urinary Analysis. (Lorenz.) xamo, x 00 

Appltcation of dome (Jeneffai Reactions to Investigations in Organic 

Chemistry. (Tingle.) xamo, i oo 

Leaeh'a The Inspection and Analyste of Food with Special Reference to State 

ControL 8ro, 7 50 

L8b's Elsctrolysis and Electrosynthesb of Organic 0>mpoands. (Lorenz.) xamo, i o» 

Lodge's Hotea on Assaying and MetaDorgical Laboratory Experiments 8vo, 3 00 

Liinge's Techno-chemical Analysis. (Cohn.) xamo, x 00 

Mandel's Handbook for Bio-chemical Laboratory xamo. x 50 

• Martlflli Labosatoty Guide to Qualitatiye Analysis with the Blowpipe . . xamo, 60 
Maaon's Water-eupply. ((^nsidered Principally from a Sanitary Standpoint) 

3d Edition, Rewritten 8vo, 4 00 

Examination of Water. (Chemical and BacteriologicaL) xamo, x 25 

Matthews'b The Textile Fibres. 8vo, 3 so 

Meyei^ Determination of Radicles in Carbon Compounds. (Tingle.). . xamo, x 00 

Mmef'a Maxmal of Assaying xamo, x 00 

IDxfesff'aBlBmentary Text-book of Chemistry xamo. i '50 

Morgan's Outline of Theory of Solution and its Results xamo, x 00 

Elements of Physical Chemistry xamo, 2 00 

Motss^ CalcnlationB used in Cane sugar Factoriea x6mo, morocco, i 50 

MttOikan's (General Method for the Iden ti fication of Pore Organic Omipounds. 

VoL L Large 8vo, 5 00 

CBfine's Laboratory (Julde in Chenkal Analysia 8vo. 2 00 

O'Driscoll's Notes on the Treatment of Gold Ores 8to, a 00 

Ostwald*s Conversations on Chemistry. Part One. (Ramsey.) (/n prew.) 
* Penfleld's Notes on Detenninative Mineralogy and Record of Mineral Tests. 

Svo, paper, 50 

Pktefa The Alkaloids and their (^lemical Constitution. (Biddle.) 8vo. 5 00 

Pinner'a Introduction to Organic Chemistry. (Austen.) xamo, i 50 

Poole's Calorific Power of Fuels 8vo. 3 00 

Fnscott and Winslow's Elements of Water Bacteriology, with Special Refer- 
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4 



* Reidg's Guide to Piece-dyeing 8vo, a5 oo 

Kichtrds and Woodman's Air,Water» and Food firom a Sanitary Standpoint. 8vo» 2 00 
Rictaards't Cost of Living at Modified by Sanitary Sciaace lamo x 00 

Cost ol Food a Study in Dietariet *. tamo, t 00 

* Richards and Williams's Tha Dietary Computer 8yo» i 50 

Ricketts and Russell's Skeleton Hotes upon Inoxganlc ChemlBttT. (Part I. — 

Kon-metallic Elemants.) 8vo# iBarocco» rS 

Rtckettt and Miller's Hotes on Assaying 8to« 3 00 

Rideal's Sewage and tha Bacterial Purification of Sewage 6to» 3 9i 

DUnfection and the Preservation of Food. 8vo« 4 00 

RIgg8*s Elementary Manual for the Chemical Laboratory 8vd» x 35 

Rostoski's Serum t>iagno8i8. (Bolduan.) xamo. z on 

Ruddiman's Incompatibilities in Prescripliona. 8vo» a 00 

Sabin's Industrial and Artistic Technology of faints and Varnish. 8*0* 3 00 

Salkowski's Phy8iok>gicaI and Pathological Chemistry. (Omdorff.). . . .Svo* a so 
Schimpf's Text-book of Volumetric AnaTysis lamo* a 50 

Essentials of Vofaunatric Analysis imm* x as 

Spencer's Hanabook for Chemists of Beet-sugar Houses. x6nio, manu^, 3 00 

Handbook for Sugar Manvfactofsn and tiiair GhamMs. .x6iiio» moroeco» a 00 
Stockbridge's Rocks and Soils 8ro» a 50 

* Tillman's Elementary Lessons in Hsat 8ro» i 50 

* Descriptive General Chemistry 8vd» 300 

TreadweD's Qualitative Analysis. (HaO.) /. 8vo» 3 00 

Quantitative Ana^fsis. (HaUX 8vo» 4 00 

Tomeaureand Russell's Public Water-euppHes Svo^ 5 00 

Van Deventer's Physical Chemistry for Be^^nnexB. (Bolkwood.) tamo* z 50 

* Walke's Lectures on Explosives 8tOi 4 00 

Wasliington's Manual of the Chemical Analysis of Rocks. 8vo» a 00 

Wassermann's Immune Sera: Hssmolysins* Cytotozins, and Precipltiiia. (Bol- 
duan.) lamo, z 00 

Wells's Laboratory Guide in Qualitative Chemical Analysis. ....-» 8vo, x 50 

Short Course In Inorganic Qualitative Chemical Analysis for Engineering 

Students xamo, z 50 

Whipple's Microscopy of Drinking-water. Svo, 3 5o ' 

WIechmann's Sugar Analysis Small 8vo. a 50 

Wilson's Cyanide Processes. xamo, x 50 

Chlorination Process xamo. i 50 

WttUing's Elementary Course In Inorganic Pharmaceutical and Medical Chem- 
istry xamo* a 00 

CIVIL BHOIREERIIIO. 

BRIDGES AUD roofs. HYDRAULICS. MATERIALS OF BBGIHEERniO 
RAILWAY BNGUfEERIBG. 

