Mathematics Of Modern Engineering Vol III

TEXT FLY WITHIN
THE BOOK ONLY

Text Cross book

CO >; DO

8]<OU 166381 g

^ DO

OSMANIA UMVERSrTY LIBRARY

Author

A'W'fvk , '

Title

This book should t>c returned on '>r before the date last marked below.

MATHEMATICS OF MODERN ENGINEERING

GENERAL ELECTRIC SERIES

FIELDS AND WAVES IN MODERN RADIO

By Simon Ramo and John R. Whinnery. 502
pagi-s. 214 figures. 5J by S?J. Cloth.

MATERIALS AND PROCESSES

Edited by J. F. Young. 628 pages. 410 figures.
5^2 by 8?a. Cloth.

MODERN TURBINES

By L. R. Newman, A. Keller, J. M. Lyom,
L. B. Wales. KcliU-fl by L. R. Neuman. ' 175
pages. 93 figures. 5JSby8?a. Cloth.

CIRCUIT ANALYSIS OF A-C POWER SYSTEMS

Volume I by Edith Clarke. 540 pages. 167 fig-
ures. SJSby 8%. Cloth.

ELECTRIC MOTORS IN INDUSTRY

By D. R. Shoults anil C. J. Rife. Edited by
T. C. Johnson. 389 pages. 2 19 figures. 6 by 9.
Cloth.

A SHORT COURSE IN TENSOR ANALYSIS FOR
ELECTRICAL ENGINEERS

By Gabriel Kron. 250 pages. 146 figures. 6 by
9. Cloth.

TKNSOR ANALYSIS OF NETWORKS

By Gabriel Kron. 635 pages. 330 figures. 6 by
9. Cloth.

TRANSFORMER ENGINEERING

By L. F. Blwnc, G. Caimlli, A. ftoyajian, and
V. M. Montsinger. Eiliti-d by L. F. Blumc. 496
pages. 348 fl^uieb. 6 by 9. Cloth.

MATHEMATICS OF MODERN ENGINEERING

Volume I by Robert E. Doherly and Ernest G.
Keller. 314 pages. 83 figures. 6 by 9. Cloth
Volume II by Ernest G. Keller. 309 pagrs. 91
figures. 6 by 9. Cloth.

TRAVELING WAVES ON TRANSMISSION SYSTEMS

By L. V. Bewley. 334 pages. 133 figures. 6 by
9. Cloth.

VIBRATION PREVENTION IN ENGINEERING

By Arthur L. KimbalL 145 pages. 82 figures.
6 by 9. Cloth.

PUBLISHED BY JOHN WILEY & SONS, INC.

MATHEMATICS OF
MODERN ENGINEERING

VOLUME II
(Mathematical Engineering)

BY
ERNEST G. KELLER

M 5., I'k I) (I'ntvemty ol
Consulting Mathematician, O/r/m-M r/g/// Corporation

One of a Series written in the interest

of the General Electric Advanced

Engineering Program

NEW YORK

JOHN WILEY & SONS, INC.

LONDON: CHAPMAN & HALL, LIMITED

IN THE REPRINTING OP THIS BOOK, THE RECOMMEN-
DATIONS OF THE WAR PRODUCTION BOARD HAVE
BEEN OBSERVED FOR THE CONSERVATION OF PAPER
AND OTHER IMPORTANT WAR MATERIALS. THE
CONTENT REMAINS COMPLETE AND UNABRIDGED.

COPYRIGHT, 1042
By THE GENERAL ELECTRIC COMPANY

All Rights Reserved
Tim book or any part thereof must not
be reproduced in any form without
the written pcrnnmon of the publu>htr.

THIRD PRINTING, NOVEMBER, 1945

PRINTED IN THE UNITED STATES OF AMERICA

PREFACE

The two purposes of this hook are implied in its title and subtitle.

Its first purpose (along with that of the first volume and a third,
which is in preparation) is to present those aspects of mathematics
which the experience of a large manufacturing organization, in dealing
with electrical and mechanical investigations, has found to be of most
value to engineers. The mathematical material treated is not the
selection of one or two individuals of what they consider mathe-
matically useful in the engineering work of a large engineering organi-
zation, but the composite opinion based on extensive experience of
engineers and physicists who apply themselves to the abstract prin-
ciples of engineering in daily engineering research and practice.

The second purpose, which is even more important, is to present
an introduction to the methods of mathematical engineering by the
analysis of discrete physical systems. There h.is developed during
the last two decades a phase of engineering which may properly be
called mathematical engineering. Its analogue in phxsics is mathe-
matical physics. Many similarities exist in the two fields. The sub-
ject materials of mathematical physics and mathematical engineering
are respectively physics and physics and engineering; the tool of both
fields is mathematics, regardless of how simple or advanced the mathe-
matics may be. Mathematical physics is not restricted to one branch
of physics. Neither is mathematical engineering confined to one branch
of engineering, because the fundamental method of analysis in mathe-
matical engineering remains the same regardless of which branch of
engineering is practiced. The similarities of the two subjects are
pointed out in the introduction.

The material and methods of this book have evolved during the
last decade out of the research engineering work and the Advanced
Course in Engineering * of the General Electric Company. Although
prepared for that course, the book should be just as useful in graduate
engineering work at universities since the material has been tempered
by use in the instruction of students not only in the course of the

* A. R. Stevenson, Jr., and Alan Howard, "An Advanced Course in Engineering,"
Trans. A.I.E.K., March, 1935. A. K. Stevenson, Jr., and Simon Ramo, "A New
Postgraduate Course in Industry in High- Frequency Engineering," Electrical Engi-
neering, July, 1940.

vii

viii PREFACE

General Electric Company but also in the courses for graduate students
in more than one university. It is thus a joint product of the engi-
neering office and the university classroom.

I am indebted to all whose publications have been used or cited,
but, in particular, to nine friends it is a pleasure to express my
gratitude.

Without the encouragement of Dr. A. R. Stevenson, Jr., Staff
Assistant to the Vice President of Engineering of the General Electric
Company, Volumes II and III would not have been written. As one of
the two originators of the Advanced Course he has continually guided,
directly and indirectly, the preparation of the manuscript in order
that it mi^ht be adapted to the needs and methods of the course.

I am grateful to Dr. Stevenson, to Dr. Saul Dushman, Assistant
Director of the Research Laboratory, and to Mr. P. L. Alger, Staff
Assistant to the Vice President of Engineering, of the General Electric
Company, for their generous aid in numerous projects of which this
text is one.

In Chapter II much use has been made of certain papers of Mr.
Gabriel K-on, consulting engineer of the company. His generosity is
deeply appreciated.

Valuable suggestions regarding both form and content (for Vol-
umes I. II, and III) have been made by Messrs. Alan Howard, B. R.
Prentice, and T. C. Johnson, who, during the preparation of the manu-
script, have been in succession in charge of the Advanced Course in
Engineering of the General Electric Company.

I thank Mr. A. B. Chafetx for his aid in checking numerical calcu-
lations and drafting and Mr. Dclbtrt Zilmer for his careful reading of
the galley proofs.

ERNEST G. KELLER

Burbank, California
Apnl, 194Z

INTRODUCTION

Mathematical engineering consists of those parts of all branches
of engineering which can be formulated mathematically.

The fundamental method of mathematical engineering consists of
the two processes- (a) reduction of the physical phenomena involved
to a mathematical system, (b) solution of the system. The two
processes in this text are called, for brevity, set-up and solve. The first
process requires, in addition to a knowledge of mathematical physics
and engineering, originality and inventive ability in thought. The
second process requires mathematical knowledge. In general, the
first process is a difficult one.

Mathematical engineering naturally resolves itself into two divi-
sions. The first division may be called the analysis of discrete engi-
neering systems. It consists of those problems which involve a finite
number of variables or a finite number of degrees of freedom. Fre-
quently, these problems reduce mathematically to systems of a finite
number of ordinary linear or non-linear, differential or integral equa-
tions. Examples of problems of the first division are the analyses of
linear and non-linear networks, rotating electrical machines, airplane
motions, locomotive oscillations, and vibrations of motors and
machines. Problems in probability, statistics, and applications of
number theory to machine windings also belong to this division. The
second division may be entitled the analysis of continuous engineering
systems. Field problems in aerodynamics, hydrodynamics, electro-
dynamics, and elasticity belong to this division. Frequently, such
problems reduce mathematically to systems of partial differential
equations.

This volume is concerned with the analyses of discrete engineering
systems. An attempt has been made throughout to place equal
emphasis on the two processes, set-up and solve.

CONTENTS

PAGE

PREFACE. ... . vii

INTRODUCTION.. . ix

CHAPTLR I
ENGINEERING DYNAMICS AND MECHANICAL VIP, RATIONS

SECTION

1. Calculus of Variations . . 1

2. Hamilton's Principle. . IS

3. Lagrange's liquations. . 25

4. Lagrangc's Equations and the Theory of Vibrations 58

5. Lagrange's Equations and Ilolonomic S> stems 79

6. Non-holonomir Systems 82

7. Energy Method and Kayleigh's Principle 83

8. Additional Methods and References . 93

CHAPTER II

INTRODUCTION TO TENSOR ANALYSIS OK STATIONARY NETWORKS
AND ROTATING ELECTRICAL MACHINERY

PART (A)
TENSOR ANALYSTS OF STAHONARV N REWORKS

1. Preliminary Description . 95

2. Matrices and Linear Transformations 103

3. Preliminary Concepts of Tensor Analysis 114

4. Stationary Networks . .124

PART (/*)
INTRODUCTION TO TENSOR ANALYSIS OF ROTATING ELECT RIC\L MACHINERY

5. Non-mathematical Outline of the Nature of the Theory of Rotating Electrical

Machinery .. 153

6. Primitive Machine with Stationary Reference Axes . 156

7. Derived Machines with Stationary Reference Axes (Constant Rotor Speed) 168

8. Primitive Machine with Rotating Reference Axes 180

9. Derived Machines with Rotating Reference Axes 185

10. Machines Under Acceleration . 190

11. Tensorial Method of Attack of Engineering Problems. 197

12. References.. .. . 198

xi

xii CONTENTS

CHAPTER III

NON-LlNEARITY IN ENGINEERING
SECTION PAGE

1. Differential Equations Analytic in Parameters 201

2. Non-linear Systems by Variation of Parameters. ... 221

3. Solutions of Systems by Method of Successive Integrations 232

4. Solutions of Systems by Matrix Methods 234

5. Elliptic Functions 246

6. Hyperelliptic Functions 264

7. Method of Collocation 284

8. Galerkin's Method 288

9. Method of Lalesco's Non-linear Integral Equations 294

10. Solutions by the Differential Analyzer . . . 301

1 1. Additional Methods and References 303

INDEX 305

MATHEMATICS OF MODERN ENGINEERING

CHAPTER I
ENGINEERING DYNAMICS AND MECHANICAL VIBRATIONS

(1) Calculus of Variations, (2) Hamilton's Principle, (3) La-
grange's Equations, (4) Lagrange's Equations and the Theory
of Vibrations, (5) Lagrange's Equations and Holonomic Sys-
tems, (6) Non-holonomic Systems, (7) Energy Method and
Rayleigh's Principle, (8) Additional Methods and References.

All engineering problems in Volume I were reduced to mathematical
systems by means of (a) Newton's laws of motion, (b) Kirchhoff* s laws of
electric circuits, and (c) the laws of vector analysis. There exist more
general principles which include the above principles as special cases.

Mathematical physicists have long sought a single principle from
which all other physical principles can be drawn. The most funda-
mental single principle of mathematical physics is Hamilton's principle.
The fundamental equations of dynamics as well as the Maxwell field
equations, equations of elasticity, and other basic systems are derivable
from Hamilton's principle. The fundamental equations of dynamics
and Rayleigh's principle yield, as a special field, the theory of vibrations.

Hamilton's principle is most easily understood and derived in the
notation of the calculus of variations. The calculus of variations itself
has many applications in engineering aside from its use in establishing
Hamilton's principle, but the proof of Hamilton's principle for the field
of dynamics is sufficient justification for the study of this branch of
mathematics.

a)

Calculus of Variations

The calculus of variations deals with problems in maxima and
minima. It is recalled from the calculus that in the elementary theory
of maxima and minima the problem is to determine those values of the
independent variables (x\, x 2t ,#n) for which the function y =

1

2 CALCULUS OF VARIATIONS

f(x\, Of2, , x n ) takes on cither maximum or minimum values. In the
elementary calculus of variations a definite integral, whose integiand is
a function of one or more unknown functions and their derivatives, is
given. The problem then is to find the unknown function (or func-
tions) which will render the definite integral a maximum or minimum.
Because the easiest problems in the calculus of variations are con-
cerned with geometrical properties this section begins with a simple
geometrical problem.

1-1. Introductory Problem. Let it be required to find, by the
calculus of variations, the equation of the shortest curve joining two
points PI and P%. It is not necessary to consider a curve existing in
three-space and joining P\ and P%, since the projection of such a curve
onto a plane containing P\ and P% is shorter than the curve itself.
Thus the shortest curve is sought only among curves which lie wholly
in a plane containing P\ and P< 2 . Moreover, as possible shortest curves,
only single- valued curves (functions) and curves which are continuous
and on which the tangent turns continuously need be considered. Such
curves in the calculus of variations are said to be of class C prime. Any
single-valued function which is continuous and possesses a continuous
first derivative is defined to be of class C'. The curves, among which
the curve is sought which minimi/es the given integral, are called ad-
missible arcs or curves. For engineering purposes the properties so far
specified for the admissible arcs may be viewed as assumptions under
which the solution is sought. The assumptions may be changed. In
the calculus of variations this actually happens; the curves admissible
for one problem may not be admissible for another. In general, more
restrictions put on the admissible arcs render the analysis easier, but
the results are accordingly restricted in value.

When a function is said to be of class C f it is understood to be of
class C in the interval xi ^ x ^ x>>. To find an extremum (minimum or
maximum) means in this chapter to find only Euler's necessary condi-
tion. No sufficiency condition is implied.

The analytical statement of the problem now is: Let it be required
to find among the admissible arcs joining P\ and P-2 that one, y = y(x),
which minimizes the integral

where x\ and x 2 are the abscissas of PI and P 2 , respectively, and
y = dy/dx. The admissible curves joining PI and P 2 (Fig. 1 I) may
then be represented analytically by the equation

y = y(x) + *(*), [1]

INTRODUCTORY PROBLEM 3

where a is a parameter independent of x and y(x) is an arbitrary func-
tion which vanishes at x\ and x%. If the value of y from Eq. (1) is
substituted in the integral of the problem, there results

/() = Al + [/(*) + ar,'( X )T}*d X ,

Jx\

where the primes denote derivatives with respect to x. Since the limits
of the integral are constants, I(a) is a function of the single parameter

FIG. 1-1

a. If /(a) is to be a minimum for a = 0, it is necessary that dl(ci)/da,
denoted by /'(a), vanish for a = 0. Since the limits x\ and oc 2 are
constants, it follows, by the rule for differentiation of an integral with
respect to a parameter, that

/'(a)

{1 + [/GO + <*r,'( X )]*}"d X

and

-jfir

Integration by parts and use of the fact that /'(O) must vanish yield

* 2 f* 2 d (

I J

xi Jxi dx [ [1 H

4 CALCULUS OF VARIATIONS

By the definition of vj(x) it follows that 17(^2) = *?(#i) = 0. Conse-
quently

It can be shown without much difficulty (see Ex. 7) that, since rj(x) ia
an arbitrary function, the last integral can vanish only if

d\ y 9

y'

The first integral of this equation is *-.-=== = C\. A second inte-
gration gives

[3]

When the arbitrary constants C\ and C 2 have been determined so
that (3) passes through the points PI and P^ the required minimizing
curve has been found.

1-2. Euler's Equations: First Necessary Condition for Simplest
General Case. The integral which was minimized in 1 - 1 is a special
form of the more general integral

F(x, y, y')dx. [4]

Let it be required to find among the admissible arcs joining P\P%
(Fig. 1-1) that one, y y(x), which minimizes /. An admissible arc,
in this more complicated theory, has precisely the properties l pre-
scribed in 1-1. If y = y(x) be the minimizing curve and Eq. (1)
be substituted in (4), evidently

r*

J 1

/xi

/(a) = / F\x, y + ary(jc), / + aif(x)]dx. [5]

In order that /(a) take on a minimum (or maximum) value for a =

it is necessary that

rxi

- 0, [6]

1 Usually the admissible arcs are taken to be curves which are continuous and
consist of a finite number of arcs on each of which the tangent turns continuously,
i.e., the curve may have corners. The results, however, are sufficiently general for
the present purpose.

EULER'S EQUATIONS 5

where F y and F u > denote respectively the partial derivatives of
F(x, y, /) with respect to y and /. If the formula for integration by

parts, / u dv = uv / vdu, is applied to (6), where
u = F y , v = i7(*)

du = - y - dv
dx

there results

-f'(O) = / F v ri(x)dx + Fy'rify

J Xl
Since

rjfe) = 17(3*1) = 0,
and

d

the reasoning following Eq. (2), or Ex. 7, yields

!>.-*, -0. [7]

Now /''y (*, y, y') is a function of the three variables x, y, y', and y and
y' are in turn functions of *. By the formula for total derivative

dy' dy

~~dx Fv ' v '

Thus Eq. (7) may be written

/W/'
where

and

-

FV " ~

Equation (7), or (8), is known in the calculus of variations as Euler's
equation. This equation is the first necessary condition which y =
y(x) must satisfy in order that this function may render / a minimum,

6 CALCULUS OF VARIATIONS

There is the theorem: Every function y = y(x) which minimizes or
maximizes the integral

/ = / Ffa y y y')dx

must satisfy Eq. (7). It is recalled from the calculus that /'(a) = is
not sufficient to insure the existence of a minimum of f(x) at x = a. It
is necessary also that f"(a) > 0. The two conditions are both neces-
sary and sufficient to insure a minimum. Hence in the calculus of
variations, it is natural to expect that v = y(x) must satisfy further
conditions in order thai it maximize or minimize (4). These conditions
are more complicated than Euler's equation and are not essential for
our purpose. In many practical problems these additional conditions
may be waived, at least in a first treatment. Equation (8) is a differ-
ential equation of the second order. Its general solution contains two
arbitrary constants and hence represents a two parameter family of
curves. These solutions arc called extremals and the curves of the
family extremal arcs.

EXAMPLE. Plane curves of constant density join the points PI
and P-2 which subtend an angle of less than 60 at the origin. Find
the equation of the curve of class c v which has the least moment of
inertia with respect to the origin.

Let the two points be P\(r\, { ) ami P(r*, 0o) and the polar equa-
tions of the curves joining these points be r = r(Q). The integral to
be minimized evidently is

= f r 2 ds = ( 'V [I + r0' 2 ] 1 * dr.

That particular r = r(0) which minimizes / must satisfy Euler's
equation

Fe - -f F 9 , = 0,
dr

where

F = r 2 [l + r 2 e' 2 ]*,
and thus

F e = 0,
and

r 4 6'

Euler's equation then reduces to
d

EULER'S EQUATIONS
The first integral of this equation is

or

Tlie last equation may be written

The integral on the right ib easily evaluated by means of the substitu-
tion r* = c sec z. The relation between and r then is

i
= tan" 1 ------ h o,

$ c

r* = c see 3(0 - fi).

By proper choice of r and Ci the graph of the last equation passes
through PI and /V This function of is a .solution of Miller's equation
and moreover, the last equation is the equation of the minimi/ing
curve required.

EXERCISES AND PROBLEMS I

1. Show that the minimum surf tire of revolution generated by revolving about
the x a\ih a curve of cldbs C' joining PI and P% is a catenary. The integral to be
minimized is

and

Remembering that both > and / are functions of x and differentiating out, there is
obtained for Euler'b equation

yy - (/)2 -1=0. Set / = p.
Then

and the differential equation is

_/> dp_ __ dy

- 27T (! + / 2 ) H dx

J-ti

F w -(\+ y*)* ?*

CALCULUS OF VARIATIONS

/*' 2

2. Minimize the integral / (w 2 *' 2 + n~x~)dt
<J 'i

3. Show that the minimum line upon a sphere joining two points of the surface is
the arc of a great circle. First show that the integral to be minimized is

/* 2
*

remembering that the spherical surface coordinates R, 0, *? are related to x, y, z by
the relations

x R cos tf> cos

y = R cos <p sin
s = I? sin *>.

4. Show that the minimum line upon a circular cylinder is a helix.

5. A particle of mass m falls from rest on a curve joining the points P\(x\, y\) and
^2(^2, yz)- It is assumed that the particle moves without friction on the curve. Find
the equation of the curve for which the time of descent is a minimum. The integral
to be minimized is

where a
V

yi and g is the acceleration of gravity.

6. Find the minimum line on a cone
of revolution.

7. If M is a function of x which is
continuous in the interval JCi ^ x ^ *2
and if

f

J *

Mri(x)dx -

for all functions 17 which vanish at Xi and
xz and which are of class C', then show
that M = in .ri ^ x ^

FIG. 1-2

y'(x)

1-3. Euler's Equation by
Means of Variations. Equation

x (7) will now be obtained by the

so-called method of variations. In
Fig. 1-2 let C be the arc, whose
equation is y = ^(.v), minimizing (or maximizing) the integral (4). Let
6 V , a neighboring curve, be given by y = y(x) + T?(JC), where ri(x) is the
increment in y on passing from C to C", x being fixed. Then

(a) y' becomes y' + ?/(.Y) on passing from C to C ; ,

(b) ij(x) is called the variation of y and is denoted by 8y,

EULER'S EQUATION BY MEANS OF VARIATIONS 9

(c) v'(x) = ty, and

(d) ri'(x) = change in the slope of y on passing from C to C' (x

dy

being fixed) = 8
dx

Thus (c) and (d) gi\e the very important relation

1 8y = 5 J! or ddy = 5dy. [9]

ff.V GJC

That is, the derivative of the variation is the variation of the derivative.
It is remembered from the calculus that the symbol d applies to changes
taking place along a particular curve. From (</), (6), (c), and (d) it
is evident that 5 applies to changes which occur on passing from the
curve C to a curve 6 V . It is natural to expect that 8F(y), 8F(y,y') t
etc., are computed by the same formulas as dF(y) t dF(y t y'), etc. In
fact the proof of the variation formulas follows the proof for the dif-
ferential formulas.

It is easy to establish the formulas:

F(y, y) = -^ Sy + , y = F y 8y + Fy, y, [10]
3;y 9y

5F(#, y, y) = 5v H ; 5v' = F 63; + /?' 6y.

3^ 33^

Now 8F(y) is defined by the equation

8F(y) = F(y + dy) - F(y).

By Taylor's series F(y + 8y) = F(y) H 6y + higher powers 2

in 5y. Thus

8F(y) = /''(y) H 8y + F(y)

= dy + higher powers in dy which are

dy
neglected as in the case of the differential d F(y).

2 The neglect of the higher powers in dy restricts the nature of the admissible curves.
However, the results are sufficiently general for the purpose at hand. See Calculus
of Variations by G. A. Bliss and Lectures on the Calculus of Variations by Oskar
Bolza.

10 CALCULUS OF VARIATIONS

Likewise

dF(y, /) = F(y + dy, y' + by'} - F(y, /)

= F y dy + F y . dy'.
Also

*F(x, y, y') = F(x + dx, y + 5y, y' + /) - F(x, y, y')

= F x dx + F y dy + F u > 5y'.

In F(x, y, y') it is understood that y and y' are functions of x and that
when the variation of F(x t y, y') is taken x is held constant. Conse-
quently dx and

By comparing formulas (10) with the formulas for total differen-
tials, it is seen that 6 operates like the symbol d. Euler's equation can
now he expressed in a different form. It is recalled from the calculus
that

~ rV[*. y(a), y'(a)]dx = C*- [F(x, y(a), y'(a))]dx,
aaj xi J X1 da

where x\ and x% are constants. Likewise

5 / V(.v, y f y')dx = / * *F(x, y, yf)dx.

J*l Jxi

It is now easily seen that the equation

5 f V(*. y, y')dx = r\F y Sy + F v , Sy']dx = [11]

AI Tri

yields Eulcr's equation. Since

/ 2

d

dx

an integration by parts applied to the last integral yields

/** d ** f x * d

(F y , - dy) dx = F y . dy - dy-F yf dx.
ax Xl J Xl dx

Thus Eq. (11) becomes

8 rV(.r, y, y')dx = f*\F v - - FJiy = 0.

m/Xl */X\ UK

EULER'S EQUATION BY MEANS OF VARIATIONS

li

But dy = rj(x) is an arbitrary function of .v and by the same reason-
ing employed following Eq. (2) it follows that

This is Euler's equation. Beginning with this equation and tracing
backward the steps displayed, we obtain Eq. (11). Thus Eqs. (7)

and (11) are equivalent, i.e., each implies the other.

r*
If 67 = / dJ f (x 9 y,y')dx = 0, then the integral / is said to be

Ai

stationary. Stationary integrals play a very important role in mathe-
matical physics.

EXAMPLE. Let it be required to find, by the method of variations,
the equation in polar coordinates of the shortest arc connecting
Pi Ol i Oi) and P*(r2, #2)- The integral to be minimized is

r,
VS + ,
.

/ = vV +

The first necessary condition to be satisfied is 81 = 0. Accordingly

= f 'V 2

Jet

r \
"^ \.(i*

r 2 + r > 2) H + (r a + ^Kj *'
In the formula ludv = uvl v dn, let

d0, v = dr.

Then

12

CALCULUS OF VARIATIONS

The variation dr is an arbitrary function of 6 and by the usual reason-
ing it follows that

_*_ *\ *'

or performing the indicated differentiation, there results
rr" - 2r' 2 - r 2

V +

0.

The negative of the left side of the last equation is the formula for
curvature in polar coordinates. The extremals of the problem, then,
are arcs of zero curvature and that one which passes through P\ and P<2
is the minimizing arc required.

EXERCISES AND PROBLEMS II

1. Obtain the required differential equations for cadi of the first six problems of
problem set I by means of setting the first variation equal to zero.

1-4. Generalization of Simplest Case : More than One Independ-
ent Variable. Let there be given in the xy plane the curve C:f(x, v)

= 0. Let it be required to find
the surface z = g(x, y), Fig. 1-3,
passing through f(x, y) = which
shall minimize the integral

(x,y,z,p,q)dxdy,

u ^ j c-

where p = , q = , and o is

e 9' v 3y

the area in the jry-plane bounded
by C.

The partial differential equation defining z = g(x, y) is obtainable
either by substituting in I

z = g(x, y) + arj(x t y)

and proceeding as in 1-2 or by setting 67 = as in 1 -3. By the
second method, remembering that both x and y are independent varia-
bles and employing Eqs. (10), we have

67= / / F(x,y,z,p,q)dxdy= I I (F z 5z+ F p dp + F q dg)dxdy = 0.
J Js J Js

GENERALIZATION OF THE SIMPLEST CASE 13

r

Since by Eq. (9), Sp = Ss and 8q - - Sz, it follows that

dp dx dy =
The last integral is converted by an integration by parts into

/ F f Ss dy - I f 4- FP dx dy te,
*/ i J j $ dx

where A and B are points on /(.v, v) = whose ordi nates are equal.
Since 5s = at these points it follows that

/ / F p dpdx dy = - I I ~ Fp dx dy 8z.
J Js J Js dx

In an identical manner

/ / F q 8q dx dy =- I I ^- F q dx dy dz,
J Jg J Js dy

and thus Eq. (12) becomes

It can be shown (see Ex. 2) that, since dz is an arbitrary function of
x and 3/, the last integral vanishes only in case

F--F --F = [13]

1 dx p dy Q L J

Equation (13) is the first necessary condition which z = g(x, y) must
satisfy in order that this surface render the integral / a minimum.
The integral / is said to be stationary for z = g(x, y) when Eq. (13)
holds.

1-5. Generalization of the Simplest Case: More than One De-
pendent Variable. Of great importance in the calculus of variations
from the viewpoint of applied mathematics is the minimi/ing of an
integral whose integrand is a function of more than one dependent
variable and their derivatives. Accordingly, let it be required to
minimize the integral

/= (*,?,*, *',/,

14

CALCULUS OF VARIATIONS

where x t y, z are functions of /. The number of dependent varia-
bles is finite. We proceed as in 1*2 by letting

[15]

where T;I(/) = S.Y, 772(0 = 5v, and Xi(t), .Y 2 (/), are minimizing
functions. When the values of .r, v, z - from Kqs. (15) are substi-
tuted in Kq. (14) the integral / is evidently a function of a\, 2 , ;j
-. From the elementary theory of maxima and minima it follows
that for /(ai, c*2 :* ' ' ') to possess an extremum (i.e., maximum or
minimum) at i = a 2 = a = a tl = 0, it is necessary that

Accordingly we have

= o,

= 0.

where

3/

= ()(* = 1,2, --,).

Integrating by parts and applying the familiar reasoning of 1 1-1-3
we have

or

d

dt

L

dt

[16]

GENERALIZATION OF THE SIMPLEST CASE 15

Equations (16) are the Euler equations of the problem. These
equations can be derived more quickly by the variation method, i.e.,
by setting 51 = 0.

r*

51 = / (F t dx + /v tot + F y 5y + /v / + )< = 0.

Jti

The usual integration by parts yields

Since 8x, 6v, are arbitrary, it follows that 57 = only in case
Kqs. (16) are valid.

EXERCISES AND PROBLEMS III

1. By means of Eq. (13) show th.it the partial differential equation of a minimum
surface is (1 + q 2 )r -f (1 + fr)t - 2 pq\ - 0, \\hm

dp dq tfz

r -, / = f , and 5 =

dx ay 3y fix

The integral to be minimi/ed is

A =ff(* + I> 2 + f^dydx.

2. The difference of the kinetic and potential energies of a dynamical system
(two-dimensional automobile, 1-10) is

w - (s' 2 -f- k*0' 2 ) - - (z 2 +

) J,

where m, g, /, and pare constants. Find the curve z z(i) and 0(t) whi( h renders
the integral

/ = f F( Zt o, z', O')dt

J 'i
stationary.

(2)
Hamilton's Principle

An understanding of the three simplest problems (1-2-1 -5)
in the calculus of variations leads naturally to Hamilton's principle.
As previously noted, this principle is the most important single one
in mathematical physics, since it holds not only for nearly 3 all clynami-

3 Appell has shown that the < onstraints, if any, of the motion considered must be
independent of the velocities.

16

HAMILTON'S PRINCIPLE

cal systems, but is also valid in its applications to electrical phenomena,
theory of elasticity, wave mechanics, and other divisions of engineering
and physics.

1-6. Statement of Hamilton's Principle. One form of Hamilton's
principle, stated in the language of the calculus of variations, is

f

Ji\

(T - V)dt = 0,

[17]

where T and V are respectively the kinetic and potential energies of a
physical system and t\ and / 2 are two instants of time during the mo-
tion of the system. In this form V is the
negative of a function U such that the partial
derivatives of U\r\ any direction give the force
in that direction. Equation (17) is the usual
form in which Hamilton's principle is encoun-
tered. However, a more general statement of
this principle is

//i

V, [18]

FK. 1 -4. Simple Pemlu
lum.

where T is the kinetic energy of the physical
system; A',, Y^ Z, l are forces acting during the

I motion of the system and dx t , dy t , 6c t are varia-

tions of coordinates of the system.
y Before proving this principle for dynamical

systems we employ it in the solution of an
elementary problem. Let it be required to
write the differential equations of motion of

the simple pendulum of Fig. 1-4. The kinetic energy of the system
is T = 2/M k?6'~ where m is the mass and k is the radius of gyration.
From the figure the potential energy evidently is

V = mgh(\ - cos 0). [19]

By Hamilton's principle we have

5 r\%m k 2 6' 2 - mgh(l - cos *)]< = 0,
//i

or

or

/

Jt\

[m k 2 0' 56' - mgh sin 6 dB]dt = 0,

[m k 2 B' - dd - mgh sin 6 bB]dt
at

STATEMENT OF HAMILTON'S PRINCIPLE

17

By application of the usual integration by parts and the familiar rea-
soning of 1-2 the last equation becomes

/'

//i

m[gh sin 6 + k 2 0"]BOdt = 0,

or

k *0

- gh sin S.

This is the required equation.

To understand more fully Hamilton's principle, let a system of n
particles experience a change of position according to Newton's laws

8

. Hypothetical
^ Paths i

FIG. 1-5

of motion. Let the particles wi, w 2 '" w have the coordinates
(xu y t , z t ), (i = 1, 2, , ). Let the forces acting on the particles be
F g and their components along the coordinate axes be X lt Y t , and Z t
(i = 1,2, -,#). In nature the motion will take place according to
Newton's laws of motion, i.e.,

m

n

X,

*.,

[20]

m l z

To fix the ideas let the Uh particle m* be at PI at time t\ and at P 2 at
time fe and let the path described in the interval /2 h be P\ A P 2 in
Fig. 1-5. Among all the mathematically possible paths which iw t -
might have described, is the actual one the most economical one in the
sense that the integral

r

I

/PI

m, Vi ds, = action

[21]

18 HAMILTON'S PRINCIPLE

was a minimum? In the integrand i\ is the velocity of mi and r/s t an
element of distance. (The statement that the action as given by (21)
is a minimum for the actual motion of a particle or system is known as
the principle of least action.) It was the above question asked by
Maupertius (1698-1 759) which led eventually (150 years later) through
the works of Lagrange, Jacobi, and Hamilton (1805-1865) to Hamil-
ton's principle. Lagrange in 1788 and Jacobi later answered this
question in the affirmative for certain types of motion. Hamilton
encountered difficulty in understanding Lagrangc's proof that the
action is a minimum for the actual motion and derived instead the
related principle which bears his name. It can be shown from Hamil-
ton's principle that the actual motion, i.e , motion according to
Newton's laws, which takes place between two points (positions) in
the time /o t\ is such as to render the integral

//2
(T-
_

V)dt

a minimum when compared with any other infinitely near motion
between the same two points provided the time interval is the same in
both motions. Hamilton's principle does not stale so much. It states
that the actual motion renders the above integral stationary, i.e., that
the first variation vanishes.

1 - 7. Proof of Hamilton's Principle. To see that Kq. (18) holds for
the actual motion of the particle (or system), it is necessary to consider
a field of paths near PI A I> 2 (Kig. 1 -5) in which the actual path or
Newtonian path is imbedded. Motion according to Norton's laws
can follow only the actual path. Consequently, all other paths are
fictitious or hypothetical. Let .v t = *,(/), v, = J't(0 z i = Z M denote
the Newtonian path and

x, = *,(/) + *,(/),

y> = yM + **(0. [22]

*, = *.(/) + MO

be the neighboring hypothetical paths of the particle m l where 6x t , 6v l?
8z t are arbitrary and independent but small variations of x l9 y it z t .
Hamilton's principle is now easily established.

The kinetic energy of the system is T = \/2^,m l (^ + $ + 2")
where the summation is taken over the u particles. 4 Multiply Eqs.
(20) respectively by dx lt dy lt and 5z t and add. The result is

SifLfo dXi + yi 8 yi + *, 8 Zi ) = Z(.Y< tot + Y l 5 yi + Z, te,). [23]

4 The notation x is frequently used for

at

POTENTIAL ENERGY OF DYNAMICAL SYSTEMS 19

It is easily verified by means of Eqs. (9-10) that

Xi to, = to to,)' - *,(to,)' = to **,)' - *? [24]

Equation (23) may be written
S{iff.[*5-Ge? + J? + 2") - (*. ^ + y t By. + z t &0']

+ (JY, to, + 7, ty. + Z t as,)} = 0. [25]
If Eq. (25) is integrated from t\ to A> we have

/"'
/

//I

[5T + 2(A', 5.v, + Y, 5y t + Z, 5s,)] dt

- S(., 8.v, + jf t 5.v, + 5, 53,)

The last term of the left member is xero since to, = 6v t = 5s, = at
/ = /i and / = / L >. Finally then

/

S(-Y, to, + K, 5j'. + ^, &,)>// = 0. [26]

Equation (26) is the general form of Hamilton's principle, Eq.
If instead of a system of particles we have a continuous body, then

It should be noted that ^(X f dx t + Y t by l + Z t dz,) is the work done
in an infinitesimal displacement of the system by the forces X t , Y lt Z t .
It is not difficult to obtain Eq. (17) from Eq. (26).

1 8. Potential Energy of Dynamical Systems. A discussion of po-
tential energy is of value before deriving Eq. (17) from Eq. (26).

The value of Fin Eq. (19) is the potential energy of the pendulum
in configuration B with respect to configuration A (Fig. 1 -4). Poten-
tial energy is the amount of work done against gravity in bringing the
pendulum from A to B. Likewise the potential energy of a system of
bodies in a configuration B with respect to A is the work which must
be done against the forces acting on the system of bodies to bring the
system from A to B.

Let W be the work required to move a system of bodies from con-
figuration A to configuration B against a system (or field) of forces F.
Next let the system return to configuration A. If the work done by
the forces Fon the bodies during this return is also Wthen the system
of forces and also the dynamical system are said to be conservative. If
a system is not conservative it is called dissipative. If a system is con-

20 HAMILTON'S PRINCIPLE

servative a function called the potential always exists and is defined to
be the negative of the potential energy. It easily follows from the
definition of a conservative field of force that the potential energy
is independent of the path by which the system attained a given
configuration.

An equivalent definition of the potential function is the following.
If X, F, Z are single- valued functions of x, y, z which do not contain /
explicitly and if there exists a function U(x t y, z) such that

, z _ f [27]

3* dy 3*

then U is called the potential function. To see that this definition is
equivalent to the definition of the potential function as the negative of
the potential energy, multiply Kqs. (27) respectively by dx, dy, dz, and
add. We then obtain

Xdx+ Ydy + Zdz = -

d y

The left member of this equation is the work done by the forces in an
elementary displacement. The right member is an exact differential
dU. Consequently, the value of dU integrated along all paths from
#o yoi Z Q to x, y, z is the same and is the total work done. Thus

W = U(x, y, z) U(x Q , y (} , z ) = U(x, y, z) - constant.

U is thus the work done by the forces and U the work done against
the forces or the potential energy.

1-9. Derivation of First Form of Hamilton's Principle. Equation
(17) is now obtained from Kq. (18). The work done by the forces
X, Y, Z in an infinitesimal displacement in the general form of Hamil-
ton's principle is

X 8x + Y8y + Z dz.

If a potential function U exists, this same work is dV or 6V.
Consequently, substituting this value for the elementary work in (18)
we obtain

5 /

//l

(T - V)dt = 0.

1 10. Engineering Applications of Hamilton's Principle. Hamil-
ton's principle is useful in deriving the differential equations of motion
of holonomic 5 dynamical systems.

' Holonomic systems are defined in Sec. 3.

ENGINEERING APPLICATIONS OF HAMILTON'S PRINCIPLE 21

EXAMPLE 1. Two-dimensional automobile. A uniform beam of
mass m and length 21 is supported on two equal springs as shown
in Fig. 1-6, and such that the beam has but two degrees of freedom:
one a small oscillation of the center of gravity in a vertical line, and
the other a small rotation about a line through the center of gravity
and perpendicular to the plane of the figure. Write the differential
equations of motion.

Evidently, by Koenig's theorem, 6 the kinetic energy is T = (ni/2)
(z 2 + k 2 6 2 ), where /;/ is the mass of the beam and k is its radius of
gyration about the i enter of gravity
of the beam. The potential energy
consists of two parts, V\ and V% .

Vi = work done against the

springs and by gravity in

a vertical displacement.
2 = work done by an angular

displacement about the

center of gravity.

21

FIG. 1 -6. Two-climcnsionul Automo-
bile.

Let X and e be respectively the
spring constant and displacement of
an end of the spring in equilibrium position under the force of gravity.

nig

If the beam
is given a vertical displacement, the elementary work done is

Then since each spring bears half of the weight - = Xe?.

or

\ r i = -mgdz + 2\(e + z)dz t
V l =

Taking FI to be xero in equilibrium position, i.e., z = 0, we find
C\ = \e 2 . Remembering that 2\e = mg, we have

The work done in an infinitesimal rotation is

dV 2 = \(e + x)dx - \(e - x)dx

6 See 1 17 for Koenig's theorem.

22 HAMILTON'S PRINCIPLE

If V 2 = for x = 0, then

The total potential energy V is

v = v, + v 2

n f e

By applying Hamilton's principle we have

--(z8z + 1' 2 50) dt

/ *wz (s te - ~
.//, L

52)

L/ = 0.
J

By the procedure of 1.5 the last equation becomes

By the usual reasoning the differential equations are

+..

KXAMPLE 2. Simple acceleromcter. A simple accelerometer is
constructed of a mass M, a spring S, and two identical carbon-piles

x^

/~~:XT- M

X^

r i

^

L^

^x

FIG. 1-7. Accelerometer.

A and J5 as shown in Fig. 1-7. The combined spring-constant of one
carbon-pile and the spring 5 is X. The mass M possesses one degree of
freedom and the compression of each carbon-pile is e when M is in
equilibrium position. Obtain the natural period of the instrument.

ENGINEERING APPLICATIONS OF HAMILTON'S PRINCIPLE 23
The elementary work done in displacing AI is
dV = \(e + x)<lx - \(e - x)dx,

whence

V = 2 ( + *) 8 + \ (' - *) a + C.

Since
and

7=0 for x = 0, C = -\e 2

V = \x 2 .
The kinetic eneigy is Afx 2 /2. Hamilton's principle,

/'2
(Mx 2 /l - \x 2 )dt -= 0,
-.

yields the difTerential equation

MX + 2\x =
whose general solution is

x = A sin Vl\/Mt + K cos V~1\/ML

The period of oscillation is 2jr V.I//2X.

KXAMPLK 3. Simple scibmognipli. A ^ate hun^ on an inclined
sui)port together \\ith a recoirling device is a simple seismograph. Let

Circle
= b sm i

]/ A COS ^

Ellipse

FIG. 1-8. Simple Seismograph.

the mass, length, width, and inclination be respectively, m, 2a, 2b,
and a. Obtain the natural period of a small oscillation of the instru-
ment.

The kinetic energy is /^? 2 /2 = 2w& 2 2 /3. The potential energy V
for an angular displacement ^ from equilibrium configuration is

V = mg(b sin a. BC)

sin a -/1J3 tan a) = mg(b sin a y cos a tan a).

24 HAMILTON'S PRINCIPLE

From the equation of the circle y = b cos <p, x = b sin <p, the potential
energy reduces to

V = mgb sin a(\ cos <p).

Hamilton's principle yields the differential equation

V? + -.7 sin a sin <p = 0.
40

For a small oscillation the last equation becomes

f*K \

<p + 1 sin a ] v = 0.

The period of oscillation is 4?r v ,

* 3g sin a

large majority of the exercises and problems of the remaining
eleven problem sets of this chapter reduces to systems of ordinary differen-
tial equations with constant coefficients. If it is desirable on the part of
the instructor and students to solve completely each problem as soon as the
differential equations are derived, then 1-26 and 1-31 can be studied
simultaneously with the material of / 11-1-26 and the solutions of the
systems of differential equations obtained. The mathematical technique
for solving many systems of differential equations which are non-linear
or otherwise difficult is found in Chap. III.

EXERCISES IV

Solve the following exercises by means of Hamilton's principle.

1. Neglecting air resistance, obtain the differential equations of the motion of a
projectile. Assume the projectile to be a particle of mass m.

2. A weight 4U r is attached to a string which passes over a fixed pulley. The
other end of the string is attached to a pulley of weight IT. A second string, to which
weights \V and 2W have been fastened, parses over a second pulley as shown in
Fig. 1 -9. Wiite the differential equation of motion of the weight 4W.

3. Obtain the differential equations of the oscillations of the double pendulum
shown in Fig. 1-10. The bobs have masses /HI and mz and the strings have lengths
a and b. Assume there is no damping.

4. Three circular discs can oscillate only in horizontal planes as shown in Fig. 1-11.
Their masses and radii are respectively m\ t m^, MS; r\, r^ rs. The torque coefficients
of the rods are k\ t 2. k%. Obtain the differential equations of motion. Assume no
damping.

5. Three uniform steel discs are mounted on a horizontal shaft as shown in
Fig 1 12. The radii and \veights of the discs are respectively r\ = 40 in., r$ = 10 in.,
rj = 40 in.; W\ = 4000 lb., W z = 1000 lb. f H r 3 = 4000 Ib. The torque coefficients
are fci = fa = 30 X 10 6 . Wiite the differential equations of the free torsional oscil-
lations of the discs.

GENERALIZED COORDINATES; HOLONOMIC SYSTEMS 25

Fll.. 1-10

FlO. Ml

FIG. M2

(3)

Lagrange's Equations

Lagrange's equations are readily derived from Hamilton's principle.
These equations are a system of n simultaneous differential equations
whose dependent variables are Jie n coordinates r/ lf </ 2 ; , t/ n specify-
ing the configuration of a dynamical system at any time.

1-11. Generalized Coordinates ; Holonomic Systems. If the con-
stitution of a dynamical system is given, its configuration can be speci-
fied by means of a definite number of quantities which vary when its
configuration is r hanged. These quantities (denoted q\, q 2 , , <7n)
are called generalized coordinates because of their general nature. In
example 1, 1-10, the configuration of the beam is specified, subject
to the restricted motion designated, by the two generalized coordinates

26 LAGRANGE'S EQUATIONS

z and 0. The first coordinate is a linear displacement, the second an
angular displacement. In exercise 4, 1-10, the coordinates arc the
three angles lt 2 , ;} . Generalized coordinates may also be volumes,
charges, currents, etc. A system is said to possess n degrees of freedom
if the least number of generali/ed coordinates necessary to specify a
general position of the system is n.

Dynamical systems are divided, for analytical treatment, into two
classes, holonomic and non-holonomic. If a configuration of a system
is specified by the n generali/ed coordinates </i, r/ 2 , -,</ and an
adjacent configuration is specified by q + 5r/i, q 2 + &/ 2 , ',</ + &/
where 5(/i, 5(/>>, , bq n are arbitrarily independent infinitesimal quanti-
ties then the system is said to be holonomic. For the holonomic sys-
tems of the present section, n denotes the number of degrees of freedom.
As a simple example of a holonomic system consider a sphere to move
on a smooth plane. The sphere, since the plane is smooth, can both
roll and slide. The quantities giving the position and orientation of
the sphere are .Y, y, the coordinates of the point of contact with the
plane and 0, ^>, ^, the orientation of the sphere about its center. In an
adjacent position the coordinates are x + dx, y + 5v, + 50, tp +
5^, ^ + 5^, and the increments 5x, 5v, 60, 5<^>, 5^ of the coordinate?,
are all mutually independent.

In the case of a non-holonomic system the increments Sr/j, 5q 2 , ,
dq n of the n generali/ed coordinates </!, </ 2 , -, q n are not independent.
There exist constraints on the system which are expressible in the form*
of the m < n equations

Cmi dqi f C m2 dq 2 + + C mn dq n + T m dt = 0,

where the C's and 7"'s are, in general, functions of q\, q 2 , , q n , t
and where (28) are non-integrable equations. As an example, suppose a
sphere, resident on a rough plane, to move so that x and .y change si-
multaneously. Since sliding is impossible, non-integrable relations
must hold between the two increments 5x, dy and the increments
50, 8(f> t 6\l/.

1 12. Lagrange's Equations. A rigid body is an aggregation of
particles connected in such a way tluit their mutual distances are
invariable. A dynamical system is regarded as a number of particles or
bodies subject to connections and constraints.

LAGRANGE'S EQUATIONS 27

Let a system consist of // particles. Suppose the coordinates of the
/th particle, whose mass is w/, f of the system to be related to the ;/
coordinates (/ lf e/ 2l , </ by the relations

*t = /*(</!, </2f ' ' 'f (7/4 1 Of

3'. = K.(<;i,<r-2, -,</,.;0, [29]

* = Ai((/it </j '. Vul 0-
The kinetic energy 7' of the system is, by definition,

where the summation extends over all particles of the system. The
total derivatives with respect to the time of Kqs. (2 ( )) are

3^* - , 3?' - , , 3^'- . , 3J?-

~ ffi + ^ V2 + + v/t + -

dt/i 3^2 3'7/i dt

3** . , 3*i . , , 3/'i . , 3/'.

and hence

:)']

9*. . ,
~q n +

Thus the kinetic energy is a function of the coordinates, their deriva-
tives, and the time.

The expression ^(.V, Bx % + Y, dy t + Z t 8z t ) in Hamilton's principle
(see Kq. 26) is the total work done on the particles of the system by
the forces acting on the system. It is required to express this work as a
function of the generalized coordinates and their increments. The
variations of Eqs. (29) are

o<ln

28

LAGRANGE'S EQUATIONS

_
d<Ji

i .
8q n .

dq,i

(The variation dt = 0; sec 1-3.) Substituting 8x lt dy t , dz t in the
expression for the total work we have

J. 7 l ^

Finally, substituting T(q\,q^ , f/ /M qit Q'a. -^
expression for the total work in Eq. (26) we obtain

and the above

By taking the variation of T with respect to </i, go,
$ n the last equation becomes

.
8q n

, q nt qi, q 2 ,

Performing the usual integration by parts and applying the reasoning
following Eq. (11), since Sqi, Sq->, , Sq n are arbitrary, we obtain

dt

QT

Qi,

[31]

LAGRANGE'S EQUATIONS 29

where

The Qi, Q 2 , - - - , Q n are called generalized forces. liquations (31) are
the equations of Lagrange for holonomic systems. The above deriva-
tion is both for systems and for a single particle.

If the motion of a system is described by s equations

3^ ^ / 4 \

plus the m(m <" s) Eo,s. (28), where (28) are integrahle, tluMi the system is
holonomic and possesses 5 wi = n degrees () f freedom. In all systems
of Sec. 3, Kqs. (28) are supi^essc^l (i.e., m 0) by choi(xi of coordi-
nate systems such that the number of <y's introduced equals the num-
ber of degrees of freedom.

Equations (28) are employed in both Sees. 5 and 6.

The ()'s are not necessarily forces, but are quantities such that
Q r 8q r is the work done by the forces acting on the system during the
displacement 8(/ r while all other displacements of the system arc zero.
For example, if bq r is

(a) a linear displacement, Q T is a force,

(/;) an angular displacement, Q, is a moment,

(c) an increase in volume, Q r is pressure.

If all of the forces acting on the system possess a potential function
U, which is the negative of the potential energy V (see 1-8), then

Q r = -- , r = 1, 2, , H. If some of the forces acting on the sys-

3?r

tern (for example the forces due to springs on which the system is
mounted) possess a potential then

where (? r (0) are those independent additional forces which arc not con-

* A- d V
tamed in --

9<Zr

30 LAC) RANGE'S EQUATIONS

1 13. Illustrative Examples. The kinetic and potential energies
and generali/cd forces in the following problems are computed by
means of elementary principles of mechanics.

KXAMPLK 1. A particle of mass m moves
without friction on a straight line inclined at
an angle a with the vertical. The line rotates
about the vertical line with constant angular
\elocity w. Find the equation of motion.

The position of the particle m is given at
time / by
>( '**) x = r sin a. cos ?/,

y = r sin a sin wt,
I K.. 1 13 z = r cos a,

where r is the variable distance of m from O. The kinetic and poten-
tial energies are respectively

T = \m(x 2 + f + z 2 ) = \m(r 2 + w 2 r sin 2 ),

V = constant mgz = constant wgr cos a.
Since

- = wr, - - = mw 2 r sin 2 a, = mg cos a.
substitution of these quantities in Lagrange's equation

gi\es the differential ecjuation

/' (?; sin a) 2 r = ^ cos a
whose solution is

r = Ae wtsina + Be-* 1 * - -

- -.
w sin" a

P3XAMPLE 2. A particle moving in free space is subject to a force
F(r, 0, ^>), where r, 0, ^> are spherical coordinates of its position. The
coordinates r, 0, # are related to rectangular coordinates A", >', 2 by the
relations

x = r cos ^? cos 0,

y = r cos ^> sin 0,
z = r sin ^>.

lU.USTRATlYK KXAMPLKS
The kinetic energy is

Substituting in Eqs. (31) we obtain

m[r r(v 2 + 6 2 cos 2 v?)] = Q\,
ml(r 2 <p)' -\- 6 2 r sin ^ cos ^>1 =

Let F be resoKcd into the perpen-
dicular eoniponiMits:

F,(r, 0, v?) along /-,

/*Vf r ^ ^) p^ipendimlar to r and
in the meridian pi, me,

I\)(r, 0, vO perpendicular to r and

/V<MW).
Then

!'!<.. 1-M

KXAMI'LK 3. A uniform rod of length 11 and mass m is free to slide
without friction in a holder of negligible mass. The axis A li of the
holder is inclined at an angle to the vertical. The rod is turned until
it is nearly hori/oiual and released. Assuming no friction obtain the
differential equations of motion of the rod up to the time it leaves the
holder.

Koenig's theorem 7 is useful in computing the kinetic energy of a
rigid body. This theorem is: the total kinetic energy of a rigid body of
mass M ionsi\h of two parts: (a) the kinetic energy of a /wrtide of mass M
moving with the center of gravity of the body; (b) the kinetic energy oj
motion relative to the center of gravity, considered as fixed.

Let the angular displacement of the holder from equilibrium posi-
tion be (Fig. 1 IS.) The first and second parts of the kinetic energy,
as described in Koenig's theorem, are respectively

7> = \
T, = J

7 See 1*17 for proof of KuenigS theorem.

32

LAGRANGE'S EQUATIONS

where k is the radius of gyration of the rod about the line A'B'. (The
angular rotation of the rod about A'B' is the same as the rotation
about AB.) The total kinetic energy is

T = ] m [r 2 + (r* +

The potential energy of the rod is

V = mgr(l cos 0) sin a. (See example 3, 1 10.)

FIG. 1-IS
Substituting T and V in Eqs. (31) we have the 1 differential equations

r rff 2 + g(l cos 6) sin a = 0,
(r 2 + k*)8 + 2rf6 + rg sin sin a = 0.

EXERCISES AND PROBLEMS V

1. Solve, by means of Lagrange's equations, the five exercises of 1 10.

1 14. Systems Subject to Dissipation Forces Proportional to Ve-
locities. Suppose that there act on a system external resisting forces
opposing the motion of the system and that each force is proportional
to the first power of the velocity of its point of application. Then there
exists a function,

F = *S(a,*?

DISSIPATION FORCES PROPORTIONAL TO VELOCITIES 33

called the Rayleigh dissipation function, such that Lagrange's equa-
tions become

jr + 8?_ ft (,-,,2...,.,. P 2]

dq r dq r

Equations (32) are established as follows. Let the relations be-
tween .v,, v,, z, and (/i, </j, - t q n be given by Eqs. (20). Let the work
done against tlie motion of the system, i.e., the energy lost be written
w(a t .v z 5.v t + (3, v, 5y, + 7,:;, 5s,), where the summation is over all parti-
cles of the system. Then Hamilton's principle for the system is

/ [8T + 2(X t 5x t + Y t 8y t + Z, 8z t - a,.v, dx t

Jt,

- frv, 8y t - 7 A &;,)>// = 0, [33]

where A",, Y t , Z t are the components of .ill forces other than the dissi-
pation forces. In view of (31) it is necessary to examine only the
expression ^(aj.Yj 6.x, + /3,v, &y; + 7,2, 8z,). Recalling from 1-12
the expressions for x n y t , z n 5.v,, dy lt and dz t we have

-2(a t x t dx t + p t y t dy t + y,z t dz t )

. ' - . < ' , , . ' *

+ ^ - ^ + ^ J U d(/i ^ + -/ 8(/n

E/9A, . . j_ 3A, . , 9/iA /DA, , , , OA. 4 \1
7. 1 ; - 4t + + - - 1* + - -7 ) I 9i + + - f/ ) ,
Vtyi 3tf a/ / \97i (3^/t /J

, UKi . ,
1 ' + ' (In +

\oqi oq

/^/;. Ttf

+ 7*

_!_ . K ' -L ' X

+ 7* I 31+ + q n + ) -' &<J
oq n ot / o

34

LAGRANGE'S EQUATIONS

To justify the last equal sign in (34) substitute the values of x t , y lt and
Zi in F and compute the partial derivative of F with respect to q r
obtaining

" dfi . dfi

Substituting in (33) the value of IS (a A Bxi + &;y t 63; t - + y t Zi 5z t )

obtained in Eqs. (34) and proceeding as in 1 12 we obtain Eqs. (32).

EXAMPLE. It is desired to obtain
uniform rotational motion by means
of three heavy discs DQ, D\, and D%
suspended as indicated in Fig. 1-16.
Disc DQ is driven by means of a
worm gear at as near uniform speed
as possible. It is desired that D%
rotate at a more uniform angular
velocity than DQ. The discs are
connected by thin rods of torque
constants Xi and X 2 . Damping is
effected between D\ and D% by vanes
immersed in fluid. Find the differ-
ential equations of motion of D\
and Z>2

Let the angular displacement of
^2 -^i> and DQ be 2 , ^i> and OQ
= COQ/ H~ S (ci n cos /Q?O 4" b n sin wcooO ,
where the Fourier series represents
the variation in velocity due to im-
perfections of the gear, and WQ is the

average angular velocity. If I\ and /2 are the moments of inertia of

D\ and Z?2 the kinetic energy is

The potential energy is
V

FIG. 1-16

X,(0i -

DISSIPATION FORCES PROPORTIONAL TO VELOCITIES
The dissipation function for the relative velocities is

F = i(*i - * 2 ) 2 .
Substitution in Eqs. (32) gives

Ii&i + otdi + (Xi + X 2 )0i - a6 2 - X 2 2

0.

35

EXERCISES AND PROBLEMS VI

1. (Dynamic Vibration Absorber) A synchronous generator is driven by an engine
which produces a component of pulsating torque Tsin co/. The distribution of mass
of the rotating parts and the torque constant of the coupling shaft are such that there

Generator Engine Absorber

FIG. 1 17. Dynamic Vibration Absorber.

exist forced torsional vibrations of the rotor of the generator and the flywheel of the
engine. These undesirable forced vibrations can be eliminated (or at least greatly
diminished in amplitude) by what is known as a dynamic vibration absorber. In
the present system this consists of extending a shaft in line with the coupling shaft
and attaching thereto a disc /i as indicated in Fig. 1*17. If k\, the torque constant,
and /i, the moment of inertia of the absorber, po&sess the proper values relative to
the torque constant of the coupling shaft and the moments of inertia of the rotating
parts of the generator and engine, the undesirable vibrations are eliminated.

Obtain the three differential equations of motion of the free torsional vibrations
of the system shown in Fig. 1-17. The solution of these equations is reserved for
problem set XI.

2. (Relative Damping) If the resisting forces acting on two particles mi (xi, yi, 21)
and W2 (*2, ?2, 22) are

-fcitei - * 2 ), -k 2 (yi - yz) t -* 3 (si -22)
and

then the dissipation function is

tti - 2 2 ) 2 ].

Deduce Lagrange's equations corresponding to Eqs. (32) for relative damping, i.e.,
for the case where the dissipation forces are proportional to the differences of the
velocities of the points of application.

36

LAGRANGE'S EQUATIONS

3. (Damped Dynamic Vibration Absorber) For machines which operate at one
speed only, such as the synchronous generator of problem 1, the dynamic vibration
absorber can be tuned sharply to operate at one frequency. In machines in which
broad tuning is necessary, damping may be required in the system. Accordingly,
let a damping device be introduced which acts on the coupling shaft between the
engine and absorber. (Fig. 1*17.) Let the damping be relative damping and pro-
portional to the difference between angular velocities of engine and absorber.

Write the differential equations of motion of the free torsional vibrations of the
system. (Solution is required in problem set XI.)

1 IS. Energies of Systems Possessing Several Degrees of Free-
dom. The kinetic energies of the systems thus far analyzed in this
chapter have been easily obtained because the motions have been, for
the most part, either motions of particles or the rotations of rigid
bodies about fixed axes. Likewise, the potential energies of these
systems have been found with little effort by the simple principle of
elementary work. If, however, a rigid body possesses six degrees of
freedom (three of translation and three of rotation) and if in addition
the body is in any way connected to similar bodies, the calculation of
the energies is difficult by the elementary methods employed thus far.
For these more complicated motions recourse is had to vectors. This
use of vectors eliminates all difficult visualization of relative motions
and confusing projections of velocities. In 1-16 sufficient formal
theory of vectors is developed to render the calculation of energies a
routine process.

1 16. Addition 9 Multiplication, Line Integrals, and Differentiation
of Vectors. A vector is a quantity which possesses direction as well as
magnitude; a scalar is a quantity which possesses magnitude only.

Vector algebra is similar to scalar
algebra. Zero and unit vectors are
those whose magnitudes are respec-
tively zero and one. Two vectors
are equal, if and only if, they have
the same magnitude and direction.
By che negative vector A, we
mean A with its direction reversed
but its magnitude unchanged. A
vector A can always be considered as A a, where a is a unit vector and
A is the magnitude of A.

(a) Addition and subtraction. C, the sum of A and B, is defined as
the vector obtained by placing the initial point of B in coincidence with
the terminal of A and taking C with its initial point coinciding with
that of A, and its terminal point with that of B. From Fig. 1*18, evi-
dently A + B - B + A. The sum of three vectors E = A + B + D

FIG. 1

ADDITION, MULTIPLICATION, LINE INTEGRALS 37

= C + D, where A + B = C. The subtraction of A is defined as the
addition of A.

(b) Vector components. A vector is uniquely determined by giving
its projections on the three coordinate axes. These projections are
A x = A cos (Ax), A v = A cos (Ay) and A z = A cos (Az) t where (Ax)
denotes the angle between the positive x-axis and A. If A + B = C, it
is apparent geometrically that

A x + B x = C xt
A v + By = C y ,
A Z + B Z = C z .

Let i, j, and k be unit vectors coinciding with the x, y, and z axes
respectively. By the definition of addition

A = A x i + A y j + A z k.

(c) Scalar and vector products. The scalar product of A by B (or B
by A) is a scalar defined by the equation A-B = AB cos B, where is
the angle between the positive directions of A and B. The scalar
product is thus the product of one vector by the projection of the other
vector upon it. Hence A-B = B-A. Also

i-i = j-j = k-k = 1, and i-j - j-k - k-i - 0.
It can be shown that A- (B + C) = A-B + A-C; thus we may write
A-B = (Li, + jAy + *A z ).(iB x + ]B V + kB z )
= i-i A X B X + i-j A X B V + i-k A X B Z

+ j-i AyB X + j-j AyBy + j'k AyB,

+ k-i A Z B X + k- j A Z B V + k-k A Z B Z
- A X B X + AyBy + A Z B Z . [35]

The vector product of A by B (not B by A) is a vector defined by
the equation

A XB = 45 sin 0,

where B is the angle between the positive directions of A and B and is a
unit vector perpendicular to the plane of A and B. The positive direc-
tion of A X B is defined to be perpendicular to the plane of A and B
in the sense of advance of a right-handed screw from the first to the
second of these vectors through the smaller angle between their posi-

38

LAGRANGE'S EQUATIONS

tive directions. (See Fig. 1 - 19a.) Consequently, i X i = j X j =
k X k = and i X j = k, j X k = i, k X i = j. Also, A X B =
B X A. It is evident that the vector product of A and B can be con-
sidered as a vector with magnitude equal to the area of the parallelo-
gram having A and B as sides and with the direction of the normal to
the plane of A and B.

It can be proved that the distributive law of multiplication, namely
(A + B) X C = (A X C) + (B X C), holds for vector products as
well as for scalar products. In view of this and the above relations be-

BxC

(a)

FIG. 1-19

tween i,
nents as

j, and k, we can express A X B in terms of its i, j, k compo-
follows:

A X B = (iA x + jA y + k^) X (iB x + jB y + kB,)
= i X iA x B x + i X JA X B V + i X kA f B z
+ j X iAB x + j X \A V B V + j X *A y B,
+ k X iA,B x + k X JA Z B U + k X *A t B,
= i(A v B t - A t B v ) + j(A,B x - A x B t )

The vector product can be written as the determinant

i J

AXB =

+*-Z

[36]

(d) Triple scalar product. The product A (B X C) is a scalar called
the triple scalar product. Inspection of Fig. 1-196 shows that this
product is equal to the volume of a parallelepiped with edges A, B,
and C.

ADDITION, MULTIPLICATION, LINE INTEGRALS

39

Since interchanging the terms in a scalar product does not change
the sign of the product whereas interchanging the terms in a vector
product does change the sign of the product, it follows that

A-(BXC) = (B XC)-A = -(C XB)-A =- A-(CXB).

Since the volume of the parallelepiped remains the same, no matter
which face is considered as base, it follows that

A-(B X C) = (A X B)-C = B-(C X A) = C-(A X B), etc. [37]

V

FIG. 1-20

Thus the dot and cross may be interchanged at will and the sign of the
product remains unchanged so long as the cyclic order of the vectors
remains the same. The triple scalar product can be written as

A z
A-(BXC) = (BXC)-A

B X

By

*s

B z

[38]

(e) Triple vector product. The product A X (B X C) is defined as
the triple vector product. The vector product of B X C should be
formed first, and then the product of A with this result. The final
result is

A X (B X C) = B(A-C) - C(A-B). [39]

(See example 5, problem set VII.)

r*

(/) Line integrals involving vectors. The integral / F- dr is a line

JA

integral. The vector dr is taken along the tangent to the curve AB as
in Fig. 1 20, and the vector F may vary in both magnitude and direc-
tion along the curve.

40 LAGRANGE'S EQUATIONS

Alternative forms are

r* r*

I f-dr = / FcosOdr,
JA JA

and

* = r

JA

(iF x + }F y + kF z )-(idx + jdy + kdz)

*

(F x dx+ F y dy + F z dz).

f B

If F represents a force on a body, then / F dr is the work done by

JA

the force as the body moves over the specified path from A to B.
EXAMPLE 1. To fix the ideas more clearly, let F be the force of

gravity. Let the curve AB (Fig. 1-21) be one-quarter of the circum-

ference of a circle. Determine the
work W done in moving a mass m
against the force of gravity from
A to B along the curve AB in the
yz-plane. Consider no friction.
Then F = mgk. (The minus sign
indicates that the force is in the
direction of negative k). Then

B

FIG. 1-21

and

dr = j dy + k dz,

z = r cos 0,

dz = rsmSde,

W

'- /
JA

- (*(-
JA

//2
.

r/2
sin dO = mgr,

mgr

This is, of course, the work done in raising the mass m a vertical dis-
tance r. If F varied both in magnitude and direction and AB were a
complicated curve, the integrations would be more complicated but no
additional principles would be involved.

ADDITION, MULTIPLICATION, LINE INTEGRALS

41

(g) Derivatives of vectors. Let r = \x + jy + kz, where x =
y ~ y(0 ^ = z(t), and / is any real parameter, usually the time. If the
initial point of r is fixed at the origin, the terminal point of r varies and
describes a space curve as t varies. Let A and B be two near-by points
on this curve. (Fig. 1 22.) Then

Ar = AB = TI r fa is not a unit vector.)
= i*i + jyi + kzi - \x - jy - kz

= iA* + jAy +

Dividing by A/ and taking the limit as A/ approaches zero, we have
dr = .<to .<ty rfa
itt *(// J rf/ <#'

It is evident that, as B approaches A (Fig. 1 22), the vector represent-
ing Ar approaches the position of the tangent to the curve at A. Hence,
dr/dt is a vector tangent to the space
curve described by the terminus of r.
Thus, it follows that the derivative of
a vector having constant magnitude
but variable direction is a vector per-
pendicular to the differentiated vector.
By procedure similar to the above.

FIG. 1-22

Formulas for differentiating P Q and

P X Q can be obtained by expressing

each product in its expanded form (Eqs. 35-36) and taking derivatives

of these forms. Thus,

dt

dt

dt

dt

and

42

LAGRANGE'S EQUATIONS

Both products are differentiated by differentiating the factors just as in
the case of scalar products, paying no attention to the dot or cross. It
is important to notice, however, that in taking the derivative of the
vector product that the order of the vectors must not be changed unless
the sign is changed.

EXERCISES VH

1. Compute both the scalar and vector products of the pairs of vectors

A - 3i + 0.4J + 6k,

B = 0.4i + Oj + 8k.
JC - 0.6i - 7j - 8k,
ID - i - j - k.

2. Find the projection of the vector A = - i + 2j - 3k on the line passing
through (a) the origin and the point (-2, 3, 7), (b) the points (-2, 3, 7) and (1, 2, 3).

3. Compute by vector methods the area of the triangle whose vertices are (3, 4, 2),
(1,0, 5), and (-1, -2,3).

4. If TI = x\i + yij + 2ik and = B\i + 2 j -f 0sk compute the projection of
X TI on the line passing through the points (* 2 , ?2t 22) and (#3, ?3t 23).

5. Let A = oil + a 2 j + ask, B = bii + 6 2 j + &sk, C = c\i + c 2 j -f csk. Per-
form the expansions A-C, A-B, and A X (B X C), and show that

also

A X (B X C) = B(A-C) - C(A-B),
(A X B) X C = (C-A)B - (C-B)A.

6. Let the curve joining the points A and B in Fig. 1-21 be the hypocycloid
jpH .f yK = r K 9 Compute, by vector methods, the work done against gravity in
moving the mass m from A to B along this curve.

7. If the length of the tangent to the
curve x = a sin J, y = b cos J, 2 = cfl is a
at the point / = w/4 find the projection of
this tangent on the line through the origin
and the point (a, b, c).

1-17. Kinetic Energy of a Rigid
Body. Consider a rigid body B
which has angular speed of rotation
co about a point which in turn has
velocity v with respect to the fixed
axes X, Y, Z. If the body is rotat-
ing about the instantaneous axis
OA with angular speed co then the
linear speed of rotation of P is pco.

The angular speed becomes a vector if it is assigned a direction. Ac-
cordingly, let the vector to coincide with OA as shown in Fig. 1 23. The

FIG. 1-23

KINETIC ENERGY OF A RIGID BODY 43

linear velocity of P with respect to the axis of rotation is co X P and the
total velocity V of P with respect to the fixed axesJf, Y, Z is

V = t o + X p

= * o + X (r - a) [40]

= f o + X r.

Let axes #, y, z fixed in the body (and rotating with the body) be so
taken that the origin of coordinates is at the point which is not neces-
sarily the center of gravity of the body. The vectors r and , expressed
in components along the #, y, z axes, are r = xi + yi + zk and =
coi + o>jj + o>sk. The kinetic energy !T of a particle of mass mi at P is

T = \nn V 2 = f w,(f + X r;). (f + X r)
and the kinetic energy T of the rigid body is

T = Sm,(f + X rO - (f + X r,-), [41]

where fo is the velocity of the point with respect to-Y, F, Z and the
summation is over all particles of the body.

Carrying out the vector operations indicated in Eq. (41) we have

T = \rnrl +Swrf -(> X r t ) + %2m t (< X rj^o) X r),

where m is the mass of B. The last summation in the expression for T
is simplified as follows.

}Zm<( X !;)( X rO = ^2m l o>.[r l - X (o> X r<)]

= |Sw t o>- [(r t -rO - r l (r l -o>)] (see Eq. 39)

+ (4 + 3^ ) w *
ZPwxWy - 2E
where the constants A, B, C, D, E, and ^, for a continuous body, are

D =* I xycrdv, E = I xzadv, F = I yzedv,

/ / /

and <r is the volume density.

44 LAGRANGE'S EQUATIONS

Finally, the total kinetic energy of a rigid body, when the point
is chosen at random in the body, is

(a>Xr i ). [42]

This general expression for T is simplified if the point is properly
located in the body. For example, if (a) any point of the body is fixed
relative to X, Y, Z and if the origin of the axes x, y, and z is taken at
this fixed point the 1 = and

T - \(A< + B<*1 + C<*\ - 2Du** w - 2&M*. - 27^,), [43]

(b) no point of the body is fixed relative to X, Y, Z, but the origin of
the axes x, y, and z is taken at the center of gravity of the body then

[44]

This value for T is Koenig's theorem.

The constants A, B, C and D, E, F in Eq. (42) are respectively the
three principal moments of inertia and the three products of inertia. If
the axes x, y, and z are taken along axes of symmetry of the body then
D = E - F = 0.

1 18. Work Done on a Rigid Body. Let the external force acting
on the ith particle of a rigid body be denoted by F t \ Let the displace-
ment relative to fixed axes X, Y, and Z (Fig. 1 23) and due to the
force F t - be dRf. Then the work done on the body by the forces F,-
(i = 1, 2, ) in the displacements dRi(i = 1, 2, ) is

&W - ZFj-dR, - 2F< **, [45]

where V- is the velocity of the ith particle during the time dt and the
summation is taken over all points of the body. (The symbol 5W,
instead of dW, indicates that the work is not necessarily an exact dif-
ferential of the coordinates of the system.) In view of Eq. (40) the
work is

SW = 2F r (f +XrO*

- (2F,)-* dt + -S(r< X F t -)<tt [46]

where F = SF- is the resultant of the external forces and L = S(r t - X F.)
is the resultant external torque acting on the body. The nature of the
computation of F and L in the general Eq. (46) obviously depends upon
the system.

POTENTIAL ENERGY OF SPRING-MOUNTED SYSTEMS 45

1 19. Potential Energy of Spring-mounted Systems. In certain
cases d W is an exact differential in the coordinates of the system or is
sufficiently close to an exact differential that a useful approximation
which is an exact differential can be obtained. Such an approximation
to dW is obtainable in the important engineering case of spring-
mounted systems.

In Fig. 1 24 let X , Y , Z be the coordinates of the center of
gravity of the cylinder (or any rigid body). Take axes x, y> and z fixed

FIG. 1 24. Mounting of Refrigerator Unit.

in the body and with origin at the center of gravity. Let r be the
vector from O to a general point P of the body. It is desired to obtain
a formula for the general displacement S of P during a small translation
and rotation of the body such as occurs in vibratory motion. Let the
unit vectors along x, y, z be denoted by I, J, K.

Suppose the body is in equilibrium (dotted) configuration at / = 0.
At time t the displacement of P is

S =fvdt -/Vo + * X r)<8 (see Eq. 40)

= S + / [(ay, - ry)I + (w*r* - ^.)J + (! - <a v r x )K]dt,

where S is the displacement of the center of gravity from its equilib-
rium position. Now r x , r v , r z are the projections on the x, y, z axes and
consequently are constants. The unit vectors I, J, K are along the
axes x, y, and z and they do change in direction with the time. Sup-

46 LAGRANGE'S EQUATIONS

pose these unit vectors are assumed constant by replacing them by the
unit vectors i, j, k along the axes X, Y, and Z. The error made in this
assumption involves the cosines of the angles between I and i, J and j,
etc., and in vibratory motion these cosines are approximately unity
since the changes in the cosines of the small angles are much smaller
than the variation in the small angles themselves. The total displace-
ment of P at any time is

S G I (Q <* ___ /) n \\ I f Q M __ /) /* \ I //) iff __ A nm \\f
DO T \yy* z """ "z~yj* ~t~ \"z*x ~~" "x'zjj i~ v"x~i/ v y rx)K

i J 1

Q a a
"x "y "z

[47]

r* r y r z

where the positive directions of X , 8 y , and 6 Z are indicated in the figure.
Evidently the X, Y, and Z components of S are respectively i- S, j S,
k-Sor

X = X + (O y r z O z r y ),

Y = FO + (O z r x O x r z ),
Z = Z + (B x r y - y r x ).

Suppose that Hooke's law holds in compressing a spring; then the
force in a compression in the z direction is

F\ r 7 i fn n \"11-

= ALZ/O -f~ (y x r y "y^x)}*

and the work done (Eq. 45) against the spring at P is

Wp * = *^ dt = x[z

J[ /x

- \ fa + (O x r y - 6 y r x )] 2 , [48]

where X is the spring constant. The total potential energy is the
work done against all the springs in a displacement from equilibrium
position.

1-20. Differential Equations of Oscillations of Spring-mounted
Motor. A motor of mass m is mounted on four identical springs as
shown in Fig. 1 25. The spring constants of a single spring in a vertical
displacement is k and in any horizontal displacement is k Q . The dis-
tances between the springs SiS^ and 5 2 5a are respectively 2a and 2b.
The center of gravity of the motor is located at its geometrical center
and at a distance R above the upper end of the springs. The moments
of inertia of the system about the shaft of the motor and about any

DIFFERENTIAL EQUATIONS OF OSCILLATIONS

47

line perpendicular to the shaft and through the center of gravity are
respectively / and IQ. Under the assumption that the gyroscopic
effect of the rotor on small oscillations of the motor is negligible, write
the differential equations of these small oscillations.

Take the origin of the axes x, y, and z at the center of gravity of the
motor and their direction at time t as shown in the figure. Denote the
coordinates of the center of gravity relative to the fixed axes X, 7,
and Z by XQ, F , Z . Let the generalized coordinates defining the posi-
tion of the motor at any time be X Q , F , Z , = 0*, i? = & y , and = 0,

r

FIG. 1 -25. Spring-mounting of Electric Motor.

and let the positive directions of the small rotations be as indicated in
the figure.

From the figure the values of the position vectors are

ri = al + bj - RK = a \ + b j - Kk,
r 2 - - al + b J - RK = - ai + bj - Kk,
r 3 =-aI-6J- RK-ai-bj - JRk,
r 4 = al - bj - KK = ai - b\ - k.
By Eq. (47) the vertical displacements of the upper ends of the springs

are

Si. = k-S

s t .

i j k =Z

v w

S V S

fl &-

Z + (b$ + ari),
Zo + (-& - a< n)-

48 LAGRANGE'S EQUATIONS

Similarly, the two X and F horizontal displacements of the tops of the
springs are

Si* - * + (-** - *f)t s *y = F O + (*6 + Of).

, 5s, - Fo + (Rt - of),

By Eq. (48) the potential energy due to the vertical compression
of Sig is

(Z + 6{
and the total potential energy of all four springs, or of the system, is

V = \ Mo + - on) 2 + (Z + $ + o

+ (Z - ft* - a,) 2 ]
+ 1 [(X - Rn - if) 1 + (*o - -Rn

+ (JT - Rri + ftf J 1 ]
+ j K^o + ** + af) 2 + (Ko + *6 - of) 2 + (Fo + ^ -of) 2

+ (F + Rt + of) 2 ].

The kinetic energy by Eq. (44) is

T = J[Jf(*g + F + Zj) + (7| 2 + 7 , 2 + /of 2 )]-
The axes in the motor arc taken so that the products of inertia vanish.

Equations (31), where Q r = , 31 = ^o. <b ^bi 2s = ^01

dffr

ff4 = { ffs = fl. S = f yield the differential equations of motion
M X + 4k (X - Rri) = 0,
MY + 4k (Y + R$ =0,
MZ + 4feZ = 0,

0,
0,
/of + 4* (o 2 + ft 2 )? = 0.

POTENTIAL ENERGY OF ELECTRIC LOCOMOTIVES

49

Obviously, not all of these equations are independent. Suitable
methods of solving equations of this type are given in Sec. 4 (Theory of
Vibrations) and Sec. 7 (Rayleigh's principle) of this chapter.

1-21. Potential Energy of Electric Locomotives. The forces acting
on an electric locomotive fall into four groups (a) spring forces, (b)
creepage forces, (c) flange forces, and (d) damping forces. The first

FIG. 1 26. Electric Locomotive, Type 2 - B 2.

set is easily found by the method of 1 -19. Figure 1-26 shows the
spring arrangement of one of the mechanically simplest types of high-
speed electric locomotives, denoted as type 2 B 2. One guiding
truck is independent of the driving truck while the other is articulated
with the driving truck in such a way that the entire spring system
(four nests of springs) is equivalent to three-point support on three

FIG. 1 -27. Schematic Three-point Support of Electric Locomotive.

springs as indicated in Fig. 1.27. The journal construction is such
that the springs are subject to compression only. The locomotive is
equipped with a quill drive so that practically all the weight of the
locomotive is spring-mounted. The mass of the locomotive is M and
its dimensions are shown in Fig. 1-27. Let it be required to find the
potential energy stored in the springs.

Let the height of the center of gravity of the spring-borne mass
above the upper ends of the front and rear springs in equilibrium be
respectively, h\ and A 2 - The distance measured, parallel to the track
from the center of gravity to a point above the front spring is 61 and
from the center of gravity to a point above and midway between the

50

LAGRANGE'S EQUATIONS

rear springs is 62- The distance between the two rear springs is 2c.
Let the origin of the coordinate system XYZ be taken (Fig. 1-27) at
the center of gravity of the spring-mounted mass. Let the angular
displacements about the x, y, and z axes be respectively , if, f and the
spring constant of the front spring be \i and of each of the two rear
springs be \2- By Eq. (47) the vertical components of the displace-
ments of the upper ends of the three equivalent springs are

Si.

i J

61 -

= Z - M,
= ZQ + c% +

= ZQ Cf +

By Eq. (48) the total potential energy V of the cab is
V = (Z -

y (Z + cl-

y (Z -

The differential equations of motion for a locomotive arc set up in
1-32.

1-22. Differential Equations of Motion of a Gyroscope. The mo-
tion of a gyroscope is an example of the motion of a rigid body about a
point which is both fixed in the body and in space. Let the origin O
of the system of axes X, Y, and Z fixed in space be taken at the fixed
point of the rigid body. Let the origin of the axes x, y, and z fixed in
the body but moving with respect to X YZ be taken also at the fixed
point. If the position of the axes x, y, and z with respect to XYZ can
be found at time / then obviously the position of the rigid body is
known. A coordinate system is desired which relates, in a simple way,
a general position of the moving axes #, y, and z to the fixed axes X, Y,
and Z. One such system is Euler's angles (Fig. 1 28). A selected point
of the rigid body can be brought from any initial to a given final posi-
tion by means of three angular displacements. To fix the ideas sup-
pose that the two sets of axes initially coincide. Beginning with the
axes coincident (a) rotate the axis ox through the angle ^ to the posi-
tion OXi ; (6) next, rotate the x, y, and z axes (and the rigid body)
through the angle about the line OXi or Ox; (c) finally, rotate the
x, y t and z axes through the angle <p about the line (axis) Oz. These

DIFFERENTIAL EQUATIONS OF MOTION OF A GYROSCOPE 51

three angular displacements give the final position of the axes x, y, and
z with respect to the axes X, Y t and Z.

In computing by means of Eq. (43) the kinetic energy of the body,
it is necessary to know the projections *, a yv and co z of the angular
velocity about the moving axes x,
y, and z as functions of ^, 9, <p and
their derivatives with respect to
the time. To obtain these projec-
tions, first resolve the vector GO
onto the lines OZ, OXi, and Oz.
The vector sum of these projec-
tions is, of course, w. These com-
ponents are

^ about OZ, 6 about OX it
and about Oz,

where the dots indicate deriva- FIG. 1-28. Euler's Angles and Gyroscope,
tives with respect to the time.

From Fig. 1-28 the projections of these angular velocities onto the x, y,
and z moving axes are easily seen to be

. = \l/ sin sin <p + 6 cos ^>,
= \l/ sin cos <p 6 sin <p,

[SO]

COS +

The kinetic energy of a rigid body rotating about a point of the
body fixed with respect to the axes X, Y, and Z and having moments
and products of inertia A, B, C\ D, E, and F is given by the substitu-
tion of (50) in (43).

The derivation of the differential equations of motion of a gyroscope
is now merely routine computation. The instantaneous angular veloc-
ity o& does not necessarily coincide with the axis of spin of the gyroscope.
In fact w may be entirely outside the rotating body. Let Oz be the
axis of spin of the gyroscope. Let the x and y axes be taken parallel
to the plane AB of the gyroscope (Fig. 1-28) and let the moments of
inertia about these axes be A, and about the z axis be C. Then by
Eqs. (43) and (50) the kinetic energy T is

sn

Co 2 ,)
0) +

cos 6) 2 ].

The only external force acting on the gyroscope is the force of gravity
mg acting at the center of gravity (Fig. 1-28).

52 LAGRANGE'S EQUATIONS

Since

- A$ sin 2 + (Ccos 0)( + J> cos 0),

cos

= A$ 2 sin cos C(<p + $ cos 0)^ sin 0,
30

and the torque Q e about 0-X"i is Q 9 = wga sin 0, it follows that Eqs. (31)
yield by routine substitutions the following differential equations of
motion of the gyroscope

4 [A^ sin 2 + (C cos 0)(^ cos + )] = 0,

-45 A^ 2 sin cos + C(^ cos + )^ sin = mga sin 0, [51]

CyMcos0+] =0.
a/

1-23. Euler's Equations for a Rigid Body Containing a Fixed
Point. Euler developed three important differential equations of the
motion of a rigid body containing a fixed point. These equations give
for every instant the time variation of the angular velocity components
a?*, a> y , a*, about the principal axes of the body (axes x, y, and z, 1 - 22)
in terms of the external moments L 9 M, N acting respectively about
the axes x, y, and z and products of the same velocity components.
Consequently, if the moments about the principal axes of a rigid body
are known, then w x , a> y , and co can be expressed in terms of these
moments. Thus <0, the instantaneous angular velocity at any time,
can be found. Conversely, if co is known then the moments L, M, N
can be found.

SUMMARY OF SECTION 3 S3

The kinetic energy T of the body by Eqs. (43) and (50) is
T = %(A<& + BJ U + C<)

sin sin <p + 6 cos <p) 2 + B($ sin cos p 6 sin <p) 2

where A, B, and C are the three principal moments of inertia. Now

|? - C(j> + $ cos 0) = Co>,,
op

ftT

= Aa)y(\ls sin 9 sin p + cos 0>) -Bo)* (^ sin cos p

9p

6 sin p)

Substituting in Eqs. (31) we obtain the Euler equation

dt y

The remaining two equations are obtained in a similar manner. Thus
the three Euler equations of the motion of a rigid body containing a
fixed point are

dt

B + (A - Q UtUx - M, [52]

at

C*^ + (B - A^&y = N.
at

1-24. Summary of Section 3. The systems under consideration
consist of single particles or of a rigid body. Once the kinetic energy,
potential energy, if it exists, and the external generalized forces have
been computed the derivation of the differential equations of motions
by means of formulas (31) is merely a routine matter. The following
summary relates to the computation of the above three quantities.

(a) Single particle. If there are no constraints a single particle pos-
sesses three degrees of freedom and consequently requires three coordi-
nates to define its position. The rectangular coordinates of the posi-
tion of the particle are related to other coordinates (spherical, cylindri-
cal, toroidal, etc.) by means of three equations such as those of example

54 LAGRANGE'S EQUATIONS

2, 1-13. The kinetic energy is given by T - (m/2)(x 2 + jr 2 + z 2 ).
If the particle has gravitational potential energy this energy is equal to
the work done in moving the particle against gravity from some refer-
ence position (equilibrium position if it exists) to its current or general
position. In finding expressions for the generalized forces Q r (r =1,2
, n) care must be taken to make sure that the product of Q r by the
corresponding displacement 6q r is work done on the system.

If constraints exist then the particle possesses less than three
degrees of freedom. The rectangular coordinates x t y, z then are
expressible in terms of less than three generalized coordinates such as
in the equation of example 1, 1-13. The statements regarding the
energies and generalized forces made for three degrees of freedom hold
also for one or two degrees of freedom.

(b) Rigid body. If the constraints are such that the body has few
degrees of freedom such as in example 3, 1-13 the kinetic energy
can be obtained by the application of Kocnig's theorem 1-13. The
gravitational potential energy of a rigid body is the work done against
gravity in bringing the body from a reference position to a current
position.

If the body has six degrees of freedom, then the kinetic energy is
given by either Eqs. (42) or (44). The latter is preferable. If x t y t
and z are taken along axes of symmetry in the body, then D = E =
F = 0. If the body possesses potential energy V and if the only ex-

dV
ternal forces acting on the body are , (r 1, 2, , n) then

9</r

dW = dV in (46) and the potential energy is found at once. In the
important engineering case of spring-mounted bodies the potential
energy is obtained with sufficient accuracy by the routine method of
Eqs. (47-48). In the use of this method one vector is drawn from the
center of gravity of the rigid body to the upper end of each spring.
If external forces, other than the forces of gravity and of springs, act
on the system these forces are (? ( ? of 1-12. If dissipation forces,
proportional to the velocities of their points of application, act on the
body then Lagrange's equations for the body are Eqs. (32).

If the body contains one fixed point then its kinetic energy is given
by the substitution of (50) in (43). The external torques are taken
about the lines OZ, OX\, and Oz. The position of the body is then
given by the solution of the Lagrangian equations for the angles ^, 0,
and <p as functions of the time. An alternative method of studying the
motion is by means of Euler's equations. The solution of (52) for the
components o> x , w y , and o>* give the direction of the instantaneous axis
of rotation and the magnitude of the instantaneous velocity as func-

SUMMARY OF SECTION 3

SS

tions of the time. When these components are substituted in Eqs. (SO)
the angular position is given by the solution of the resulting system of
differential equations. The torques L t Af, and N in Euler's equation
are taken about the moving axes .r, y, and z.

EXERCISES Yin

1. Two particles m\ and w*, connected by a rod of negligible we : t$ht, move on a
smooth vertical circle. Kind the differential equation of motion.

2. A triangular lamina ABC of Miles a, b, c is suspended by the vertex A. The
lamina swings in its own plane under the influence of gravity. Find the length of the
equivalent simple |>endulum.

3. A rough uniform circular cylinder of radius r and moment of inertia / has coiled
around its middle section a flexible inextcnsiblc string. The string is rolled up until
the cylinder in a horizontal position touches a fixed point P to which the string is
attached. The cylinder is made to revolve in a horizontal plane with angular veloc-
ity <> and then released. Find the differential equations of motion.

4. Two masses mi and mi are connected and susjrcnded by inextensiblc strings of
lengths a and b as shown in Fig 1 -29. The masses m\ and mi are pulled aside in
opiwsite directions from the plane A BCD

and released. Write the differential equa-
tions of motion.

5. A heavy uniform rod is mounted in
a frame such that one end of the rod is
constrained to move without friction in a
horizontal plane, the other end without
friction in a vertical groove of the frame.
The frame is rotating with constant angular
velocity about the vertical groove as an axis.
Write the differential equation of motion
of the rod.

6. The foot of a ladder is resting on a smooth horizontal plane and its top leans
against a lamp post. The top of the ladder slides down the post while the foot of the
ladder is free to move in any horizontal direction. Write the differential equations of
motion of the ladder.

7. Show that the expression J^2m, ( X r,)-( X r t ), obtained in the reduction
of Eq. (41), can be written w4-/2, where o> = | Wj + jw v -f- fcw and

FIG. 1-29

- ji D + jj B - jk F

where i-(ii) = i-i(i) = i, i-(ji) = i-j(i) = 0, etc. The quantity * is known as a
dyadic in nonion form.

8. Two masses m\ and mi (mi > m\) are suspended from a wheel and axle of radii
r i and n ('2 < 'i)- The moment of inertia of the combined wheel and axle is /. Find
the acceleration of mj.

56

LAGRANGE'S EQUATIONS

PROBLEMS IX

1. A mass m\ is supported by a wheel and elastic tire, and a mass m* is supported
above mi by a spring. (Fig. 1 -30.) Constraints permit vertical motion only and the

FIG. 1 -30. Spring,
Tire, and Shock-
absorber.

FIG. 1*31. Compound Seismograph .

wheel is not allowed to rotate. A shock-absorber, which acts equally for either direc-
tion of motion of its piston is placed in parallel with the spring. The force exerted by

FIG. 1-32

the shock-absorber is always proportional to the difference of the velocities of mi and
f2. The system is set in motion. Write the differential equations of motion of mi
and W2.

2. Obtain the differential equations of motion of the seismograph shown in
Fig. 1-31. The dimensions, inclination, and masses are shown in the figure. Assume
there is no damping.

SUMMARY OF SECTION 3

57

3. Solve problem 2 with the additional condi-
tions that there is relative damping in the system
such that the motion of the second plate is damped
relative to the first, and the third (lowest) plate is
damped relative to the second. Let the damping
be proportional to the differences of the first powers
of the velocity.

4. Given that the angular displacements from
equilibrium position of a spring-mounted mass do
not exceed 5, show by examination of the inte-
gral leading to Eq. (47) that the maximum error
in the potential energy as given by use of Eq. (47)
is less than 2 per cent.

5. The four coiled springs of an automobile are
alike in pairs and all obey Hooke's law. Let the
spring constant of a rear spring be \2 and that of
a front spring be \\. The distances, measured
parallel to the length of the car, from the center
of gravity of the car to a point above and midway
between the front and rear spring supports are re-
spectively &i and 62- The lateral distances between
the springs is c. The height of the center of gravity
above the tops of all four springs is h. Compute
the potential energy stored in the springs under
the assumption that the angular motions are small,
10.

6. Suppose the forward component of the ve-
locity of the center of gravity of the car in problem
5 is V, a constant. Let the car travel over an
undulatory road and each undulation be a sine wave
of length L and amplitude ?o. Let the principal
moments of inertia about axes through the center
of gravity be A, B, and C. Neglecting the effect
of the tires and assuming the angular displace-
ments small, write the differential equations of
motion of the car.

7. Electric locomotives of the type 2 -CdbC-2
possess six driving axles and two guiding trucks.
Each half of the spring-borne weight of the loco-
motive rests on three driving axles and on one
guiding truck by means of three-point support as
shown in Fig. 1-33. The locomotive cab rests on
two king-pins shown. Very slight lateral rolling
motion of the cab is possible before the springs
are appreciably acted upon. Let the equivalent
spring constants of each of the guiding-truck
springs be Xi and the spring constants of each of
the other equivalent springs be X 2 . Neglecting the
small lateral rolling motion described above, com-
pute the potential energy of the spring-borne mass

Q

I

U

-H

u

I

O

3

58 LAGRANGE'S EQUATIONS AND THEORY OF VIBRATIONS

of the locomotive. No angular displacement exceeds 3. (The differential equations
of motion are required in problem set XII.)

8. Solve Ex. 4 of set VIII when the inextensible strings have been replaced by
elastic bands which obey Hooke's law. Let the modulus of the elastic bands be X.

9. Suppose the motor of Fig. 1-25 to be mounted on three springs, the two rear
springs as shown, and a third front spring under the shaft of the motor. Obtain the
differential equations of motion.

10. Obtain from Eqs. (51) the single equation in

.. (L - Ca cos 0) (L cos - Cot)

Ae . . - mga sin 9 = 0,

A sin 3

where a and L are constants of integration.

(4)

Lagrange's Equations and the Theory of Vibrations
(Normal Coordinates)

Lagrange's equations are of use in writing the differential equations
of motion of small oscillations or vibrations of a rigid body about either
equilibrium position or about steady motion. Motion about equi-
librium configuration is the more important in engineering applications.

1 25. Potential and Kinetic Energies of Oscillating Systems. Let
01 02 ' > On be the n generalized coordinates of a holonomic dynamical
system. Let 0[ 0) , 0) , , 0i 0) be the values of 0i, 2 , , fti when the
system is in equilibrium position. Make the change of variables of
position

0,- = 0* (0) +<Zi (t- 1,2, -,*),

where now all <? vanish in equilibrium position. Denote by VQ the
potential energy of the system in equilibrium configuration. Then the
potential energy V in a general position can be written, by aid of
Taylor's theorem, as

[531

3 V

where the coefficients , 2", , are evaluated at

oai 30i 302301

that is, in equilibrium position. The forces acting on a system in equi-
librium position are zero. From 1 8 the forces acting in the directions

POTENTIAL AND KINETIC ENERGIES OF OSCILLATING SYSTEMS 59

9 V

of possible displacements of the system are - (* 1, 2, ,).

90*

Consequently,

ar.ar 9I = .

301 902 90n

If the zero of potential energy is taken at equilibrium position, then
VQ = and if all motions are small (vibrations or small oscillations)
then terms in powers of the q's higher than the second can be omitted
and Eq. (53) becomes

V - ifaifl? + 2b 12 q iq2 + + b nn &), [54]

where &(i f j = 1,2, -,#) are constants.

Suppose Eqs. (29) do not contain the time explicitly. Then remem-
bering that a general position of the system is denoted by 0i, 2 , , n ,
the kinetic energy T, by the reasoning following Eqs. (29), is

+^*' +

In general, the coefficients of #1, 32* > 5n are functions of q\ 9 q%, ,
<7n but since the motions are small we may regard their values at q\ = q%
= ... = q n = o as being their values at any time. Consequently,

T = J(an# + 2a 12 hq 2 + + a nn &), [55]

where the a# are constants.

dV

If no forces act on the system other than , (i = 1, 2, , n) %

dqi

then Lagrange's equations are obtained by the substitution of Eqs,

60 LAGRANGE'S EQUATIONS AND THEORY OF VIBRATIONS

(54-55) in Eqs. (31). Equations (49) are an example of the systems
in question.

1 26. Solution of Differential Equations of Vibrations with Damp-
ing. The method of solution of a system of homogeneous linear differ-
ential equations with constant coefficients is made clear by the solution
of three equations in three unknowns. (A review of determinants and
simultaneous linear homogeneous algebraic equations may be advisa-
ble.) 8 Let the equations be

- 0,

where

Z 32 (p)q 2
+ b ijt and

0,

The substitution of

9i

[57]

in Eqs. (561 and the division of each of the resulting equations by
e mi yield

2n(w)Ci + z l2 (m)C[ + s 13 (w)C'; = 0,

z 2 i(m)d + z 22 (m)Ci + z 23 (m)Cl = 0, [58]

231 WCi + z Z2 (m)C[ + 2 3 s(w)Ci = 0.

In order that Eqs. (58) have a solution in Ci, Ci, C\ other than the
trivial solution C\ = Cj = C\ = 0, it is necessary and sufficient that
the determinant

A =

z 2 i(ni) z 22 (m) z 23 (m)
231 W 2 32 (w) 233 (w)

vanish. Let the s roots (s = 6) of the characteristic equation A
be iwi, m 2 , , We. Then

q\

[59]

8 L. E. Dickson, Elementary Theory of Equations, pp. 138-149; also Vol. I, pp.
55-69.

SOLUTION OF DIFFERENTIAL EQUATIONS

61

where C,-, Cj, C" are arbitrary constants in a solution of Eqs. (56)
for j equal to any one of the integers from 1 to 6. Moreover,

C 6

22 =

C' 2

[60]

is a solution of Eqs. (56).

The number of arbitrary constants contained in the solution of
Eqs. (56) is equal to the order of the system of the differential equa-
tions, or what is the same thing, equal to the degree of the characteris-
tic equation A = 0. Thus in Eqs. (60) only six of the eighteen C's
are independent. It is necessary to eliminate twelve of the C's. The
unprimed C's will be retained and all the primed and double-primed
C's will be eliminated. Since Eqs. (59) satisfy Eqs. (56) we have,
on substituting the former in the latter and dividing by e m i*

= 0,

" = 0,

= 0,

[61]

where j = 1, 2, , 6. To solve Eqs. (61) for the primed C's in
terms of the unprimed C's, rewrite the equations with the unprimed
C's on the right side of the equations and re-order the equations, if
necessary, so that a non-vanishing determinant of order 2 appears in
the upper left-hand corner of A, i.e.,

[62]

The first two equations of Eqs. (62) can be solved for Cj and C"
by Cramer's rule in terms of C/. By a well-known algebraic theorem
the values so obtained will satisfy the remaining equation. Thus

D

D

[63]

62 LAGRANGE'S EQUATIONS AND THEORY OF VIBRATIONS
where

D -

In view of Eqs. (63), Eqs. (60) now become
ffl = df* + C*f* + +
q 2 - */ C^ + k 2 C 2 e"* + - - - + fc' C 6 ^ f [64]

which contains only six arbitrary constants and is the general solution
of Eqs. (56).

If all of the roots of A = are real then all the quantities in Eqs.
(64) are real and the solution of the system as given by Eqs. (64) is
complete.

If, on the other hand, A = has complex roots, then not all
k'j and k" are real. In this case it remains to remove the apparent
complex quantities from Eqs. (64). In vibration problems the roots
of the characteristic equation A = are, in general, all complex. Let
these roots be mj = ry /*, (j = 1, 2, 3). Here, since the roots
are complex the arbitrary constants in Eqs. (64) must be complex in
order that qi, g 2 , 93 be real quantities. The method of eliminating
imaginary quantities from Eqs. (64) is made clear by the consideration
of one pair of complex roots. Accordingly, let m\ = r\ + u\i and
m 2 = r\ wii. Equations (64) then, by use of the relations

r fl '(cos Wl j + i s ; n Wl j),

become

qi = e~ rit [(Ci + C 2 ) cos w^ + (Ci - C 2 )i sin i/] + C 3 e mt H

q 2 = e' rit [(k\Ci + k' 2 C 2 ) cos coi/ + (fcid - k' 2 C 2 )i sin o>i/]

+C 6 fe^, [65]

cos !/ + (tJCi r

If Ci - l ~ 2 , C 2 ll then Ci + C 2 = 4 t and (Ci - C 2 )i

- BI, where AI and J5i are real numbers. The number k\ (Eqs. 63)
is a complex number a\ + p(i and it is evident from (63) that k' 2 is k\
with i replaced by i. Consequently, if k\ = d\ + ff\i then )fe 2

ILLUSTRATIVE EXAMPLES 63

= oli 0i*- Substituting these values for k\ and k 2 and the above
values for C\ and 2 we obtain the real quantities

+ k' 2 C 2 =

f66j
- k' 2 C 2 )

If &" = i + f[[i then 2 ' = i 0T*- In the same manner the real
quantities

*7Ci + *JC a = aMi + tfS lf

f(*ICi - *5C a ) = d(B - /&*!.
are obtained.

When the values given by Eqs. (66-67) are substituted in Eqs. (65),
then qi, q 2 , <? 3 are real quantities as far as the roots r\ it u\i are
concerned. If r\ = 0, the above procedure yields the correct result,
but for this a simpler procedure is given in the second illustrative
example of 1-27.

To evaluate the six arbitrary constants of the solution it is neces-
sary to know the values of </ lf q 2 , </ 3 and q\, q 2 , fa for some value of the
time.

In engineering work the frequencies of the oscillations are more
often required, because of possible resonance with applied forces, than
the solution of the differential equations. To obtain the frequencies
of the oscillations only the solution of the characteristic equation is
required since <o,/27r computed from m; = TJ fyi, (j = 1, 2, 3)
gives the frequencies of the oscillations. If the characteristic equation
is factorable, the roots are, of course, found by elementary methods.
At all times GraeftVs method 9 yields all the roots. If there is no
damping then YJ = and the roots are pure imaginaries wyi. In this
case substitute m = wyi in A(m) = and all the roots of the resulting
equation are real. If the roots are real they can be found graphically,
by guessing, or by Gracffe's or Horner's method. 10

1-27. Illustrative Examples. Two illustrative examples are now
solved; one is numerical, the other literal.

EXAMPLE 1. Obtain the general solution, by the method of 1 26,
of the system

(P 2 - 9)<Z! + (p - 1)(Z 2 + 0-33 - 0,

(P + 3)ffi + 0-02 + (P 2 + 16) 33 - 0,

0-ffi+ 32 + (P 2 + 9) 5 3 = 0.

9 J. B. Scarbough, Numerical Mathematical Analysis, p. 198; E. J. Berg, Heavi-
side's Operational Calculus, p. 140; also Vol. I, p. 105.

10 L. E. Dickson, Elementary Theory of Equations, p. 115.

64 LAGRANGE'S EQUATIONS AND THEORY OF VIBRATIONS
The characteristic equation is

A = -2(p + 0.101 + 3.59i)(p + 0.101- 3.59i)(p - 2.202)(/ + 3) = 0.
The roots of the characteristic equations are

mi = - 0.101 - 3.59, m 2 = - 0.101 + 3.59,
m a = 3, j 4 - 2.202.

The general solution is

ffi = Cje" 1 ' + C 2 <?* + W + C^,

It remains to eliminate complex quantities and the primed and double-
primed constants from this solution.

Equations (62), for this example, are

(p - 1)C; + 0- C" = -(/>- 3)(p + 3)Q,

0-C; + (p* + 16)C? = -(p + 3)C,,

whence

(p - 3)( + 3)

(P + 3) (p 2 + 16)

C" =
c *

16)

(p-l) (p- 3)(p + 3)
(/> + 3)

-(/> + 3)

(#-3)
1 1

(P-l)

1 (p-3)
1

(p - l)(/> 2 + 16)

16

*

In the expressions for Cj and C^ let j = 1, i.e., * mi = 0.101
- 3.59*. Then

Ci - (-1.53 + 5.66*)^ =
C? - (-0.626 + 1.29534

1 + ffif)d,
(? +

ILLUSTRATIVE EXAMPLES 65

Next let j = 2, i.e., p = w 2 = -0.101 + 3.59i in C' and C". Then

C 2 = (-1.S3 - 5.66*)C 2 - k' 2 C 2 = (cl 2 + &i)C 2

C' 2 - (-0.626 - 1.295i)C 2 = AgC, - (o4' + &i)C 2 - (a? -

For j = 3, i.e., p = w 3 3 in Cj and Cj

C 3 = 0-C 3 = AgCa - 0, Cg - 0-C 3 - %C 3 - 0.
For j = 4, i.e., # = m 4 = 2.202

C' 4 = 3.45 C 4 = ^C 4f Cl = -0.25 C 4
Equations (66) and (67) are

i ^i + /3i 5i - -1.53.4! + 5.66J9 lf
i ^i - ]8i 4i = -1.53 B l - 5.66 -4i,

and

<*iAi + & B l = -0.626^! + 1.295 J5 lf

ai^! - & A l = -0.626^ - 1.295 ^
The final substitution in Eqs. (65) gives
qi = e"' i0lt (A l cos 3.S9/ + #1 sin 3.590 + C 3 *~ 3 ' + C 4 e 2 ' 202/ ,
fl2 = e-' m '[(-153Ai + 5.66^0 cos 3.S9/

+ (-5.66^! - 1.53Bi) sin 3.S9/] + 3.45 C 4 e 2 ' 202 ',
fia . e- aio "[(-0.626^i + 1.29550 cos 3.59/

+ (-1.295^! - 0.626^0 sin 3.59/]-0.25C 4 ^ 202< .

EXAMPLE 2. Let it be required to find the general solution of
Eqs. (49) by the method of 1-26. The third and sixth equations of
Eqs. (49) are independent of the remaining four and can be solved at
once. The four remaining equations form two independent systems of
two each, that is

(Mp 2 + 4* )*o ~ 4/Wfy - 0,

- 0,
and

(Mp 2 + 4Jfe ) Y + 4*o^ = 0,

+ 4 (* j2 + *o* 2 )]* - 0.

These systems are solved independently of each other. By the substi-
tution of

C\e ,

66 LAGRANGE'S EQUATIONS AND THEORY OF VIBRATIONS
in the first system above its characteristic equation is found to be
Mm 2 + 4 k Q - 4 k R

/ O w 2 + 4(fcz 2 + k<>R 2 )
4(Mka? + Mk R 2 + IMm 2 + 16k Q ka? - 0.

The four imaginary roots m\, w 2 , ma, and w 4 of A (w) = are =bio>i,
dbio> 2 , where

W1

and

Mk R* + I k , TI = (165? -
The solution of the first system is

X ^_
^

-"

1

or what is the same thing (see Eqs. 65)

^

XQ = 2 (Aj cos j/ + Bj sin Wj/),

17 = 2* (Aj cos ,/ + Bj sin WjO-

Among the eight constants of this solution only four are independent.
The substitution of

XQ = Aj COS 0>j/

17 = A^ cos o?j/
in the system in question and division of the results by cos w^/ yield

-4koRAj + [4(ka 2 + k Q R 2 ) - I<p>*]A'. - 0.
Applying the theory of Eqs. (61) to the last equations we have

ILLUSTRATIVE EXAMPLES 67

In precisely the same manner the substitution of

XQ = Bj sin Wjt
i\ = B*. sin Wjt
in the same system yields eventually

Finally, die solution of the first system is

XQ = A i cos wi/ + B\ sin coi/ + A 2 cos w 2 / + B 2 sin 2 J,

^i cos o>i/ 1 sin

(-4 2 cos w 2 / + B 2 sin co 2 /).

The solution of the second system is found in an identical manner
and the general solution of Eqs. (49) is

XQ = AI cos wi/ + B\ sin u\t + A 2 cos co 2 / + B 2 sin co 2 J,
YQ = Ci cos co 3 / + DI sin wa/ + C 2 cos o>4/ + Z) 2 sin o>4/,
Z = 1 cos co 5 / + E 2 sin o> 5 /,

.
cos -3/ + Asm 0,30+

(C 2 cos o> 4 ^ + D 2 sin

, . J .

^i cos Wl/ + Bl sin Wl/ )

(A 2 cos w 2 / + 5 2 sin
f = FI cos co 6 / + F 2 sin we/,

where there are twelve arbitrary constants since the system was of
order twelve and of six degrees of freedom.

EXERCISES AND PROBLEMS X

1. In the differential equations derived in Ex. 3, problem set IV, 1-10, let both
the angular displacements and velocities be small. In this case the approximations
sin 0i = 0i, cos 0i = 1, sin (0i 02) = 0i 02, 0f = 0102 = 0, etc., can be made and
the differential equations become linear with constant coefficients. Obtain the gen-
eral solution of this linear system. Evaluate the arbitrary constants for the initial
conditions 0i (0) - (small), 02 (0) - 0i (0) = 02 (0) - 0.

68 LAGRANGE'S EQUATIONS AND THEORY OF VIBRATIONS

2. Obtain the general solution of the system of differential equations derived in
Ex. 5, problem set IV, 1-10.

3. Obtain the general solution of the system of differential equations derived in
problem 1, problem set IX, 1-24.

4. Obtain the general solution of the system of differential equations derived in
problem 9, problem set IX, 1-24.

1-28. Forced Vibrations. The vibrations thus far considered are
free vibrations. In contrast there exist forced vibrations, which are
caused by application to the system of external forces which are func-
tions of the time. Let the work done, in an infinitesimal displacement,
by these applied forces be

Then Lagrange's equations are

+^w (r = 1 - 2 ' -"> C68]

where V and T are given by Eqs. (54-55). In engineering work Q r (f)
are developable in Fourier series

00

(MO = /\ (a cos 5w r / + b r8 sin

1 29. Solution of Differential Equations of Forced Vibrations. The
solution of Eqs. (68) consists of two parts. The first part is called the
transient solution. It is obtained by solving Eqs. (68) with all Q r (t)
= 0. The transient solution is obtained by the method of 1 26.

It remains to obtain only the steady-state solution. First consider
Q r (f) = for (r = 2, 3, n) and Qi(t) = E sin /. Write Eqs. (68)

[69]

H ---- + *nn(p)q n = 0,

where

atj p 2 + dij p + bij.

In solving Eqs. (69) we sh^ll first solve
*ii(/>)2i H ---- + *m(P)q n
......... [70]

*nl(/>)2l H ---- + *nn(/>)2n = 0.

DIFFERENTIAL EQUATIONS OF FORCED VIBRATIONS 69

The substitution of qj - Qj e"* (j = 1, 2, - , ) in Eqs. (69) yields

E

where

+ id t ,<a + bij,

[71]

= 0,

The solution, by determinants, of Eqs. (71) for any Q (say (?*) is

ft. lk ^ **' [791

Vk 2*A(w) ' L J

where A(tw) is the determinant of the coefficients of Qj in the system
(71) and Aik(iw) is the cofactor of Xi^ in A(z'co). If A(*'co)/4ifc(iw) is
denoted by Zifc(fco) then

T? _tot

[73]

To obtain the solution of Eqs. (70) with - replaced by H

Li

t is necessary only to replace i by * in Eq. (73). Then the steady-
state solution of Eqs. (69) is the sum of (73) and (73) with i replaced
by i, or

" e* e-

L J

2iLZ lk (iu) Zu(-f

Since A(iw) and -4ifc(ico) are both polynomials in io>, both are complex
numbers. Hence Zi k (iw) is a complex number (say a + W). The
complex number a + bi can be written re w where r is its modulus and
<p its argument. But since Z^( iw) is obtained from Zi&(iw) by
replacing i by i, it follows that Zik(-iw) and Z\k(i<*>) are conjugate
complex numbers. Thus if

Z u (fco) = re**,
Zik(-io) = re"^*.
When these values for Zifc(t')and Z\k{iu) are substituted in Eqs. (74)

70 LAGRANGE'S EQUATIONS AND THEORY OF VIBRATIONS

EXAMPLE. Obtain the complete solution of the system of differ-
ential equations

(p 2 9)gi + (p 1)^2 + 0'<Z3 3 sin 5/,

(P + 3)0! + 0-02 + (P 2 + 16)03 = 0,
n /7 i n \ ( j$ i o\/7 n

The complementary function or transient solution is given in 1-27.
It remains to find the particular integral or steady-state solution.
From Eq. 75

3 sin (5/ #2)

3 sin (5/ <pi)

3 sin (5/

where

|z u (so| ' q

2 1-

Z 12 (5t) | '

(Si) 2 - 9

S- 1

it (5*) s=

(50 +3

(SO* + 16

1

(S0*+ 9

'- 754 + 160*,

A n (5i) = 9, A l2 (Si) = 16(3 + 5i), A 13 = (Si + 3),
Zu(Si) = -83.8 + 17.W, Z w (5) = -2.69 + 7.8 f

Zis(Sf) - -43 + 125*,
17.8 __ _, 7.8

tan

-i .

-83.8

168 C

= tan

i .

-2.69

108

Finally,

85.7

= tan

_3_

132 '

i

125
-43

22

= 108.

8.26

sin (St - 109),

Let the above values of q\, q^ ffa be denoted by qi 8 , q 28 , qz, and those
of 1 26 be denoted by qu, q%t, q^t- Then the complete solution of the
illustrative example is

Si = 2i< + ^i ffi = Q2t + ff2. 3s = fls< + fl3-

1*30. More General Q 8 (t) and Resonance. If Q 8 (t) = sin 5wl
and all other Q f s are zero then the steady-state solution of 1 29 is
given by replacing co throughout by sw. If Q 8 (t) = cos w/, then the
solution is given by Eqs. (75) but with sin (w/ p*) replaced by cos
(w/ ^). If Q r (f) is a Fourier series the steady-state solution is the

NORMAL COORDINATES 71

sum of the separate solutions obtained by employing sequentially the
terms of the Fourier series.

Suppose next that no Q r (f) is zero. The procedure is as follows:
First solve Eqs. (68) under the restriction that Q r (f) = 0, (r = 2, 3,
, n) and Qi(t) 5* 0. Next let Q r (t) = 0, (r = 1, 3, , n) and Q 2 (t)
7* 0. Carry on this process, finally solving Eqs. (68), for all Q r (f) =
except Q n (f). The n values obtained for q k are then added giving the
complete steady-state solution for q k .

If the number of dependent variables is large it is more convenient
to abandon the classical method of solution of 1-26-1 -29 and to
resort to operational methods. 11

If, in computing the steady-state solution, A(*o>) = then resonance
is said to exist between the applied force or voltage E sin wt and the
system on which the force or voltage acts. In this case Eq. (75) does
not give the steady-state solution. In fact the resonance solution will
contain / at least linearly.

1-31. Normal Coordinates. The potential and kinetic energies of
a vibrational system are both definite quadratic forms in g lf g 2 (Zn
and gi, $2 ' ' ' <Z respectively. By a well-known algebraic theorem 12
there exists a real linear transformation of coordinates and velocities
ffit ff2t > ffn and Ji, 2 , <In which changes Eqs. (54-55) to the forms

--- + Mn*), [76]

.+g), [77]

where /*i, /*2 Mn are real constants. The coordinates 1, & ' > fn
are called normal or principal coordinates of the vibrating system. In
these new coordinates Lagrange's equations are

or

l+nt = (Jb- 1,2, ..-, n). [79]

The solutions of the n independent differential equations are

& = A k sin V^(/ - a k ) (* - 1, 2, - - -, ). [80]

The solutions (80) are simple. However, the linear transformation
reducing V and T to the forms (76) and (77) is tedious and involves a
knowledge of the roots of the characteristic equation. The natural
frequencies of the vibrations are the same as already obtained in 1 -26.

Vol. I, Chap. IV.

12 L. . Dickson, Modern Algebraic Theories, p. 74; E. T. Whittaker, Analytical
Dynamics, p. 181.

72 LAGRANGE'S EQUATIONS AND THEORY OF VIBRATIONS

EXERCISES XI.

1. Obtain the steady-state solution of the illustrative example of 1*29 with
3 sin 5/ replaced by 5 sin 3f.

2. Obtain the complete solution of the system

vT3ga = 5 sin 2t,
(p + I) = 0.

3. Obtain the complete solution of Ex. 1, set VI, Chap. I.

4. Obtain the complete solution of Ex. 3, set VI, Chap. I.

5. Write the differential equations of motion and obtain the complete solution of
problem 5, set IX, Chap. I.

6. Obtain the complete solution of problem 6, set IX, Chap. I.

1-32. Electric Locomotive Oscillations. As a general example il-
lustrating both the dynamical principles thus far developed and the
method of engineering analysis described in the introduction of this
textbook, the motions of an electric locomotive are analyzed. 18

(a) Factual information. Experience classifies the five oscillatory
motions of an electric locomotive as pitch, roll, plunge, nose, and rear-
end lash. The last two are especially important because their pro-
nounced existence in a locomotive signifies a tendency to derail. Con-
sidered superficially, characteristic oscillations of an electric locomotive
would seem to be very similar to those of an ordinary vehicle such as an
automobile, but experimental data and observation indicate the exist-
ence of dangerous nose and rear-end lash which are not oscillations
common to an automobile. If the tendency to nose exists in an electric
locomotive and if the locomotive noses for a given speed VQ then it will
nose more violently for all speeds greater than VQ. Consequently,
nosing is not a resonance phenomenon and cannot be avoided by run-
ning at a slightly different speed. It might be supposed that nosing is
due to the coning of the wheels or to the staggering of the rails or to a
combination of these two possible causes. However, such causes would
produce resonance frequencies for definite discrete values of V instead
of instability for all values of V exceeding VQ. Rails on European rail-
roads are not staggered and yet locomotive nosing persists. The tend-
ency to nose and the violence of the oscillation increase with the
weight and power of the locomotive. Nosing usually starts as a roll
induced by the locomotive rounding a curve onto straight track, but
unlike the oscillations of roll, pitch, and plunge, once it is set up it is
not damped until the speed of the locomotive is reduced. The pulling
of a train has only a second order effect on the nosing of a locomotive.

19 From unpublished work of B. S. Cain and E. G. Keller.

ELECTRIC LOCOMOTIVE OSCILLATIONS 73

This dangerous oscillation of a locomotive occurs most frequently on
straight track at high constant speed. When rounding a curve the
flanges of the wheels remain in contact with the outside rail and nosing
is not pronounced.

(b) Theory of performance. The postulated theory of performance
is that the energy of nosing oscillation is transferred from the motors of
the locomotive to the mass of the locomotive through the creepage
action of the driving wheels.

(c) Assumptions. It is assumed that (1) impacts can be replaced by
continuous forces acting through finite intervals of time ; (2) the driv-
ing wheels roll and creep, but do not slide; (3) the creepage forces are
functions of the velocities and displacements.

(d) Choice of principles. The derivation of the differential equa-
tions is based on Lagrange's equations of dynamics.

(e) Derivation of the equations of motion. Although the method can
be extended to locomotives of any type, we shall for simplicity set up
the differential equations of motion for locomotives of type 2 C 2.
(Two-axle guiding truck three driving axles two-axle rear truck.)
The three groups of forces acting on the spring-borne mass of a loco-
motive are (1) spring, (2) creepage, (3) flange, and (4) damping
forces.

(1) Spring forces. The spring arrangement of the 2 C 2 type is
the same as that of the 2 B 2 locomotive described in 1 -21 and its
potential energy is given by the last equation of the same article.

(2) Creepage forces. The action of a locomotive driving wheel,
because of the creep of metal at the region of contact of wheel and rail,
is not one of simple rolling. Instead, forces exist at the treads of the
two wheels of a driving axle, which, if referred to the center of the
driving axle, constitute a torque about a line through the center of the
axle and perpendicular to the plane of the track and lateral and longi-
tudinal forces acting at the same point. A creepage force F is defined
by the equation

F - -/<*, [81]

where

_ displacement rolling displacement - -

rolling displacement
and /is the coefficient of creepage which is calculated by the formula

In this formula r is the radius of a driving wheel in millimeters, W is
the weight expressed in kilograms borne by one wheel, and A is an
empirical constant equal to 800.

74 LAGRANGE'S EQUATIONS AND THEORY OF VIBRATIONS

Let the following symbols have the significance indicated:

2b track gage,

2 61 = lateral play between flanges of the driving wheels and rails,
25 2 = lateral play between flanges of guiding trucks and the rails,
r =s radius of driving wheel,
A = tangent of the angle of coning of tire,
6 = angle through which a wheel has turned in a rolling displace-

ment,

<p = angle the driving axles make at any time with the horizontal
perpendicular to the track or the angle the frame makes
with the center line of the track (Fig. 1-34),
(re, y) = coordinates of center of driving axle (Fig. 1-34).

The meaning of h\, h%, b\, &2> and
2c is given in 1-21.

To obtain the force F it is
necessary only to compute d by
the substitution of the various dis-
placements in Eq. (82). Let fixed
axes be taken as indicated in Fig.
1-34. Let A and B denote the
points of contact of the driving
wheels with the rails. The coordi-
nates of A and B are

FIG. 1-34. Creepage Displacements for M) ( x _ W y + b),

Driving Wheels of Electric Locomotive.

(B) (x + b<p,y- b).

The rolling displacements, to the accuracy required, are

(A) [(r + y)dB t r 9 ddl

(B) [(r-y)d0,r<f>d0].

dx

(B) [(r-y)d0,r<f>d
The components of creepage at A and B are

(A\\ dx 1 ( bd <f>i^y

w - 1 - + x

The component forces are

ELECTRIC LOCOMOTIVE OSCILLATIONS 75

The component forces acting at A and B are equivalent to the torque
GI and forces X\ and FI acting at the center of the driving axle

If V is the constant forward velocity of progression then V dt == rdB
since / is extremely large in comparison with X\. The last equations,
in view of this approximation, are

= a constant, [83]

The second of Eqs. (83) implies constant forward velocity which is the
only case of interest. All driving axles are attached rigidly to the
frame of the locomotive with the exception that vertical motion of the
frame with respect to the axles is possible. Equations (83) are to be
summed over all driving axles.

(3) Flange forces. The flange forces FI, F 2 , /i, and /2, which act
at the points NI, N 2 , NQ, and N^ shown in Fig. 1-35, are non-linear
functions of the displacements of the points of application. To the
accuracy required

F l = //i hr

where y\ 9 y^ ?3, and 3> 4 are displacements of the center points of the
driving axles and guiding trucks from a vertical plane passing through
the center line of the track. The flange forces on the middle driver can
be neglected. The constants //i, /i, /i, *i, *i, j\ are determined from
force curves.

76 LAGRANdE'S EQUATIONS AND tHEORY OF VIBRATIONS

Inspection of Figs. 1-35-1 -36 and use of Eqs. (83-84) yield the
following table of creepage and flange forces. The locomotive is sup-
posed displaced in the direction of positive <p and the Roman numerals
refer to the parts, shown in Fig. 1 -36, of the locomotive on which the
forces and torques act.

FIG. 1-35. Dimensions for Electric
Locomotive of Type 2-C=fc2.

II

FIG. 1 36. Driving Truck for

Electric Locomotive of Type

2-C2.

Ill IV

Y:

Z:

H:

H:

2ft -

-(?*+?)

ELECTRIC LOCOMOTIVE OSCILLATIONS 77

Let the origin of coordinates X YZ be a point in the vertical plane
passing through the center line of the track. This point is at the
height of the center of gravity and has the same forward velocity as
the locomotive. When the locomotive is in equilibrium position the
center of gravity coincides with the origin and # , yo *o & ^ anc ^ ?
all vanish. It should be noted that, because of the constraints of the
journals, f of Fig. 1-27 is identical to <p of Fig. 1-34. Moreover,
XQ, yo so of 1 32 are identical to X Q , F 0f Z of 1 21.

Lagrange's equations are

._ + (r=12 ... 6)

a<zr as, a ( ' ' ' }>

where

* 2i = *o, 22 = yo, & = so, 24 = , 2s - i?, 26 = fi

and Cr^ are the forces given in the preceding table, and T and V are
given in 1-20-1-21. The complete differential equations of the
problem are

Mx Q = 0,

My = - F 2 -/, - 2/? - * - 2/ -

= 0,

\%C(ZQ + Cl; + &2 1 ?) ~~ ^2 C ( 2 "~ c H~ ^2 7 ?) H~ ^2s ^ rorl

LOOJ

- 26,

+ X2&2(20 + 4 +

0,

78 LAGRANGE'S EQUATIONS AND THEORY OF VIBRATIONS

The number of dependent variables in the differential equations
is 11, but y\, y2, yz, yt, and y are expressible in terms of y$, and f
by means of the relations

i J

j-so

-

Thus the number of dependent variables of Eqs. (85) is reduced to six
which is the number of differential equations of the system. The points
of application of F it F 2 ,/i, and/ 2 are taken with sufficient accuracy to
be points in the plane of the track and directly beneath either the mid*
points of the driving axles or the center points of the guiding trucks.

(4) Damping Forces. In an electric locomotive there are two kinds
of mechanical damping forces, structural and creepage. The latter are
functions of the speed; the former are not. Motion is stable or
unstable according as the total damping is positive or negative.

(/) Solution of the system of differential equations. Equations of the
form of Eqs. (85) are solvable by the methods of Chap. Ill and in par-
ticular by Cotton's method indicated in Ref. 1 1 of Chap. III. The only
purpose of the solution of the differential equations is a check on the
theory of performance because a useful and simpler criterion of the sta-
bility of the locomotive is obtainable by very little labor.

The differential equations (85) are non-linear equations, the non-
linearity being introduced by the flange forces. Derailment of the
locomotive is, of course, prevented only by the flanges. Yet the motion
defined by the linear terms of Eqs. (85) may be either stable or unsta-
ble. If any roots of the characteristic equation of Eqs. (85) with
FI = F 2 = /i = /2 = possess positive real parts the motion is unsta-
ble and the locomotive is said to be unstable. The roots of the charac-
teristic equation are a function of V. The problem is thus reduced to so
specifying the constants of the locomotive (particularly spring con-
stants) that the roots in question do not possess positive real parts
except for excessively large values of V.

It is unnecessary to solve the characteristic equation since there
exists a criterion by which it is possible to determine the number of
roots of a characteristic equation which have positive real parts without
obtaining these roots. 14

14 E. J. Routh, Advanced Rigid Dynamics, p. 170; or Vol. I, p. 129.

MODIFICATION OF LAGRANGE'S EQUATIONS 79

It is beyond the present purpose to solve Eqs. (85). The calculation
of the characteristic equation for the motipn described by the linear
terms is left as Ex. 1.

(g) Experimental checks. The periods of oscillations calculated from
the solutions of Eqs. (85) were approximately checked experimentally
by test runs on the Erie test tracks of the General Electric Company.
Confidence was gained in the theory of performance which was
postulated.

EXERCISES AND PROBLEMS XII

1. Compute the characteristic equation for Eqs. (85) with FI = F* =/i =/2 = 0.

2. Is the variation in the height of the center of gravity during motion taken into
account in computing the potential energy of the locomotive in 1-21?

3. Derive the differential equations of motion for the more complicated loco-
motive of Fig. 1-33.

4. Develop another mathematical theory of locomotive oscillations which takes
into account impacts between wheel flanges and rails. (Consult Ref. 6 at end of
chapter for Lagrange's equations and impulsive motion).

(5)
Lagrange's Equations and Holonomic Systems

The dynamical systems analyzed thus far possessed precisely the
same number of degrees of freedom as there were dependent variables
in Lagrange's equations. That is, the system possessed n degrees
of freedom. In a more general situation m relations exist between
Si i #2 "'Sn in addition to the differential equations of Lagrange.
These relations are expressed by Eqs. (28). If Eqs. (28) are integrable
then the dynamical system is said to be holonomic, if not, it is said to
be non-holonomic.

1-33. Modification of Lagrange's Equations for Holonomic Sys-
tems. Let the m constraints be expressed by the equations

C M 8ji + C M asa+ +Cfa,8s-0 (* - 1, 2, -, m) [86]

where the C's are functions of q it q 2 , - , q n . In this section (86) are
integrable. Thus the dynamical system possesses exactly n-m degrees
of freedom. From Eqs. (30) we have

.a [87]

Multiplying the first, second, etc., of (86) respectively by the undeter-

80 LAGRANGE'S EQUATIONS AND HOLONOMIC SYSTEMS

mined multipliers Xi, \2, , \m and adding the results to (87) we
have

The m Eqs. (86) contain the n unknowns 8q\ t $<? 2 , 8q n . From
the theory l6 of such equations the values of n-m unknowns (say
&7m+i Stfn) can be assigned arbitrary values and the equations
then solved for 8qi, , bq m . Next let the m undetermined multipliers
Xi, -, Xw be chosen so that the m equations

(r-1,2, ...,m) [89]
are satisfied. Then Eq. (88) reduces to

., <Zr = 0. [90]

If 5^ m+ i = constant 9* and 8q m+2 = $g m + 3 = ' ' ' = *0* =
then (90) becomes

0. [91]

dq m +i

2 = constant ^ Oand 6g w+ i = 6g m+3 = = dq n = Othen (91)
is obtained with m + 1 replaced by m + 2. Continuing this process
n-m equations similar to (91) are obtained. These n-m equations,
along with (89), form the system of n equations

(r-l,2 f ... f fi.) [92]

When the n-m multipliers Xi, X2, , X OT have been eliminated from
(92) n-m equations in q\, q 2 , , q r remain. These equations along
with the m Eqs. (86) furnish n equations for the determination of

<Zli <?2 ' ' ' (Zn-

EXAMPLE 1. A homogeneous and perfectly rough sphere of mass
m and radius r rolls on a fixed sphere of radius R. The only external
force is gravity. Obtain the differential equation of motion.

"Vol. I, p. 64.

MODIFICATION OF LAGRANGE'S EQUATIONS 6i

Let the coordinates and dimensions be represented in Fig. 1-37.
Evidently,

V = mg(r + R) cos q 2 .

Since the contact is rough, 8qi and dq 2 are not independent. To obtain
the relation between bq\ and 8q 2 it is necessary only to note that, at
the point of contact of the two spheres,

rqi - Rq*

from which, by integration
rqi - Rq 2 .

From the last equation, by taking varia-
tions, the equation corresponding to Eqs.
(86) is

r dqi = R dq 2 ,

where Cn = r and C 12 = -R. The
equations corresponding to Eqs. (89)
and (91) are respectively

FIG. 1-37

882

_ R _

9ft l

These equations correspond to Eqs. (92).

Eliminating Xi between the last two equations and substituting
the values of T and V we obtain

+ (r + R) 2 q 2 - (r + R)g sin q 2 = 0.
has been eliminated, by means of the relation rqi = Rq 2 , the

After

final equation is

[(2/5)R 2

-R) 2 ]& - (' + Rh sin q 2 - 0.

EXERCISES AND PROBLEMS

1. A hemisphere rocks on a rough plane. Obtain the differential equation of
motion using the coordinates & and #o shown in Fig. l*38a.

2. The flywheel, rods, and horizontal piston represented in Fig. 1 -386 assume an
equilibrium position when there is no steam in the cylinder. Taking q\ and qi as

82

NON-HOLONOMIC SYSTEMS

generalized coordinates, obtain the differential equation of motion of the system
when displaced from equilibrium position. (NOTE: dqi and dq% are not independent
and the problem has one equation of constraint.) Show that if the engine is statically
balanced it is not dynamically balanced.

3. Obtain Lagrange's equations of motion for the governor represented in Fig.
l38c. Employ as coordinates the angles 6 and <p shown, (NOTE: the problem
involves no constraints.) Hint:

T = CP/2 + Ij?/2,

where C and 7 are functions of 6 and / includes the moment of inertia of both the
engine and the machinery driven. Denote the potential energy of the governor by

(a)

V and let $ be the generalized force representing the excess driving torque over
resistance.

(6)
Non-holonomic Systems

The dynamical systems of this section differ from those of Sec. 5 only
in the nature of the constraints. The m equations of constraint

C k2

[93]

are w0n-integrable and thus the system considered retains n degrees of
freedom corresponding to the w-coordinates g lf q 2t ,g n . Non-
holonomic systems can be regarded as holonomic systems by taking into
consideration certain reactions of the constraints.

1-34. Reduction to Holonomic Form. To the generalized forces
Git (?2. Qn of Eqs. (31) let there be added n additional generalized
forces (/i, 62. , Q' n - The latter are forces exerted by the constraints
which compel the system to fulfil the kinematical conditions of the
dynamical system. The constraints may now be considered removed

GENERAL AND NORMAL MODES OF VIBRATION 83

and replaced by the forces @i, <, , (&. Consequently, the system is
now holonomic and the equations of motion are

It remains to describe the generalized forces ( r . The equation

Ci 5<zi + & *& + ' ' ' + & to - [95]

and the m equations

C k i dqi + C k2 dq 2 + + Cfcn$<Z = (* - 1, 2, -, m) [96]

state that the work done (left member of Eq. 95) by the additional
forces of constraint in displacements permissible by the constraints
(Eq. 95) is zero. Multiplying the first, second, etc., of Eqs. (96)
respectively by the undetermined multipliers Xi, \2, , X TO and adding
the results to (87) where Q r has been replaced by Q r + Qf r we obtain

By means of (95) Eq. (97) reduces to (88). The reasoning from (88)
to (92) of Sec. 5 is repeated.

When the m multipliers Xi, Xa, , X m have been eliminated
from Eq. (92) then n~m equations in qi, q^ , q n remain. These equa-
tions along with the m equations

Cfclil + Cfc2<?2 + ' ' ' + CknAm + ^fc 0,

furnish n equations for the determination of q\ 9 q%, , q n .

<7)
Energy Method and Rayleigh's Principle

In Sec. 4, 1 26 and 1 31 , two methods are given for obtaining the
natural periods of vibration of an elastic system with a finite number of
degrees of freedom. The labor involved by either method is considera-
ble; in the first it is necessary to solve the characteristic equation
A = ; in the second the successive transformations introducing normal
coordinates are required. Rayleigh's principle is frequently not only
more easily applied, but it is also applicable to continuous systems with
infinitely many degrees of freedom.

1 35. General and Normal Modes of Vibration. The simultaneous
Eqs. (80), 1 -31, describe in normal coordinates the most general vibra-
tion of an elastic system possessing n degrees of freedom. Seldom are

84 ENERGY METHOD AND RAYLEIGH'S PRINCIPLE

the most general vibrations of interest. Instead there exist natural or
normal modes of vibration characterized by the fact that the motion of
each particle is simply periodic and given by one of Eqs. (80) ; the other
Ak being zero. There are thus, in general, n distinct norme! modes.
The mode of lowest (smallest) frequency is called the fundamental
mode. The frequencies of the normal modes are called natural fre-
quencies. The smallest of these is the fundamental frequency. A
frequency when multiplied by 2w is called a pulsatance. By the intro-
duction of frictionless constraints (consider one side of the motor
analyzed in 1-20 to be constrained by a hinge) each particle of an
elastic system can be compelled to vibrate with frequency w/2ir or
pulsatance a> according to the equation # = Bi sin w/, where w is not
necessarily a natural pulsatance of the system.

1-36. Energy Method for Systems with a Finite Number of De-
grees of Freedom. This method gives the n natural frequencies. Let
the holonomic conservative elastic system be specified by the coordi-
nates ji f g 2 , , <Zn and the potential and kinetic energies be given by
Eqs. (54-55) respectively. Let the system describe, by introduction
of frictionless constraints, simply periodic motion according to the
equations

qt = Xi cos (at (i = 1, 2, , n) [98]

where w is, in general, not a natural pulsatance. If Eqs. (98) are sub-
stituted in Eqs. (54-55) then

V = (&ll*l + 2&i2*l* 2 + ' ' + b nn X 2 n ) COS 2 *,

T = 5 (an*? + 2ai 2 xix 2 + + a nn xl)o) 2 sin 2 10*.

Since the system is conservative it is evident, from the last two equa-
tions, that the coefficients of cos 2 co/ and sin 2 w/ are equal. Equating
these coefficients and solving for w 2 we obtain

' y/. + * nr j - = P9]

Obviously, w 2 is a function of the amplitudes #1, x%, , * n ofjthe

y
motion. A necessary condition, from the calculus, for w 2 = = =

/(*i> * *n) to be maximum or minimum is that

t 2 ^v j i^ i^ ol/ f\ Pirtnl

3*1 x a* n Xn "" '

or

of _ 9/ ^ 9/ _ * noil

9i ~ tea ~ " ' ~ ~ ~ '

STATEMENT OF RAYLEIGH'S PRINCIPLE

85

When Eqs. (100-101) are satisfied then w 2 is stationary. (See 1 -6.)
From (99-101)

or since

co 2 F

^E _ 2 9T - n r

dXi Qxi

Equations (102) when rearranged are

(&12 &12CO )#2 "T" " "

E102]

(bin -

0,

, [103]

*2 + ' ' ' + (bnn - 0nn 2 )*n = 0.

A necessary and sufficient condition that (103) possess a non-trivial
solution for x\, x& , x n is that the determinant A of the coefficients
vanish, but this A is identical to that of 1-26. Thus A(w) = Ois
the characteristic equation for Lagrange's equations and its roots
i 2t 't divided by 2v are the natural frequencies of the system.

Although the energy method displays no advantage, in determining
the natural frequencies, over the method of 1 -26 yet it furnishes the
very important result that the amplitudes characteristic of the normal
modes, i.e., the values of x% satisfying (103) render w 2 = V /T stationary.

1-37. Statement of Rayleigh's Principle. Rayleigh's principle is
sometimes stated: The distribution of the potential and kinetic energies,
in the fundamental mode of vibra-
tion of an elastic system, is such
as to render the frequency a)/2ir a
minimum.

It may be of aid to interpret this
principle with reference to a par-
ticular problem. Let the problem
be to find the frequency of the
fundamental mode of vibration of
three . equal masses attached to a

light elastic string as shown in Fig. 1 39. The string is under tension
S. If the system vibrates in its fundamental mode the form of the
string is represented by the continuous line of the figure and B\, B%,
and J5 3 are the displacements characterizing the fundamental mode.
It is supposed that the fundamental mode is unknown. Instead, it is
known by observation that the mode resembles a parabola or a sine
curve as represented by the broken curve. The displacements charac-

FIG. 1-39. Actual and Approximate

Displacements in an Application of

Rayleigh's Principle.

86 ENERGY METHOD AND RAYLEIGH'S PRINCIPLE

terizing these curves are xi, x 2t and #3 and these values specify a dis-
tribution of energy of the system. The quantities x\, x 2 , and #3 also
specify a constrained motion of the system. Applying the methods of
energy we obtain the two formulas

2
and Wa

The frequency wi/27r is the minimum frequency since BI, B 2 , and B%
characterize the fundamental mode. The frequency w tt /27r is a con-
strained frequency characterized by x\, x 2 , and #3 and since the curve
*ii x 2t x 3 is almost the curve BI, B 2 , Ba the values xi, x 2 , and x$ will
almost minimize o>J. Rayleigh's principle states that wi < o> . There
are as many values of w a as there are curves resembling the continuous
curve in Fig. 1-39. The only restrictions on #1, x 2t #3 are that they
must satisfy a possible initial displacement of the system. Rayleigh's
principle is important not only because coi < w a but because w a is a
good approximation to coi.

EXAMPLE 1. Obtain approximate values for the fundamental
pulsatance of the problem pertaining to Fig. 1 39. If q\, q 2t q$ are the
coordinates of the system then

V = ~ l<& + (22 - i) + (23 - 22) 2 + 2],
2 a

If j,- = xt cos cat is substituted in V and T the energy method yields

2 (X 2 -

If the three masses are estimated (guessed) to be on a parabola during
the fundamental mode then #1 = #3 = 3#2/4 and

<4 = 0.5882-
ma

If the three masses are estimated to lie on the sine curve x = h sin
ir//(4a) then *i = # 3 = -\/2 h/2 and x 2 = h and

w^ = 0.5970 -
ma

The exact value for cof is

? = 0.5858 --
ma

PROOF OF RAYLEIGH'S PRINCIPLE 87

EXAMPLE 2. Two heavy discs, whose moments of inertia are
/i and 7 2 are supported vertically as indicated in Fig. 1 40. The con-
stants of the mechanism are: I\ = 4 slug-ft. 2 , /2 = 6 slug-ft. 2 , k\ =
1 Ib. ft./radian, k 2 = 2 Ib. ft./radian. Find the
pulsatance of the fundamental mode of angular
vibration.

The energies of the system are

V -

If the system vibrates with frequency co/27r, i.e.,

according to the equation 0,- = Xi sin wt then, by

the energy method, w will be a natural pulsatance for those values

of x\ and # 2 which render

2 *1^1 T" K2\%2 ~"~ #l) #1 i 2(#2 #j)

Wa = /i? + / 2 *1 4*? + 6*1

stationary. If a fairly accurate estimate of the ratio of x\ to x 2 in the
fundamental mode of vibration can be made, these values will render
w 2 a minimum. By observation of the system it seems that # 2 = 4^/3.
Substituting these values in w 2 we obtain w 2 = 1/12. This is a good
estimate since the exact value of wf = 1/12.

1-38. Proof of Rayleigh's Principle for Systems with a Finite
Number of Degrees of Freedom. For systems with a finite number of
degrees of freedom Rayleigh's principle is also stated: The distribution
of the potential and kinetic energies, in the fundamental mode of vibration
of an elastic system, is such as to render the frequency a minimum and
moreover the frequency of any simply periodic vibration lies between the
greatest and least natural frequencies of the system. The first part
of this theorem, as stated in 1-37 has already been established in
1-36, i.e., the distribution of energies as represented by Eq. (99)
is such as to render w 2 a minimum for w = <*>i the fundamental
pulsatance.

The second part of the theorem is best established by the use of
normal coordinates. Of course the natural (normal) modes of vibration
of an elastic system are independent of the coordinate system employed
in its analysis and consequently the use of normal coordinates does not
impair the generality of the second part of the proof.

It is recalled from 1-31 that in normal coordinates

+ + b n q%), T i(aii + +

88 ENERGY METHOD AND RAYLEIGH'S PRINCIPLE

and the solutions of Lagrange's equations are

qt - Ai sin eo t * (i = 1, 2, - , w),

where w? = bi/at. A general pulsatance w of a simply harmonic vibra-
tion, given by the energy method, is

j> _ 61*1 + 62*2 + + b n xl
ai#i + 02*2 + + a n xl '

where the amplitudes #1, #2, , x n may or may not belong to a natural
frequency.

Since & = a t a>f the last equation reduces to

2

If wi and co n are the least and greatest of the natural pulsatances a>i, co 2 ,
-, w n then it follows from the last equation that

2 _ 2

Wl
and

2 _ 2

^ "

Since all the terms in parentheses in the last two equations are positive
it follows that wf < w 2 < ov

1-39. Rayleigh's Principle and Continuous Systems. Rayleigh's
principle as stated in 1-37 is true for continuously distributed sys-
tems. 16 ' 17

EXAMPLE. Obtain, by means of Rayleigh's principle, approxima-
tions to the fundamental pulsatance of vibration of a uniform string
of length /, linear density p, and under tension r. The potential and
kinetic energies are

If the manner of vibration is given by y = z(x) sin wt then, by the
energy method,

- v -Ml

dx

/

&*dx

16 G. Temple and W. G. Bickley, Rayleigh's Principle.

17 D. Prescott, Applied Elasticity.

ORTHOGONALITY CONDITION 89

(a) If the string is assumed to vibrate as a sine curve then z = sin
vx/l and

2 / cos 2 vx/l dx o
Tir 2 ./> 7 Tir 2

o
./> _ 7 Tir 2 9.87r

Wl " ^ T. 2 /;,, " i? " 7T'

/ sm 2 ir#// dx

This is the exact value of the fundamental pulsatance given by the solu-
tion of the partial differential equation of the vibrating string.

(b) If the string is assumed to vibrate in the form of the parabola
z = (1 - 4* 2 // 2 ) then

. M ~ -.- / w (LX

W a = =

//2
64x 2 // 4
_

^ 9 r (l - s^+16^- /2 >

/ x ]& ' n '

*/0 II

(c) If the string is assumed to vibrate as two sides of a triangle, the
equation of one side being z = (1 2x/l) then

2 J*

, 12r

^ =

1*40. Orthogonality Condition. Rayleigh's principle gives the
pulsatance of the fundamental mode. The second natural pulsatance
can be found with but little additional labor by means of the so-called
orthogonality relations. Let x\, x' 2 , , x n and x", x%, , a denote
the amplitudes characterizing respectively the fundamental and second
smallest pulsatance of the elastic system. Then x( , *, -,* satisfy
(103) and the equations

^ = and > >i'^ = 0, [105]

where V and T are given by (99).

Equations (105) are established for a system ofjhree degrees of
freedom as follows. For this case V - constant and T = constant are
equations of ellipsoids whose centers are at x\ = #2 = *3 ^ _On
any line ^i = C& x% = c 2 t, x 3 = c 3 t (t a parameter), the ratio V/T is
a constant. In one particular direction this ratio is a minimum.

90 ENERGY METHOD AND RAYLEIGH'S PRINCIPLE

The equation of the plane tangent to w 2 = V/T at the general
point *io, *20, *ao is

/ \ . / N . / N A

- - (*i - #10) + r- -- (*2 - *2o) + r (*3 - *so) - 0.
o# 10 0*20 C* 30

It is recalled from analytic geometry that the partial derivatives are
proportional to the direction cosines of the normal from the origin to
the plane. If r" = jc/oi + JC 2 oJ + x-^k is any direction perpendicular
to the normal to the plane then

. , Sco 2 \ A
, + ^ kj =

or, in view of the equation preceding Eq. (102)

The last equation is true for infinitely many values of w 2 .
Consequently,

* , " /i n/^T

and >.^ - =0. [106]

Now *i, .Y2t -v-i and .v'/, .\o, .vj lie on perpendicular axes. Letting .v l( , ==
x( and XM = JT* Kqs. (106) reduce to (^05) for n = 3. Moreover, since

<~\ -tr "\nr

for a natural frequency -- == w'f (see Eq. 102), Eqs. (106) are

9-v O^'i

dependent. Thus either the first or second of Eqs. (105) is the orthogo-
nality condition.

KXAMIM.K 1. Obtain the second lowest pulsatance of illustrative
example 2, 1.37. Equations (105) for this example become

k\x\x\ + *a(jr a - JfOCva x\) = and I\xix\ + /o^.vi = 0,
(The primes have been diminished by one.) By the energy method

Substituting ,r 2 = 4^i/3 in the second orthogonality relation we obtain
#2 ** -vi/2. When this relation is substituted in the expression for
o2 we obtain w 2 = 1. This is the exact value for the second pulsatance.

SUMMARY 91

EXAMPLE 2. Obtain wo for the illustrative example of 1.39.
Referring to the expression for V and 7" and (105) we have for the
orthogonality conditions

T

The latter is

/ - 9C dx = and p / zz' dx 0.
9.v Ov J

p I (sin wx/l)(sT)dx = 0.

A value of s' satisfying this equation and the conditions of a possible
initial displacement is z' = sin //TT.Y//, (// = 2,4, ). This \alue of
z' is now to be used in the first expression for a> 2 in 1 '.W. Letting
n = 2 and making this substitution, \\e have

cos 2 2?r.v// dx 2

27T.Y// dx P

I sin 2 2?r.Y,

1-41. Summary. The procedure in the application of Rayleigh's
principle is:

(a) Obtain expressions for the potential and kinetic energies of the
system relative to its equilibrium position.

(b) If the system has a finite number of degrees of freedom substi-
tute </ t = x t sin /; if a continuous system let y = z(x) sin at.

(c) Solve for or = 7/7'.

(d) Endeavor to minimize co 2 by the substitution x l = ( t x or
z = z(x), where c t x or z = z(x) characterizes either the fundamental
mode of vibration or what is thought to be the fundamental. For this
estimate of the fundamental mode the engineer is dependent upon
knowledge of physical principles, intuition, experiment, and experience.

(e) If the system is one of a finite number of degrees of freedom the
value of m obtained is an approximate or exact root of A(w) = 0. (In
general, it is easier to verify the solution of an algebraic or transcenden-
tal equation than to solve it.)

(/) The orthogonality conditions, leading to the second lowest fre-
quency, are written by reference to Eqs. (105) or by analogy with the
illustrative example of 1 40.

(g) If T and V denote the mean values of T and V taken over a
cycle, the results of Sec. 7 are unchanged.

92 ENERGY METHOD AND RAYLEIGH'S PRINCIPLE

EXERCISES AND PROBLEMS XIV

1. Obtain, by Rayleigh's principle, approximations to lowest and second lowest
frequencies of vibration of the double pendulum of Ex. 3, I 10, under the assump-
tion that 0i and 02 are small.

2. Find, by Rayleigh's principle, an approximation to the fundamental frequency
of vibration of the accelerometer of illustrative example 2, 1 10.

3. Obtain, by Rayleigh's principle,* approximation to the fundamental frequency
for the transverse vibration of a stretched uniform string having a mass M attached
at the mid-point of the string. The mass (>er unit length of the string is p and the
tension of the string is S. Show first that

4. Show that the orthogonality conditions for Ex. 3 are

S f~ d *~ dx - and f W dx + Mz\z\ - 0,

J a* a* J

and obtain the second lowest frequency.

5. A revolving shaft is subject to transverse forces owing to its loading and
impressed torque. When the shaft is deflected from its position at rest its motion
consists of (a) revolution about it* axis and (b) rotation about its un del lee ted axis at
rest. The frequency of revolution dei>ends u|K>n the impressed torque. The fre-
quency of rotation depends upon the distribution of kinetic and potential energies
of the distorted shaft. If the^e frequencies coincide undesirable resonance exists.
If the lateral displacement of the axis of the shaft is y at a distance x from one end
then the potential energy 1M due to bending is

where / is the length of the shaft, E is Young's modulus, and / is the moment of
inertia of the area of a cross-section.

If an element of shaft has mass m dx and its velocity of rotation is 2*wy then the
kinetic energy of the shaft is

f

/o

Part of the bending of the shaft may be due to end thrust />. The shaft possesses
potential energy due to this distortion, but it is not available for translation into
kinetic energy. The expression for this energy is

The total potential energy is V - T 6 - V p .

11 A. L. Kimball, Vibration Prevention in Engineering.

SUMMARY 93

Obtain the frequency of the fundamental mode of lateral vibration (i. c., the
fundamental rotational frequency) in case the shaft is mounted:

(a) in short bearings at both ends, [y =* El <Py/dx l = at the ends of the shaft
* 1/2. The origin is taken at a point midway between the bearings. Assume
y - c(P/4 - **) (5/2/4 - r 8 ).]

(6) in long bearings at both ends, [y = </y/</x at x =b //2, and y -
c (/V4 - r 5 ) 2 .]

(c) in one long and one bhort bearing at each end.

(d) in one long bearing at one end, other end free.

(8)
Additional Methods and References

A brief description of additional methods and a list of references to theory and
applications follow.

1 -42. Equations of Appell and Bghin. The equations of Appell arc a generali-
zation of the equations of Lagrangc. The treatment of both holonomic and non-
holonomic physical systems arc reduced to a single system of equations of dynamics.
(Kef. 8.) The equations of Beghin are an extension of the equations of both Lagrange
and Appell. The extension is important with reference to service mechanisms ("aux
mecanismes comportant un asservissement"), in particular to gyrostat ic compasses
of Anschiitz and Sperry. (Rcf. 9.)

1-43. References. Only a very limited number of names and references are
given in this article because most of the pa|>crs and l)ooks cited contain bibliographies
covering a portion of the field. References are arranged according to topics.
In the final section of each chapter the elements of a reference to a paper are:
name of author, title of paper, journal, scries number [ ] if it exists, volume, page
(year).

1. Calculus of Variations. For results in parameter representation and additional
conditions for an extremum see G. A. Bliss, Cal(ulus of Variations, Carus Mathe-
matical Monographs, University of Chicago Press, 1925. Oscar Holza, Lectures on
the Calculus of Variations, University of Chicago Press, 1904; reprint G. E. Stechert
and Company, New York, 1931. For Eulcr's equations and extremals of multiple
integrals see A. R. Forsythe, Calculus of Variations, Cambridge University Press,
London, 1927.

2. Purely Theoretical Treatment of Dynamics. G. D. Birkhoft, Dynamical Sys-
tems, American Mathematical Society, New York, 1927.

3. Limitations of Hamilton's Principle in Dynamics. In the use of Hamilton's
principle in the analysis of dynamical systems the contraints, if any, need not be inde-
pendent of the time, but the contraints must not depend upon the velocities. See
Paul E. Appell, Mecanique Rationale, Gauthier-Villars, Paris, 1918.

4. Gauss' Principle. Gauss' principle in dynamics is applicable under still more
general conditions than Hamilton's principle. W. D. MacMillan, Statics and Dynam-
ics of a Particle, p. 419, McGraw-Hill Book Company, New York, 1927.

5. Constraints in Dynamical Systems. W. D. MacMillan, op. cit. t p. 306.
N. W. Akfmoff, Elementary Course in Lagrange's Equations, Chaps. I, II, III, Phila-
delphia Book Company, 1917. Horace Lamb, Dynamics, p. 301, Cambridge Uni-
versity Press, London, 1914.

94 ADDITIONAL METHODS AND REFERENCES

6. Impulsive Motion. Lagrange's equations were modified by Lagrange for
impulsive motion. E. T. Whittaker, Analytical Dynamics, Third Ed., p. 50, Cam-
bridge University Press, 1927. J. H. Jeans, Theoretical Mechanics, p. 344, Ginn and
Company, Boston, 1907.

7. Equations of Impact E. T. Whittaker, op. cit. t p. 234.

8. Appell's Equations. Paul E. Appell, op. cit.

9. Bgghin's Equations. M. II. Bcghin, "tude Theoiique des Compas g\ro-
statiques," Ann. llydr., p. 259 (1921).

10. Velocities as Coordinates, Quasi-Holonomic Systems. Chap. II. 1C. T.
Whittaker, op. cit., pp. 4^, 215.

11. Vibration Theory of W. Ritz, Ref. U, Chap. III.

12. Solutions of Non-linear Equations in Dynamics. Chap. III.

13. Suddenly Impressed Velocities. Harold Jeffreys, Operational Method* in
Mathematiial ]'hy\us, Cambridge University Press, 1927.

14. Damping Proportional to Square of the Velocity. Lord Raylei^h, Theory of
Sound, Second Kd., Vol. I, p, 77, MacMillan and Company, London, 1.X94. M. V.
Ostrogradsky, Memoirei de I'Atad. des Sciences de St. Peter^boiirg [6], 3 (1840).

15. Books on Vibrations. \Vilheltn Hort, Tethnisrhe Sihmn^un^lchre, Julius
Springer, Berlin, 1922. A. L. Kimball, Vibration Prevention in Engineering, John
Wiley and Sons, 1932. J. P. IVn I faring, Me(haniil Vibration*, McGraw-Hill
Book Company, 1934. S. Timoshenko, Vibration rroblems in Engineering D. Van
Nostrand Company, 1937. Karl Klottcr, Einfuhrung in Die Technisihe Silnvtn-
gun^lehre, Julius Springer, Berlin, 1938.

16. Vibration Measuring Instalments. J. Ormondroul, "The l r se of Vibration
Instrument son Klcrtric.il Machinery," Tran\. A.I.E.E., 45, 443 (1926). II. Steud-
ing, "The Measurement of Mechanical Vibrations," V.I) /., 71, 605 (1927).

17. Balancing Machines. K. L. Thcarle, "A New Txpe of Balancing Machine,"
A.SM.E., Applied Afafaniti, 54, 131 (1932). O. K. Ksval and C. A. Frischc,
"Dynamic Balancing of Small G\ row-ope Rotors," Tnn\. A.I.E E., 56, 729 (1937).
J. G. Baker and K. C. Rushing, "Balancing Rotors by Means of Klcctricil Networks,"
Franklin /WA/., 222, 186-196 (1936).

18. Electric Motor Pulsations. A. L. Kuuball and P. L. Alger, "Single Phase
Motor Torque Pulsations," Tram. A.I.E E., 43, 730 (1924). F. H. Bro\\n, "Lateral
Vibrations of Ring-Shaped Frames," Franklin /wv/., 218,41 (1934).

19. Flywheel Calculations. A. R. Stevenson, Jr., "Short Method of Calculating
Flywheels," Central Eleitru Review, 28, 580, 731 (1925).

20. Aeronautical Dynamics. J. K. Younger and B. M. Woods, Dynamics of Air-
planes and Airplane Strmtures, John Wiley and Sons, 1931. For sudden action of
elevator and gun recoil on airplane see N. W. McLachlan, Complex Variable and
Operational Calculus, Cambridge University Press, 1939. W. F. Durancl, Aero-
dynamic Theory, V, Julius Springer, Berlin, 1935.

21. Noise Measurements. P. L. Alger, "Progress in Noise Measurements,"
Elcc. EUR. 52, 781 (1933).

22. Airplane Flutter Analysis. Theodore Thcodorsen, "General Theory of Aero-
dynamic Instability ami the Mechanism of Flutter," National Advisory Committee
for Aeronautics, Rejwrt 496 (1935). (The alxive analysis is a two-dimensional one.)
William M. Blcaknev, "Three-Dimensional Flutter Analysis," Journal of the Aero-
nautical Sciences, (1942).

CHAPTKR II

INTRODUCTION
TO

TENSOR ANALYSIS OF STATIONARY NETWORKS AND
ROTATING ELECTRICAL MACHINERY

(1) Preliminary Description, (2) Matrices and Linear Trans-
formations, (3) Preliminary Concepts of Tensor Analysis, (4)
Stationary Networks; (a) General Theory, (b) All-Mesh Net-
works, (c) Mesh Networks, (d) Interconnection of Networks,
(5) Non-mathematical Outline of the Nature of the Theory of
Rotating Electrical Machinery, (6) Primitive Machine with
Stationary Reference Axes, (7) Derived Machines with Sta-
tionary Reference Axes, (8) Primitive Machine with Rotating
Reference Axes, (9) Derived Machines with Rotating Ref-
erence Axes, (10) Machines Under Acceleration, (11) Ten-
sorial Method of Attack of Engineering Problems, (12)

References.

This chapter is an introduction to methods of reducing electrical
engineering problems to mathematical form by means of tensor analy-
sis and the theories of Kron.

PART (A)
TKNSOR ANALYSIS OF STATIONARY NKTWORKS

Part A, consisting of Sees. (1-4), is concerned with the elementary
theory of matrices, tensors, and the development of stationary network
analysis.

(D
Preliminary Description

This section is a brief non-mathematical description of the theories
of the whole chapter. No mathematical knowledge is presupposed.

95

96 PRELIMINARY DESCRIPTION

2-1. Historical Note on Tensors. Although the applications of
tensors in engineering is of very recent date, 1 tensor analysis itself is by
no means new. The study of tensors was begun by Christoffel in 1869
after the foundations of the subject were laid by Gauss and Riemann
two or three decades earlier. The study was greatly advanced by Ricci
and Levi-Civita in 1901 by the paper, "Methodes de Calcul Differen-
tiel Absolu." In 1916 Einstein called attention to the usefulness of the
work of Ricci and Levi-Civita and since that date tensor analysis is
often referred to as the "Mathematics of Relativity." The body of
theory of tensor analysis is extensive and its applications in other
branches of mathematics and physics are exceedingly numerous.
Among the most important are the applications in differential geome-
try, calculus of variations, quantum mechanics, dynamics, elasticity,
and thermodynamics.

2 2. Scope of Kron's Theories. The applications of Kron's theo-
ries are so numerous as to be bewildering. The methods of thought
and analysis seem destined to extend to mechanical engineering as well.
So many fields are already opened up that a generation may be required
for their complete exploration. Some fields to which the methods have
been applied are (a) all linear (stationary or moving) networks with
lumped parameters, (b) every type of rotating electrical machine, \c)
communication and transmission systems, (d) magnetic and electro-
static networks, (e) multi-electrode vacuum tube circuits, (/) intercon-
nected systems of similar and dissimilar apparatus and machines, (g)
generalizations of Maxwell's field equations, and (h) mechanical engi-
neering 2 problems. It is not here feasible to catalogue exhaustively the
multitudinous applications of the theory. It is preferable to obtain an
impression of its partial scope and its various branches and their mutual
relations from the outline of the table in Fig. 2-1.

2-3. Nature of the Theories. A non-mathematical description of
the nature of selected portions of Kron's work may be of value before
engaging in the detailed mathematical analysis of the theory.

Just as the theory of relativity is a physical theory distinct from ten-
sor analysis and from any single or group of principles of advanced

1 Gabriel Kron, Tensor Analysis of Rotating Machinery, I, 1932; II & III, 1933,
mimeographed; "Non-Kiemannian Dynamics of Rotating Electrical Machinery/'
Journal of Mathematics and Physics, 1934, pp. 103-194; "Analyse Tensor idle Appli-
qu6e a 1' Art de I'lngenieur," Bulletin de f Association des Ingenieurs Eleetrititns,
Liege, Belgium, Sept., Oct., 1936; Feb., 1937. (Prize paper of Fondation George
Montefiore.)

* C. Concordia, "The Use of Tensors in Mechanical Engineering Problems,"
General Electric Review, July, 1936.

NATURE OF THE THEORIES

97

Equs of voltage torqut
Won- invariant transformations
Mined reference frames

Transient operation
Balanced polyphase operation

Law of transf of Z
Maxwellian equs
Christeffel voltage

Slip -ring machines
Non -sinusoidal currents
Interconnected systems

Fquation of motion

Holonomic ft non - holon frames

Quasi - holonomic frames

Christoffel symbol

Non -holonomic object

Torsion tensor

Non - Rtemanman spaces

Solvable cases

Step -by -step solutions

Equ of small oscillations
Motional - impedance object Z
Law ol transf of Z
Riemanman - Christoffel tensor

Individual machines
Control systems, etc
Combined mechanical systems

Uniformly moving frames
Accelerated frames
Generalized field equs.
Equ of motion of electrons
Small oscillation of electrons

Ultra -high frequency tubes
Field problems

Dynamics
Hydraulics
Thermodynamics
Optics

Types of energy transformations
Equations of constraint as C
Invarianct of power input i.
Laws of transformation
Geometric obtects and tensors
Theory of groups
Spmors

Interconnection of coils as C i
Symmetrical components as C*
Neglecting magn currents as C 3
Successive transf CC ( Cj C 3 C
Multiwindtng transformers
Reactance calc of windings
Voltages in dc windings
Star -mesh transformations
Interconnection of networks as C

Twotypesofvanablesl &C
Dual tensors of equations
Power- series development

General junction networks
N- electrode tubes
Amplifier ft oscillator circuits
Dielectric networks
Small, non - linear oscillations

+ EZ<i+I> ft it's dual eqv
Dual equations ol Lagrange
Non -singularity of all C
Dual metric tensors
Raising and lowering indices
Topological concepts

Networks with impressed &!
Magnetic networks
Interlinked networks
Transmission lints
Generalized per -unit systems

Sub -division of tensor equations
Compound tensors
Multiple tensors
Reduction formulas
Criterion* of performance

Three -phase networks
Compound networks
Generalization of networks theorems
Analysis of networks
Synthesis of networks

FIG. 2-1

98 PRELIMINARY DESCRIPTION

physics so the epoch-making researches of Kron are much more than
tensor analysis and advanced electrical engineering.

This achievement is such a discovery, generalization, and organization
of those intrinsic physical entities common to a wide variety of similar and
often seemingly totally dissimilar electrical and mechanical systems as to
disclose the frequently multiple parallelism between the performances of
these systems and to describe the behavior and relations between the entities
of a system or systems by means of general mathematical equations whose
forms are independent of the reference frames employed. The mathemat-
ical language of this work is tensor analysis.

Certain general features characterize this newest development.

(a) Derivation. It is primarily a systematic method of setting up
and manipulating systems of differential and integral equations of per-
formance of those problems in electrical and mechanical engineering
which are expressible in terms of systems of differential and integral
equations. 3

(b) Discovery and organization. It has discovered and so organized
the concepts of advanced electrical engineering that the derivation of
the equations of performance of an unlimited number of physical sys-
tems and their general analysis and synthesis are reduced to routine
manipulations and errors are largely precluded by extensive mathe-
matical symmetry.

(c) Power. It is a method of great power. Its power consists in its :

(1) (Generalization) It unifies electrical engineering by substituting for a
great multiplicity of separate and distinct theories of electrical devices certain
broad general principles which supersede a patchwork of theories.

(2) (Routine operations) The quick reduction by routine methods of intricate
problems to mathematical form which otherwise, if they can be reduced at all,
are so reduced by the expenditure of great thought, waste of time, and toilsome
effort.

(3) (Analogies) The generalization reveals analogies leading to the develop-
ment of new machines and disclosing new relations between engineering and pure
science.

(4) (Notation and generalized reference system) Throughout the analysis of
any system only one equation of performance is required. The reference system
is generalized in the sense that the equation of performance and other equations
are valid without change for an infinite variety of coordinate systems. Thus,
after the general equation has been derived for a simple coordinate system that
special coordinate system can be selected which is most suitable for the solution
of the problem at hand.

(5) (Modern algebraic theories) The analysis itself and the resulting equa-
tions of performance make available in engineering the power of modern algebra :

3 If the system is linear and has constant coefficients the system is immediately
solvable by tensor methods. It is solvable in numerous other cases.

NATURE OF THE THEORIES 99

matrices, group transformations, substitutions, elementary divisors, invariants,
etc.

(6) (Modern analysis) The equations of performance are capable of physical
interpretation and are of forms adapted to the methods of modern analysis and
newly developed integrating machines.

It is impossible to give, in a few paragraphs, a clear, detailed, and
comprehensive description of this achievement. It is, however,
possible to sketch the construction and modus operandi of the new
methods as restricted to the material of sections (3) and (4) of this
chapter.

(a) Stationary networks. The differential equations of performance
of a passive network of k meshes are

- f , d, 3 = 1, 2, , *) [1]

where

Gwti* = Lf P i + <> ^ +

4 0) i 4\ 4 0> are properly chosen mesh currents, and e ( , e 2 (0) , ,
4 0) are mesh voltages.

The differential equations of performance of the same network can
also be written

Kj} -* 1 ", (i,j= 1,2, .-.,*) [2]

where

ti 1 *, 4 !) 4 1 * are k branch currents, not necessarily identical with

4 0) ,4V--,e

Equations (1) and (2) are equations of performance of the same
identical network. Equations (1) and (2) are similar in/orw, but are
not identical. The quantities (if, if, - , 4 0) ) and (4 1} , 4. , 4)
are two different sets of dependent variables. This raises the question
as to whether there is anything intrinsic or invariant regarding the net-
work and its behavior, i.e., anything which remains unchanged under
change of variables not only from (if, 4\ -, 4 0) ) to (4 1} , 4. ,
4 1J ) but under all possible changes of sets of variables.

The following questions are suggested: May there exist a hypo-

100

PRELIMINARY DESCRIPTION

thetical current i (a vector quantity 4 ) which can be expressed by many
sets of components; one set of which is (4 0) , 4 0) ,4 > ) another
(*i 1} i 4 i -i 4 l) )? May there exist a hypothetical voltage e (a vector
quantity 4 ) which can be expressed by many sets of components; one
set of which is (ei 0) , e$\ - , 4 0) ), another (#>, \ -, #>). May
there exist a quantity z, labeled the impedance quantity of the net-
work, which can be expressed in many sets of k 2 quantities; one set of
which is zf (i,j = 1, 2, ,&), another $(i,j = 1, 2, ,*)? Does
the scalar product P = e-i yield the power consumed in the circuit
independently of the reference axes of e and i ? The answer to all these
questions is in the affirmative.

It is of course obvious that any network can be disconnected or
broken up into n (n finite) distinct coils, where a coil is defined to be a

Jfj_

<b e

, FIG. 2-2. Primitive Mesh Network.

portion of a circuit possessing an impedance which is independent of
any component of i or e. We imagine any mesh network of physics or
engineering so disconnected and the n component coils of the given
network arranged in a linear configuration or sequence as in Fig. 2 2.
The self-impedances of the n coils are denoted by z l i (i = 1, 2, , n).
Whatever mutual impedances exist between the coils of the given net-
work are indicated on the coils of the linear configuration. The mutual
impedances are 0,7 (i, j = 1, 2, , n), (i j& j) which in the general case
are asymmetrical. The n coils are each short-circuited. It is further
supposed that there exists a voltage in series with each coil as indicated
in Fig. 2 '2. These n voltages are e (i = 1, 2, , n). This linear
sequence of n coils just described is called the primitive network for
all-mesh network systems.

It is clear that a very large number of different prescribed stationary
networks of engineering can be built by the proper connection of n
coils, the only restriction on n being that it is finite. This raises the
fundamental question of the entire matter: Does there exist a mathemat-

4 The definition of vector in this case is not the definition of 1 16, but is that
given in 2*17.

NATURE OF THE THEORIES 101

ical transformation or process, representable as a simple operator or sym-
bol, which corresponds to the physical connection of n coils into any
prescribed mesh network and does the application of this process to the
proper function of the parameters of the primitive network (and also, of
course, parameters of the prescribed network) yield the differential equa-
tions of performance of the prescribed network into which the n coils of the
primitive network are connected? The answer is in the affirmative. The
operator or symbol is Kron's connection tensor.

The full significance of this question and its answer must be under-
stood. The primitive network for the given network and its differential
equations are at once written down. The connection tensor is set up.
The application of this tensor yields the differential equations of per-

FIG. 2*3. Primitive Junction-pair Network.

formance of the given network. Three concepts stand out: (1) primi-
tive network and its differential equations that are easy to establish; (2)
interconnection of coils and its mathematical representation as a trans-
formation; (3) given (or derived) network and its differential equations
of performance which are to be found. The simplicity of the rules for
these steps and the symmetry of the notation preclude the necessity of
difficult thought and largely preclude the possibility of errors in alge-
braic sign or symbolism.

It is necessary, or at least convenient, to view certain given net-
works as junction-pair networks. This is true, for instance, with
vacuum-tube and dielectric networks. Here the elements of the primi-
tive network are not n short-circuited but n open-circuited coils. The
variables in this case are not currents, but the n differences of potential
existing across the n junction-pairs; the admittance tensor 'Y* is the
dual of the impedance tensor Z v - of mesh networks. The coils are
arranged in a linear sequence as represented in Fig. 2-3. A connection
tensor exists which is the dual of that of the mesh network and the pro-
cedure is similar to that for mesh networks.

The most general possible stationary networks are orthogonal net-
works. These are a combination of coils in which both currents and
voltages are impressed. This generalization is effective also to provide
a most general basis for the interconnection of electrical and mechanical
networks into larger or super-networks.

102 PRELIMINARY DESCRIPTION

Finally, it may be necessary to connect together, in an arbitrarily
prescribed fashion, m such networks as described above. Each of the
m networks may be separately analyzed. Next a connection tensor can
be found which corresponds to the physical connection prescribed. The
original work on the m individual networks is preserved. The final
result is the differential equations of performance of the entire network,
composed of the m smaller networks.

The theory finds its greatest usefulness when hypothetical reference
frames (such as symmetrical components, magnetizing and load cur-
rents) and hypothetical design constants (bucking reactances) also are
introduced. It should be mentioned that, in the analysis of a given
network, just one equation of performance results. This is a tensor
equation and it is the equivalent of a system of n differential equations.
Thus the results are sometimes referred to as the equation or equations
of performance.

(b) Rotating electrical machinery. The objective in this theory may
be partially characterized by a comparison with the work of Lagrange
in dynamics, 5 although the latter is comprised by the former in its
broader aspects. Lagrangc's equations, when adapted for constraints
and holonomic and non-holonomic coordinates render the derivation of
the system of differential equations of motion of dynamical systems
largely a matter of routine. The equations of Lagrange formulate the
dynamical problem as a system of equations to be solved; the method
of Lagrange does not solve the differential equations. Kron's re-
searches perform this same function for electrical networks, stationary
or in motion; for vacuum-tube circuits; for every kind of rotating elec-
trical machine under every kind of operating condition ; and finally for
all such systems interconnected. His work also establishes a routine
procedure for the formulation of the equations of complicated physical
systems by the aid of equations established first for simpler systems,
the so-called primitive systems.

Hitherto a large portion of electrical engineering was given over to
multitudinous diverse and independent theories of many machines.
From previous points of view these machines were all different. Kron's
work shows that all types of electrical rotating machines (whether di-
rect current or alternating current) are mathematically identical except
for interconnection of the windings and reference frames assumed.
The primitive machine (there are in fact two, depending upon whether
the reference axes are stationary or rotating), startling in its simplicity,
is (discovered and) defined which includes, when the proper connections

5 See Sees. 5 and 6, Chap. I.

NATURE OF THE PRESENT APPROACH 103

have been made, all types of rotating machines as special cases. The
primitive machine possesses all the fundamental physical and geomet-
rical entities possessed by each individual type of machine, such as
induction motor, compound direct-current motor, synchronous alter-
nator, Schrage motor. The mathematical entity corresponding to the
winding connections of the machine is the connection tensor. The
application of the proper connection tensor (easily set up) with refer-
ence to the physical constants of the primitive machine produces, by a
mathematical process no more difficult than matrix multiplication, the
differential equations of performance of the specific machine under
analysis. The analysis is complete for alternating-current or direct-
current, symmetrical or asymmetrical machines under balanced or
unbalanced loads, for steady-state or transient solutions, and with
constant or accelerated rotor speed.

In addition to all this, the theory then passes on to the interconnec-
tion of rotating machines both with other machines and with other
electrical and mechanical apparatus.

The work of Kron, because of its generalizations, power, synthesis
of apparently diverse phenomena, symmetry and beauty of notation,
and its interrelations with other branches of advanced mathematics
and modern physics should be most pleasing to mathematicians and
engineers.

The results and analysis of this new development are also expressi-
ble in the general language of multidimensional geometries.

2-4. Nature of the Present Approach. The approach to the oper-
ational calculus of Heaviside in Vol. I, Chap. IV, was a mathematical
one, i.e., by means of ordinary linear differential equations and the.
theory of functions of a complex variable. This approach was rapid
and necessitated no knowledge of engineering. It has been justified by
the response from readers of the first volume.

The method of approach to the material of this chapter is likewise a
mathematical one in the sense that the prerequisite pure mathematics
employed is explained prior to entry upon the theories of the chapter.
It is a mathematical approach also in the sense that the minimum engi-
neering knowledge is presupposed.

(2)
Matrices and Linear Transformations

A knowledge of matrices and linear transformations is prerequisite
for an introduction to tensor theory.

104 MATRICES AND LINEAR TRANSFORMATIONS

2 5. Definitions. The rectangular array

f

011 012
021 022

\
CL\fL

' ' 02n

\ 0ml 0m2

' ' a mn l

or

011 012 ' ' ' 01n
021 022 " * " 02n

or

a n

012

...

am

a 2 i

22

...

"2*

0ml

0m2

0mn

composed ofmXn numbers or functions a,-/ is called a matrix. Abbre-
viated notations for the above matrix are A or (fl t -y). The w X n num-
bers are the elements of the matrix. As a special case m may equal n.
In this special case the matrix is not a determinant, although the matrix
and the symbol of a determinant may be identical.

Two matrices, (a#) and (&#), each with m rows and n columns are
equal only in case all corresponding elements are equal, i.e., <z# = &#.
A zero matrix is one, all of whose elements are zero. A unit matrix is a
square matrix such that 0,7 = 1, i = j and ay = 0, i ^ j. A square
matrix (a#) such that a t j = 0, i ^ j and a# 5^ for i = j is called a
diagonal matrix. A diagonal matrix each of whose diagonal terms is /
is a scalar matrix. The determinant of the square matrix

011 012 ' 01n
021 022 ' ' ' 02n
0nl 0n2 ' ' ' 0nn

is the determinant

011 012 0i

021 022 ' ' ' 02n

0nl

0n

The matrix (a,-,-) is said to be singular or non-singular according as the
determinant \ a^ \ does or does not vanish.

2-6. Rank, Adjoint, Transpose, Inverse, Sum. From the matrix
(0*v)> possessing more than one element, other matrices may be formed
by striking out of (a#) certain rows and columns. The determinants
of the square matrices so formed, are called the determinants of (&;,)
A matrix is defined to be of rank r if there exists at least one r-rowed
determinant of (a^) which is not zero while every determinant of
order (r + 1) of (a#) is zero.

The adjoint of the matrix A is defined as the matrix

Adj. A

AH

12 -

2n

RANK, ADJOINT, TRANSPOSE, INVERSE, SUM

105

where Ay is the cof actor of the element ay in the determinant
The cofactor Ay = (- I)'" 1 "' Af#, where Af t -y is the minor of the element
ay. The minor Mij is the ( 1) rowed determinant formed from
| &ij | by deleting the ith row and jth column.

The matrix A, formed from A by employing the successive rows of
A as the successive columns of A/, is called the transpose of A.

The inverse A" 1 of the square matrix A is defined by the equation

A- 1 -

a

dl=

a

where AIJ are the cof actors defined above and a is the determinant of
The inverse of a non-square matrix is not defined.

A rule for rapid computation of A"" 1 is:

1. Write down the transpose A of A.

2. Replace each element of A by its minor.

3. Divide each element of the matrix in (2) by the determinant
a of A.

4. Give to each element of the final matrix in (3) an algebraic sign
according to the checkerboard array

The sum of two matrices (each m X n) is defined to be an m X n
matrix each of whose elements is the sum of the corresponding elements
of the two given matrices. Likewise, the difference of two m X n
matrices is an m X n matrix each of whose elements is the difference
of the elements of the two original matrices. For example,

012

n

012

021 =b

106

MATRICES AND LINEAR TRANSFORMATIONS

EXERCISES I

1. Determine the rank of the matrix

"2

3

4

6

7"

1

7

-3

sin x

4

1

2

4

e

3

4

6

8

12

14

_2

3

-1

7

2_

2. Compute the adjoint of each of the matrices

"235

1 3 -1

46-5
J 7 9

3. Compute the inverse of each of the above matrices.

4. Compute the inverse of the diagonal matrix

4"

"-1

2

3"

2

and

1

4

-1

2

3

3

4

6

1

Ll_

7

5

2

3_

B

C

D

5. A matrix A is symmetrical if a,, = a,,. A matrix A is said to be skew-sym-
metric if a t j = dji, i T& j and a t ; = 0. Show that a general matrix A can be expressed
as B -f- C where B is a symmetric matrix and C is a skew-symmetric matrix.

2 7. Linear Forms, Linear Transformations. Matrices are impor-
tant in the expansion of functions, the solution of systems of ordinary
differential equations (Sec. 4, Chap. Ill), and in making transforma-
tions of variables. We begin with the simplest cases. The equations

[3]

a m \i\

define m linear forms in the n variables i\, , i n . Suppose that the
n variables ii, , i n are linearly expressible in terms of $ new variables
*i4 "** that is,

4 tf-i,2,

[4]

LINEAR FORMS, LINEAR TRANSFORMATIONS
Substituting these values of ij in Eqs. (3) we have

107

where

n s 3

2D Z^ aij bjk *

4** (- 1,2, ,) [S]
(- 1,2, ..,; *- 1,2, -,) [6]

Since i ranges from 1 to m and k ranges from 1 to 5 the elements c&
may be written out in the form of an m by s matrix C which is

It is evident that this expression can be obtained from the matrices
A and B by a routine manioulation. By inspection of matrix C and
the two matrices

a i2

*mn_

and B

bn biz 61,

it is evident that the element in the first row and column of C can be
obtained from A and B by multiplying the successive elements of the
first row of A by the successive elements of the first column of B and
adding the resulting n products. Likewise, the element in the ith row
and Jfeth column of C is obtained by multiplying the successive elements
of the ith row of A by the successive elements of the kth column of B
and adding the n products.

The matrix C is defined to be the product of A and B. This product
will be written A-B.

Finally, we have the important theorem that a linear transforma-
tion (Eqs. 4) with matrix B replaces a set of linear forms (Eqs. 3)
with matrix A by a set of linear forms (Eqs. S) with matrix A-B.

This theorem is as useful in change of reference axes both in linear
network analysis and in the theory of vibrations as in the study of
projective geometry.

108 MATRICES AND LINEAR TRANSFORMATIONS

EXERCISES n
1. If

Li - 2*2 + 3* 2 - 4*3 + 7*4,
2 - -2*i + 7*2 - *3 + 3*4,
3 = 7*1 + 5*2 - 9*3 + *4,

4 = 11*1 + *2 *3 + 3*4,

and

*2 yi 3^ + 7^3 + 3^4,
*s = 2yi 3^2 + 5ys + 7y4,
*4 3yi -|- 5^2 + ya + 5^4,

express, by means of the theorem of 2 7; LI, 2, 3, 4 as functions of yi, yi, yz, yi-

2-8. Multiplication of Matrices. The product A-B of two matrices
A and B was defined in 2-7. If the positions of A and B are inter-
changed above and the multiplication indicated by the product B- A
is performed, then it is evident that A-B 7* B-A. Thus the multipli-
cation of matrices is, in general, not commutative.

The multiplication of matrices is associative. Let A = (0#), B =
(by), C = ($ be any m Xn, n X s, s X t matrices respectively. To
see that multiplication is associative it is sufficient to show that a
general element of (A B) C is identical to the same element of A (B C) .
By the rule for the product of two matrices (Eq. 6) the element il of

(A-B) -C is

* / \

VM V 1 * ir m

/ J \ /. a>ij ojk ] CM- 17 J

tt\7=* I
The element fl of B C is

and the element il of A- (B-C) is

Since the finite sums (7) and (8) are identical (A-B)-C = A- (B-C).
The arrangements of the rows and columns, i.e., m X n, n X s, s X t
should be noted. (See Ex. 4, problem set III.)

The reasoning above can be applied to any finite number of ma-
trices, as long as the order of the matrices is preserved.

DIFFERENTIATION AND INTEGRATION

109

The ik element of A-B + B-C is

The multiplication of matrices is distributive. To show that
A-(B + C) = A-B + A-C it is sufficient to show that a general ele-
ment of the matrix A-(B + C) is equal to the corresponding element
of the matrix A-B + AC. (The sum of matrices is defined in 2-6.
The i& element of A- (B + C) is

+

+ a>ij Cjk).

In the same manner it is shown that

(B + C)-A = B-A + C-A

and the proofs are extensible to any number of matrices.

2-9. Division. Division by a non-singular (two 6 ) matrix A is de-
fined to be multiplication by A" 1 . The product of A"" 1 -A and A- A"" 1
is the unit matrix I.

2 10. Differentiation and Integration. The derivative and integral
of a matrix are defined as follows. The derivative with respect to a
single variable t of a matrix is found by differentiating each component
separately. For example,

*-

dt

' t 2 sin /"

f t cos t

sin t 4 cos

" 1 cos/"

2/ 1 -sin /

cos/ sin /

The derivative of a matrix is, of course, a matrix.

A matrix is integrated with respect to a single variable by integrat-
ing each of its components separately. The integral of a matrix is a
matrix.

EXERCISES m
1. Compute the product A-B-C, where

246-3"
-7892

6310
11 9 7 3

B =

' 6 -1
7
2 4

-3 3

3 9 11
2 1 -1

6 Division for more general matrices is not defined.

110 MATRICES AND LINEAR TRANSFORMATIONS

2. Verify that:

(a) (A-B), = (B,)-(A,),

(*) (A-B)- - (B-'HA-*),

(c) (A-B-C), = (C<)-(B,)-(A).

(d) (A-B C)- 1 = (C- 1 ) (B- 1 ) (A- 1 ).

3. Prove that scalar matrices are the only matrices commutative with every
n X n matrix.

4. The proof given in 2-8 for the validity of the associative law in matrix
multiplication was for the three matrices A, B, and C of dimensions respectively
m X n, n X s, s X /. Show that

if A and B are n X n and e is I X n matrix.
5. Prove that, for m a scalar,

man

6. The equation

_jna 9 i

,(X) = I \I - [a] | - 0;

where [a] is an w-rowed square matrix whose elements a tj - are constant, 7 is a unit
matrix, X is a parameter, and | A/ [a] | is a determinant, is called the character-
istic equation of [a]. If [a] is a square matrix and <f>(\) = is its characteristic
equation then ^>([o]) = 0. Verify this theorem for [a] a square matrix of order
three.

7. Compute the derivative of the matrix

"1 2 sin x '

2 2e sin 2x

_sin x sin 2x 3e

8. Compute the derivative of the determinant

1 2

2 2e

sin x sin 2x

sin x

sin 2x

3e

9. Compute the reciprocal, in terms of the determinant, of the determinant itself
in Ex. 8.

2-11. Three-matrices. The matrices of 2-5-2-10 are 2- or 1-
matrices, i.e., the number of elements in such matrices is m X n or
1 X n and the elements are arranged in either a rectangle or line. In
the applications which follow the elements of a matrix are parameters

INDEX NOTATION

111

belonging to some circuit or physical system. It is necessary to iden-
tify these elements with the physical system. For example, if i is the
current written

a b c k

then i a may signify the component of i or the current associated with
the a component or a mesh of the circuit. Likewise, if z is a matrix of
impedances and is written

a b c n

m

%aa

Z a b

Zac

Zan

Zba

Zbb

Zbc

Zbn

Zca

Zeb

Zee

%cn

Zma

Zmb

Zmc

Zmn

then z^ may denote the mutual impedance between mesh b and mesn c.

A 3-matrix possesses m X n X r elements. These may be arranged

in a box. The symbol A a bc, for example, denotes a specific component

b ' c

Zaa

Zab

Zac

Zan

Zba

Zbb

*%i

Zbn

Zea

Zcb

Zee

Zen

Zma

Zmb

Zmc

Zmn

Aaaa

Aaba

Abaa

7

FIG. 2-4. Two-matrix.

FIG. 2 5. Three-matrix.

or element of A, namely, that component located in the shaded volume
of Fig. 2-5.

2 12. Index Notation. Evidently, for higher dimensional matrices
a more convenient notation is imperative. Accordingly, the symbol

112

MATRICES AND LINEAR TRANSFORMATIONS

Aap will denote a 2-matrix such as shown in Fig. 2-4. The Greek
letters a and arc known as variable indices. It is understood that
they assume all values a,b,c t . Thus A^ denotes all the m X n
components of a 2-matrix and these are arranged in a rectangle, i.e.,
the matrix itself. On the other hand, A^ denotes one definite element
of Aafi, namely, the element shaded in Fig. 2-4. The Roman letters
a,b,c, are fixed indices. The symbol A a i> contains one variable
and one fixed index. This symbol denotes the m elements of the b
column of the matrix of Fig. 2-4.

The symbol A^y denotes the 3-matrix of Fig. 2S. The symbol
Aafta denotes the 2-matrix forming the face, nearest the reader, of the
cube in Fig. 2-5. A 3-matrix can be represented on paper as a set
of 2-matrices. For example, the 3-matrix of Fig. 2-5 can be repre-
sented as the set of r matrices

Actpat -^-aftbt ' * "

EXERCISES IV

1. Represent, geometrically, i.e., by drawing rectangles and labeling with proper
notations, the 4-matrix A a ^ y (a = a,b, c, , m; |3 = a, b, c, , n\ d = a, b, c, -,
r; 7 = a, b, c, ,$) as a set of (s X r) 2-matrices.

2. Represent, geometrically, i.e., by drawing cubes and labeling with proper
notations, the 5-matrix Aapyh (a = a, b, c, , m\ = a, b, c, ,; y = a, &,
c, , r; 5 = a, b, c, , s; e = a, b, c, -, /) as a set of (/ X s) 3-matrices.

3. Represent, geometrically, i.e., by drawing rectangles and labeling with proper
notations, the 5-matrix of Ex. 2 as a set of (t X s X r) 2-matrices.

2 13. Applications of Matrix Notation, (a) Solution of linear,
non-homogeneous, algebraic equations. The system of equations

[9]

= e n ,

can be written
where

[10]

2

, e

e 2

SOLUTIONS BY MATRICES 113

Multiplying both sides of Eqs. (10) by A"" 1 we have

which is the complete solution of Eqs. (9) for the variables ii, i%, , i n .

(b) Stationary circuit equations. Equations (1) can be written

Z-i = e. [11]

This equation is of the form of Eq. (10).

(c) Differential equations of vibrations. The potential and kinetic
energies of a discrete dynamical system are given by Eqs. (54-55)
Chap. I. Lagrange's equations (1 12) for such a system are

ft

5<

(r = 1, 2, , n). [12]

These equations are written as the single matrix equation

WBfl + MM - 0. [12a]

EXERCISES V

1. Solve, by the method of this article, the equations

*i + 2*2 + 3*3 + 4*4 - 34,
-*i + 3*2 + 7*3 + 2*4 - 36,
4*i + 8*2 + 5*3 + 6*4 - 69,
4*i + 7*2 + 3*3 + 4*4 = 39.

2. Express Eq. (49) Chap. I in the form of Eqs. (12) of this chapter.

2 14. Solutions by Matrices. Sec. 4, Chap. Ill is devoted to the
solution of systems of linear differential equations by means of matrices.
Further theory of matrices such as raising of matrices to high powers
and the expansion of analytic real functions in matrix form is found
in the above section.

The solution, by Cramer's rule, of many linear equations in equally
many unknowns is often cumbersome. A method of Kron, employing
compound tensors, 7 reduces both the complexity and the probability of
errors.

7 Gabriel Kron, Tensor Analysis of Networks, Chap. IX. John Wiley and Sons,
1939.

114 PRELIMINARY CONCEPTS OF TENSOR ANALYSIS

(3)
Preliminary Concepts of Tensor Analysis

We shall need in Sec. 4 a preliminary knowledge of: (a) Definitions
of tensors, (b) algebra of tensors, (c) inner multiplication and contrac-
tion, and (d) quotient law of tensors.

2 15. The Summation Convention. In sums, such as

*y

*P7- *

^T 7^T

the summation signs may be omitted and the expressions written

if it is understood that whenever an index (subscript or superscript)
appears twice in a term that term is to be summed for certain definite
values (usually n) of the index. Thus

+ e n i n ,
+ Z an i n .

The index which appears twice is called a dummy index. A dummy
index may be denoted by a different letter even in the same equation,
e.g., Z a pip = Z ay iy. The remaining indices of the equation are called
free indices. In Z a 0ip, is the dummy index and a is a free index.

The convention is not a mere abbreviation, but a valuable tool
indicating and performing automatically certain operations in the
analysis.

2 16. Definitions. Matrices of dimensions 1,2, , n have been
defined in Sec. 2 of this chapter. Suppose henceforth that the elements
of the matrices are functions of general independent coordinates 8
X 1 1 x 2 t ,#". Let there be a change from the coordinate system
x l t x 2 , , x n to the system x 1 ', # 2/ , , x n ' 9 where the re*' are defined
by the equations

x if = * V, x 2 , - , x n ) (i = 1, 2, , n) [13]

and where the n functions tf are independent real functions of jc 1 , x 2 ,
, x n . Equations (13) define a transformation of coordinates of an

8 The superscript n in x n is not to be confused with an exponent. The symbol # w
is an abbreviation for x (n) .

TENSORS OF VALENCE ONE 115

w-dimensional space. Since the n functions <?* are independent, the
x's are expressible in terms of x l ''s as

a'-fV',* 2 ', ,**') (t-1,2, ..- f n). [14]

A tensor is sometimes defined as a matrix A plus a definite law of
transformation for the components of the matrix when the coordinate
system or reference frame is subjected to very general transformations.
We shall see presently these definite laws of transformation whereby
the components A a , of the matrix A are changed to the components
A a> of a new matrix when the independent variables are changed from
x 1 , x 2 , , x n to x 1 ', x 2 ' j , of 1 ' by means of Eqs. (14). In the follow-
ing, the change of coordinates from x 1 , x 2 , , x n to x l ' 9 # 2 ', , x n '
is understood to be by the general transformation of Eqs. (14).

2-17. Tensors of Valence One. If under change of coordinates
from x 1 , x 2 , , x n to x 1 ' ', x 2 ', ,x n ' the components of the 1-matrix
A m (m = 1, 2, , n) are changed from A m to A m ' according to the
law 9

m '

m , ('- 1,2, ,) [IS]

then the 1-matrix is called a contravariant tensor of valence one, or a
contravariant vector.

If under change of coordinates from x 1 , x 2 , ,#" to x l ' t # 2 ', , x n '
the components of a 1-matrix A m (m = 1, 2, , n) are changed from
A m to A m ' according to the law

m , (m- 1,2, .-.,) [16]

then the 1-matrix is called a covariant tensor of valence one, or a
covariant vector. The definitions of vectors just given are those re-
ferred to in 2 3. From these definitions it is to be noted that a vector
does not necessarily possess either magnitude or direction. A vector is
merely a matrix whose components obey, under change of reference
system, one of the laws of transformation (15) or (16). Vectors of
vector analysis possess physical significance, but their components are
fictitious quantities. The vectors now considered are, in general, fic-
titious quantities but their components possess physical significance.
However, the vectors of conventional vector analysis satisfy Eqs. (15)
and (16) and occur as exceptional and special cases of the more general
definitions of vectors here given. The restriction that a vector possess
magnitude and direction is no longer imposed.

9 For a clearer and more useful expression of this law see Sec. 4, 2*25.

116

PRELIMINARY CONCEPTS OF TENSOR ANALYSIS

EXAMPLE 1. The components of the vector A are A 1 and A 2 or
A 1 ' and A 2 ' according as the reference axes are (x l 9 x?) or (x v , # 2 ').
(See Fig. 2-60.) It is shown in this example that the new components
A v ', A 2 ' of A, under a particular change of coordinates, are computed
from the components A 1 9 A 2 of A by Eqs. (IS). The equations of the

FIG. 2-6

particular transformation of coordinates under consideration are, from
Fig. 2-66,

x l = Oc be = x 1 ' cos 6 x 2 ' sin 0,

or

x v =
# 2 ' =

From Eqs. (15)

A = a A

1 cos + x 2 sin 0,
x 1 sin + # 2 cos

Jl I A2

* + ^

By inspection of Fig. 2-66 it is evident that the components of A m ' of
the vector A in terms of the components A m are given by the last
equations.

If the new components of A 1 ', A 2 ' can be computed, by means of
Eqs. (15), from A 1 , A 2 , not merely for the particular transformation

x v = x 1 cos 6 + i? sin 0,

but for all changes of coordinates x*' = <? (x l 9 x 2 9 , x n ) possible rela-
tive to A then the vector A is a tensor of valence one.

EXAMPLE 2. The concept of covariant vector does not lend itself
so easily to graphical illustration. Let F(x l , *?, , x n ) be a scalar

TENSORS OF VALENCE TWO 117

function of position such that its value remains unchanged at a fixed
point of space regardless of the coordinate system employed. Then
the n quantities

9F 9F 9F

a* 1 ' a* 2 '*"' a**'

under transformation (14), are transformed according to Eqs. (16) and
constitute the n components of a covariant vector or tensor of valence
one. (See example 2 2 18 for an illustration possessing more obvious
physical significance.)

2-18. Tensors of Valence Two. If under change of coordinates
from x 1 , y? t -, x n to x 1 ', x 2 ', , x n> the components of a 2-matrix
are changed from Y& to Y*'? according to the law

n 71
L J

then the set F^ is a contravariant tensor of valence two. The varia-
bles in YP on the right side of Eqs. (17) are changed from x 1 , x 2 , , x n
to x 1 ', # 2 ', -, x n ' by means of Eqs. (14).

If under change of coordinates from x 1 , x?, , x n to x v ,x 2 ', , x n '
the components of a 2-matrix are changed from Z^ to Z a >p according
to the law

then the set Z a $ is a covariant tensor of valence two.

EXAMPLE 1. Covariant tensor, valence 2. A rereading of the par-
agraph on stationary networks in 2-3 will be of aid in the following
example.

Consider the circuit of Fig. 2-7. If i l and i 2 are the mesh currents
of the network shown in Fig. 2 7a, then by Kirchhoff 's second law the
differential equations of performance are

Z n i l

or Z mn i n = e m or Z-i - e, [19]

where

RH,

L 2 ip, Z 22 = (L 22 + Li 2 )p + R 22 + TTT*

C 22 p

118 PRELIMINARY CONCEPTS OF TENSOR ANALYSIS

Let us again write the differential equations of performance by using
branch currents i 1 ', i 2 ' where, from Fig. 2-76

[20]

where

_ in #u _. LnCuX*

Hgywr WM, i @^nrrH!

(a) (6)

FIG. 2 - 7. Two Reference Frames.

The differential equations, by KirchhofTs second law are

=

Z'l ; i ^ *2'
2'!'^ "T ^2'2'^ ==

where

= -(

) ' ^2'2'
(^22 +

and

^

Consider the matrices

* 1 Z 12 ; 2 1
i 1 Z 22 i 2 J

[21]

[22]

Equations (20) are a special case of Eqs. (14). The quantities Z n
and Zi/i>, for example, correspond to Z\\ and 7,\'\> of Eqs. (18). We

TENSORS OF VALENCE TWO

119

shall compute Z vv by means of Eqs. (18) and compare the values
obtained with those of Eqs. (21-22). We have

R 22

CzzP

L 22 )p

R 22

C 22 p

This is the same value for Z r i/ as that obtained from Eqs. (21-22).

By the same procedure it is easily shown that Z^i, Zi^, and Z^i
for the particular equations of transformation given by Eqs. (20), are
computed by Eqs. (18).

It must be pointed out that Eqs. (20) are only one special case of
Eqs. (14) relative to Figs. 2-7. In Figs. 2 7 only two reference frames
are shown. Many others exist. Twelve of these are shown in Fig. 2 8.

1

1

1

1

FIG. 2-8. Additional Reference Frames.

The remaining twelve are obtained by interchanging i l and i 2 on the
diagrams of Fig. 2-8.

EXAMPLE 2. Co/variant tensor, valence one. It is evident from
Eqs. (21-22) that the 1-matrices [e\, e 2 ] and [0r, e 2 '] can be viewed as
functions of i 1 , i 2 , i 1 ', and i 2 '. Let the components of e\> and ey be

120 PRELIMINARY CONCEPTS OF TENSOR ANALYSIS

computed by Eqs. (16) where the equations of transformation of
coordinates are Eqs. (20). We have

9* 1 _L. 9*' 2

e l' = Z^p 01 + T^p 02 = 01+ 02,

_L

02' T^/- 01 + -& 02 = 02-

o* o*

These values agree precisely with the values of e\ t and e y given by the
last of Eqs. (22).

2 19. Mixed Tensors and Tensors of Higher Valence. If under
change of coordinates from x l ,x 2 , , x n to x l ',x 2 ', , x n> the com-
ponents A of A are changed to A$ by the law

"- [23]

then A% is a mixed tensor of valence two.

The concept of tensor is generalized to those of higher valence. For
example,

9*" 9*" a*' i

is the law of transformation for a tensor, contravariant of valence one
and covariant of valence three.

2 20. Collection of Laws of Transformation. The classification of
the formulas of 2 17-2 19 are as follows :

/ 9# m/
Contravariant tensor, valence one A m = A m

dx m
Covariant tensor, valence one A m -

Contravariant tensor, valence two 7*'*' = Y mn -^ [27]

Covariant tensor, valence two Z a *p = Z
Mixed tensor, valence two Aa* A

Mixed tensor, valence four A^ = A^ ^^"^P ~^T '

[30]

ADDITION AND SUBTRACTION OF TENSORS 121

In formulas (25-30) x l , x 2 , , x n are old coordinates, whereas
x l ' t jc 2 ', , x n ' always denote new coordinates. By inspection of the
above formulas we have

f below
When the indices are \ on the left side of the equation of trans-

formation of components the new coordinates (#*') are \ on the right

{above

side of the equation.

EXERCISES VI

1. Write down the 24 matrices referred to in example 1, 2-18. Show that:

(a) If A and B are any two of the 24 matrices then A-B always yields a third
matrix of the set.

(b) If A, B, and D are any three of the 24 matrices then (A-B) -D * A-(B-D).

(c) One C matrix is unit matrix, which belongs to the set.

(d) Each matrix of the set has an inverse which belongs to the set.

2. The components of the matrix Amnr form a mixed tensor contravariant of
valence 2 and covariant of valence 3. By analogy with the law of transformation of
Eq. (30) write the law of transformation for the above tensor. If m, n, r, s t / each
range over the integers 1, 2, 3, 4, how many components has the above tensor?

3. Express the rule for the multiplication of two 2-matrices in index notation.

4. A 3-matrix A a ^ can be split up into 2-matrices A a p\, A a pi, -, one for each
fixed value of y. The product of A a fo times A * e is a 3-matrix. This product can be
formed by multiplying <4 a 0i A a ^ sequentially by A fa and arranging the result-
ing product in a cube. Indicate these operations in index notation.

5. In example 1 of 2 17 let A be a vector (sense of conventional vector analysis)
in 3-dimensional space. Let the transformation be from rectangular coordinates
x, y, z to polar coordinates r, 0, <p, where the equations of transformation of coordi-
nates are

x l - * 1 ' sin 9? cos T f *' - [(.r 1 ) 2 + (* 2 ) 2 +

2 I' - * - ^ 2' , -1 R* 1 ) 2 *^ 2 ) 2 "]*

* 2 = x l sin y? sin *% y? = tan l - 5 - ,

x 3 = x 1 ' cos x* t **' = tan"" 1 -j ,

where x l - *, x 2 = y, x 3 s, x l> '- r, x* = 0, * 3 '= *>. Obtain the components A*'
of A in the new system geometrically and by means of Eqs. (25) and establish their
identity.

6. Compute, by means of transformation formula (28) the components Z\v, Zv\,
Zw, (Zi'i* was computed in 2*18) for the circuit shown in Fig. 2*7.

7. Matrix Y is the inverse of matrix Z. Solve Eqs. (19) and (21) for i anrf '
obtaining i = Ye and 1' = Y'e'. Compute F 1 ' 1 ' by means of the value F 11 ,
formula (27), and the equations of transformation e[ = e\ + e%, e% = e%. Show
that this value of Y 1 ' 1 ' is identical to that obtained in the equations i' Y'-e'.

8. Compute, as in Ex. 7, F 1 ' 2 ', F 2 ' 1 ', and F 2 ' 2 '.

2*21. Addition and Subtraction of Tensors. Two tensors are of
the same valence if they possess the same number of indices. They are

122 PRELIMINARY CONCEPTS OF TENSOR ANALYSIS

of the same kind if they possess the same covariant and contravariant
character.

The sum or difference of two tensors of the same kind is a tensor
of the same kind. It will suffice to display the proof for two tensors
each of valence four. Let the two tensors be

and

9**' 9* 7 ' 9* mi 9***

Adding and subtracting these equations we have

AW 4-

-fla'0' T

^ , ^/ ^

Since this equation is identical in form to the equation of transforma-
tion of each member of the sum, the sum itself is a tensor of valence
four, covariant of valence two and contravariant of valence two.

2*22. Inner Product and Contraction. It is recalled from 1-16
that the scalar product

A-B = A X B X + AyB y + AgBg,

where A = i A x + j A y + k A z and B = i B x + j B y + k B z . It is
pointed out that here the product of two vectors turns out to be a
function of lower valence than either of the multipliers, i.e., turns out
to be a scalar.

An analogous product exists in tensor analysis. In the last of Eqs
(29)if/3 7 - a' or/ = t' then

where k is a dummy index. This simplification is due to the relation

^1'^j.^^ij. o.^

a** a* 1 ' + a** a* 2 ' + " ' + a**

3*' | if ^

' a** " 1 1 if /

[33]

since x l and of are independent coordinates. If a function A(x 1 ', of',
,#*') is obtainable from a function A(x l , a; 2 , , x n ) by means of
the equations of transformation Eqs. (14) without employment of par-

QUOTIENT RULE 123

tial derivatives, the A is a scalar. Evidently, in Eq. (32) A\ is a
scalar. Since no derivatives appear in the transformation formula of
a scalar, a scalar is also called a tensor of valence zero.

The product of two vectors represented in Eq. (32) is called the
inner product. This product is called also contraction because a tensor
of valence zero is obtained from a tensor A] of valence two. The
process of contraction is applicable not only to tensors of valence two,
which are built as the product of two tensors of valence one, but to all
mixed tensors of valence two. Moreover, it is applicable to all mixed
tensors. Before establishing this fact it is convenient to introduce
the concept of Kronecker deltas.

From Eqs. (13) the #*''s are functions of the #*'s and from Eqs. (14)
the x l 's are functions of the x l ''s. It is recalled that the x % '*s and x l 's
each form a set of n independent coordinates. Thus

9^ = 9^9^ = k = fl if k = j
8**' a* l 'a*>' ' JO lik^j

and [34]

3*^ do^_dx^ (l if k' = f
dx>'' ~ Qx* &?' '" 10 if k'*j'.

The deltas defined above are called Kronecker deltas.

In the mixed tensor Ai m the expression Af k is the sum of n compo-
nents of A\ m . That this sum is a tensor of valence one is shown as
follows. From Eqs. (18)

* for m = *'

In the two examples above it is evident that contraction has pro-
duced in each a tensor which is two valences lower than the original
tensor. This result is true regardless of the valence of the original
mixed tensor.

2-23. Quotient Rule. The physical dimensions of an unknown
quantity occurring in a physical equation can be determined by the
conditions that each term of the equation possess the same dimensions.
In like manner the contravariant and covariant valence of an unknown

124 STATIONARY NETWORKS

quantity entering a tensor equation can be inferred. For example, if

then (A) must be a tensor of the form A".

In fact there exists the theorem: A quantity which, when subjected
to inner multiplication by an arbitrary covariant (or any arbitrary
contravariant) vector, always yields a tensor is itself a tensor.

2 24. Summary. The results of 2 15-2 23 are sufficient formal
theory for the study of stationary networks. Additional formal parts of
tensor analysis are developed as required in later sections.

An important aspect of tensor analysis is the fact that if a tensor
equation holds in one system of coordinates, it continues to hold under
any possible change of coordinate system. Example 1, 2-18 illus-
trates this important fact and anticipates, with a very simple and
special example, the more general results of 2-40, Sec. 4.

New tensors are recognized by investigating directly their trans-
formation laws, by the fact that the sum, difference, and product of two
tensors is a tensor and by applications of the quotient law. Use is made
of the second generalization postulate (2-28) in establishing the
tensor character of a mathematical entity.

EXERCISES VII

1. Show that if the Kronecker deltas are taken as the components of a mixed
tensor of valence two in one set of coordinates then they are the components of a
tensor in any set of coordinates.

2. Show that, by multiplication and contraction, the tensor A^ B jst of valence
three can be obtained from AIJ and B rst .

3. Note that it is possible to have an inner product of two non-zero tensors equal
to zero.

4. Represent on paper, by means of cube and rectangle, the processes carried out
in example 2.

5. Show that, by multiplication and contraction, the scalar A^ can be obtained
from the tensors A* and A y &.

6. Show that the order of the factors in Exs. 2 and 5 is immaterial.

(4)
Stationary Networks

Sections 2 and 3 contain those elements of the classical theory of
matrices and tensors prerequisite for the study of stationary networks.

Section 4 is an introduction to the study, by means of tensor analy-
sis, of stationary networks.

KRON'S FORM OF THE TRANSFORMATION FORMULAS 125

(a)
GENERAL THEORY

2-25. Kron's Form of the Transformation Formulas. Kron ex-
presses the transformation formulas (Eqs. 25-30) in a very convenient
form. Let us re-examine formulas (25-30). Equations (13) and (14),
namely,

#*' = v*(x l , of, -, x n ) (New in terms of old variables) [35]
x i = ^(x 1 ' , *'i , * n/ ) (Old in terms of new variables) [36]

with the conditions imposed in 2 16, remain the equations of trans-
formation of variables. In Eqs. (25) and (26), if the indices m and a

both range from 1 to n then both sets of quantities - and

L

may be defined in matrix form. The defining matrices are

a* 1 "

and

r m' _ p-l

<^a ^

a^'
a*"

La* 1 "'

.a* 1 a* 2

'a* 1 a* 1
a* 1 ' a* 2 '

a* 2 ax 2
a* 1 ' a* 2 '

[37]

3^1

a*'
a* 2

.a* 1 '

a?.

[38]

The matrix of Eq. (37) is denoted by Q* (or C" 1 ) and that of Eq. (38)
by C%' (or C). The inverse of C is not, in general, calculated as the
inverse of other matrices. (See Ex. 2.)

In view of the notations above, transformation formulas (15-16)
become respectively

126

STATIONARY NETWORKS

A*-A*Cf, or A'

A m

or A'

C *-A (Components of A being con- [39]
travariant),

= A C (Components of A being co- [40]
variant),

where the matrices [A a ] and [A a ] are components of contravariant and
covariant vectors.

In the classical theory the attention is focused on Eq. (13). In this
theory formulas (39-40) are the formulas for transformation of vector
components. In Kron's work the attention is directed on Rqs. (14) and
the equivalent 10 of (39-40) which are

*m r

A"

A a

C m .A m
C? A m >

or i"
or e a

^m' *
?' *m'

or i
or e

C-i',
e'-C- 1 ,

[39a]
[40a]

where A" = i a and A a = e a and i a and e a are usually components of cur-
rent and voltage respectively. If there is a most fundamental equation, it
is Eq. (39a).

In the defining of Eqs. (37-38) for C"" 1 and C the arrangement of
old and new variables is displayed in the partial derivatives them-
selves. It is much more convenient to display the reference frames
or old and new variables by labeling the rows and columns of the
tensors, i.e.,

a' b'-.-n' a b n

a

ct

c*

b

c*.

ct

ct

n

cj

a-

c;.

A" =

A*

A b \---

A n

[41]

[42]

10 Attention must be given Eqs. (37, 38, 39, 40). The symbol C is a more gen-

3jc a
eral symbol than - The variables in Eqs. (35, 36) denote holonomic coordinates.

9*

The corresponding electrical variables are charges. Since many electrical net-
works and machines are non-holonomic dynamical systems (See Sec. 6, Chap. I and
2-47.), in the general case no such relations can be established between the charges.

3t a
However, in linear stationary networks, the C tensor can be found formally by ,

3*

where the i and '' are currents (velocities). (For quasi-coordinates see Ref. 10 at
end of Chap. I.)

KRON'S FORM OF THE TRANSFORMATION FORMULAS 127
The n Eqs. (390) are now written as the single equation

a' b' . n'

a

i a

a

C'

ci-

c%.

a

i"'

b

i b

b

c" a -

f^b

c b >

Lr W '

b

i b>

.

:

=

'

'

'

n

i n

n

CV

n

i-'

[43]

The computation of the entire set of new components (new vector) is
accomplished by the formal process of mere matrix multiplication.
It is emphasized that thc/orwaJ manipulation is matrix multiplication,
but the analysis is not matrix analysis. The analysis is tensor analysis.
In the same manner formulas (27-30) are expressed

- Y mn

i-l

= Cj'Z'C

_ At
A

-m /~-n /~-s /-TJ
^t I C, Cj .

[441
[45]
[46]
[47]

I below
The rule of 2-20 is transcribed to read: When the indices are }

[ above

on the left side of the equation of transformation of components, the same

| below \ below

indices are \ , on the C's. Whenever a dummy index appears } .

[above {above

on the components of the old tensor it appears \ on the C's.

[fix old 1
new I i Which

is given by Eq. (37). The fundamental equation of transformation
to keep in mind is (39a) which corresponds to Eqs. (14) of 2- 16.
Equations (13) and (14) are the equations of transformation of the
.classical theory. The advantages of (39a) will appear in the applica-
tions which follow.

128 STATIONARY NETWORKS

Equations (45) written explicitly are

a' b' -.. n' a b n

a'
b'

a
b

z a . a ,

z, b .

z tt , n .

a'

c>.

ct

...

C

z b . a .

Zb'b f

Zb'n'

b'

<*

ct

...

cj-

t

Zn'a'

z n . b ,

...

z n , n .

n'

c a ,

L, n *

ct

O

a' b'

Zaa

Z a b

Zan

Zto

Zbb

z bn

z na

Znb

z nn

c a ,

^a'

ci-

C a i

U w '

c b .

^a'

ct

C 6 ,

C w '

C n ,
< a'

Cn
b'

r n ,

tx w '

[48]

EXERCISES VIII

1. Write one C^ for each of the illustrative examples of 2-17-2-18, Sec. 3.
Compute the corresponding Cj 1 '.

2. If the equations of transformation are linear, x i = fl/* 7/ (*', ;' = 1,2, ',)
show that C' the inverse of Cm' can be computetl by the rule for computing the
inverse of a matrix given in 2-6, Sec. 2.

2-26. Geometric Objects. A mathematical concept called a geo-
metric object is now introduced which is more general than the con-
cepts of matrix or tensor. Following Eqs. (22) appear the two matrices

i 2

i 2 '

n

and

i 1 '

i 2 '

The first results from a choice of Maxwell currents, while the second
obtains from a choice of branch currents.

GEOMETRIC OBJECTS

129

If reference frames different from the two introduced in Figs. 2-7 are
chosen, additional matrices appear such as

i :

Each of these matrices is a different representation of a single under-
lying entity.

Analogous matrices exist for a general linear network. The totality
of all possible matrices (one for each possible reference frame) indicate
the existence of a quantity called a geometric object.

A geometric object is defined if:

(a) A particular w-matrix is given along one reference frame.

(6) All axes of this particular reference frame are specified.

(c) All possible reference frames are defined.

(d) The formula, that is, the "law of transformation" for finding the
n-matrices along any possible reference frame is given.

Henceforth Z( a )<0) (or any other letter, say -4 ()(#) will denote a
2-matrix. The symbol Z^ will stand for a geometric object having
components in a large or possibly infinite number of reference frames.
For example, with reference to 2-18, example, %# denotes the geo-
metric object whose representations are 24 matrices.

In contrast then: (a) A matrix is an array of ordered components.
(6) A tensor is a geometric object whose transformation formula, re-
ferred to in (d) above, is one of Eqs. (25-30). (c) A geometric object is
characterized by the four defining specifications above and its transfor-
mation formula may be more general than those given in 2-25. The
formula of transformation of a geometric object may be, but is not
restricted to one of Eqs. (39, 40, 44-47). Thus a tensor is always a
geometric object; a geometric object may, as a special case, be a tensor.
In the study of stationary networks all geometric objects are tensors,
but such is not true in the study of rotating machines.

Obviously, in the mathematical representation of a geometric
object it is usually not possible to display all the matrix representations
of the object. Instead there is given:

(a) An w-matrix showing the components of the geometric object in
one specific reference frame.

(b) The specific reference frame is given.

(c) All possible reference frames are defined.

130

STATIONARY NETWORKS

(d) The formula of transformation is expressed.
The number of dimensions of the w-matrix is the same as the valence of
the geometric object.

In the example above Z mn is a matrix showing the components of
Z0 in one specific reference frame such as

i 1 i 2

i 2

or Z n

A symbol Z aa , say, represents one component of Z mn .

2-27. First Generalization Postulate. In Sec. 2, Chap. I, Hamil-
ton's principle was proved for dynamical systems. It can be estab-
lished by separate individual proofs for certain other systems. There
exists no single general proof establishing simultaneously its validity for
all physical systems. In new situations it is assumed to hold and its
validity checked by experiment. When used in this manner, Hamil-
ton's principle is employed as a postulate.

In the remainder of this chapter two ll important postulates are
employed, which are called generalization postulates. The first gen-
eralization postulate as stated by Kron is: "The method of analysis and

the final equations describing the performance
of a complex physical system (with n degrees
of freedom) may be obtained by following step
by step those of the simplest but most general
unit of the system, provided each quantity is
replaced by an appropriate n-dimensional
matrix. The simplest unit of the system may

j \/^ r h ave one or more degrees of freedom"

I U^ EXAMPLE 1. Vibrating meclianical sys-

tem. As an illustration of the first pos-
tulate consider vibrating mechanical dis-
crete systems. By inspection of Fig. 2-9
the differential equation of the motion of mass M is seen to be

aq + cQ + bq = 0, [49]

where a, 6, and c are respectively the mass, spring, and damping con-
stants and q is the displacement, at time /, of M from equilibrium posi-
tion.

I Equilibrium
L Position

FIG. 2 9. One Degree of Free-
dom.

11 In all, four exist.

FIRST GENERALIZATION POSTULATE

131

Lagrange's equations of motion (1-12, 1-14, 1-25, Chap. I) of a
vibrating dynamical system are

d O^ ) 9* 9-*^ ov , * * i---.

T . r- =-~-, (r = 1, 2, , n) [50]

where T and V are respectively the kinetic and potential energies and
F is the Rayleigh dissipation function. The expressions for 7\ V, and
Fare

h---+6.S) f [51]

On substituting Eqs. (51) in Eqs. (50) Lagrange's equations can be
written

a-0 + c-q + b-q = or a mn q n + c mn q n + b mn q n = 0,

(m = 1,2, ...,w) [52]

where

' ' ' a 2n

J>nlb rt 2 ' '

, c

C\\C\2 * * ' C\n
c 2 1^22 * ' ' C 2n

The system of Fig. 2 -9 is characterized by the constants a, b, and c
whereas the system of Eqs. (52) is characterized by the matrices a, b,
and c. Equation (52) is obtained from Eq. (49) by replacing a, 6, c,
and q by a, b, c, and q.

EXERCISES IX

1 . Reduce the differential equations of problems 5 and 7, problem set IX, Chap. I,
to the form of Eqs. (52) of this chapter.

2. The partial differential equations for a single- wire (ground return) transmis
sion line are

and

where

,

d* a*

- m 2 i = 0, 2 - m 2 e 0,
- + L/, F - C + C/>, m 2 - ZY.

Consider a transmission system with n parallel conductors, electrostatic and elec-
tromagnetic coupling existing bet \veen the conductors. Arrange the resistances and

132

STATIONARY NETWORKS

self- and mutual inductances in the matrix Z shown below. Likewise arrange the
leakage conductances and self- and mutual capacitances in the matrix Y.

Zaa

Zab

...

Zan

Zba

Zbb

...

Zbn

Zna

Znb

...

z nn

Y M

Y at

...

Y an

Y*

Yu,

...

Y bn

Y na

Ynb

...

Y nn

The i and e matrices are
a b

e =

(a) Show that the partial differential equations for the current and voltage on the
n wire system are

d*

and =-

9*

(b) By differentiation and substitution obtain the equations m?e = and

= m?i = from the equations = Z-i and = Y-e.

a* 2 * dx dx

(c) In a manner identical to that in (b) obtain the equations

3 2 e

ZY-e

and j ~" Y-Z'i = from the equations

-Z-iand - -Y-e.

(The results in this problem arc given by G. Kron in "The Application of Tensors
to the Analysis of Rotating Klcotncal Machinery," General Electric Review, April,
1935.)

2 28. Second Generalization Postulate. The second postulate is a
statement regarding the permanence of form of the tensor equations of
physical systems. Its statement, as formulated by Kron, is: (a) "The
new system (under change of coordinates or reference frame) has the same
number and types of n-matrices as the old system (namely y e, z, and i)
but they now Itave different components, (b) The equation of the new sys-
tem in terms of n-matrices is exactly the same as the equation of the old
system, e.g., e = z -i. (c) The n-matrices of the new system may be estab-
lished from those of the old system by a routine transformation." That
is, the matrix equation of a physical system is valid for an infinite
number of analogous systems of the same type if each n-matrix is
replaced by an appropriate geometric object having a permanent law

STATIONARY NETWORK 133

of transformation. The C's, transforming the various systems into
each other, must be known.

EXAMPLE 1. Network. The differential equations of the two-
mesh network of Fig. 2-7 is a very simple illustration of the second
postulate. The equations, for two reference frames, are

Z-i = e and Z'-i' - e'.

The equations are identical in form. The matrices Z', i', and e' are
established from Z, i, and e respectively by the routine transformation
formulas of Eqs. (39, 40, 45).

EXAMPLE 2. Dynamical system. In example 1, 2-27, let Eqs. (51)
be subject to a change of variable from q\, q 2 , *i <Z to q\ 9 q 2t ,
q n by means of the equations of transformation

' ' + d 2n qn , [S3]

q n = dniq'i + d n2 q 2 + - + d nn q n >

whose matrix is d.

Evidently, V = -2(61191 + 2b V2 qiq 2 + + b nn ql), although a
quadratic form, can be viewed as a special case of a bilinear form.
Accordingly, the bilinear form V with matrix b is replaced by a bilinear
(quadratic) form V with matrix drb-d when the q's are subject to the
linear transformation (53).

The forms T and F can be transformed in a similar manner. Thus
Eq. (52) becomes

a'.q'+c'-q' + b'.q' = 0, [54]

where

a' = dra-d, c ; = d r c-d, b' = d r b-d [55]

and q is the column matrix (qi, q 2t , q n )* Equations (52) and (54)
are identical in form and Eqs. (55) furnish the routine formulas of
transformation.

2-29. Stationary Network. The equation of performance of the
network of Fig. 2 10 is

Z mn (p)i n = e m (m-1,2, -,*), [56]

where Z mn (p)i n - L mn (p)F + R^r + -- f i n dt,

^mn J

i 1 , i 2 , ,!* are k properly chosen mesh currents and c\, e^ , to
arc mesh voltages.

134

STATIONARY NETWORKS

If the network is extremely simple and k very small and the ele-
ments of the network merely wound coils, resistances, and capacitances
the equation of performance can be easily written down by the direct
application of Kirchhoff's laws. On the other hand if k is large, the
network complex, and the elements are, in addition to the above ele-
ments, vacuum tubes, rotating machines, or if hypothetical currents

V

FIG. 2 10. Network, Sub-network, Junction-pair, Mesh.

(symmetrical components, magnetizing and load currents, etc.) are in-
troduced, no such simple procedure will suffice. In such a simple net-
work as that shown in Fig. 2 10 it may be difficult to determine even
the minimum number of meshes or variables to be used.

2 30. Component Parts of Networks. In the networks considered
the parameters are lumped. A network consists of two kinds of compo-
nent parts: coils and junctions. No limitation is imposed on the
physical nature of a coil. A coil may be a wound coil, a capacitance,
vacuum tube, rotating machine winding, saturated reactor, etc. Elec-
tromagnetic and electrostatic couplings may exist between some or all
the coils of the network. With each coil is associated certain numbers
Zaa, Y ab , etc. No limitation is imposed upon the nature of the numbers

MESH, JUNCTION-PAIR THEOREM 135

%aa, Y ab , etc. These numbers associated with the coils (most fre-
quently an impedance or admittance) may be real or complex numbers,
functions of the time, or operators.

The two ends of a coil where it is joined to other coils are called
junctions. When two or more junctions arc joined with an impedance-
less wire they are considered to be one junction. In Fig. 2-10, a$ and
an are one junction. The number of coils in the network of Fig. 2 10
is 31. There are 14 junctions.

A complete network may consist of a number of sub-networks. If
between the pieces of a network there exist no physical connection*
(electrical connection) then each such piece is called a sub-network.
However, magnetic or dielectric couplings may exist between the sub-
networks. The network of Fig. 2-10 consists of 3 sub-networks.

2-31. Analytical Units of Network. Any closed circuit in a network
is called a mesh. The path a^a 6 a 7 is a mesh. The path a7e3 is a
mesh and it is the negative of the mesh a^a^aj.

Any two junctions located on the same sub-network are called a
junction-pair. In Fig. 2* 10 the junctions <* 3 , a Q or a a , 9 constitute a
junction-pair. The junction-pair 63 is the negative of the junction-
pair a36

Meshes and junction-pairs constitute the analytic units of a net-
work. There exists an important theorem by which the least number
of meshes, and the least number of junction-pairs required in the
analysis of a network, are obtained by the mere process of counting.

2-32. Mesh, Junction-pair Theorem. From 2-30-2-31 we
have the concepts of coil, sub-network, junction; mesh and junction-
pair. The number of each of these entities is not independent of the
number of the others. Denote by 5, /, Af, P, and N the number of
sub-networks, junctions, meshes, junction-pairs, and coils respectively
of a given network. There exist the following theorems. 12

/. The least number of junction-pairs of a network is equal to the
number of junctions minus the number of sub-networks. In symbols

P = J - S. [57]

II. The least number of meshes (required in the solution of a network)
is equal to the number of coils minus the number of junction-pairs. In
symbols

M = N - P. [58]

Thus

M=N-(J-S) = N+S-J. [59]

"O. Veblen: "Analysis Situs," American Mathematical Society, 1931, pp. 15-18.

136 STATIONARY NETWORKS

The determination of the least number of meshes is reduced by Eq.
(59) to the process of mere counting.

EXAMPLE. In the network of Fig. 2.10

M = 31 + 3 - 14 = 20 meshes.

2-33. Types of Stationary Networks; Variables in Networks. It

has been pointed out in 2-3 and also in Table I that the most general
type of stationary networks, subject to the most general operating
conditions, are orthogonal networks. However, a large class of elec-
trical networks, subject to very general operating conditions, can be
analyzed as mesh networks. The analyses of mesh, junction, 13 and
orthogonal 14 networks are not unrelated. If the analysis of either of
the first two is understood, a knowledge of the other is acquired without
difficulty.

In example 1, 2-18, the variables of the simple network were
taken first as mesh currents. Next branch currents were taken as vari-
ables. This is partially indicative of the choice of variables in complex
networks. Hither mesh currents, branch currents, or a combination of
mesh and currents or hypothetical currents or differences of potential
existing across coils of the network, or a combination of all of these
quantities, may be taken as variables or coordinates in a network.

A network may be viewed as a configuration or arrangement of
meshes. A network may be viewed also as a collection of junction-
pairs. In the last case the variables are the differences of potential
existing between the two junctions of the junction-pair.

In the most general network operating under the most general con-
ditions a network must be viewed as both a collection of meshes and
junction-pairs. The variables thence consist of the currents flowing in
the meshes and the differences of potential existing across the junction-
pairs.

Simple passive networks, multi-winding transformers, transmission
lines, and rotating machines are primarily mesh networks. Multi-
electrode vacuum tubes 15 are primarily junction -pair networks. A
combination of vacuum-tube and transformer networks would produce
a complete or orthogonal network.

2-34. Sign Conventions for Mutual Inductance. Two coils, with
mutual inductance between them, can be connected in series in two dif-
ferent ways. If the connection is such that the flux coming from the

18 Gabriel Kron, Tensor Analysis of Networks, Chap. XIV.
" Ibid., Chap. XVI.

11 Gabriel Kron, "Tensor Analysis of Multielectrode-Tube Circuits," Electrical
Engineering, November, 1936. Also Tensor Analysis of Networks, Chap. XV.

INTERCONNECTION OF COILS

137

first coil links the second in the same direction, then the connection is
called series aiding. In going around a closed circuit in any direction
the ends of the two coils are numbered 1-2 and 1-2 or 2-1 and 2-1 if
the connection of the two coils is series aiding as shown in Fig. 2 1 la.
If the connection is such that the flux coming from the first coil
links the second in a direction opposite to the linkage in the first coil,
then the connection is called series opposing. In going around a closed
circuit in any direction the ends of the two coils are numbered 1-2 and

(a) (6)

FIG. 2*11. Series-aiding and Series-opposing Connections.

2-1 or 2-1 and 1-2 if their connection is series opposing as shown in
Fig. 2-116. (See 2 -39.)

(6)

ALL-MESH NETWORKS

An all-mesh network is one having the same number of coils as
meshes.

2-35. Interconnection of Coils in All-mesh Networks. The net-
work of Fig. 2 13a, 2-39, is an all-mesh network. Each coil of an all-
mesh network is short-circuited upon itself, the ends of the coils being
joined through a source of impressed voltage with the remainder of
the network by means of impedanceless wires. With each coil is joined
in series an impressed voltage. There are n coils, n impressed volt-
ages, and n meshes. Some of the impressed voltages may be zero.

Imagine the all-mesh network broken up into n individual circuits,
there being no electrical connections between the n simple circuits.
Each simple circuit consists of a coil and an impressed voltage. (See
Fig. 2 136, 2 39.) A fact of great importance in the theory following
is: The same currents flow through each of the n coils whether existing
as a set of n individual coils or whether connected into an all-mesh net-
work. The reason for this is that in either case the same voltage is in
series with the coil and the circuit is then short-circuited upon itself
by means of an impedanceless conductor. For example, the current
through the coil Z^ (Fig. 2 130) is identical to the current through
the coil Z aa (Fig, 2*136). Moreover, since the same value of current

138

STATIONARY NETWORKS

flows, since the same voltage is impressed, and since the same imped-
ance exists in both, the total instantaneous power input is identical.

All-mesh networks are not the practical networks of engineering.
Practical networks, in general, contain more coils than meshes. Such
networks are called mesh networks. However, the theory of the all-
mesh network is first developed. This theory is then modified and ex-
tended so as to be applicable to mesh networks.

2-36. Primitive System of Mesh Networks. To establish the
equation of performance of a given network the following procedure is
used:

(a) Establish first the equations of another network, whose analysis
is simple.

(b) Next change these equations into the equations of the given
(hereafter called the derived) network by a routine transformation.

FIG. 2 12. Primitive Mesh Network.

The network, whose equations are the simplest to establish and
which serves as a standard reference network, is called the primitive
network. A given network, whose equation of performance is required,
will be called the derived network. The derived network will consist of
n coils and k meshes where, in general, n j* k. Figures 2-13a and
2 '140 represent derived networks.

The primitive network consists of the following objects and rela-
tions: (a) n physically separate coils, each short-circuited upon itself.
Figures 2 13b and 2 14& represent primitive networks, (b) In series
with each coil is an applied voltage. Some of these voltages may be
zero, (c) Each of the n coils is labeled (an ordered set) so that it may
be identified in the new network, (d) The ends of each coil of the
primitive network are numbered 1 or 2. The positive direction of cur-
rent and the positive direction of voltage in the primitive network is
taken to be from 1 to 2. (e) If mutual impedances exist between k ^ n
of the n coils of the new network this fact is indicated by arrows on
the diagram of the primitive network. In the general case Zy 5* Z/,-.

By virtue of the first generalization postulate 2-27 the definition
of the primitive network is a natural one. The simplest unit of a k mesh

DERIVATION OF THE DIFFERENTIAL EQUATIONS

139

network is one isolated mesh with impedance and impressed voltage.
The equation of performance of this simplest unit is Z(p)i = e. In view
of the first generalization postulate the equation of performance of the
primitive network is Z-i = e or Z mn i n = e m , where i, e, and Z are
properly chosen matrices.

The properly chosen geometric objects of the primitive network are
as follows. The n real currents i a , i b , , i n in the coils of the primi-
tive network will be considered the real components of a fictitious le
current vector i. Its mathematical expression is

i =

*

i"

*"

[60]

The n impressed voltages e a , e^ , e n in series with the coils repre-
sented in Fig. 2-12 will be considered the real components of the
fictitious vector e. Its mathematical expression is

e =

[61]

The geometric object

Z =-

7

*aa

Z a b

%an

z ba

Zbb

...

z bn

Z na

Z n b

z nn

[62]

The equation of performance is

Z-i = e or Z mn i n = e m .

[63]

2-37. Derivation of the Differential Equations of All-mesh and
Mesh Networks. In 2-41 it will be shown that e, i, Z; e', i', Z', the
geometric objects of the primitive and new networks respectively, are
tensors. However, in this section we shall assume the tensor character

16 More advanced concepts of tensor analysis show that the vector i is not fictitious
but it represents the instantaneous stored magnetic energy in the whole system.

140 STATIONARY NETWORKS

of the above objects and explain the procedure for the setup of the dif-
ferentia! equations of performance of the systems. Moreover, since the
mere rules for obtaining the equations of performance of all-mesh and
mesh networks differ in only a few details we shall reduce the procedure
to one set of rules applicable to both types of networks.

The setting up of the differential equations of the derived network
consists of three steps: (a) Establish correctly labeled diagrams of the
derived network and its primitive. This is a purely descriptive step.
(b) Obtain the transformation matrix C showing the difference between
the two networks. This is the only analytical step involved. The step
employs Kirchhoff s laws, (c) Establish the equations of the derived
network from that of the primitive network with the aid of C. This
step involves only routine calculations.

Step (a). Before the analysis may begin it is necessary to draw a
correctly labeled diagram of the given network. This step is necessary
in any method of analysis. The rules for this step are as follows:

(1) Draw a diagram of the derived network and label the separate
coils Z aa , Zbb , , Z nn . (See Figs. 2 13a-2 14a.)

(2) Examine the derived network for mutual impedances. Number
the ends of a coil, selected at random, in the derived network with the
numerals 1-2. If the next coil, in tracing out a closed circuit in the
derived network, is connected series aiding (see 2-34), number its
ends 1-2. If it is connected series opposing, label it 2-1. If no mutual
inductance exists between the two coils they may be labeled arbitrarily
1-2 or 2-1. Label the ends of all coils.

(3) Indicate one impressed voltage in series with each of the n coils.
Some of these voltages e a , et, - t e n may be zero. Indicate by means
of arrows the direction in which these impressed voltages (battery, gen-
erator, etc.) act. If these arrows are from 1 to 2 of the respective coils,
the numerical voltage is positive, otherwise negative. (See examples
2-39.)

(4) Count the junctions, coils, and sub-networks and apply Eq.
(59). By means of Eq. (59) the least number of meshes in the networks
of Figs. 2- 13a-2- 14a are respectively 5 and 3.

(5) New variables. Introduce as many new variables i a ' 9 i b ', , i n '
as there are least number of meshes by drawing as many arrows. These
arrows may be drawn along a coil or along an impedanceless branch.
The direction of each arrow is arbitrary. Label the arrows i a> , i b ' ,
, * w/ . The only restriction on the assignment of i', i 6 ', , f 1 ' is
that they be independent, i.e., they must be sufficient to determine
all the currents flowing in the remaining branches. In a mesh network
n 1 k ? n.

DERIVATION OF THE DIFFERENTIAL EQUATIONS 141

(6) Old variables. It is helpful (though not necessary) to draw a
diagram of the primitive mesh network having n coils Z 00 , Z&&, , Z nn
with mutual impedances between some of them, n currents i a , i b , , i n
and n impressed voltages e a , e b , , e n . The positive direction of the
voltages is always from 1 to 2.

When in the derived network no impedance appears in series with
an impressed voltage, the primitive network assumes an impedance Z
with zero value in series with it. Similarly when in the derived net-
work no voltage appears in scries with an impedance, the primitive
network assumes a voltage e with zero value in series with it.

Step (b). To establish the connection tensor C the steps are as
follows :

(1) On the diagram of the derived network write along each coil the
new current that flows in it. To do this apply Kirchhoff s first law,
which states that the sum of all currents entering a junction is zero.
(SeeEqs. 64a, 2-39.)

(2) Relations between the old and new variables. On the two dia-
grams there now exist two expressions in terms of different variables
for the current through each coil. Equate the two expressions (that is,
the two currents flowing in each coil) giving n equations in k unknowns,
* /(*') It must be remembered that the positive old currents
i a , i b , , i n flow from 1 to 2. For the all-mesh network of Fig. 2-130
examine Eqs. (64a-64), 2-39. For the mesh network of Fig. 2-14a
examine Eqs. (64A), 2 '40.

(3) The C matrix. The matrix composed of the coefficients of the
new variables is called the connection matrix or the C matrix or C-
matrix. The C matrix for the networks of Figs. 2 13a-2 14a are given
by Eqs. (64c) and (64i), 2-39 and 2-40 respectively.

With the establishment of the C matrix the set of equations t" 1 =
f(f n ') may be written as

i = C-i' or * w = C.?"'

representing the relations between the old components i and the new
components i' of the current vector. These relations for the networks
of Figs. 2 13a-2 14a are given respectively by Eqs. (646) and (64h).

Step (c). To establish the equations of the given system the steps
are as follows:

(1) Geometric objects of the primitive network. The three geometric
objects e, Z, and i of the general primitive mesh network are given in
2-36. For illustrative examples represented in Figs. 2-13a-2-14a
the geometric objects of the primitive network are given by Eqs. (64d)

142 STATIONARY NETWORKS

and (64j), 2-39 and 2-40 respectively. The equations of the primi-
tive network are

Z-i = e or Z mn F = e m .

(2) The impedance tensor Z' or Z m n > of the derived network is
found by the formula

Z' = CfZ'C or Z TO / n / = Z mn CriC"t.

These relations for the networks of Figs. 2 -130-2 -140 are given by
Eqs. (64*) and (64Jfe).

(3) The impressed voltage vector e' or e m * of the derived network is

e' = C r e or e m , = C%e m .

These relations for Figs. 2-13a-2-14a are Eqs. (64/) and (64/).

(4) The equation of voltage or equation of performance or differ-
ential equations of the derived system are

Z'-i' = e' or Z*^ f '-*,'-

This set of differential equations may be subjected to various manipu-
lations depending on the problem at hand.

2 38. Solution of Equations of Performance. If the components of
e' are known, the unknown currents are found by i' = (Z')"" 1 ^'.
Once the components of i' have been found then : (a) The currents in
each coil are found by i c = C-i'. (b) The differences of potential
appearing across each coil are e c = Z-C'i', where Z-C has already
been calculated as a step in finding Z'.

2 39. Illustrative Example : All-mesh Network. Obtain the equa-
tion of performance of the network represented and described in Figs.
2 '13. The explanation of the solution is given in the rules of pro-
cedure in 2*37. The mutual inductances are: Z aa and Z&& series
aiding, Z bb and Z cc series opposing, Z aa and Zdd series aiding. The
absolute values of the impressed voltages e a , e bj e e , *, e/ are 3, 4, 7,
sin /, cos /. Their directions are indicated on Fig. 2*13a. e a = 3,
6b = 4, e c = 7, ed = sin /,/= cos /.

In three of the coils of Fig. 2 13a new variables have been assumed.
Kirchhoff's first law applied to Fig. 2-13a yields Fig. 2-13c. From
Fig. 2 13s the current in coil

oa s

Z// is i f ' - i c ' - i*', [64a]

Z bb is i b ' + i* + i*'.

ILLUSTRATIVE EXAMPLE: ALL-MESH NETWORK

143

Remembering that i a , i b , i c , i d , and i f flow from 1 to 2 and equating
the currents flowing in each coil (compare Figs. 2 13ft and 2-13cand
use Eqs. 640), we find that the current in coil

Z aa is i a - i a ' - i b ' + +0 - *'',

Z bb is i b = - i b ' - i c ' - i* + 0,

Z ce is i c = +0 + i c ' +0+0,

Z dd is i* = + + - i d ' + 0,

Z fff is f' = + - i c/ - i d ' + i ft .

[646]

FIG. 2 13. All-mesh Network.

The C-transformation tensor is found by taking the coefficients of the
new currents. It is

a' b' c ; d' f

a
b
C = c

d
f

1

-1

-1

-1

-1

-1

1

-1

-1

-1

1

[64c]

144

STATIONARY NETWORKS

The current, voltage, and impedance tensors of the primitive net-
work are

a b c d f

abed

i"

*

i e

i*

if

a b c d f

-3

-4

7

sin/

cos/

a b c d f

a
b

Z = c
d
f

z-

Xab

Xad

Xba

Zbb

Z cc

-

Xda

z dd

Zff

The impedance tensor Z' = C< Z C = Z
work is

of the new net-

a'

c'

a'

Z'-c'
d'
f

~"Z a

Xab

X a d

X a b

z bb

"T X ab

Zbb + Z cc

+ z ff

Zbb +

Xab Zff

X a b

X ad

+

+*//

^ ab "i ^ ad

i

Xab Zff

-Zff

Zaa +

[fAe]

CONSTRAINTS
The voltage vector of the new network is

145

e' = C t -e

1

-1

-1

-1

1

-1

-1

-1

-1

-1

1

e*

a'
b'
c'
d'
f

-3

3 + 4

4 + 7 - cos t

4 + sin t cos t

3 + cos J

[64/]

The differential equations of performance are
Z'-i' = e' or Z^f"'

to

MESH NETWORKS

2 40. Constraints. In 2 35 it was mentioned that most practical
networks contain more coils than meshes. A network possessing more
coils than meshes is called a mesh network. Mesh networks can be
viewed as special cases of all-mesh networks. A mesh network can be
considered as an all-mesh network with certain meshes open-circuited
(frictionless constraints). The theory is introduced by means of an
example. Obtain the equation of performance of the mesh network
represented and described in Fig. 2 14. The explanation of the solu-
tion is given in the rules of procedure in 2-37. The coils Z mm and
Z nn are wound such that f 1 and i n produce additive values of the

146

STATIONARY NETWORKS

flux. The absolute values of the impressed voltages e m , e n , e p , e q are
7, 1, 2, 4. Their directions are indicated on the figure.
By Kirchhoff 's first law and Fig. 2 14a we have

"-"' + + 0,
i? = o + "' + 0,

_ o + o - ',

where it is remembered that *"*, i", i p , i 9 flow from 1 to 2.

errYi

[64*]

(a) (6)

FIG. 2 14. Mesh Network with Sub-network.

The C tensor is

m' p' q'

m

1

1

1

1

-1

[64*1

The current, voltage, and impedance tensors of the primitive net-
work are respectively

m n

m

e =

e m

CONSTRAINTS

m n p
m

147

mm

<X mn

X n rn

z nn

Zpp

z qq

[64/1

The impedance tensor Z' = C<-Z-C or Z a >p - Z mn C" % of the
new network is

m'

P'

m'
P'

q'

z mm

x nn

x mn

x nm

Znn + Z, p

z nn

x mn

Z nn

z nn + z gq

The voltage vector e' = C^e or ^ a / = C%e a of the new network is

e m

-

-7

e n + e p

3

e n - e q

-3

[64*]

The differential equation of performance or the equation of per-
formance is

Z'-i / = e' or

EXERCISES X

1. Obtain the differential equations of performance of the network shown in
Fig. 2-15.

The mutual inductances are:

Z aa and bb series aiding,
Zib and Zee series opposing,
Zaa and Zdd series aiding.

The absolute values of the impressed voltages are e a =* 1, ^6= /(Of *c = Esint,
c* 0, / = 4. Their directions are indicated on the figure.

148

STATIONARY NETWORKS

2. In the example of 2-40 let the values of the coil impedances be

Zmn - L nm p = p,

- R

pp

FIG. 2-15. All-mesh Network.

Zbb Lbb P

= 0.4/>,

'0.2, X mn = M m

1
0.008^'

Obtain the matrix solution for i' of
Z'-i' = e, i.e., Eqs. (64m).

3. The voltages induced in the indi-
vidual coils of the network are given
by Z-i = Z-C-i'. This matrix giving
these voltages is denoted by e. Com-
pute e for the network of 2-40.

4. In the illustrative example of
2-39 let the values of the coil imped-
ances be

Z// = Lf f p

X a b = M a b P '
Xbc = Mbc P
Xad Mad P

Obtain the values of the voltages induced in each coil of the network.

5. Set up the differential equations of performance for the network of Fig. 2- 16.

6. Obtain the differential equations of performance of the network of Fig. 2-17.

ft:

T j

:*

FIG. 2 16. Mesh Network.

FIG. 2 17. Mesh Network.

The mutual impedance relations are as follows: The pairs of coils Zaa
Zee Zddt Zdd Zhh, Zhh Zn, Zkk Z gg are connected series opposing. The
coils Z/f Z gg are connected series aiding. The absolute value of e a is unity. All
other voltages are zero.

7. If no mutual impedance exists between any of the coils of the network of Ex. 6,
write the differential equations of performance. The direction of the impressed
voltages are shown in the figure. The absolute values of the voltages are e a = 1.
The remaining voltages are zero.

2 41. Transformation Formulas for i, e, Z. The geometric objects
i, e, Z are now shown to be tensors. In 2 35 it has been pointed out
that the instantaneous power input P of the primitive network has
(he same value as the instantaneous power input P' of the new all-

TRANSFORMATION FORMULAS FOR i, e, Z 149

mesh network. This fact gives the relation e-i = e'-i' or e m f* =

em'f".

To prove that i, e and Z are tensors we have the following relations:

1. i = C-i' or $*- CSri"', [65]

2. e-i = e'-i' or e m f n = e m *f n ', [66]

3. Second generalization postulate.

We now readily obtain :

(a) Current transformation formula. Equation (65) is the transfor-
mation formula for the current. For an all-mesh network C is non-
singular and we have

i = C-i' or t" = O m '. [67]

(6) Voltage transformation formula. Substituting the value of i
from Eq. (65) in Eq. (66) we have

e-C-i' = e'-i' or e m CZ,f" = e m ,i m ',
or

e-C = e' or e m C, - e m ,. [68]

(c) Transformation formula of Z mn . The equation of performance
of the primitive network is

e - Z-i'.

Substitution of the values of i and e from Eqs. (67) and (68) in the
above equation yields

Cr 1 e' = Z C-i.

Multiplying the equation by C* we have

e' - CrZ-C-i'.

By the second generalization postulate e' = Z'-i'. From this equation
and the expression for e' above it follows that

Z' = C ( -Z-C or Z.v-CSCfrZ*,. [69]

This is the transformation formula for Z mn .-

INTERCONNECTION OF NETWORKS

The methods, thus far developed, are of outstanding value in the
interconnection of networks. If a large network is composed of a finite
number of simple or complicated networks each of the smaller networks
may be analyzed separately and use of these analyses made after the

150

STATIONARY NETWORKS

smaller networks are interconnected into the super-network. This is
especially advantageous if the network can be'divided up functionally,
i.e., in such a manner that all circuits performing similar functions can
be grouped together.

It is emphasized that if each of the smaller networks have been
analyzed it is not necessary, by the present methods, to start ab initio

<tr

II I

FIG. 2 17a. Interconnected Networks.

in the analysis of the super-network. All analyses of the smaller net-
works can be employed without change.

The procedure is evident from the solution of an example.

2-42. Description of Illustrative Example. It is required to set up
the equation of performance of the network of Fig. 2 1 7a. The compo-
nent parts of this network are the two networks shown in Figs. 2- 13a-

vj

"1

J

9

1

o

-1

o

ii c

r* i

o

9

A

* i

--i'

r i

i

FIG. 2-176. Primitive Network for Interconnected Networks.

2-14a. The interconnections are as shown in the figure. The mutual
impedances in Fig. 2-17a are the same as in the networks of Figs.
2 13a-2 14a. The same statement holds regarding the voltages.

The primitive network of Fig. 2-17a is shown in Fig. 2-176 con-
sisting of the two original networks (Figs. 2-13a-2-14a) placed side
by side.

GEOMETRIC OBJECTS OF PRIMITIVE NETWORK

151

2-43. Geometric Objects of Primitive Network. The impedance
tensor of the network of Fig. 2 - 13a is given by Eq. (640). The imped-
ance tensor of the network of Fig. 2-14a is given by Eq. (64k).
Denote these two tensors by Zi and Z' 2 respectively. The impedance
tensor of the primitive network (Fig. 2-176) is the sum of these two
tensors.

a' b' c 7 d' f m' p' q 7

z;

a 7
b 7
c'
d 7

:

f

m'

P'

q'

Xft

t a t

Xd'b*

Xfw Xf, c ,

X C fdt

Xf,dt

Z C fft

Xdfff

Zf,f,

X

qfpf

pfgt

[70]

The value of the components of Z' in terms of the components of the
individual networks is found by comparison of Eq. (70) with Eqs.
(64e-64&). For example

Z aa X ab and

= Z n

The voltage vectors of networks (2-13a-2-14a) are given by Eqs.
(64/-64/). Denote these by ei and e 2 respectively. The voltage vector
of their primitive is their sum,

e'

el + e' 2

a 7 b' c' d' f m' p' q'

tof

e c >

e*

e f ,

em,

e q ,

[71]

where, for example, e' t is identified by means of Eq. (641) to be e n + e p .
Likewise the current vector is

a' b' c' d' f m' p' q'

'

i v

V

*<"

if'

f

i*

if

[72]

152

STATIONARY NETWORKS

2-44. The Transformation Tensor. We are now ready to inter-
connect the two networks as indicated in Fig. 2 1 7a. Introduce five
new currents i b " , i c " , i r " t i 8 ", i 1 " shown in the figure as there are only
five meshes. From the figure it is clear that

in coils where formerly i 5 ' and i qt flowed, now i r " flows,

in coils where formerly i a ' and F 1 ' flowed, now i 8 " flows, [73]
and

in coils where formerly i d ' and i p ' flowed, now i l " flows.

It is evident from Eq. (73) that the nine currents i a> ', i b ' ', i c ', i d/ , i e ', i f ',
f^ 9 i n ', $ can be expressed in terms of five new currents. Relations (73)
and the figure yield the relations of Eq. (74) from which the C trans-
formation tensor is found

b" c" r" s" t"
' . o + + + **" +
p = p' + o + + +
' = + i e " + 0+0 +0

r" s" t"

i d ' =0+0+0 +0+ i 1 "

i f> - + + i r " +0+0
"' = + + - i'" +
,T' =0+0+0+0- *'"
,v = o + + * r " + 0+0

[74] C" =

b'
c ;
d'
f
m'
P'

1

1

1

1

1

-1

-1

1

[75]

2-45. Geometric Objects of New Network. The voltage vector
of the new network is

| "6 f

where * and e a are given respectively by Eqs. (75-71). (See Ex. 1.)
The impedance tensor Z a p is given by

Z a "0" = Zm'n' CH Cpu = C| -Z'^C". [76]

The equation of performance is

GEOMETRIC OBJECTS OF NEW NETWORK 153

The solution of Eq. (76a) yields i b ", i c ", i r " ', i* n ', 2". These values
substituted in Eq. (74) give i a ', i b ', i e ', i* ', *'', i" f , i p ', i qt . Finally, these
currents when substituted in Eq. (646) and Eq. (64A) yield the currents
passing through the individual coils of the network.

EXERCISES XI

1. Compute e a " by carrying out the multiplications indicated in e a = C%" e a f >

2. Compute Z a "0" by carrying out the operations indicated in Eq. (76).

3. Solve Eq. (76o) for *".

PART (B)

INTRODUCTION
TO

TENSOR ANALYSIS OF ROTATING ELECTRICAL
MACHINERY

Sections (1-4) of the present chapter consist of an introduction to
certain parts of tensor analysis of general linear networks where the
nature of the coils has not been considered. They may have been sta-
tionary or rotating coils. Sections (5-10) of this chapter are devoted
to a brief introduction to certain portions of tensor analysis of rotating
electrical machinery.

(5)

Non-mathematical Outline of the Nature of the Theory of
Rotating Electrical Machinery

Physically, a rotating electrical machine is but two electromag-
netic-mechanical configurations composed of non-magnetic and mag-
netic materials and mesh networks (with their coexisting magnetic
fields) such that relative rotary motion of various velocities is possible
between the configurations. The configurations differ in many details
and consequently there are many different types of machines. Exam-
ples of types of machines are: shunt direct-current motors, synchronous
generators, induction motors, repulsion motors, etc. However, when
analyzed from the proper point of view (tensor viewpoint) the many
different types of machines (called derived machines in this chapter)
are strikingly similar and may be considered mere aspects of one primi-
tive machine.

154 NON-MATHEMATICAL OUTLINE

2-46. Scope. Tensor analysis of rotating electrical machinery is a
large field. The published work in this field is very extensive. In Sees.
(5-9) the motion of the rotor is assumed to be known and the analysis
is for purely electrodynamic systems. The amount of tensor analysis
required in these sections is restricted to definitions, tensor transforma-
tion formulas, certain properties of transformations, tensor addition,
subtraction, inner product. The theory is applied to a number of
derived machines.

The theory in Sec. 10 is more condensed than the work in the pre-
ceding sections, but many references to the original papers of Kron are
given. In this section the motion of the rotor is, in general, unknown.
The system (or systems) is an electrodynamic-mechanical one and ad-
vanced tensor analysis and advanced geometrical concepts are em-
ployed. The most general equation of performance of rotating elec-
trical machinery is developed. This equation is valid for the most
general situation possible in the analysis of one or a system of rotating
electrical machines. One of the purposes of the development of the
general equation of motion is the study of acceleration in all types of
rotating electrical machines. Another use of the general equation is the
analysis of hunting of machines. Closely associated with the last
analysis is the study of stability of machine systems.

2 47. Preliminary Description of Primitive Machines. It has been
noted in Sec. 4 that every mesh network consists of n coils intercon-
nected. The connections may be electromagnetic or conductive. No
matter how complex the mesh network, it has been made to depend
upon the primitive network of 2 36 consisting of n distinct coils pos-
sessing no electrical connections between them. They may have mag-
netic or dielectric connections between them. Any particular network
(called a derived network) can be built physically from the n coils of
the primitive network and the equation of performance of the derived
network can be obtained from the tensor concepts and the C connection
or transformation tensor. The number of types of mesh networks is
very large. A classification of such according to function, characteris-
tics, or application would be a tedious task. Examples are: bridge cir-
cuits, two- and multiple-winding transformers, auto-transformers,
transmission and filter networks, and armature windings. From a ten-
sor viewpoint all these are but aspects of one primitive network.

It is then anticipated that Kron's tensor analysis of rotating ma-
chines will proceed along similar lines. As reasonably expected the
analysis is much more complicated than that of stationary mesh net-
works because there enters the complexities introduced by various
relative motions between magnetically coupled circuits. There are two

DERIVED MACHINES 155

sets of axes: one pertaining to the stator of the machine, the other be-
longing to the rotor. At least one primitive machine is expected. In
fact because of computational exigencies there are two primitive ma-
chines. The distinction between them is one of difference of preferred
reference frames and the consequences resulting therefrom.

The first primitive machine is called the primitive machine with
stationary reference axes. The reference axes on both stator and rotor
of this machine are stationary in space, i.e., fixed relative to the base
of the machine. For reasons later evident, this machine will be called
also the non-holonomic (or rather quasi-holonomic) machine. (See
Chap. I, Sec. 6, for non-holonomic dynamical systems and coordinates.)
There are associated with this primitive machine, as for the primitive
mesh network, certain fundamental tensors called the resistance, in-
ductance, and torque tensors. The components of these tensors for the
quasi-holonomic machine are constants. This fact reduces many analyses
of many rotating machines to the simplicity of linear stationary net-
work analyses. It is possible to base the derivation of the equations of
performance of the non-holonomic machine on Maxwell's field equa-
tions. It is also possible to establish its equations of performance by
means of ingenious physical concepts of Kron. The latter method will
be followed here.

The holonomic primitive machine or second primitive machine dif-
fers from the non-holonomic machine in the following respects: The
reference axes on the stator remain fixed, but the reference axes on the
rotor are rotating axes. The speed of rotation of the rotor and axes are
identical. The components of the fundamental tensors associated with
this machine are, in general, no longer constants but functions of 0,
the angular displacement of the rotor. The equations of performance
of the holonomic machine can be derived directly from the equations of
Lagrange, provided the rotating axes move at the same speed as the
rotor.

The equations of performance of either primitive machine are de-
rivable from the equations of performance of the other primitive
machine by change of reference systems.

2-48. Derived Machines. Derived rotating electrical machines
are classes or types of rotating machines such as: salient-pole synchro-
nous motors or generators, direct-current motors or generators, repul-
sion motors, squirrel-cage motors, and Schrage motors. Derived ma-
chines are analogous to derived stationary linear mesh networks of
Sec. 4. The equations of performance of a derived machine are ob-
tained from the equations of performance of one or the other (some-
times both) of the primitive machines by routine manipulations (ma-

156 PRIMITIVE MACHINE WITH STATIONARY REFERENCE AXES

trix multiplications) involving the use of a transformation or connection
tensor, called the C tensor. The C tensor is determined by an inspec-
tion of and a comparison of the windings of the derived machine with
the windings of a primitive machine. There are at least as many C
tensors as there are derived machines. When the derived machine is
obtainable from both the non-holonomic and holonomic machines, then
there are two C tensors for each derived machine. For each machine
also as many additional C tensors may be introduced as there are types
of artificial or hypothetical reference frames employed. Hypothetical
reference frames appear with the use of symmetrical components,
magnetizing and load currents, and with other labor-saving con-
cepts.

The old theory or theories of rotating electrical machinery consists
of a large number of individual and largely mutually independent theo-
ries of each machine based on some original physical picture invented
by a specialist in the theory of one derived machine. In general, more
than one theory exists for each derived machine, so that the total num-
ber of theories closely approximates the total number of specialists.
All these piecemeal theories are replaceable by Kron's tensor theory of
rotating electrical machinery. Clearly, only a brief introduction to
this theory and its application to but few machines can be attained in a
single chapter.

(6)
Primitive Machine with Stationary Reference Axes

In this section the general equations of performance of the non-
holonomic or quasi-holonomic machine are derived from the physical
pictures of Kron. The material is analogous to that of 2-36 on the
primitive system of mesh networks.

2-49. The Primitive Machine with Stationary Reference Axes.
The first primitive machine has the following characteristics:

(a) The stator has two salient-poles. (See Figs. 2 18a and 2 186.)

(ft) The rotor is smooth.

(c) All slip-rings, commutators, and electrical connections between
any windings of the machine are considered removed. (Compare primi-
tive network 2-36.)

(d) The rotor windings are symmetrically distributed around the
circumference. There may be any number of them arranged in layers
and each winding may have different constants.

(e) On the stator, windings exist in the axes of the salient poles and
in axes midway between the salient poles. There may be any number

TWO-REACTION COORDINATES

157

of windings in each of these axes and each winding may have different
constants.

(/) Along each winding of the stator and of the rotor are two refer-
ence axes. One is called the direct axis, denoted by d; the other the
quadrature axis, denoted by q. (A description of direct and quadrature
axes as applied to the special case of a synchronous machine is given in
2-50. The d and q axes on both rotor and stator are fixed in space,
i.e., their origin and directions are fixed relative to the base of the
machine.

(g) Associated with each winding of the rotor are two unit vectors
d t and q,. These unit vectors on the stator windings are denoted by

'*

com row

(a) (6)

FIG. 2 18. Generalized Rotating Machine.

dtii d2, ', dn and q,i, q, 2 , , q, where d al and q sl belong to the
stator winding nearest the air gap of the machine. The unit vectors
on the rotor windings are denoted by d r i, d r2 , , d rn and q r i, q r 2, ,
q rn , where d r i and q r i belong to the rotor winding nearest the stator.
(See Fig. 2-186.)

(h) In the theory saturation and iron losses are neglected. It is
assumed that the inductance between a stator and rotor coil is a sinus-
oidal function of the position of the rotor. However, all formulas
developed are valid for any number of harmonics provided only that the
tensors and geometric objects are enlarged by the addition of proper com-
ponents, i.e., by the addition of rows and columns.

2 - 50. Two-reaction Coordinates. (A Digression from the Tensor
Theory to Direct and Quadrature Quantities Relative to Synchronous
Machine Analysis.) The reason for the choice of direct and quadrature
coordinates is evident from their application to a salient-pole synchro-
nous machine. Various m.m.f.'s of armature and exciting windings of a
machine can be combined vectorially only in case they act upon the
same magnetic circuits. It is obvious (Fig. 2-190) that the magnetic
circuit formed by armature phase A and by the rotor when it is in posi-

158 PRIMITIVE MACHINE WITH STATIONARY REFERENCE AXES

tion 6 = is much different from that formed by the same circuit and
by the rotor when it is in position = ir/2. For vectorial addition
(and for other reasons) it is advantageous to choose a reference axis
pointing from the center of the rotor along the central line of a field
pole. This is called the direct axis d. At right angles (90 electrical
degrees) ahead of the direct axis is the quadrature axis q. In the holo-
nomic approach to synchronous machine theory these axes are moving
axes, they being attached or fixed to the field or moving rotor. Of
course, due to symmetry existing in all machines it is sufficient to con-
sider the machine to be a two-pole machine. The armature windings of

Axis
phase B

r

1

1
j

A j

a i

?
\

c

n

F

is
i
i

"7 c

*

i ]
i i

N

' S

i
i

i
i

"? 1

1

i

T

1
1

FIG. 2-181. Direct and Quadrature Quantities.

a three-phase winding arc represented in Fig. 2-181a. Denote the
three-phase currents and the three-phase voltages of the machine by
iaj ib, i c and e at e^ e c respectively. Denote the magnetic linkages (self-
and mutual) in phases A , B, and C by \l/ H , \l/ b , and &.. The nine variables
i*, ibt i c , a, e& e c , $a, ^61 ^ c arc not fictitious quantities, but actual
physical quantities existing in the machine. However, for the reason
already stated and for greater mathematical simplicity, which is a
consequence of the previous reason, it is convenient to employ nine new
variables

id* tfa fa direct-axis current, voltage, linkage

* *fl $q quadrature-axis current, voltage, linkage
io eo, ^o zero-axis current, voltage, linkage.

These nine new variables are defined in terms of the old by the equa-
tions:

id = f [ia cos <f> + i h cos (? - 120) + i c cos (y> + 120)].

*' - - |[H sin ? + i b sin (*> - 120) + i c sin (^ + 120)]. [77]

EQUATIONS OF A MOVING WINDING 159

*0 = Ilia + ib +*"e].

e d = [>a cos <p + e b cos (<p - 120) + e c cos (<p + 120)].

* = - f lea sin p + * 6 sin (p - 120) + e c sin (*> + 120)]. [78]

*o = -3-fo + e b + e c ].

td = f [*. cos *> + *& cos (? - 120) + f cos (> + 120)].

* = ~ f I>a s^ ^ + ^ 6 sin (? - 120) + f . sin fo + 120)]. [79]

The values ^, ^ in Eqs. (77, 78, 79) need not concern us here.

Equations (77) can be solved for i a , ib, i c in terms of i^ i q , to. Like-
wise Eqs. (78) and (79) can be solved respectively for e at e^ e c and

^a, tb, tc>

The performance of the machine can be described by means of a
system n of differential equations some of whose dependent variables
are id, i q , ea, e qj eo, io. When these differential equations can be solved
for the direct-, quadrature-, and zero-axis quantities, then the actual
physical quantities, phase voltages, currents, and linkages, are imme-
diately obtained from the inverses of Eqs. (77, 78, 79).

In general, the direct and quadrature quantities are not actual
physical quantities. Yet under certain modes of operation and in ma-
chines of certain design they may be physical quantities. For example,
if the only winding in the field of a synchronous machine is the main
field winding then Id = 1, the field current of the machine.

2-51. (Tensor Theory Resumed) Equations of a Moving Winding.
Let an instantaneous voltage e be impressed on a closed winding moving
with instantaneous velocity pB in a magnetic field which is produced by
an outside current flowing in a stationary winding. At time / all cur-
rents vary and the moving winding is accelerated.

It is desired to obtain a differential equation relating to the physical
phenomena present in the moving winding. It is necessary to select
a space- time reference frame in order to specify the quantities which
are to be measured. In the reference frame chosen the observer (meas-
urer) is electrically stationary relative to the moving coil. If a volt-
meter is connected to a moving coil through slip-rings RI and R% then

17 R. H. Park, "Definition of an Ideal Synchronous Machine and Formula for the
Armature Flux Linkages," General Electric Review, 31 (1928); R. H. Park, "Two-
Reaction Theory of Synchronous Machines, Part I, Generalized Method of Analysis,"
Trans. A.I.E.E.,4& (1929); B. R. Prentice, "Fundamental Concepts of Synchronous
Machine Reactances," Trans. A.I.E.E., 56 (1937); P. L. Alger, "Calculation of
Armature Reactances of Synchronous Machines," Trans. A.I.E.E., 47 (1928).

160 PRIMITIVE MACHINE WITH STATIONARY REFERENCE AXES

the observer reading the meter is electrically stationary relative to the
winding or moving coil.

Consider the difference of potential measured between two points
PI and P 2 fixed in space and such that PI and P% are in contact with R\
and R 2 respectively. At any instant four voltages are measurable be-
tween PI and P 2 - These are the (a) impressed voltage e, (b) resistance

dtp

drop Ri, (c) voltage , induced in the winding due to change of flux
at

linkages ^>, (d) voltage ^ p0, generated by the moving winding, as if all
currents were steady and the winding moving with velocity pO.

The differential equation expressing the relation between these four
voltages is

e = Ri + + (pff)t. [80]

at

Equation (80) is the equation of voltage of a moving winding. If Eq.
(80) is multiplied by i, the resulting equation

ei = Ri? + ^i + (peWi [81]

at

is the equation of power flow, where,

d<p . ...

ei = instantaneous power input, i = rate of increase of

at

stored magnetic energy,

Ri 2 = power heat loss, (pff)^ i = mechanical power output
(torque X velocity).

The torque upon the coil is i\l/.

By the first generalization postulate 2-27 it has been shown that
the equation of performance, Z -i = e, of a network of n meshes can be
obtained as a generalization of the equation of performance, Zi = e, of a
single mesh. In an analogous manner, the method of procedure is to
modify Eqs. (80-81) (equations of voltage and power flow) for a moving
winding so that these equations become the equation of voltage and
equation of power-flow of a rotating machine. The final equations,
which will be developed in 2 52-2 56, for the performance of the
first primitive machine are

e = R-i + ^ + />0 (Equation of voltage) [82]

at

e - R-i + Lp-l + p6G-i - (R + Lp + pOG)-i

(Equation of voltage) [83]

REPLACEMENT OF ROTOR 161

i-e = i-R-i + i-Lp-i + pOi-G-i (Equation of power) [84]

/ = i . * = i . G i (Equation of torque) , [85]

or in index notation

= Rnn? + L mn ^ + pe 9 m [86]

^ =***" + L mn + pe G mn *, [87]

t-V, [88]

[89]

The tensors to be developed for the primitive machine are: e, R, i,
*, , L, and G or in index notation e m , R mn , A <? m , tm, L mn , and G mn .
The constructions of these tensors are based on Kron's physical con-
cepts and pictures explained in 2-52-2-56, although they can be
derived from purely dynamical considerations.

2-52. Replacement of Rotor (Armature) by Two Sets of Coils at
Right Angles. A stationary reference axis on a rotor winding can be
replaced by a set of stationary brushes, which may be real or fictitious.
The line joining the brushes (Fig. 2 19c) is a brush axis. Since the cur-
rent flows in through one brush and out through the other the current
flows in only one direction AB on one side of the brush axis and in
the opposite direction CD on the other side of the same axis (Figs.
2-19a, a'). Evidently, the flux produced by the q axis current extends
in the direction of the q axis. Thus a set of brushes and consequently
a set of reference axes may be considered as a coil (Fig. 2 19&). Figure
2- 19c represents the brush axis. Since each rotor layer has both a d
and q axis the two reference axes are replaced by two sets of
brushes and these in turn by two coils at right angles. (Figs. 2 19a',
&', c'.)

A final diagram of a primitive machine having two layers of wind-
ings on both the stator and rotor is represented in Fig. 2-186. A four-
layer primitive machine thus consists of four sets of coils. The first two
sets belong to the stator and these two sets are arranged at right angles
as shown in the figure. As many coils exist in each set as there are
separate layers of winding on the stator. The stator coils are stationary
in space. The second two sets of coils belong to the rotor and they also
are arranged at right angles shown in the figure. As many coils be-

162 PRIMITIVE MACHINE WITH STATIONARY REFERENCE AXES

long to each rotor set as there are layers of windings on the rotor. The
rotor windings have instantaneous velocity pO relative to the stator coils.
It may be pointed out that although the physical windings of the rotor
have velocity relative to the stator the rotor reference axes remain
fixed in space since it is evident from Figs. 2 19a, a! that the flux of the
rotor remains constant in direction. Again it is stated that the brushes
may be real or they may be fictitious, depending upon the derived ma-
chine. There may be no brushes on the derived machine. The con-
cept of a brush merely furnishes a reference axis.

(b)

(c)

r

(a') < 6 '> (c')

FIG. 2 19. Representations of a Rotor Axis and a Rotor Winding.

2 53. Current, Voltage, Resistance, and Inductance Tensors of the
First Primitive Machine. In the theory of stationary networks (Sec. 4)
there was only one current and it possessed n components. In the
primitive rotating machine there is only one current and it possesses
2m + 2n components, where m is the number of layers of winding on
the stator and n is the number of layers of windings on the rotor. For
simplicity in writing tensors it is assumed in 2-53-2-58 that m =
n = 1.

The generalized current is the contravariant vector

i<"

i dr

i gr

if

[90]

INDUCED VOLTAGES; FLUX LINKAGE VECTOR
The generalized voltage is the covariant vector
e a = e = e ds d 8 + e<i r d r

d s d r q r

163

e,i a

tdr

e qr

e qa

[91]

whose four components are the terminal voltages of the machine, some
of which may be /cro.

The resistance and inductance tensors representing respectively the
resistances and the self- and mutual inductances of the four windings
are

a d r q r

8 d r q r q.

d,
d r

rds

r r

r r

r q *

[92],

JL

L d

M d

M d

L dr

L v

M q

M q

L q .

[93]

The resistance drop is R-i. Each rotor winding is symmetrical and
consequently r& r = r qr = r r for each rotor layer.

2-54. Induced Voltages; Flux Linkage Vector $. The linkage
vector $ is given by the equation

d, d r q r q

where

<t>ds

<t>dr

<t>qr

*,.

+ 0,
+ 0,

=

PS]

+

+ i*>L g ..

The component $d is the numerical value of the linkages in the d a
axis due to currents in all axes except those at right angles to d*. The
unit vectors in the column at the left of L indicate the axis in which the

164 PRIMITIVE MACHINE WITH STATIONARY REFERENCE AXES

linkages are taken. Of the four components of $ two are linkages in
the stator and two are linkages in the rotor axe3. Consequently * can
be written

* = L-i = L,-i + L r -i, [96]
and

* = L-i= (L. + L r )-i, [97]
where

d, d r q r q, d d r q r q.

[98]

L r =

[99]

The construction of the tensors for the first three terms of each of
Eqs. (82-84) or (86-88) are now complete. It remains only to obtain
the tensors for the last term of these equations.

2-55. Generated Voltages; Rotor Flux-Density Vector . The
vector represents the resultant (due to all currents of the machine)
flux density cut by the rotor conductors. In 2 -54 the linkage vector
represents the resultant flux linkages of all windings of the machine.

First it is desired to derive relations between and $ r . In this
derivation, by means of physical concepts, it is assumed that the flux-
density waves are sinusoidally distributed around the circumference in
each rotor layer. In order to specify vector directions it is assumed
that: (a) the primitive machine is a motor; (#) the rotor rotates clock-
wise ; (c) negative values of the induced and generated voltages appear
in Eqs. (82-85). Suppose (Fig. 2-20a) that the rotor is stationary and
that the resultant flux due to all windings alternates in time. Consider
first the direction of % in a single layer winding of the winding of the rotor.
Now a two-pole sinusoidal wave in a winding can be represented by a
vector drawn from the axis of the rotor to the positive maximum value
of the wave. The maximum induced voltage results in coil AB since
this loop links the maximum number of lines. Consequently, the vector

OL (Fig. 2 20a) represents $. The directions of $ and are the same.

at

The direction of in a single layer winding of the rotor is next con-
sidered. Suppose that the rotor moves at speed pQ and that the flux
is stationary and constant in time. The maximum voltage is generated
in the coil CD (Fig. 2-206). The vector OF represents . The indi-

TORQUE TENSOR

165

cated relations (Fig. 2 206) between the direction of and the direc-
tion of the generated voltages in the rotor conductors arc correct be-
cause the negative of the generated voltages are required for Eqs.
(83-84).

It is evident (Figs. 2 20) that the vectors and $ r in a single layer
winding are perpendicular. Moreover, their magnitudes are equal.
From these two relations W is obtainable from $ r . It is evident from
Figs. (2 20) that to obtain from r it is necessary only to replace the

(a) (6)

FIG. 2-20. Direction of Waves Inside Machine.

(a) Direction of * and d&/dt.

(b) Direction of and V pO.

q axis of $ r by d and the d axis of $ r by q. For comparison, in a
single layer,

$ r = (i d *M d + i dr L dr )d r + (i qr L qr + i qa M q ^ rj [100]

* = (i**M d + i dr L dr )(-q) + (i*L qr + i*M q )d. [101]

The orthogonality of $ r and ^ is verified by $ r -^ = 0.

2-56. Torque Tensor. The vector $ is expressible as the product
L-i. It is desirable to express ^ as a similar product. Accordingly,
define G, the torque tensor, by the relation

d d r q r q

q*

L qr

M q

-M d

-L dr

[102]

166 PRIMITIVE MACHINE WITH STATIONARY REFERENCE AXES

The above tensor represents the mutual inductances between windings
on axis d and those on axis q due to the existence of rotation. The
extension (scarcely a generalization) of this definition to a machine
with any number of layers is obvious. (See Ex. 2, problem set XII.)
From Eq. (101) it is clear that

and

[103]

In the general case, when the flux-density wave is not sinusoidal in
space, the components of G differ from those of L. The derivation of
Eqs. (82-85) or (86-89) is thus concluded.

2-57. Transient Impedance Tensor. Equation (83) has been
written

e = (R + Lp + pOG)-L [104]

Denote R + Lp + p6G by Z. The matrix Z is called the transient
impedance tensor of the first primitive machine. It is of central im-
portance in the sections which follow. For a two-layer machine it is

d r

d.

d r

Z =

'da + L da p

M d p

M d p

r r + L dr p

L gr pe

Mgpd

-Mdpe

-L drP

r r + L vr p

M q p

M q p

r <l* + L <l>p

[105]

2 S8. Direction of Rotation. Since only the relative rotation of the
stator and rotor members determine the equation of voltage, all the

above tensors are valid without any
change if the salient pole rotates
in the opposite direction and the
smooth member is stationary, as
shown in Fig. 2*201. The reference
frames now rotate together with the
salient poles, as in a synchronous
machine. Instead of stator and rotor
subscripts s and r the members may

be called field and armature (subscripts /and a). When the direction
of rotation of the armature or field changes, then pB in Z changes sign.
Otherwise all tensors remain the same.

FIG. 2-201. Direction of Rotation.

ZERO-PHASE-SEQUENCE QUANTITIES

167

2-59. Zero-phase-sequence Quantities. When there are zero-
phase currents i* and i n in both stator and rotor windings, as in case
of unbalanced three-phase machines, two additional rows and columns
exist in Z, also in R, L, and G.

Z =

d.
d r

qr
q*

0,
O r

d.

0.

Or

fd,+Ld,P

M d p

M d p

r r +LdrP

L qr pe

M q pB

-M d pe

-L dr pO

r r +L qr p

M q p

M q p

r qa +L qa p

r 8 o+L*oP

r ro +L ro p

EXERCISES XII

1. The notation in Eqs. (90-93) is easily extended to machines having any number
of windings on both rotor and stator. For m = n = 2 the unit vectors are d,2 f di,
dri, dr2, q/2t 4ri> qi q2- Write out in detail the tensors e, i, R, L, L r , L for a machine
for which m = n = 2. See Ex. 2 for G.

2. For m = w = 2 write out the equations corresponding to Eqs. (94-95). For
m = n = 2, obtain

G =

d r2

qn

q.2

Mqr

Lrql

M qn

MM

L q *

M qr

M&

M02

-Md22

-Mm

-M dr

-L dr2

-M d i2

-M d n

-L dr i

-M dr

168 DERIVED MACHINES WITH STATIONARY REFERENCE AXES

(7)

Derived Machines with Stationary Reference Axes
(Constant Rotor Speed)

The equations of performance of most derived rotating machines
running at constant rotor speed can be set up using stationary reference
axes. Exceptions are pointed out in Sec. 9. The equations of per-
formance and their solution for a few derived machines are given in this
section.

2 60. Equations of Performance of Derived Machines. In 2 36
the method of obtaining, from the primitive network, the equations of
performance of any derived mesh network was explained. It consisted
in obtaining necessary relations between n old and s new currents or
between m primitive and 5 derived currents. From these relations the
C-transformation tensor was written down.

The procedure in obtaining, from the primitive machine with sta-
tionary reference axes, the equations of performance of a derived
machine with stationary reference axes is similar. It is necessary to
obtain relations between the m + n old currents of the primitive ma-
chine and s = m' + n' new currents of the derived machine. From
these relations the C transformation is obtained. Definite rules for the
procedure are given in 2-62.

The tensors of the primitive machine with stationary reference
axes and constant rotor speed are i, e, R, L, G, and Z. The correspond-
ing tensors for the derived machines will be denoted by i', e 7 , R', I/, G',
and Z'. By means of the second generalization postulate 228 it
easily follows that

i = C-i', e' = C r e, R' = C r R C, L' = C r L-C,

[106]
G' = C r G-C, Z' = C r Z-C.

The currents are found by i' = Z^-e'; the torque by/' = i'-G'-i'.

Before stating general rules 2-62 for the determination of C for
derived machines the equations of performance of the repulsion motor
will be derived in detail.

2-61. Introductory Example ; Single-phase Repulsion Motor. Both
the derivation and solution (for stationary axes) of the equation of per-
formance are routine processes. Moreover, these routine processes are
the same for all derived machines with stationary reference axes. The
analyses of types of machines which differ as much among themselves
as a compound direct-current motor, an alternator, a double squirrel-

SINGLE-PHASE REPULSION MOTOR

169

Repulsion Motor.

cage induction motor, or a repulsion motor is substantially one analy-
sis. For this reason a general idea of the routine process of obtaining
and solving the equations of performance of derived machines can be
obtained by the application of the tensor theory to a specific example.
We shall employ a single-phase repulsion motor.

It may be stated to the mathematics student that a single-phase
repulsion motor is one of the simplest alternating-current motors. The
field or stator winding consists of a single layer which is supplied
by single phase alternating-current
voltage. The rotor winding is also a
single layer winding. It is a corn-
mutated drum-armature winding
similar to that of a direct-current
motor. In Fig. 2-21a, CD' repre-
sents the plane of the stator coil; CD
is the line of the brush axis inclined
at an angle to CD'. The brushes at C
and-Dare externally short-circuited.
That a torque is exerted on the rotor for < a. < 90 can be seen as
follows. If a = so that the brush axis is in line with the field poles,
then currents are induced in the coil C'eD'. The torques exerted in
both directions of possible rotation are numerically equal but oppo-
sitely directed and the resultant torque is zero. If a = 90 no currents
are induced in C'eD". Finally, if a has any intermediate value be-
tween and 90 then the planes of the stator and short-circuited rotor
coil intersect in the rotor axis. A force of repulsion acting in the
direction of the displacement exists between the two coils. This is in
accordance with the experimental fact that conductors carrying cur-
rents in opposite directions repel each other. This repulsion furnishes
the operating torque.

(a) The C transformation. To obtain C for the repulsion motor it is
necessary only to compare the windings described above (Fig. 2-216)
with the windings of the first primitive machine (Fig. 2-186). Evi-
dently, the relations between the primitive currents i and the derived
currents i' are (from Fig. 2-216)

- i d> + 0-

<*

0- * + (cos a) * whence C - d r

+ (sin a) t

q,

I

cos a

sin a

[107]

170 DERIVED MACHINES WITH STATIONARY REFERENCE AXES

(6) Primitive and derived tensors. By comparison of the winding
layers of the derived machine (Fig. 2-216) with those of the primitive
machine (Fig. 2-186) the Z transient matrix (Eq. 105) is found to be

d.

d.

d r

d.

r, + L d ,p

M d p

M d p

r r + LdrP

LgrpO

-M d pO

-L dr pO

r r +L gr p

d,

r.+L,p

Mp

Mp

r r +L r p

L r pO

-MpO

-L r pB

r r +L r p

[108]

The equality sign in (108) is justified by the fact that the machine has a
smooth air gap resulting in L dr = L qr = L T .

The transient impedance matrix of the derived machine is

d.

Z' = C-Z-C

d,

r, + L.p

M (cos a)p

M[(cos a)p (sin ct)p8]

r r + L r p

[109]

The induced metric tensor of the derived machine is

d. a

d.

If - C t L-C

a

L,

M cos a

M cos a

L r

[110]

The voltage vector of the primitive machine is

d d r q r

e -

[111]

The voltage vector of the derived machine is

d, a
e 7 = CfQ =

[112]

SINGLE-PHASE REPULSION MOTOR 171

The admittance matrix of the derived machine is

d. a

d.

(r r + L T p)\/D

(-Mcosa)p\/D

1113]

- Jl/[(cos a)p - (sin a)pe]l/D

(r, + L t p)l/D

where

D = (L 8 L r - M 2 cos 2 a)p 2 + (r r L 8 + r 8 L r + M 2 sin a cos a pff)p + r r r 9 .
(c) Equation of performance. The equation of performance is

Z'-i' = e' or i' = Y'-e'.
The symbolic solution of the last equation for i' is

where

D = (L 8 L r - M 2 cos 2 a)p 2 + (r r L 8 + r 8 L r + M 2 sin a cos a pO)p + r r r 8 .

(d) Transient current solution. Equation (114) is the symbolic cur-
rent solution under all conditions. The transient solution due to
suddenly impressed constant voltage can be obtained by the Heaviside
operational calculus as if the network were a stationary linear network.
This startling fact is true not only for the repulsion motor, but for all
rotating machines with stationary reference axes and constant rotor speed.

Under constant rotor speed pQ = vu where w is constant synchro-
nous speed and v is a proper fraction. To obtain the transient currents
replace, in Eq. (114), ea by lea where i is the Heaviside unit function,
and substitute the symbolic expression for the current in the Bromwich
line integral. 18 The substitution yields

' - ! H C gX ' JX -u *L C

1 " 2* bo d Vo (X + a) 2 + f + A) *Vo X

X[(X+ a) 2 + ft 2 ]
B

where A, B, C, E, D , a, and ft are constants.

(e) Steady-state current solution. The steady-state solution for
terminal voltage (e sin o>0d a is obtained as in the case of linear sta-
tionary networks.

18 Volume I, p. 262.

172 DERIVED MACHINES WITH STATIONARY REFERENCE AXES

As a general procedure convenient for all rotating machines it is
systematic to write the steady-state equation of performance for the
machine in question. This equation is Z 8 -i a = e where Z 8 is the
steady-state impedance tensor of the derived machine obtained from
Z by the substitution p = jw, pB = z>, coZ, = X, and wM = X m . The
impedance matrix Z' s for the repulsion motor is

r.+jX.

jX m cos a

X m (j cos a

v sin a)

r r + jX r

[116]

The steady-state admittance Y ( of the repulsion motor is

d. a

d.

where

(r r +jX r )l/D.

-jX m cosal/D 8

^Y m (sin a v j cos a)\/D 8

(r a +jX 8 )l/D 8

[117]

D 8 = (r r r t + X 2 m cos 2 a - X 8 X r ) + j(r r X s + r 8 X r + vX* m sin a cos a).
The steady-state currents are

" __ v' / " Ov+.7^r)d . * v (sin av j cos a)a ri101
i a i a 'C = tf r A m [lloj

(/) Transient and steady-state currents. If the suddenly impressed
voltage is (ej sin w/)d, then both the transient and steady-state currents
are obtainable by the substitution of the results of Eq. (115) in Du-
hamel's superposition formula. 19

(g) Transient torque. The torque tensor G' for the derived ma-
chine is

G' = Ci-G-C. [119]

The transient torque ft is given by

ft = i'-G'-i ;
where the instantaneous current i' is given in Eq. (115).

[120]

19 E. J. Berg, Heaviside's Operational Calculus, p. 67; V. Bush, Operational Circuit
Analysis , p. 56.

SINGLE-PHASE REPULSION MOTOR

The torque tensor for the repulsion motor is

d.

G' = CrO-C =

M sin

173

[121]

(A) Transient and steady-state torque. The total torque is given by
/,. = i'-G'-i' [122]

where i' is the current of heading (/) above. The result will contain
(1) transient torque, (2) steady part of the steady-state torque, (3)
oscillating component of steady-state torque. Important torque cal-
culations are, however, more easily carried out as shown in headings
(i) and (j) following.

(*) Steady part of steady-state torque. The steady part of the
steady-state torque is given by taking the real part of

/. p -i'*-G'-i' [123]

where i'* is the complex conjugate of i'as given in Eq. (118). The quan-
tity fsp for the repulsion motor is

X m (sin a v + j cos a) . e(r r + jX r )
fsp = e ^r* ( X m sin a)

When the torque is computed in synchronous watts the torque tensor
is to be multiplied by co. The co has already been multiplied into G' in
the expression above since coM has been replaced by X m . The real
part of f ap is easily written out.

(j) Oscillating component of steady-state torque. The oscillating
component of the steady-state torque is the non-transient and non-
steady part of the torque in heading (h).

EXERCISES XIH

1. The winding diagram of a single-phase induction
motor is shown in Fig. 2-22. It will be shown in 2-62 on
the C tensor that the C tensor for a single-phase induction
motor with stationary reference axes is

d, d, q r

d.
C = d r

Qr

1

1

1

FIG. 2-22. Single-phase
Induction Motor.

174 DERIVED MACHINES WITH STATIONARY REFERENCE AXES

Compare the windings indicated in Fig. 2 22 with those of the primitive machine
in Fig. 2*186 and by inspection fill in the transient impedance matrix, Eq. (105).
Compute Z' = C r Z-C. Carry out headings (b) and (c) of 2-61 and obtain the
equation of performance corresponding to Eq. (114).

2. The winding diagram of a compound direct-current motor is shown in Fig. 2 23.
As shown later the C-transformation tensor (or
transformation matrix which is one manifestation
of the transformation tensor) is

1

-*

1

-*.

qn

FIG. 2-23. Compound Direct- Compare the windings indicated in Fig. 2 23 with

current Motor. those of the primitive machine of Fig. 2 18& and by

inspection fill in the transient impedance matrix,

Eq. (105). Compute Z' = CrZ-C. Carry out headings (b) and (c) of 2-61 and
obtain the equation of performance corresponding to Eq. (114).

2 62. The C-Transformation Tensor for Machines with Stationary
Reference Axes. The construction of the C-transformation matrix
(one manifestation of the C tensor) can be broken up into three steps.

(a) Winding diagram. The first step is a sketch of the winding
diagram of the derived machine in question. Such a diagram for a
single-phase induction motor occurs in Fig. 2-22. The diagram must
show the (1) number of layers of windings on the stator and their rela-
tive positions, (2) number of layers of windings on the rotor and their
relative positions, (3) electrical connections between the different rotor
windings and between the rotor and stator windings, (4) physical or
fictitious brushes. (See 2-52.)

(b) Comparison of winding diagrams. The second step is a com-
parison of the winding diagram of the particular derived machine with
the winding diagram of the primitive machine shown in Fig. 2-186.
The currents of the primitive machine are denoted by unprimed letters,
i d * 2 , i dal , i drl , i dr2 , etc. The currents (really components of the current
vector) of the derived machine are denoted by primed letters, i', i 6 ', i c ' ,
etc. (See Eq. 107.)

(c) Current relations: equations. The third step consists in express-
ing the old or primitive currents in terms of the new or derived currents

THE C-TRANSFORMATION TENSOR 175

of the derived machine. In general, the number of derived currents is
smaller than the number of primitive currents. The primitive currents
are written on the left-hand side of the equal signs; the derived cur-
rents on the right. The C matrix is the matrix of the coefficients of the
derived currents. (See Eq. 107 and the C matrix for the single-phase
repulsion motor.) In writing the current equations four important
principles are employed.

(1) If a winding of the derived machine is in the same position as in
the primitive machine, then i d = i d ' (see Fig. 2-22 and the C matrix).
Let the magnetomotive force of some winding whose current is, say i d ',
be taken as standard. If the number of turns in the winding i d is n
times that of i d ', then i d = n i d ' . (See Fig. 2-23 and the adjacent C
matrix.)

(2) If two windings are connected in series their currents are re-
placed by one current and the number of columns in C diminishes.
The second half of principle (1) may necessarily be employed. (See
Fig. 2-23.)

(3) If a set of brushes on a rotor winding is shifted through an angle
a the winding in the brush axis can be considered to lie on the original
layer of winding before the shift took place. This is because the rotor
windings are symmetrically distributed. If i f ' is the current in the
brush axis the relations between i f ' and the primitive currents i dr and
i qr are

dr = /'

i r = i fl sin a.

If there are two sets of brushes and the first is shifted through the angle
a, the second through the angle 0, then the relations between the primi-
tive and derived rotor currents are

i dr = if cos a i g ' sin j8

For examples of single and double sets of brushes, see Eq. (106) and the
C matrix belonging to Fig. 2-27 respectively.

(4) If a winding on the stator is shifted through an angle a then it
is assumed in the analysis that it lies on a different layer from the other
stator windings. (See Fig. 2 26 and the C matrix.)

The four principles above pertain to the machines with stationary
reference axes. Additional principles for machines with moving refer-
ence axes are given in Sees. 8-9.

176 DERIVED MACHINES WITH STATIONARY REFERENCE AXES

Kron
FIG. 2-231.

Park
Sign Conventions.

2-63. Performance Calculations for Machines with Stationary
Axes. In 2-61 are the performance calculations of the single-phase
repulsion motor. They consist of the ten steps (a), (6), , 0) of
2-61.

The performance calculations for each type of derived machine,
possessing stationary reference axes and constant rotor speed, are
identical; only the tensors and the C-transformation matrix differ.
Most machines can be analyzed by means of stationary reference axes
under the conditions given in the last sentence. (See Sec. 8 for moving
axes.) The equation of performance is linear with constant coefficients
and can be solved in terms of time functions either by means of the
Heaviside operational calculus or by the principles of 1 - 26-1 29.
These two characteristics of only a very small part of Kron's analysis of

rotating electrical machinery are alone
sufficient to rank it as an important
achievement.

2-64. Sign Conventions and Ma-
chine Constants. In analyzing the syn-
chronous generator, Park assumed a
sign convention which differs from
that used here. In this chapter the sign

conventions are the same as those used by induction motor engineers
and which also follow, from the dynamical equations of Lagrange.
Park's sign convention differs in two respects: (a) Assuming that the
salient pole of the primitive machine rotates (2-50) and the armature
is stationary, in this chapter the salient pole rotates from d to q
though with Park it rotates from d to q. Hence, to check Park's
results all pO of the present chapter should be replaced by pQ. (V)
In this chapter every term in the equation of voltage represents an
impressed voltage, although Park uses generated voltages which are
the negative of the impressed voltages.

(Kron) e imp = Z-i, (Park) e gen = -Z-i,
i.e., assuming zero speed for a single coil

(Kron) e lmp - Ri + L j , (Park) e gen - - - L %
at at

The convention of Park differs also from that universally used in sta-
tionary network analysis.

SIGN CONVENTIONS AND MACHINE CONSTANTS

177

In addition to sign conventions, Park also differs in symbolism from
that of this chapter in two respects: (a) Park uses a per unit system,
hence among others he denotes inductances L by X (since numerically
a reactance is equal to an inductance in the per unit system). (6)
Park assumes the field winding and the amortisseur winding to be
permanently short-circuited, so that the remaining equations contain
short-circuited inductances X(p) instead of open-circuited induc-
tances.

Of course, by eliminating the variables of the same two windings by
the method of problem 3, problem set XVII, the equations of this chap-
ter are reducible to Park's results.

EXERCISES XIV

The following exercises pertain to machines
having stationary reference axes and constant
rotor speed.

1. Show that the winding diagram for the
salient-pole synchronous machine is that shown
in Fig. 2-24. Show that

d*2 di d r q r Q

Qr

1

1

1

1

1

FIG. 2-24. Salient-pole Synchro-
nous Machine.

2. Show that in a synchronous motor running at synchronous speed and under
balanced conditions the applied voltages are all constant.

3. Obtain the equation of performance (Eq. 114) for the synchronous machine of
Ex.1.

4. Obtain, from the equation of performance, the symbolic solution for i dr and
t* r in Ex. 1.

5. Express i dr and i qr in Ex. 4 as functions of the time by Heaviside's operational
calculus.

6. Compute G' = Cr G C for the salient-pole synchronous machine.

7. Compute the steady-state torque of the salient-pole synchronous machine.
Obtain expressions for the transient torque, under three-phase short-circuit, of a
synchronous machine. Compare with R. E. Doherty and C. A. Nickle, "Three-Phase
Short-Circuit of Synchronous Machines/' Trans. A.I.E.E., 49, April, 1930.

178 DERIVED MACHINES WITH STATIONARY REFERENCE AXES

8. Show that the winding diagram for the double squirrel-cage induction motor is
that shown in Fig. 2-25. Show that

d d r i dr2 Qr2 Qrl Q

qn

1

1

1

1

1

1

FIG. 2*25. Double Squirrel-cage Induction Motor.

9. Show that for the double squirrel-cage induction motor running in steady-state
operation all applied voltages are sinusoidal.

10. Obtain the equation of performance (Eq. 114) for the double squirrel-cage
motor.

11. Solve Exs. 7, 8, 9 for the asymmetrical squirrel-cage induction motor.

12. Obtain the symbolic solution for i dr and i qr of the asymmetrical induction
motor.

13. By means of the Heaviside operational calculus solve for i dr and i qr in Ex. 12.
The applied voltages are sinusoidal.

14. Solve for steady-state i dr and i qr in Ex. 12. The applied voltages are sinus-
oidal.

15. Obtain the torque tensor for the asymmetrical induction motor.

16. Compute the steady-state torque of the machine in Ex. 11.

17. Obtain the symbolic solution for i f and i da * for the machine of Ex. 2, problem
set XIII.

SIGN CONVENTIONS AND MACHINE CONSTANTS

179

18. Obtain by means of the Heaviside operational calculus transient i f and t d *a.
19 Show that the winding diagram for a shaded-pole motor is that shown in
Fig. 2-26. Show that

C =

d.

Qr

q.

1

cos a

1

1

sin a

FIG. 2 26. Shaded-pole Motor.

20. Show that the winding diagram for the shunt polyphase commutator motor
is that shown in Fig. 2-27. Show that

FIG. 2-27. Shunt-poly-
phase Commutator
Motor.

da
d r
Qr

q

1

cos a

sin a

sin a

cos a

1

PROBLEMS XV

1 . Derive the equations of performance of the single-phase induction motor.

2. Sketch the winding diagram and obtain the C matrix for the Schrage motor.
Engineering reference for description of winding layers of the machine, A. S. Langs-
dorf, Theory of Alternating- Current Machinery, p. 752.

3. Sketch the winding diagram and obtain the C matrix for the Deri motor.

4. Sketch the winding diagram and obtain the C matrix for the phase advancer.

5. Sketch the winding diagram and obtain the C matrix for a frequency converter.

180 PRIMITIVE MACHINE WITH ROTATING REFERENCE AXES

(8)
Primitive Machine with Rotating Reference Axes

In this section the equations of performance of machines with rotat-
ing rotor axes are derived. Rotating reference axes are necessary for
machines possessing only one slip-ring (such as a single-phase alterna-
tor) and for machines possessing slip-rings across which the load is
unbalanced (for example, the single-phase short circuit of an alternator

m

FIG. 2-28. Rotor Reference Axes FIG. 2-29. Rotor Reference Axes Ro-
Rotating at Same Instantaneous tat ing at Instantaneous Velocity Dif-
Velocity as the Rotor. ferent from that of the Rotor.

and for balanced polyphase machines where axes may be assumed ro-
tating with the revolving field, thereby reducing the analysis to that
of a direct-current machine (important in hunting studies). For these
cases stationary reference axes on the rotor cannot be employed.

2-65. Second Primitive Machine. The second primitive machine
can be described by contrast and comparison with the first primitive
machine.

In the first primitive machine all reference axes are stationary in
space. In the second primitive machine the stator reference axes are
stationary axes, but the reference axes on the rotor move with the rotor
conductors. (See Fig. 2 28.)

The components of the metric tensor L for the first primitive ma-
chine are constants. The components of the metric tensor for the
second primitive machine are functions of rotor position, i.e., of the
time. (For example, see Eq. 126.)

Items (a), (6), (c), (d), (e), and (h) of 2-49 are identical for both
machines.

STARTING POINTS IN DERIVING THE VOLTAGE EQUATION 181

2-66. Starting Points in Deriving the Voltage Equation of the
Second Primitive Machine. The voltage equation of the second primi-
tive machine can be derived in at least three different ways. These are
characterized by the starting points or fundamental underlying equa-
tions. The underlying equations are: (a) holonomic equations of La-
grange, (b) Maxwell's voltage and torque equations, (c) equations of
the non-holonomic machine, 2-53. Although only the last method is
employed in this chapter, it is instructive to sketch briefly the first two
methods.

(a) Lagrange's equations. (See Sec. 3, Chap. I.) The holonomic
equations of Lagrange can be used as a starting point for the equations
of the second primitive machine if the rotor reference axes move with
the rotor conductors. The instantaneous stored kinetic energy, the
dissipation function, and the potential energy are respectively T =
1/2 L mn t n i n , F = 1/2 R mn ?"i n , and zero. Let x k denote the total num-
ber of charges that have passed through any winding and the angle
described by the rotor during some definite time interval. Then
dx k /dt = i k . The voltages applied to any winding and the instantane-
ous applied shaft torque arc denoted by e k . By substitution in the
equation

<B\a*V 3** 3**~

and the performance of certain simplifications, the equation of motion
of the second primitive machine is

e k = R mk i m + L mk + [mn, k]^i n [124]

where [mn, k] = - (-~ + -^ - ~^)' The geometric object of

rank 3, [mn, k] is the holonomic Christoffel symbol of the first kind.
The manipulations described above in obtaining Eq. (124) from
Lagrange's equations are left as an exercise.

Much analysis remains in constructing the forms of L mn and R mn
adaptable to rotating electrical machines and in deriving C transfor-
mations for winding connections.

(6) Maxwell's voltage and torque equations. Maxwell's equation of
voltage for a system of conductors is

e = R-i + or e m = R mn i n + ~> [125]

182 PRIMITIVE MACHINE WITH ROTATING REFERENCE AXES

where * = L*i is the flux-linkage vector. When L as a function of
angular position of the rotor and R are known, the voltage equation of a
machine can be established. Its equation of torque is

97;^!. 3L .
1 36 2 1 'a0' 1
or

As an example consider the two-phase salient-pole alternator. The
components of the inductance tensor L are found by test by measuring
the self- and mutual inductances of the field and armature as a function
of the angular position of the rotor relative to the stator. For this
machine L explicitly is

d a b q,

d.

a
b

q.

L d .

Md cos 6

Ala sin 6

Aft cos

LS + LD cos 26

LD sin 26

Mq sin 6

Md sin 6

Li)Sin 26

LS - LU cos 26

M q cos 6

M q sin 6

Afq COS 6

L v ,

[126]

where L s = (L dr + L qr )/2 and L D = (L dr - L qr )/2. The tensor R
has the same form as in the first primitive machine.

It is possible to start with either of the equations of voltage (124)
or (125) and to obtain Eq. (83) which is the equation of voltage of the
first primitive machine. This is accomplished by changes of variables
by means of quasi-holonomic relations between the currents of the
second and the currents of the first primitive machine.

Since Eq. (83) can be derived * from Eq. (124) or (125), it is rea-
sonable to suppose that the equation of voltage for the second primitive
machine can be obtained from Eq. (83). This supposition is correct.
We shall follow this method in 2-67.

2 67. Equation of Voltage of Machines with Axes Rotating at Any
Speed. Let the reference axes on the stator or rotor of the first primi-
tive machine (Fig. 2 29) be rotating with any instantaneous velocity

" See Ex. 1, problem set XVI.

EQUATION OF VOLTAGE OF MACHINES 183

pBi different from the instantaneous velocity of the rotor p0 2 . The
equation of voltage for the first primitive machine is

e = R-i + Lp-i + p0 2 G-i or e m - tf mn i n + L mn + p8 2 G n i n . [127]

at

Let the currents of the new primitive machine be denoted by i' and let
the relation between i and i' be given by i = C -i' where C is a function
of 62 and where C is such that the power input i-e is invariant. The
substitution of i = C-i' in Eq. (127) yields

e = R C-i' + L-/>(C-i') + p0 2 G-C i'
or

e = R-C-i' + L-~i' + L-C-^-' + p0 2 G-C-i'.

at at

The multiplication of the last equation by C t gives

C r e - C r R-C-i' + C r L- -i' + C r L-C-

O"l Ot (it

or

e' = R'-i ; + L'-^-- + p0 2 G'-i' + pOi V'-i' [128]

(it

where

e' = C,-e

R' = CrR-C

L' = CrL-C [129]

G' = CrG-C

Equation (128) is the voltage equation for machines with axes rotating
with a velocity different from that of the rotor. The matrix V is called
the Christoffel object.

To obtain the transformation formula for V, let Eq. (127) be written

e - R-i + Lp-i + pd 2 G-i + ft l V-i
where V = O. Making the substitution i = C !' in this equation, mul-

184 PRIMITIVE MACHINE WITH ROTATING REFERENCE AXES
tiplying through by C/ f and using Eqs. (129), we have

e' - R'-i' + L'-pi' + p0 2 G'.i' + pe l crV-C + C r L-

or

e' - R'-i' + !//>!' + />0 2 G'-i' + /^V'-i' [130]

where

V'-CrV.C + CrL.|. [131]

oVi

Equation (131) is the transformation formula for V. Evidently V is
not a tensor.

If pdi and p6 2 are constants, then Eq. (130) can be written

e' = (R' + Up + pd 2 G' + p6 l V) -i' = Z'-i'
where

Z' = R' + Up + p0 2 G' + pOi V.

2-68. Equation of Voltage for the Second Primitive Machine (or
for the Machine with Reference Axes Attached to the Rotor). If the
reference axes rotate at the same velocity as the rotor then pdi = p0 2
and Eq. (130) reduces to

e' - R'-i' + L'p-i' + peTX'-i' [132]

where N ; = G'+ V and the subscript of 6 has been deleted. Evidently,
the transformation formula of N is the same as that of V.

IT - CrN-C + CrL- -

90

If p0 = a constant, then Eq. (132) can be written

e' - Z'-i ; [133]

where

Z' - (R' + L'p + p8W).

SIS 3L'

It can be proved that N 7 = ^ Since ^ p$ = pL' t Eq. (132) for

90 90

the second primitive machine reduces to

e' = R-i f + p(L'-i'). [134]

THE C MATRIX FOR ROTATING AXES 185

2 69. The Transformation Formula of Z' for Machines with Rotat-
ing Axes. To find the transformation formula for Z we have

Z' - R 7 + L'p + p0 2 G' + pe l V

= (CrR-C + CfL-Cp + p0 2 CrG-C)

3C

= CfZ-C + C|-L -- pQ\ since V = for stationary axes.
90i

Since the transformation formula for Z is

Z' = (c t 'Z'C + Crl'^ t*i) [135]

\ D0i /

evidently Z, for machines with rotating axes, is not a tensor.
The torque tensor for machines with rotating reference axes is

G' = CrG-C [136]

and the torque is

/' = i'-G'-i'. [137]

PROBLEM XVI

1. Derive the voltage equation (Eq. 83) of the first primitive machine from
Maxwell's equation

e =R-i + p(L-i)
where L is given by Eq. (126). '

Hint: Take the C transformation to be the inverse of the C given by Eq. (138).
Replace i in Maxwell's equation by C-i' and carry out operations somewhat similar
to those of Eq. (127-134).

Derived Machines with Rotating Reference Axes

The equations of performance of derived machines are obtained in
much the same manner as explained in Sec. 7.

2 70. The C Matrix for Rotating Axes. If slip-rings exist on the
machine instead of brushes, the C matrix is the same as in 2-62, with
the important difference that the constant angle must be replaced by the
variable angle 0, where 6 defines the position of the rotor at time I.
The steps in 2-62 apply in the order enumerated.

186 DERIVED MACHINES WITH ROTATING REFERENCE AXES

2-71. Representative Example: Two-phase Synchronous Ma-
chine. The winding diagram and the C matrix for the two-phase syn-
chronous machine are respectively

FIG. 2-30. Two-phase
Synchronous Machine.

1

COS0

sin0

sin 6

COS0

1

[138]

The metric tensor L and the transient impedance matrix both of
the first primitive machine are given by Eq. (93) and (105) respectively.
(The subscripts in Kq. 93 may be omitted.) The transient impedance
matrix for the two-phase synchronous machine with moving axes is com-
puted by Eq. (135). (Subscripts on 6 will now be omitted.) The C ma-
trix for the computations is given by Eq. (138). The computations yield

d a a b q.

Z'

r d , + L da p

A/d(cos 6p
sin BpB)

A/d(sin Op
+ cos OpO)

A/ d (cos Op
sin OpO)

[r r +(L dr co$?0
+L qr sm 2 0)p
+ 2(L qr

-L dr )

sin 6 cos BpB]

(L qr - L dr )
[sin B cos Bp
+ (cos 2 0-
sin 2 0)pO]

M q (sinOp
+ cos OpO)

M d (sin Op
+ cos OpO)

(L qr - L dr )
[sin B cos Op
+ (cos 2
- sin 2 B)p6]

[r r +(L dr sm 2
+L qr cos?0)p
+ 2(L dr

--Z.fr)

sin cos OpO]

M q (cos Op
sin OpO)

M q (sin Bp
+ cos OpO)

M q (cos Op
- sin OpO)

r qa + L q ,p

[139]

STEADY-STATE CURRENT SOLUTION 187

where p refers only to i. (See Ex. 1, problem set XVII for simplifica-
tions.)

The voltage vector e' is given by the first of Eqs. (129).

The voltage equation is given by Eq. (133).

The torque tensor G' is found by G' = <VG-C or by selecting all
terms in Z' containing p6.

The instantaneous torque / is given by

/- i'-G'-i'. [140]

2-72. Transient Current Solution. In general, the voltage equa-
tions are linear differential equations with periodic coefficients. Such
equations cannot be solved directly by the operational methods of
Heaviside. Two cases obtain relative to the transient solution.

(a) Rotating axes unessential. If in the derivation of the equation of
voltage stationary axes could have been used instead of rotating axes,
then by changes of variables or by Heaviside shifting formulas the dif-
ferential equations can be reduced to forms to which Heaviside's
methods are applicable. In this case it is preferable to derive ab initio
the equations of performance employing stationary axes.

(b) Rotating axes essential. If rotating axes are necessary in the
derivation of the voltage equations, again two cases obtain relative to
the transient solution.

(1) A sufficiently accurate transient solution may be obtained by
simplifying assumptions based on physical principles. One specific tool
relative to such simplifying assumptions is the constant linkage the-
orem. 21 An application of this theorem has been made to the set of
differential equations defining (subject to certain assumptions) the
transient currents of a single-phase short circuit of a synchronous gen-
erator with moving reference axes. 22

(2) For an accurate mathematical solution recourse may be had to
the advanced methods for solving analytic differential equations in
Chap. III.

2-73. Steady-state Current Solution. With the exception of item
(ft) (1), the statements of 2.72 are true relative to the steady-state
current solution.

For the steady-state current solution item (b) (1) should be replaced
by the statement that it is possible under many practical conditions to

R. E. Doherty, "A Simplified Method of Analyzing Short-Circuit Problems,"
Trans. A.I.E.E., 42, 849 (1923).

" R. E. Doherty and C. A. Nickle, "Synchronous Machines IV; Single-Phase
Phenomena in Three- Phase Machines/' Trans. A.I.E.E., 47, 457-492.

188 DERIVED MACHINES WITH ROTATING REFERENCE AXES

derive the steady-state impedance matrix from the transient impedance
matrix. For this derivation the reader is referred elsewhere. 28

PROBLEMS XVH

1. In the impedance matrix of Eq. (139) make the following obvious replace-
ments:

Md cos $ pi by M<t(p cos i + sin $ pOi) t

(cos pi sin pOi) by Mdp cos i
and reduce Z' to the simpler form

d. a b

Td + Ldsp

pMd cos

pMdsin

pMd cos e

r r + P(L a + L D cos 20)

pLo sin 20

pjl/gsin

pMdsin

pLo sin 20

r r + />(s - D cos 20)

pM q cos

/> Jlfg sin 9

pAfg cos

r fl . + L q9 p

where LS = (Ldr + L qr )/2, L/> = (Ldr L qr )/2, and p refers to both cos and *".

2. Write the torque tensor for the two-phase synchronous machine.

3. The winding diagram of the three-phase synchronous machine is shown in
Figs. 2 -31 and 2 -32.

The C matrix is

df ft b c q

Qr*

qn
q.

1

cos

cos(0 + 120)

cos(0 - 120)

sin(0 - 120)

sin(0 + 120)

sin

1

M G. Kron, The Application of Tensors to the Analysis of Rotating Electrical
Machinery, pp. 73-74. General Electric Review, 1938.

STEADY-STATE CURRENT SOLUTION

189

The inductance tensor L is given by Eq. (93) where L is enlarged to eight rows
and eight columns. The elements of L are constants.

FIG. 2-31. Generalized Machine.

FIG. 2-32. Three-phase
Synchronous Machine.

4. Compute I/ by means of the third equation of Eqs. (129).

5. In L' of Ex. 4 make the substitutions

Ldr cos 2 -f L qr sin 2 - A + B cos 20
M dr cos e cos(0 + 120) + M qr sin 9 sin(0 + 120) - -~ ^SlJ!fe

2

M dr -

: cos(20- 120)

and replace

L r byJdr + Zo/2)
Jlfrbyf(L,-L )
Jlf by fJlf.

Check the linkages given by L'-i' with those given by Park. 24

6. Derive, by the method of Sec. 9, the voltage equations of single-phase short
circuits and compare with the Doherty-Nickle equations. 26

7. Derive, by the method of Sec. 9, the voltage equations of three-phase short
circuits and compare with the Doherty-Nickle equations. 20

24 R. H. Park, "Definition of an Ideal Synchronous Machine and Formula for the
Armature Flux Linkages/' General Electric Review, 31 (1928); "Two- React ion Theory
of Synchronous Machines/' Part I, Generalized Method of Analysis, Trans. A.I.E.E.,
42 (1929).

R. E. Doherty and C. A. Nickle, op. tit., 2-72.

* Ft. E. Doherty and C. A. Nickle, "Three-Phase Short Circuit/ 1 Trans. A.I.E.E.,
49 (1930).

190 MACHINES UNDER ACCELERATION

(10)
Machines Under Acceleration

In Sec. 5-9 inclusive it has been assumed that the rotor runs at
constant speed. In Sec. 10 accelerated motion of the rotor is taken into
account.

2 74. Equations of Voltage and Torque. When electrical machines
run at a constant speed p6 2 = v 2 , two invariant equations are used in
their analysis; namely, the equation of voltage

e m - R mn f l + L mn + pe 2 G mn F + pe l V mn F [HI]
at

(where pB\ is the speed of the reference frame, if rotating) and the
equation of torque (impressed)

f=-G mn f n i n . [142]

Each of these equations has been established separately.

When machines have an accelerated motion (during starting or
during small oscillations, etc.), the friction R and moment of inertia L
also play a part in the analysis and the above equation of torque be-
comes (for a single machine)

f = Rv + L - - G mn i m i n . [143]

at

In order to study accelerated motions more conveniently, it is neces-
sary to replace the two invariant Eqs. (141) and (142) by a single in-
variant equation, the so-called equation of motion, which splits up
conveniently into its component equations of voltage and torque. The
establishment of a single invariant equation also facilitates the analysis
of rotating machines with complex structure, also the analysis of any
number of interconnected machines with any type of actual or hypo-
thetical reference frame.

2 75. The Equation of Motion. In order to establish the equation
of motion, new types of geometric objects will have to be introduced
whose components contain both electrical and mechanical quantities.
(In Eqs. 141 and 142 each tensor contains either electrical or mechanical
quantities.) For instance, for the primitive machine, the quan-

THE EQUATION OF MOTION

191

titles that are not due to motion are arranged in the following four
tensors

a d r q r q s

a d r q r q,

Cdr

f

o\ d a d r q r q a s

d,
2.

5

*.

r r

^r

^.

/?

f

\p

a \
d,

dr
Oaf = 9r

S

f

i dr

;-

i"

pe

d.

d r q r

q. s

I'd*

M d

Ldr

7/ f

/<

L

[144]

The tensor a a p is called the metric tensor.

In general there are as many geometrical axes 5 as there are me-
chanical degrees of freedom in the system. Since the shafts of the
various machines may also be interconnected by couplings, the trans-
formation tensor C/, also contains geometrical axes. For the repulsion
motor (Fig. 2-33)

d,

rf a a s

1

cos a

cos a

1

[145]

The quantities which are due to the existence of motion, namely, the
torque tensor G mn (that occurs twice in the equations, once giving
generated voltages, the second time torques) and V mn are arranged into
a geometric object of valence three, r a p, y called the affine connection
as shown in Fig. 2-34.

192

MACHINES UNDER ACCELERATION

In terms of the five geometric objects e a , i", Raft t a a ft, and Taft, y the
equation of motion of all electrical machines (and in general all electro-
mechanical or electrical systems) is

, * * + a ^+r fff F1461

^a == -*Hr/9 * i Q(xft ~~\ T *-fty,a * * L^^"J

at
The equation of power is

( O/V . ja . vv

e a a

[147]

<*.{

FIG. 2-33. Repulsion Motor. FIG. 2-34. Affine Connection, T a 0,y.

2 76. The Metric Tensor a a/J . The equation of motion introduces
three geometric objects T a ft, y , a a p, and Raft (in addition to the vectors
e a , i a and the scalar /) which play a basic part in the study of dynamics
and geometry. The metric tensor a a p plays a part in the definition of
the magnitude of a vector, while the affine connection and resistance
tensor Raft play a part in the definition of its direction. (In the invari-
ant equations of stationary mesh networks e a = z^ the vectors i a ,
and e a have neither magnitude nor direction. They have only compo-
nents, that is, an existence.)

One of the most important concepts is the metric tensor a a ft repre-
senting the self- and mutual inductances and moments of inertia.
When a vector A a is given, its magnitude is defined as

\A

[148]

If the vector is the generalized current vector i a , then its magnitude is
equal to \/2T where T is the total kinetic energy stored instantane-
ously in the system.

THE COMPONENT PARTS OF THE AFFINE CONNECTION r AY 193

With the aid of the metric tensor a a $ it is possible to raise or lower
the indices of tensors. If the inverse of a^ is a^, then

^ =.* a 7 or R ft a ay - R0 y . [149]

The indices of F a/ 3 t7 (not being a tensor) cannot be moved. An excep-
tion is the last index, so that

IVM^ 1 -!*,. [150]

The flux-linkage vector <p a is also the covariant form of i a and vice
versa, since

i" a a p = <f>p = ifi and <? a a a& = <f = fi. [151]

Also

2T - a+Ft = *;<* = i a i = <t>a<p = | v>| 2 - |* | 2 .

That is, the current vector i a and the flux-linkage vector <p a are the con-
travariant and the covariant representation ("extcnsity" and "inten-
sity" factors) of the same physical entity, the "stored kinetic energy"

r.

Tensors having the same base letters but having indices in different
positions, as R a p or R? ft or R* or R"? arc called associated tensors.
The components of R"p, however, do not represent resistances but
"decrement factors" 5, and the components of rj y do not represent
self- and mutual inductances but "leakage coefficients" where

r resistance , self-inductance

and X

L inductance mutual inductance

The use of ratios (generalized per unit quantities) in place of actual
design constants facilitates the comparison of machines of different
sizes, supplies a ready-made method to find the locus-diagrams graphi-
cally, and in general simplifies the algebraic and numerical calculations

In terms of associated tensors, another form of the equations o/
motion is

> [1S3]

where

f - *1
1 " dt

2 77. The Component Parts of the Affine Connection r a 0. 7 . The
affine connection r^^ appears in the equation of electrical machines
because of the existence of mechanical motion and it contains all the
additional self- and mutual inductances that appear between the ter-

194 MACHINES UNDER ACCELERATION

minals by the presence of these motions. (In general these inductances
are independent of those of the components of a^, that appear because
of the motion of electric charges.)

In general there are at least three different types of motion in a
system of rotating electrical machines (besides the motion of the elec-
trical charges) : (1) Conductors rotate. (2) The magnetic paths rotate.
(3) The reference frames (real or hypothetical) rotate.

Each of these motions introduces a different set of self- and mutual
inductances that are arranged in each reference frame into a cube,
forming part of r/3, v Kach of these component parts forms a separate
geometric object, so that r0, 7 is itself a sum of three different types of
geometric objects. In particular:

(a) The inductances due to rotation of the conductors may be ar-
ranged into the "torsion tensor" S a p y . It is a tensor of valence three
skew-symmetric in its first two indices

since it contains only G mn and G mn .

In machines in which the flux-density waves in the rotor are not
sinusoidal in space (such as in direct-current and alternating-current
commutator machines), the components of S a $ y are independent of the
components of a aft and there is no relation between S tt 7 and a a/ j. How-
ever, ia machines with sinusoidal rotor-flux densities (such as synchro-
nous and induction machines), the components of S a p y may be found
(for the primitive machine only) from those of a by the formula

where " changes the slip-ring axes to direct and quadrature axes and
C*' is a function only of the displacement #" of the rotor or rotors.

(b) The inductances due to the rotation of the magnetic paths
(salient poles) may be arranged into a cube da a $/dx y . This quantity can-
not be denoted by one symbol since it is not a geometric object in
general.

When the flux due to the rotating magnetic paths is non-sinusoidal
in space, the inductances due to the additional non-sinusoidal portions
may be arranged into a tensor of rank three, Q a p y . This tensor has no
special name. It is symmetrical in its last two indices.

[156]
Hence the magnetic paths contribute dttap/dx" 1 and

DEFINITION OF THE AFFINE CONNECTION r A7 195

(c) The inductances due to the rotation of the reference frames are
arranged into a geometric object, the non-holonomic object. It is
defined by a formula analogous to Eq. (155), namely,

nc7i
[157]

where now the components of C> are functions of the displacements .r*
that differ from those of the rotors.

2-78. Definition of the Affine Connection T afjtj . The affine con-
nection is built up from four sets of inductances:

(a) The motion of the conductors introduces S a p y .

(b) The motion of the magnetic paths introduces da a p/dx y and

(c) The motion of the reference frames introduces 8 a 0, 7 .

In the definition of r tt 0, 7 each of the above four quantities occurs
three times, with their indices arranged in the same even permutation
a0y, Pya, and yap. That is,

This is the most general form of r a f7 that is used in tensor analysis
(in affine differential geometry). Its formula of transformation is

The expression in parenthesis is called the Christoffel symbol. It is
also a geometric object of valence three

Its transformation formula is the same as that of r a/ j f7 of Eq. (159).
It is customary to include the non-holonomic object ft a /3,7 in the defini-
tion of [afry] and call it the non-holonomic form of the Christoffel
symbol as

Its law of transformation is still the same as that of

In special cases F a 0, 7 assumes simpler forms. In all problems of
classical mechanics 5^ 7 and 0*y ar e zero and in most problems

196 MACHINES UNDER ACCELERATION

is also zero, so that fl/3, 7 is identical with the holonomic Christoffel
symbol [a0,y] Eq. (160). In most standard electrical machines with
stationary axes [afi,y], Q a y and Qa0 iT are /cro, but not Sa0 y .

2 79. Permanent Magnets. When permanent magnets are present
in the system (as in the case of the numerous types of subsynchronous
motors), they introduce an additional flux-linkage vector <p a that is not
a function of the currents i a . Their presence introduces a skew-
symmetric tensor of valence two:

which may be combined with the symmetrical resistance tensor Rap to
form a general tensor of valence two:

[162]

In any reference frame the symmetrical part of B a p gives R^ and its
skew-symmetric part gives Af a p-

2-80. The Most General Form of the Equation of Motion. Al-
though in the definition of F a/ g, 7 each of the four quantities da a p/dx y ,
S a 0y, Qapy and 12 tt 0, 7 occurs three times, in the equation of motion r o/ 3, 7
appears multiplied by i a twice as r a/ 3, 7 i a $ and consequently some of
the quantities disappear or simplify. That is, the equation of motion
of rotating electrical machinery may be written as

+ ([oft?] - 23*. + ft* - I ft*,) f t, [163]

at

where i a = d(f/dt and [a/9,7] is the non-holonomic Christoffel symbol of
Eq. (160a) simplified to

[1606]

This is the most general form of the equation of motion that is used
in affine differential geometry where it represents (when / is replaced
by the arc length s and e y = 0) the equation of the shortest lines
between two points (paths) in a curved affine space.

It is emphasized that the physical interpretation given above
(namely, that each term in the equation represents the voltages and
torques due to the motion of some particular medium) is valid only if
the first primitive machine is used as the primary reference machine
to find the equations of some particular machine. However, if some
other machine, say one whose reference axes are connected to the mov-
ing conductors, is assumed as the primary reference machine (which is,

DERIVATION 197

of course, allowable), then the above equation is still valid, but the
physical interpretation of each term is far more complicated. Each
term then represents the voltages and torques due to several of the
moving materials instead of one.

Sections 1-10 of this chapter are but an introduction to the tensor
theory of networks and rotating electrical machinery. The literature of
the field is extensive. A number of references are given in Sec. 12.

(11)
Tensorial Method of Attack of Engineering Problems

The question arises as to how the engineer can put tensor equations
to practical use. This question has already been answered by engineers
having put them to use. However, it may be helpful to summarize
briefly the process.

2-81. Derivation. Only the derivation of equations of performance
is here considered. Suppose the engineer is called upon to analyze a
complicated engineering structure such as the hunting of a turbine gov-
erning system or an electric drive. The steps in the tensorial method of
establishing the equations are as follows.

(A)

1. Do not attempt to analyze immediately the given system, since
it is too complex.

2. Instead, subdivide the complex system into smaller component
parts, the primitive system (for a simple example, see 2-42) where
(a) the equations of each component have already been established
before (see 2-39, 2-40), or if not previously analyzed, then (b) it is
comparatively easy to establish the equations of each part either by
further subdivision or by assuming more convenient reference frames,
or by any other means. The subdivision may be accomplished in one or
more steps depending upon the complexity of the resultant and the
component structures. In addition to subdividing the system, new
and more easily analyzable reference frames can be assumed.

There is no necessity to assume the existence of reaction forces
acting at the points or planes where the original structure was broken
up. That is, each component system is analyzed as if the other com-
ponent systems are non-existent. (See 2 39, 2 40.)

3. Establish the tensor equations of the primitive system consisting
of several isolated structures. (See Eqs. 70, 71, 72.)

198 REFERENCES

(B)

Set up the connection tensor (transformation tensor) showing how
the component parts are interconnected into the actual system and also
how the actual reference frames differ from the simplified ones. (Eqs.
74.)

(O

Transform each tensor of the primitive system with the aid of the
connection tensor. (Eq. 75, for example.) Since the tensor equation
of the original complex system is the same as that of the simpler primi-
tive system, the equations of the given engineering structure have been
established.

The material of 2-39-2-42 illustrates, in a simple case, the pro-
cedure described. For a complex illustration the reader is referred to
Kef. 2 of 2- 82.

(12)
REFERENCES

The following bibliography is short because most of the references contain bibliog-
raphies relating to their resjx^ctivc fields.

2-87. Applications in Physical Problems. The ph>sical applications are most
numerous in physics, electrical and mechanical engineering.

1. Matrices. A. I*. Sah, "Dyadic Algebra Applied to Three-Phases Circuits,"
Elec. /?//., 55, 872 (1936). Louis A. Pijxjs, "Matrices in Engineering," Elec. Fng.,
56, 1177 (1937). R. S. Hurington, "A Matric Theory Development of Symmetrical
Components," Phil. AhiR. [7], 27, 605 (1939). M. B. Reed, "Properties of Three-
Phase Systems Reduced with Aid of Matrices," Elec. /iwg., 57, 74 (1938). Louis A.
Pipes, "Transient Anal) sis of Symmetrical Networks by the Method of Symmetrical
Components," Trans. A.I.K.E., 59, 457 (1940). I. H. Summers, "Vector Theory of
Circuits Involving Synchronous Machines," Trans. A.I.E.E., 51, 318 (1932). See
particularly G. Kron's discussion in Trans. A.I.E.E., 51, 325 (1932) of the Sum-
mers' paper.

2. Tensors in Mechanical Engineering. C. Con cord in, "The Use of Tensors in
Mechanical Engineering Problems," General Elettric Review, 39, 335 (1936). J. L.
Synge, "Applications of the Absolute Differential Calculus to the Theory of Elas-
ticity, 11 Proc. Land. Math. Soc. [2], 24, 103 (1925). W. K. Boice, S. B. Crary, G.
Kron, and L. XV. Thompson, "Direct- Act ing Generator Voltage Regulator," Trans.
A.I.E.E., 59, 149 (1940). This reference is to an electromechanical system.

3. Tensors and Stationary Networks. G. Kron, Tensor Analysis of Networks,
John Wiley andTSons, 1939. This book is a comprehensive encyclopedia and synthesis
of work on the subject of stationary networks. Vacuum tube networks are included
and an entirely general theory of networks is developed. L. V. Bewley, "Tensor
Algebra 10 Transformer Circuits," Elec. Eng. t 55, 1214 (1936). S. A Stigant, "A-c

REFERENCES 199

Circuits, Symmetrical Components, Determinants, Tensors, and Matrices," Elec-
trician ,119, 433 (1937).

The solution of the characteristic equation of a high order system of differential
equations with constant coefficients is lalwrinus. Fur the method of relaxation of
constraints, see R. V. Southwell, Relaxation Methods in Engineering Science, Oxford
University Press, 1940. For solution of such equations by machines, see S. L. Brown
and L. L. Wheeler, "A Mechanical Method for Graphical Solution of Polynomials,"
/. Franklin Institute, 231, (1941).

4. Tensors and Rotating Electrical Machinery. G. Kron, "The Application of
Tensors to the Anal>sis of Rotating Electrical Machinery," General Electric Review,
a serial beginning in April, 1935, and running in 38, 39, 40. Sixteen parts of the above
serial have been published in book form, The Application of Tensors to the Analysis
of Rotating Electrical Machinery, Parts I-XVI, General Electric Review, Schenec-
tady, 1938. Norbert Wiener, "Notes on the Kron Theory of Tensors in Electrical
Machinery," J. Elec. Eng., Nos. 3 and 4, China. A. H. Lander, "Salient Pole Motors
Out of S>nchronism," Elec. Eng., 55, 636 (1936). G. Kron, "Non-Kiemannian Dy-
namics of Rotating Electrical Machinery, 11 J. Math, and Phys., 13, 103 (1934).

5. Physics. G. Kron, ' 'Quasi- Holonomic Dynamical Systems," Physics, 7, 143
(1936). G. Kron, "Invariant Form of the Maxwell-Lorentx Field Equations," /.
Applied Physics, 9, 196 (1938). Applications of the Absolute Differential Calculus,
Blackie and .Son, London, 1936. F. D. Murnaghan, "On the Application of Tensor
Analysis to Physical Problems," Phil. Mag., [7], 6, 779 (1928). H. Van Dijl, "The
Application of Ricci-Calculus to the Solution of Vibration Equations of Pie/o-clcctric
Quartz," Physica, 3, 317 (1936). G. Birkhoff, Relativity and Modern Phytits, Harvard
University Press, 1923. A. S. Eddington, The Mathematical Theory of Relativity,
Cambridge University Press, 1924. R. Becker, Theorie der Elektrizitat, Teubner,
Leipzig, 1933. R. C. Tolman, Relativity, Thermodynamics, and Co\nwlogy, Clarendon
Press, 1934. E. T. Whittaker, Analytical Dynamics, p. 47, Cambridge University
Press, 1927. F. D. Murnaghan, Vector Analysis and the Theory of Relativity, Johns
Hopkins Press, 1922.

2-83. Theory. The following are references to tensor analysis books and papers
and the applications of tensors to geometry.

6. Tensor Analysis. T. Y. Thomas, The Elementary Theory of Tensors, McGraw-
Hill Book Co., 1931. T. Levi-Civita, Tlte Absolute Differential Calculus, Blackie and
Son, Toronto, 1927. Weatherburn, An Introduction to Riemannian Geometry and the
Tensor Calculus, Macmillan, 1938.

7. Tensor Analysis and Differential Geometry. J. A. Schoutcn and D. J. Struik,
Einfuhrung in die Neueren Methoden der Differentialgeomelrie, I*. Noordhoff, Gro-
ningen, 1935, 1938. D. J. Struik, Grundzuge der Mehrdimensionalen Differentialgeome-
trie in Directer Darstellung, Springer, Berlin, 1922. O. Veblcn and A. N. Whitehead,
Foundations of Differential Geometry, Cambridge Tracts in Math, and Math. Phys. t
Cambridge Press (1932). L. P. Eisenhart, Riemannian Geometry, Princeton University
Press, 1926. N. Coburn, "A New Approach to Kron's Work," J. Math, and Phys.,
17, 112 (1938).

CHAPTER III
NON-LINEARITY IN ENGINEERING

(1) Differential Equations Analytic in Parameters, (2) Non-
linear Systems by Variations of Parameters, (3) Solutions of
Systems by Method of Successive Integrations, (4) Matrix
Methods, (5) Elliptic Functions, (6) Hyperelliptic Functions,
(7) Method of Collocation, (8) Galerkin's Method, (9) Method
of Lalesco's Non-linear Integral Equations, (10) Solutions by the
Differential Analyzer, (11) Additional Methods and References.

The first two chapters of this text were concerned with the analyti-
cal development of certain fundamental principles of mathematical
engineering and the reduction of engineering problems to mathematical
systems by means of these fundamental principles. Solutions of the
resulting discrete systems may or may not depend upon advanced
mathematics. If the solutions required no mathematics beyond the
domain of elementary differential equations, Heaviside's operational
calculus, or the elementary theory of matrices the solutions were com-
pleted in Chaps. I and II.

Kngiiieering problems of considerable difficulty may lead to mathe-
matically discrete systems whose solutions depend upon advanced
mathematics. Such problems frequently reduce to systems of non-
linear differential or non-linear integral equations.

In general, a non-linear problem is one which, when formulated
mathematically, reduces to (one or) a system of differential, integral, or
integro-differential equations such that at least one of the three quanti-
tives, a derivative, an integral, or a dependent variable, is involved tran-
scendentally or in some manner to a power higher than the first in at
least one equation of the system. From Part I it is evident that analy-
ses of investigations in circuits, electrical machines, heat-flow, elas-
ticity, and dynamical systems lead more and more to systems of
differential and integral equations whose dependent variables and (or)
their derivatives are involved to a power higher than the first. The dif-
ferential equations present such a variety of types that the so-called
standard forms of differential equations studied in a first course in dif-

200

SYSTEMS OF DIFFERENTIAL EQUATIONS TO NORMAL FORM 201

ferential equations are of slight use for the simple reason that they fail
to arise in difficult problems in engineering practice. Engineering non-
linear problems are most often reducible mathematically to the solution
of systems of non-linear differential equations and non-linear integral
equations. It is the purpose of this chapter to explain briefly the theory
of these systems and, what is more important from an engineering point
of view, to apply them in the solution of practical problems in engi-
neering.

(1)
Differential Equations Analytic in Parameters

The general theory of differential equations analytic in parameters
is, in general, conveniently applicable to equations in the so-called
normal form.

3-1. Reduction of Systems of Differential Equations to Normal
Form. The normal form consists of a system of simultaneous differ-
ential equations, the left members containing a single first derivative,
while the right members contain no derivative. The number of
equations in the normal form of the system equals the order of the sys-
tem, i.e., the number of constants in the solution. Reduction to the
normal form is merely a routine process. One new dependent variable
must be introduced for each derivative of order higher than the first
which occurs. The process is illustrated in the example following.

EXAMPLE. The differential equations of motion of a projectile,
under proper conditions, are

-_
dt* ~ dt

where x, y are the coordinates of the projectile, t is the time, and

The constant C is the ballistic coefficient dependent upon the shape of
the projectile, H(y) is a function of the height of the shell above ground
and G(v) is a function of the velocity.

202 DIFFERENTIAL EQUATIONS ANALYTIC IN PARAMETERS
Reduce these equations to the normal form. Let
x = *i, y = #3, .

dx _ dy

Then

dt "~ * 2 ' dt " **'

2 __ ,, *4 _ _ , 2 _

A " 2> * " 4 fr

The last four equations are the normal form of the two second-order
differential equations of the motion of a projectile.

3-2. Equations of Type II. Let the system of differential equations,
as given by the physical problem, be reduced to the normal form

*', = ft(*ii *2, , *; = Fi(*j\ ) (ij = 1, 2, - , ri)

i. ^ [1]

^(/o) = fl< where * f =

In Eqs. (1) the second system of n equations, namely, Xi(to) = ai are, of
course, the n initial conditions. If (1) contain a parameter r or if a
parameter r can be introduced by change of variables in such a way
that (1) are reducible to the form

x t = fi(xj\ /) + r gi(x jt r; f) (i,j = 1, 2, - , w)

[2]

OCi(to) = fli

then the system is said to be of type II. The properties of the func-
tions fi and gj are described presently.
First consider the system of equations

* = /.(*r. 0. (*. 3 = 1, 2, , n)

where the f unctions / are analytic in Xj and / for all x$ and / which sat-
isfy the relations

|*,-o,|Sr* |/-fc|S7o. [4]

Equations (4) state merely that the n functions / are analytic in the
interior and on the boundary of some (n + 1) dimensional region.
This condition is usually satisfied in engineering problems. A function
f(xi,x 2 , x n \t) is analytic in the region specified by (4) if it is
uniquely expansible in a power series in the (n + 1) variables (x$ a ; -)

NATURE OF THE SOLUTION OF TYPE II 203

and (/ /o) and if the series is convergent in the region defined by
(4). The method of obtaining this expansion is given in 3-5.

It is provable * that (3) possess a unique continuous solution
Xi = *- 0) (/). If, in (2), the ,(# r\f) are analytic in Xj #J 0) and r
uniformly with respect to t and are continuous in / for all #y, r, and t
in the region

| X j - *< 0) | g r, f r g p, to^t^T^ T [5]

then there exists a formal solution 2 of (2) of the form

*i = . TO (0 + i l) (Or + *, (2) > 2 + ,(- 1, 2, ...,) [6]

where ^ 0) , x[ l \ are determined in 3-5.

The notation in (6) may require explanation. The functions
fi(xj\f) are functions of (n + 1) variables of which n of them are
x\i %2> #n- Each Xj, by (6), is the sum of infinitely many explicit
functions of /. Hence either an additional subscript or some other
device must be employed to arrange in order the set of functions for
each Xj. It is convenient and customary to use superscripts.

3-3. Nature of the Solution of Type II. Systems of type II are
useful especially in solving engineering problems in which rg*(ry,r;/) are
less in absolute value then/ 4 (#y,r;/). Series (6) is then usually rapidly
convergent and the terms linear and quadratic in r furnish sufficient
accuracy. If the system contains no parameter r, one can frequently
be introduced by change of dependent or independent variables. If
it is evident that the solution Xi = # t (0) (/) f which is called the generating
solution of (2), is not even an approximate solution of (2), then the
method may yield such complicated results that they may be of little
engineering value. It is then necessary to resolve F t into/ t and g in a
different manner or resort to methods of the sections which follow.
The guide in resolving Fi into the sum of two functions is the physics
of the problem.

During transient performance of rotating synchronous machines 8
the effect of field and armature resistance on fluxes is small if the time
is sufficiently small. In this case the constant leakage theorem 4 may

1 E. L. Ince, Ordinary Differential Equations, Chap. III.

2 A formal solution is one which merely satisfies the differential equations when
substituted therein. The solution may be a divergent series of such a nature that it
does not define a function. A formal solution may be worthless.

8 R. E. Doherty and C. A. Nickle, "Synchronous Machines IV," Trans. A.I.E.E.,
47 (April, 1928).

4 R. E. Doherty, "A Simplified Method of Analyzing Short-Circuit Problems/ 1
Trans. A.I.E.E., 42 (1923); "Short-Circuit Current of Induction Motors and Gen-
erators," ibid., 40 (1921).

204 DIFFERENTIAL EQUATIONS ANALYTIC IN PARAMETERS

furnish the generating solution Xi = #| 0) (/), i.e., the solution of (3).
Steinmetz has given the rate of build-up of field flux 6 of a compound or
shunt generator running at constant speed. This solution can be used
as the generating solution for the more general case of a machine run-
ning at variable speed. In the differential equations of dynamic brak-
ing of a synchronous machine the most complicated term in the
differential equations contributes only ten per cent of the solution.
(That is, a solution computed without the complicated term gives a
result which is 90 per cent accurate when an oscillogram is used as an
answer book.) The equations in this case are reducible to type II.

Non-linear problems are problems of great difficulty. In problems
of this type, solutions in closed form (a closed solution is a non-series
analytical solution) are not to be expected. Indeed it is sometimes
provable that no solutions in closed form exist. Series solutions may
not possess the elegance of form that solutions of differential equations
with constant coefficients possess, but if the solution contains the
parameters of the problem in such a way that performance of the
physical system can be predicted, then it is sufficient.

3-4. Introductory Example. Obtain the solution, as far as the
terms linear in r, of the system of differential equations

x\ = # 2 f

[71

' I 2 L J

* 2 = *i + r *2,
with the initial conditions

*i(0) = 0,

*a(0) = -1.

There exists a solution of (7) of the form of (6). The substitution
of (6) in (7) yields

or equating coefficients of like powers of r

-*r,
4 o> ,
& C8]

UP" = 4 + (*?')',

* C. P. Steinmetz, Transient Phenomena, p. 32.

INTRODUCTORY EXAMPLE

205

The first pair of (8) is equivalent to *( 0) " + *i 0) = whose general
solution is

*i } = A Q sin / + BQ cos /.
From

there follows

AQ COS t + BQ sin t

90 180 270

FIG. 3-1. Solution and Generating Solution.

360*

The initial conditions, when substituted in the general solution, give
*F> - sin /,
4 0) = - cos /.

Substituting x^ in the second pair of Eqs. (8) and solving the result-
ing equations, we have

x[ l}

AI sin / + BI cos t - | + | cos 2/,
-^i cos / + BI sin t + \ sin 2t.

Since the initial conditions of the problem have been satisfied by
^ (0) (0) = 0, and *f(0) = -1, it follows that ^"(O) - ^(O) - 0.
Applying these conditions to the general solution for x[ l) and x^, we
obtain

5 (sin t + sin 2f).

206 DIFFERENTIAL EQUATIONS ANALYTIC IN PARAMETERS
The required solution of (7) is

xi = sin / - |(3 - 2 cos / - cos 2/)r + .

# 2 = cos / + 3( s * n 1 + sin 2t)r +

The graphs of both the generating and complete solution for r = J
are plotted in Fig. 3-1.

EXERCISE I

1 . Reduce the equation x" -f (a + rbx)x' + ex = to the normal form. Obtain
a formal solution, by the method of 3-4, of the resulting normal system under the
assumption that a and c are very large relative to r and b. The motion is oscillatory
in the physical problem. (The equation is a simplified equation of hunting.) Take
as initial conditions that the displacement x\ = at / = and the velocity #2 = k,
a small quantity, at / = 0. Two terms of each of the two series are sufficient.

2. Obtain, by the method of 3-4, a formal solution of the system

x{ = #2 ,+ r #1*2,

where r is less than unity. Choose the initial conditions such that the solution is
simple. Two terms of each of the series are sufficient. Find a physical system of
which the above differential equations are the equations of performance.

3. Obtain a formal solution, subject to the initial conditions *(0) =* k t t'(0) =
and for the interval ^ / ^ m < 1, of the differential equation

" + i + rf + rVT^T* i 5 - 0,

where < r < %. Three terms of the series in r are sufficient. Find a mechanical
system of which this differential equation is the equation of motion. Find an elec-
trical system of which this is the equation of performance.

3 5. General Theory of Equations of Type II. The success of the
method of integration in powers of a parameter depends upon the
resolution of FI of (1) into f t and gi such that (3) are integrable in
suitable form. It is supposed then that the solution of system (3) has
been obtained. This solution xt = ^ (0) (/) is the generating solution
of (2).

In complicated problems it is necessary to expand /i and gi as power
series in (x$ # ; (0> ) and r. A proof of Taylor's expansion of a function
of several variables is recalled in order to emphasize the distinction
between the expansion of a function in powers of (x 3 oy) where aj are
constants and in powers of (xj xf } ) where x^ are functions of /.
A function of two independent variables and one parameter is suf-
ficient to display the reasoning.

GENERAL THEORY OF EQUATIONS OF TYPE II 207

First let/(*i, x 2 , r) be expanded in powers of Xj ay and r. Sup-
pose that /(*i, x 2 , r) and its first n partial derivatives are continuous
in the region | Xj ay | ^ hj, and < r : U. Let

*y = ay + kjS

and [10]

r = ps

where ay, Ay, and p are constants and 5 is a variable which lies in the
interval 0^5^ 1. Define a new function

The expansion of ^(s) in a Maclaurin series is

F(s) = F(0) + F'(Q)s + F" ^- + F'" ^ f- + [11]
The total derivatives of F'(s), F"(s), are

ds ds ds

. x 2 , r) + h 2 f xtxi (xi, x 2 , r) + pf rxi (x\, x 2 , r)]

r \\lff \ \_ f f M ""^2

xi, x 2 , r) -f- ntfxyc^xi, x 2 , r) -r pj rxt (xi t x 2 , r)\

dr
** r ds

\ i Y 2 1* / \ i 2_r / \

X 2 i *) T" n 2 f X 2Xz\Xli X 2t T) + p J rr (Xi, X 2j T)

:i/X2\^l X 2 , T) + 2hipf rxi (Xi 1 X 2t T)
\\ x lt X 2 , ^)J,

where ** ^ ^

The values of F(0), ^'(0), F"(0) are
= y(ai, a 2 , 0)

= hifxixifoi, &2, 0) + h 2 f xtxt (di t a 2 , 0)
+ P 2 /rr(i, 02, 0) + 2hih 2 f xix ,(ai, a 2 , 0)
+ 2hipf, Xl (ai, o 2 , 0) + 2A 2 P/r*(oii <*2 0)

208 DIFFERENTIAL EQUATIONS ANALYTIC IN PARAMETERS
When these values are substituted in (11)

F(s) -/( fllf a 2 , 0) + [hif X} (a lt a 2 , 0) + h 2 f x ,(a it a 2 , 0)

r (ai,a 2 , 0)]5 + [h 2 J riXl (ai, a 2 , 0)

/M(I. 2, 0) + p*/ rr (0i, a 2 , 0)

/Wi. 02, 0) + 2h 1P f rXl (ai, a 2 , 0)

1 , 2 , 0)> 2 + - -
The last equation may be written

(3 8 3V

**i + sh 2 + SP ) /(*i, ^2, r)
O%1 OX 2 Off

(3 3

shi --- h sh 2
3^i Qx 2

+ sp I has been raised to the power indicated and the partial
3r/

derivatives taken, then the variables x\ 9 x 2l r in/are replaced by a\, a 2 ,
andO.

From (10) shj = Xj fly and 5p = r. If these substitutions are
made in the value for F(s), there results

2 , r) = /(a lf a 2 , 0) + [ 2 fo ~ /) ^. + r

If the independent variables are JCi, x 2 , , x n then the sum-
mations in (12) range fromj = 1 toj = n.

The development (12) is valid in the vicinity of the point (ai, a 2 , 0)
and holds for all points within the parallelepiped \ Xj a } \ g Ay,
< r g -R. The solution X{ = ^, (0) (/) of (3) defines a curve in space.
If in (12) ay is replaced by x*(t) and the function /and its derivatives
satisfy continuity conditions then the expansion becomes

r

C13]

GENERAL THEORY OF EQUATIONS OF TYPE II 209

where (xi, x 2l r) in /are replaced by (#} 0) , 0) after the indicated opera-
tions have been carried out.

Upon expanding /(x,;0 and g,(*y,r;0 of (2) in powers of (xj xf*)
and r by (13); replacing (x y - xf*) t in these expansions, by x^ l) (f)r +
*} 2) (0^ + from (6); substituting these results and (6) in (2) and
finally equating corresponding powers of r on the two sides of the
equations, it is found that

[ul

where Xj and r in fa and g t are replaced by xf* and after the differentia
tions have been performed.

Criteria for the convergence of the series of (6) arc given in 3 7.

The ff, (0) in (13) is a function of t in the interval fo ^ * ^ *i If the
series (13) converges for all values of r and for every value of / in
/ g / g /! then (13) is uniformly analytic in the interval /o ^ t g /i.

EXERCISES AND PROBLEMS II

1. Expand the function

x sin x + s 2 + r cos * + (r 2 -f- r 8 ) sin 2 x

as a power series in x # (0) and r where # (0) = sin t. The third powers of x * <0)
and r are sufficient to illustrate the process.

2. In the functions

- x d .)[x 2 q +

+ x dXq f 5 4 [(r */*

E 1

let J = 5oc~* and / = + !&"*, where 5, /, 2, and y are variables and all other
R

letters represent constants. Expand /i, gi, and ft in power series in */ 2/ 0) and
y - y (0) where s, (0) = ai/ + aa/ 2 and y (0) - &i/ + brf*. Two terms for each function
are sufficient.

210 DIFFERENTIAL EQUATIONS ANALYTIC IN PARAMETERS

Instead of the single parameter r the system (2) may contain m
parameters r\,r^ , r m in which an expansion is possible. In this
situation Taylor's expansion formula, (13), can be extended to the
n + m variables x$ xf* and r\ 9 r%, , r m . It is, however, preferable
to write r,- = c t r and obtain the expansions in Xj xf* and r alone.
In the final answer c % r is then replaced by n, i = 1,2, ,*.

The parameter r may occur in gi in two ways. In some terms (or
expressions) it may occur simply, whereas in others it may occur in a
highly complicated manner. When this is the situation, the parameter
r can be set equal to r Q in those terms in which it appears in a com-
plicated way. The expansions can then be carried out in powers of
Xj #J 0) and r, whereas r Q is being treated as a mere constant. After
the mathematical solution is completed, it must be remembered that
the mathematical solution belongs to the physical or engineering prob-
lem only if r = r . Physics is a guide in the designation of the param-
eter as r or as r in the expressions of the system. Understanding of
the behavior of a physical system frequently diminishes its mathe-
matical difficulties.

3-6. Synchronous Motor Operating Below Synchronous Speed
with Field Unexcited. We shall now illustrate the method of analysis
set forth in 3 '1-3 -6 by the integration of the differential equations
of performance of a synchronous machine operating as a reluctance-
induction motor. The differential equation 6 of hunting of a synchro-
nous motor of design such that the electrical transients, due to switching-
on of the field voltage, do not appreciably affect the steady-state elec-
trical torques, is

^| + k(l - b cos 26) + r sin 26 + sin = T, [15]

a-r ar

where

T = P L /P mt a 2 =Pj/P mt r = P r /P m , k = P d /VRf^

and where PL, Pm Pj, Pn and Pd are constants defined elsewhere. The
independent variable r is given by T = \/a where X is time in seconds.
Before the field voltage is switched on the motor may operate below
synchronous speed as a reluctance-induction motor. The equation of
performance 7 of such a motor is Eq. (15) with the term sin deleted.

8 For the derivation of Eq. (15) see H. E. Edgerton and P. Fourmarier, "The
Pulling into Step of a Salient Pole Synchronous Motor," Trans. A.I.E.E., 50 (June,
1931). For a more general differential system see D. R. Shoults, S. B. Crary, and
A. H. Lauder "Pull-in-Characteristics of Synchronous Motors/' Elec. Eng., 54
(December, 1935).

7 H. E. Edgerton and P. Fourmarier, loc. tit.

MOTOR OPERATING BELOW SPEED WITH FIELD UNEXCITED 211

If change of dependent variable in (15) is made by the relation 26 = x
then the required equation is

i

2 + k(l - b cos x) + 2r sin x - 2T [16]

*

2 -

dr* dr

or, in normal form,

dr "'

[17]

- = 2T &# 2 2r sin #1 + kcir x 2 cos #1,
ar

where c\r = b and x\ = #.

Representative values of the parameters are

< b < 0.5, 0.028 < k < 0.11 (k for electrical degrees),
0.3 < T < 0.8, 0.25 < c < 0.50.

From physical considerations it is known that the solution of (17)
consists of an oscillatory component superimposed upon a constant
component of slip. Both the period and magnitude of the oscillatory
component are unknown. However, it is known that both the period
and magnitude of the oscillatory component are affected by and affect
the constant component of slip. This physical situation frequently
arises in certain types of engineering problems. Accordingly, the
desired solution of (16) will illustrate, in addition to the principles of
3- 1-3-5, a method of solving this type of problem.

The procedure is as follows. First in (16) make the change of
independent variable

T - (1 + )< f 6 - tir + b^ + ^ + -, [18]

where 5i, $2, ^3, - are determined by subsequently imposed periodic-
ity conditions. Next, (a) write the equation in / in the normal form,
(b) expand sin x\ and cos x\ in power series in *i 4 0) > W substitute

(*' - 1, 2) [19]

in the differential equations resulting from steps (a), (i), and (c).
Finally, in each of the two differential equations obtained thus far
equate to zero the coefficients of each power of r. The final equations
corresponding to Eqs. (14) are

* r = * f

L J

212 DIFFERENTIAL EQUATIONS ANALYTIC IN PARAMETERS

I* 1 " 2 ' [196]

I ~W 1(* ~() _L ~0) r -v ( ) ,^o V<0)\ O : ~(0) I J^ * L J

(#2 = ' K\Q\X2 ~T #2 Cl%2 COS *j ^ * Sin 5Ci -f- *-/ GI,

f*i e) ' - 4",

h 5 1(^2 ^1^2 cos 5fi ) -f- #2 ~h #1 ^2 ^i sin #f
. cos *1 0) ] - 2*i cos 4 0) - 4i sin 4 } + 2r(8? + 2 2 ),

The above sets of equations are now integrated sequentially. The
general solution of (19a) is

[20]

(0) >i kt i

#2 = -4o e + e>

where e = 2T/k. Only a steady-state solution is desired. Conse-
quently, the initial conditions are chosen such that A = C = 0. Thus
*i 0) (0) - 0,* 0) (0) - eand

(0) /j\ ^_ _j A(0) / /\ * r9O/rl

The substitution of (20a) in (196) yields the differential equations

x^ 1 ' = k(die + X2 l) c\e cos et) 2 sin et + 47" 61,
whose general solution is

e(kci + 2) sin et k(c\e 2 2) cos ei\ A\e~~ ht

e(e 2 + k 2 ) J ife

" Jk^i - 2) sin l + e(Jfe 2 d + 2) cos et] ^06]

Choose ^4i = Ci = 0. Then

The value of 61 is now to be determined by the periodicity conditions.
If (206), with AI = Ci = are substituted in (19c) and if the solution
of the resulting differential equations carried out with BI 9* then
terms of the form / sin et, t cos et appear. From physical considerations
guch terms cannot appear. Consequently, 6j must vanish.

MOTOR OPERATING BELOW SPEED WITH FIELD UNEXCITED 213
Write (20t) in the form

*> = DI sin et + D 2 cos et, 4 = ^i sin et + E 2 cos et.

Substituting these values of x^ and x in (19c) and integrating the
resulting differential equations we have for the general solution

(feg - 2ea) sin 2e* - (fea + 2fte) cos 2ef

@ - 2eo) cos let + (kg + 2$e) sin 2et

F+4?

where

- 2) - (k 2 Cl + 2) ^ fe^^fe 2 ^ + 2) + k( Cl e 2 - 2)

The linear term in t in #i 2> must vanish and consequently

2 cje 2
52 = e 2 (c 2 + * 2 )'

If A 2 = C 2 = 0, the initial conditions are

The entire solution as far as terms in r 2 , when the relation x = 26 is
employed, is

_ et r [(k 2 Ci + 2)g sin et k(e 2 Ci 2) cos et\

f 2 [ (fej3 2ea) sin 2et (ka + 2pe) cos 2e/1 , P/%/% _

I ^ ^ I -r 1 22

2 L 2*( 2 + 4* 2 ) J L J

c r T

62 = 2 + 2 L

- 2) sin */ + efak 2 + 2) cos e/1

^f f (feg 2gg) cos 2et + (ka + 20e) sin 2et\
2L * 2 + 4e 2 J '"'

where ' = ^/. . x - ~:rT^ . * 2 . N x b 6111 ^ in seconds.

214 DIFFERENTIAL EQUATIONS ANALYTIC IN PARAMETERS

The periodic component of the slip is periodic of period 2?ra(l + d\r +
+ ) in X. The graphs of the angular displacement B\ and the

-0 t (angular
displacement)

T-0.8
fc-0.832
5=05
r-05

r t (sl.p>

"0 1 2 3 4 v 5 6 7 8

M not in seconds)

FIG. 3 2a. Slip and Angular Displacement for Rotor of Induction-reluctance Motor.

slip 2 are shown in Fig. 3-2a. The graph of the slip plotted against
the displacement is shown in Fig. 3 2b.

T-0.8 b>05
fc-0832r=05

^

^

^

jp

i*

X

^B

s

^

^

01234567
Ot (angular disolacement)

FIG. 3*26. Slip Plotted Against Angular Displacement for Induction-reluctance

Motor.

It is needless to state that the solutions for 0i and 02 can be con*
tinued to as high a power in r as is desired.

MOTOR OPERATING BELOW SPEED WITH FIELD UNEXCITED 215

EXERCISES AND PROBLEMS III

1. Carry out the solution (22) of 3 -6 as far as the terms in r 3 and compute $3.

2. The differential equations of the field current / and speed s of a synchronous
machine during dynamic braking are

dl_ (RI - E)[(rso/s)* + Xd x q ]
dt L[(f5 A) 2 + x d ,x q ]

ds KPrI*

=
'dt ~Jsll [(rso/s)* + x d x q ]*

The range of the constants for a typically small and typically large
machine are

P = Rating of the machine kva = 15 or 400,

/ = Moment of inertia, pounds 2 feet = 0.330 or 7.215,
SQ = Initial speed, radians per second = 125.8 or 9.93,

t = Time in seconds,
K = Constant = 735.5,

5 = Speed at any time, radians per second,
Xd = Direct synchronous reactance, per unit = 1.104 or 0.64,
x q = Quadrature synchronous reactance, per unit = 0.767 or 0.46,
x^ = Direct transient reactance, per unit = 0.654 or 0.29,

r = Shorting resistance plus armature resistance, per unit = 0.682

or 0.277,

= Field voltage, volts = 25 or 76.67,
/o = No load field current, amperes = 6.5 or 57.5,
/i = Jump in field current on short circuit, amperes = 5.82 or 32,

/ = Field current at time /, amperes,
R = Field resistance, ohms = 3.54 or 0.802,
L = Field inductance, henrys = 0.512 or 0.67.

Inspection of speed curves and oscillograms of the field currents of
typical machines suggests the change of dependent variables

/-!+/,*-.,

E
where s(0) = $ and 7(0) =- + /!.

216 DIFFERENTIAL EQUATIONS ANALYTIC IN PARAMETERS

(a) Obtain the resulting differential equations in z and y.

(b) Note that the solution of

^
dt ~ L'

dt Jllsle-** [x d x q + (re*) 2 ] 2

can be used as a generating solution. Obtain this solution y

2KPr 2
(c) Note that #d# g #d'# g and 2T~ #(*<* ~~ #<*') * n the equa-

tions obtained in (a) are small. Call the first r\ and the second r 2 .
Let r i = Ci/i and r 2 = CM.

(d) Expand, by Eqs. (13), the right members of the equations
obtained in (a) in powers of (y ;y (0) ), (z s (0) ) and /*.

(e) To illustrate the method of this section (Type II) compute the
solution, as far as and including the terms y (1) and z (l \ (A better
method of handling this particular problem is given in Sec. 9 of the
present chapter.)

3 7. Convergence of the Solution (6). Thus far the solution (6) of
(2) may be merely a formal solution and of no value. It remains to in-
vestigate the convergence of the series (6). It will be shown, in this
article, by means of the well-known method of dominant functions that
the series (6) converges for certain domains of r, (xj * ; (0) ), and /.

The gist of the method of dominant functions in establishing the
existence of solutions of differential systems in normal form now fol-
lows. Some details in the method are left as exercises in problem set IV.
The right members of the differential equations in question are ex-
panded by (13) in series of the required type. Next, the right members
of the differential equations are replaced by functions which, if ex-
panded in series, are greater term by term (i.e., dominant) than the series
of the right members of the given differential equations. Moreover, the
dominant series must be such that the dominant system can be integrated.
In general, certain restrictions will be imposed on the parameters of
the dominant system in order that the solution of the dominant system
converge. Since the solution of the dominant system converges, the
solution (6) also converges because the solution (6) is less term by term
than the dominant solution.

CONVERGENCE OF THE SOLUTION (6) 217

Explicitly, then suppose that the solution #, *P(fl of (3) has
been found. If (3) are subtracted from (2) there results

7,(*;-*} 0) ) -/<fe;0 -/<(^sfl + ri<<wrs0, (w - 1, ,).
* [23]

* - *< 0) = for / = /o-

Suppose that the right members of (23) are expanded by (13) as power
series in x$ #y 0) and r and then make the change of variables Xi
#, (0) = Xi(t). The initial conditions for the new system in Xi are
Xi(to) = 0. The right members of (23) are expansible in powers of
Xj xf } and r within the region | Xj xf* \ g py, | r \ ^ <r, and
k^ t ^ T provided /,- and gi are analytic within this region. In
engineering problems these conditions are always satisfied, if not
over the complete interval / ^ ' 2i T , then at least over each
of a finite number of subintervals into which (Q, T) can be
divided.

Let Mi be an upper bound of !/(*>;/) -/i(^ 0) ;/) + ffoft(*y;';fl|
in the region specified above. The quantity O-Q satisfies the relation
< | r | < (T < <r. Let Af be as large as any M-. It is not difficult to
see that (Ex. 1) the right members of (23), when expanded by (13),
are dominated by the expansion of the right members of the equations

M p. + -+*. + j: I + *L

^i + JL]

P 0"oJ

* f, *i.+ ^ll*. r| '[24]

= 0,

[y ,

..itii

where p < p/. Since the right members in the n Eqs. (24) are all
identical and since Xi(t ) = for = 1,2, , it follows that

^ = jf 2 . . . . . X n . Set Xi - - (^ - -) in (24). Then AT must

<TO

satisfy the differential equation

dX _ nM X(l + X)

dt ~ p 1-X ' [25]

218 DIFFERENTIAL EQUATIONS ANALYTIC IN PARAMETERS
The solution of (25), subject to the initial conditions, is

. Qd + rM

where [26]

(See Ex. 2, problem set IV.)

The right member of (26) is expansible as a power series in r. Since
the solution of (24) as a power series in r is unique, this solution is
identical to the expansion of (26) as a power series in r.

It is next necessary to examine the region of convergence of the
series resulting from (26). By the theory of functions the series in
question converges interior to a circle whose center is zero and whose
radius is the distance from the origin to the nearest singular point
of the function X. The only finite singularities of X are the branch-
points 8 defined by the equation

O)/P _ o.

The two roots of this equation are
r ^[-i + 2*""-/'

the smaller of which is the one with the negative radical. From the
smaller root

(See Ex. 3.)

By the reasoning of the preceding paragraph the region of conver-
gence in r of the solution of (24) as a power series in r is given by (26a).

The steps necessary to complete the proof of the existence of a solu-
tion of (2) in the form of (6) are as follows. (The details of the steps are
left as Ex. 4.) Expand the right members of (23) as power series in
Xj #J 0) and r. Substitute in these expansions

xj - *< 0) - r*< !) + r 2 *f + . [26ft]

Equate corresponding powers of r, obtaining a sequence of differential
equations. Next, expand the right members of (24) as power series
in Xj and r . Substitute in these expansions

8 See Vol. I, Chap. IV, or J, Pierpont, Functions of a Compkx Variable, pp. 95,
235-238.

DIFFERENTIAL EQUATIONS OF TYPE I 219

Equate corresponding powers of r, obtaining a sequence of differential
equations. Show, by expressing the integrals of the two sequences of
differential equations, that the series in the right member of (26c)
is greater term by term than the series in the right member of (266).
The radius of convergence of (26c) is, however, given by inequality
(26a) and consequently (26V) converges in the same domain. In fact,
(26&) will usually converge for a larger value of r than indicated by
(26a).

It is unfortunate that there exists no method in all mathematics
of determining the true radius of convergence of (6) without first finding
the series. By true radius is meant a value of r, say, r such that for
T ^ r the series converges and for r > r o the series diverges. This fact
brings out an important engineering observation. In engineering inves-
tigations the value of r as given by (26a) is usually smaller than the
value required in the problem under solution, but in important elec-
trical problems there exist oscillograms and in important mechanical
problems there exist very frequently differential analyzer solutions
which may serve as answers or checks on analytical solutions and by
these an idea of the convergence of (6) can frequently be ascertained.
Often such electrical or analyzer solutions are of aid in the choice of the
Mi and in the choice of dominant functions, Eqs. (24).

3 8. Differential Equations of Type I. The system of differential
equations

x'i = rMxj'Sfi, (ij =!,-,)

Xi(h) = a>i

is known as a system of type I. Although systems of type II are of
much wider applicability in engineering and applied science than are
those of type I the latter are of considerable industrial importance.
It is sufficient for our purpose if the functions fi(xj\r\t) are analytic in
Xj and r and continuous in / within the domain | Xj a/ 1 ^ ry, | r \ <r
for fo ^ * ^ T. For if these properties of the f unctions /(#y;r;/) do not
exist for the entire interval of t for which the solution of (27) is desired
they will exist at least over each of a finite number of subintervals into
which (/o, T) can be resolved.

Under a criterion subsequently stated (Eq. 29a) there exists a solu-
tion of (27) of the form

*(/) - <* + x?\t)r + aPW + ' '. (*' - 1, -,*), [28]

where the aj(/) are determined by solutions of Eqs. (29). To obtain
(29) letfi(xj;r;t) be expanded as power series in Xj 0y and r. Sub-
stitute the values of Xj ay from (28) in Eqs. (27) after the right mem-

220 DIFFERENTIAL EQUATIONS ANALYTIC IN PARAMETERS

bers of (27) have been expanded. Equate like powers of r of the right-
and left-hand members of the equations and obtain

dt

[29]

Equations (29), like (14), can be integrated sequentially.

EXAMPLE. In the theory of the series non-linear circuit (335)
there is the following system of differential equations:

% - -r(l + 3b 3 yt + 5b 6 y\)u cos 2 (/ + *) f
at

^ = r(l + 3^1 + S6 6 yJ) sin (t + v) cos (t + ),
at

with the initial conditions u(0) = CQ, v(0) = 0, where r, 63, and 65 are
constants ; r is of the order of 0. 1 ; u and v are dependent variables ;
and y\ = u sin (J + v). From physical considerations in non-linear
circuits and (28) there exists a solution of the form

u = e + u\r + U2* 2 + i v = v\r + v&* + .

The quantities -, are determined in 3-35.

Of course, series (28) do not converge for all values of r. Existence
proofs, by means of dominant functions, yield theorems which specify
conditions under which (28) is a solution of (27). One of the most
useful of these theorems is: Let T\ be an arbitrary value of t such that
/o < TI ^ T. It is possible to determine a value of\r\, say, <T O such that
(28) will converge for all values of r and tfor which \r\<v^t^^t^T\.
The above theorem follows as a consequence of inequality (29a).
Inequality (29a) is established by means of dominant functions in much
the same way that inequality (26a) was established. Let the functions
be analytic in Xj and r in the region | Xj a, \ ^ pj < p 9
r | g <r. In choosing Mj the inequality | r \ < <TO < <r is satisfied. The
common upper bound offi(xj ; r ;/) is denoted by M. The inequality cor-
responding to (260) is

[29 a]

f;

1 + 2nM (t - to)
P

The details of establishing (29c) are left as a problem for the student.

GENERATING SOLUTION 221

EXERCISES AND PROBLEMS IV

1. Show that the right members of (24), when expanded in powers of Xj and r,
dominate the right members of (23) when expanded in series.

2. By separation of variables, solve (25) subject to the initial conditions X(t$)
r/<r .

3. Obtain inequality (26a) from the equation which precedes it.

4. Fill in the analytical steps in the reasoning employed from inequality (26a) to
the end of 3-7.

5. By the method of Sec. (1) obtain a formal solution of the differential equation

with the initial condition y(0) - yo < 1. The ranges of variables in the physical
problem are 0^#^l,02y^jl.

6. Obtain a solution by the method of isoclines 9 of the differential equation of
problem 5. Let the initial conditions and the ranges on the variables be the same as
in problem 5. Determine the largest value of r for which the analytical and isocline
solutions are in good agreement.

7. By the method of dominant functions obtain a value of r (say <TO) in problem 5
such that a solution in the form of (6) converges for all r in the interval r ^ <TO

(2)
Non-linear Systems by Variation of Parameters

The solutions of systems of non-linear equations are most conven-
iently carried out when the systems are expressed in normal-form.

3-9. Generating Solution. Suppose the system reduced to the
form given by (1). If any of the F % consist of more than one term then
(1) can be written in the form

In general, the resolution of Eqs. (1) into the form (30) is not unique.
The first part of the construction of a solution is to break up the Fi so
that (a) x'i = /,(x/;0 represents the greater part of the system and (b)
at the same time is solvable by either the elementary theory 10 or by
the methods of 3-1-3-8 or of 3-13. Suppose then that a solu-

tionof

For method of isoclines, see Vol. I, p. 170.

10 Any text on a first course in differential equations.

222 NON-LINEAR SYSTEMS BY VARIATION OF PARAMETERS
has been obtained and let it be denoted by

Xi = <f>i(a it a 2 , , a n ;/) (i = 1, 2, , n) [32]

We may think of a as constants, since they are the n arbitrary con-
stants of the solution, or as variables which in turn have constant
values for some specified values. Suppose we consider them as vari-
able parameters 3> and write the solution of (31) as

Xi = <f>i(yi t y2, , y n \i). [33]

Equation (33) may be used as equations for change of variables in
system (30). By the formula for total derivative,

_ . _ ^ d<t>i dx n 9g?,-

dt ~~ 9*1 dt " " dx n dt 9/ '

equations (31) become

9*>i dyi d<t> n dyi d<pi r ,

^ + " <+ i7^~ + ~a7 = /lC ^ (y *'' ) :/] + gl[Vi(yklt) '
................ [34]

3*n dyi 3<pn dy n . dvn , r , .. .-, . r / A <i

^"^ t '" + ^;^" + ^" = ^^^^ :/] + gnMk>t} :/]t

where ft, j = 1, 2, -, n.

9^t

Now indicates the derivative of <p % (yi> y^ , yn ;0 with respect
9*

to / where / occurs explicitly, the yy being considered as constants.
Thus the functions Xi = <f> t (yi t y%, , y n m fy satisfy the equations

and Eqs. (34) reduce to

dy n

[35]

Equations (35) can be solved for y\, y' 2 , , y n under the same condi-
tions that a set of n linear non-homogeneous equations in n unknowns
can be solved. By Cramer's rule the solution is

dy,

where

SOLUTION OF EQUATION OF HUNTING

dy*

223

yi 3y

and A ra is the cofactor of the element in the rth row and 5th column
of A(/).

Before discussing the nature of the solution of (36) it may be helpful
to employ the method in the solution of an illustrative example.

3 10. Solution of Equation of Hunting. To illustrate the method
of variation of parameters we shall solve Eq. (15). The normal form
of (15) is

<P2 = T f sin 2^i sin <p\ k(\ b cos 2<pj)<p2

[37]

where primes denote derivatives with respect to T and where, upon
correlation with Eq. (30),

0,

/ 2 = T r sin 2^i sin ?i,

The initial conditions at the time of switching-on the exciter are given
by Eqs. (22). Let these conditions be written ^i(/o) = #01 and

The change of variables vi = ^i + 0o ^2 = #2 in (37) and sub-
sequent division of the second by the first of the resulting equations
yield

d02 _ T - r sin 2(0! + ) - sin (0 t + 0p)

*/01 02

- k[i - b cos 2(0! + )].

The solution, which satisfies

<% T - r sin 2(0 t + ) - sin (0i + 0p)

and the boundary conditions 0i(/o) = 0> ^2(^0) =: 3S is
(0 2 ) 2 = 2r0i + r cos 2(0! + 0o) + 2 cos (0i + ) - r

cos 20 - 2 cos + y 2 , [38]
where y is an arbitrary constant or a new variable. Equation (38)

224 NON-LINEAR SYSTEMS BY VARIATION OF PARAMETERS
corresponds to (33) of the theory. The equation corresponding to (34) is
302 d)P , 202 T - r sin 2(Bi + ) -.sin (e l - )

whence

2

- b cos

- k[l - b cos 2(0! + )],
)]0 2 , [39]

where 02 is given by (38). To obtain 2 from (38) it is necessary to
extract the square root of the right member of (38). To do this expand
the right member of (38) as a power series in BI obtaining

where the , are known constants. Set

where the/* are determined by squaring the right member and equating
like powers of BI on the two sides of the equation. Equation (39) now is

frr 4+r

b cos 2(0i + )][/o +/i#i +/20? + ] [40]

By reference to 3 -6 evidently* < 1.
The above differential equation is of
type I (3-8) and there thus exists a
solution as a power series in k.

Since the derivative in (40) is
always negative the quantity y 2 is
always a decreasing function until
the slip 02 is zero. The graph of
(38) for y 2 a constant is shown in
Fig. 3 3. The curve labeled S is the
FIG. 3-3. Pulling-into-step of Syn- complete solution of (37) up to the

* v

chronous Machine.

first time that 02 = 0.

EXERCISES AND PROBLEMS V

1. By the method of variation of parameters, solve the following differential
equations.

(a) x' - ax + e', (d) x' - ax + c mt ,

(W ^-
(c)

- 1,

W (/> 2 -2

(/) *' - (sin /) + cos /.

SOLUTION OF EQUATION OF HUNTING 225

2. Solve, by the method of variation of parameters, the differential equation
x' + P(t)x = <?(*), where P(t) and Q(t) are functions of t.

3. Solve, by the method of variation of parameters, the differential equation
(P 2 + ap + b)x = /(/), where a and b are constants and /(/) is a function of /.

4. Solve the differential equation j" + k*s = g, where k and g are constants.

5. Obtain, by the method of variation of parameters, the solution of the system

where z r (p) - Lr./* 2 + # ra p + C ra and L ra , r ,, C r , are constants.

6. Solve Eq. (40) as a power series in k inclusive of the term in ft 2 .

7. The equation of a simple pendulum, where the damping force is proportional
to the square of angular speed, is

The algebraic sign depends upon the direction of motion. By the method of variation
of parameters, solve the differential equation for 0'.

8. The elastic law for a certain non-linear spring is / = kx + rx 3 , where x is the
elongation and r is small relative to k. The differential equation of motion of a mass
m attached to the spring is

If x(0) = 0, x'(O) = a, find a periodic solution of the differential equation and deter-
mine approximately the period of the solution.

HINT: The solution is by the method of 3-6. Let / = (1 + 5)r and write the
differential equation

m jZ + (1 + d)*kx + r(l + )V = 0,

where * = 8ir + fa* + . Substitution of x - x co) + * (1) r + - in the differ-
ential equation and the equating of like powers of r yields a sequence of linear differ-
ential equations. The imposed condition of periodicity determines sequentially

*i, *2, - -

9. Solve problem (8) when a periodic force F =* E sin nt acts on the mass m and
the differential equation becomes

*2

m TT + kx + rx* = E sin nt.
at*

10. The differential equation

du b sin at

* ;

arises in the study of muffle chamber discharge. The variable u is a measure of the
pressure within the muffler. Figure 3-46 shows the nature of the variation of u in
the operation of the muffler. At the point in the cycle when at = wfo 125 there
is a discharge into the cylinder at the intake port A. The pressure then builds up to

226 NON-LINEAR SYSTEMS BY VARIATION OF PARAMETERS

the value i* at / = 180 and then decreases, by exhaust through the port B, to the

du

value o according to the equation -7- = a.

at

The problem is to determine vo the minimum positive pressure, so that v\ = 02 in
the steady-state operation of the muffle chamber. The unknowns are i>o, vi, ife. The

(a)

FIG. 3-4. Pressure in Muffler.

constants of the differential equation for a typical machine are w = 1 77.8, a = 1009,
b - 1.89 X 10 3 .

HINT: If the independent variable is changed from t to B by the equation / = kO
where k = 1/6, then the differential equation becomes

du sin &cod a sin i
dO u b u

r.

du

This equation holds for wfo ^ k8u ^180 and the equation = r is valid for

ad

kOw ^ OJ/G. Since r is small, set

w = MO + Mir -f 2f 2 +
in ' = sin kuQ r and get, by equating like powers of r,

2 - -

The solution of the first equation of this sequence, subject to the condition o
for = 0o is the generating function

fo 2 J*

MO = o + 7~ (cos * w ^o cos kuO)

I k<* J

A simple function approximating the generating function is

where a\ is so determined that the values of MO as given by the last two equations are
identical at kuO = 180.

ANALYTIC IMPLICIT FUNCTION THEORY 227

Continue the solution as far as the term in r and find the value of t>o, subject to
the condition v\ = tfe.

11. With this value of t>o integrate the differential equation by means of the
method of isoclines until a value of VQ is obtained which yields v\ = ife to two decimal
places.

12. Obtain a better analytical solution of problem 10 than the one suggested
above.

3-11. General Theory Resumed. If in Eqs. (36) A(/) vanishes for
values of / for which the solution of the system is desired, then difficul-
ties are introduced into the solution and other methods may then be
preferable.

In some problems the gtfo(y*;0] f (35) may vary either rapidly or
slowly due to the presence of / explicitly, but slowly due to slowly
changing y& In this case the yk can be replaced by constant values
yk(to) in the gi[<f>j(yk',t)] without modifying appreciably the solution.
The right members of (35) are then explicit functions of the time, and
the difficulty of the problem is greatly reduced.

This naturally raises the need for a criterion for the possibility of
setting yk = y*(/i) in the g. (a) Frequently, from engineering knowl-
edge and Eqs. (33) the range of yk is known. The yk may then be set
first equal to their least values and then to their greatest values in their
domain and (35) solved for both sets. If the two solutions are approxi-
mately equal then either solution is satisfactory, (b) Recourse may
be had to the differential analyzer or numerical integration for repre-
sentative values of the parameters involved. These numerical solutions
will serve as a check on the substitution in question.

3 12. Analytic Implicit Function Theory. When the generating
functions are implicit functions of the dependent variables it may be
possible to express the dependent variables as explicit functions of the
independent variables. When the generating functions are implicit
functions of both dependent and independent variables it may be pos-
sible to express the dependent variables as explicit functions of the
independent variables or as an explicit function of some parameter r.

The reversion of series is the simplest case of the theory desired.
Suppose that the generating function F(x, /) = is of the form t = /(#),
where f(x) is an analytic function of x in the interval | x a \ p.
Then f(x) is expansible in the convergent Taylor's series

t - do + ai(x - a) + a 2 (x - a) 2 + . [41]

It is supposed further that a\ j* 0. Then (x a) can be developed as a
power series in (t OQ) which is convergent for t a sufficiently

228 NON-LINEAR SYSTEMS BY VARIATION OF PARAMETERS

small. If (/ flo)/0i x a, and a/ai are replaced respectively by
r, X, and A the series is

T - X + 4 2 * 2 + 4 3 * 3 + . [42]

Assume that

X - T + 6 2 2T 2 + 6 3 ^ 3 + [43]

and substitute this value of X in the series for T. If the resulting series
is rearranged according to powers of T and if coefficients of correspond-
ing powers of T on both sides of the series are equated, the following
relations are obtained.

7, A a *

02 = A% = -- ,
Ol

A z = 2 4--, [44]

/VV*

5 I I
\ai/

. 04

H -- g ---

Finally

&2 ft 0Q
dl \ fli ,

The coefficients 6 2 ^s> &4 have been computed to the thirteenth
term. 11 The series (45) can be tested by the usual methods. If a\ =0
it is still possible, under certain conditions, to reverse the given series.
Suppose next that the n generating functions are

^i(*i, ,*n;r) = 0,

[46]

Fn(x\, ,^;r) - 0,

where the functions Fi are analytic in the region | Xj oy | g p ; - and
< r g r - The functions Fi are expansible in powers of Xi a t - by
(13). It is further supposed that Xi = a, and r = satisfy Eqs. (46),
i.e., that the curve defined by (46) passes through the point (0i, a 2 ,
a n , 0). There is no loss in generality in choosing the origin of coordi-
nates so that ai = 02 = = a n = 0. The Fj are then analytic
for | Xj | ^ pj and < r r .

C. E. van Orstrand, "Reversion of Power Series," Phil. Mag. [6], 19 (1910).

ANALYTIC IMPLICIT FUNCTION THEORY 229

The Maclaurin expansions of (46) by (13) are

+ + amXn = cir + PI(XI, , x n \ r),

[47]
+ a nn x n = c n r + p n (xi, , x n \ r),

where 0,7 and Ci are constants and pi are power series in x, containing
no terms lower than the second degree in Xj and r .
If the determinant

does not vanish, Eqs. (47) can be solved for x\, , ff n in terms of the
right members of these equations. Let the solution be

Xi = ap + w(xf,r) (ij = 1, 2, ,) [48]

where the w are power series in Xj and r of degree two or greater.
To obtain a formal solution of (47) in powers of r substitute

*,-o?>r + a? ) r 2 +... (t - 1, 2, .--,*) [49]

in (48) and equate coefficients of corresponding powers of r on both
sides of each equation. These equalities determine uniquely the coeffi-
cients a?>, ai 2) , in (49).

It can be shown, 12 by means of dominant functions 3-7, that the
series (49) converge for all values of r for which

[SO]

where r' Q < r , p < p/, n = the number of equations, and M is a
constant upper bound of | w(x/;r)| for | Xj \ ^ PJ and | r \ g r . However,
the domain of convergence of (48) is usually greater than that given
by inequality (SO).

Those cases for which the determinant in a*y vanishes are discussed
elsewhere. 18

11 F. R. Moulton, Differential Equations, p. 81.

11 F. R. Moulton, Periodic Orbits, Carnegie Publication 161, Chap. I; W. D.
MacMillan, Mathematische Annalen, 72, 157-202.

230 NON-LINEAR SYSTEMS BY VARIATION OF PARAMETERS

EXERCISES VI

1. Reverse the following series so that the results contain terms in the fifth power
of/.

(c) t = 2x + 3* 2 + 4s 3 + 5x 4 +
2. Obtain the solution of

for * as a power series in r to terms in r 3 .

3. Solve the equation

= 2x - r + 'a 2 + *r + Jr 2 + *

for ac as a power series in r to terms in r*.

4. Solve the equations

*i -f *2 = 0,

X2 + 3r _|_ ^r^a-2 -2=0

as a power series in r as far as the terms in r 2 .

3 13. Generating Functions in Series Form; Additional Observa-
tions on Convergence. It may be impossible to resolve (1) into the
system of (2) such that the solution of system (3) shall resemble the solu-
tion of Eqs. (2) and at the same time be integrable by the elementary
methods explained in a first course in differential equations. More-
over, it may be impossible to introduce into (3) a suitable parameter in
powers of which a series solution can be obtained. Under these circum-
stances and as a last resort a solution as a power series in the independ-
ent variable may be attempted. For the technique of power series
solutions in the independent variable, the reader is referred elsewhere. 14
In engineering work, power series solutions in the independent variables
very frequently fail due to lack of convergence or due to complexity. In
both cases the evident properties of the solution are lost.

The methods of Sec. 1-2 are methods of great power. Even more
difficult problems are solvable when both methods are used sequen-
tially in either order and with any number of repeated applications of
the methods.

Additional observations on the question of convergence may be of
value. If, in (26a), / / is sufficiently small then a value of r always

14 Any text on a first course in differential equations.

GENERATING FUNCTIONS IN SERIES FORM 231

exists for which the solution given by (6) converges. In many engi-
neering problems a solution is necessary for all values of the time and
not for the time in a restricted interval. If / /o = oo in (26a), then
r = 0. This is no cause for alarm, because (26a) does not give the true
radius of convergence. In fact, the value of r can be much larger than
zero and the series converge in the infinite interval t / -

Another observation is important. It may be known from the
physics of a problem that a periodic solution exists. An example is
problem 8, set V. If the substitution / = (1 + 6)r is not made but if x
is replaced by x = x (0) (f) + x (l \f)r + and if the solution of the
differential equation is reduced, in the usual manner, to the solution of
a sequence of linear differential equations, then it will be found that
powers of / will appear in the solution. This solution is valid for r and T
sufficiently small in the interval / < * ^ T. This solution resulted
from an attempt to force on the differential equation a solution whose

period is the period of the solution of the equation m z + kx = 0.

at

In the application of the methods of Sec. 1-2, skill must frequently be
employed if suitable solutions are to be found. The physics underlying
the problem is the guide in finding suitable solutions.

PROBLEMS VII

1. Solve the differential equation (16) or (17) by substituting x = * (0) (r) +

+ # <2) (T)r (2) + directly in the differential equation. The computation
of * (0) (r), * (I) (T), and a (2) (r) are sufficient.

2. The differential equation of the free torsional vibrations of a flywheel with
variable moment of inertia is

j (id) + ko = o,

at

where k is the torque constant of the shaft on which the flywheel is mounted. Let
/ be represented by / = /o (1 + rsin ut), where r is small relative to unity. The
differential equation then is

/orw cos tat B k0 _

1 + r sin col 1 + r sin wt

or, if damping be neglected and obvious approximations made,

0.

Obtain a solution of the differential equation by the following steps:
(a) Use as a generating function = A cos nt + B sin nt t n 2 = k/I, which is the
general solution of

e[ - 2f d = - kei.

232 METHOD OF SUCCESSIVE INTEGRATIONS

(6) Employ the method of variation of parameters.

(c) Express A and B by the equations

A - A Q (t) + Ai(t)r + , B = 5 + J?i(/)r +

and complete the solution for A and B as power series in r.

(d) Note that the solution is for the interval to < t ^ T and that it contains
powers of /.

(e) Try to obtain, by the methods already explained, a periodic solution for all
finite values of t.

(3)
Solutions of Systems by Method of Successive Integrations

The method of successive integrations is frequently of value in ob-
taining an approximate solution of a system of differential equations.
Moreover, it is basic in the development of the matrix methods of Sec. 4
and in the integral equation method of Sec. 9 of this chapter.

3 14. Approximating Sequences. Let the system of differential
equations be reduced to the normal form

*'* = MX;',*), (ij = 1, 2, , n)

where the properties of the f unctions fa are specified in a closed region
-n Xi -a,^n, ^ / - /o ^ P, (* = t, 2, ..-,). [52]

The f unctions / of this section may be more general functions than the
fi of Sec. 1. However, engineering functions possess, at least in a finite
interval, the properties specified for the/i of Sec. 1. An, engineering
problem can be solved for each of the finite intervals over which the /
are analytic. It will be sufficient for the validity o'f the method of this
section to assign to the / here employed, the properties of the / of
Sec. 1.

The method consists formally in determining sequences of func-
tions Jc, (1) , ff, (2) , , (i = 1, 2, , n) the limit of which constitutes
the solution of (51). The sequences are defined by the equations

(* " ^ 2 ' ' ' ")'

USE OF APPROXIMATE SOLUTION 233

The proof 16 exists that the sequences defined by (53) possess limits
and that these limits constitute the solution of (51).

EXAMPLE. Solve by the method of successive approximations the
system

x' 2 = *i + x 2 ,
subject to the initial conditions #(()) = a,.

Let / ( )dt be denoted by Q and xl k) by #?. Then the sequences
corresponding to (53) are

I

/o

I

oci = fli + 200*1 + 2Qa,) = a x (l + 2t + 2/ 2 ),

*2 = a 2 + Q[(d! + a 2 ) + (3ai + a 2 )f] + 2a^ 2 ,

x\ = a x + 2Q ai (l 4- 2/ + 2/ 2 ) = Ol (l + 2t + 2t 2 + -J * 3 ), [54]

1 + i + f/ 2 + i^ 3 + M
2/ + 2/ 2 + / 3

The solution by inspection is x\ aie 2 ', ^ 2 = &&* + i( 2 0-
3 15. Use of Approximate Solution. The principal weakness of the
method of this section is the slow convergence, in many engineering
problems, of the sequences defined by (53). The successive approxima-
tions, after the second or third step, may become too cumbersome.
This difficulty is sometimes avoided if an approximate solution #* a
+ w(f) is known. It can be rigorously shown that the limit of the
sequences (53), where the first Eqs. in (53) are taken to be

*P } = ^ + P/ifa + <pj(t) \f\dt (i - 1, 2, - - - , n) [S4a]

//o

15 E. L. Ince, Ordinary Differential Equations, p. 63; F. R. Moulton, Differential
Equations, p. 189.

234 SOLUTIONS OF SYSTEMS BY MATRIX METHODS

and where dj + <pj(t) are continuous functions of / in the region defined
by (52), is the solution of (51). This device is employed in Sec. 9 of
this chapter.

EXERCISES VHI

Solve by the method of successive integrations:

1. x{ = "Xi - 2X2, Xz - Xi - * 2 .

) -I- *?y = 0, subject to initial conditions y(0) = 1, y'(0) = 0.

3. x' - ax.

j

4. Ry = 5, where R and 5 are functions of x.

ax

5. *" + ax' + bx - e where a 2 - 4a6 < 0.

6. #" + R*' = S where R and 5 are functions of /.

(4)
Solutions of Systems by Matrix Methods

The method of solution by matrices is largely the method of suc-
cessive integrations recast in matrix notation. However, it differs in
the following respects. The matrix method is more convenient than the
method of successive integrations. The method of successive integra-
tions is applicable to both linear and non-linear systems. The method
of matrices is, at present, adapted only to linear equations. When the
matrix method is applied to systems of equations possessing coefficients
which are functions of the independent variable it yields a convenient
method of numerical integration superior to the method explained in
Chap. I, Vol. I. The method of this section does not pertain to non-
linear systems. Before explaining the method it is necessary to state
and illustrate certain theorems regarding matrices in addition to those
theorems of Chap. II.

3-16. Certain Definitions and Theorems on Matrices. The
equation

= \\I [a] = 0;

where [a] is an n-rowed square matrix whose elements a# are constants,
/ is unit matrix, X is a parameter, and | X/ [a] \ is a determinant, is
called the characteristic equation of [a]. The n roots of the character-
istic equation are called the latent roots of [a].

FUNCTIONS OF A MATRIX 235

The following theorem is an important theorem of matrix theory.
If [a] is a square matrix and <f>(\) = is its characteristic equation then

*(M) = o.

In the theory of functions of a complex variable the definitions of
the calculus were extended to the case where the independent variable
was the complex variable z = x + iy. It is here desirable to extend the
definitions of functions so that the independent variable is the matrix
[u]. The following theorem is basic in these definitions. If P(\u\) is
any polynomial of the square matrix [u], whose latent roots are Xi,
^2 "i ^ then

[55]

where the matrix [Z r ] is

n (A./ - [])

For a proof of this theorem see Ex. 7.
EXAMPLE. By means of Eq. (55) express

u] = [ j J '

P(W) = [u] 2 + 3[w] + 7, where [u] = I I , as a matrix.

In this case the characteristic equation reduces to (X 1) (X 2)
= and the latent roots are Xi = 1, X 2 = 2. Then

(2)/

-[! 3 , <>'-[; 3

2-1 ' L JJ 1 - 2

This result is easily checked by squaring [u] and adding to the square
3[] + I.

3 17. Functions of a Matrix. Since a polynomial P(x) can be used
to approximate a function /(*) of elementary mathematics, Eq. (55)
with P([u]) replaced by /([>]) can be used as the definition of a function
of a matrix.

EXAMPLE. Express /([>]) = e [u] , where [u] =^ Q J, as a matrix.
The characteristic equation is (X - 1)(X - 2) = and the latent roots

236
are

SOLUTIONS OF SYSTEMS BY MATRIX METHODS
i = 1, \2 = 2. The expressions for [Z r ] are

ri o]_ri oi [2 <n_ri oi
.._.. to iJ Lo 2] Lo 2\ Lo 2] fi ol

[Zi] -- xT^ rn Lo o!

n

o I =

ZJ

o i

o 2

I" 1 ]-\ l

Lo iJ Lo 2

- X 2

1-2

fo ol

1 I

LO 1 J

Equation (55) gives

3 18. Derivative and Integral of a Matrix. The derivative and in-
tegral of a matrix are defined by the equations

dW

dt

dt

dt

*ln

_ dt dt

where Quij = / (uq) dt.

EXERCISES IX

1. By means of Eq. (55), express as a matrix sin [], where

Qu nn .

w

i o o

220
343

2. Evaluate e [u] , approximately where []

-[::]

by means of the series

[an 012 1

21 022 J

4. Prove the first theorem of 3- 16 for [o] an w-rowed square matrix.

5. By means of Eq. (55), express as a matrix tan" 1 [w], where

M

Da-

6. Express log [u] as a matrix, where [u] has the same value as in Ex. 5.

7. Establish Eq. (55) by filling in the details in the following outline of a proof.

HIGH POWER OF A MATRIX 237

Let />([]) be any polynomial of degree m in the square matrix []. Let ^(X)
X n + ai\ n ~ l + + o n = be the characteristic equation of []. From ?(A)
we have

X n - -fliX"- 1 - 02X W - 2 ----- a n ,

-ai\ n - 02X"- 1 ---- - a n X.
Substituting the value of X n from the first equation in the last we have

By a continuation of this process it is possible to express p(\), a polynomial of degree
m, as a polynomial P(X) = PiX"" 1 + P 2 X n ~ 2 + + P n . But [u] satisfies its
own characteristic equation and thus all the relations written for X are valid when X
is replaced by [u].

Lagrange's interpolation formula 18 for the n points [01, P(ai)], [^2,
[a nt P(a n )] is

- 02) (ai - a 3 ) (ai - a n )

(X - fll ) (X - a,) (X - m) (X -

(a n ai) (a n - a 2 ) (o n -

where a\ t 02 , a n are arbitrary. If a n X rf where X r (r = 1, 2, , n) are the
latent roots of [11], then

'P(M)'

where [Z r ] is given in Eq. (55).

8. Evaluate e [v] , where the latent roots of [u] are a pi.

3 - 19. High Power of a Matrix. An approximate value of a matrix
raised to a high power is easily obtained from Eq. (55). By Eq. (55)

II (\J - [a])

Let the latent roots of [a] be Xi > X 2 > > Xn- If m is very large

II (X.J - [a])

[56]

n (x, Xi)

3^1

[1 Ol 238
EXAMPLE. Find an approximate value of I I . By Eq. (56)

M J. B. Scarborough, Numerical Mathematical Methods, p. 72.

238 SOLUTIONS OF SYSTEMS BY MATRIX METHODS

3-20. Matrizant. Tl solution of systems of simultaneous linear
differential equations is based on a function called the matrizant. The
matrizant is defined by the equation

rf'M - I + GM + G 2 M + Q'M + -,

where Q= /( )*, Q 2 =/{()/( )dt}dt, . Useful properties

Jl* JtQ JtQ

of the matrizant are displayed in the following theorems.

Theorem I: J2' 0/0 [V] = /. Proof is mere inspection of the definition.

Theorem II: -- Q tQt [u] = [u]Q M [u]. This result is evident by dif-
dt

ferentiation of the defining equation.

Theorem III: tf'[a] = I e {a](t ~~ to \ where [a] is a constant matrix.
This result is established directly from the definition, i.e.,

= / + I[a](t - / ) + I[a] 2 (t - / ) 2 /2 +

3-21. Solution by Matrices. The system of differential equations
is first reduced to the normal form of 3-1. The general method of
solving simultaneous systems is easily understood from the solution of
a simple system. Let it be required to solve the system

* = 011*1 + <*12*2, d . . . .. .

or [*] = [a][>],

022*2, dt

where the initial conditions are #i(/ ) = *? and x 2 (to) = x%. From the
method of Sec. 3 and the definition of the matrizant the solution of

I M = MM

evidently is r -,

and by theorem III of 3-20,

[*] = Ie [a]T [x Q ], where T = / - / .

It remains to compute e la]T . The characteristic equation and latent
roots of [a] are respectively

X QH &i2

a 2 i X a 22

Xi = I { (011 + 022) + V(an - 022) 2 + 4a 12 a 2i } = a + A
*2 - 2 { (011 + 022) - V(a n - o 22 ) 2 + 4a 12 a 21 } = a - A

VIBRATIONS BY MEANS OF MATRICES 239

The values of [Z t ] and [Z 2 ] (sec Eq. 55) f e 1 ' 1 are

and

e "" r = ^ r( M - ( - 0J) - -*[(W - (a +
Finally, the solution is

[*] = Ie aT j/cosh 0r + J ([a] - a/) sinh /*r}
or, in non-matrix notation,
. *! = ^j*? cosh 0F + [^ (a n a? + a la *3) - ?] sin 07 j - 0,

* 2 = <j r jsg cosh /57 1 + I ^ (a 21 .r? + fl 22 .v2) - a.tS J sin jSrf = 0.

EXERCISES X

1. Solve, by the matrix method, the system x{ = 2i, *J = xi + x* subject to
the initial conditions *,(0) = a,.

2. Solve, by the matrix method, the system x{ = x\ 2^2, *2 = ^i #2 sub-
ject to the initial conditions # t (0) = a,.

3. Solve, by the matrix method, the equation

dx , d n ~ l x , d n ~*x

^F + ai d^ + ^^ + ''' +anXss0t

subject to the initial conditions x (0) - x, x' (0) -= scj, -, ^"-U (0) *(-.

3 22. Vibrations by Means of Matrices. A good approximation to
the frequency of the fundamental mode of vibration of a conservative
dynamical system with n degrees of freedom can be found by Ray-
leigh's principle (see 1-39). It is possible to obtain an equally good
approximation by means of matrices.

The potential and kinetic energies of a discrete dynamical system
are given by Eqs. (54-55) Chap. I. 17 Lagrange's equations (sec 1-12)
for such a system are

r.3. - - a r &(r - 1, 2, -,)
*-i *-i

[b](q] = -
" Or, see E. T. Whittaker, Analytical Dynamics, Chap. VII.

240 SOLUTIONS OF SYSTEMS BY MATRIX METHODS

which reduces to

M - -[DM,

[57]

where [D] = [6] -1 [o].

If g, = x, cos ut (see 1 -37) is substituted in Eqs. (57) we obtain

or

[58]

The determinant A ( -% J of the system (58), homogeneous in xi, x 2 ,
x n is

G)

1

~~2 DH
a)

1

" " " 2 *^ln

or

3-A.

1

[59]

The determinant is also the characteristic determinant of the system of
differential equations (57). If, in Eqs. (59), l/o> 2 is replaced by X then
the resulting A (A) is the characteristic determinant of the matrix [D],
and Xi > Xg > > X n are the latent roots of [D]. Evidently l/v'Xi
= MI, where wi is the smallest root of Eq. (59) and i/2ir is the fre-
quency of the fundamental mode of vibration. (See 1 '36.)

It remains only to obtain a simple method of finding Xi. It is
possible to obtain a close approximation to Xi from the formula for a
high power of a matrix. Equation (56), 3-19, becomes

II (X.7 - [D])

n (x. - xo

or

where the significance of x is given later. Obviously,

ra-vi * x

Dividing the last equation by its predecessor we obtain

X l * rmr.J>i ' C 60 1

VIBRATIONS BY MEANS OF MATRICES 241

The elements of the matrix [#] are the values of the coordinates in
the estimated fundamental mode. Equation (60), with [x] deleted,
will ultimately give the value of Xi. However, reasonable values of
[*] decrease the value of m which must be employed in Eq. (60). In
evaluating [D] m formula (56) is not used, but the value of [D] m is
obtained by m multiplications.

EXAMPLE. Obtain the period of the fundamental mode of vibra-
tion of the double pendulum of Ex. 3 (see 1-10), where <* b 10
ft, mi = 1 slug, m 2 = 2 slugs, and Q\ and 2 are small.

The differential equations are

= 0,

or

a0j + 602 + #02 = 0,

which become, on substituting numerical quantities,
300i' + 2002 + 96.6 0! = 0,
1001' + 1002 + 32.2 2 = 0,

_30^ 20 "
96.6 96.6

JO JO
32.2 32.2.

Let us estimate that the displacements in the fundamental mode are
o& = 10 sin 10 - 1.73 and xl = 20 sin 10 = 3.46. We then have

"30 20 "

[*] =

96.6 96.6
10 10

_32.2 32.2_

0.3105 0.2070

0.3105 0.3105

1.73

3.46

1.253

1.611

[0.3105 0.2070

[0.3105 0.3105

3105 0.2070

3105 0.3105

ro.

LO,

] ri.253"| _ ["0.7229"!

J [l.61lj ~ [0.8903 J '

1 ro.7229"| _ |"0.4086~|

J [0.8903J " [o.5004j

From [>]V] and

[]VJ

0.564.

242 SOLUTIONS OF SYSTEMS BY MATRIX METHODS

We shall carry the process an additional step.

I" 0.3105 0.2070] fO.4086] [0.23051
[0.3105 0.3105] [O.S004J [0.2822J

From [>] V] and [D]*[x*]
Xi

Evidently, the process has been carried sufficiently far. The approxi-
mate value of wi = l/\/Xi = 1.332. The accurate value of i is 1.34.

EXERCISES XI

1. Three equal weights each of mass m are attached to a light elastic string which
is then under tension S. In equilibrium position the length of the string is 4a and the
three weights are respectively a, 2a, and 3a units from one end of the string. If the
coordinates of the three masses are qi, qz ,43. which denote the perpendicular dis-
placements of the three masses from equilibrium position, then the kinetic and poten-
tial energies are

T = " (ql + & + 1),

V - [ql + (qz - fil) 2 + (23 " 22) 2 + !

Find by the method of 3*22 the period of the fundamental mode of vibration.

2. Two heavy discs, whose moments of inertia are I\ 4 slug-ft. 2 and /2
6 slug-ft. 2 are supported on a vertical shaft which is attached to a horizontal plane.
The constants of the shaft from the horizontal plane to the first disc and of the shaft
between the two discs are respectively k\ = 1 Ib. ft./radian and fe = 2 Ib. ft./radian.
The energies are

T - (/i$ + '201), V - J[*i0?

Find by the method of 3-22 the period of the fundamental mode of vibration.

3. Two pendula, formed by equal masses m and by two rods attached to a hori-
zontal plane, execute vibrations. The two rods are connected by a spring which is
attached to the rods a distance h below the two points of support of the rods. The
spring constant is k. The length of each bar is /.

Neglecting the weight of each rod and of the spring, calculate, by the method of
3*22, the period of the fundamental mode of vibration of the system.

3 -23. Solution by Matrices of Linear Equations with Coefficients
Which Are Functions of the Time. From 3- 14 it is evident that the
solution of the linear system of differential equations

#1 - UllXl + ' + UinX n ,

........ or

X n = U n iXi + + U nn X n ,

SOLUTION BY MATRICES OF LINEAR EQUATIONS 243
where Xifo) = #?, is given by

fl*M - / + Qu + C[V| + Q*M + . [61]

The cumbersomeness, in general, of this formula has been pointed out
in 3-15. However, Eq. (61) can be modified as a useful method of
numerical integration greatly superior to the method of 65, Vol. I,
provided the system of equations is linear.

Let the interval / ^ t ^ t n be divided into n lengths t a / a _i
= h 9 (s = 1, 2, , n). For simplicity all lengths will be taken equal.
Over each interval h s we shall suppose the matrix [w] to be a constant
matrix [a a ] or [a], the elements a lj of which are the average values with
respect to / of w# over the interval h 8 whose right end point is s.

The initial conditions for the differential equations at the beginning
of the first interval hi are # t (/ ) = #? At the beginning of the 5th
interval they will be xl" 1 , these values being computed by integrating
over the interval whose right end point is 5 1. Over the 5th interval,
since u is assumed constant, Eq. (61) reduces, in view of 3-20 to

tf-*[] = JW r , [62]

where [a 8 ] is a constant and T = /, / 8 _i.

EXAMPLE. Integrate, by the method of this article, Legendre's

equation

<Px 2t dx , m(m + 1)

dt 2 " 1 - / 2 dt + l-P * " '

subject to the initial conditions x (0) = -H. *'() = - Lct w = 2.
If x = 3Ci and ^i = #2 the normal form of the equation is

6X1 2t or [*'J= _ 6 21

M = MM- [63]

For the interval ^ t g 0.1, Eq. (63) is replaced by

M = f l ] M

I #21 #22 J

where L J

-6

244 SOLUTIONS OF SYSTEMS BY MATRIX METHODS

The solution of Eq. (64) is

M = e [a] V; [65]

where #? = -J^ and & = 0.

The latent roots of [ai] are complex. The value of e^ T where the
latent roots of [a] are a (li is found, by the theory of 3-21 to be

e *r

e [a]T - { 08 cos PT - sin 0r)7 + sin pT[a\}.
p

The values of x\ and rc 2 at / = 0.1 (i.e., x\ and x\) are

where T = 0.1, a db j9t = 0.05 db 5.95i, jc? = - Ji, ^ = 0,

fl 0] f 1 ]

7- , and [m] =

[0 ij [-5.95 0.1J

Numerical substitution yields ac} = 0.42 and x\ = 0.28.
For the interval 0.1 g t g 0.2 Eq. (63) is replaced by

[

W =

022 J

-6
021 =

= r* tdt = n7
=[ i ]

"* ~ [-7.5 1.1?J
The values of xf and a| are

r*n <!*< ,r*n

, = { (ft cos 0r a sin 0r)J + sin 0r[a 2 ]} ,

L*sJ ^ L*2j

where T - 0.1, a (8* - 0.15 + 4.91, ai = -0.42, ^ = 0.28.
The numerical values are *? 0.37 and a^ = 0.49.

SOLUTION BY MATRICES OF LINEAR EQUATIONS
Continuing the process we complete the table of values

245

t

0.000

0.10

0.20

0.30

0.40

0.50

x\

-0.500

-0.42

-0.37

-0.31

-0.23

-0.13

*2

0.000

0.28

0.49

0.73

0.99

1.27

-0.1

'a~' 2
-03

-0.4
-0.5

FIG. 3-5

The curves in Fig. 3.5 show both the approximate numerical and
also the exact solution P 2 = 3(/ 2 - J^)/2 over the interval ^ / g
0.5. The approximating solution
would have been closer to the exact
solution if h = 0.05 instead of 0.1.

This naturally raises the ques-
tion as to the magnitude of h if no
exact solution is known and, of
course, in general in practical prob-
lems no exact solution is known. If
two numerical solutions are carried

out and in one of these the interval h is half its value in the other and
if in addition no appreciable difference exists between the two result-
ing solutions then h is sufficiently small.

The interval h need not be constant throughout the range of the
solution. If, in some regions, it is evident that the dependent variable
is changing very rapidly as / increases it may be necessary to reduce the
value assigned h until a region is reached in which the solution changes
more slowly.

It has been emphasized previously that recourse to numerical inte-
gration is a last resort. The answers so obtained are merely curves and
the parameters of the problem are lost from the solution. If the system
contains many parameters and the system is integrated for a series of
values of each parameter, either by the method of this section or by
means of a mechanical or electrical differential analyzer 18 the solutions
will be a book of curves. To express the data thus obtained it is usually
necessary to integrate the system of differential equations in an analyti-
cal solution.

The most common systems of differential equations whose coeffi-
cients are functions of the independent variable and which arise in engi-
neering are those whose coefficients are periodic. (See Ex. 3.) Such
equations, even when very simple, may present most formidable diffi-
culties. For certain analytical methods of treating equations of this
type see Ref. 14, 3-47.

"See Sec. 10.

246 ELLIPTIC FUNCTIONS

EXERCISES Xn

1. Solve the illustrative example of 3-23 employing h = 0.05.

2. Integrate, by the method of this article, the differential equation

^ + 06,rV 2 ' - D* = 0,

subject to the initial conditions *(0) - 1, x(0) = 0.5. Take the range of / to be
t 2. Let h = 0.2. It is easily verified by substitution that x = e e/2 cos (4ire~')
is the exact solution for the boundary conditions imposed. Use this solution as a
check on the accuracy of the matrix method.

3. Mathieu's equation

is of use in two-dimensional wave motion, vibrations of elliptical membranes,
astronomy, and free vibratory motion in which there occurs either variable moment
of inertia or periodic spring stiffness. The equation possesses periodic or non-
periodic solutions dependent upon the values of a and q.

Integrate, by the method of 3-23, Mathieu's equation where q = 0.1 and 0=1 +
Sq 8g 2 Sg 3 TJ<? 4 + insignificant higher degree terms in q. Let the initial con-
ditions and interval of integration be respectively (0) = 0, w'(0) = 0.5 and
^ / ^ 2r.

It may be advantageous to change the independent variable in the differential
equation from t to T by the relation 2t = r.

4. The differential equation

x + 2mx + (k* - 2n sin 2t)x =

is the equation of the free vibrations of a system possessing one degree of freedom,
variable spring stiffness, and damping proportional to the first power of the velocity.
Integrate, by the method of 3-23, the above equation for m = 1 and n =0.1.
Let the initial conditions and interval of integration be respectively x(Q) = 0,
x'(0) = 0.5 and ^ * g 2r.

PROBLEM XIII

The matrix method of 3-23 is applicable to linear differential equations only.
Originate a matrix method which is valid for systems of non-linear differential equa-
tions.

(5)
Elliptic Functions

Elliptic and hyperelliptic functions are of increasing importance in
engineering investigations. Problems involving non-linear forces and
oscillations whose periods are functions of the initial conditions lead to
elliptic functions. Integrals whose integrands contain the square root

INTRODUCTORY PROBLEM 247

of a polynomial of the third or fourth power of the variable of integra-
tion are reducible to elliptic integrals. A few of the many elementary
applications of elliptic integrals are the length of an ellipse or hyperbola,
area of a right elliptic cone, determination of the field intensity at a
general point within a circular loop of wire carrying a current, 19 equa-
tion of the elastica, 20 equation of a jumping rope, 21 path of a particle
moving subject to a central force which is proportional to the inverse
fifth power of the distance. 22

Theories of non-linear springs, non-linear circuits, advanced
Schwarzian transformations, and synchronous machines employ ellip-
tic and hyperelliptic functions. (See
3-31 and 3-35.) An introduction to
elliptic functions is necessary for the
study of hyperelliptic functions.

3-24. Introductory Problem. El-
liptic functions are introduced by the
study of the simple pendulum. Let ra
be the mass of the spherical bob, h the
pendulum's length measured from 0,
the point of suspension, to the center
of gravity of m, and (Fig. 3-6) the
angular displacement of the pendulum FIG. 3-6

at time /. If damping is neglected,
the differential equation of motion of the pendulum is

i + a 2 sin = 0, [66]

where a 2 = g/h and g = the acceleration of gravity. Integration of
(66), after first multiplying the equation through by 20', yields

tf* = 2a 2 (cos B - cos )i

where the constant of integration has been so chosen that 0' = for
6 = 0o- The maximum angular displacement OQ is supposed less than
IT. By the identity cos 8 = 1 2 sin 2 0/2 the last equation can be
written

0' - 2a Vsin 2 /2 - sin 2 0/2.

11 1. S. and E. S. Sokolnikoff, Higher Mathematics for Engineers and Physicists,
p. 13.

20 W. D. MacMillan, Theoretical Mechanics, p. 195.

" E. B. Wilson, Advanced Calculus, p. 511.

W. D. MacMillan, Theoretical Mechanics, p. 297.

248 ELLIPTIC FUNCTIONS

Change of dependent variable from to y in the above equation, by
means of the relation sin 8/2 = (sin /2) sin <p = k sin ^, yields

- a Vl - k 2 sn

*> - 2

or

A- d *

aVl-fe 2 sin 2 v>

If / = /o when the pendulum is at its low point, integration of the last
equation gives

a(t /o) :

The integral, which is the right member of the last equation, is called an
elliptic integral of the first kind. It cannot be evaluated in terms of a
finite number of elementary functions.

3-25. Definitions and Derivatives of the Jacob! Elliptic Functions
of a Real Variable. The above equation expresses / / as a function
of (p. It is desirable to express <? as an explicit function of a(t / )-
In so doing we are led to the definitions of elliptic functions. For sim-
plicity in writing, denote a(t /o) by u. In the equation

r

Jo

F671
C ]

the upper limit <p is defined to be the amplitude of , or in symbols
<p = am u. The elliptic functions, sine amplitude, cosine amplitude,
and delta amplitude of #, are denoted respectively by sn u, en u, and
dn u and arc defined by the equations

sn u 5 sin am u = sin <?,

en u s cos am u s cos ^, [68]

dn M s A ow M s A^> == Vl - * 2 sin 2 *> = VI - * 2 sw 2 .

It may be pointed out that the definitions of sn u and en u are very
similar to the definitions of sin u and cos 2* if the latter definitions are
expressed in terms of an integral. That is, if

Jo Vl -

then u = sin l x or w = cos l x and, consequently, x = sin M or x
cos M.

ELEMENTARY PROPERTIES 249

The derivatives of sn u, en u, and dn u are easily obtained. Evidently,

d d . dtp dtp

sn u = sm 9 = cos <p = en u
du du du du

The value of = Vl k 2 sin 2 v = dn u is obtained by differenti-
ae

ating Eq. (67).
Finally,

- sn u = en u dn u.
du

In a similar manner

en u = sw u dn u, [69]

and

d ; L2

dn u = k* snucnu.
du

3-26. Elementary Properties of Elliptic Functions of a Real Vari-
able. It is evident, from Eq. (67), that am = 0, and consequently
sn = 0, en = 1, and dn = 1. If in Eq. (67) <p is replaced by <p
then u changes sign. Thus am ( u) am u, and from this fact
and the definition of sn u, en u, and dn u it follows that

sw( u) = snu, cn(u) = cnu, dn( u) = rfww.

The functions am u and src u are odd functions; in u and #*# w are even
functions.

The introductory problem of 3-24 can now be completed. From
the equation sin 0/2 = k sin <p and Eqs. (68)

6 = 2 sin- 1 ^ sw w) = 2 sin"-^ k sn a(t - *>)] [70]

If damping is neglected in the pendulum's motion then the motion will
be purely periodic. Equation (66) contains no damping term and
consequently 6 as given by Eq. (70) is purely periodic. We can study
the periodicity of elliptic functions in obtaining the period of the
pendulum.

When the pendulum is at its highest point <p = ir/2 and the quarter
period is

. i r 12 d *

4 * a Jo Vl - * 2 sin 2 .

250 ELLIPTIC FUNCTIONS

If * < 1 then

l2 do

2-4-6

*

1
J

The value of the above integral is denoted by K . Thus the period of
the pendulum is 4K(h/g)**. If k is very small, then an approximate
value of K is ir/2.

To obtain the real periods of sn u, en u, dn u, it is necessary to
examine the integral

r*dv r 7 *^ , /**

I T" = / T~ + /

^0 A^) ^0 Afp y,/ 2

for w a positive integer. Each integral in the above series is of the form

rd<p
~T~
,... -/2) ^^

or

,... -

where m is a positive integer. If in the first integral <p = mv and
in the second ^ = mv + 6, then each integral becomes

/ 2 ja

Thus

and, from the definition of the amplitude function,

am(nK)
Consider next the integral

where

am(nK) = = n am K.

ELEMENTARY PROPERTIES 251

From the definition of the amplitude function
am(2nK + u) = WIT + 0.

But me + ft = 2(T/2) + - 2n am K + am u. Thus the important
relation

am(2nK + u) = 2n am K + am u [71]

is obtained. Similarly, the examination of the integral

r-'fe, r

Jo &<f> JQ &<f>
yields the formula

am(2nK u) = 2n am K am u. [72]

Taking the sine of both sides of Eq. (71) we have

sin [am(2nK + u)] = sin [2n am K + am u}
or

sn(2nK + u) = sin (nw + am u)

= sin HIT cos(am u) + cos rnr sin(am u)
= cos nif sn u.
From the last equation, if n = 2,

sn(u + 4K) = sn u.

Thus the period of sn u is 4K.

In a similar manner Eqs. (71-72) give the relations

sn(u + 2K) = -sn u, sn(u + 4K) = sn u,
cn(u + 2K) = -en u, cn(u + 4K) - en u,

dn(u + 2K) = Vl - k 2 sn 2 (u + 2K) - Vl -

From the last equation the real period of dn u evidently is 2K.

The values of the elliptic
integral of Eq. (67) were tabu-
lated by Legendre for values of
k less than unity. A five-place
table appears in Pierce's Short

Table of Integrals. From such FIG. 3-7

a table u is given as a function

of <p. To obtain the graph of sn u it is necessary only to plot sin ^

against u as the independent variable. Figure 3-7 shows the graphs

of sn u, en u, and dn u.

252 ELLIPTIC FUNCTIONS

EXERCISES

1. A pendulum beats seconds when swinging through an angle of 6. How many
seconds a day will the pendulum lose if it swings through 10?

2. The period of a pendulum when swinging through an arc of 72 is two seconds.
Find the time required for the same pendulum to swing from 72 down to 52.

3. In the first integral of the differential equation of a pendulum let the constant

2
of integration be + & 2 , where b 2 > 0. In this case the angular speed of the pendu-

lum never vanishes. Find the period of revolution.

4. The defining equations for tn u, ctn v, nc u, and ns u are

snu cnu 1 1

tn u = ..... , ctn u = , nc u = - - - , ns u = -
cnu snu cnu snu

Obtain the derivatives with respect to u of these four functions.

5. Differentiate

dn u

(a) log sn u, (d) ---- ,

sn u

, 2

J (en u sn ur

6. The functions u sn~ l x, u = en" 1 x, u = tn" 1 x, and u = ctn~ l x, are
defined to be the inverse of x - sn , x en u, x = tn u, and x = ctn u. Obtain the
derivatives with respect to x of the inverse functions.

7. Show that

dx

r

JQ

3-27. Elliptic Integrals. Applications of elliptic functions fre-
quently arise in the form of elliptic integrals.

From the integral calculus it is known that any integral of the type

R(t, V// 2 + gt + h) dt,

where -R is a rational function of / and of the radical V// 2 + gt + h,
is expressible in terms of elementary functions.
It can be shown that integrals of the forms

+ bi? + cit + d,) dt
and [73]

R(t, a V/ 4 + be* + cP + dt + e) dt,

ELLIPTIC INTEGRALS 253

where J? is a rational function of / and of the radicals can be evaluated
in terms of elementary functions and elliptic functions at most. It is
supposed that the radicands do not contain multiple factors. The
integrals (73) can be reduced to integrals of the elementary calculus
and three elliptic integrals:

/dx

(6)

r dx

Legendre's three elliptic integrals, expressed in canonical form, arc:
(a) Elliptic integral of the first kind :

jf

(6) Elliptic integral of the second kind :

E(k, x) or / &<p dp = E(k, <p), [74]
/o

-k 2 x 2
-x 2 l ./o

(c) Elliptic integral of the third kind:

/* dx . _ . r d^

I . =n(n,fe,jc) or I n

JQ (1 + WJC 2 )V(1 x 2 )(lk 2 x 2 ) /<> (1+wsm"

where A^ = Vl k 2 sin 2 ^, < & < 1 , and n is a real number.

The coefficients in the polynomials / 3 + b\t 2 + c\t + d\ and
/ 4 + &J 3 + c/ 2 + dt + e are real, / is real, and each polynomial is
assumed positive for some value of / within the interval of integration.

The second forms of Eqs. (74) are obtained from the first by the
change of variable of integration x = sin <p. Integrals (74) have been
evaluated, by numerical integration and other methods, for all values
of k in the interval < k < 1 and for < <p < 7r/2.

In the introductory problem of 3-24 the elliptic integral was
readily reduced to the canonical form of the first kind. This was
unusual. In non-linear circuits and Schwarzian transformations the
reduction is often tedious. The general reduction is now given.

254 ELLIPTIC FUNCTIONS

Let the roots, real or complex, of J 4 + bfi + cf* + dt + e be a, j3, 7,
and 5. The real transformation / = (p + qy)/(l + y) transforms
the second integral of Eqs. (73) into

fRi[y,aV~Y](<l-p)dy, [75]

where

Y =[-+ (q - a)y][p - + (q - &)y]

[p - T + (q - 7)y][(/> - 5) + fe a);y].

If the first two factors in Y are multiplied together and the coefficient
of the linear term in y set equal to zero there results

(P - <*)(q - ft + (p - fi(q - a) = 0. [76]

Treating the last two factors in Y in the same manner we obtain

(P - 7)fe -*) + (- )(ff - 7) = 0. [77]

If real values of p and q can be so determined that (76-77) are satisfied
then integral (75) will reduce to

, a V( m 2

where the real quantities m z , 2 , r 2 , / 2 are
m* = (/> - ct)(j> - ft), r 2
2 = (j - a)(g - |8), ^ = (q - y)(q - 5).

(Explicitly, in numerical calculation if (/> )(/> |9) = 7 then
m 2 = 7 and the symbols db m 2 is written - 7. If (p a)(p ft) = 7,
then w 2 is written +7.) From (76-77)

pq

M + y* - y (7 + )
From the last two equations

+ g <3-7g . |8(y + g) - -y8(a + fl)

-^- = a + ^_ 7 _ 5 - ^ -- + /J- 7 -d ' [78]

which in turn yield

q-p (a ~ >)( - g)(^ ~ 7)03 -

2 * a + j8 - 7 -

ELLIPTIC INTEGRALS 255

From Eqs. (78-79) the real values of p and q are

- 6)

The case of most frequent occurrence is

" VY '

where Ro(y) is a rational function of y. The rational function Ro(y) is
the sum of an odd function Ra(y) and an even function J^GO- Thus the
integral is expressed as the sum of two integrals. The integral

is integrable, by means of the substitution y 2 = u, by the methods of
the calculus. The integration of the integral

tttfdy

VY

leads to elliptic integrals. The function R%(y 2 ) can be resolved into an
integral and a fractional part. The fractional part can be broken up
into simple fractions, and by integration by parts, the integration is
made to depend upon the terms

dy y 2 dy dy

7= t 7=^ t and

VY ' VY f (i +n y 2 )Vv

We shall carry out in detail the evaluation of the integral

where T = J 4 + o>& + bt 2 + ct + d and the value of F is given above.
The denominator of the integrand in (80) can be written

256

ELLIPTIC FUNCTIONS

where g = n/m and h = l/r. If g < h, then the substitution hy
reduces the right member of (80) to

Ndx

[81]

where c 2 - (g/Kf < 1 and N - ( -

The eight combinations of sign in the radical of (81) result in eight
cases, but the combination of signs + + need not be considered
because the polynomial / 4 + a* 3 + bt 2 + ct + d, which is by hypothe-
sis positive for some range of / within the interval of integration, can-
not be transformed by real transformations into a function which is
always negative. There exist real transformations which transform the
integral in (81) into the integral

Lf

MJ vn

dtp

k 2 sin 2

where both Mand k are real and < k < 1. The following table indi-
cates the transformation for each combination of sign and gives the
corresponding values of k 2 and M.

Sign

Transformation

Value of k*

Value of M

+ - -

x = sin <p

c*

N

+ - +

X = COS <p

cVd+c 2 )

-Nk' = -N(l -k*)

+ + -

x = (cos *)/c

1/tt + c*)

-Nk

+ + +

x = tan <p

1 -c 2

N

+

x = sec <t>

1/0 H-c 2 )

Nk

- + -

x - (sec <p)/c

c 2 /(l + c 2 )

Nk'

# = sin 2 v + (cos 2 v )/c 2

1 -c 2

-N

If in (74) the upper limit <f> is w/2, then F(k, ir/2), E(k t v/2), and
H (n> *, ir/2) are called complete elliptic integrals of the first, second,
and third kinds respectively. In the integrals F(k, x), E(k, x), and
II(n, *, x) the upper limit may be any real value. Consequently, these
integrals may be complex quantities. The explanation of complex
values for these integrals is reserved for 3-28.

ELLIPTIC INTEGRALS 257

EXERCISES XV

1. Evaluate by the method of 3-27 the integrals
/* i

- 2)<* - 3)<* - 4) '
dx

2. If T is of the third degree and if its roots a, 0, 7 are real and a > > 7, show
that the transformation / = 7 + (ft 7) sin 2 ^> transforms

dt . 2 d<

,-- into

y --- ^ y

Va -7 VI -sn <f>
where

and 7 < / < ft.

a - 7

3. (Reciprocal modulus transformation.) Show that the transformation sin ^ =*
(sin 0)/c, where c > 1, transforms

d<f> . \ d6

t^= into -

, -- - -

Vl-8ln* v
4. Plot the integrands of the integrals

/* rf^ /* .

-7====lf=rT= , (W / V 1 -
- V 1 ~ jfe 2 sin 2 > ^o

sin 2

for k = 1/2. Let ?> be taken as abscissa. The areas under the curves give the values
of F(l/2, ?) and (1/2, ^).
5. Express the integral

/ fl sin 2 ^dd
/ 9 . .
- V I - k 2 sin 2

as the sum of elliptic integrals.

6. Express as elliptic integrals, by proper changes of the variable of integration,
the integrals

/* dO r*/ 2 dO f*/ 2 dO

. ,(>!), ( / 7T-^Z, W / 7 - 7712-

- Vc - cos ^o (sm fl Jo (cos 0)

7. If the four roots of T = are a > ft > 7 > 5. show that -T= is transformed
into

2

258 ELLIPTIC FUNCTIONS

where

-(- 7 )(U-a)' T "'"

by the substitution

j _ 7(0 - a) - a(0 - 7 ) sin 2 e

(ft 5) (ft 7) sin 2

8. Establish the Maclaurin developments

(a) sn u - * - (1 + * 2 ) Jj + (1 + I** 2 + * 4 ) ^

w 2 w^ 4/

en M - ~~ 2~! 4!

2
(c) dn u * 1 ^ 2 -h fe 2 (4 + & 2 )

2! ^: u:

9. Given that the addition formula for sn(u + v) is sn(u + v) - - (JH ucnvdnv
+ snvcnudnu), where Z? = 1 ^ 2 sn 2 w 5W 2 , show that

en (u + v) *= ~-(cn u cnv snudnusnvdn r),

dn ( -f v) = (dw u dn w Jfe 2 5n u en u sn v en v).
D

3 * 28. Elliptic Functions of a Complex Variable. Let it be required
to examine the integral

/!/* fi

~777=====T^7 P2]

.

where k < 1. Evidently

u = / / +

/o

/o
To transform the last integral write

1

where V (called the complementary modulus) is denned by the equa-
tion k' a + * 2 = 1. The above transformation changes

rdx f 1 dz

- V(i - (i - ftV) int *-( V(i - O(i - A'V)

ELLIPTIC FUNCTIONS OF A COMPLEX VARIABLE 259

or, if z = sin 6, then into

/* de =

Jo Vl - k' 2 sin 2

where K 1 is given by the series preceding Eq. (71) if k is replaced by V,
Equation (82) can now be written

n "/* dip

===*=== = K + iK'.
VI - k 2 sin 2 <p

From the definitions of the elliptic functions 3-25

am (K + iK 1 ) = sin- 1 I/A
and

sn (K + iK 1 ) = I/A.

In this particular example the sn function of a complex argument yields
a real value. We now proceed to the study of elliptic functions of a

general argument. In the integral / <W/A(0, k) make the substitution

cos 6 cos <p = 1. [83]

Then sin = i tan v and

If

so that
then

C'__de_ = .
Jo A(,*) *

IU

Jo W,k)
and

f-^-r- = u [84]

JQ A(<>, R )

p de .

Jo A(,*) *"

(itt, k). [85]

From relation (83) there follows immediately

sin 6 = i tan p,
cos = I/ cos ^ f
tan = i sin ?.

260

ELLIPTIC FUNCTIONS

Substituting the values of <p and 6 from (84-85) in the last equations
we have, from sin 6 = i tan <f>, the relation

or
Likewise

sin am(iu, k) = i tan am(u, k') 9
sn(iu, k) = i tn(u, k').
cn(iu, k) = l/cn(u, k') 9

, ,. ,. dn(u,k')
dn(tu, k)

[86]

cn(u,k')

If in (86) u is replaced by v + 4K' then the last equations yield
sn[i(v + 4X ; ), *] = i tn [(v + 4X'), k'] = i tn (v, k') - sn(iv, k).

cn\i(v + 4X0, k] - cn(iv t k), [87]

] =dn(iv,k).

If in (87) v is replaced by iv (and this is a possible substitution by in-
spection of the definitions of u and v) there results

sn(v + 4iK' t k) = sn(v, k),
cn(v + 4iK', k) = cn(v, k),
k) =dn(v,k).

[881

r-o

v-o

V-co

lf-1

V oo
y-0

f 1

y0

y0

K 2K 3K 4K

FIG. 3 - 8. Rectangl^ of Elliptic Function sn u.

It can now be shown that the periods of sn u 9 en u and dn u are re-
spectively (4K and 2iK'), (4K and 2K + 2iK') and (2K and UK').
Thus all values of y = sn u are given in the rectangle shown in Fig. 3 8.

INTEGRATION OF ELLIPTIC FUNCTIONS 261

EXERCISES XVI

1. Fill in a rectangle, similar to that shown in Fig. 3-8, for the function y = en u,

2. Show that

3K iK'\ (1+tVF')

-

3. Show that

. 3K 1 ... 3K VF . . 3K

4. Show that

sn

5. Express

dx

as a complex number A + Bi.

3-29. Integration of Elliptic Functions. The methods of evaluation
of integrals whose integrands contain elliptic functions are very similar
to the methods of the elementary calculus. From

it follows that

du = d<p/A<p, or dp = dnu du t or t/(am u) = dnu du.

From the last equations,

d(sn u) = cnudnu du 9
d(cn u) = snudnu du,
d(dnu) = k 2 snucnudu.

Some methods of integration are illustrated by the following examples.
1. Evaluate the integral / snudu. (Omit the arbitrary con-

stant.) The integral

dv

= ,
- V 2

/I C k 2 snucnu , 1 C

sn u du = - 75 I du = - 7 / -
k 2 J cnu k J -

262 ELLIPTIC FUNCTIONS

where v = dnu. The value of the last integral is

2. The evaluation of / dto/sn u is as follows:

/d _ C sn u cnu dnu _ 1 C
sn u J sn 2 ucnu dnu 2 J

v V(l v)(l

where v = sn 2 u. Evaluating the last integral by the methods of the
calculus and expressing the result in terms of u it is found that

/du_ _ j [ en u "I
snu Lcnu + dn u]

3. Evaluate the integral / sn~ l u du. In the integrand make the
substitution sn" 1 u = v or u = sn v. Then

/ sn" 1 udu = I v en v dn v dv.
Integration by parts yields

/sn" 1 udu = / v en v dn v dv = v snv + 7 cosh"" 1 1 77 )
J k \k /

-i , * ,.-iA"" 1 A
= $ *M + -cosh M 7; )

4. Evaluate the integral / dn 2 u du. By definition E(k, tp)

= / AV? (2^. We have p = am u and d(am u) = d<p = dnu du. Recall-
Jo

ing that A^ = dn u, and substituting for A^> and rf^ in the last integral
we have

/

^o

dn 2 u du = (Jfe, aw w).

5. Evaluate the integrals / sn 2 u du and / en 2 u du. From the

JQ JQ

d dn u the relations
sn 2 u + en 2 u = 1,

Q

definitions of sn u, en u, and dn u the relations

INTEGRATION OF ELLIPTIC FUNCTIONS 263

obtain. By means of these relations the evaluation of the two inte-
grals in question is made to depend upon the integral of example 4.
The results are

/ sn? u du = Tg l u -E(<M u, k)],
f*cn 2 udu = ^ [E(am u, k) - k'*

/* dx

.

.

If the substitution x = tn(u, k') is made then

' cn\u, k 1 )

1 + * 2 -

(, k') ' cn\u, k')

= dn*(u, k')
~ cn\u, k') '
and the integral becomes

dx dn(u, k')cn z (u, k')

/"

Jo

_ _
o V(l + * 2 )(1 + *V) J cn 2 (u,k')dn(u,k')

*

sn~ l ( . * , k') = F(k', tan' 1 *).
\V 1 + x 2 '

A rather extensive table of integrals for elliptic functions is found
in Elements de la Thtorie des Fonctions Elliptiques IV, Tannery and
Molk.

EXERCISES XVn
Establish the following formulas:

du E(u) tfsnucnu

' ~

fc' 2 fc' 2 dtt '

o /" ^ ! i <fa + y

2. I tnudu ~ - log ,

/o * cw i*

264 HYPERELLIPTIC FUNCTIONS

\-dnu

3. / ctn udu = log -

r snudu 1 dnu

J l^T --JS^'

5. I dnu i log (en u * sn u),

_ f cnudnu

6. / dtt = log sn ft,

J snu

r dnudu

. I = - /n M
y cn 2

t ' dn u du
J c

r du r en u _ , r sn u ,

8. I I du+ I du.

J snucnu J snu J en u

Evaluate, by substitution of elliptic functions (see 3-29), the integrals
dX (0 < k < 1). Let x

t

V( - a) [(x - r) 2 + s 2 ]

Let y = and obtain under the radical sign an expression having

x a

,hree real factors. Then use Ex. 2, problem set XV.

(6)
Hyperelliptic Functions

Some of the uses of hyperelliptic functions have been given in the
introductory paragraph of Sec. 5. It is the purpose of this section to
develop the theory, sufficient for the applications considered, of
hyperelliptic functions. Integrals of the form

ELLIPTIC FUNCTIONS IN SERIES FORM 265

where T is a polynomial of degree higher than four and R is a rational
function of / and VT, lead in general to hyperelliptic functions.
It is not easy to generalize the classical theory of elliptic functions as
given in Sec. 5 so as to obtain a theory of hyperelliptic functions. A
theory of elliptic functions is now developed which is based on the
solution of differential equations by the methods of Sec. 1 of this
chapter. This theory will then be extended so as to include hyperellip-
tic functions.

3 30. Elliptic Functions in Series Form. Consider the differential
equation

=--(1+^ + 2^*3, 0<* 2 <1. [89]

atr

If Eq. (89) is multiplied through by 2 dx/dt and the integration per-
formed there results

-k 2 x 2 ) [90]

provided *'(0) = 1, and *(0) = 0. The solution of Eq. (90), satisfying
these initial conditions, is

x = sn(t, k).

In view of the theory of Sec. 1 it seems reasonable to suppose that
the solution of Eq. (89), subject to the initial conditions #'(0) = 1 and
ar(O) = 0, is obtainable as a power series in k 2 . Accordingly, let

Substituting this value of x in Eq. (89) and equating the coefficients of
like powers of k 2 we have the sequence of equations

*6 + *o = 0,

*4 + #4

The solution, subject to the initial conditions, of the first of (91) is

X Q = sin /.
The substitution of XQ in the second equation above yields

+ ** - I s" 1 * - \ sin 3t -
The solution, subject to the initial conditions # 2 (0) - # 2 (0) = 0, is

X 2 - 77 sin / - - cos / + sin 3/.

JO 4 10

266 HYPERELLIPTIC FUNCTIONS

The solution of Eq. (89), as far as the terms in k 2 , is

k 2
x = sin / + (sin t - 4* cos / + sin 30 + -. [92]

Evidently, the solution (92) is not satisfactory. The term (/ cos t) /4
is not periodic. Moreover, the remaining terms of (92) are periodic of
period 2w, whereas the solution x = sn(t, k) is known to be periodic of
period 4K in /.

A solution must be devised that displays the period of 4K in /.
Accordingly, let a change of independent variable from / to r be made
in Eq. (89) by the relation / = (1 + 5)r, where 5 is a constant later
determined. Since

^ _ ^L^L _ * dx d?x _ 1 d?x

dt " dr ~dt " (1 + ) dr an di? "~ (1 + d) 2 d^ '
Eq. (89) becomes

d?x

J* _ _(1 + j)>[(i + fe 2 )^c - 2k 2 * 3 ]. [93]

From 3-26 the value of 4K is

If the solution of Eq. (93) can be expressed in a form which is periodic
of period 2ir then by the relation /==(! + 5)r, the solution in / will
be periodic of period 2?r(l + 5).

Since 5 is a function of k 2 it is reasonable to write 5 = S 2 k 2 + S 4 fc 4 +
, where 621 ^4, are constants to be determined. The substitution
of this value for $ and x = * M + x 2 (r)k 2 + * 4 (r)* 4 in Eq. (93)
gives

k*[x 4 + (28 4 + a + 2S 2 )*

By equating coefficients of like powers of k 2 we obtain the infinite
sequence of linear differential equations

S + *o - 0,

4 + *8 - - (1 + 25 2 )*

[94]
2)* a + 6^*2 + 42

where the derivatives are with respect to r.

ELLIPTIC FUNCTIONS IN SERIES FORM 267

The solution of the first of (94), subject to the initial conditions
*o(0) = 0, *o(0) = 1, is #o = sin r. When sin r is substituted for XQ
in the second equation above, there results

x 2 + #2 = ( 2 2$ 2 ) sin r ^ sin 3r.

In order that no term of the form r cos r appears in the solution, it is
necessary that | 25 2 = or d 2 = .

Before integrating the next differential equation and the remaining
equations of (94), it is necessary to determine the initial conditions
for #2, x*t From / = (1 + 5)r it follows that

dx dx dt dx 2 4 o*

= j- = (1 + 6) = (1 + 6 2 k + 6 4 fc + )
or a/ or a/ a/

<fo

From Eq. (90) it is evident that = 1 for x = 0, t = 0. Thus

at

~ 1 + 5 2 ^ 2 + 5 4 fe 4 + -, for r = 0.
or

From ^ = XQ + x 2 (r)k 2 + # 4 (r)& 4 + , a second value for is

ar

dr dr dr dr

Since these two values are identical for all values of k when r = 0, it
follows that, at r = 0,

dxQ dx 2 dx 2n

dr dr dr

The solution of x 2 + x 2 = J^ sin 3r, subject to the initial condi-
tions x 2 (Q) = and x' 2 (G) = d 2 is

x 2 = ipg' (sin r -f- sin 3r).

By substituting X Q and x 2 in the third equation of (94) and imposing
the condition that all sin r terms vanish from the right member of the

equation, 5 4 is found to be ( ) . Moreover, it is evident that in

each successive equation just one additional 5 2n enters. Thus the 8 2n
can be determined. The integration of the equation in * 4 gives

# 4 = 2 (7 sin r + 8 sin 3r + sin Sr).

Finally, the value of sn(t, K) as far as the terms containing k* is

sn(t, k) = XQ + x 2 k 2 + x 4 k* + , [95]

268 HYPERELLIPTIC FUNCTIONS

where

The solution of (89) given by (95) is periodic of period 4K and is
of a form satisfactory for computational purposes.

EXERCISES XVm

1. Solve, by the method of 3-30, the differential equation

subject to the initial conditions g(0) = 1, g'(0) = and thus obtain a series expan-
sion, similar in form to (95), for en t.

2. Obtain the differential equation whose solution is dn t. Show that the solution,
satisfying the initial conditions #(0) = 1, x'(0) = is

3-31. Non-linear Spring. Let it be required to find the period of
oscillation of the mass m supported by a non-linear spring as illustrated
in Fig. 3-9. The displacement from equilibrium position at time / is
denoted by x. Let the restoring force of the spring be given by

where b\ and 63 are positive empirical constants.
The differential equation of motion is

or

NON-LINEAR SPRING 269

The first integral of the differential equation is

Denote the maximum displacement of the spring from equilibrium posi-
tion by c. Since the velocity vanishes at maximum displacement
Ci = a\<? + a^c 4 /2. The differential equation becomes

FIG. 3 9. Non-linear Spring.
The substitution x = cy reduces the last equation to

where

2 3 . ..

k = - - - < 1.

When y ranges from 1 to +1 the mass has executed a half -period
P/2. Thus

p = ( I - i dy

A 2 /)
4fe/2\ /' 1 (l + *V)- M jf

C Vffls/ ^A Vl y 2

TT?

270

HYPERELLIPTIC FUNCTIONS

The trigonometric substitution y = sin 6 and Wallis' integration for
mulas yield

It should be noted that the period, unlike the period of a linear spring,
is a function of the maximum displacement of the mass.

EXERCISES XIX
1. Solve by the method of 3-31 the differential equation

subject to the initial conditions y(0) = 0, y'(0) = aa/4& 2 . Show that the action of
the spring is rougher than a linear spring.

3-32. Hyperelliptic Functions. Hyperelliptic functions may be
introduced by the equation

d 2 z

2 = b o + biz + b 2 sr + + &n-i 2 , [96]

where &o b\, b 2 , &n-i are real constants and the right member is a
polynomial of degree greater than four or a convergent infinite series.
(Ref. 4, end of chapter.) A first integral of Eq. (96) is

[97]

00

FIG. 3-10

where a Q , a\, -, a n are all real. Let the
n roots, real or complex, of f(z) be ori,
2. * <*n- In any physical problem the
variation of z will lie between fixed values
Zi and Z 2 where Z\ and Z 2 are real or com-
plex. Let ai, cx2, , a r be those roots of
f(z) which lie within the ring (Fig. 4-10)
LI ^ z ^ L 2 . Let /() be written

/(*) - (z - aO(2 - a,)(s - en) (i - r )foW, [98]

where /o() is finite and does not vanish within the ring.

PERIOD OF THE SOLUTION 271

3-33. Period of the Solution. If n = 4, it is evident from 3*28
that the periods of the solution of (96) will be one real and one imag-
inary. Thus in the general equation n > 4, both real and imaginary
periods are expected, but in the problems of this section, only real
periods need be considered.

The real periods in the general case are obtained in much the same
way that the period of the non-linear spring was obtained in 3-31.
Let us suppose that during the motion or variation of current z varies
from a* to a,-+i where z, -, and a + i are real and that /(*) > for
oti^z ^ a,+i. If the change of variable

(<Xj + Qf t -+j) , (Ctj+l - Ctj)

Z = 2 + 2 *

is made in/(2), then

x

- **> n

y-l

where

< 1

because a,-+i and a f - are consecutive roots of f(x). If bj I 1 + x ) is

f *

complex root, then its complex conjugate (say) fy+i 11+ 7

\ bj+i

272 HYPERELLIPTIC FUNCTIONS

is also a root. The coefficients dj and e$ in the product

are real numbers and B 2 is a positive quantity. Finally,

(I + <rp*)(l + d t x + !**) (l+d l x + eix*)g(x),

where the number of real roots of /(z) in the ring L\ <\x\ < L 2 is p,
the number of complex roots is 21 and A 2 is such that g(0) = 1. The
change of variable from z to x reduces (97) to

/:

dx

AV(l-x 2 )(l+<r lX )- (l+v p x)'-'(l+d 1 x+e l x 2 )---(l+d,x+eix 2 )g(x)
The period T(cti+i, ,-) is

(1 +!*)

(1 + d,x +

Since g(jc) does not vanish within the interval of integration [g(x)]~^ is
expansible in this interval as a convergent power series in x. Since
<TI, (r 2 , , <r p are less than unity (1 + <r\x) (!+ <vO i s a ' so expansi-
ble as a power series in x for the interval in question.
Two cases now obtain :

Case (a). If \x\ < l/VJVl (J = 1, 2, -,/) then each factor
V(l + djX + e 3 x 2 ) can be expanded as a power series in x. Thus
) becomes

. 2

where Z>i, Z^2, are constants. By the trigonometic substitution
# = sin the integral is easily evaluated and the period obtained.

Case (ft). If any or all of the | Cj \ are greater than unity the pro-
cedure is more complicated. Consider first that one of the | j \ > 1
and the absolute value of each of the others is less than unity. Let the
corresponding factor be written

e,[x - (a + #)][* - (a -

SOLUTION OF THE DIFFERENTIAL EQUATION 273

where a and are real. Let the integral r(a,+i t a,) be computed from
1 to a and then from a to + 1. Over the first range make the change
of variable

- 1 + 1

2 2 y '

Then

[ej[x- (a-

4
The last factor is expansible in a convergent power series in y as long as

I < V

H -- " Since ^is inequality is satisfied for the inter-

val 1 < y < a

r(a, f a)-/
*/ i

where the (? are constants.

For the interval a g x ^ 1 the transformation is

+ 1 ( - o v

* 2 2 y

and the integral is

dy,

where the P< are constants.
The total period T is

rrnt \ I rp/ \

= T(oti, a) + T(a, ai+i).

If the absolute values of two of the e, are greater than unity then
the interval 1 g x g + I can be broken into three intervals and the
above process applied. The method is extensible to / such factors.

3-34. Solution of the Differential Equation. Equation (96), by
means of a first integration, the change of variable from z to x, and the
reductions of 3-33 becomes

- A[(l - *)(! + *!*)(! + *>*)

(1 + <r p *)(l + d lX + *!* 2 ) - (1 + dix + e l ^)g(x)]^. [98a]
In Eq. (98a) make the substitutions

<ri = a(k, di dfik, i = e' t k 2 , gi

274

HYPERELLIPTIC FUNCTIONS

where the & are the coefficients in the expansion of g(x). The periods
of Eq. (98a) can be obtained by the method of 3-33 and are of the
form

Q(K) =

[99]

To set in evidence the period in question of the solution of (96) and
(98a) change the independent variable from t to r by the substitution
t ** Q(k)r/A. Equation (98a) then becomes

dx
dr

- <?(*)[! - **)(!

(! +

(1 + fox + J*V)(1 + gikx +

)(! + fox + c',

+ iff if + )]*. [100]

This equation is of the form of Eq. (2) of this chapter. Consequently
there exists a solution of the form

which is convergent for k sufficiently small or what is the same thing for
k = t and cr',, (/*,, /,, g', sufficiently small. This solution is periodic of

period 2?r in r and of period T = Q(*) in /. The method is employed

^ra

in non-linear circuit analysis in 3-35.

E tin cut

-o-

C132

20

40 60 80

RMS. Volts

100 120

FIG. 3*11. Non-linear Series
Circuit.

FIG. 3 12. Volt-ampere Characteristic
of Non-linear Series Circuit.

3*35. Resonance in Series Non-linear Circuits. The resonance
theory of series non-linear control circuits illustrates the principles of
Sec. 6. The theory here developed is applicable to series circuits pos-
sessing variable inductance, capacitance, and resistance.

Many experimental facts regarding these circuits appear in the
literature. The three most pertinent are the following.

(a) Volt-ampere characteristic. The volt-ampere characteristic of
the circuit in Fig. 3 1 1 is shown in Fig. 3-12. The values of current and

RESONANCE IN SERIES NON-LINEAR CIRCUITS 275

voltage displayed are root-mean-square values. In the regions ab and
cd the current response is approximately linear with the voltage whereas
in the region be the current is critical with respect to the voltage, i.e.,
a slight increase in voltage produces a large increase in current. The
value of the applied voltage for which this increase in current is greatest
is called resonant voltage.

(b) Resistance limited. The maximum value of the current in the
region be is resistance limited, i.e., the peak of the current is piven by
i - E/R.

(c) Phase agreement. At resonant voltage, the voltage and current
are nearly in phase.

The B-H function is of course many-valued, hut this function tends
to become single- valued at large magnetizing forces for nicalloy, perm-
alloy, and low-loss steels. Moreover, the numerical integration of
the differential equations by means of the integraph shows that the
graph obtained for the current in circuits of variable inductance is not
changed by employing a single-valued B-II function. Accordingly the
equation of the magnetization curve is taken to be

7/-M-*- as* 3 + a fi x 6 , [101]

where // = magnetizing force in gilberts per square centimeter,
i = current in amperes,

x = B/BQ where B is the flux density in gausses per square
centimeter

and B is the slope of the magnetization curve at the origin. The
quantities k, <z 3 , and a f , are positive constants. More terms may be
added to Eq. (101) if necessary.

The differential equation for the current in the circuit shown in
Fig. 3-11 is

- E cos co(r - T O ), [102]

dr
uhich, by means of Eq. (101) and the substitution = wr, Ixicomes

dx r

M + R(x aa* 3 + a&x 6 ) + x e I (x a&? + a^de
ad J

- -Ek cos (0 - ), [103]

where M kuNAB 10"^ f C capacitance in micro-farads,

x e 1/C, /? resistance in ohms,

w = 377, A = area of cross-section of coil in

square centimeters,
r time in seconds, N number of turns.

276 HYPERELLIPTIC FUNCTIONS

Differentiating Eq. (103) we obtain

M T* + * (1 "

- Ek sin (8 - ). [104]

To investigate the resonance between the applied voltage and the cir-
cuit it is necessary to determine the natural period of the circuit, i.e.,
it is necessary to integrate Eq. (104) for E = 0. This integration is ac-
complished in two steps. First Eq. (104) is integrated for E = R =
and then the solution is modified to take care of the resistance. Accord-
ingly, the first equation to be integrated is

d?x
M*-z + x c (x - a^x 3 + a 6 ar) = 0. [105]

Performing a first integration of (105) we have
where

_ _ \S Q M.\Jl *\t ~~ \J

M

The integral of Eq. (106) is hyperelliptic and we obtain first the period
of the solution. The period of the solution is dependent, as in the
elliptic case, on the amplitude of the flux.

The right side of Eq. (106) vanishes for only one real value of #,
i.e., at the maximum value of c of the flux density. Accordingly, let
(106) be written

= 0> [107]

^r 2 i

where (a 2 + 6 2 ) 2 = \ , (a 2 + 6 2 ) 2 - 2(6 2 - a 2 )* 2 . -

as,c o 5

[108]

RESONANCE IN SERIES NON-LINEAR CIRCUITS 277

If x = cy, a/c = |8, and b/c = 7, Eq. (107) becomes

Y . Al c*(\ - ;y 2 )[G5 2 + 7 2 ) 2 + 2( 7 2 - 0V + y 4 ], [109]

where A* = a& c /ZM.

The solution of Eq. (109) possesses but one real period T e which is

T _ _- / +1 *

* c o ..

c .2^ / J ( t _ y 2 )[(p 2 + 7 2 ) 2 + 2(7 2 - /3 2 )/

- 7*2 + TS),

where

V (1 - /)

-|_ fl^ 2 4_ ^, 2 ll~H

- dy,

V(l - y 2 )
rs = ^ >L (y - g) 2 + ^+ ft . . ., ^

Evidently, TI = T 3 . By the substitution
7"i is reduced to

[110]

where

'

[(1 + 3 2 + 4r 2 ] H " ' ~ [(1 + 3/S) 2 +
(1 - fl* 1-1?

Since <Jn<l, < x f < 1

*?)- - 1 + h i P 1 (x i )

278 HYPERELLIPTIC FUNCTIONS

where P\,Pz, -,?* are Legendre polynomials. The expression
(1 o<w)~^ is expansible in a rapidly convergent series. Substitution
of these expressions in Eq. (110) and the carrying out of the integra-
tions yield TI + T 3 = 2irQ Q S Q , where

" ,{[(1 + 30) 2 + 4 7 2 ][(1-
So - 1

4 - A1 .

By means of the substitution y = jSr; and the method employed in
evaluating T\ and T 3 , the value of T 2 is T 2 = vQ\S\ where

Si - 1 + $(2a* - 03
g

Qr9 =SS . _ T>

3 03 2 + 7 2 )*

The series 5 and S\ are so rapidly convergent that two terms are suf-
ficient in all computations. Finally,

rp __ "f xxx r* . ** r* *. *>^ n t -.O Ov ^^

* c "T"

The equation is now integrated for zero resistance. Differentiating

Eq. (109) with respect to 6, canceling out -~, changing the independent

dO

variable from 6 to / by means of the substitution 6 = -~ = /,

A A

d?y

and writing -- y", we have
at

y" + (1 + ) 2 { [C8 2 + 7 2 ) 2 - 2(> 2 - ft]y

+ 2[2( T 2 - ft 2 ) - I]/ + 3/} - 0. [Ill]

Equation (111) by the aid of the identities (108) can be written
y" + (1 + *) 2 {[l + aw* 8 + ai4M 4 ]y

+ [a 3 2M 8 + OJ4M 4 ]/ + 064MV} = [112]

RESONANCE IN SERIES NON-LINEAR CIRCUITS 279

where

- p cJ k* 9 a 52 = 0,

032 = 4*2, a 34 = - -|a 5 -2 * 4 54 =

and /i is a parameter to guide the integration. The parameter /z is
subsequently set equal to unity. If the core of the reactor is any kind of
magnetic material with a moderately sharp 72-77 curve the ranges of the
constants employed either directly or indirectly turn out to he

2 < c < 5, 2000 < BQ < 5000, 0.4 < ft < 0.7,
0.4 < 7 < 0.8, 0.3 < a 3 < 3, 0.01 < a 5 < 0.5,
0<* <1, 0<&<1, -1 <$< + !.

Now the solution of Eq. (112) is periodic of period T in and of period
2ir in /. This solution is now obtained. By 3 2 there exists a solution
of (112) of the form

y = yo + 72M 2 + y*n 4 + , [114]

which converges for /x 2 sufficiently small, or which converges for /u 2 = 1 ,
provided both k% and k 2 are sufficiently small. Write

5 = 6 2M 2 + 6 4 M 4 + -. [US]

Substituting Eqs. (114-115) in Eq. (112) and equating like powers of
M 2 we obtain

^0+^0 = 0,

y 2 + y* + 2d' 2 y + a i2 yo + a 32 yo = 0,

y"* + y* + (62 + 25t)y Q + a 14 :yo + 03470 + fl64>5

0,

for the determination of yo, y 2 , yi

To determine the initial conditions in /, substitute - - in
Eq. (109) and let y = 0. Then

/(O) - (1 + d)(p* + 7 2 ) = (1 + 2M 2 + 4M 4 + -Mil [H7]
from which ^o(O) = AH. The solution of the first of Eqs. (116) is

yo = An sin /.

280 HYPERELLIPTIC FUNCTIONS

When this value of y is substituted in the second of Eqs. (116) it
follows that the solution for y 2 cannot be periodic unless 6 2 is so chosen
that the linear terms in y vanish. From this condition

The initial condition for y 2 is j4(0) * $2-4 n- Thus the solution for y 2 is

y% = AZI sin / + A 23 sin 3/,
where

A 2l - 4 n *8(l - ^11). ^23 = I*o4ii.

When y and y 2 are substituted in the third equation of (116) the coef-
ficient of the linear terms in yo must vanish in order that y\ be periodic
in /. Thus the 5's and y's may be found sequentially as far as desired.

Owing to the arrangement of the problem the maximum value of y
is unity and the maximum value of the instantaneous flux is B c. The
flux B is given by the relation B = B$cy. Substitution in Eq. (101)
gives the instantaneous value of i.

The solution can now be extended so as to include resistance. The
solution of Eq. (104) for K = is, of course, not periodic. The solution
decays; its periodjncreasing and its amplitude decreasing as c dimin-
ishes until the circuit has become linear. The solution after the circuit
has become linear is, of course, a damped sinusoid of fixed period. But
we are interested eventually in those cases where the applied voltage
E sin (0 ) maintains in the steady-state the maximum value of B
at BQC. From physical considerations it is obvious that the time of
oscillation is increased by resistance. From Eq. (104) it is evident that
the effect of resistance on the flux is a non-linear one.

It is now desired to obtain the natural period of the circuit with
resistance when the circuit is operated at maximum flux B c. By means

of Eqs. (108) and the relation = -j (1 + 6)/, Eq. (104) can be trans-

A

formed to

y" + r(l + a&ay 2 + 5ftyV+ (1 + 8) 2 (6' iy + 6^ + W) = 0, [118]

where

r - -AQ(\ + o, 2M 2 + a, 4M 4 ) - -

XG X

&

03 "* 77

RESONANCE IN SERIES NON-LINEAR CIRCUITS 281

Equation (118), written in the normal form, is

The solution of system (119) for r = is

y = C^ii + 4 2 i) sin / + A 23 sin 3/ +

On passing to numerical results y is observed to be practically sinusoi-
dal. Thus let the solution of Eqs. (119) for r = be

- sin (/ + ,), [j2o]

y2 = w cos (/ + v).

In completing the solution of Eq. (119) we shall employ the method of
differential variation of parameters. (See Sec. 2, this chapter.) Evi-

dently, , ,

dyi du dyi dv dyi ^

du dt dv dt dt y *'

[121]
9^2 du dyz dv dy? = /

Qu dt dv (It Qt y *
Equations (121) yield

bu co* 2 (t + *),

t) sin (t + v) cos (/ + v).

By substituting Eqs. (120) in (122) and carrying out the expansions we
obtain

u' = r(>o + *2 cos (/ + v) + e 4 cos 4(/ + v) + e^ cos 6(/ + )],
^
where

+ |ft 3 3 + AM 5 , /2 - ^ + f W +

A -

/6 -

By 3-8 there exists a solution of Eqs. (123) as a power series in r.
It is clear that u = (An + A^i) = e and v = for r = 0. Accordingly,

write , ,

-* + t r

+

282 HYPERELLIPTIC FUNCTIONS

Now r, for control circuits, is very small. Consequently, v is small.
Expanding cos n(t + v)(n = 2, 4, 6) in Eqs. (123) as a power series in
v t substituting Eqs. (124) in (123), and equating, in the resulting equa-
tions, the coefficients of like powers of r we obtain

b - 0,

v\ = /2 sin 2/ + / 4 sin 4/ + /o sin 6/,

w'l = - fco + *2 cos 2/ + e 4 cos 4/ + e 6 cos 60, [125]

t4 = (#2 sin 2/ + 4 sin 4/ + go sin 6f)n

+ 2t/i (/ 2 cos 2t + 2/4 cos 4/ + 4/ cos 60,

where

- (2*

3

Equations (125) are solved sequentially subject to the initial conditions
o = c, u t = v, = for / = 0, / = 1, 2, . The solutions are

- cos 20 + 5/i(l - cos 40 + J/ 6 (l ~ cos 60],

MI = (c^t + 2^2 sin 2t + \e\ sin 4/ + \e$ sin 60,

i i t l26 3

[2 COS 2/ + 2 #4 COS 4/ + 3/frj COS 60

4" 2 r 4 + 3ti<*o) + 2(/2 +/I +/eVi

where the/'s, f's, and ^'s of Eqs. (126) are the same symbols previously
defined, excrpt // is replaced by ?.

Substituting Eqs. (126) in (124) and (124) in turn in (120) the solu-
tion of Eqs. (118) as far as the terms in r 2 is obtained. The immediate
objective is to find the period of a half-cycle when the applied voltage
maintains the circuit in operation with maximum flux B^c. It is then
necessary to solve Vi(/) = for /. Evidently Vi vanishes for / + v\(t)r
+ 1*2 (')r 3 +=*. A solution of this equation by Newton's method
(an approximate root is IT) is

/-(! + g S ), [127]

where
$ = i -f 2.24& 3 e 2 + (2.086 5 + 1.4&)<? 4 + 0.366 3 & 3 e 6 + 1.44J* 8 . [128]

RESONANCE IN SERIES NON-LINEAR CIRCUITS

283

Finally, by means of the transformation = - / and equations

A

T c = -7- Q and (127) the period, taking into account resistance, is

T __

* cr ,

[129]

The period in 6 of the applied voltage, Eq. (104), is 2w. The condition
for the circuit to be in resonance with the applied voltage is 7' = 2ar.
Substituting this value in Kq. (129), neglecting the term containing r 4 ,
and solving for x e we obtain

' The final results are: (a) proof of the physical principle that the
sudden increase in current in the region /;<* (Kig. S-12) is due to the
circuit being in resonance with the applied voltages (/>) formula (I.W)
giving the amount of capacitative reactance required to pnxluce reso-
nance at a prescribed voltage. The theory checks accurately experi-
mental results. 23

EXERCISES XX

1. Prove that the equation

has two fundamental real periods and one imaginary period Taking the initial con-
ditions to be x(0) = 0, x'(0) = 1, and < It 1 < kl < 1, obtain the^c |>eriodh.

2. If a hemisphere rocks so that its motion re-
mains in a plane, the differential equation of
motion is

(r 2 -f- n* 2rw cos 0)6 -f gn sin = 0,

where g is the acceleration of gravity, and the re-
maining quantities are shown on I ; ig. 3-13. Intc
grate the differential equation subject to the initial
conditions 0(0) = 0o, 0'(0) = 0.

3. Solve the non-linear spring problem of 3-31, FIG. 3-13. Rocking Hcmi-
taking as the expression for the force F sphere.

Let 63 and 65 be large compared with b\.

"E. G. Keller, "Re.sonance Theory of .Series Non-linear Control Circuits, 1
J. Franklin Institute, 225, 561-577 (1938;.

284 METHOD OF COLLOCATION

4. Let the non-linear spring of 3-31 be subject to a periodic force a cos w* such
that the differential equation of motion of m is

* .

m i ( *i* i ^s* ) ~ cos CM.

Deduce the condition for resonance between the applied force and the natural period.

PROBLEMS XXI

1. In Ex. 4 above let u have a value such that neither resonance nor beats occur
in the motion of m. Under these conditions obtain a solution of the differential equa-
tion of Ex. 4 which will give the amplitude of the motion. There is an electrical
analogue. 24

2. Set up the differential equations for the primary and secondary currents of a
transformer where the saturation curve is ki = x a&? 4- a&x 6 and x = B/BQ.
The primary impressed voltage is E sin / and there is a condenser in Ixrth the primary
and secondary circuits.

3. The differential equation of a simple series circuit with sinusoidal impressed
voltage, constant inductance, and thyrite resistance is

M * + R(i)i = Esinfl,
dO

where R(i) K i~ 7l/1 l and K is a constant. Solve this equation.

4. Read 2S the paper listed below and then by means of a non-linear inductive
circuit with thyrite resistance, design a lightning arrester such that when the current
reaches its peak in the inductance the resistance of the thyrite element is near zero.

3-36. Advanced Schwarzian Transformations. The theory of
elliptic functions is employed in the study of two-dimensional field
problems 26 by means of the Schwarzian transformation. More ad-
vanced Schwarzian transformations become hyperelliptic. Even these
can be carried out if the hyperelliptic functions are expressed in the
form of those of 3-32.

(7)
Method of Collocation

The method of collocation is primarily one of solving systems of
linear differential equations. However, it is extensible by means of

14 E. G. Keller, "Beat Theory of Non-Linear Circuits," J. Franklin Institute,
228 (September, 1939).

11 K. B. McEachron, *Th>rite: A New Material for Lightning Arresters,"
General Electric Review, February, 1030, p. 92.

I. A. Terry and E. G. Keller, "Field-Pole Leakage Flux in Salient-Pole Dynamo-
Electric Machines," Journal of tlic Institution of Electrical Engineers, 83, 845-854
(1938).

THEORY OF METHOD OF COLLOCATION 285

implicit function theory (3-12) to non-linear systems. Essentially,
the method consists in setting up a sequence of functions which satisfy
precisely the boundary conditions of a system of differential equations
and which satisfy the differential equations to a prescribed decree of
approximation. The prescribed degree of approximation is specified in
the next article. The success of the application of the method is deter-
mined largely by the skillful choice of the functions which form the
sequence. The guide in the choice of the functions is a thorough
knowledge of the physical problem.

3-37. Theory of Method of Collocation. In 3- 1 it waj indicated
that any system of differential equations can be reduced to the normal
form of Kqs. (I). It is evident also that any system of differential
equations may be reduced, by repeated differentiations and elimina-
tion of variables, to a single differential equation of the same order as
the order of the system. Accordingly, in the method of collocation, we
shkll take as the normal form of a system of linear differential equations
the single wth order differential equation

/oW + /lW 'I--' + " ' +/a(x)y ~ * w = 0>

or

f(P)y - *(*) = o,

where

Let the interval of the solution, i.e., the range of tin- independent
variable be a ^ x ^ b. The boundary conditions arc expressed in a
form different from that used in the classical solution of 3-2, 3-9,
and 3-14. In the present method the boundary conditions consist of
the n equations

fo(* Q )p n y |x-x + + /(*oM*o) - , (t - 1 , 2, - , ), [132]

where /j(jc ) and B f are constants and not all B t are /cro. The n values
of #o satisfy the relation a g JT O ^ b. The n values of XQ are not neces-
sarily identical for the n equations (132).

The first part of the construction of the solution consists in setting
up 5 + 1 linearly independent functions F , Y\, , F. of x such that
y - Y (x) satisfies Eqs. (132) where the,- have definite values not all
of which are zero and each of the functions Y\, Y 2 , , Y 9 satisfies
(132) with every B> replaced by /ero. Evidently then the function

Y,, [133]

286 METHOD OF COLLOCATION

where the Cj arc arbitrary constants, will satisfy the boundary condi-
tions (132). In a physical problem the Y 3 (j = 0, 1, 2, , s) are deter-
mined by principles of physics and engineering. The functions must
be such that a linear combination of them will permit the behavior of
the dependent variable anticipated. Experience with the system, or
oscillographic or differential analyzer solutions of the equations may
serve as a guide in the choice of the Fy.

The second part of the construction of the solution consists in deter-
mining the Cj such that Eq. (133) will satisfy (131) at least approxi-
mately. Denote by t(x) the result of substituting Y of (133) for y in
(131). The result of the substitution is

(*) =f(P)Y(x) - v>(*)

[134]

where

Zt-SWY, (7-1,2, .,*).

In the method of collocation the condition is imposed that the
Cj shall be such that e(jc) shall be zero at s different values of x. [That
is, the exact solution of (132) and the approximate solution (133) shall
have identical locations at 5 points.] From this condition Eq. (134)
becomes the s equations

2, ,*). [135]

Since Eqs. (135) are linear in c. 3 the 5 equations are readily solvable for
Cj. When the values of r, so determined are substituted in Eq. (133)
the value of F there defined is the approximate solution desired. It
is frequently convenient to solve for r, by matrices.

EXAMPLE. The method of collocation is illustrated by the solu-

dy

tion of -T- y = subject to the initial conditions y = 1 for x = 0.

Let the interval of the solution beO^#gl. If the choices are made
that

F = 1, F,=*> (j= 1,2,3,4)
then 4

F = Fo + r,F, - 1 + c lX + c 2 x 2 + c 3 * 3 + c 4 x 4 ,

and

Z - - 1, Zi - - i *> = jx*- 1 - *>', *,(*) = 0.

METHOD OF COLLOCATION FOR A NON-LINEAR PROBLEM 287

Since there are four unknown <r's it is necessary to use four values of
x k . Let these be xi = 0, .v 2 = J 3 ', *3 = % and x\ = 1. Equations (135)
now become, in matrix notation,

"10 0"

"o"

~\~

1 A U

02

1

I * -If 3?

05

1

_0 1 2 3

.f\_

_!_

The solution of thcsr equations is c\ = l,r a = 0.5078, r 3 - 0.1406,
4 = 0.0703. The approximate solution of the* differential filiation is

Y = 1 + x + 0.5078.V 2 + 0.1406.x*' 1 + 0.0703.V 4 .

EXERCISES XXII

1. Solve by the method of collocation (/> 2 -f l)v - T subject to the initial condi-
tions y(0) SB 0, /(()) = 1. Let the interv.il, in which the solution is desired, be
IF 2g x TT. The functions 1 o and F ; are .sunK^ st ed by

. czYz x(ir 2 X 2 )(l/ir 2 H- CiX 2 + <"2* 4 )-

2. Solve exercise 1 by employing Fo = sin x, KI = cos x so that Y sin * -f*
c\ cos x.

3. Solve, by the method of collocation, Lcuendrc's c( ju.it ion

d z y 2x dv m(m -f \)y

",; ., "- -f- ~ -- ="

</x 2 1 x 2 dx 1 x*

subject to the initial conditions >-(0) =* - ! 2, y'(0) = 0. Let m 2 anrl let the inter-
val for the solution l>e ^ x fC 0.5 Compare the ac curacy and the total labor done
in obtaining the solution with that done for the matrix solution of 3-23.

3-38. Method of Collocation for a Non-linear Problem. The

method of collocation is applicable to non-linear problems. Let it be
required to solve dy/dx = y 2 /2 subject to the initial condition y(0) = 1.
[The exact solution is y = (1 - x/2)~~ l .\ Let

s

Y " F O + J^ CjYj - 1 + cix + c 2 x 2 + - + c.x'.
a- 1

Substituting Y in the differential equation we have
c\ + 2c^Xk + + 5 c t x* k ~ l = |(1 -

288 GALERKIN'S METHOD

where k = 1,2, , s. If s = 3 and x\ = 0, # 2 = 0.5, * 3 =* 1, the last
equations become

Ci - 0.5,

- + 2c 2 + 3c 3 =

One solution of the above equations is c\ = 0.5, c 2 = 0.0207, 3 =
0.5004. A closely approximate solution of the differential equation is

y = 1 + 0.5* - 0.0207* 2 + 0.5004* 3 .

EXERCISES XXIII

1. Solve, by the method of collocation, the differential equation - = y L % subject

ax

to the initial conditions ?(0) - /(O) y'(0) - 1.

(8)
Galerkin's Method '

Galcrkin's method is of s[>ecial value in the solution of problems in
dynamics and elasticity. The method differs from the method of
collocation only in the conditions imposed on *(#). [See Eq. (134)
for definition of e(.v).]

3-39. The Galerkin Equations. Galerkin's theory is identical to
the theory explained in ^-37 as far as Eq. (134). The interval for
which the solution of the differential equation is sought is a g x ^ ft.
Either or both a and 6 can be infinite. The condition imposed on t(x) is
that

shall be a minimum. A necessary condition for J to be a minimum is
=0, (k = 1,2, ,5).

We shall show that the 5 equations above reduce to 5 equations of

the form

rb

(*) Fjk (x)dx = 0, (k - 1, 2, , 5). [136]

THE GALERKIN EQUATIONS 289

Evidently,

where

97 o r ** *

= 2 I dx,

&k Jo 90fc

or

where

and

(.v) = Z () -
Since ZA(JC) arc expansible in the form

Z k (x)

where the gkj are constants and 77^ are remainders in the expansions,

O t (* = 1,2,...,j) [137]

Since the / 6^ dx are negligible for properly chosen F* for the

physical problem in question and since the F* arc linearly independent
the last equations yield

/(*) F! dx - 0, , / t(x) Y. dx - 0. [138]

Jo

Equations (138) are the required Galerkin equations.

EXAMPLE. Solve, by Galerkin's method, the differential equation
dy/dx y = subject to the initial condition ;y(0) ** * Let the
interval for the solution be 3 x g 1.

If Fo - 1, Fi - Jtr, F 2 - * 2 , F 3 - Jt 3 then

290 GALERKIN'S METHOD

and

c(jc) " ~ Y " (ci ~ !)

The Galerkin equations, for the present problem, are

/tx dx = / X 2 dx = I e* 3 r/jc = 0.
/o /o

The substitution of the value of e(jc) and the carrying out of the in-
dicated integrations yield

25c 2 + 33c 3 = 30,
5Ci + 18^ 2 + 26a = 20,
21f, + 98c 2 + lSOf 3 = 105.

The values of c\, r 2 , and c a are respectively 1.03,0.388,0.301 and the
approximate solution is

y K - 1 + 1.034* + 0.388* 2 + O.JO I* 3 .

3 40. Torsional Oscillations of a Uniform Cantilever by Galerkin* s
Method. The partial differential equation governing the torsional mo-
tion of a uniform cantilever is

3.v\9.v

where = angle of twist per unit length,

C torsional stiffness per unit length,
/ = moment of inertia per unit length,
/ = time.

Suppose C and / are constant throughout the length of the shaft.
If is set equal to X- T, where A' is a function of .v alone and T is a
function of / alone, and = A r T is substituted in Eq. (139) then

CTd*X #T

~i d* " A ~d

or

C\_d 2 X ^^d^T

!X"dx 2 ' ~ T d?*

The left-hand member of the last equation is a function of x alone and
the right-hand member is a function of / alone. This equality holds for

TORSIONAL OSCILLATIONS 291

infinitely many values of .v and /. Consequently, each member of the
equation is equal to a constant and we have

C \ = < a

IX~dx* T dt* "
or

d i + k *T = 0, [140]

where k 2 is a constant yet to be determined.

The partial differential equation (139) has thus been reduced to the
two ordinary differential equations (140) and (141). The general
solution of (140) is

T = A sin kt + B cos fe/,

where A and B are arbitrary constants. Likewise the general solution
of Eq. (141) can be written down at once. However, we shall solve it

Ik 2 !' 2
by Galerkin's method. By the substitution x = / and = w 2

Vx

Eq. (141) and the boundary conditions become respectively

+ ' 2 -V - 0, ,Y(0) = 0, = forf - 1.

First it is necessary to choose functions which satisfy the boundary
conditions. A binomial in scums a reasonable approximation to the
possible displacement in X. Accordingly, set

Evidently, X r satisfies the boundary conditions for all r. For r 1,2,
X = Cl Xi + c 2 X 2

The Galerkin ccjuations (Eqs. 138) for the problem under consider-
ation are

o>

292 GALERKIN'S METHOD

which, on substitution of the binomials above and subsequent integra-
tion, become

= 0.

c 2 (-f + fm 2 ) = 0.

The necessary and sufficient condition that the above homogeneous
system possess a non-trivial solution is that the determinant

shall vanish. The roots of A = are m 2 = 2.4680 and 23.5625. From
the homogeneous linear system we have

(+1.33 - ^ftm 2 ) _

From this relation c 2 = 0.0281ci for m 2 = 2.4680 and c 2 = - 0.77ci
for w 2 = 23.5625.

Two particular solutions or natural modes of vibration satisfying
the partial differential equation are

= T-X = ci\A sin Jk/][(2{ - 2 ) + 0.0281(3| 2 - 2{ 3 )],
B _ r-^T - ^[4 sin tt][(2J - $ 2 ) - 0.7700(3{ 2 -

Cw 2
where * 2 = -z^p and in the first solution w 2 = 2.4680 and in the second

solution m 2 = 23.5625.

If more than two X functions are employed, additional natural
modes of vibration can be obtained. (Compare Rayleigh's method
Sec. 7, Chap. I.)

EXERCISES XXIV

1. In the application of Galerkin's method it is sometimes possible to add addi-
tional boundary conditions in addition to the necessary and sufficient conditions of
the problem. These additional boundary conditions are called secondary boundary
conditions, whereas the necessary and sufficient boundary conditions are then called
primary boundary conditions. The aid of the secondary conditions is that they
insure a more accurate solution with the choice of fewer X functions and thus reduce
the labor required to solve the problem in question. To illustrate the principle let
it be required to solve the illustrative example in 3-39. Obviously, there must be
no contradiction between the primary and secondary conditions.

GALERKIN f S METHOD 293

Solve, by Galerkin's method, - -- y = subject to the primary condition y 1
ax

for x = and subject to the secondary condition 1 for x 0. Then

ax

5

The numerical work in carrying out this solution consists in solving four linear equa-
tions in Ci, C2, 3, 4. This is the same work involved in the solution of the illustrative
example, but in the present problem greater accuracy is possible since a term in ** is
available in the approximate solution.

dy

2. Solve, by Galerkin's method, --- - y - subject to the conditions y(0) -

ax

dy

and = 1 for x = 0.
dx

3. Solve, by Galerkin's method, the uniform cantilever problem of 3 40 employ-
ing three functions X\ t Xz, and ^3.

4. The partial differential equation governing the flexural oscillations of a uni-
form cantilever is

where E = Young's modulus,

/ = moment of inertia of normal cross-section,
m = mass per unit length,

y = lateral displacement of point whose distance from the end of the beam is x,
x distance from the fixed end, which is taken as the origin of coordinates,
i = time.

Take the boundary conditions to be

y . Q = for x = 0, -? - ^ - for x - /.
dx dx* dxr

By Galerkin's method, obtain the periods of the two lowest modes of vibration.
Hint: Set x = l and take for the appropriate functions

Y r - | (r + 2) (r + 3) f* 1 - |r (r + 3) f+* + \r (r + 1) r** (f - 1, 2).

3*41. Galerkin's Method Extended to Non-linear Problems. The

method of Galerkin is extensible to at least simple non-linear prob-

dy

lems. Let it be required to solve = ry 2 subject to the initial condi-

dx

tion y(0) = 1. Suppose r g 1/2 and let the interval of the solution be
* 1.

If Y - 1, F! - x, and F 2 - * 2 then

294 LALESCO'S NON-LINEAR INTEGRAL EQUATIONS

The result of substituting y = F in the differential equation is e.
The conditions

3

2 dx = 0, (i = 1, 2)
GCi JQ

yield

/'

Jo

When the indicated integrations are performed the last equations are
evidently of the form of Eqs. (47) with x\ and x 2 replaced by c\ and c 2 .
They may be solved for ci and c 2 as a power series in r by the method
of 3 -12.

EXERCISE XXV

1. Carry out in detail the solution indicated in 3-41.

Method of Lalesco's Non-linear Integral Equations

The solution of non-linear problems by means of non-linear differ-
ential equations requires a knowledge of the theory of linear differen-
tial equations. On the other hand the use of the non-linear integral
equations of Lalcsco does not require any knowledge of linear integral
equations.

3-42. Lalesco's Equation. Lalesco's non-linear integral equation is

*>(*) = /(*) + f K[x t *;*({)]#, [142]

/o

where <p(x) is the unknown function or solution which is to be found
and/(#) and K(x, #>) are explicit known functions of their arguments.
The functions and quantities involved are subject to the following
restrictions:

(a) The variables x and f and the function <p are real.

(b) K[x, f #>({)] is a function of the real variables x, f, and the
unknown real function <p.

(c) I *(*,<*>) I <M [143]
and | K [x, {;*>,] - K[x, ;?,] |<tf|w-tt|forO<(<c<aand
A 6<v><.4 + ii where M, N, a, and b are positive constants.

LALESCO'S EQUATION 295

It can be shown 27 that the limit of the infinite sequence of functions

(*)-/(*) +

(*) = /(*) + /"**[*,*!
JQ

*.(*) = /(*) +

7

is the solution of Eq. (142).
EXAMPLE 1. The equation

di

h r(i + b i ) = E cos o/

at

is the equation of an inductive series circuit possessing a non-linear
resistance. Let the initial condition be i($i) = 0.

Integrating, with respect to /, both sides of the differential equation
we have

t

^ ^

i = - sin at-r I (i + b i*)dt,
a JQ

which is an integral equation of the form of Eq. (142). Consequently,
there exists a solution given by the limit of the sequence defined by
Eqs. (144). Identifying the quantities of the illustrative example with
those of Eqs. (144) we have

E

<t>o = - sin a/, K = - r(i + b i 3 ),
a

E C l f E /Z\ 3 1

<Pi = sin at r I sin at + b I ) sin 3 at dt,
a Jo La \a/ J

E rE rb /E\ 3

= sin at -- o" (1 cos at) I I
a a 2 ' 4 \a/

- (1 - COB at) - ;r (1 - cos3a/) ,
La 3a J

27 V. Volterra, Leqons sur les Equations Integrates, p. 90; E. Picard, Traitt
<f Analyse, H, p. 340.

296 LALESCO'S NON-LINEAR INTEGRAL EQUATIONS

If b is small in the problem in question v?i above is an approximate
solution for a finite interval of time /.

It should be noted that, by conditions (143), the interval of the
independent variable t (or x) is finite. It should be pointed out that it
is the limit of the sequence that is the solution. Hence <p n for n suffi-
ciently large is an approximation to the solution desired. If b is very
small in the present problem then <p\ above is an approximate solution.

EXAMPLE 2. A different approach, which is extensible to other
circuit problems, is possible in example 1. Write the differential
equation in the form

7 + ri = E cos at -rW 3 .
at

The indicial admittance A(f) for the linear circuit whose differential
equation is (p + r)i = 1 is

-

By Duhamel's superposition theorem the total current is

_ - (1 _

r

- - [ (1 - e - r(t -) (aE sin a\ + 3rb? ^ ) d\.
r J Q \ at t -\/

This equation is of the form of Eq. (142). Identifying the quantities
involved with those of Eqs. (144), we have

aE
sina\d\+ e" n

- (1 ~ O, K = - - (1 - <T r( '- x) ) (aE sin a\ + Srbi 2 ^
r r \ at

E aE C*

-(l-O -- / si
T T JQ

(f

( r cos

36 (1 - - x ) 2 e-** dX + 36

77

2 i g
- *

Obviously, the values for <pi as given in examples 1 and 2 are differ-
ent. Neither is a solution of the differential equation since only the
limit of the sequence is the solution. Both <p\ are merely approxima-

SYSTEMS OF NON-LINEAR INTEGRAL EQUATIONS 297

tions. The weakness of the method is due to the rapidly increasing
complexity of the & as i increases. In certain problems the conver-
gence of the sequence to a very approximate solution is greatly hastened
by the following fact. It can be shown that the sequence (144) will
converge in the limit to the same value if ?o(g) /(#) in the problem is
replaced by a function which is known from physical considerations to
resemble the final solution of the problem. (See 3 -15, this chapter.)

EXERCISE XXVI

jjj

1. Solve, by the method of this article, the differential equation 7-5 + r (# 4- fa 8 )

00*

J5 cos 0, subject to the initial conditions x(0) = C, #'(0) = 0.

3-43. Systems of Non-linear Integral Equations. If a problem is
reducible to the system of non-linear integral equations

i() - *i() + f *i[*. ; Witt), -, "ntt)]#,
/o

/o

then there exists a solution which is the limit of the infinite sequences

o

(),

"* [146]

S )) (*) = ,(*),

298 LALESCO'S NON-LINEAR INTEGRAL EQUATIONS

rx

Jo

In general, the carrying out of the sequences (146) leads to too great
a complexity. However, if sufficient physical insight into a problem
has been attained by means of engineering principles, oscillograms,
speed curves, etc., then devices may be employed which insure suffi-
ciently rapid convergence of the sequences (146) that two or three
members of each sequence are ample for the accuracy required. Fre-
quently, mathematical expressions of the dependent variables as func-
tions of independent variables are not required in an engineering
problem, but instead only an upper or lower limit to the range of certain
quantities must be known. The present method is of value in such
cases. To illustrate the principles and facts set forth in this paragraph
we shall apply Eqs. (145) to the problem of dynamic braking of a
synchronous machine.

EXAMPLE. The differential equations of dynamic braking (see
problem 2, set III, this chapter)

dl (E - IR)[(rs Q /s) 2

dt L[(rs Q /s) 2

* - 2 a

L J

ri481
L J

E
under change of dependent variables s = s Q e *,/ = + /ia" w (sug-

R

gcsted by an oscillogram of the field current and by a speed curve) are
reducible to the non-linear integral equations

' L J

SYSTEMS OF NON-LINEAR INTEGRAL EQUATIONS 299

The integral equations (149) and (150) are of the form of Eqs. (145).

The problem in dynamic braking is to find an accurate expression
for the number of revolutions before the rotor, running at full speed
when the braking is applied, comes to rest. It will suffice if we find an
upper limit to the number of revolutions provided this upper limit is
within a few per cent by test of the actual stopping time. This upper
limit is obtained with little labor by the method of this section although
the complete integration of the differential equations by the method of
3-5 is indeed laborious. By this reduced method we shall also illus-
trate the principles stated in the paragraph immediately preceding the
illustrative example.

It is necessary in that which follows to keep in mind that it is
known from an oscillogram of the field current and from the speed curve
of that rotor that both / and 5 arc decreasing functions of the time.
Consequently, both y and z are increasing functions of the time.

Identifying the notation of Eqs. (149-150) with Eqs. (146) we have

2(*) = 1fc(*) = 0.

Write Eq. (150)

(KPr\R _

==0+ jJ ~

The value of z satisfying

A1 fKPrfE , , _W /A*-*?\] dt ....

=0+ l jm-R +Iie V \[ I -(A^)\A*T? [1S2]

is smaller than the value of z satisfying (151) because the two values in
question are identical at t = and z from (152) is smaller than the
exact value from (151) for all values of t greater than zero. This is
satisfactory since an upper limit of the solution is desired. The silbsti-

Rt

tution of u\(x) = for y in (152) yields
L

KPr( + r 2 )

+

2JB/
; 2R (e

[153]

300 LALESCO'S NON-LINEAR INTEGRAL EQUATIONS

For R/L large this value of z is approximately z = A\t where

Preparatory for a similar treatment of Eq. (149) it is noted that
(Al + r 2 ) <?* \A\ + (re*) 2 ],
(A 2 + r 2 ) P * [A 2 + (re*) 2 ], [154]

(A 2 + r 2 ) 3 * 6 * ^ [A 2 + (r**) 2 ] 3
for all z > 0.

If the values z = Ait and those from Eqs. (154) are substituted in
Eq. (149) an approximate value for y is obtained which is less than the
true value in the differential equation. This approximate value for y is

r

r"), [155]
where

2KPr a E 3 y*

ce a = *-

+ r 2 )^ 8 + r 2 ) 3
The integral in Eq. (150) is rewritten as

SYSTEMS OF NON-LINEAR INTEGRAL EQUATIONS 301

The integral in Eq. (156) is

r 2 * + (1 - <r 2 *) - A 2 log [1 - X,(l - e~**)]

where

(*3

The integral of the right member of Eq. (157) is evaluated by
numerical integration for any particular machine giving z as a function
of /. The quantity y is determined from Eq. (155) for use in (157).

The answer is

N = number of revolutions = / e * dt.

/a
.

The integration in the last equation is performed numerically. The
test results indicate that the value of N is sufficiently accurate.

In the above example all parameters are carried nearly to the end
of the solution. Thus the expressions carry information for improve-
ment of design.

The process just completed is frequently representative of a certain
type of engineering solutions. The solution of a system defining an en-
gineering problem may fail because (a) its mathematical processes
cannot be carried out, (b) if carried out they may be so complicated for
computational purposes as to be practically worthless. The above
method may, as in the present problem, furnish a simple answer suffi-
ciently accurate and greatly superior to a completely accurate and
complicated solution.

EXERCISE XXVII

1. Rework the illustrative example of 3-43 obtaining greater accuracy by
choosing less liberal inequalities than those employed in Eqs. (154).

(10)
Solutions by the Differential Analyzer

Only very fragmentary ideas of the nature of the differential
analyzer and its solutions can be given in a page. References to the
literature are given in 3-44.

302 SOLUTIONS BY THE DIFFERENTIAL ANALYZER

3-44. Differential Analyzer. The relations expressed between the
independent and dependent variables in a system of ordinary differen-
tial equations may be viewed as merely constraints imposed upon the
behavior of the variables. A machine possessing parts whose motions
or electrical variations represent the behavior of the variables and
whose interconnections (mechanical or electrical) represent mathemat-
ical operations and relations is a differential analyzer.

The invention and design of such a machine calls forth the highest
ingenuity and inventive skill.

One of the very first, if not the first, differential analyzer was in-
vented and built at the Massachusetts Institute of Technology by V.
Bush and H. Hazen. Since 1927 new and ever improved machines have
been continually under development by the staff M of the Institute and
others. 29 The latest differential analyzer of the Massachusetts Insti-
tute of Technology is nearing completion. This machine will integrate
eighteenth order systems of differential equations. The integrator
units are mechanical but most of the interconnections are electrical.
Even the system of differential equations and their initial conditions
are impressed electrically upon the machine. Although earlier ma-
chines were approximately one hundred times more rapid than analyti-
cal processes the new machine is still much more rapid. Complete de-
scriptions of the new machine will appear presently in the literature. 30

3-45. Solutions. A differential analyzer solution of a system of
ordinary differential equations is a graph of the solution. The graph
(or graphs) may actually be drawn by the machine or it may print a
table of values from which the graphs may be drawn. Before the
equation or system can be set up on the machine all parameters (letters)
are replaced by numerical quantities. Initial conditions are introduced
by the initial settings of the entities which represent the variables.

28 V. Bush, F. D. Gage, and H. R. Stewart, "A Continuous Integraph," /. Frank-
lin Institute, 203, 63 (1927); V. Bush and H. L. Hazen, "Integraph Solution of Differ-
ential Equations," /. Franklin Institute, 204, 575 (1927); K. E. Gould, "A New
Machine for Integrating a Functional Product," /. Math. Phys., 17, 305 (1929);
H. L. Hazen, O. R. Schurig, and M. F. Gardner, "The Massachusetts Institute of
Technology Network Analyzer Design and Application to Power System Problems"
(not the differential analyzer), Trans. A.I.E.E., 49, 872 (1930); V. Bush, "The Differ-
ential Analyzer. A New Machine for Solving Differential Equations," /. Franklin
Institute, 212, 447 (1931); T. S. Gray, "A Photo-Electric Integraph," J. Franklin
Institute, 212, 77 (1931).

"D. R. Hartree, "Differential Analyzer," Nature, 135, 940 (1935); I. Travis,
"Differential Analyzer Eliminates Brain Fag," Machine Design, 7, 15 (July, 1935).

10 S. H. Caldwell and Staff, forthcoming articles in J. Applied Physics and J.
Franklin Institute.

REFERENCES 303

(ID
Additional Methods and References

Descriptions of additional methods and a list of references follow.

3 46. Systems of Differential Equations with Periodic Coefficients.
Systems of ordinary linear differential equations with periodic coeffi-
cients frequently arise in engineering problems. An example is the
system of differential equations of the armature and field currents of a
synchronous machine under short circuit when all resistances are taken
into account. Reference to an approximate solution is given in 3-3.
Other examples are the equation of Ex. 4, problem set XII and the
equation of problem 2, set VII. The last two equations have analogues
in electrical engineering. Often such equations are solvable by the
methods of Sec. I. However, this is not always the case. Equations
with periodic coefficients have long been of astronomical importance
and consequently a large body of theory has been developed for the
integration of such systems. See Ref. 14 of this article.

3 47. Non-linearity in Continuous Systems. Non-linear problems
in continuous fields usually lead to non-linear partial differential equa-
tions. The methods of Poritsky and Ritz attack such problems. See
Ref. 13.

3-48. References. The titles of papers are not always given in the following list
when the topic heading amply identifies the subject matter.

1. Systems of Differential Equations Solved as Power Series in Parameters.
F. F. Tisserand, Mecanigue Celeste, Vol. Ill, Chap. 6, Gauthier-Villars et fils, Paris,
1889. E. Picard, Traite d' Analyse, Vol. II, pp. 255-260, Gauthier-Villars, Paris, 1883.
F. R. Moult on, Introduction to Celestial Mechanics, pp. 264-265, Macmillan Com-
pany, New York, 1923.

2. Variation of Parameters. J. Lagrange, Nouv. Mem. Acad. Berlin, 5 (1774),
6 (1775), p. 190. John Bernoulli, Acta Erud. (1697), p. 113.

3. Differential Variations. H. Poincare, Les Mcthodes Nouvelles de la Mecanique
Celeste, Vol. I, Chap. 4, E. Flammarion, Paris, 1908. F. R. Moulton, New Methods in
Exterior Ballistics, Chap. IV, University of Chicago Press, 1926.

4. Hyperelliptic Functions. F. R. Moulton, Am. J. Math., 34, pp. 177-202. The
hyperelliptic functions in this publication are in a form suitable for applications in
engineering.

5. Method of Successive Approximations. E. Picard, Traite d' Analyse, Vol. II,
p. 340, Gauthier-Villars (1905). E. Picard, Journal de Mathtmatiques [4], 6, 197-
210 (1890).

6. Series Solutions in Independent Variables of Non-linear Equations. E. T.
Whittaker, A Treatise on Analytical Dynamics, Chap. XVI, Cambridge University
Press, 1927. W. O. Pennell, J. Math. Phys., 7, 24 (1927). The method given by
Pennell is an operational one and is similar to the methods of E. J. Berg which are to
appear presently in book form.

304 ADDITIONAL METHODS AND REFERENCES

7. Method of Collocation. R. A. Frazer, W. B. Jones, and S. W. Skan, R. and M.
No. 1799 (2913), A.R.C. Technical Report (1937), Air Ministry, London: His Maj-
esty's Stationary Office.

8. Galerkin's Method. W. J. Duncan, R. and-M. 1798 (3287), A.R.C. Technical
Report (1937), Air Ministry, London: His Majesty's Stationary Office.

9. Operational Method. "An Operational Treatment of Nonlinear Dynamical
Systems," L. A. Pipes, Journal of tlie Acoustical Society of America t 10, 29 (1938).

10. Existence Theorems. E. L. I nee, Ordinary Differential Equations, Chap.
XIII, Longmans, Green and Co., London, 1927.

11. Non-linear Integral Equations. M. Lalesco, in the text, Lemons sur les Equa-
tions Integrates by V. Volterra, Gauthier-Villars, Paris, 1913. E. Cotton, "Quasi-
non-linear Differential Equations," Bull. Soc. Math. Fr., 38, 144 (1910). H. Galajikian,
Amer. Math. Soc. Bull. 19, 342 (1913); also Ann. of Math., 2, 16 (1915). E. Schmidt,
Math. Ann., 65, 370 (1908). Lewi Tonks and I. Langmuir, "General Theory of the
Plasma of an Arc," Physical Review, 6 (1929).

12. Mechanics. S. J. Mikina and J. P. Den Hartog, "Forced Vibrations with
Non-linear Spring Constants," Trans. A.S.M.E., 54, A. P.M. 157 (1932). E.
Trefftz, "Stability of Non-linear Systems," Math. Ann., 95, 307 (1925). J. G. Baker,
"Subharmonic Resonance," Trans. A.S.M.E., 54, 162 (1932).

13. