Mining dept
LIBRARY
OF THE
UNIVERSITY OF CALIFORNIA.
Class
THE CONSTRUCTION
OF
GRAPHICAL CHARTS
Published by the
McGrmv-Hill Book. Company
Nev/ York
\Svicc e.s.sor\s to the BookDepartment* of the
McGraw Publishing Company Hill Publishing Company
Publishers of Rooks for
Electrical World The Engineering and Mining Journal
The Engineering Record rower and The Engineer
Electric Railway Journal American Machinist
THE CONSTRUCTION OF
GRAPHICAL CHARTS
BY
JOHN B. PEDDLE
PROFESSOR OF MACHINE DESIGN, ROSE POLYTECHNIC INSTITUTE
OF "HE
UNIVERSITY
McGRAW-HILL BOOK COMPANY
2.?9 WEST 39TH STREET, NEW YORK
6 BOUVERIE STREET, LONDON, E. C
1910
2
cc
COPYRIGHT, 1910
BY THE
MCGRAW-HILL BOOK COMPANY
Printed and Electrotyped by
The Maple J*rfss
York, Pa.
PREFACE
Much of the work of calculation done by the engineer or designer is in
the repeated application of a limited number of formulas to a variety of
different conditions, which involves merely the substitution of different
variables in identical equations.
Any mechanical means for performing this operation expeditiously
will not only lead to a saving of time and mental wear and tear, but will
also minimize the chances for error.
Such a device is the calculating chart, or nomogram, and the increasing
frequency with which it is employed in the more recent technical publica-
tions is a good evidence of the growing recognition of its value.
Many excellent examples of these charts have appeared of late years
and are available for use, but it is evident that to realize their full value
as useful instruments the engineer should have a sufficient acquaintance
with their underlying principles to construct charts suited to his individual
needs.
Some of the chart forms employed to-day have been known and used
for many years, but it is only within recent times that any systematic
study has been made of the subject as a whole or any attempt to properly
classify and correlate the different types.
In this work the French have been pioneers, and it is to one of them,
Maurice d'Ocagne, that we owe what is probably the most thorough and
comprehensive text on the subject, his "Traite de Nomographie."
Although books on nomography have been published in many foreign
languages, there does not appear to have been anything written on the
subject in English outside of a few scattered magazine articles which
have covered only restricted portions of the field. Books in English on
graphical calculus and computation are by no means uncommon, but
this is generally looked upon as something different from nomography,
although a strict line of demarcation between the two subjects would be
somewhat difficult to trace.
It was with the idea of supplying an elementary English text in this
neglected field that the following chapters (originally contributed in
serial form to the American Machinist) were written.
210341
VI PREFACE.
Believing that the subject should be particularly useful to the practising
engineer, who is often a trifle rusty in some parts of his mathematics, an
effort has been made to simplify the mathematical treatment. A series
of illustrative problems has also been worked out in detail for nearly all
the chart forms which are here described, as it was thought that a study of
these would afford a clearer insight into the methods and a better under-
standing of the difficulties likely to be encountered than would be possible
from a purely theoretical analysis.
The desire for simplicity in mathematical treatment has made it
necessary to restrict the application of the charts to the simpler forms of
equation. Equations of the more complex types may be and have been
charted, but the mathematical difficulties are such as to make a discussion
of the methods used out of place in the present volume.
The processes described here, if thoroughly understood, should be
sufficient to cover a large proportion of the formulas in common use.
Those of my readers who wish to pursue the subject further are referred
to the more ambitious works of d'Ocagne, Soreau, and others.
JOHN B. PEDDLE.
August, 1910.
CONTENTS.
CHAPTER I. — CHARTS PLOTTED ON RECTANGULAR CO-ORDINATES, i
The simplest form of chart. Charts plotted on rectangular
coordinates. Chart for the proportions of band brakes.
Charts with irregular scales. Chart for focal distance of a lens.
Logarithmic charts.
CHAPTER II.— THE ALINEMENT CHART . . , . 15
The alinement chart. Chart for areas. Chart for collapsing
pressure of tubing. Chart for twisting moment of a shaft.
Doubled or folded scales. Alinement chart with curved
support.
CHAPTER III. — ALINEMENT CHARTS FOR MORE THAN THREE
VARIABLES :.•:.; .-.. ..... . 31
Chart for helical compression spring. Chart for strength of
gear teeth. Chart for strength of rectangular beam.
CHAPTER IV.— THE HEXAGONAL INDEX CHART 43
The hexagonal index chart. Modification of the preceding
type.
CHAPTER V.— PROPORTIONAL CHARTS 48
The proportional chart. Chart for strength of thick hollow
cylinders. The rotated proportional chart. Chart for re-
sistance of earth to compression. Charts with parallel axes
for sums or differences. Chart for centrifugal force. Chart
for piston-rod diameter. The Z-chart. Chart for polar mo-
ment of inertia. Chart for intensity of chimney draft. Chart
for safe load on hollow cast-iron columns
vii
Vlll CONTENTS.
PAGE
CHAPTER VI.— EMPIRICAL EQUATIONS 68
Empirical equations. Finding the equation of a straight
line. Another illustration of finding the equation of a straight
line. Finding the equation of a curve. Method of selected
points. Another illustration of the method of selected points.
Value of logarithmic cross-section paper in determining form
and constants of an equation. Method of areas and moments.
An alinement chart method. Another illustration of the
alinement chart method.
CHAPTER VII. — STEREOGRAPHIC CHARTS AND SOLID MODELS . 98
Three dimensional charts. Axonometric charts. The solid
model. Cardboard substitute for solid model. The tri-axial
model.
OF THE
UNIVERSITY
CONSTRUCTION OF
GRAPHICAL CHARTS
CHAPTER I.
CHARTS PLOTTED ON RECTANGULAR CO-ORDINATES.
THE SIMPLEST FORM OF CHART.
The simplest form of graphical chart is that which is frequently used
to compare different systems of units of the same character with each
other. It is often used, for instance, to show the relative values of tem-
peratures as measured on the Centigrade and Fahrenheit scales.
It is exceedingly simple to construct and to use.
If an equation containing but one variable and its function is to be
represented, one side of a straight line is graduated to represent one of the
variables, and the equation solved to give as many corresponding values
of the other variable as are needed. These are laid off on the other side
of the line, and in order to read the chart we have merely to run across
the line from one scale to the other to get corresponding values of the
variables. It may be used for a variety of equations, such as y = ax+b,
v=\/2 g h, s= 1/2 at2, a= — d2, y = log. xt y = sin. x, etc.
4
For purposes of illustration I have plotted the two charts shown in Fig.
i to represent the corresponding values of the diameter and area of the
circle. Such a chart is of very little practical value, since a table of circular
areas will give the desired results with much greater accuracy and con-
venience. I have introduced it here partly to illustrate the type of chart,
but mainly for the purpose of discussing the relative merits of the two
systems of graduation which are shown.
It will be noted that in Chart A the diameters are expressed in equal
scale divisions, and the areas by divisions which diminish in size as the
areas increase. In B the areas are represented by equal divisions and the
diameters by divisions which increase in size as the diameters increase.
i
2 CONSTRUCTION OF GRAPHICAL CHARTS
The accuracy with which we can read such charts will evidently depend
upon the size of the divisions. In general, the conditions represented in A
are preferable for, although the absolute error in reading the upper part
of the unequal scale will be greater than in the lower part, the percentage
of error throughout the scale will be more nearly equal with A than with B.
0123456 Diameter
I I II I T~l I I I I II I I II II I II I I I II I I I I II I I II I I I III I II I I III I I I I I I I
I ' ' I '1""| I I I | I I 1111111 I | I II I I I | II | I I I I I ||
0 123456789 10 15 20 25 Area
A
0 1 1.5 2 2.5 Diameter
zbm..i..i. .j..i. ..i... i.. .1.
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0123456 Area
B
FIG. i. — Plotted scales of the diameter and area of circles.
On the other hand, if most of our readings are to be about the upper
part of the scale, it may pay us to use the B arrangement in order to take
advantage of its larger divisions.
CHARTS PLOTTED ON RECTANGULAR CO-ORDINATES.
Let us take an equation of the form
y = b±ax. (i)
This equation, when plotted on rectangular coordinates, gives us a
straight line. That is, if we lay off values of y on the vertical or Y-axis
and of x on the horizontal or X-axis and erect perpendiculars to these axes
at corresponding values of x and y, these perpendiculars will intersect at
points which lie on the same straight line. Thus in Fig. 2, line 7 corre-
sponds to the equation
y=io+i/2x.
If we erect a perpendicular to any point on the X-axis, say 40, find its
intersection with line 7, and then run horizontally to the Y-axis, we will get
the corresponding value of y as 30.
If we give b different values, say 15, 20, and 25, leaving a the same, we
get the parallel lines 6, 5, and 4, which intersect the Y-axis in the new
values of b. If we change a, we change the slope of the line; if we make it
negative, we get the downward sloping lines 3,2, and i.
Suppose we make a in the equation equal to i. Our sloping lines will
now run at an angle of 45 degrees. Taking a new chart to avoid confusion
we will have something like Fig. 3. Two sets of diagonals are shown: one
UNIVERSITY
OF
CHARTS PLOTTED ON RECTANGULAR CO-1
sloping up as we move to the right and the other sloping down. The first
corresponds to
y = b + x (2)
and the other to
y = b-x (3)
According to the first equation, we have y as the sum of b and x. If,
therefore, we enter on the^ X-axis at, say 24, run up as indicated by the
heavy line to diagonal 15, and thence to the Y-axis, we will read the sum, or
39. Subtraction would be performed by going in the opposite direction or
by using the b lines designated by negative values.
50
10
10
20
30
40
50
FiG. 2. — Lines plotted from the general equation y=b±ax.
Right here it might be well to suggest that the quantities represented
by the diagonal lines in this or any other chart should be such as are not
likely to vary much, and are capable of being expressed in round numbers.
Fractional values can be much more easily picked off of the scales on the
axes. A large number of diagonals on the drawing is very likely to cause
confusion in reading, and will certainly entail additional labor to construct.
Let us now consider the other set of diagonals, corresponding to
y = b —x.
This may also be written
b=x+y. (4)
4 CONSTRUCTION OF GRAPHICAL CHARTS
It indicates that if we enter the X- and Y-axes with two numbers to be
added and run the perpendiculars out to their intersection, this intersec-
tion will be found on the diagonal numbered with the sum. Thus entering
the X- and Y-axes at 26 and 44, and running as indicated by the heavy
lines, we find the intersection on diagonal 70. Next suppose £ = o, and
give a different values. The diagonals will now be a series of radiating
lines from the intersection of the X- and Y-axes. This is shown in Fig. 4.
50
40
20
10
^J
^
&
0 10 20 30 40 50
FIG. 3. — Lines plotted from the general equation y=b±x.
Here, as our equation informs us, the chart may be used for multiplication.
Entering the X-axis at 2, running up to the diagonal 3, and from there to
the Y-axis, we read the product, 6. Division is, of course, performed by
going through the chart in the opposite direction.
This chart, while simple in appearance, is not very practical where
the multipliers differ greatly in value. It is easily seen that if we wish to
multiply any number on the X-axis by 10, it will be necessary to have the
chart 10 times as high as it is wide. Moreover, the intersection of the
vertical lines with the diagonals near the ic-line is very acute and neces-
sarily difficult to read accurately. The best position for the diagonal for
this purpose is on or near the 45-degree angle.
CHARTS PLOTTED ON RECTANGULAR CO-ORDINATES 5
These difficulties may be partly overcome by changing the scale values.
If we renumber the diagonals from o . i to i making their values 10 times
as great, as shown in the parentheses, and also give the graduations on
the Y-axis a double set of numbers, we may be able to keep the dimensions
of the chart within reasonable limits and also use diagonals which are
more favorably disposed for accurate reading. In any case, however,
there will be an unavoidaMe crowding together of the diagonals near the
0123456789
FIG. 4. — Lines plotted from the equation y=ax.
origin which will make the readings about the low numbers difficult, if
not impossible.
There was no real need to suppose that b in the equation was zero. It
was done merely for convenience in illustrating the point I wished to
explain. If b had had any value, positive or negative, we should have
had the same set of radiating lines, but their point of intersection would
have been shoved up or down the Y-axis by the value which we give to b.
Let us now investigate another form of chart for multiplying, writing
our equation
a=xy (5)
If we give a a definite value and find corresponding values of x and y,
6 CONSTRUCTION OF GRAPHICAL CHARTS
it will be found that perpendiculars erected at these corresponding points
will intersect on a curve called the equilateral hyperbola. For each
different value of a we will have a different curve.
A chart constructed with them, like Fig. 5, could therefore be used
for multiplication and, of course, for its converse, division. We have
only to pick out the numbers to be multiplied on the two axes, follow up
their perpendiculars to their point of intersection, which will be found
on the curve numbered with the product. Should this point fall between
two curves, instead of on one of them, the product must be interpolated
01234
FIG. 5. — Chart for multiplication and
5 6 7 8 9 10
division, plotted from the equation a =xy.
by eye. It will readily be seen that this method is not at all suited
to any case in which the desired number of products is large, since the
labor of drawing in the curves would be prohibitory.
Note that curves i or 10 might be used as tables of reciprocals.
Next, let us consider a case in which some power of one of the quanti-
ties is involved. We will select a case involving several multiplications
in order to show how some of the principles already discussed are applied.
This will be done in some detail in order to clearly show the process of
attacking a simple problem.
CHARTS PLOTTED ON RECTANGULAR CO-ORDINATES
CHART FOR PROPORTIONS OF BAND BRAKES.
Take the formula for the band brake
in which P represents the resultant tangential pull on the brake, A the
area of the cross section of the brake band in square inches, T the tension
in the tight side of the band in pounds per square inch, /the coefficient of
friction (o. 18 in the case of iron on iron) and a the arc of contact of the
band in degrees.
Inside of the parenthesis in our equation there is only one quantity
which need be considered as a variable, a the arc of contact; / will be
constant for any given materials for band and drum and, as indicated
above, will be taken as 0.18. Under these circumstances, instead of
drawing a separate line or set of lines for each quantity inside the parenthe-
sis, we need only draw one line for the parenthesis as a whole, getting the
different values for plotting this line by letting a vary. We will have to
assume the limits within which this variation is to take place. Suppose
we take these as 200 and 300 degrees. Then solve the parenthesis for
every 10 degrees between these limits.
In Fig. 6 the results of these calculations are shown plotted as ordinates
on the chart, while the corresponding arcs of contact are taken as abscissas.
In laying off the latter, one small scale division on the horizontal scale is
used to represent two degrees of arc. The vertical scale will need to be
large as the values of the parenthesis only vary from 0.4666 to 0.6103,
and this, if plotted to a small scale, would make a very flat and therefore
undesirable curve.
Suppose we make one small scale division on the vertical scale equal to
o. 01. This has been done on the chart, and the curve drawn through the
points thus found. These values must now be multiplied by the assumed
values of T, the tension per square inch in the band. According to one
authority, the safe values for T will range from 4500 to 6500 for wrought
iron, and from 8500 to 11,500 for steel. We have therefore to provide for
a total range of 7000 pounds and we will cover this by steps of 500 pounds.
We will adopt the multiplying method shown in Fig. 4, making the radiat-
ing lines stand for the different tensions. They must converge to a point
somewhere on the zero line of the curve just drawn, and this point may
be chosen at will. In reading the chart we must run up or down a vertical
line until we strike the curve, and then go horizontally until we reach the
desired T-line. It is evident that all the jT-lines must be in such a position
that they may be intersected by any horizontal drawn from the curve.
8
CONSTRUCTION OF GRAPHICAL CHARTS
They must be so drawn that the tangents of the angles they make with the
vertical will be proportional to the tensions they represent. Let us run
up ten of the large divisions from the zero line and then horizontally 41/2,
5,51 /2, 6, etc., of the large divisions, corresponding to tensile stresses of
200° 210' 220
Arc of Contact in Degrees
230° 240° 250° 260' 270°
290' 300'
FIG. 6. — Proportions of band brakes.
4500, 5000, 5500, 6000, etc., so as to get the lines well spread out. If we
take the point of convergence at 14 large divisions to the right of the left-
hand edge of the chart, the conditions we have imposed above will be
fulfilled, and this has accordingly been done. The results of this multipli-
CHARTS PLOTTED ON RECTANGULAR CO-ORDINATES Q
cation will be read on some horizontal axis, and they must next be multi-
plied by the assumed values of A, the area of the cross section of the band.
We could use the same point of convergence for the A -lines as for the
T-j but inasmuch as this would cause some confusion in reading the dia-
gram, it will be better to use some Other center, which, however, must be
located on the vertical line passing..through the T center. According to
the authority quoted above, the thickness of the band for ordinary cases
should vary between 0.08 inch and o. 16 inch, corresponding, roughly, to
No. 12 and No. 6 Brown & Sharpe gage. If our bands are not to be
less than i inch nor more than 3 inches in width, the maximum variation
in area will be between 0.08 square inch and 0.48 square inch. For
convenience let the areas vary by steps of o . 04 square inch, although any
other size of step might have been chosen. This will give us n lines
which must be so drawn that the tangents of the angles they make with
the horizontal will be proportional to 0.08, 0.12, o. 16, etc.
We must now determine the limits within which our results, the desired
values of P, must fall. For the least area, 0.08 square inch; the least
tension, 4500; and the smallest contact angle, 200 degrees, we have P=i68.
For the largest values of the same quantities we have P = ^^6g. These
values of P will be read on a vertical scale. It \vill be found that if we
allow i large division on the vertical scale to represent 200 pounds it
will give us a convenient scale length and readings may be made with an
accuracy which is sufficient for all practical purposes. The length of
the vertical scale will thus be about 17 of the large divisions.
Therefore, going up 1 7 large divisions from the zero of the curve, we
locate the center for the radiating A -lines on the vertical line which passes
through the center of the TMines. From this center we go 10 large divi-
sions to the left and, going down 2, 3, 4, 5, etc., large divisions (propor-
tional to 0.08, o. 12, o. 1 6, o. 20, etc)., we locate the points through which
the A -lines must be drawn from their center. By this arrangement they
will cover the desired length on the scale of P. Our chart is now complete
except for lettering the lines and scales. The left-hand scale must of
course be lettered so as to make each large division represent 200 pounds'
pull.
To read the chart, enter at the bottom or top at the assumed arc of
contact and run up or down to the curve, from there go horizontally to the
desired tension in the band, then vertically to the area line, and then
horizontally to the vertical scale representing the tangential pull. Or, if
the pull, arc of contact and tension are known, enter as before at the arc
of contact, run vertically to the curve, thence to the tension line, and the
10 CONSTRUCTION OF GRAPHICAL CHARTS
intersection of the vertical through this point with the horizontal drawn
from the desired pull will be on or near one of the area lines, thus giving
the necessary size of the band.
It is obvious that for all practical purposes our chart might have been
trimmed off on the right-hand side at the end of the curve so as to omit
all of the diagram not sectioned with the small divisions, also that there
is no need of continuing the 2"-lines above or below the curve.
CHARTS WITH IRREGULAR SCALES.
There is no necessity in these charts for having the scale divisions
equal, as has been the case in all the charts except the first. If we admit
this, there is a distinct advantage in many cases in having them irregular.
CHART FOR THE FOCAL DISTANCE OF A LENS.
For instance, take the formula connecting the two foci of a lens with its
principal focus
where / and /' are conjugate focal distances and p the principal focal
distance.
Make
The above equation becomes x+y = b which is identical with equation (4)
above.
We have, in this case, to lay out on the X- and Y-axes the reciprocals
of/ and/' and draw in the diagonals as shown in Fig. 7, just as we did
in Fig. 3. Knowing the principal focal distance of our lens, we select the
diagonal corresponding to it, enter the X-axis, say, at the distance of the
object from the lens, run up to the diagonal, from there to the Y-axis, and
read off the distance at which the object will be in focus.
LOGARITHMIC CHARTS.
A more important case is where the divisions are laid off to a logarith-
mic scale. Paper ready ruled in this way may now be had from dealers
in mathematical instruments and is valuable for many purposes. On it
many problems which would have to be solved by tediously drawn curves,
may be worked with ease by straight lines.
CHARTS PLOTTED ON RECTANGULAR CO-ORDINATES
II
Let us return to equation (5) a = xy. This may also be written log. a =
log. x+log. y which is identical with (4), the equation for a straight line.
The paper in question is graduated on its horizontal and vertical axes
so that the lengths from the origin are equal to the logarithms of the num-
bers placed opposite the graduation marks.
If in Fig. 8 we connect 2 on the vertical axis with 2 on the horizontal
axis, 3 with 3, and so On, we get a chart similar to Fig. 3, which was
used for addition, but in this case is for multiplication. It also bears some
resemblance to Fig. 5, the equilateral hyperbolas used there being
replaced by straight lines. To use the chart enter at the X- and Y-axes
with the numbers to be multi-
plied and follow out the perpen-
diculars at these points to their
point of intersection, which will
be found at the diagonal num-
bered with the product.
We might also draw the diag-
onals so as to slope upward from
£
§30.
« VI.
\
\\
\i\
X *
r_K%
$s&&
^°^
\
s^
Distance of Object from Lens
FIG. 7. — Chart for focal distances of a lens.
left to right instead of downward,
as shown on the same chart.
This is identical with the second
form of addition chart of Fig. 3,
and may also be used for multi-
plication. Thus, entering on
the X-axis with the multiplicand we run up till we strike the diagonal
numbered with the multiplier, and thence over to the product on the
Y-axis. Such paper is also very convenient for handling equations con-
taining powers and roots of the variables, and especially where these
powers and roots are fractional.
For instance, y=x2 may be written log. y=2 log. oc.
This indicates that a line drawn so that its tangent with the horizontal
is 2 could be used for squaring numbers on the X-axis, or conversely for
extracting the square roots of numbers on the Y-axis. This is shown in
Fig. 9. The top of the diagram is bisected, and a line drawn to this
point from the origin, enabling us to find any square not exceeding 10.
Entering at 2 on the X-axis and running up till we strike this line and
from there to the Y-axis, we read 22, or 4.
To get squares greater than 10 we should have to extend our chart
above the lo-line. It would be exactly similar, however, to the part
below, and it is therefore only necessary to lower our squaring line so as to
12
CONSTRUCTION OF GRAPHICAL CHARTS
1 2 3 456789 10
FIG. 8. — Logarithmic chart for multiplication.
1 2 3 4,56789 10
FIG. 9. — Lines of powers, roots, etc., on logarithmic paper.
CHARTS PLOTTED ON RECTANGULAR CO-ORDINATES 1$
cut the base of the chart in the middle, and make it pass through the upper
right-hand corner. We thus get a chart which may be used for getting
the square or square root of any number, the only thing to be noted in the
latter operation is that we must use one or the other section of the line
according to the position of the decimal point. If the number whose
square root is desired has one, or three, or five places (any odd number)
Fig. 10. — Logarithmic charts plotted from the equation Z—
before the decimal point, use the first section of the line; if it has two, or
four, or six places (any even number), before the decimal point, use the
second section.
From what has been said it is plain that the cube line should be drawn
by dividing the upper and lower edges of the diagram into three parts so
as to make the tangent of the angle of slope 3. Here there will be three
14 CONSTRUCTION OF GRAPHICAL CHARTS
lines crossing the diagram. For getting cube roots the first section should
be used where the number of places before the decimal point is i, 4, or 7,
etc., the second section where the number of places is 2, 5, or 8, etc.,
while the third section is used where the number of places is 3, 6, or 9, etc.
For getting fractional powers or roots the tangent of the angle of
the slope must, of course, be equal to this fractional exponent. Equa-
tions such as pvn = c are easily solved. In Fig. 9 the line representing
pv1'41 = 10 has been drawn for purposes of illustration, v being read on the
horizontal, and p on the vertical axis. On the same chart a line has been
drawn for getting circular areas, showing the extreme simplicity of the
method. Diameters are read on the horizontal and areas on the vertical
axis.
In Fig. 10 is shown the application of this paper to the formula for
the section modulus of a beam of rectangular section
In this chart, values of h are read on the base line, b on the diagonals,
and Z on the vertical or Y-axis. For the sake of clearness only two of the
diagonals representing b have been drawn. They are for b = 2 and 6 = 4.
The intersections with the vertical or Z-axis are found by letting h = i .
The tangent of the angle of slope is 2.
In reading any of the logarithmic charts here given, significant figures
only will be found. No definite rules need be given for finding the position
of the decimal point. As with the slide rule, it needs only the application
of a little common sense.
CHAPTER II.
THE ALINEMENT CHART.
.
A type of chart which has received considerable attention of late years
and which differs radically from those already described is that known as
the alinement chart. In the charts hitherto examined the necessary
lines were plotted on what are known as rectangular coordinates; that
is, the axes on which the values of x and y were plotted met at a right
angle. This is by no means a necessary condition. The axes may be
parallel, and, in fact, I have a little book in which the author has de-
veloped a system of coordinate geometry based on parallel, instead of
rectangular coordinates.
To aid us in understanding this form of chart, let us take an equation
of the form
au+bv = c (6)
where u and v are variables and a, 6, and c are constants. TLis is the
equation of a straight line where rectangular coordinates are used. To
illustrate, let us assume a = 4, b = 6,
and c = 60, and draw the line repre-
sented by the equation as shown
in Fig. ii.
Since u and v may have any
values, let w = o, then 6 v=6o, and
v=io. Again, letting v = o, 411 =
60, and ^=15. This gives us the
coordinates of two points on the
line, one on each axis. Lay off u= 15 on the Y-axis and v=io on the
X-axis and join them. Then for v = 4, ^=9, as shown by the heavy line
on the chart.
Now let us lay off the same quantities on the parallel axes in the
second chart of Fig. n. On the axis marked A u lay off 15 and join it
to v = o on the B v axis. Lay off v— 10 on the B v axis and join it to
u = o on the A u axis. These two lines meet at the point marked p. It
will be found that all lines joining corresponding values of u and v, as
found from the equation, will pass through the same point. Or, if we
15
FIG. ii. — Comparison of a rectangular
and alinement chart.
1 6 CONSTRUCTION OF GRAPHICAL CHARTS
•
take 4 on the B v axis and join it with p, this line prolonged will cut A n
at 9, giving us the same result that we got by the other chart.
Thus, what was the equation of a line with the rectangular system
becomes now the equation of a point. Keeping a and b constant and
changing c gives us, with the rectangular system, a series of parallel
lines. With the parallel coordinates this merely moves the point p up or
down on the line C p. This line is called the "support" for the points
of intersection p. On the other hand, changes of a or b will shift p to the
right or left of the support C p. To establish the point p it will generally
be sufficient to solve the equation for a few easily determined values ~biu
and v, lay them off on their axes, and join corresponding points, as has
just been done. Or it may be worked out analytically as follows:
Let the position of the point be supposed to be located by reference
to rectangular coordinates, of which the line A B represents the X-axis,
and the line through O midway between A u and B v and parallel to them
the Y-axis. Draw a horizontal line through p. It will intersect the
lines A u and B v at the same height above A B asp. Call this distance y.
Equation (6) may now be written
au+b v = a y+b y = c,
or
(7)
a+b'
This gives the distance of the point p above A B.
If oc'j y', and x" y" are the rectangular coordinates of two known points
on a line, analytic geometry teaches us that
x — xf y—yf
x'-xff=yf-y"'
Let us apply this to the two lines originally drawn to locate p, and call the
distances O A and OB,— d and + d, respectively. The coordinates
of the points at the two ends of one of the lines are :
x' = - d, y' = ---, and x" - d, f = o.
a
Therefore
-d-d c
o
a
For the other line xf = - d, y' = o, and x" == d, y" = — - are the co-
THE ALINEMENT CHART 17
ordinates of the ends, and its equation will be
x + d y — o
-d-d'~ ~~c~*
°"
Combining these equations so as to eliminate y we get
*r
thus giving us the distanced the point from the vertical line through O.
It shows, also, that this distance is independent of c, and that, therefore,
however c varies (if a and b are constant), p will always lie on the line C p.
The actual location of the various points on this line may be found
by solving the equation for y for different values for a and b, or, as said
before, by joining up corresponding points on the A u and B v axes by
lines whose intersection with the locus, or support, of p will give us the
desired points.
A practical example worked through will, perhaps, give us a better
idea of the methods used than we should gain by a purely abstract
discussion.
CHART FOR AREAS.
A man engaged in making blueprints asked me to make him a chart
for calculating the areas of his prints in order to aid him in fixing his
charges. As it makes a very simple problem for this purpose I will
use it in this demonstration, merely remarking that the diagram furnished
him differs in some particulars from the one used for illustration here,
in order to better adapt it to his needs.
The formula used was
WL
A = --,
144
in which A is the area in square feet, W the width in inches and L the
length in inches.
Write this in its logarithmic form
log. W + log. L = log. A + log. 144.
Let us plot log. W on the A u axis, log. L on the B v axis and log. A
on the intermediate support.
We must first decide upon the scales by which these lengths are to be
measured on their axes. For instance, u, the measured length of log. W
on the A u axis, is obtained by multiplying log. W by some " modulus"
18
CONSTRUCTION OF GRAPHICAL CHARTS
or coefficient to get the desired length in inches. Let us call this modulus
/j for the A u axis, /2 for the B v axis, and /3 for the intermediate support.
Then
u = I, log. W,
v = 12 log. L,
and
Mb
2ft.-:
'ir,-
10".-
5 -
log. W = ~,
*i
%. L = ^-.
^2
10
.1-
of
FIG. 12. — Alinement chart for areas.
Calling log. A + log. 144 = c, we have
u v
or
/2 w + /, v = /j /2 c.
From equation (8) we have
|-5ft.
Uft.
hsft.
-2ft.
10
9"
8"
7"
6"
5"
4"
THE ALINEMENT CHART 19
from which
/i _ d+x C A
It d-x ~~~~~CB' (9)
thus locating the support for the product.
From the equation (7) we see that
/• / c
c must therefore be multiplied by
in order to give the measured lengths along the third axis, or support,
and this quantity must be its modulus, or
/! -M.' (10)
The graduated lengths along the different axes may be anything we
choose to make them. In general, they should be about equal and as
long as possible while keeping the size of the chart within reasonable
limits. The largest size of print which was called for in this problem
was 40 x 60 inches. The logarithm of 40 is 1.602, and of 60 is 1.778.
These numbers are so nearly equal that it will not pay us to use different
scales in laying them out in order to represent them by exactly equal
lengths on the axes, and /t will accordingly be made equal to 12.
The best scale to use for a practical problem would probably be i
inch for a logarithmic value of o. i, thus making the length of the longest
axis about 18 inches. In the drawing made for this article the twentieth
scale was used, giving a chart half this size. This does not refer to the
cut which, of course, has been reduced.
Since /t and 12 are to be equal, equation (9) shows that the distances
C A and C B are equal, or the support for the areas must be midway
between the outside axes. For the modulus on the third axis we have
from equation (10)
*i k J,
or we must use a scale of fortieths on the middle axis, on which the areas
are plotted, if we use twentieths on the outside axes. The distance
between the outside axes may be anything we wish. If the axes are
too close we get a compact chart, but the intersection of the index line
with the axes may, in some positions, be so acute as to make accurate
reading difficult. The farther the axes are apart the better this condition
20 CONSTRUCTION OF GRAPHICAL CHARTS
will be, but we must not make the distance so great as to get a chart
which will be awkward to handle.
Perhaps the best arrangement for average conditions will be to have
the chart about square, in which case the index line will never make a
smaller angle with the axes than 45 degrees; this is not objectionable.
The two outside axes are now to be graduated so as to represent the
logarithms of the desired lengths and widths expressed in inches. Start
with i inch (whose logarithm is o) on the A B line.
On the middle axis instead of putting i on the A B line we must
remember that logarithm A is to be added to logarithm 144 (which is
2.158), and we therefore run up 21.58 measured with the fortieth scale
before beginning to graduate. Calling this point i, we lay off from it
the logarithms of 2, 3, 4, etc., and such subdivisions of them as may be
necessary, till we reach 17, a trifle beyond the limits of our other scales.
The chart is now complete with the exception of the lettering.
To read it, lay a straight-edge or draw a fine thread tightly across the
chart so as to join the points representing the length and width of the
print, and the intersection of the line with the middle axis will give the
area in square feet. Better, perhaps, than either the straight-edge or
thread is a piece of glass or thin celluloid with a straight line scratched
on its under surface.
Such charts as this will ordinarily show a very marked advantage
over those previously described. They are usually much simpler to
construct, and they avoid the confusing tangle of lines so often found
with the rectangular type. Moreover, since we do not have to draw a
separate line for each value of the variable, as is sometimes necessary with
the other form, it will be easier to get close readings by interpolation.
The scope of this chart might have been somewhat enlarged, without
much trouble, had it been thought desirable. The prices corresponding
to the different areas might have been marked on the other side of the
area line in something the same manner as was done in Fig. i. Also
the chart might have been extended to give the total area or price of a
number of prints of given size. To do this we should merely have to
consider the area line as the outside axis of a new diagram, the other out-
side axis being graduated to represent any desired number of prints, and
the product would be read off on a new intermediate axis. The A B,
or base, line need not have been left on the chart, as it is of no use after
the construction is once made, and it will generally be omitted.
THE ALINEMENT CHART 21
CHART FOR COLLAPSING PRESSURE OF TUBING.
Another formula charted on this plan is shown in Fig. 13. It is
//Y
P = 5oy2io,ooo I — 1,
and will be recognized as "Stewart's formula for the collapsing pressure
for bessemer-steel tubing, to be applied to pressures not exceeding 581
pounds or to values of — not exceeding 0.023.
In it P is the external pressure in pounds per square inch, t the thick-
ness of the tube in inches, and D the external diameter of the tube, also in
600-
500-
400-
300-
200-
100-
13
-.12"
-.ll"
-.10"
-.07
-.06'
.04
6"
-5
-4
-2"
FIG. 13. — Alinement chart for Stewart's formula for collapsing pressures
of Bessemer tubing.
inches. It is very similar to the case we have just worked out, but there
are one or two practical points in which they differ which will make it
worth our while to hastily run through the construction.
The formula may also be written
P D3 = 5o,2io,ooo/3
or
log. P+3 log. D = log. 50,210,000 + 3 log. t.
Its essential similarity with our fundamental equation will be readily
seen.
22 CONSTRUCTION OF GRAPHICAL CHARTS
Suppose we take the range in tube diameters from i inch to 6 inches,
and let our pressures vary from 100 pounds to 600 pounds, the latter a
trifle above the 581 pounds for which the formula is supposed correct.
Log. i is o and log. 6 is 0.778; log. 100 is 2 and log. 600 is 2.778. This
gives us a range of 0.778 in the value of the logarithm in each case. Let
us make the length of line corresponding to this range the same on the
two outside axes, say 7.78 divisions on whatever scale may be convenient.
On account of this equality we may write for the maximum values
of P and D
I, (log. P - 2) = 3 12 log. D
or
O
Then ^ (log. P — 2) = l^ log. D, showing that the two scales are iden-
tical so far as graduation is concerned.
The logarithms of the values of D will be laid off from the horizontal
base line; the logarithms of P, above 100, from the same line. But it
must be remembered that the real zero for the P-line is 20 divisions
(on the scale we have chosen) below the base line, and that consequently
the line corresponding to A B of Fig. n will slope up from this point
to the point marked i inch on the Z)-line. There is no need to draw it,
however.
The location of the /-line is given by
h =CA A
/2 C B i '
Next let us determine the modulus /3 for the support, or axis, for t.
According to equation (10)
i *-*' A.
~*iH "4"
Inasmuch, however, as the log. t is multiplied by 3, it will be con-
venient, to consider its modulus as j llt and graduate log. t directly with
this scale instead of using a modulus of J and laying off the values of
3 log. t. The other quantity log. 50,210,000 laid off on this axis will,
however, only be affected by the modulus J /, since the coefficient
of this logarithm is i instead of 3.
In graduating the /-line note that we must add log. 50,210,000 or
7.7007 to log. /, and that the zero from which the graduations are meas-
ured must be on the sloping A 5-line referred to above. If the left-hand
end of the line, corresponding to point A , is 20 divisions below the hori-
THE ALINEMENT CHART 23
zontal base line, the zero for the /-line will be 5 of these same divisions
below. Now, using a scale one-fourth the size of that used on the outside
axes (since /3 = i /J, lay up 77.007 divisions. This will give us the point
corresponding to i inch on the /-line. Our values for /, being less than i,
will all fall below this.
For example, take / = ©.i inch;"70g. / = — i. This will be measured
down from point i on the /-axis, the length being 30 divisions on the one-
fourth scale or 10 divisions on the three-fourths scale; or, what is the
same thing, we may go up 47.007 divisions from the zero. The other
points on this axis may be located in the same way or by joining up suit-
able points on the outside axes. The chart now needs only to be lettered
to be complete.
A simple modification of the alinement chart as already described
is sometimes of value.
Let our general equation have the form au+ bv = o.
In this equation c has been made zero, and, since this is so, y in equa-
tion (7) is also zero. This shows that the support for the points of inter-
section is now the line A B. In order to have the points of intersection
lie between the points A and B it will be necessary that Au and Bv axes
lie on opposite sides of the line A B. As indicated in the last example,
there is no necessity that A B should lie perpendicular to the axes, and it
will evidently be to our advantage to make it sloping, since in this way
the chart can be made to occupy less room.
CHART FOR TWISTING MOMENT OF A SHAFT.
The methods followed in constructing this diagram will be shown by
working out another practical example. For this purpose let us take
the equation for the twisting moment in a cylindrical shaft
M = 0.196 D3f,
or
-0.196 D3f+M = o,
where M is the twisting moment, D the diameter of the shaft in inches,
and /the fiber stress in pounds per square inch.
Let
u = — /!/ and v = 12M,
then
u v
f= — — and M - — -—,
24
and
CONSTRUCTION OF GRAPHICAL CHARTS
0.196 D3u
= o
or
0.196 D3 12u + /j v = o
Now from equation (8)
^-0.196 £>3 /
~
+ 0.196 D3 L
(n)
-190000
- 180000
- 170000
- 160000
- 150000
- 140000
- 130000
- 120000
- 110000 -e
§
- iooooo I
- 90000-3
.2
r- 80000 H
70000
60000
i- 50000
r 40000
30000
20000
10000
0
15000
FIG. 14. — Alinement chart for the twisting moment in cylindrical shafts.
This is the equation for graduating the support for D. The two
axes must be graduated according to the equations u = — l^f, and
v = l2M, which show that the divisions on each axis are to be equal among
themselves, or that the graduation is regular. Let us assume that the
greatest fiber stress we shall need is 15,000 pounds and that our largest
shaft will be 4 inches in diameter. Our maximum moment will then
be about 188,200. If we make i inch equal to 1000 pounds on the/-ajcis,
THE ALINEMENT CHART 25
this axis will have to be 15 inches long. Making i inch equal to 10,000
pounds on the moment axis will give us a length of about 19 inches; /x
will, therefore, equal 10 12. Suppose we say that 20 inches will be a
convenient length for the diagonal, then d will equal 10 inches.
Now graduate the outside axes, into inches and tenths, taking as the
zero point on each the intersection of the axis and the diagonal. The
graduations for the Z)-axis or diagonal will be determined by solving
our equation for x. Let us find the point corresponding to the 4-inch
diameter. From equation (u)
10 — o.i 06X64 X i
x= 10 = —i. 1 1.
10+0.196X64X1
The division mark for the 4-inch diameter will, therefore, be placed
1. 1 1 inches to the left of the middle of the diagonal. As many other
points as may be considered necessary are found and laid off in the same
manner. In Fig. 14 this has been done for every quarter inch from
i inch to 4 inches. To save work, the graduations on the fiber stress line
need not have been extended below, say, 8000. The line on which the
diameters are laid off need not extend beyond the 4-inch graduation,
but for the sake of clearness it has been retained here.
IV
in
DOUBLED OR FOLDED SCALES.
When an alinement chart is intended to cover a considerable range of
values we are confronted with the difficulty that it must be large, and
therefore awkward to handle, or we must have scale divisions which are too
small for accurate reading. These
difficulties may be overcome with
but little additional trouble by a
system of double graduation of
the axes.
In Fig. 15 let A and C be the
outside axes of an alinement chart,
and B the support on which the
results are to be read. Say we
wish to graduate the A -axis for a
length equal to a-c, and that this
length is too great for our chart
if we use a desirable scale unit. Take a length a-b, equal to about
half of a-c and lay this off on the left-hand side of A and graduate it.
On the right-hand side of A lay off the rest of the length, or b-c. Call
A B B' c c'
FIG. 15. — Diagram of an alinement chart
with doubled scales.
26
CONSTRUCTION OF GRAPHICAL CHARTS
the first scale I and the second II. On the C-axis do the same,
graduating the first half of the desired length (which we will call d-e)
up the left-hand side of the axis, and the second half, or e-f, on the other
side. Mark them I and II to correspond with A. The location of the
central support and its scale unit, or modulus, is determined as previously
8*-
.5-
8*-
-8*
.4-
3-
p-3
r-5Ft, 5 Ft. -
r-
7"-
- -
-7"
.3-
- r
6"-
E}| 2~-
~-l 6"-
^4 Ft. 4 Ft. ^
-6"
'
=•14
5 5
6"-
-11
5*-
~ —
-5"
-10
i ~
3 3 Ft.
L9 I
-3 Ft. 3 Ft. -
: 8 !-
-1
I
4"-
— 7 «9 ~
-.9 4'-
—
-4"
.1-
6.0
-.8
-~
— .
.7-
-.7
-
—
I 5 ,6-
-.6
~i
3"-
-2 Ft,
r4 .5-
-.5 3"-
- 2 Ft. 2 Ft. -
-3"
— •
I3 .4-
-.4
: :
" Scale of Widths
Scale of Areas
.3-
\ScaleofAreas
-.3
Scale of Lengths
2"-
2"-
— —
-2"
_
.2-
-.2
I
II
in
"iv i
Iii m
IV I
II Ilf
IV
-IFt.
-i
-IFt. IFt.-
.9
- 11"
.8 .1-,
-.1
— 11* 11*-
-10*
.7
- 10* 10*-
.6
- 9*
- 9* 9"-
-.5
i-
-8*
- A
r-
- 8* 8 -
-r
FIG. 16. — An alinement area chart with doubled scales.
explained for the simple alinement chart. The left-hand side will be
graduated with values, say from g to h, corresponding to I and I on A
and C, and marked I, while the right-hand side will be graduated from
h to i, corresponding to II and II on A and C, and marked II.
So long as we wish to get values on B corresponding to I and I, or to
II and II on A- and C-axes, we evidently have no trouble, but if we
THE ALINEMENT CHART 27
attempt to combine I on A with II on C we find no place on B where the
result can be read. We are, therefore, compelled to use two new axes,
one for values of B and the other for C. Call these new axes B' and C' '.
On C' graduate the left-hand side exactly the same as the right-hand
side of C, or from e to/, and the other side like the opposite side of C, or
from d to e. Mark these scales III and IV, respectively, and since the
III side of C' is to be combined with the a-b length on A, the latter had
better also be marked III. For the same reason mark the right-hand
side of A, IV.
The central axis, Bf, must be located in the same relation to A and C'
as was B to A and C, and will be graduated on the left to correspond
with the combination Ill-Ill on A and C', and on the other side to corre-
spond with the combination IV-IV on the same axes.
At first sight this diagram is a little confusing and there is always a
chance for mistakes in connecting up wrong pairs of axes. If a little
care is taken, however, to see that the readings are made on axes bearing
the same Roman numeral, the seeming confusion will disappear and the
liability to error will be small.
The process of constructing this chart is so simple that further expla-
nation seems unnecessary. For purposes of illustration the area chart
shown in Fig. 12 is reproduced in Fig. 16 by this method. A comparison
with Fig. 12 will show that while the new chart is somewhat more com-
plex in appearance, it permits the use of divisions which are so much
larger that they compensate in a large measure for the additional
confusion.
ALINEMENT CHART WITH CURVED SUPPORT.
All of the alinement charts dealt with so far have had straight-line
axes or supports for the different scales. This is by no means necessary
since any one or all of them may be curved.
A case in which the intermediate support is curved will next be con-
sidered. Let the equation take the form
5 = m^
2
This is the equation for the space passed over by a body falling
under the influence of gravity and starting with an initial velocity. In
it 5 = the space moved over in the time /, V = the inital velocity, and
g=the acceleration of gravity, which we will call 32.
The formula is chosen not so much for its practical value as because
28
-250
-200 -
— 150-
-100-
-50-
CONSTRUCTION OF GRAPHICAL CHARTS
+ 50-
+ 100 -
150-
+ 200-
+ 250-
-10
i of Time, t.
-5
Scale of Initial
Velocity, V.
+ 5
Scale of Space, S._
-250
— +200
— +150
— +100
— + 50
200
250
FIG. 1 7. — An alinemcnt chart with a curved support, solving the equation S = Vt + -
THE ALINEMENT CHART 29
its form is a good one for the purpose of illustrating this type of chart.
Let us write it
S= F/ + i6/2,
or
-F/-!-S=i6/2.
Make
u = — l^V and v = 12S,
then
V= -M7-and5=-^-,
— /j /2
Substituting above we have
or
This is evidently identical with the fundamental equation for the
alinement chart. From equations (7) and (8) for y and x we have
_
~
and
These are the equations of the points constituting the support for t.
The choice of the scale units is of little or no importance in this case,
since we are not obliged to work between any definite limits. For sim-
plicity in calculation, then, let us take /1 = /2.
Then
and
i—t
x = d-
Here our formula does not have a logarithmic form, and we can,
therefore, graduate our scales in lengths proportional to the numerical
values of the quantities involved and not of their logarithms. This has
been done on the scales for V and S. It should be observed that since
the modulus for F, or /p is negative, the positive values of that quantity
are measured down from the base line. The distance between the axes
30 CONSTRUCTION OF GRAPHICAL CHARTS
may be anything we like, but to simplify our calculations we will make
it 20 of some unit in order that the half distance, or d, may be 10.
Solve the equation
i—/
#=io — -
! + /
for as many values of / as are wanted. In the chart shown in Fig. 17 the
values taken for / were o, 1/2, i, 2, 3, 4, 5, 6, 7, 8, 9, 10, and u. The
corresponding values of x are 10, 3.33, o, —3.33, —5, -6,- 7. 15, -7. 5,
— 7.78, — 8, — 8.18, — 8.33. For the same values of / we have for y: o, 2.6,
8, 21.3, 36, 51.1, 66.6, 82.2, 98, 113.8, 129.5, i45-2» l6l-4-
Plot the curve for these values of x and y, and letter it to correspond
with /. The construction for the point £ = 3 has been indicated by dotted
lines.
The horizontal axis on which x is plotted is only used for the con-
struction of the curve and may be omitted in the completed chart. It
is retained in Fig. 17 in order that the process may be clearly indicated.
If we connect two points on the outside axes by a straight line, the
intersection of this line with the curved support will give /, or by connect-
ing the initial velocity V with the time on the curved support we read on
the 5-line the distance passed over. This has been done in the figure
for F=3O and / = 3, giving the value for S as 234. By making the index
line pass through V = o and the given time we get a case corresponding
to the simple law of falling bodies. If V be taken negative we may get
two intersections with the /-line, and either of the times thus found will
satisfy the equation.
CHAPTER III.
ALINEMENT CHARTS FOR MORE THAN THREE VARIABLES
CHART FOR HELICAL COMPRESSION SPRING.
So far the alinement charts as described have only taken account of
three variables. This is not a necessary limitation and we will next con-
sider a case in which the'number of variables is four. For illustration
we will use the formula for the load supported by a helical compression
spring
d3
P = o. igS—/,
where P is the load, d the diameter of the wire, r the mean radius of the
coil, and / the fiber stress. Say we wish to have our chart cover wire
from No. 10 to No. oooo B. S. gage, or from 0.102 to 0.46 inch diam-
eter. Let us assume that the mean radius of the smallest spring will be
1/2 inch and of the largest 2 inches, and that / may vary between 30,000
and 80,000 pounds. Put the equation into its logarithmic form
log. P = log. 0.196+3 log. d + log.f — log. r.
We will have to make two steps in getting our solution, and in each
step but three variables must appear. Therefore let us say
log. 0.196 + 3 log. d — log. r = log. q
and
log. P = log. q + log. f.
These two equations are evidently of the same form as those pre-
viously treated by the alinement chart, and will be charted by exactly the
same methods. The quantities d and r, or rather their logarithms, we
will plot on the outside axes and read q on the intermediate support. See
Fig. 1 8. Since log. d and log. r are affected by opposite signs, the positive
values of these quantities will be laid off in opposite directions from the
base line. As previously explained, the base line may be made sloping,
and for convenience we will suppose that this has been done here. Our
former constructions depended upon a knowledge of the position of this
line, but once the matter is understood there is no real necessity for ac-
tually locating it, and in the present instance it will be disregarded.
We have assumed that the values of r are to lie between 0.5 inch and
2 inches. The logarithm of 0.5 is —0.301 and of 2 is +0.301, making
a total range of 0.602. Choosing a suitable scale unit, this length is laid
off on a vertical line at the right of the paper. The middle point will
32
CONSTRUCTION OF GRAPHICAL CHARTS
.460%- No. 0000
.410?- -No. 000
- -No. 00
.325- -No. 0
- - No. 1
.258-
--No. 3
No. 2
- - No. 5
- - No. 6
144- - No. 7
114 --No. 9
1000
900
700 —
500
400
-30000
-Spring Load in Lbs., P.
Scale of Coil Radius, r .
r, d Support
Scale of Wire Sizes
Diam. and B. & S. No.
- .6
- .7
-1.1"
-1.2"
-1.3"
-1.4*'
-1.5"
-1.6"
-1.7"
-U8"
-1.9"
-2.0'
FIG. 18. — Alinement chart for determining load supported by a helical spring.
ALINEMENT CHARTS FOR MORE THAN THREE VARIABLES 33
evidently be lettered i, and will be the point at which this axis is inter-
sected by the base line — a matter of no importance, however, in the pres-
ent instance. If we call directions upward positive and downward nega-
tive, and remember that log. r has a minus sign, we will see that the point
corresponding to 2 will be at the lower end of the line and that correspond-
ing to 0.5 at the upper end. Graduate the intermediate portions for as
many values as are desired, of course, in their logarithms. The other
outside axis, on which d is to be laid off, is drawn to the left of the axis
just constructed and may be placed in any convenient position. The
values of d called for lie between 0.102 and 0.46 inch for which the loga-
rithms are 1.0086 (or —0.9914) and 1.6628 (or —0.3372). Log. d is to
be multiplied by 3, however, and therefore these values become —2.9742
and — 1.0116. Their difference is 1.9626 which, after multiplication with
the scale unit, gives the graduated length of the d-axis. If we use a scale
unit of 1/3 the size of that used on the r-line we will get substantially the
same length for the two axes. It will be convenient, in graduating this
line, to take the r-line unit and graduate the logarithms of d directly
from it rather than use the 1/3 scale and then multiply by three, since log. d
is to be multiplied by 3.
The position of the zero on this line (corresponding to i) will evi-
dently be beyond the upper end, since all the logarithmic values are
negative. The point marked 0.46, being nearer i than 0.102, will be at
the upper end and the other at the lower. Having chosen the positions
for the limits of this line, we proceed to graduate it.
The logarithm of 0.102 is 1.0086. Lay your engineer's scale on the
line so that the point chosen to represent 0.102 is opposite 0.86 on the
scale. Then with the aid of a table of logarithms pick off the intermedi-
ate points up to 66.28. Our formula shows that log. 0.196 should be
added to 3 log. d. The method of making this addition was explained
in the previous problems where we worked from the base line. In the
present case where we are ignoring the exact position of the base line we
disregard the log. 0.196 since its only effect is to change the distance of our
indefinite base line from what we must look upon as the fixed position of
the (/-line graduations.
While speaking of the J-line I should like to call attention to the re-
markably regular appearance of the graduations. The nearly equal spac-
ing means that the diameters of the wire increase by approximately a
geometrical progression.
We must next locate the position of the r-d or q support and deter-
mine the value of its scale unit. The scale units on the r- and d- lines are
3
34 CONSTRUCTION OF GRAPHICAL CHARTS
Jin the ratio of i to i /3. If we take the unit for r as the standard of refer-
ence, we find from equation (10) that the unit for the r-d support will be
and from the ratio of the unit lengths on the outside axes we find that the
intermediate support should divide the distance between them in the pro-
portion of 1/4 to 3/4. Equation (9.) This line may now be drawn and
might be graduated in the unit we have determined if there were any need
to have the numerical result of the r-d operation. As this will not usually
be wanted, we will save ourselves the trouble.
Take now the second of the equations started with,
log. P = log. q + log.f
which shows that P is the product of the multiplication of q and/. Their
scales will be the outside axes of a new chart and P will be graduated on
an axis between them. We have assumed a variation of / from 30,000 to
80,000. The logarithm of 30,000 is 4.4771 and of 80,000 is 4.9031.
The difference, 0.4260, multiplied by the scale unit chosen, gives the
length of the axis. In the chart made for this article the scale unit selected
for the /-axis is i / 2 that of the reference standard used on r. It would
have been better on some accounts if the unit had been made larger in
order to get greater scale lengths on the different axes. I found, however,
that any larger scale unit that I could use would give a unit for graduat-
ing theP-axis which would be utterly impracticable with the ordinary engi-
neer's scale. The graduation of the P-scale might, of course, be made by
a series of projections from the other axes had there been any pressing
need to have the /-scale long, but this is a tedious operation. In the
problem we are considering the values of /will generally be expressed in
round numbers, and there will be no need of minute subdivision — the
chief advantage of a long scale. Accordingly, the unit value of 1/2
was chosen for/
In locating the /-line it was simply placed as far to the right as it would
conveniently go without interfering with the r-line, and, as with the other
axes, the graduations are located on it in any position we please. Begin-
ning at the lower end, which we mark 30,000, we graduate up with the
logarithms of the desired fiber stresses until we reach 80,000.
Lastly, we must locate and graduate the P-axis. The scale unit for
the r-d support has been found to be 1/4; that of the /-line is 1/2.
Therefore, substituting in formula (10), we get for the scale unit for P
i+i I
ALINEMENT CHARTS FOR MORE THAN THREE VARIABLES 35
The P-axis will divide the distance between the/- and <?-lines into parts
which have a ratio of 2/3 to 1/3, since the units on the side lines are 1/2
and 1/4. Equation (9). The range over which we must suppose the
values of P to vary is a trifle indefinite. It will not do to substitute the
values of the variables already settled upon, which give the minimum and
maximum values of P, for 4his woufd lead to absurd combinations. It is
not probable, for instance, that a spring would be made of No. 10 wire and
a 2-inch coil radius, and it is still less likely that wire 0.46 inch in diameter
would be used in a spring whose coil radius was 1/2 inch. Taking average
conditions, I find that the range for P should be somewhere in the neigh-
borhood of from 10 to 1000 pounds. To make sure of being on the safe
side, I have extended the limits a little beyond each of these values.
Now, when it comes to starting the graduations onPwe ought, properly
speaking, to know the location of the base line; but we have completely
lost track of this, and it cannot, therefore, serve us. We may easily locate
one point on P, however, if we run through a trial calculation. Let
d= 0.102 inch, r = i inch, and/ = 50,000 pounds. Then
. O.I023
P = 0.196— —50,000 = 10.4.
On the chart join up d = 0.102 with r = i, and find the intersection
with the r-d support. From this point draw a line to 50,000 on the
/-line and find its intersection with the line chosen for P. This must be
the point corresponding to 10.4, whose logarithm is 1.017. We have thus
found a starting point for our graduations, and the other marks may easily
be located with the proper scale, i /6 that of r. The chart is now com-
plete except for lettering. For convenience in reading I have given the
rf-line a double set of numbers, one for the diameters and the other for
the corresponding gage numbers.
To read the chart draw a line between the selected values of r and d
(say 0.8 and 0.204) and get the intersection with the r-d support. Con-
nect this point with the chosen fiber stress (say 80,000). The intersection
with P, which is at 166, gives the load the spring will carry.
CHART FOR STRENGTH OF GEAR TEETH.
Next, let us take a formula containing five variables instead of four.
The principles involved are precisely the same as those already discussed:
we merely carry the process one step further. For the sake of variety
a slight change will be made in the disposition of the axes. The formula
36 CONSTRUCTION OF GRAPHICAL CHARTS
chosen for charting is the well-known one by Lewis for the strength of
gear teeth
W = spfy
where W is the pitch-line load, s the fiber stress, p the circular pitch,/
the face width, and y a constant corresponding to the number of teeth.
Let us separate the right-hand side of the equation into two parts and
construct a separate chart for each, one giving the product of s and y,
and the other the product of p and/. Then, if we take the resulting prod-
uct lines as the outside lines of a new chart, we will find the value of W
(their product) on their intermediate support. See Fig. 19.
We must impose the customary limits on the variables in order to
determine the size of the chart. Suppose we let p vary between i /2 and
2 inches and / between i and 6 inches. According to the tables which
usually accompany the formula, s may vary between 1700 and 20,000
and y from 0.067 f°r a i2-tooth pinion to 0.124 for the rack. For the
sake of simplicity we will suppose the application of the chart to be
limited to the i5-degree involute teeth.
Take first the values of y. The logarithm of 0.067 is 2.8261, and of
0.124 it is 1.0934, giving a difference between the extremes of 0.2673;
this multiplied by the scale unit chosen gives the graduated length of the
axis. Pick out the values of y from the table, find their logarithms, and
lay down the latter on the axis, making the lower end of the line the loga-
rithm of 0.067. Lewis gives a formula
0.684
y-o.124-—
for calculating the value of y from the number of teeth. I have made these
calculations and laid off the results on the other side of the line for purposes
of comparison. It will be noted that the tabular values are spaced some-
what irregularly as compared with the calculated. This is a matter of
passing interest, but the chief point to which I wish to direct attention is
the ease with which empirical constants, which are connected by no known
law, may be handled by these diagrams. There is no need of trying to
force them to fit some arbitrary equation for they may be inserted in the
chart exactly as they were obtained from experiment. In lettering this
line we place opposite the graduations the numbers of teeth corresponding
to the different values of y, which we have plotted, instead of the ^-number
themselves. The former we know from our given gear, while the latter
is of no special interest. This line from now on will be called the n- instead
of the y-axis.
Opposite and parallel to this line we draw the axis for s, the fiber stress.
ALINEMENT CHARTS FOR MORE THAN THREE VARIABLES
37
The logarithm for its lowest value, 1700, is 3.2305, and for the highest,
20,000, is 4.3010, giving a difference of 1.0705. If we take a scale-unit
value of one-fourth that used on the w-axis the two lines will be approxi-
mately equal. Taking the lower end of the line at any convenient point,
mark it 1 700 and graduate up to the top in the logarithms of the desired
values of s. In lettering this line it might be well, in case the gears for which
38 CONSTRUCTION OF GRAPHICAL CHARTS
the chart is to be used are all to be of the same material, to place opposite
the fiber stresses the appropriate speeds as shown by the table, thus making
the chart entirely self-contained. Where several different materials are
to be used this would probably cause a considerable amount of confusion,
and it has therefore been omitted here.
The lengths of the scale units on the outside axes being i and i /4, we
find the unit length for use on the intermediate support to be
and the support will divide the distance between the outside axes into
intervals whose lengths are 1/5 and 4/5 of this distance. We do not
graduate the intermediate support, since the numerical results of the
multiplication are not wanted.
Next take the values of p and /; p varies from i /2 inch to 2 inches.
The corresponding logarithms are — 0.301 and +0.301, making a total
range of 0.602. The lowest value of /is i inch (log. = o) and the highest
6 inches (log. = 0.778), making a total range of 0.778. Since these two
lengths are so nearly equal we might as well use the same scale unit for
each, and it will be found convenient to make it i /5 that used on the w-axis.
The support for the product will have a scale unit i /2 the size of that
used on the outside axes, or will be equal to i /io the length of that which
was used on the w-line. This support and the one previously located
are to be used as the outside axes for the last multiplication, whose product
is W. The size of the scale unit on the TF-line, since those on its outside
axes are i /io and i /5, is
and the line itself will divide the distance between these axes in the ratio of
i /3 to 2 /3.
It will be convenient to have the PF-line fall between the diagrams used
for the preliminary multiplications in order to avoid confusion. There-
fore, locate it somewhat to the right of the w-axis and then draw a vertical
for the support for the p-f product so that its distance from the TF-line is
1 1 2 the distance of the latter from the n-s support. At convenient equal
distances from the ^-/support draw the p and /-axes, and graduate them
with the logarithms of p and /, using a scale unit i /5 the size of that we
used for the w-graduation. As before, we may locate the graduated parts
of these lines anywhere we please on them.
The W-axis is now to be graduated, and its graduations, unlike
the others, must start at some definite point. Solve the equation for
ALINEMENT CHARTS FOR MORE THAN THREE VARIABLES 39
any values within the prescribed limits. Take, for instance, w=i2,
y = 0.067, 5 = 1 7°°> /= l > and P=i I2- Then
w = 1700 x i / 2 x i x 0.067 = 56.95.
On the chart join 1700 on the 5-line with 12 on n, and get the inter-
section with the intermediate axis, which will be at the product (unknown)
of the two. Join i on the /-line with 0.5 on the />-line, and get the inter-
section with their intermediate axis, giving again the product (unknown).
Join the the product of n and 5 with that of p and/, and the intersection
with the PF-line must be the point corresponding to 56.95. Its logarithm
is I-7555- Lay an engineer's scale with the proper-sized graduations
(1/15 that used on the w-line) on the PF-line so that 1.7555 on it is at the
point we have located, and graduate the rest of the line from a table of
logarithms. The method of using the chart should be obvious from what
has preceded, but may be briefly recapitulated. Join the desired values
on n and 5, say, 27 and 10,000, by a straight line and mark its intersec-
tion with the support. Join the desired values of p and/, say, i and 3,
by a straight line and get its intersection with their support. Join these
two points by a third line, and its intersection with the W-line gives
the load, 3000 pounds, which the gear will carry safely.
I believe that a comparison of this diagram with others which have
been published for the solution of this equation will show that it has some
very marked advantages over them in point of clearness of reading and
simplicity of construction. The only point which gave any trouble in
construction was the selection of scale values for the different lines so
that they might all be read from an ordinary engineer's scale. Several
trials were necessary before they were finally settled.
Enough has been said, I think, to indicate the general method to be
followed in cases where the equation to be charted contains more than
three variables, and there should be no difficulty in extending the method
to any case where more than five — the largest number treated here — are
involved. Before leaving this part of the subject, however, I wish to take
up briefly another case, differing slightly from those which have gone
before, and which is occasionally serviceable in special problems.
CHART FOR STRENGTH OF A RECTANGULAR BEAM.
Suppose we have an equation of the form
WL _bh-
This is the equation for a rectangular beam, supported at the ends
40 CONSTRUCTION OF GRAPHICAL CHARTS
and uniformly loaded. In it W is the total load, L the length of the beam
in inches, b the breadth, and h the height of the rectangular, cross-
section of the beam, both in inches, and /the fiber stress.
Let us suppose for convenience that the beam is of white oak or long-
leaf yellow pine for which the "Cambria" pocket book gives a safe fiber
stress of 1200. Our formula may then be simplified to read
WL=i6oobh\
For our limits let us say thatZ, varies between 10 and 24 feet, or 120
and 288 inches, b from 2 to 10 inches, and h from 4 to 14 inches. Then
W will vary from about 178 to 26,100. Suppose, now, we construct
two charts, one for multiplying W andL and the other for 1600 b times
h2, Fig. 20. The two products are to be equal. We may, therefore,
use the same line as the support for the product for each chart if the
scale units on the two supports have the same value. The base lines for
the two charts may or may not coincide, but it is essential that they inter-
sect the intermediate support at the same point if we expect the two index
lines to cut it at a common point. This must be the case if the products
of the two multiplications are to be equal as we have supposed. As in
the previous illustrations, there is no necessity for actually drawing the
base line. The general method of procedure in constructing this diagram
is so similar to what has gone before that it will not be described in detail.
After finding the range of values required for the L-line we choose a
convenient unit length and graduate the line in the logarithms of the
desired values. The PF-line is placed opposite it at any convenient
distance and graduated with a scale unit whose length is one-quarter
that used on L. The support for the product of these quantities must,
therefore, divide the distance between them in the ratio of 1/5 to 4/5, and
its scale unit will be
iXt=1
For the b- and /z-lines it will be found that a scale unit of the same size
as the standard used on L may be taken for b, and one of one-quarter the
standard for h. These will give convenient lengths for the two axes, and
the intermediate axis will also have a scale unit of 1/5, since again
This is essential if, as remarked above, the products of the two multi-
plications are to be represented by equal lengths on the common support.
Remember that when h2 is plotted the lengths of the logarithms of h
must be multiplied by 2. The scale units chosen for the b- and /z-lines
ALINEMENT CHARTS FOR MORE THAN THREE VARIABLES
being i and 1/4, the support must be distant from these lines in the ratio
of 4/5 to 1/5. Lay off the b- and ^-lines at any convenient distances
from the W-L support which will satisfy this ratio.
10
r-
5"-
4"-
3"-
2-
30000-
-
Common Support
for all Scales.
rll
10
FIG. 20. — An alinement chart plotted from the equation W L=i6oo bh2.
Graduate the 6-line in the logarithms from 2 to 10 with a scale unit of i.
The Mine is to be graduated in twice the logarithms of the numbers
between 4 and 14. The position of the graduations on b is chosen arbi-
42 CONSTRUCTION OF GRAPHICAL CHARTS
trarily, but for h must be found by a simple trial calculation, since the
location of the base line is unknown.
Assume 6 = 2 inches, h= 5 inches, andL = 1 60 inches (13 feet 4 inches).
Then
W=i6oo — I = zoo.
1 60
Join 500 on the PF-line with 160 inches (13 feet 4 inches) on theZ-line
and mark the intersection with the intermediate axis. Through this
point of intersection draw another line so as to pass through 2 on the Mine.
Where this line intersects the h-\me must be the point numbered 5. Its
logarithm is 0.699. Knowing this and the proper scale length, we may
easily find the other points on this line. To read the chart draw a line
between the chosen values of W and L and mark its intersection with
the support for the product. Any line drawn through this point to the b-
and /£-axes will intersect them in values which will give the necessary
strength to the beam. Thus, on the chart, the solution has been found
for the case where ^ = 3200 andL = 200 inches (16 feet 8 inches). It is
found that a beam 4 x 10 inches will satisfy the conditions as to strength.
CHAPTER IV.
•r
THE HEXAGONAL INDEX CHART.
• A type of chart of quite a different character from any of those pre-
viously described will now be considered. Suppose we have a diagram
like Fig. 21, where A O C is any angle whatever, and O B its bisector.
Measure equal distances Oa and Oc on the OA- and OC-axes and erect
perpendiculars ab and cb. They meet, of course, on O B. The length
Oa = Oc = Ob cos. A OB, orOa+Oc=2 Ob cos. A O B.
Now suppose b moved out to V on the perpendicular b bf. Project
b' to a' and c' '. Then since b b1 makes the same angles with O A and
O C, its projections on these two axes will be of equal length, or a a' will
equal c c' '. Therefore,
Oa+Oc = Oa'+Oc' = 2 O b cos. A O B. \
We have here, evidently, a new form of addition chart. If the scale
values on O A and O C are equal, and that on O B is this unit times
, the length O b measured to this unit is equal to the sum of
2 cos A OB
Oa! and Oc' . If A O C is 90 degrees the unit length for the O B-axis is
that used on O A multiplied by— 7^, and if A O C is 120 degrees the unit
\/2
lengths on all three axes are the same.
If we were to graduate the three axes with their proper units and then
erect perpendiculars to the axes at the division points we could find the
sum of Oa' and Oc' by finding the perpendicular from O B which passes
through the point of intersection of the perpendiculars from a' and c1 '.
It will readily be seen, however, that this would entail a very confusing net-
work of lines, and it is, therefore, customary with this form of chart to
use what is known as a transparent index. It consists of a transparent
sheet, preferably of thin celluloid, on the lower side of which are ruled
three lines meeting at a point; each line is perpendicular to one of the axes.
The axes having been properly graduated, the index is laid on the chart
(care being taken that the index lines are perpendicular to their respective
axes) and is so adjusted that one perpendicular passes through the se-
lected value on O A and the second through that on O C. The third
43
44
CONSTRUCTION OF GRAPHICAL CHARTS
perpendicular will then intersect O B at the sum of the two quantities.
The angle A O C may be anything we like, but since we get equal scale
units on the three axes with an angle of 120 degrees, it is advantageous,
in general, to use that value. Where this is done the arrangement is
known as the "hexagonal" type. The whole thing is so simple and self-
V"3
FIG. 21.
FIG. 22.
a' b'
FIG. 23. FIG. 24.
Diagrams illustrating the hexagonal index chart.
evident that it scarcely seems to call for an illustrative example, and I will,
therefore, not attempt to do more than refer to some of its more important
peculiarities.
Like the other forms of addition chart already examined, it may be
turned into a chart for multiplication by graduating the axes in the log-
arithms of the numbers instead of the number themselves, Fig. 22.
When the graduation of the middle or O 5-axis is identical in general
form (not necessarily in length) with those on the side axes, it may be
THE HEXAGONAL INDEX CHART 45
projected from them by parallel lines whose angle with O A or O C is
the supplement of the angle which O B makes with them. This is a con-
venience in case the angle O A C is, say, 90 degrees, as it does away with
the necessity for a scale whose unit length is — -^ times that used on O A
V2
and O C. It will also be noted by reference to Fig. 23 that the gradu-
ated lengths on the three axes may be moved as far as we please in a
direction perpendicular to these axes without changing the points at
which the index line cuts them. This is sometimes an advantage in
that it allows us to shift our scales so as to get a more compact and con-
venient arrangement of the chart than is always possible if the axes are' to
meet at O. For instance, suppose we wished to arrange the three scales
on the sides of an equilateral triangle, shown dotted in Fig. 23. It is
plain that we get precisely the same results with the lines a! V ^c' d', and
e' '/' that we do with the lines a b, c d, and ef; i.e., if a b+ef=c d it is like-
wise true that a' b' ' -\-e' f ' = c' d' . It is also advantageous in case any of
the quantities is affected by a number of different coefficients. In this
case it is only necessary to draw a separate parallel line for each value
of the coefficient multiplied into the variable, and graduate it with the
product of the two. Then, taking the line affected by the desired coeffi-
cient, pick out the required point on it and pass the index line through
this point.
This form of chart may be arranged easily to take care of a larger
number of variables than three. On Fig. 24 the product of O A and O C
will be found on O B. If we draw a new axis O D, making an angle of
1 20 degrees with O B, we have a new diagram on which we may obtain
the product of O B and O D. The product will be read on O C, or any
line, as O E, parallel to it. This operation may be repeated an indefinite
number of times, and it is here that the advantage of being able to move
the scales in a direction perpendicular to themselves becomes most
apparent. It enables us to handle a large number of variables and have
a separate scale for each one of them.
In problems of this sort it is an advantage to have a transparent index
made in the shape shown in the same figure. It is a hexagon with the
sides parallel to the index lines. This chart takes its name from the shape
of this index sheet. After setting the index to get the product on O B,
place a straight-edge against the side parallel to the O B index line, and
it is easy to slide it into position for the next reading without losing its
orientation, and at the same time always keep the index through the
point last found on O B.
46
CONSTRUCTION OF GRAPHICAL CHARTS
A MODIFICATION OF THE PRECEDING TYPE.
Personally, I must confess, the method of the transparent index does
not appeal to me very strongly. It has the disadvantage of not being
self-contained, and unless we provide a special index for each chart the
two are not likely to be found together when they are wanted. In the
second place, it is easier to " fudge,"
or force the index to give the desired
results than with most of the other
types. Still it must be admitted that
it has its ad vantages in certain cases,
and I have had one or two problems
— A to chart which it seemed impossi-
FIG. 25.— Diagram illustrating a modification ble to handle with any approach
of the preceding type. . ,. . .
to simplicity by any other method.
A form of chart which is related to both the hexagonal and alinement
types is shown in Fig. 25 In it the axes O A and O C make any
angle, and O B bisects it. Draw any line a c. Then from similar tri-
angles we have
Od _Oc-bd
~Oa ' Oc '
or
Od bd^
Oa Oc
and
Od bd
— = i.
Now
Ob
2 cos. A O B
i 2 cos. A O B
Oa Oc Ob
The simplest case is where the angle A O B is 60 degrees; then cos.
A OB=i/2 and
iii
Oa + (te ™.O*"
This is in reality the " reciprocal" form of the type just described.
The equation we have derived is of the same form as that which was
used in plotting the chart shown in Fig. 7,
THE HEXAGONAL INDEX CHART
47
As a matter of interest, this formula has been recharted by the new
method. In Fig. 26 / and /' are graduated on the outside axes and p
on the middle. To read the chart join up points on two of the axes which
are known and get the intersection of the line with the third. This will
be the value necessary to satisfy the equation.
The 6o-degree arrangement of the axes is not quite so satisfactory
in this type of chart as in the last de-
scribed, since when we are working
out toward the limits, the index
line is likely to cut some one of the
axes at an angle which is disagree-
ably acute. For this reason it is
generally considered that the ad-
vantage lies with a smaller angle
even if the work of graduating is
somewhat more difficult.
Where the two outside axes are graduated alike, the central axis may
be marked off without much difficulty by simply joining like points on
the outsides. The marks thus found on the middle axis will have num-
bers whose values are one-half those on the outside lines. This form of
chart might be used for multiplication by plotting the reciprocals of the
logarithms of the numbers to be multiplied on the outside lines and of
their products on the middle. The advantages of such an arrangement
are not very apparent, however, and it has but little practical interest.
F'«-
plotted b^method illustrated
CHAPTER V.
PROPORTIONAL CHARTS.
A family of chart-forms of great structural simplicity is that which is
known under the general name of the "proportional" or "parallel aline-
ment" type. The ease with which they may be laid out and the fact that
they may be used with certain forms of equations which cannot be handled
so conveniently by those types previously described are strong recom-
mendations for their use in these cases.
Take any two lines meeting at any angle and lay off the distance a and
b, as shown in Fig. 27. Connect the points at the ends of these lengths
by a straight line and draw a parallel to it. This parallel intersects the
axes at the lengths c and d. From similar triangles we have
a c
r
i
jj
If, therefore, we lay off on one side of the vertical axis a scale for the
values of a, and on the other side for c, and similarly, on the horizontal
, _ axis, the scales for b and d, we have
a chart which takes account of four
variables. Knowing a, b, and c, for
instance, we join a and & by a
straight line and draw a parallel to
it through c. This line intersects
the d axis at the required value of
that variable.
It may be advantageous, in certain
cases, to have the scales graduated
on separate lines instead of doubling
up as was done with a and c or b and d. This is also shown in Fig. 27
where two lines parallel to the original axes have been drawn. The solu-
tion d' is found by drawing through c' a parallel to the original a b line.
CHART FOR STRENGTH OF THICK HOLLOW CYLINDERS.
As an illustration of this type of chart take the Lame formula for the
strength of thick hollow cylinders subjected to internal pressure
IJ+7
o —
-d—
FIG. 27. — Diagram of proportional chart.
where D is the outside diameter of the cylinder, d the inside diameter
PROPORTIONAL CHARTS 49
(both in inches) , / the fiber stress in the material, and p the internal pres-
sure (both in pounds per square inch).
Squaring both sides of the equation we have
This has the same form as the "fundamental equation. Plot on the
horizontal and vertical axes the desired values of D2 and d2. On the
same axes plot as many values oif+p and/— p as may be deemed neces-
15"-
-10000
- 9000
- 8000
-7000 £
3
-6000 | Key- JdwithD
| Connect Jf.pwithf+p
5000
^4000 ^^
,,
_1000 -^ JScale^f + p
J.. ' .1 j i i i, i i i i, i i, i i ,i f i i i 1 7^1 i ,T . i>^i .1,1.1.1.1
!_-":* :^ "^ ^. "_ * a * *1 * •= •* ' *' TT
00 A O H 01 eo V* «
!-»r-li-lr"(t-*i-| r-i ,-| J5
Scale for External Diameter D.
FIG. 28. — Proportional chart for the strength of thick hollow cylinders.
sary. The scale units used for corresponding quantities on the two
axes may be equal or not, as we please. In this case if we use equal scale
units the horizontal axis will be considerably longer than the other, and
the index lines are likely to cut it at a disagreeably acute angle. Accord-
ingly the values of D and/+/> are laid off with a scale unit whose length
4
X
50 CONSTRUCTION OF GRAPHICAL CHARTS
is 2/3 that used for d and/— p. On the chart the solution is shown for
/=8ooo, p = 4Qoo and d=io inches, giving ^=17.3 inches.
The only objection which might be raised against the chart just shown
is the fact that a preliminary calculation — the addition and subtraction
of the quantities / and p — is necessary before the chart is used. This,
however, is not the fault of the chart but of the equation which was pur-
posely chosen to bring up this point. A makeshift of this sort should, of
course, be avoided where possible, but is often not objectionable. In this
case where the values of /and p will usually be given in round numbers
the necessary computations otf+p and/— p are easily made mentally
and no serious difficulty will result. I have, however, seen this scheme
used on some charts where it involved quite a little calculation or con-
sultation of tables and where, on account of the complexity of the equa-
tion, it was evidently the only method which permitted it to be charted
at all.
THE ROTATED PROPORTIONAL CHART.
This type of chart is susceptible of a slightly different arrangement
which is sometimes considered advantageous. Suppose the lines carrying
the quantities c and d, Fig. 27, to have been rotated about the origin, O,
through 90 degrees. We will have a diagram like Fig. 29. In making
this rotation the line joining the points c and d will likewise turn through
90 degrees, and will be at right angles
instead of parallel to that joining a and b.
In reading such a chart it is generally
customary to have a transparent index
consisting of a sheet of thin celluloid with
two lines, at right angles to each other,
scratched on its lower surface. This is
laid on the chart in such a way as to have
one of the lines pass through a and b and
the other through c. The intersection of
the latter with the d scale then gives the
required value of that quantity. The same result may be obtained, of
course, by a pair of draftsman's triangles laid against each other.
With this chart, as with the first one described, there is no need that
the axes carrying a and d, or b and c should coincide. Every condition
will be satisfied if the lines are separate but parallel. The advantage of
this arrangement of chart over the other is not very marked, and I do not
FIG. 29. — Diagram of rotated pro-
portional chart.
PROPORTIONAL CHARTS 51
incline much toward its use. Some authorities, however, seem to look
upon it with considerable favor and that is my main reason for referring to
it at all.
CHART FOR RESISTANCE OF EARTH TO COMPRESSION.
One formula will be worked Out showing its application. For this
purpose let us take the formula for the resistance of earth to compression,
used in calculations for foundations. It is:
7 / i + sin. (f>
P = wh{ ! : — I-
\ i — sin. 0
where P is the ultimate load on the earth in pounds per square foot,^ is the
weight of the earth in pounds per cubic foot, h is the depth in feet and
(f> the angle of repose of the earth.
The expression
/i + sin.
\i-sin.
may be treated as a single variable and the equation arranged
P _w
h /i — sin. </>
i -f sin. (f>/
This gives us the simple proportion we need for this type of chart.
The limits were determined as follows: The friction angles given by
Rankine for different conditions lie, roughly, between 15 and 45 degrees,
though they exceed this in a few cases. To cover them all the gradua-
tions on the (f) scale will be run up to 60 degrees, though it is probable that
most of the values wanted will lie below 40 degrees. The extreme value
of h was arbitrarily taken as 1 5 feet. The values of w given in the pocket-
books range from about 70 to 130 pounds. Taking h as 15 feet, w as
130 pounds, and <j> as 40 degrees, we find P to be about 40,000.
Next let us choose our scale units. If we take the scale unit for h
(which we will call /J as 1/4, then 15 X 1/4=3 3/4 inches, which is about
the length wanted in the original drawing. For w let the scale unit (/4) be
taken as -fa. Then 130 X i /4Q = 3 i /4 inches, again a convenient length.
For the 0-axis let the unit length (/8) be 0^4 The maximum value of
the parenthesis containing 0 is 0.347 when </>= 15 degrees. Then 0.347 X
-5^4-= 8.675 inches, which will be about right.
Now the scale units should be in the same ratio as the quantities they
affect. Hence, calling the scale unit for P 12 we have
1
"0.04
CONSTRUCTION OF GRAPHICAL CHARTS
Then
Multiplying the maximum value of P by this unit we get,
40,000 Xi--<nnr=I° inches
Connect
I22
"
25X
34 -
36
40 -
45 -
50
55
GO
-25000
-20000
150QO
\ *
V'
-10000
- 5000
Weight, w, of 1 Cubic Foot of Earth
in Pounds.
t- oo o>
Depth, h.
if \n
FlG. 30. — Proportional chart for earth resistance in compression.
UNIVERSITY
PROPORTIONAL CHARTS ^
N^CALlfC^
as the length of the P axis. This was a trifle greater than I wanted for
the limits I had placed on the size of the chart and I arbitrarily reduced
it to about the same length as the <£-line, making the maximum value for
P, 35,000. This would correspond to an angle <£ of about 38 degrees
with h and w at their maximum, and would probably cover most cases.
It was, however, entirely a matter of convenience and there is no reason,
in a practical case, why the scale should not extend as much further as
the conditions in the problems likely to be encountered would seem to
require. The graduation of the different scales is now an easy matter
and the completed diagram is shown in Fig. 30.
The broken lines show the position of the index forw=i2O. ^ = 30
degrees, and h=i$ feet, giving the load P— 16,200 This is, as stated
above, the ultimate strength of the soil. If a fixed factor of safety may
be used for all cases likely to be met with, it might easily be introduced
when P is plotted; that is, the numbers placed opposite the graduation
marks on this scale would be divided by whatever factor we chose. Then
the diagram would give us safe, instead of ultimate loads.
CHARTS WITH PARALLEL AXES FOR SUMS OR DIFFERENCES.
Next let us take a case like that shown in Fig. 31. Here the quan-
tities are laid off from an arbitrary zero line on two axes which are
parallel. Draw a transversal between the ends of the lengths a and 6,
and another parallel to it cutting the axes in the lengths c and d. An in-
spection of the diagram shows that
a — b = c — d,
or the difference between the lengths on the two axes cut by any system
of parallel lines is constant. If one pair of corresponding quantities had
been laid off below the zero and the other above it we should have had
a constant sum instead of a constant difference. We may even get a case
corresponding to
a — b = c + d,
if we lay off the quantity d below the zero and the others above. This
will be referred to later. As in the previous type of chart, there is no
need to have the values a and c, or b and d laid off on the same axes.
They may be laid off on parallel axes if the distance between each pair of
axes is the same. This distance might be varied if the need for it arose,
but it would require an alteration in the scale units to correspond.
To save referring to it again, it might as well be noted here, once for
all, that this chart, and all of those yet to be described involving four
54
CONSTRUCTION OF GRAPHICAL CHARTS
variables, has the same rotational property as was indicated in Fig. 29
for the first type. This is shown in Fig. 32, where the c and d axes have
been turned through 90 degrees without altering their relative positions.
The position of the index is shown by the fine lines, and the construction
is sufficiently clear, I think, to render any further explanation unnecessary.
FIG. 31.
Diagrams of proportional charts.
A chart of the kind we have been examining is of little importance if
we are only to use it for addition and subtraction; but it acquires an
added value if, instead of plotting the numbers themselves on the axes,
we plot their logarithms. This transforms the chart into one for multi-
plication and division.
CHART FOR CENTRIFUGAL FORCE.
An example of the use of it is given in Fig. 33. The formula used is
that for centrifugal force,
C =
w v
where C is the centrifugal force in pounds, w is the weight in pounds, g
is the acceleration of gravity, v is the velocity in feet per second, and r is
the radius in feet of the path of the weight. Rewrite the equation
Then
log. C — log. w = 2 log. v — log. gr
which is identical with the fundamental equation given above. The
limits between which we are to work are of no special importance here,
since the chart is not supposed to be applied to any particular problem.
We will have to fix some conditions, however, so let us say that w varies
PROPORTIONAL CHARTS
55
from i pound to 100 pounds, v from i foot to 50 feet, and r from o.i foot
to 10 feet. The maximum value of C will be 776.4, and of its logarithm
2.89. The maximum value of log', v2 is 3.398, of log. w is 2, and of log.
32.2 r is 2.508.
800
700
600
500
45
40
35
30
25
--20
200-
0 100
<o 90
JSn
•3 w
% 50
5 40
<§ 30
•s
"3 20-
^-T10
9
8
T
6
5
4
100-
90-
80-
70-
60-
50-
40-
1:
7-
6-
5-
4- 2
f]
•s
•3 10"
7
6
5
4
FIG. 33. — Proportional chart for centrifugal force.
In graduating the axes the same scale unit must be used throughout.
All except the r-scale commence with mark,i at the zero point, and are
laid off in any convenient sized divisions from a table of logarithms.
The C-scale was extended to 800 instead of stopping at its exact upper
56 CONSTRUCTION OF GRAPHICAL CHARTS
limit, 776.4. In the case of the r-scale we must place mark i at a distance
of 1.508 (log. 32.2) above the zero, and graduate above and below this as
desired. Fig. 33 shows the completed diagram with transversals drawn
to indicate a solution for w = 30, v = 9, and r = 10; C should then
equal 7.55.
CHART FOR PISTON-ROD DIAMETER.
With this type of chart it is not necessary that the equation be in the
form of a simple proportion, though it should be capable of being placed
in that form by a little manipulation. For instance, take the formula
given by Kent for the diameter of the piston rod of a steam engine,
d = 0.013 \/ D lp,
where d is the diameter of the rod, D is the piston diameter, and / the
length of the stroke, all in inches, and p the maximum steam pressure in
pounds per square inch. Squaring, this becomes
D I p
d2 = 0.000160 D I p =— - .
5917
In its proportional form it is:
I
D 59*7
~T'
or in logarithms,
2 log. d - log. D = log. I - (log. 5917 - log. p),
which agrees with the fundamental equation for this type of chart.
Here we plot the logarithms of d2, D and / as usual. In the case of
p, however, we first lay off the log. of 5917 ( = 3.772) from the zero and
from that point plot the logarithms of p downward, since we use the recip-
rocal of p and not p itself in the last member of the proportion. For the
sake of compactness it is well to have all four scales on about the same
horizontal zone, and since those of / and p are much higher than the others
we drop their zeros by equal distances below those of d and D. The
zeros are not shown in the chart, Fig. 34, since none of the graduations
go down that far, and only the working parts of the scales are needed.
No error is introduced by this shifting of the scales since the slopes of the
lines joining them are the same before and after the transfer. The units
used in graduation must be the same for all scales unless a different one
is indicated by the exponent of the quantity. Therefore, D, I and p are
plotted with one unit, and d (since it is squared) with one twice as large.
The resulting chart is shown in Fig. 34, and the broken parallel lines give
PROPORTIONAL CHARTS
57
the solution for the case where D = 20 inches, / = 30 inches, and p =
100 pounds. Then d = 3.18 inches.
It may be mentioned here that this type of chart may be applied to
equations containing but three variables. If, for instance, in our equa-
tion
5.5-
6" -I
4.5"-f
4" -_
3.5-
"8
« 2.5-
1.5-
15-
12-
-50
--40'
SOLbs. .
A<
90 rf
-100 | |
-110 3 g
120 y ^
.130 W |
- 140
-150 W
15
-12"
FIG. 34. — Proportional chart to determine piston-rod diameter.
a, b and d are variables and c a constant, the c graduation is reduced to
a single point through which all lines referring to c must pass. The
method of using such a chart is precisely the same as for those just de-
scribed, and it is hardly of sufficient importance to merit more than a
passing notice.
5» CONSTRUCTION OF GRAPHICAL CHARTS
THE Z-CHART.
The examples which have been given will illustrate sufficiently well,
I think, the general methods to be followed in cases involving a simple
proportion, and we will now proceed to examine a new type which,
while it bears a family resemblance to some of those previously described,
differs from them in several important particulars.
In Fig. 35 we have three axes arranged in the form of a letter Z.
Draw a transversal across them. From similar triangles we have
bd
= — , or a —
a c c
Add d to each side of the equation and we have
hd d
bd d
d = - -+ d = — (b + c).
0 C/
FIG. 35. FIG. 36.
Diagrams illustrating Z-charts.
Now (b + c), the length of the diagonal of the Z, is a constant which
we may call k.
.'. a + d = — k.
c
Draw a second line parallel to the first transversal. Then
d
(12)
and the original equation becomes
a + d =
k.
(13)
If then the equation which we are to chart has the form
x
w
PROPORTIONAL CHARTS 59
we lay off u on the upper horizontal, v on the lower, x also on the lower
and y on the diagonal. Joining the values of u and v corresponding to
a and d by a transversal, and drawing a parallel to it through the value
of oc corresponding to e we get the resulting value of y, or /, on the
diagonal.
Similarly, we may get the solution of a problem where the difference
of two quantities is used instead of their sum. Fig. 36 shows the arrange-
ment. Here, as before,
a b bd
and
— - = — or a =
d c
c c c
.\a-d = --k.
I r
The selection of the scale units is of some importance with this chart
and a brief discussion of their mutual relation is necessary. It is under-
stood, of course, that the numerical values of the quantities u, v, x and y
are to be multiplied by certain scale units in order to get their measured
lengths, a, d, e and /on the axes. Let these lengths be llt /2, /3 and 14
for «, v, x and y, respectively. For u and v the scale unit must be the
same (1^=1^), since otherwise parallel lines joining their scales would
not indicate a constant sum, but /3 and /4 may be chosen at will. Now
since a = l1u) d = llv) e = l3x and /= /4 y we have by substitution in
equation (13)
or snce
x
u+v ..-,
and
k=(b + 0 = -y-,
*3
which gives the necessary length of the diagonal of the Z.
For the subtraction formula this becomes
(15)
60 CONSTRUCTION OF GRAPHICAL CHARTS
f
Sometimes we wish to make /3 = lr In this case
c) -= It, (16)
or the diagonal is the same length as the scale unit used in graduating it.
It should be noted here that the a and d scales may be shifted along
their axes the same amount and in the same direction as far as we please,
without changing the direction of the transversal joining them and that,
therefore, no error will be introduced. This sometimes permits us to
make a more convenient arrangement of the scales as will be shown later
in connection with the chart for chimney draft.
CHART FOR POLAR MOMENT OF INERTIA.
As the first illustration for the construction of the Z-chart I have
chosen the formula for the polar moment of inertia of a flat rectangular
plate about an axis perpendicular to its plane and passing through the
center. It is sometimes used in the power calculations for the draw spans
of bridges, the assumption being that the span may be taken as having
approximately the same polar moment of inertia as the flat plate. The
formula is:
W
I = -(B>+L>),
where 7 is the polar moment of inertia, W the weight of the plate, and
B and L its breadth and length. The weight will be expressed in pounds
and B and L in feet. The engineer who wishes to have the forces in his
final results in pounds instead of poundals, will usually prefer to divide
at once by g, instead of doing this at the end of his calculations; in which
case the formula becomes
This may be written
386.4
and we evidently have an equation suited to the Z-type of chart.
When planning this chart my intention was to give it something like
a practical form by taking the maximum values of B and L as about 10
and 80. However, since these quantities are to be squared and L2 =
6400, while B2 is only 100, it is evident that if we lay them off to the
same scale and use any practicable length for 6400, 100 would be so small
(after the necessary reduction by the engraver) that its subdivisions for
PROPORTIONAL CHARTS
6l
90000-J-30'
FIG. 37. — Z-chart plotted from formula for polar moment of inertia.
62 CONSTRUCTION OF GRAPHICAL CHARTS
smaller values of B would be illegible in the cut. In view of the fact,
however, that the charts which illustrate this book are intended pri-
marily as examples of methods of construction and application, I
have not hesitated in many cases to sacrifice a practical chart for the sake
of getting one which showed a process clearly, and this I shall do in the
present instance. The conditions are assumed to give clear reading scales
in the cut, but the chart in its present form will have little practical value
for the bridge designer.
Let us say then that the maximum value for W is to be 35,000, that
the maximum value for B is 10 feet, and for/,, 30 feet. Then the maxi-
mum value for I' is very nearly 90,000. The scale units and scale lengths
must next be fixed. I wished to keep the original drawing inside of a
length of 10 inches. By making the scale unit for /' y-g-g-g- <j- 1 get a gradu-
ated length of
90,000 X TTroinF = 9 inches.
This is the scale unit we called /3 in the preliminary explanation. For
the 1^-line it will be convenient to make the scale unit, /., equal to —
5000
Then
35,000 386.4
^- — X - = 7 inches
386.4 5000
is the graduated length of this axis. This unit makes it possible to plot
W directly from the 50 side to an engineer's scale without bothering about
the coefficient - . For the L and B scales let us take /x = i /ioo.
386.4
Then for!,2 we have a graduated length of 900 X i /ioo = 9 inches, the
same as for I', and for B2 ioo X i /ioo = i inch. Substituting the scale
units thus found in equation (14) we get for the length of the diagonal of
the Z,
n * 3*864,000
(b + c) = 7 = - * - = - - = 7.728 inches.
Timnr
Having drawn our axes (the diagonal making any convenient angle
with the parallels) we have only to graduate them, and this is a simple
matter, B and L being plotted in the squares of the desired values with a
scale unit of i /ioo, while the W- and /'-lines are plotted directly from
the innnr and f ooflT scales. The parallel broken lines show how the
chart is read for the case where B = 8, L = 20, W = 30,000, which gives
for /' 36,000.
PROPORTIONAL CHARTS
CHART FOR INTENSITY OF CHIMNEY DRAFT.
The next formula which I have charted is one for the intensity of
chimney draft
7-95
!
600°-
f-h\hr- T
^ L 2 l I
where/ is the draft expressed in inches of water, h the height of the chim-
ney in feet, Tl the absolute temperature of the chimney gases, and T2
the absolute temperature of the external air.
The formula will be seen at once to belong to the second type of Z-
chart where we have a difference instead of a sum of two variables. The
general method of procedure is
identical with that just described,
but there are a few differences of
minor detail which require a brief
description. Thus the variables Tl
and T2 appear in the denominators
of the fractions instead of the
numerators, which indicates that
the plotted values are proportional
to the reciprocals of these quantities
and not to the quantities them-
selves. For our limits let us take
h as varying between 50 and 150
feet, Tl from 761 to 1161 degrees
absolute (or from 300 degrees
Fahrenheit to 700 degrees Fahren-
heit), and T2 from 461 to 561
degrees absolute (or from o degrees
Fahrenheit to 100 degrees Fahren-
heit). Then / will have a maxi-
mum value of 1.46 inches and we will graduate it from o to 1.5 inches.
The scale unit on the /-line (/3) I took as ~ which gives a length of
1.5 X ^r = 7.5 inches for its graduations; /4, the /t-scale unit was taken
as i /4O. This gave a graduated length from zero of 150 X i /4Q = 3.75
inches; lv the unit used for Tl and T2 was made i /o.ooi. The extreme
length of the T^-line from its zero will then be
I
'ft
y 0.2*
0.3"
0.4"
"V"^
•3
JO,"-
•9 0.8'-
£
Jl.2"
™?(Xl$
m- 1
^"l>3L*
70° g
60" 7,
1.4"
50° 1
40' ^
1.5*
30 s
20° i
10° "
«•«',) i
FIG. 38. — Z-chart to determine intensity
of chimney draft.
1 6.6 inches,
64 CONSTRUCTION OF GRAPHICAL CHARTS
too great for the size of chart planned which I wished to keep within a
length of 10 inches. The lower limit of the graduations on this line is
7.64
—7- X o.o oi^ I3-^2 inches
from the zero. This is an empty space which is of no advantage, and the
chart will be improved in appearance and compactness if we slip the
graduations along the axis toward the zero point or the point where the
diagonal intersects this $xis. In my drawing the graduations were
shifted a distance of 8 inches, which brought them within the prescribed
limits. The 7\-graduations were shifted the same amount in the same
direction, and thus no change was made in the direction of the trans-
versals joining them and no error introduced.
The length of the diagonal, from equation (15) is
/ / i v *
fL \ 1 4 o.OOl ^40 02 V.
~l ~r~ = 0.04 = 5 inches.
As many values of -J— and -^— as are wanted are now calculated
•*- 2 •* 1
and plotted, remembering that their zeros are 8 inches beyond the points
where the diagonal intersects their axes, h is plotted on the diagonal
and/ on the same axis with T2.
In lettering the 7\- and TVlines it will be a convenience for the person
who uses the chart to have the temperatures marked in the Fahrenheit
scale instead of from the absolute zero. This has been done on the chart,
Fig. 38, but the absolute temperatures have been retained at each end of
the scales as an aid to a clearer understanding of the construction. The
position of the parallel index lines shows the application of the chart to
the case where the chimney temperature is 400 degrees Fahrenheit, the
temperature of the outside air 60 degrees Fahrenheit, and the height of
the chimney 100 feet. The draft gage reading should then be a trifle
over 0.54 inch.
CHART FOR SAFE LOAD ON HOLLOW CAST-IRON COLUMNS.
An interesting application of the Z-type of chart is to certain equations
where a variable appears twice. Since each time it appears it occupies
a scale, the number of variables we can handle is reduced from four to
three. Suppose the fundamental equation to be of the form
v
u+v =
y
PROPORTIONAL CHARTS
24
FIG. 39. — Z-chart to determine safe load of hollow cast-iron columns.
66 CONSTRUCTION OF GRAPHICAL CHARTS
This evidently refers to equation (12) used in demonstrating the
Z-chart. Here one index line instead of two is used in making a reading.
This gives a particularly useful chart since equations of this type are
by no means uncommon and are awkward things to handle by any of the
methods hitherto described. A formula which will serve as an excellent
illustration is the one given below which is taken from the "Cambria"
pocket book:
P- 5
U
SooD2
It is the formula for the safe load on hollow round cast-iron columns
with flat ends. In it P = the safe load in tons (of 2000 pounds) per
square inch of column section, D is the outside diameter of the column
in inches and L the length of the column also in inches. The successive
steps required to put the formula into working shape are indicated below
L2 5
800 D2 P
and
0.2 P
Let us take the limits for D as 6 and 15 inches, forZ, as 72 inches ( = 6
feet), and 288 inches ( = 24 feet). Then P will vary between 1.3 tons and
4.86 tons. The maximum value to be laid off on the L-line will be 288*
or 82,944, on the D-line 8ooX I52= 180,000, and on P, 0.2X4.86 = 0.972.
The scale units for L and D being the same, it is evident that the value
180,000 will control the choice of the scale unit if we are planning a chart
of a certain size. Suppose the scale unit /x is made 1/20000. Then
1 80,000 X i / 20000 = 9 inches,
which is about right. For L we have
82,944X1/20000 = 4.147 inches.
The scale unit /4 used in graduating P on the diagonal will be taken as
i /o.i and the graduated length will, therefore, be
0.972X1/0.1=9.72 inches.
Since /3 = lt the length of the diagonal intercepted between the parallel
axes is, according to equation (16), 14 or i /o.i = 10 inches. On the chart,
Fig. 39, Z>2 has been graduated for every inch between 6 and 15 inches,
and from its zero the diagonal, 10 inches long, has been drawn in any
PROPORTIONAL CHARTS 67
convenient direction. On it we may consider that we are graduating
0.2 P with a scale unit of i/o.i or P with a scale unit of 1/0.5. Lastly,
the second parallel is drawn from the end of the diagonal and graduated
for!,2. This has been done for every 12 inches and the points marked
with the corresponding values in feet.
As noted above, but a single transversal or index line is required for
reading this chart.
By joining 15 feet on theL-line with 10 inches on the D-line, we find
the safe load per square inch on the column is 3.56 tons.
Before leaving this subject it might be well to call attention to the fact
that another way of treating three variables by the Z-chart is to imagine
that one of the four which normally belong to it is replaced by a constant.
The scale belonging to it then reduces to a point through which all of the
lines pertaining to it must pass.
CHAPTER VI.
EMPIRICAL EQUATIONS.
In the previous chapters I have discussed some of the methods used in
plotting curves and charts from given equations. The present one will
be devoted to the reverse process, namely, the derivation of equations to
fit a given set of empirical data when these data are plotted in the form of
a curve or chart.
The subject is one which is full of difficulties, and, so far as I know, no
systematic general method has ever been devised which will give the
correct form of equation to be used. The discovery of the equation's form
is to a large extent a matter of intuition which can only be acquired by
long experience. Some persons seem to be peculiarly gifted in the ability
to pick out the proper kind of equation for use in compensating a particular
set of observations, but for the rank and file of the men engaged on experi-
mental work this is, and probably always must be, a matter of pure guess-
work, which must be verified by cut-and-try methods.
In getting an algebraic expression to show the relations between the
components of a given set of data there may be two entirely distinct objects
in view, one being to determine the physical law controlling the results
and the other to get a mathematical expression, which may or may not
have a physical basis, but which will enable us to calculate in a more or
less accurate manner other results of a nature similar to those of the
observations.
To attain the first result it will generally be necessary to have as a
starter some soft of hypothesis as to the physical relations of the data in
question, although in a few isolated cases it has been possible to arrive at
hitherto unknown laws by a fortuitous treatment of the observations.
In such a case as this, questions as to the intricacy or convenience of the
formula in use are considered subordinate to correctness of form.
In the second case, where we want an expression which will enable us
to calculate results of the same general character as the observations,
form will generally be sacrificed to convenience of handling and no pre-
tense will be made that the derived formula conforms to any physical
law. This condition is one very commonly met with in engineering
practice, and will be the one with which this chapter is chiefly concerned.
68
EMPIRICAL EQUATIONS 69
It has been a common matter of complaint among the so-called " prac-
tical" men that the "theorists" who are responsible for the formulas are
very prone to unnecessary complication, and that the formulas they offer
are in many cases no more exact, than others of a much simpler type.
It cannot be denied that there is some justification for these charges, due,
perhaps, to a popular impression fhat a complicated formula presupposes
brain work of a high order for its production.
That this is not necessarily true needs no special proof, but, on the
other hand, we should be carefully on our guard lest we be led by a desire
for simplicity into devising mere rules of thumb, applicable, perhaps,
to the very special conditions in which they originated, but nowhere else.
As an example of this, take the numerous formulas which have been pro-
posed in the past for the strength of gear teeth; formulas giving results
which in some instances differ from each other by several hundred per cent.
A few words of caution may be necessary at the start to prevent the
reader from expecting too much of the processes described. Except in
some of the simplest cases where the line connecting the plotted data is
straight, it will generally be possible to fit a number of very different
forms of equation to the same curve, none of them exactly, but all agree-
ing with the original about equally well. Interpolation on any of these
curves will usually give results within the desired degree of accuracy.
The greatest caution, however, should be observed in exterpolation, or
the use of the equation outside of the limits of the observations.
If the form of the equation is known at the start to be correct and the
observations are merely used to determine the constants, exterpolation
will generally be safe. If, on the contrary, the form of the equation has
been guessed at, exterpolation is hazardous in the extreme, and, if an
attempt is made to use the formula much outside of the range of the
observations on which it is based, serious errors may be committed.
The whole subject is full of pitfalls against which one must constantly
be on guard.
About the only process for getting empirical equations which is dis-
cussed in the text-books is that known as the method of least squares. It
will yield satisfactory results where a good equation has been chosen at
the start, but it is tedious and laborious in the extreme even under the
most favorable circumstances, while for certain forms of equation the
difficulties of the method are so great that it can hardly be considered as
practicable. On this account, and because it can be found fully described
in the ordinary text-books, I shall not touch upon it here, but confine my-
self to a number of graphical or semigraphical methods with which I am
7O CONSTRUCTION OF GRAPHICAL CHARTS
acquainted. Some of these at least are but little known. Nevertheless,
there are some very decided advantages in their use, as I hope to show
later.
Square Feet of Belt Surface per Minute.
J
7
°°/
/
<j
/
0
0
7
I
/
/
9
/
O
/
D °
/
'
0
A
00
/
h
/
0
2
°y
».°
X
y
c
3?
*>
/o<
>*
r
^
#
r"
$*
/
100 200 300 400 600 COO 700 800 900 1000 1100
indicated Horsepower
FINDING THE EQUATION FOR A STRAIGHT LINE.
To begin with, let us take a very simple case where the relation between
the variables in the equation is linear; that is, where the plotted results
fall upon a straight line. The literature of engineering contains numerous
examples of this type, and I have chosen as illustrations two charts taken
from Bulletin No. 252 of the
University of Wisconsin, entitled,
"Current Practice in Steam
Engine Design," by O. N.
Trooien. Fig. 36 of this bulletin
is reproduced here in Fig. 40,
and is intended to show the rela-
tion between the indicated horse-
power and square feet of belt
surface per minute for Corliss
engines.
The necessary data for plot-
FIG. 40. — Chart showing relation between ting this diagram were obtained
moving belt surface and horsepower of Corliss en- , , , .,,
gines. Equation of middle line is y = 2ix + 1000. trom a number Ot engine builders.
The rim speed of the belt pulley
as given by them was multiplied by the width of the belt and the result was
used as an ordinate, while the horsepower of the engine was taken as the
abscissa. A point was thus charted for each engine, as shown in Fig. 40.
In this case, as in many others which occur in practice, the chart looks
as if a charge of bird shot had been fired at it, and it is manifestly impossi-
ble to find a line which shall even approximately pass through all of the
points.
If we have any reason to suppose that a rational formula connecting
the belt surface and horsepower would be of a simple linear type, all we
have to do is to draw a straight line which will coincide as nearly as possible
with the "axis" of the group of points and take its equation as the best
representation we can get for the data. In the case of horsepower and
belt surface it is generally assumed that there is a rough proportionality
between them; hence a straight line is used here. The points in Fig. 40
show two fairly distinct groups of points, and judging by eye, Mr. Trooien
EMPIRICAL EQUATIONS 7 1
appears to have favored the lower group in drawing his line of average
values.
This proceeding is in many instances not only justifiable, but impera-
tive, if we wish to have our line represent the best probable values. It often
happens that certain observations are known to be more accurately made
than others, and hence^should be given greater weight in determining the
final result. In the least-square method the better observations are
affected by coefficients corresponding to their greater accuracy and in
the graphical method the same end is attained by causing our line to pass
closer to the points representing the better observations. Just what reason
Mr. Trooien had for giving greater weight to one group than to the other
is not stated. It may be that the builders had a better reputation or the
results may have been more in conformity with theoretical considerations.
Our average line being located (and it will generally be found advan-
tageous to use a fine thread stretched through the points for this purpose)
its equation is easily determined. The general form will, of course, be,
y = ax+b,
where b is the height of the intercept on the Y-axis (in this case at 1000)
and a is the tangent of the angle made by the line with the horizontal.
The Y-axis, as just noted, is cut at 1000, and the ordinate, through 1000
horse power is cut at 22,000. The difference is 21,000. Dividing this by
1000, the horizontal distance, gives 21 as the value of a.
Our formula then reads,
y = 2i x+ 1000,
or, as y represents belt speed and x the horsepower,
5=21 H. P. + 1000.
Where the points representing the observations scatter as badly as is
the case here, the formula must be looked upon as a very rough approxi-
mation, and considerable deviation from it may be allowed in practice
when for any reason this seems desirable. To indicate the limits within
which this deviation may be made without departing from common
practice, Mr. Trooien draws two lines to include the extreme cases and
derives the constants for them in the same manner as before. Since all
the lines meet at the same point on the axis, the value for b is 1000 in each
case, while a varies from a maximum of 35 to a minimum of 18.2.
The quantity laid out on the X-axis does not have to be of the first
power, as in the case just discussed, and may even be itself a product of
several variables. In such a case we must lay off not x itself, but x2, x3,
, etc., as the case may be, or x z if it is a product.
CONSTRUCTION OF GRAPHICAL CHARTS
ANOTHER ILLUSTRATION OF FINDING THE EQUATION FOR A STRAIGHT LINE.
This may be illustrated by the chart for the connecting rods of Corliss
engines shown in Fig. 16 of the same bulletin and reproduced here in
Fig. 41.
If the Euler formula for struts be taken as correct for the connecting
rod, it may be reduced to the
expression.
30 40
Values of \/T)L
FIG. 41. — Chart showing relation between
diameter of connecting rod and square root
of piston diameter times the length of rod for
Corliss engines. Equation of middle line is
d = 0.092^ DL.
ordinates and of \/ DL as abscissas,
straight and pass through the origin, and the angle with the horizontal
gives the desired value of C as 5.5/60, or 0.092, for the mean and 0.104
and 0.081 as the maximum and minimum values.
where d is the diameter of the
rod, C a constant whose value
is to be determined, D the
diameter of the piston (supposed
to be acted upon by a standard
steam pressure), andL the length
of the rod.
From the data furnished by
the engine builders the points in
the chart shown in Fig. 41 were
plotted, using the values of d as
The resulting line should be
FINDING THE EQUATION OF A CURVE.
Next let us consider the case where the line connecting the observations
is curved. Here we have no ready-made equation as with the straight
line, requiring merely the discovery of a couple of constants. The general
form of the equation must be guessed at if the physical law is unknown,
and here we encounter one of the greatest difficulties connected with the
subject and one for which it is practically impossible to offer much real help.
The appearance of the curve may or may not afford a clue, and in this
connection it is suggested that a book like Frost's "Curve Tracing,"
may be useful for reference. It contains a large number of curves plotted
from various equations and their shapes will sometimes suggest a good
form of equation if we are at fault.
It has sometimes been suggested as a solution of this difficulty that we
plot a considerable number of functions, such y = x2, y = x3, y = log. x,
y=i/x, etc., on a straight line and then from any pole draw a series of
EMPIRICAL EQUATIONS 73
radiating lines through the points thus found as shown in Fig. 42 where
y = x? has been used. The observed results are plotted on a similar
straight line for equally spaced values of the variable. This graduated
line is then laid on the radiating lines and shifted around until we get
the plotted points falling on them. Such an agreement would indicate
at once the proper function to use, V v \ / /
p .«* ^JXZ \3 \< 5/ ^
and a measurement of its distance
from the pole would indicate the
coefficient.
While this looks promising, my
Own experience leads me to accord FlG- 42.— Trial diagram of a known function.
. In this case y=x2.
it but little practical value. The
observation points can hardly ever be made to agree even approximately
with the trial function.
Many experimenters assume that an equation of the parabolic form,
y = a+b x+c x2. . . . etc.
may be used for almost any class of observations with good results, and
it is surprising sometimes how closely it may be made to fit unpromising
conditions.
It should not, however, be blindly used for all cases, for while, on the
one hand, it may be forced, with a sufficient number of terms, into the
semblance of an agreement with almost any set of data, on the other hand,
a large number of terms is detrimental to its subsequent use in calculation
and in many cases a far simpler equation may be discovered which will
not only be easier to handle, but may even give more accurate results.
For instance, the crest of a sine curve may be made to agree quite closely
with a parabola, but the longer this arc is the greater is our difficulty in
getting a fit.
In fitting an equation to a given set of observations the first step is. to
draw through the plotted points a smooth curve. If the experimental
work has been carefully and accurately done the curve may be made to
pass through, or close to, almost all the points. If not, the curve must be
drawn in such a way as to represent a good probable average; that is,- so
as to leave about an equal number of points at about equal distances on
either side of it, these distances, of course, being kept as small as possible.
Such a curve is assumed to represent the most probable values of the
observations, and we then attempt to get its equation.
It may be stated that it is always possible to get an equation which
will agree exactly with a given curve at any desired number of points,
providing we use an equal number of constants in our equation.
74
CONSTRUCTION OF GRAPHICAL CHARTS
7
10 11 12 13 14 15
Speed in Knots
17 18
FIG. 43. — Chart showing relation between indicated horsepower and speed in knots for the
battleship "Maine." Equation of the dashed curve is y = 440.5* — 82. T,2x2 + $. 6$x3.
EMPIRICAL EQUATIONS 75
METHOD OF SELECTED POINTS.
This is called the method of selected points and will be described
first as it is the simplest and quickest method and, if a good equation
has been chosen at the start, we may get results of a very satisfactory
character.
For purposes of illustration I have chosen a curve given in the
Journal of American Society of Naval Engineers for November, 1902.
It shows the relation between the speed in knots and the indicated horse-
power for the battleship "Maine" and is reproduced by the solid line
in Fig. 43.
The data from which the curve was plotted are not given and there is
no means of knowing how accurately it represents the results of the test.
It will, therefore, be taken as it stands and an attempt made to find the
compensating equation. As to the form of the equation, we will disre-
gard all theoretical considerations and assume it to be parabolic since
it has most of the ear-marks of this type. ... .
The curve stops at about eight knots and we have nothing to guide
us as to its shape below this point. The assumption will be made,
however, that the horsepower and speed became zero together; that
is, that the curve passes through the origin. If this is so the first
constant in the general parabolic equation (the one unattached to a
variable) vanishes.
Let us assume that the equation contains the first three powers of x,
or that
y = ax+bx2 + ex*
where y represents the indicated horsepower and x the speed in knots.
We have here three constants whose values must be determined. To
do this take three points on the curve, one at about the middle and the
others at or near the ends, and form three equations, inserting in them
the values of y and x for these points taken from the curve.
In the case in question I have selected the points at 9, 13, and 17
knots. The corresponding values for y (the horsepower) are 1400, 4180,
and 11,350.
Inserting these in the chosen equation we have:
1,400= ga+'&ib+ 729 c,
4,180=13 0+169 6+2,197 c,
11,350=17 0+289 b + 4^3 c.
These equations are solved by the customary methods for a, b, and c,
giving us 440.5 for a, —82.32 for b, and 5.628 for c.
76 CONSTRUCTION OF GRAPHICAL CHARTS
The equation then reads:
#-82.32 x2+ 5.628 A-3,
or
H. P. = 440.5 5-82.32 S2+ 5.628 S\
The curve for this equation has been drawn as a broken line on the
same chart as the original curve, and is seen to pass through the chosen
points exactly and to give a very fair agreement at nearly every other
point.
At the upper end, however, although the two curves are not much
separated, there is a considerable difference in the horsepower as read
from the two curves, and the indications are that this will become worse
as we overstep the limits of the chart. Up to about 171/2 knots, however,
the equation would usually be considered a passable fit. The rapid rise
in the horsepower as the speed increases at the upper end of the curve
would indicate that better results might have been reached by the use of
a higher power of x in the equation.
ANOTHER ILLUSTRATION OF THE METHOD OF SELECTED POINTS.
The above method will answer every requirement in many cases, but
too much reliance should not be placed in it without an actual test of the
results. As an example of the danger of this I have applied the method
to a series of experiments showing the variation of the coefficient of fric-
tion of straw-fiber friction drives with the slip.
The experiments were made by Professor Goss, who describes them
in the Transactions of the American Society of Mechanical Engineers
for 1907, page 1099.
To avoid a confusion of notation, I have replotted the curve from the
original paper in Fig. 44, with the ordinates and abscissas interchanged.
The small circles represent the observations and the solid curve is Pro-
fessor Goss' idea of the best representation of their average value. We
will attempt to compensate this curve by a suitable equation.
At first glance the curve seems to have some of the characteristics
of the parabolic type, enough at any rate to make it amenable to treatment
by that form of equation. It straightens out suspiciously, however, in
each direction, as it leaves the region of greatest curvature near the ordi-
nate erected at 0.4, and this would suggest the hyperbolic rather than the
parabolic type. As an experiment, however, we will run it out on the
assumption of its being a parabola and will try compensating by the
equation
y = a + b x + c cc2 -f- d x5.
EMPIRICAL EQUATIONS
77
The four constants will make it possible to get four points of exact
agreement instead of three, as in the previous example, and we should
naturally expect that the general agreement would be better on account
of this larger number of points.
Let these points be y = o.$$, # = 0.15; ^ = 0.825, # = 0.3; ^=1.42,
# = 0.4; ^ = 2.7, # = 0.45. The four equations then become:
°-55 =^ + 0.15 £+0.0225 £ + 0.003375 d, I
0.825 = 0 + 0.3 6 + 0.09 £ + 0.027 d,
1.42 =0 + 0.4 6 + 0.16 £ + 0.064 d,
2.7 = 0 + 0.45 6 + 0.2025 £ + 0.091125 </.
/
0.2 0.3
Coefficient of Frictiou
FIG. 44. — Chart showing relation between coefficient of friction and slip for straw-fiber
frictions. Equation is y = '- + 0.181.
# — 0.502
The solution of these equations gives us a = -5.89, b = 80.5,
c = - 308, d -381.8,
or,
y= -5.89+ 80.5 *-3o8*2 + 381.8 ;v3.
The values of y were now calculated for every 0.05 of x from o.i to
0.5, and the result is shown by the broken line on the same diagram. It
78 CONSTRUCTION OF GRAPHICAL CHARTS
hits the selected points with practical exactness, but it would require
a vivid imagination to say that the fit elsewhere was even fairly good. A
larger number of selected points and constants would undoubtedly have
helped materially in improving this state of affairs, but the most cursory
inspection of the diagram will show that the trouble is not due to the
small number of points, but rather to the choice of an improper form
of equation.
Returning now to the suggestion made above as to its hyperbolic form,
let us see what can be done on that supposition. We will assume that the
curve is a rectangular hyperbola of which we do not know the asymptotes.
Let us try an equation of the form
(y+a}(x+b)=c.
The three constants will demand three equations, and we will select
for our points ^ = 0.55, # = 0.15; ^ = 0.825, # = 0.3; ^ = 2.7, # = 0.45.
Substituting these values in the equation above we have,
(0.82 5 + a) (0.
The solution of these equations for a, b, and c, gives us
a = —0.1806,6= —0.5015, and c= —0.1298.
These values, in round numbers, substituted in the original equation
give us
(^-0.181)^-0.502)= -0.13,
or,
+ 0.181.
x — 0.502
If, now, we substitute values of x for every 0.05 from 0.15 to 0.45,
we get the points represented by the double circles in the chart. They
agree so closely with the original curve as to be practically identical with
it. Thus, with a less number of points we have obtained an extremely
satisfactory fit, and have given a practical illustration of the statement
made above as to the desirability of starting with a good equation rather
than trying to force a fit by the use of an unsuitable equation and a large
number of constants.
VALUE OF LOGARITHMIC CROSS-SECTION PAPER IN DETERMINING FORM
AND CONSTANTS OF AN EQUATION.
This may be a good place to say that the logarithmic paper described
in a previous chapter is often of great service in determining the form and
constants of an equation.
EMPIRICAL EQUATIONS 79
, If the equation involves only a simple product or quotient with no
addition or subtraction, its trace on logarithmic paper will be a straight
line. The tangent of the angle made by this line with the horizontal
(and this may be positive or negative) will give the exponent of the vari-
able, while the intercept on the Y-axis will give the constant by which the
variable is multiplied.
It is much to be regretted that the ordinary commercial logarithmic
paper is only laid off from i to 10 on the axes, for my experience is that
almost invariably the line will extend beyond these limits, and it then
becomes difficult to see clearly if it is rectilinear, since it must be broken
and appear in two or more places on the sheet. If such paper were
printed with graduations on each axis from i to 100 instead of from i to
10, it would greatly facilitate many of these operations. Any curve having
the aspects of the hyperbolic or parabolic type should always be so plotted,
since, if it does appear as a straight line, it saves a large amount of labor
in determining its equation.
One special case may be mentioned here, which is sometimes useful
in gas-engine work; namely, the determination of the exponent of the
v in the equation for the expansion curve. If we have an indicator
diagram we take the ordinates representing the pressures (absolute) and
lay them out on the logarithmic paper from points on the X-axis repre-
senting the volumes (which must include the clearance). The points
thus found should fall upon a line which is sensibly straight if the exponent
is constant for all parts of the curve. Otherwise the exponent must be
determined for any particular point by drawing the tangent to the curve
there.
As an illustration, I have reproduced the expansion line from the indi-
cator diagram of an old Clerk gas engine. The volumes are measured
from the clearance line in any convenient unit. The length of the diagram
made it convenient to call the clearance volume 9. From there on, the
indicator diagram was divided, as shown in Fig. 45 (a), and the logarithms
corresponding to the numbers on the X-axis were laid off on the X-axis of
the lower logarithmic diagram, (b) of Fig. 45.
The pressures from the absolute zero were then measured from the
indicator card and their logarithms laid off from the corresponding points
of the X-axis of (b).
A straight line was now drawn to indicate the general direction of the
middle set of points and then a parallel to it through 10 on the X-axis.
Its intercept on the Y-axis measured in linear (not logarithmic) units
gives the tangent of the angle of slope. In this case it is 1.32 which,
8o
CONSTRUCTION OF GRAPHICAL CHARTS
divided by i (the distance to 10 measured on X), gives 1.32 as the value
of the exponent.
The method of selected points, while accurate enough for many pur-
poses, especially where the form of the equation is definitely known at the
9 10 12 14 16 18 20 22 24 26
FiG. 45. — Expansion line of a gas-engine indicator card and logarithmic determination of
value of exponents in the equation of the expansion curve.
start, is not so satisfactory when we wish for greater refinement, and
especially when we are in the dark as to the proper form of equation. The
number of points which can influence the result is no more than the
number of constants employed, and if we wish to use a small number of
EMPIRICAL EQUATIONS 8 1
constants we cannot expect any high degree of accuracy in the fit. Some
method by which a larger number of points on our curve may enter into
the result without burdening the equation with constants is, therefore,
much to be desired.
i
METHOD OF EQUATING THE AREA AND MOMENTS OBTAINED FROM
MEASURING THE AREA UNDER A CURVE WITH THE
INTEGRATION OF THE ASSUMED EQUATION
OF THE CURVE.
Suppose that, an observation curve being drawn, we obtain its area by
any planimetric method. If, now, we find the area of the curve of the
assumed equation by integration and equate it to the area just found of
the observation curve, we evidently have a condition in which we can take
account of as large a number of points as we please without necessarily
using a large number of constants. In fact, this one equation takes care
of one and only one constant. It would, of course, be possible to have
two curves of equal area and quite different shape if the assumed formula
were not well chosen.
Suppose, however, that we get the moment of the area of the original
curve by dividing it up into a number of vertical slices, taking the area of
each slice above the X-axis and multiplying it by the distance of its center
from any arbitrary vertical axis, generally Y, and then adding the moments
thus found; we shall in this way obtain the moment about the assumed
axis of the entire area between the curve and X. Its value will evidently
depend upon the form as well as the area of the curve. The moment of
the assumed curve may likewise be determined by integration and can be
placed equal to the measured moment. This accounts for another con-
stant. Similarly we may obtain second and third moments, etc., by
multiplying the areas of the slices by the square and cube of the distances
from the assumed axis and, from each of these, form equations with the
same moments of the theoretical curve. We must, of course, have as
many of these equations as we have constants to determine.
Any of the well-known methods for getting the areas and moments
may be used, but as it will make the explanation simpler I shall get my
areas and moments in what follows by taking the mean ordinates of a
series of vertical slices in the same way that we do when averaging an
indicator diagram, and assume that all necessary accuracy can be secured
by making the strips narrow and of considerable number. As an illus-
tration of the application of the method it will be interesting for purposes
82
CONSTRUCTION OF GRAPHICAL CHARTS
of comparison to take again the curve for the speed and horsepower of
the "Maine."
The same formula will be assumed as before, having three constants
to be determined and, therefore, demanding three equations. The curve
extends practically from 8 to 18 knots and these will be taken as the
limits within which to work.
For convenience we will divide this space into 10 vertical slices. A
greater number would lead to greater accuracy, but the work of calcula-
tion is laborious at the best and for illustrative purposes this will be
amply sufficient. The height of the curve at the middle of each of these
spaces is then measured and tabulated in the column headed y alongside of
the corresponding value of x. Next to the oc column is one of x2. Then
follow columns for x y (the first moment) and#2 y (the second moment).
>'
X
X2
xy
x*y
1,260
8-5
72.25
10,710
91,077
1, 660
9-5
90.25
15,770
149,810
2,150
10.5
no. 25
22,575
237,040
2,850
ii. 5 132-25
32,775
376,920
3,670
12.5 156-25
45,875
573,440
4,730
13-5
182. 25
63,855
862,040
6,050
. 14-5
210. 25
87,725
1,272,000
7-75°
15-5
240.25
120,130
1,862,000
10,000
16.5 272.25
165,000 2,722,500
13,050 17.5 306.25
228,370
3,996,500
53,170 = Area 792,785 = Ml 12,143,327 = M2
Since the width of the slices is i, the height of the middle ordinate
gives us its area at once, and the sum of the ordinates will be the area of
the space under the entire curve.
The sum of the values in the y column gives us 53,170 as the area.
The sum of the values of x y gives us 792,785 as the first moment, and
the sum of the values of x2 y gives us 12,143,327 as the second moment.
Both moments are reckoned about the Y-axis.
These quantities must be equated to the area and first and second
moments of the assumed theoretical curve, to get which we must use a
little integral calculus. The area of a small vertical slice of height y
and width d x is y d x and since,
y d x = a x d x + b .v2 d \
EMPIRICAL EQUATIONS 83
The integral of this expression is the area of the curve, or
fx
2axdx+bx2dx + cx3dx,
x^
where x1 and x2 are the limits between which the integration is to be
performed (here 8 and 18), or
0 b c
= ;(*i-*?) + v|i*i *2)f-<*i-*i,.
This, after substituting the values of xl and x2 given above, is placed
equal to 53,170.
If we multiply the differential area y d x by x we get its moment about
the Y-axis and its integral will be the first moment of the entire area, or
i J Xia' 3 * 4 * 5 *
This is placed equal to the measured first moment, or 792,785.
Multiplying the differential area next by x2, we get its second moment
about the Y-axis and its integral will be the second moment of the whole
area, or
a 4 4 b 5 5 c
4 * 5 l 6 ^
which must be placed equal to 12,143,327.
After substituting the limiting values of x^ and x2) which are 8 and 18,
we have the three equations
1300+1773 6+25, 218 £=53,170,
17730+25,2186+371,366^=792,785,
25,218 0+371,366 6+ 5,624,977 £=12,143,327.
In solving these equations, while it may not be necessary to run all
calculations out to the last figures, it will generally be desirable to carry
them out to five or six significant figures, since we often have to take the
difference between two numbers of nearly equal magnitude, in which
case the last figures may have an important influence on the result.
The slide rule is, therefore, absolutely useless for these calculations
except as a check against large errors. After the calculations are complete
it will generally be safe to throw away all except the first three or four
significant figures in order to simplify the formula for practical use.
The solution of the above equations gives
0 = 422.8, 6= -77.98, £=5.4115,
making the equation read
;y = 422.8#— 77. 98 ^+5.4115 XB,
or, more simply,
}' = 423*-78A*2+5.4i:v3.
84 CONSTRUCTION OF GRAPHICAL CHARTS
By the method of selected points we got
^ = 481 #-88.5 x2-}- 5.853 x3.
The agreement is as close as could be expected and is really closer
than the appearance of the equation might lead us to suppose.
This is shown in Fig. 43, where the equation just obtained is
plotted with the previous one by selected points. As the curves run
pretty close together, I have not attempted to draw the last one, but
have simply indicated the value of y for each even knot by a small
circle.
The difference between the two curves is quite small, the last one being
possibly slightly nearer the curve we are trying to compensate than the first.
So small a difference would hardly make it worth while, as a rule, to use
the more laborious method of moments if we knew that the results were
going to come out this way beforehand. We have no means of knowing
this, however, and there is generally an added feeling of safety in using it
on account of the larger number of points which are taken account of.
We should probably have obtained a closer approximation to the
original curve by using a larger number of ordinates in getting our
area and moments. Whether or not this would be desirable would
have to be determined after an inspection of the calculated curve to
see if its deviation from the original was within the desired limits of
accuracy.
This method is of very general application and may be used for any
equation of integrable form.
AN ALINEMENT CHART METHOD.
The next method I propose to discuss is one based on the alinement
chart described in Chapter II.
The method is due to Captain Batailler, of the French artillery service,
who describes the process in the Revue (TArtillerie of December, 1906.
Those who are interested are referred to it if they desire fuller information
than can be given in this brief outline. The process depends on the
alinement of a series of points taken from the data or from a curve which
is assumed to represent them.
It is not easy to explain the method in a simple manner, but I hope
that I shall at least be able to make the practical application clear. This
I think can be best done by working out a practical example, explaining
each step as it is taken.
The example chosen will be the data given by Prof. R. T. Stewart as
EMPIRICAL EQUATIONS 85
the results of his experiments on the collapsing pressure of bessemer-
steel tubes, and published in the Transactions of the American Society of
Mechanical Engineers for 1906. Professor Stewart showed his results in
chart form by laying off the values of the thickness of the tube divided by
the diameter, or //</, on the X-axis and the corresponding collapsing pres-
sures as ordinates. He found that a smooth curve drawn through these
points was difficult to represent fay any simple formula and, therefore,
took two bites at it, so to speak, and derived two formulas limited in their
application to different parts of the field. This is a very common and
useful expedient where the experimental curve is rebellious to representa-
tion by a simple formula.
Let us see what can be done toward getting the whole range of
results into one equation. To start with, the averages from the
tabulated results have been plotted in Fig. 46, and are indicated by
the small circles. In doing this I have interchanged the ordinates and
abscissas as they appear in Professor Stewart's chart since, with the
form of equation I wish to try, there might otherwise be some confusion
of nomenclature.
Then a smooth curve was passed through these points so as to
represent as nearly as possible a good general average. This is not
strictly necessary for the first process I am going to describe, but I
have done it in order to have something definite to work toward as a
measure of the success of the method, and also because a definite
curve is more suggestive of the type of equation than a number of scat-
tered points. Professor Stewart assumed that the greater part of the
curve is a straight line with a sharp bend toward the origin as the lower
values are approached.
He was probably justified in doing this, as the small number of obser-
vations among the higher values make the direction of the curve in that
region somewhat uncertain.
In my chart I have drawn the line with a reversal of curvature to per-
mit it to pass closer to the higher-value observations and thus get a some-
what closer agreement with the actual tests. At the lower end of the
curve, according to Professor Stewart, the collapsing pressure seems to
vary as (//d)3, or t/d is proportional to the cube root of the collapsing
pressure. Acting on this hint, we will use *$/x in our equation (x being
taken to represent the collapsing pressure) .
Now, the curve as I have drawn it reverses its direction of curvature
as it moves away from the origin. This effect could be brought about
by the use of some power of x in the equation in addition to the root.
86
CONSTRUCTION OF GRAPHICAL CHARTS
0.08
0.07
0.06
0.05
+»|-o
> 0.04
0.03
0.02
0.01
1000
4000
5000
6000
2000 3000
Collapsing Pressures
FIG. 46. — Chart showing relations between collapsing pressure of bessemer steel tubes and the
ratio of thickness to diameter. Equation of curve is — = 0.00274 ,y/.P +o.oooooooonP2.
EMPIRICAL EQUATIONS 87
The power will have but little influence on the shape of the curve for
the lower values of x where the predominant effect of the root is felt, but
as we get to the higher values the power will overbalance the effect of the
root and cause the reversal we wish. A high power is evidently not indi-
cated as the bend upward is comparatively small, hence (as it is easily
calculated) we will try the second.
Let our trial equation then* take the form
where y represents t/d and x the collapsing pressure.
For convenience in handling let us express these pressures in units of
1000 pounds.
The general form of equation used in the discussion of the alinement
diagram was
au+b v — Cj
where u and v represent measured distances on the U- and V-axes. If
u and v are kept constant while #, 6, and c vary, we get a series of points
lying along the straight line joining u and v. Hence, if this line can be
determined, its intersection with the U- and V-axes should fix the values
of u and v.
In our assumed formula A and B are constants, therefore let us
consider that they replace the quantities u and v in the alinement
equation. Now tyx, x2, and y may be given various values, hence
let us suppose that they take the place of a, 6, and c in the alinement
equation.
In order to get the position of the points lying on the line joining u
and v or, as we now call them, A and B, we make use of formulas (7) and
(8) developed in Chapter II. There, in order to locate our points, we
used rectangular coordinates of which the Y-axis was parallel to, and
midway between, the U- and V-axes and the X-axis was the line joining
the zero points on these same axes.
The formulas for the coordinates of the various points on the third
line of the diagram were then found to be
••* ^ 5 -*• f J
b + a b + a
d being the half distance between the U- and V-axes.
In these equations we replace a by ^/x, b by x2, and c by y.
Now, x and y are the coordinates either of the points representing
the observations or of the chosen points on the curve. We will in this
instance consider them as belonging to points on the curve.
88
CONSTRUCTION OF GRAPHICAL CHARTS
Below are tabulated the quantities we shall require, x and y being
read from the curve and x being given in 1000 pound units:
y
"
x*
v-
0.015
0.15
o. 0225
0-531
o. 0215
0.5
0.25
0.794
o. 0284
I . 0
I . 0
. 0
0.0392
2. 0
4-0
.26
o. 05
3-o
9.0
• 44
o. 0616
4-0
16. o
•59
0^074
5-o
25.0
•71
o . 0808
5-5
30-25
•77
Substituting in the equations for X and Y we have for x =0.15,
X=d
0.0225 —
0.0225 + 0.531
--= -0.0188
0.5535
v=
0.015
# = 0.5
# =1.0
#=2.0
# = 3.0
0-5535
X= —0.521 d
= 0.0271.
F = o.02o6o
Y = 0.01420
F = 0.00745
F = 0.00479
Y = 0.003 5°
Y = 0.002 7 7
F = 0.002 5 2
= 0.521 d
£
4
#=5.0 ^ = 0.872^
#=5.5 ^ = 0.889 d
These points are now plotted on the alinement chart, shown in Fig. 47.
In this chart the distance between the U- and V-axes has been made 20;
hence d=io, and the values calculated for X will be multiplied by this
before laying them off. The values of Y are laid off to any convenient
scale which will give clear readings. The measurements on the U- and
V-axes are to this scale.
The points with the exception of the first two seem to be in nearly
perfect alinement, which leads us to infer that the formula chosen is a
good one. If they fail to line up in a satisfactory manner it is useless to
go further, as this is an indication that the wrong equation is being used.
Of course, if the ordinates taken from the observations themselves had
been used instead of the points on the curve, we could not expect them
to fall so nearly on a straight line, but they should be grouped close enough
to one to make it evident that the axis of the group is straight and not
curved.
EMPIRICAL EQUATIONS
The line extended cuts the U-axis at 0.0274 and the V-axis at o.oon,
which are, therefore, the desired values of A and B. Before using them
in the equation, however, we shall have to modify them slightly to take
account of the change in size of the pressure unit which is really 1000
times that which we have been working with. Thus A will have to be
divided by ^/iooo, or 10, and B will have to be divided by iooo2, or
1,000,000, and our final formula becomes, after substituting t/d for y
and P for x,
t/d = 0.002j4 ^/P +0.000000001 1 P2.
The formula was now solved for a series of values of P, and the results
are shown by the double circles on the chart. The curve could not be
drawn in a satisfactory manner as it
lies very close to the original for a
considerable portion of its length,
and this closeness is a good indica-
tion of the success of the method.
Lest I be misunderstood, let me
say here that I make no pretense at
having obtained a better mathe-
matical expression for his results
than Professor Stewart. The scarcity
of data in the region of higher values
renders it extremely unsafe to say
whether the line there is straight or
curved. What interested me mainly
in this problem was the possibility of
expressing the entire series of results
by one formula. This, I believe, has
FIG. 47. — Alinement diagram for testing
points found in determining equation for
curve of Fig. 46.
been accomplished with a very fair degree of success and by the use of a
comparatively simple equation.
The method we have been investigating is generally quite sensitive,
and if the equation is not a good one for the purpose the points will depart
markedly from the straight line. Thus the possibility of forcing an un-
suitable equation into the appearance of an agreement with the original
curve, which may be done with most of the other methods, is largely
absent here. Often a portion of the points will lie along a straight line
while the others depart from it. In this case it indicates that in a limited
field the compensation is possible and may be good, a fact which it is
sometimes desirable to ascertain.
In the example just explained, we have assumed that not only the
90 CONSTRUCTION OF GRAPHICAL CHARTS
general type of the compensating equation was known, but also the
values of the exponents of x.
Some guide to the choice of the exponents is evidently much to be
desired, since if we rely upon guesswork we may consume a great deal of
valuable time in hunting for them, and may even then not hit upon the
best values.
ANOTHER ILLUSTRATION OF THE ALINEMENT-CHART METHOD.
The Batailler method just described may be extended to do this for
many types of equation in a manner which is comparatively simple in
operation, though a little difficult to explain. The additional one or two
constants which may thus be determined are not limited to exponents,
but may also be coefficients.
As before, I shall make the explanation while working out a problem.
The example chosen will be taken from Rateau's "Flow of Steam Through
Nozzles," and is the diagram shown in Plate IV of that book for Hirn's
experiments on the flow of air through thin plate orifices. I have redrawn
the curve for this in Fig. 48. In it the abscissas represent the ratio of
back pressure p to initial pressure P, and the ordinates the ratio of
observed discharge to the maximum discharge. The abscissas on the
X-axis are numbered from i to 0.4, but I have reversed the numbering
in order to avoid confusion and will afterward insert the quantities as
they appear in the original diagram. My numbers will then be o, o.i,
0.2, 0.3, etc., instead of i, 0.9, 0.8, 0.7, etc.
The problem will be to see how nearly we can compensate this curve
by an equation of the type
y = Axp + Bxq
in which ^4, B, p, and q are all unknown and are to be evaluated.
The first step is to differentiate the curve and obtain its first and
second derivatives, y' and y". Then A and B are eliminated from these
equations, and p and q determined by a process analogous to that last
described.
The equation and its first and second derivatives are :
y = Axp + Bxq
yf == Apxp~l + Bqxq~l
y" = Ap(p - i)xp~2 + Bq(q-i)xq~2
To eliminate A and B from these equations and put them in the neces-
sary form for use, I am constrained to use determinants. Any other
method lands us in such a snarl of equations as to be very objectionable,
5 0.5
g
I
ss
2
' (H4
— • — _^
i tang nti to y
Ratio of Back I'ressnre p to Initial Pressure P.
FIG. 48. — Chart showing relation of ratio of back pressure to initial pressure, and ratio
of observed to maximum discharge of air through thin plate orifices.
1
CONSTRUCTION OF GRAPHICAL CHARTS
while by determinants we can reach the desired results by a com-
paratively simple process. ,A11 forms of equation will not demand this
treatment, and each case must be looked upon as more or less of a
special problem.
The three equations may be written in the determinant form as
follows:
Ax* Bxq
Apxp~1 Bqxq~l
Ap(p-i)xp~2 Bq(q-i)xq~2
Divide the second column by Axp and the third column by Bxq and
we have:
=o
px
qx
y
= o
Then multiply the second row by x and the third by x2, giving:
Xy'
= o
Taking the three columns as the coefficients of three equations of
the alinement type, we have:
y=xy'u + x2y"v
i=pu+p(p-i)v
i = qu+q(q—i)v
The first equation is affected only by x or its functions yf y', and yff,
the second by p only, and the third by q only.
Three supports for an alinement diagram may be constructed from
these three equations on the same U- and V-axes, giving us three curves,
one for x, one for p, and one for q.
If we join up some point on the ^>-line with another on q, the con-
-necting line will cut the #-line in what must be looked upon as a corre-
sponding value. But according to the original assumption p and q were
constants in the equation and remain so whatever the value of x. If
this is true, the desired values of p and q must be so located that the line
joining them will cut every value of x on the x-\me. This can only be
possible by having the support for x a straight line joining these constant
values of p and q.
Our next step is to plot the support for x from the equation:
y=xy'u+x2y"v
EMPIRICAL EQUATIONS 93
The coordinates for the points on this line will be found from the
equations used in the previous example, which will read here
v=
x
Here oc and y are*, of course, the coordinates of any points on the obser-
vation curve; y' and y" must, however, be determined from this primary
curve. As is well known, the tangent to a curve at any point corresponds
to the first derivative. If we get the tangents at a sufficient number of
points their values may be plotted into a second curve of which the ordi-
nates are y'. Similarly by drawing tangents to this second curve, we get
the values of the quantity y".
These values of y' and y" are then to be substituted in the equations
for X and Y.
The chief and only difficulty connected with this process is in drawing
the tangents to the curves. The "curve of error" is sometimes recom-
mended for this purpose but is, in my opinion, too cumbersome for prac-
tical use where any considerable number of points is to be operated on.
My own preference is for taking two ordinates at equal distances on
either side of the point at which the tangent is desired and draw the chord
of the curve between them. If the curve is flat these side ordinates may
be considerably separated, but if not they must be closer. The slope
of the chord will be nearly equal to that of the tangent. The greatest care
must be exercised in this part of the process, but if this is done the method
will yield results of a very satisfactory character.
To get the numerical value of the tangent, or y', a parallel to the chord
is drawn from a point on the base line at unit distance to the left of the
foot of the ordinate we are operating on and its intersection with the
ordinate gives the desired value of y' when measured by the same scale
as was used for the ordinates of the original curve.
If this unit distance is inconveniently small or large we may increase
or diminish it to a more suitable value but must remember that the read-
ing on the ordinates must then be changed to correspond.
In Fig. 48 the primary curve and its first and second derivatives are
shown, the last being represented only by points and the actual curve
omitted as unnecessary . It was convenient in laying out these secondary
curves to use a base unit on the X-axis equal to i /io; hence the readings
on the y' curve must be multiplied by io to get their true value and those
of the /' curve by 100 (since we have used the i/io base unit twice).
94
CONSTRUCTION OF GRAPHICAL CHARTS
In the accompanying table are given the quantities necessary for our
calculations, the values of y, yf and y" being read directly from the curves;
/', it will be noted, has the minus sign prefixed throughout, as its points
all lie below the X-axis.
X
y
y'
/'
X*
x?y" xy'
o. 05
o. 283
3.00
-34.5
o. 0025
— o . 0863
0.150
O. 10
0.402
1.94
-ii. 8
O.OIOO
— 0.1180 j 0.194
0-15
0.488
1.50
- 7-1
0.0225 — 0.1598 °-225
0. 20
0.556
1.19
- 5-o
O.O400 — 0.2OOO 0.238
0.25
0.613
I . 00
- 3-7
0.0625 — 0.2313 0.250
0.30
0.658 0.83
- 2.85
o. 0900 — o. 2565
0.249
o-35
0.695 0.70
- 2.35
o. 1225 — o. 2879
0.245
o. 40
0.728 o . 605
— 2. O
0.1600 — 0.3200
o. 242
o.45
0.757 0-50
- i-9
o. 2025 — 0.3848
o. 225
o. 50
o. 780
0.42
- i-7
0.2500
—0.4250
0. 210
0-55
o . 800 0.33
— 1.6
0-3025
— o . 4840
o. 1815
Now, substituting from the first row in the equations given above for
X and F, we have
X
—0.0863 + 0.15
0.283
0.0637
and for the remaining values of x,
0.0637
= 4-45
* = o.i5
A =
- 5-9x^
# = O.20
X =
-n-55^
# = 0.25
Y"
-25-75^
# = 0.30
X =
67.5^
# = 0.35
X =
12 -45^
x = o . 40
X =
j .2id
#=0.45
X =
3.8irf
#=0.50
X =
2.96i/
*=0.55
X =
2.19^/
Y= 5.3
Y= 7-5
7=14.62
7 = 32.8
F=-87.8
F= — 16.2
F= - 9.35
F= - 4.74
F= - 3.63
F= — 2.64
Slide-rule calculations are usually of sufficient accuracy for this pur-
pose, and after a start is once made they may be run off quite rapidly.
* These values must next be plotted on the alinement diagram, as shown
in Fig. 49. Since we make no use of the U- and V-axes here, we will omit
them and indicate only the X- and Y-axes from which the above quan-
EMPIRICAL EQUATIONS
95
titles are laid off. The half distance d between the U- and V-axes appears
in the equation for X, but since they are not shown, its only function will
be that of a scale unit, which we may make any size we please. Here it
32.8
-16.2
FIG. 49. — Alinement diagram for testing points found in determining an equation
for curve of Fig. 48.
was made of such a size that all of the points given above could be plotted
within the limits of Fig. 49, except the one corresponding to x = 0.30.
We do not need this point, however, as there are enough other points
without it to determine the alinement.
96 CONSTRUCTION OF GRAPHICAL CHARTS
An examination of Fig. 49 shows that, although the plotted points do
not lie exactly on any straight line, they are in very close agreement with
the one shown. Exact agreement is never expected, of course, and there
will generally be more divergence than shown here. The alinement of
the points indicates that the type of equation chosen is good for the pur-
pose. If it had not been the points would have scattered badly, or would
have had a curve as their locus.
Now, we must plot the curves for p and q for such values of these
quantities as we suppose to lie near the line just drawn.
The p and q equations for this type of formula are identical, hence
they will be represented by only one curve instead of by two, as is the
case with other types where p and q are not symmetrically disposed. In
order to get a value each for p and q, we must, therefore, have two inter-
sections between the x support and the suppport for p q.
The alinement equation for p as given previously is
i=p u + p (p — i) v.
The coordinates for the various points on the p support will then be
p
T T
y __
These equations have been solved for p = 0.5, 0.6, 0.7, 0.8, 0.9, i, 1.5,
and 2, and the points thus found plotted in Fig. 49.
A larger scale for the drawing would have made things clearer, but
it can be seen that the ^-support cuts this curve at two points, one of them
exactly at 1.5 and the other at a point between 0.5 and 0.6, which I have
called 0.53. It would have required but a slight shift of the ^-support
to have made the intersections at the points 0.5 and 2. If convenience
in use were an important factor these latter values could probably be
employed with but little less accuracy. However, we will use the original
more exact figures and call ^1.5 and q 0.53. It is immaterial at this stage
which quantity is assigned to which letter.
Having found p and q, our equation now reads
and our next step is to proceed exactly as we did in the previous example
to find A and B, using the equations
^0.53 V1.6 «.
Y_ ix ~x V= —
#o-M+*'-6' "V-"**1'1"
in order to locate the points on the test line.
EMPIRICAL EQUATIONS
97
As there is nothing novel in the process the details will be omitted
and reference merely made to Fig. 50, where the points have been plotted
and where they are seen with but one exception to lie almost exactly upon
a straight line, again indicating the adaptability of the formula to the por-
tion of the curve we have operated on. This line extended to the U-
and V-axes is found to cut them •
at the points —0.59 on U and
+ 1.425 on V.
Then A is —0.59 and B, 1.425,
and our equation reads
y= -0.59 xlf5+ 1.425 x°'53.
Reverting now to the first dia-
gram where, as was stated, the
nbmbering on the X-axis was
altered, we see that we can get
the original numbers by putting
!.**&
I.?*
1.15
J.03
l.OJS
(l.'A
o.w»
0.938
p IP being the ratio of the back
to the initial pressure and the
final equation may then be written
FIG. 50. — Alinement diagram for testing
points found in determining an equation for
curve of Fig. 48.
A series of points has been calculated from this formula and plotted
on the chart in Fig. 48, as indicated by the small circles, and the agree-
ment will be seen to be quite good.
The lengthy description which I have given of the Batailler method
has, I know, a somewhat formidable sound, but in practical operation "It
is," as Bill Nye observed of Wagner's music, "much better than it sounds."
The only operation which presents much difficulty is the graphical
differentiation which must be done with great care, or the results will be
poor. Otherwise the work is all of a simple character, and may be carried
out very expeditiously as compared with some of the other processes.
While other types of equation are developed on the same general lines as
the one explained, there are differences of detail which have to be looked
out for, and which could not even be touched upon here without lengthen-
ing this discussion beyond reasonable limits. The equation we have
worked on is, perhaps, the one most commonly met with in practice and
shows as well as any other the very decided advantage of this process for
certain classes of work.
CHAPTER VII.
STEREOGRAPHIC CHARTS AND SOLID MODELS.
THREE DIMENSIONAL CHARTS.
i
Two dimensional charts for the representation of mathematical equa-
tions or experimental data are in very common use nowadays and are
everywhere recognized as valuable devices for giving a clear conception
of the manner in which the variables are related.
Their application is generally restricted, however, to cases where there
is but one variable and its function, if the variation to be shown is continu-
ous. Nevertheless cases often arise in which there are two variables and
a function to be represented and where it is desirable to show a continuous
variation for all three.
A simple and logical extension of the two-dimensional chart, in which
the variation is represented by a plane curve, leads us to the idea of a solid,
three-dimensional chart in which the variation is shown by a surface.
It has received some attention at the hands of a number of writers on
engineering matters and graphics, but for some reason, probably the labor
and expense involved in its construction, its actual use has been rather
limited. Where it has been used it has in some instances been fruitful
in good results and has thrown much light upon obscure phenomena. In
this connection its chief value has probably come from the facility with
which we are able to detect maximum and minimum conditions and rates
of change among variables whose relationship is complex or unknown.
Often we must deal with conditions where no known equations will
connect our experimental results and where a mere tabulation of figures
will not yield the desired information without much tedious study. The
well recognized superiority of any graphical representation over an equa-
tion or table in conveying a clear impression to the mind of the way in
which a set of variables is related will often in itself be a sufficient justifica-
tion for the use of this type of chart.
Between the solid model and the plane chart there is a borderland
occupied by types which do not truly belong to either and which are
really plane projections of solid models. They may be orthographic,
isometric, perspective or, generally, axonometric, according to the taste
of the maker or the exigencies of the subject.
The orthographic projection here referred to is the topographic map
projection in which the relief of the model is indicated by a series of
STEREOGRAPHIC CHARTS AND SOLID MODELS
99
contour lines. Each line passes through a series of points at the same
elevation and is numbered to show this elevation. Only a slight effort of
the imagination is required to give a very good idea of the undulating sur-
face which they represent. The familiar weather map is a good
example of such a chart. Here points of equal barometric pressure are
connected by curved lines called isobars. Charts of this description have
been, much used to record tidal
phenomena, magnetic observa-
tions, etc., and also in the pre-
sentation of vital and financial
statistics.
Axonometric projections will
usually be found superior to the
topographic in bringing out clearly
the shape of the surface and are
not at all difficult to construct.
The special case where the pro-
jection is isometric was very fully
dealt with by Prof. Guido Marx in
the American Machinist, Volume
31, Part 2, page 701.
Any of the other well-known methods of rectangular axonometry or of
perspective may, of course, be applied to these figures. As these methods
are generally understood or may be found described in almost any good
book on projection or descriptive geometry, no attempt will be made
to discuss their principles here.
The accompanying table may, however, be convenient for reference
as indicating the proper choice of angles for the axes to conform to the
scale units most commonly used.
TABLE OF RATIO OF UNIT LENGTHS ON THE AXES AND ANGLES OF
THE AXES FOR AXONOMETRIC PROJECTIONS
FIG. 51. — Axes and angles for axonometric
projection.
Ratio of unit lengths
Tan. (j)
Tan
.0
ux : uy : uz
\ Isometric J
\
(f) = 0 = 60°
2 i : 2
8 :
8 :
7
3
i : 3
18 :
18 :
J7
4
i : 4
32 :
32 :
51
5
4 : 6
5 :
3 :
i
9 ^ : 10 ii :
2? :
8
100 CONSTRUCTION OF GRAPHICAL CHARTS
The letters in the table refer to the same symbols in Fig. 51 and the
scale values, designated by u, represent the ratio of sizes for a unit length
on each of the axes.
The question of scales in the projection of such figures as we are now
considering is, however, of relatively little importance since the units used
on the different axes have generally no relation which makes any special
scale ratio necessary. The angles for the axes, on the other hand, should
be so chosen as to agree, approximately at least, with those given in the
table, otherwise the figure may have an awkward and unnatural ap-
pearance.
AXONOMETRIC CHARTS.
But one simple illustration will be given for this type of chart which
will, howeVer, show some interesting and rather unusual features. It is
taken from the Zeitschrift des Vereines Deutscher Ingenieure for December
27, 1902, and occurs in an article by O. Lasche on the friction of journals
with high surface velocities.
Fig. 52 was redrawn from a chart given in this paper with a few
unimportant modifications to render it better adapted to purposes of
illustration. The chart was constructed from data obtained from experi-
ments on a nickel-steel journal running in a white-metal bearing and is
intended to show the relation between the temperature of the bearing in
degrees Centigrade, the surface velocity in meters per second and the
heat generated per square centimeter of effective projected area, expressed
in heat units, and also in meter-kilograms per second.
The experiments were made at a specific pressure of 6.5 kilograms per
square centimeter, but since, with the lubrication used, the product of the
specific pressure and the coefficient of friction was sensibly constant over
a considerable range, the results are said to be applicable to any specific
pressure from 3 to 1 5 kilograms per square centimeter.
In laying out a chart of this description the three coordinate axes and
their planes are first drawn and the former properly graduated between
the limits set by the experiments. From the graduations on the ground
plane axes perpendiculars, lying in the ground plane, are drawn, thus
giving a checkered surface on which points may be located as is done with
ordinary section paper. At the points thus found perpendiculars are
erected to the ground plane, their height being so taken as to represent
the value of the third variable. The tops of these lines are now connected
by suitable curves, which must lie in the surface we are seeking and which
are assumed to represent it.
STEREOGRAPHIC CHARTS AND SOLID MODELS
101
In the chart under discussion five different curves were drawn parallel
LJ the temperature-heat 'plane, and then, to bind them together and
render the shape of the surface more apparent, three more curves were
drawn at right angles to the first and parallel to the velocity-heat plane.
Taking one of these latter curves, that corresponding to 50 degrees, we see
that at this constant temperature the heat generated by friction mcreases
with increasing velocity, not exactly in direct ratio with it, however, since
FIG. 52. — Axonometric chart showing relation between journal bearing temperature,
surface velocity and heat generated.
the coefficient of friction does not remain quite constant as the velocity
changes. Keeping the velocity constant and varying the temperature,
we see that the heat generated by friction decreases as the temperature
rises, rapidly at first and then more slowly.
It is evident from the chart that with the journal in question the heat
produced by friction will be greatest when it is starting up and the tem-
perature is low. Also that at this temperature the radiation to the
surrounding atmosphere will be small on account of the small temperature
difference. The heat produced therefore goes to warm the bearing, but
as its temperature rises the heat generated becomes less and the radiation
102
CONSTRUCTION OF GRAPHICAL CHARTS
greater until we reach a point where the radiation just balances the heat
production and the temperature remains stationary.
The question naturally arises as to whether it is possible to tell where
this point will be. If the necessary experimental data are at hand it may
be done on the chart. Suppose we have this data and from it construct a
second chart on the same heat, temperature and velocity axes as before.
See Fig. 53. It shows the capacity for heat radiation per square centi-
meter of effective projected area for the bearing we are considering and is
FIG. 53. — Axonometric chart showing capacity for heat radiation per unit of
effective projected area of journal bearing.
constructed for a room temperature of 20 degrees Centigrade. When the
bearing has this temperature its radiation is, of course, zero. The radiation
is independent of the velocity of the journal and this is indicated by the
fact that the surface is a ruled one composed of straight lines parallel to
the velocity axis. Increasing bearing temperature means,- of course,
increasing radiation.
Next suppose these two charts to be combined as in Fig. 54. It is
apparent that the two surfaces will intersect along some line as h c j, the
location of which is easily found by the rules governing this form of pro-
jection.
STEREOGRAPHIC CHARTS AND SOLID MODELS
103
Any point on this line will correspond to some temperature and veloc-
ity at which the radiation just equals the heat production, the necessary
condition for constant temperature. This line projected to the ground
plane gives us the line i d j. Any point in the ground plane, projected
from the temperature and velocity axes, which falls in front of the line
will indicate that under these conditions the radiation is greater than the
FIG. 54. — Chart combining charts of Figs. 52 and 53.
heat generation or that natural cooling will be effective to keep the bearing
below the chosen maximum temperature. Points which fall beyond this
line correspond to conditions where artificial cooling must be resorted to.
Suppose, for instance, we take some point, such as c on the line h c j.
Project it to the ground and we find that it falls at the intersection of ordi-
nates from 80 degrees on the temperature axis and 5 meters per second
1-04 CONSTRUCTION OF GRAPHICAL CHARTS
on the velocity axis. Under these conditions the temperature will be
steady.
Next suppose that we arbitrarily fix the upper limit for temperature
at 80 degrees, and that we have a velocity of lo-meters per second.
Entering the radiation diagram on the 8o-degree line at a we run up
till we reach its surface at b. Then, following the surface along the line
be, we find its intersection with the lo-meter plane of the friction diagram
at e. Through this point a perpendicular is drawn to the ground. The
length /# on this perpendicular measures the heat generated by friction,
efis the amount carried off by radiation, while eg represents the remainder
which must be artificially removed by circulating a current of water, oil, or
air around the bearing.
The writer of the article from which I take this illustration goes on to
show how, after measuring eg in heat units, a very simple calculation will
give the amount of cooling fluid.
It will be apparent from the foregoing description that the axono-
metric projection has some advantages over its solid prototype from the
facility with which we can project through the figure in case of need.
Special attention should also be directed to the use which has been made
of the line of intersection of the two surfaces. It is a rather novel feature
and one which should prove valuable in many engineering problems.
THE SOLID MODEL.
Next let us consider the true solid model. It has received attention
at the hands of several eminent writers, among them the late R. H. Thur-
ston. He published a number of articles explaining its uses and advan-
tages, among which articles may be cited one on glyptic models in the
Transactions, American Society of Mechanical Engineers, for 1898. He
appears to have been much impressed by the possibilities it offered for the
solution of a certain class of problems and he illustrates its application
by a number of examples.
In spite of his optimistic views as to its value, the solid model has never
seemed to "take" well; at least there are relatively few recorded instances
of its use. This may be partly due, as was observed before, to the labor
involved in its construction, but possibly, also, to a lack of sufficient
exploitation.
These models may be made in various ways. Wood is a suitable
material where the surface to be produced is sufficiently regular, but this
is not often the case. Ruled surfaces may be produced by strings
STEREOGRAPHIC CHARTS AND SOLID MODELS 105
stretched on suitable frames, but the material most generally used is
plaster of Paris. After the first model is made, replicas may, of course,
be cast in any suitable metal or material. Cardboard, as will be shown
later, is a cheap and convenient substitute for some of the above-named
materials.
In making the plaster-of-Paris model, we first stretch a sheet of section
paper on a board and lay off on It the points corresponding to two of the
variables in the usual manner. At these points we next insert vertical
FIG. 55. — Solid model showing relation between heat units per hour per brake horse-
power, compression pressure and volume of gas mixture for a gas engine.
wires which are cut off at heights corresponding to the third variable. A
box is then formed around the whole and wet plaster of Paris is poured
into it until all the wires are covered. After it has set, the upper surface
is carefully cut and smoothed away until the tops of the wires are exposed
and the resulting surface is taken as the graphical representation of the
law of connecting the variables.
A comparatively recent example of such a model is found in an article
on the mixture ratio for gas engines in the Zeitschrift des Vereines Deut-
scher Ingenieure for September 14, 1907.
This model is represented in Fig. 55. It is based on data obtained
from a gas engine running at four horsepower on producer gas, 'and is
intended to show the relation between the heat units per hour per brake
106 CONSTRUCTION OF GRAPHICAL CHARTS
horsepower, the compression pressure in atmospheres and the volume of
mixed gas and air per 1000 heat units of lower heating value.
The front horizontal axis is graduated to represent the cubic meters
of mixture per 1000 heat units and extends from about 1.7 cubic meters
to 3. Perpendicular to this axis, and also horizontal, is the axis for com-
pression pressures graduated from back to front between 4 and 13
atmospheres.
The vertical axis is used for the heat units required per brake horse-
power-hour, the graduations beginning at 2500 at the ground plane and
running up to 5000.
The hollowed surface in the middle of the model covers the range
within which the experiments were conducted, and the cut-off portions
at the sides of the hollow have no meaning.
Without in any way attempting to discuss the conditions under which
the experimental results were obtained, we will take the model as it
stands, and see what conclusions may be reached from a simple
inspection of it.
The bottom of the valley indicates the lowest heat consumption per
horsepower-hour in any given locality.
The intersection of the valley with the back vertical plane is a curve
somewhat resembling a parabola with steeply rising sides. As we come
toward the front the curves cut by parallel vertical planes flatten out and
the vertex of the curve becomes lower, indicating a smaller heat consump-
tion as the compression increases. At the back of the model the lowest
part of the curve is tangent to a horizontal line at about 4100, while in
front it touches 3100.
It will also be noted that the slope of the bottom of the valley is steepest
in the rear and is nearly horizontal in front, indicating a more rapid gain
in heat economy from increased compression when the original compres-
sion is low than when high.
The bottom of the valley shows a tendency to drift to the left as we
come forward, indicating that with increased compression the best econ-
omy was obtained by increasing the dilution of the mixture. The flatten-
ing out of the front part of the valley indicates that as compression in-
creases the necessity for an exact mixture ratio for good economy becomes
progressively less important.
These points are all interesting, and while they might have been dis-
covered from an inspection of a series of curves or of the tabulated data,
it is clear that the model has greatly simplified the process of deduction,
and has thus justified its construction.
STKREOGRAPHIC CHARTS AND SOLID MODELS
107
CARDBOARD SUBSTITUTE FOR THE SOLID MODEL.
Reference has been made above to the cardboard model as a cheap
substitute for the solid type. The next illustration will be an example
showing its construction.
If we assume one of three variables to have different constant values,
we get a series of plane curves connecting the other two. Then, by doing
the same with one <5f the other variables, we get a second series of curves
FIG. 56. — Cardboard substitute for a solid model.
for planes at right angles to the first. Each of these curves is cut from
a piece of cardboard and slit half way up or down the lines of intersection
with the cards at right angles to it. They are then fitted together some-
thing on the principle of an egg box, and the result will be a series of plane
curves, properly spaced with reference to each other, all of which lie in
the surface we are trying to represent. It is evidently closely related to
the axonometric projection previously described.
Such a model is shown in Fig. 56. It was constructed from the curves
given in a paper in the Transactions, American Society of Mechanical
Engineers, for 1904, by E. S. Farwell, entitled "Tests of a Direct Con-
nected Eight Foot Fan and Engine." These curves were chosen chiefly
on account of the irregular hilly character of the surface to which they
belong as affording a good test of the method. They occur in Fig. 39, of
the article referred to, and are supposed to show the relation between the
108 CONSTRUCTION OF GRAPHICAL CHARTS
efficiency of the fan and the area of the outlet opening for various speeds,
the area of opening being designated as a percentage of the product of the
fan diameter by the width of periphery.
Eight different curves are given for eight different speeds, which
advance by steps of 25 from 50 revolutions per minute to 225 revolutions
per minute. The curves shown in the figure had the efficiencies plotted
as ordinates, and the outlet-opening percentages as abscissas, and were
used as they stood for one set of cards. Then taking the intersection of
these curves with one of the perpendiculars to the outlet axis we get a
series of lengths which we use as equally spaced ordinates for a curve at
right angles to the plane of the original drawing, the ordinates again
representing efficiencies while the abscissas this time are velocities. As
many similar curves as were deemed necessary were taken from the other
perpendiculars. All were cut out and fitted together, forming the model
shown in the photograph. Such a model may be applied to many, if not
most, of the purposes for which the solid type is used and has a decided
advantage in simplicity of construction.
THE TRI-AXIAL MODEL.
Before leaving this subject a brief reference must be made to an in-
genious form of solid chart described by Professor Thurston in several of
his articles. It is called the tri-axial model. By its use it is possible to
take into account four different variables
instead of three as was previously the case.
It is a necessary condition, however, that
for each set of corresponding variables three
of them should add up to a constant value,
generally 100 per cent. The fourth is unre-
stricted. These models have been very use-
ful in representing the properties of ternary
alloys, furnace slags, etc. If we have an
.Fl(f- 57-7 D?agram illustrating equilateral triangle as shown in Fig. 57, and
principle of tn-axial solid model.
from any point, O, within it we drop per-
pendiculars to the three sides, geometry tells us that the sum of these
perpendiculars is constant wherever the point may be located.
Therefore, if we wish to study the alloys composed of, say, copper,
tin, and zinc, with reference to any property such as strength, ductility,
hardness, or melting point, a large number of experiments are made with
specimens of varying composition and the value of the quality we are
studying tabulated with the composition of the alloy.
STEREOGRAPHIC CHARTS AND SOLID MODELS
This composition is expressed for each constituent as a percentage,
and the three percentages necessarily add up to 100.
Laying out a triangle whose altitude to some scale is TOO, we designate
one side as copper, another as tin, and the third as zinc. Parallels are
then drawn to the sides, distant from them by amounts corresponding to
the percentage of each metal in
the specimen. The, scale in -which
these distances are measured is
the same as that which was used
in laying off the altitude of the
triangle. These three parallels
must meet in a point which is
taken to represent the alloy in
question. Perpendicular to the
ground plane at this point we in-
sert a wire whose length repre-
sents the value of the quality we
are studying. When all the wires
are fixed the whole is covered with
plaster of Paris, as explained be-
fore, Which is then pared down to FlG- 58.-Professpr Thurston's solid tri-axial
model for copper alloys.
the tops of the wires.
The resulting model is shown in Fig. 58, and from it Thurston found
that the strongest alloy had a composition of Cu = 55 per cent., Zn = 43
per cent., and Sn = 2 per cent.
Models of this description are evidently of especial value in the study
of metallurgical problems and are by no means uncommon, particularly
in that field of work.
Often, however, instead of the solid model, a topographical chart of
it with the necessary contour lines is plotted, which answers many pur-
poses almost equally well and commends itself for use in a great many
cases.
MINERAL TECHNOLOGY LIBRARY
UNIVERSITY OF CALIFORNIA LIBRARY
BERKELEY
Return to desk from which borrowed.
This book is DUE on the last date stamped below.
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