Baker's Engineers' Surveying Instruments xamo* 3 00 

Bixb/s Graphical Computing Table. Paper X9iXa4i inchss. 35 

«* Burr's Ancient and Modem Engineering and the Isthmian CanaL (Postag»» 

a? cents additlonaL) .8vo» net* 3 50 

Comstock's Field Astronomy for Engineers. « 8vo, a 50* 

Davis's Elevation and Stadia Tables 8vd» x 00 

EUlotfs Engineering for Land Drainage ismoi x 50 

Pnctical Farm Drainage. lamot x 00 

Folwell's Sewerage. (Designing and Malntenanca.X 8vo» 300 

Freitag's Architectural Engineering, ad Edition Rewritten 9fQ, 350 

French and Ives's Stereotomy 8vo, 2 so 

Goodhue's Munfcipal Improvements xamOf z 75 

(Goodrich's Economic Disposal of Towns' Refuse .8vo» 3 50 

Gore's Elements of Geodesy 8vo. a 50 

Hayford's Text->book of Geodetic Astronomy Svot 3 00 

Bariag's Ready Refersoca Tables (Conversion Factors). x6iiio, nocooob, a 50 

5 



Bowv'aRAttlninsWftUi for Earth zinio, i 45 

Johnson's (J. B.) Theory and Practice oi Surveying Small 8to, 4 00 

johnsonis (L. J.) Statics by Algebiaic and Graphic Methods. 8to. 2 00 

Laphice's Philoeophical Easay on ProhabUities. (Tmaeott and Emory.) tamo, a 00 

Mahan's Treatise on CItU Bnginacrtng, (1873.) (Wood.) 8vq» 5 00 

* DeacriptiTa Oeometry 8vo, 150 

Xanimaa'aBlamaittiof PrediaSiinreyincandOeodesy 8to, 3 50 

Elamtntaof Sanitary Engineering Svot 2 00 

Maniman and BroolES^i Handbook for Surreyon i6mot morocco. 2 00 

Xagntfli Plane Snrvaying 8to 3 50 

Ogdao'iStwer Deaign. lamo, a 00 

PatlMi'a Treatiae on CItU Engineering 8to half leatiier. 7 50 

Saed'a Topographical Drawing and Sketching 4to, 5 00 

BideaTa Sewage and the Bacterial Purification of Sewage 8vo, 3 so 

fl ia b ert and Biggin's Modem Stone-cutting and Masonry 8vo, x 50 

Smith's Manual of Topographical Drawing. (McMUkn.)....- Svo, 2 50 

SoodarickiBr'a Oiaphlc Statica. with Applications to Trusses, Beams, and 

JUchai. A909 2 00 

Iliylor and Thompsonis Treatise on Concrete^hun and Reinforoed. {In press.) 

t Tkantwine's CItU Engineer's Pocket-book idmo, morocco^ 5 00 

Waltfa Engineering and Architectural Jurisprudence. 8vo, 6 00 

Sheep. 6 50 
Law of Operations Preliminary to Conatmctlon in Bogbiaerittg and Archi- 
tecture 8vo. 5 00 

Sheep. 5 50 

Law of Contracts 8to. 3 00 

Warren's Stereotomy — ^Problems in 8tone*cuttlag. 8to. 2 50 

Webb's Problems in the Uf e and Adjustment of Engineering Instruments. 

lOmo. morocco, z 35 

* Wheeler'* Elementary Course of CMi Engineering. Sro. 4 00 

■Wikon's Topographic Surreying 8?o, 3 50 

BRIDGES AUD roofs. 

BoOir's Practical Treatise on the Construction of Iron Highway Bridget. .Svo, 2 00 

^ Thames River Bridge !. .4tD, paper. 500 

BarfB Course on the Stresses in Bridges and Roof Trussea. Arched Riba. and 

Suspension Bridges. 8to. 3 50 

Du Bols's Mechanics of Engineering. VoL IL. Small 4to. 10 00 

Voetor'a Treatise on Wooden Trestle Bridges 4to. 5 00 

Vawler's Coffer-dam Process for Piers 8n>. a 50 

Ordinary Foundations .8to, 3 50 

Oieene's Roof Trusses 8to, i 25 

Bridge Trusses • 8n>. a 50 

Arches in Wood, Iron, and Stone 8yo. 2 50 

Bowe's Treatise on Arches 8vo. 4 00 

Design of Simple Roof-trusses in Wood and Steel 8vo. 2 00 

JdhnaooSlBryan, and Tumeaure's Theory and Practice In the Designing of 

Modem Framed Stractures. Smill 4to. 10 00 

Mairiman and Jacoby's Text-book on Roofs and Bridges: 

Part L— Stresses hi Simple Trussea 8to. 2 50 

Part IL— Oraphic Statics 8to. 2 50 

Part m— Bridge Design. 4th Edition. RewriUen 8vo. 2 50 

Part IV.— Higher Stractures 8n). 2 50 

Moriaon's Memphis Bridge 4t0k xo 00 

WaddeU's De Pontibus, a Pocket-book for Bridge Engineers. . . x6mo. moroeco. 3 00 

Specifications for Steel Bridges xamo. x 25 

Wood's Treatise on the Theory of the Constraction of Bridges and Roofs. 8vo. 2 00 
Wflghfs Designing of Draw-spans: 

Part L —Plate-girder Draws 8to» a 50 

- Part n. — ^RlTeted-truss and Pin-connected Long-span Draws 8vo. 2 9> 

Two parts in one volume 8fo» 3 90 



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Bwdn't Svperiments upon the Contraction of tha Liquid Vein leening from an 

Orifice. (Trautwine.) Svo, a oo 

Bevey's Treatise oa Bydraulica. 8vo, 5 00 

Oivrch's Merhanice ef Rngineerlng. Sro, 6 00 

DlagiaiBa of Mean Velocity of Water in Open Channeli peper, i 50 

OolBa'k Gfmpiiical Solution of Hydranlic Problems i6mot morocco, a 50 

flatliar*t Dynammneteit, and the Meaaorement of Power lamo, 3 00 

Hohrelfs Watar-sapply Engineering. Pro, 4 00 

Mieirs Water-power..... 8fo, 5 00 

Vaartea't Water and Pablic Health lamo, i 50 

Water-filtration Works lamo, a 50 

. Cangwillet and Kutter's General Formula for the Uniform Flow of Water in 

Rivers and Other Caiannels. (Bering and Trautwine.) 8to, 4 00 

Haaen'i Filtration of PubHc Water-supply 8yo, 3 00 

Haxlehnnlfft Towers and Tanks for Water-works 8vo» a 50 

HerseheTs 115 Eiperiments on the Carrying Capacity of Large, Rhreted, Metal 

Conduits 8vo, a 00 

Maaon^ Water-supply. (Considered Principally from a Sanitary Stand- 
point) 3d Bdillon, Rewritten 8yo, 4 00 

MsRlman's Treatise on Hydraulics, gtfa Edition* Rewritten 8to, 5 00 

• Michle's Elements of Analytical Mechanics 8vo« 4 00 

Schuyler's Rcferroirs for Irrigation, Water-power, and Domeatic Water- 
supply Large 8to, 5 00 

** Thomas and Watt* s ImproTcment of Riyers. (Post, 44 c. additional), 4to, 6 00 

TWoeaure and RusselTs Public Water-supplies. Svo, 5 00 

Wegmann's Deeicn and Construction of Dams. 4to, 5 00 

Water-supp]yofthe(2ityof Hew York from 1658 to 1895 4to, 10 00 

Weirtach*s Hydraulics and Hydraulic MotofiL (DuBois.) 8yo, s 00 

Wilson's Manual of Irrigation Engineering Small 8to. 4 00 

Wolffli Windmill as a Prime Mover 8vo, 3 00 

Wood's Turbines 8vo, a 50 

Elements of Analytical Mechanics , 8vo, 3 00 

MATERIALS OF EH GIHEERIirG, 

Baker's Treatise on Masonry Construction. Svo, 5 00 

Roads and Pavements. 8vo, 5 00 

Black's United States Public Works Oblong 4to, 5 00 

Bovey's Strength of Materials and Theory of Structures. 8vo, 7 50 

Burr's Bhuticity and Resistance of the Materials of Bni^neering. 6th Edi- 
tion, Rewritten 8vo, 7 50 

Byrne's Highway Construction Svo, s 00 

Inspection of the Materials and Workmanship Employed In O>nstruction. 

x6mo, 3 00 

(^lurch's Mechanics of Engineering Svo, 6 00 

Du Bole's Mechanics of Engineering. VoL I Sxhall 4to, 750 

Johnaon's Materials of Construction Large Svo, 6 00 

Fowler's Ordinary Foundations Svo, 3 50 

Keep's CtM iron Svo, a 50 

Lanza's Applied Mechanics Svo, 7 50 

Martens's Handbookon Tecting Materials. (Henning.) 2 vols. Svo, 7 50 

MerriirtoStonee for Building and Decoration Svo, 5 00 

Merriman's Text-book on the Mechanics of Materials Svo, 4 00 

Strength of Materials xamo, x 00 

Metcalf's'SteeL A Manual for Steel-users lamo, a 00 

Patton's Practical Treatise on Foundations. Svo, 5 00 

Richey'S Handbook for Building Superintendents of Construction, (/n pren.) 

{Lo^kweU^S Roads and Pavements in France lamo, z as 

7 



Sabln's InduBtrial and Artistic Technology of Paints and Varnish 8vo, 3 00 

Smith's Materials of Machines xamo, x 00 

Snow's Princii»al Species of Wood 8vo, 3 50 

Spalding's Hydraulic Cement « xamo, 2 00 

Text-book on Roads and Pavements lamo, a 00 

Taylor and Thompson's Treatise on Concrete, Plain and Reinforced. (In 

preaa,) 

Thurston's Materials of Engineering. 3 Parts. 8vo, ft 00 

Part I. — ^Non-metallic Materials of Engineering and Metallurgy ftro, a 00 

Part n. — Iron and SteeL 8vo, 3 5© 

Part m.— A Treatise on Brasses, Bronzes, and Other Alloys and their 

Constituents 8vo. a 50 

Thurston's Text-book of the Materials of Construction 8vo. 5 co 

Tillson's Street Pavements and Paving Materials. ftvo, 4 00 

Waddell's De Pontibus. (A Pocket-book for Bridge Engineers.) . . x6mo, mor., 3 00 

Specifications for Steel Bridges lamo, i as 

Wood's (De V.) Treatise on the Resistance of Materials, and an Appendix on 

the Preservation of Timber 8vo, a 00 

Wood's (De V.) Elements of Analytical Mechanics 8vo, 3 00 

Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and 

SteeL 8vo, 4 00 

RAILWAY ENGINEERING. 

Andrews's Handbook for Street Railway Engineers 3x5 inches, morocco, i as 

Berg's Buildings and Structures of American Railroads 4to, 5 00 

Brooks's Handbook of Street Railroad Location x6mo, morocco, i so 

Butts's Civil Engineer's Field-book. i6mo, morocco, a so 

Crandall's Transition Curve i6mo, morocco, i so 

Railway and Other Earthwork Tables 8vo, x 50 

Dawson's "Engineering" and Electric Traction Pocket-book. i6mo, morocco, 5 00 

Dredge's History of the Pennsylvania Railroad : (1879) Paper, s 00 

* Drinker's Tunneling, Explosive Compounds, and Rock Drills, 4to, half mor., as 00 

Fisher's Table of Cubic Yards Cardboard, as 

Godwin's Railroad Engineers' Field-book and Explorers' Guide x6mo, mor., a so 

Howard's Transition Curve Field-book x6mo, morocco, x 50 

Hudson's Tables for Calculating the Cubic Contents of Excavations and Em- 
bankments. 8vo, X 00 

Molitor and Beard's Manual for Resident Engineers i6mo, x 00 

Ifagle's Field Manual for Railroad Engineers x6mo, morocco, 3 00 

Philbrick's Field Manual for Engineers. i6mo, morocco, 3 00 

Searles's Field Engineering i6mo, morocco, 3 00 

Railroad Spiral. x6mo, morocco, x 50 

Taylor's Prismoidal Formulae and Earthwork 8vo, i 50 

* Trautwine's Method ot Calculating the Cubic Contents of Excavations and 

Embankments by the Aid of Diagrams. 8vo, a 00 

The Field Practice of Laying Out Circular Curves for Railroads. 

xamo, morocco, a 50 

Cross-section Sheet. Paper, as 

Webb's Railroad Construction, ad Edition, Rewritten. x6mo, morocco, s 00 

Wellington's Economic Theory of the Location of Railways Small 8vo, s 00 

DRAWING. 

Barr's Kinematics of Machinery. « 8vo, a 50 

* Bartlett's Mechanical Drawing 8vo, 3 00 

* " Abridged Ed 8vo, 150 

Coolidge's Manual of Drawing. 8vo, paper, x 00 

Coolldge and Freeman's Elements of General Drafting for Mechanical Engi- 
neers Oblong 4to. a 50 

Durley's Kinematics of Machines. 8vo, 4 00 

8 



HiU's Text-book on Shades aiid Shadows, and Perspective Sto. 2 00 

Tamison's Elements of Mechanical Drawing 8vo, a 50 

Jones's Machine Design: » 

Part L — ^Kinematics^of Machinery 8vo, x 50 

Part IL — Form, Strength, and Proportions of Parts. 8vo, 3 00 

MacCord's Elements of Descriptive Geometry. Bro, 3 00 

Kinematics; or, Practical Mechanism. 8to, 5 00 

Mechanical Drawing 4to, 4 00 

Velocity Diagrams 8vo, x 50 

Mahan's Descriptive Geometry and Stone-cutting. 8vo, x 50 

Industrial Drawing. (Thompson.) 8vo» 3 5o 

Moyer's Descriptive Geometry, (/n preM.) 

Reed's Topographical Drawing and Sketching 4to» 5 00 

Reid's Course in Mechanical Drawing. 8vo, a 00 

Text-book of Mechanical Drawing and Elementary Machine Design . . 8vo, 3 00 

Robinson's PrinciplM of Mechanism. 8vo, 3 00 

Schwamb and Merrill's Elements of Mechanism. 8vo, 3 00 

Smith's Manual of Topographical Drawing. (McMillan.) 8vo, a 50 

Warren's Elements of Plane and Solid Free-hand Geometrical Drawing . . i amo, x 00 

Drsftlng InstntmcihtB and Operations xamo, x as 

Manual of Elementary Projection Drawing xamo, x 50 

Ms mis 1 of Elementary Problems in the Linear Perspective of Form and 

Shadow xamo, x 00 

Plane Problems in Elementary (yeometry xamo, x as 

Primary Geometry xamo, 75 

Elements of Descriptive Geometry, Shadows, and Perspective 8vo, 3 So 

General Problems of Shades and Shadows. 8vo 3 00 

Elements of Machine Construction and Drawing 8vo. 7 SO 

Problems, Theorems, and Examples in Descriptive Geometry 8vo, 2 so 

Weisbach's Kinematics and the Power of Transmission. (Hermann and 

Klein.) 8vo, s 00 

Whelpley's Practical Instruction in the Art of Letter Engraving tamo, 2 00 

Wilson's (H. M.) Topographic Surveying 8vo, 3 50 

Wilson's (V. T.) Free-hand Perspective 8vo. a 50 

Wilson's (V. T.) Free-hand Lettering 8vo, x 00 

Woolf's Elementary Course in Descriptive Geometry Large 8vo, 3 00 

ELECTRICITY AND PHYSICS. 

Anthony and Brackett's Text-book of Physics. (Magie.) SmaD 8vo, 3 00 

Anthony's Lecture-notes on the Theory of Electrical Measurements xamo, x 00 

Benjamin's History of Electricity 8vo, 3 00 

Voltaic CeU, 8vo, 3 00 

Classen's Quantitative Chemical Analysis by Electrolysis. (Bohwood.). .8vo, 3 00 

Grehore and S<|uier's Polarizing Photo-chronograph 8vo, 3 00 

Dawson's "Engineering" and Electric Traction Pocket-book. . i6mo, morocco, 5 00 
Dolezalek's Theory of the Lead Accumulator (Storage Battery). (Von 

Ende.) xamo, a 50 

Duhem's Thermodynamics and Chemistry. (Burgess.) 8vo, 4 00 

Flather's Dynamometers, and the Measurement of Power. xamo, 3 00 

Gilbert's De Magnate. (Mettelay.) 8vo. a 50 

Hanchett's Alterxuiting Currents Explained ; . xamo, x 00 

Hering's Ready Reference Tables (Conversion Factors) x6mo, morocco, a 50 

Holman's Precision of Measurements ., 8vo, a 00 

Telescopic Mirror-scale Method, Adjustments, and Tests Large 8vo, 75 

Landauer's Spectrum Analysis. (Tingle.) 8vo, 3 00 

Le ChatcUer's High-temperature Measnrexnants. (Boudouard — ^Burges8.)xamo 3 00 

LCyg Ele cUa lj sIs axid Ele cUuayu thesb ci OnanicCompoanda.. (Lorani,) lamp, i 00 

9 



* Lyoni'sTreatiie on Blectromacnetk Phenomena. Volt. L end IL 8?o, each, 6 oo 

* lOchie. Elements of Wave Motion Relating to Sound and Light. 8to, 4 00 

Hlaodef 8 Elementary Treatise on Electnc Batteries. (FJshpack, i ismo, 2 so 

* Rosenberg's Electrical Engineering. (Haldane Gee—Kiaxhrunner.). . . .8vo, i 50 

Ryan, Horris, and Hoiie's Blectrieal Mschinery. YoLL 9vo« a 50 

Thtnton's Stationary Steam-engines .....Sto, 2 50 

e nUman's Elementary Lessons in Heat. 8to, i 50 

Tory and Pitcher's ifsnnsl of Laboratory Physics Small Sto, 2 00 

Ulks> Modern. Blsctroljtte Copper Wellning 8to* 3 00 

LAW. 

* DaTis^ Elsments of Law Sro, 3 50 

* Txeatise on the MUttary Law ot United dtatee Sro, 7 00 

* Sheep, 7 so 

Mamal for Conrts-martial i6mo, morocco, i 50 

Waifs Engineering and Architectural Jnrisprodenee Svo, 6 00 

Sheev* 6 so 
Law of Operations Preliminary to Constmction in Engineering and Asthi- 

tectore \ Sfo, 5 00 

Sheep, 5 50 

Law of Contracts Svo, 3 00 

Winthrop's Abridgment of Mlfitary Law lamo, 2 50 

HAKTJFACTURES. 

Beniadov*t Smokelsas Powder^Hitro-ceUnloss and Theory of the CeUiilose 

Molecule lamo, 2 so 

BoUand'sIron Poonder lamo, a 50 

** The Iron Foonder,** Supplement. lamo, a so 

Encyclopedia of Founding and Dictionary of Foundry Terms Used in the 

Practice of Moulding lamo, 3 00 

Bissler's Modem High EzploeiTes Sto, 4 00 

Bffronfs Enzymes and their Applications. (Prescott) Sto 3 00 

Fitzgerald's Boston Machinist iSmo, x 00 

Ford's Boiler Making for Boiler Makers iSmo, i 00 

HopUns's Oil-chemists' Handbook Svo, 3 00 

Keep's Cast Iron Sto, a so 

Leach's The Inspection and Analysis of Food with Special Reference to State 

ControL (In prepanUtn.) 

Matthews*8 The Textile Fibres Sto, 3 50 

MetcalTs SteeL A Manual for Steel-users lamo, a 00 

Metcalfe's Coct of Manufactures— And the Administration of Workshops, 

PubUc and Prirate *. Sto, 5 00 

Meyer's Modem LocomotiTe Construction 4to, xo 00 

Morse's Calculations used in Ca n ee uga r Factories. • iteio« mnrocco, x 50 

* Reisig's Guide to Pieceniyeing Srob as 00 

Sabln's Industrial and Artistic Technology of Paints and Yamish Sto, 3 00 

Smith's Press-working of Metals .Sto, 3 00 

Spalding's Hydraulic Cement xamo, a 00 

Spencer's Handbook for Chemists of Beet-sugar Houses i6aio» morocco, 3 00 

Handbook-f or Sugar Manufacturers and their Chemists.. . x6mo morocco, a 00 
Taytor and Thompson's Trsatlae on Concrete, Plain and Rdnfocced. On 

preu.) 
Thurston's Manual of Steam-boiieffB, their Designs, Construction and Opera- 
tion • Sto, s 00 

* Walke'9 Lectures on EzplosiTes Sto, 4 00 

West's American Foundry Practice lamo, a 50 

Moulder's Text-book xamo, a 50 

10 



WoUTt Windmin •■ a Prime Motw 8to» 3 00 

Woodbury*! Fin Protectioii of Milto Syo, 2 50 

Wood's RnsUen Coatiiici: Coiroiion aad Electrolytit of Iron and Steel. . .8vo, 4 00 

KATHEKATICS. 

Bakor't BOIvtie Fnactionf 8ro> x 50 

* Bali's BlenM&ts of Diffemtlal Cak wl oe tamo, 4 00 

Bricgi'fl BlBments of Piano Analytic Oeomotry lamo, x 00 

Oo m pt o n*! Ifanwal of Logaritfamic Compotations xamo, x 50 

DaTii'B Introdvction to the Lock of Alfebra 8to, i 50 

< Dkkion'k Cottege Algebra Large xamo, x 50 

* Answers to Dickson's College Algebra 8vo, paper, 25 

< Introdnctlon to the Theory of Algebraic Equations Large xamo, i 2$ 

Ha]sted*s Blements of Oeometry 8to, x 75 

Elementary Synthetic Oeometry 8to, x 50 

Rational Geometry xamo, 

* Johnson's (J. B.) Three>place Logarithmic Tables: Vest-poclcet size. . paper, xs 

xoo copies for 5 00 

* Mounted on heavy cardboard, 8 X 10 Inches, as 

xo copies for a 00 

Johnson's (W. W.) Elementary Treatise on Di£ferential Calcnhis . . . Small 8vo, 3 00 

Johnson's (W. W.) Elementary Treatise on the Integral Calculus. .Small Sto, x 50 

Johnson's (W. W.) Curve Tracing in Cartesian Co-ordinates 12 mo, x 00 

Johnson's (W. W.) Treatise on Ordinary and Partial Differential Equations. 

Small 8vo, 3 50 

Johnson's (W. W.) Theory of Errors and the Method of Least Squares. . xamo, x 50 

e Johnson's (W. W.) Theoretical Mechanics xamo, 3 00 

Laplace's Philoaophical Essay on Probabilities. (Tntacott and Emory.) xaxno, a 00 

* Lodlow and Ba«. Blements of Trigonometry and Logarithmic and Other 

Tables 8vo, 3 00 

Trigonomatry and Tables published separately Each, 2 00 

* Ludlowli Logarhhmk and Trigonometric Tables 8to, x 00 

Mmtrer's Technical Mechanics. 8vo, 4 00 

Marrimsn and Woodward's Higher Mathematics 8to, 5 00 

MsRiman's Method of Least Squares 8to, a 00 

Bios and Johnson's Blemantafy Treatise on the Differential Calculus. 8m., Svo, 3 00 

Differential and Integral Calculus, a vols, in one Small 8to, a 50 

Wood's Blexnants of Co-ordinate Geometry 8yo, a 00 

Trigonometry: Analytleal, Plane, and Spherical xamo, x 00 

MBCHAHICAL EMGCIEERIHO. 

MATERIALS OF ElfGIHEERBIG, STEAM-EBGIirES AND BOILERS. 

Bacon's Forge Practice lamo, x 50 

Baldwin's Steam Heating for Buildings lamo, a 50 

Barr's Kinematics of Machinery 8to, a 50 

* Bartletf s Mechanical Drawing 8to. 3 00 

* - -. « AbrldgedEd. 8vo. x 50 

Benjamin's Wrinkles and Reclpea xamo, a 00 

Carpenter's Exp er im e n tal Engineering 8yo, 6 00 

Heating aad Ventilating BulMings Sto, 4 00 

Cary'ft Smoke Suppcessloa In Plants using Bituminous CoaL (/n prvp- 
araiian.) 

Clark's Gas and Oil Engine Small 8to, 4 00 

Cooiidge's Msnnal of Drawing 8to, paper, i 00 

Coofldge and Freeman's Elements of General Drafting for Mechanical En- 
gineers. Oblong 4to, a 50 

11 



Cromw«Il*lB Treatise on Toothed Gearinf zamo i so 

Treatise on Belts and Pulleys. zamo, i 50 

OorleT's Kinematics of Machines 8to, 4 00 

blather's Dynamometers and the Measaremeat of Power lamo, i 00 

Rope Driving zamo, 2 00 

OilTs Gas and Fuel Analysis for Engineers lamo* i as 

Hall's Car Lubrication. zamo. z 00 

Bering's Ready Reference Tables (Conversion Factors) z6moA mo r occo, a 50 

Button's The Gas Engine 8vo» 5 oe 

Jamison's Meclianical Drawing 8vo, a 50 

Jones's Machine Design: 

Part L — Kinematics of Machinery 8vo« z 50 

Part IL — Form, Strength, and Proportiona of Parta 8ve» 3 00 

Kent's Mechanical Engineer's Poclcet-book i6mo» morocco. 5 00 

Ken's Power and Power Transmission 8vo. 2 00 

Lsooaxd's Machine Shops, Tools, and Methods. (In prsst.) 

MacCord's Kinematics; or. Practical Mechanism. 8vo« 5 00 

Mechanical Drawing 4to. 4 00 

Velocity Diagrams 8to, i 50 

Mahan's Industrial Drawing. ( T hompson.) 8vo, 3 50 

Poole's Calorific Power of Fuels 8to, 3 00 

Riid's Course in Mechanical Drawfaig 8vo. a 00 

Text-book of Merhsnifsl Drawing and Elementary Machine Design. .8yo> 3 00 

Sichards's 0>mpressed Air zamo* z 50 

Robinson's Principles of Mechanism 8to» 3 00 

Schwamb and Merrill's Elements of Mechanfsm 8vo, 3 00 

Smith's Press-working of Metals 8vc», 3 00 

Thorston's Treatise on Friction and Lost Woric in Madidnery and Mlil 

Work 8vo» 3 00 

Animal as a Machine and Prime Motor, and the Laws of Baergatics. zamo, z 00 

Warren's Elements of Macliine C^nstructior and Drawing 8ro, 7 50 

Weisbach's Kinematics and the Power of Transmission. Hezrmami-^ 

Klein.) 8vo, s 00 

Machinery of Transmission and Governors. (Herrmann — Klein.). .Svo. 5 00 

Hydraulics and Hydraulic Motors. (Du Bois.) 8vo» 5 00 

Wolff's Windmia tM a Prime Mover Svo, 3 00 

Wood's Turbines Svo. a so 

MATERIALS OF ENGINEERING. 

Bogey's Strength of Materials and Theory of Structures Svo, 7 50 

Burr's Elasticity and Reaistance of the Mjg^terials of Engineering. 6th Edition 

Reset. Svo . 7 50 

Church's Mechanics of Engineering Svo, 6 00 

Johnson's Materials of Construction Large 8vo. 6 00 

Keep's Cast Iron Svo, 2 50 

Lanza's Applied Mechanics Svo, 7 50 

Martens's Handbook on Testing Materials. (Henning.) Svo, 7 SO 

Merriman's Tezr-book on the Meclianlcs of Materials Svo. 4 00 

Strength of Materials zamo, x 00 

MetcalTs SteeL A Manual for Steel-users lamo a 00 

Sabin's Industrial and Artistic Technology of Pamts and Varnish. Svo. a 00 

Smith's Materials of Machines lamo. i 00 

Thurston's Materials of Engineering 3 vols , Svo. 8 00 

Part n.— Iron and Steel Svo. 3 50 

Part m. — A Treatise on Brasses, Bronzes, and Other Alloys and their 

Constituents. Svo 2 50 

Text-book of the Materials of O>nstruction Svo. 5 00 

12 



Wood's (De V.) Treatise on the Resistance of Materials and an Appendix on 

the Preservation of Timber 8vo, 2 00 

Wood's (De V.) Elements of Analytical Mechanics 8vo, 3 00 

Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and Steel 

8vo, 4 00 



8TBAM*£NGni£S AND BOILERS. 

Carnof t Reflectioos on tha Motiv* Power of Heti. (Thurpton.) xaiio, x 50 

Dftivion's "Enflnecrinc^ and Btoctric Traction Pocket-book.. i6ino, mcr.. s 00 

Ford's Botter Makiqg for Bolter Maken xSmo. x 00 

Ooii't Locomotive Spuks .8vo. a 00 

Bemenway't Indicator Practice and Stoam-oiiciiie Economy xamo a 00 

Hstton's Mfchanical Rnginiwiring of Powaf Planti 8vo, 5 00 

Heat and Beat-eoginea 8to, 5 co 

Ka&faSleam-bo'ler Bcoaomy 8vo, 4 00 

Knaav's Pnctica and Theory of tba Injector 8to. x 50 

KacCord'a SUde-vmtvea 8to. 2 00 

Meyer'a Modem Locomotife Conatmction 4to. xo 00 

Feabody** Manual of the Steam-engine Indicator zsmo, i 50 

Tablee of the Propertiea of Saturated Steam and Other Vapors Svo, x 00 

Tbermodynamica of the Steam-engine and Other Heat-enginei Sto, 5 00 

Yahro-geara for Steamrenginea Sto, 2 50 

Peabody and Miller's Steam-boflen Sro. 4 00 

Fray's Twenty Yeara with the Indicator Laxge Sto, 2 50 

Pnpln'a Tbermodynamica of Reveiaible CycJea in Gaaea and Saturated Vapora. 

(Oaterberg.) xamo, x 35 

Reagan's I^ocomotJTea; Simple, Compound, and Electric xamo, 2 50 

Rontgen's iPrind^lea of Tbermodynamica. (Du Boia.) 8to, 500 

Slnclair'aLocomotiTe Engine Rimning and Management xamo* 2 00 

Smart's Handbook of Engineering Laboratory Practice xaxno, 2 50 

Snow's Steam-boiler Practice Sto, 3 00 

Spangler'a Vahre-geara Svo, 2 50 

Rotes on Tbermodynamica xamo, x 00 

Spangler, Greene, and Marshaira Elements of Steam-engineering Svo, 3 00 

Thurston's Handy Tablee Sro, x 50 

Manual of the Steam-engine a vola. Sto, xo 00 

Part L — ^EQatory. Structuce, and Theory 8to, 6 00 

Partn. — Deaign, Conatmction, and Operation 8to, 6 00 

Handbook of Engine and Boiler THala, and the Use of the Ixidicator and 

the Prony Brake 8to, 5 00 

Stationary Steam-enginea 8to, 2 50 

Steam-boiler Bxploeiona in Theory and in Practice lamc, x 50 

Manual of Steam-boilerv, Their Dcaigxia, Conatmction, and Operation 8to, 5 00 

Weisbach'a Heat* Steam, and Steam-engines. (Dv Boia.) Sto, 5 00 

Whitham's Steam-engine Dsdgn 8vo, 5 00 

Wilson's Treatise on Steam-boilers. (Flather.) x6mo, a 50 

Wood's Tbermodynamica Heat Motors, and Refrigerating Machines 8to, 4 00 



HECHAmCS AND MACHIKERT. 

Barr's Kinematics of Machinery Sto, 2 50 

Bovey's Strength of Materials and Theory of Structures. . . t 8to, 7 50 

Chase's The Art of Pattern-making xamo, 2 50 

ChordaL — Extracts from Letters xamo, 2 00 

Church's Mechanica of Engineering 8to, 6 00 

13 



Church's Hotet an4 fizamples In ttecbanics. * « ftvo, d 06 

Comma's First Lessoni in Metal-worldng .* tsmo, z so 

Comirton and De Groodt's Tha Speed Latb« lamo. z so 

Cromwell't Treatise on Toothed Oearing zamo. z 50 

Treatise on Belts and PuUeys zaiao» z so 

Dana's Text-book of Elementary Mechanics for the Use of Colkces and 

Schools lamo. z so 

Dingey's Kachinery.Pattem Making zaaio» 2 00 

Dredge's Record of the Trazisportation Exhibits Building of the World's 

Cohinibian Exposition of 1893 4to half morocco. 5 00 

Do Boil's Elementary Principles of Mechanics: 

.VoL L— Kinematics Sro 3 50 

Vol. n.— Statics Svo. 4 00 

Vol. m.— Kinetics Sro. 3 50 

Mechsnics of Engineering. Vol. L SmaU 4to, 7 so 

VoL n. SmaU 4to, zo 00 

Doiley's Kinematics of Machines 8yo, 4 00 

Fitzgerald's Boston Machinist i6mo, z 00 

Flather's Dynamometers* and the Measorement of Power tamo, 3 00 

Rope Driving tamo, a 00 

Ooss*s Locomotive Spsrks 8vo, 2 00 

HalTs Car Lobrication tamo, z 00 

Holly's Art of Saw Fifing tSmo, 75 

♦ Johnson's (W. W.) Theoretical Mechanics. ^ zamo, 3 00 

Johnson's (L. J.) Statics by Graphic and Algebraic Methods. Svo, 2 00 

Jones's Machine Design: 

Part I. — Kinematics of Machinery Sro, z 50 

Part n.— Form, Strength, and Pzoportioiit of Parts 8vo, 3 00 

Ken's Power and Power Transmisilon 8vo. a 00 

Lanza's Applied Mechanics Sro, 7 50 

Leonard s Mschine Shops, Tools, and Me t h ods. (/» iirsss.) 

MacCord's Kinematics; or. Practical Mechanism 8vo, 5 00 

Velocity Diagrams Svo, z 50 

Manrer*s Technical Mechanics. Svo, 4 00 

Merriman's Text-book on tha Mechanics of Matsrialt Sro, 4 00 

•MicUs't Elements of Analytical Mechanics Bfo. 400 

Reagan's Locomotives: Simple, Compotmd, and Electric lamo, a 50 

Raid's Course in Mec h a nic a l Drawizig 8vo, a 00 

Text-book of Mechanical Drawizig and Elementary Machine Design. .Svo, 3 00 

Richards's Compressed Air zamo, z so 

Robinson's Principles of Mechanism .Svo, 3 00 

Ryan, Bonis, and Hoxis's Electrical Machinery. VoLI Svo, a so 

Schwamb and Merrill's Elements of Mechanism. Svo, 3 00 

Sinclair's Locomotive-engine Rozming and Management zamo, a 00 

Smith's Press-working of Metals Svo, 3 00 

Mateiials of Machines. zamo, z 00 

Spangler, Greene, and Mazshall's Elements of Steam-engineering Svo, 3 00 

Thurston's Treatise on Friction and Lost Work in Machinery and Mill 

Work .Svo, 3 00 

AnimalasaMachineandPrimeMotor, and the Laws of Energetics. zamo, z 00 

Warren's Elements of Machine Construction and Drawing Svo, 7 50 

Weisbach's Kinematics and the Power of Transmission, (Hetrmaim — 

Kleizi.) Svo, 5 00 

Machinery of Transmission and Governors. (Henmans— K]ain.).8vo, 5 00 

Wood's Elements of Analytical Mechanics Svo, 3 00 

Principles of Elementary Mechanics .zamo, z aj 

Turbines Svo, a 50 

The World's Cohimbtan Exposition of Z893 i 4to, z 00 

14 



MBtALLimGT. 
Bffktton^ MetftOitrKy of Sttrsr, OoU, and Mtrcury: 

VoL L— Sihrer 8vo, 7 So 

VoL U.— Oold and Mercory »vo, 7 50 

** Qm's Lead-«meltinc. (Poftaca 9 cants additionaL) lamo, 2 50 

Keep's Cast Iron 8to. 2 50 

Kunhardf 8 Practice of Ore Dreainc in Europe 8yo. i 50 

Lc Chatelier's High-temperature UeasorementB. (Boudouard — Burgess.). xamo, a 00 

MetcalTs SteeL A Kanual f or Steel-uaers xamo» 2 00 

Smith's Materials of Machines xsmo, x 00 

Thurston's Materials of Engineerinc. In Three Parts .8vo, 800 

Part II. — Iron and Steel. . . . « 8to, 3 50 

Part nL— A Treatise on Brsssei, Bronxes, and Other AUoys and their 

Constituents 8vo» a 50 

OIke's Modem Electrolytic Copper Refining 8yo, 3 00 

mueraloot. 

Baninger's Description of Minerals of Comiiierdal Valne. Oblong, morocco, a 50 

Boyd's Rasoorces of Southwest Virginia 8to, 3 00 

Map of Southwest Virginia.... PoclEet-book form, a 00 

Brash's Manual of DetermlnatiTe Mfaiaralogy. (Psnfleld.) 8to, 4 00 

Chester's Catalogue of Minerals 8vo, paper, i 00 

Cloth, X as 

Dictionary of the Names of Minerals 8vo, 3 50 

Danals System of Mineralogy. Large Svo* half leather, la 50 

Fbst Appendix to Dana^ Hew ''Sjstem of Mineralogy.**. . . .Large Sto, i 00 

Text-book of Mlnenlogy 8yo» 4 00 

Minwals and How to Stady Them tamo, x 50 

Catalogue of American Localities of Minerals Large Svo, i 00 

Maxraal of Mineralogy and Pet r ogr a phy lamo, a 00 

Douglas's Untechnical Addresses on Technical Subjects. xamo, i 00 

Bakle^MlasnaTaUes. Svo, x as 

Bgleston's CatalDgue of Minerals and Synonyms 8to, 2 50 

Hnssak*s The Determiiuition of Rock-forming Minerals. (Smith.) Small 8vo, 2 00 
MsCTflPsHonHBtiJIisMinfffals; ThaJrOccairsncsandUsss. 8vo, 400 

* Panfield's Botes on DetermlnatiTe Mineralogy add Record of Mineral Testa. 

8yo, paper, o 50 
Rosenbusch's Microscopical Physiography of the Rock-making Minerals. 

(Iddings.) 8vo, s 00 

e Tillman's Text-book of Important Minerals and Docks 8to, 2 00 

Williams's Manual of Lithology 8to, 3 00 

MnmrG* 

Beard's Ventilation of Mines xamo, 2 50 

Boyd's Resources of Southwest Virginia 8to, 3 00 

Map of Southwest Virginia Pocket-book fMm, a 00 

Douglas's Untechnical Addresses on Technical Subjects xamo, i 00 

* Drinker's Tuxmeling, BxploslTe Compounds, and Rock Drills. 

4to, half morocco, as 00 

Eissler's Modem High BxplosiTes 8to, 4 00 

Fowler's Sewage Works Afudyses xamo, 2 00 

Goodyear's Coal-mines of the Western Coast of the United States xamo, a 50 

Ihlseng's Manual of Mining 8to, 4 00 

** Iles's Lead-smelting. (Postage pc additionaL) lamo, a so 

Kunhardt's Practice of Ore Dressing in Europe Svo, x so 

O'DriscolTs Notes on the Treatment of Gold Ores 8vo, a 00 

* Walke's Lectures on Explosives I^o, 4 00 

WUson's Cyanide Processes xamo, i 50 

Chlorination Process iamo» i so 

15 



Wilson's Hydraulic and Placer Mining X2iiio, 

Treatise on Practical and Theoretical Mine Ventilation i2mo, 

SAinXARY SCIENCE. 

Fotwell's Sewerage. (Designing, Construction, and Maintenance.) 8vo, 

Water-supply Engineering 8vo, 

Fuertes's^Water and Public Health. xamo, 

Water-flltration Works xamo, 

Gerhard's Guide to Sanitary House-inspection i6n:o, 

Goodrich's Economical Disposal of Town's Refuse Demy 8vo, 

Hazen's Filtration of Public Water-supplies 8to. 

loach's The Inspection and Analysis of Food with Special Reference to State 

Control 8vo, 

Mason's Water-supply. (Considered Principally from a Sanitary Stand- 
point) 3d Edition, Rewritten 8vo, 

Examination of Water. (Chemical and BacteriologicaL) lamo, 

Merriman's Etements of Sanitary Engineering. 8vd, 

Ogden's Sewer Design. lamo, 

Prescott and Winslow's Elements of Water Bacteriology, with Special Reference 
to Sanitary Water Analysis. . . xamo, 

* Price's Handbook on Sanitation ismo, 

Richards's Cost of Food. A Study in Dietaries lamo. 

Cost of Living as Modified by Sanitary Science tamo, 

Richards and Woodman's Air, Water, and Food from a Ssnitary Stand- 
point «TO, 

* Richards and Williams's The Dietary 0>mputer Svo, 

RideaTs Sewage and Bacterial Purification of Sewage 8vo, 

Tumeaure and Russell's Public Water-supplies 8to, 

Von Behring's Suppression of Tuberculosis. (Bolduan.) xamo, 

Whipple's Microscopy of Drinking-water. 8yo, 

Woodhull's Hotas and Military Hygiene. x6nio, 

MISCELLANEOUS. 

Emmons's Geological Guide-book of the Rocky Mountain Excursion of the 

Intematioiial (ingress of Geologists Large 8to, 

Ferrel's Popular Treatise on the Winds 8vo, 

Haines's American Railway Management lamo 

Mott's Composition, Digestibility, and Nutritive Value of Food. Moimted chart 

Fallacy of the Present Theory of Sound x6mo, 

Ricketts's History of Rensselaer Polytechnic Institute, 1824- 1894. Small 8to, 

Rostoski's Serum Diagnosis. (Bolduan.) lamo, 

Rotherham's Emphasized Hew Testament Large 8vo, 

Steers Treatise on the Diseases of the Dog. Sto« 

Totten's Important Question in Metrology 8vo, 

The World's Columbian Exposition of 1893 4to, 

Von Behring's Suppression of Tuberculosis. (Bolduan.) xamo, 

Worcester and Atkinson. Small Hospitals, Establishment and Maintenance^ 
and Suggestions for Hospital Architecture, with Plans for a Small 
Hospital lamo, i 25 

HEBREW AND CHALDEE TEXT-BOOKS. 

Gretn's Grammar of the Hebrew Language Sto, 3 00 

Elementacy Hebrew Grammar xamo- i 25 

Hebrew^ Chrestomathy 8vo, 2 00 

Gesenius's Hebrew and Chaldee Lexicon to the Old Testament Scriptures. 

(Tregelles.) Small 4to, half morocco, 5 00 

Letteri^ fiebrew Bible ; .8yo, a 25 

16. 





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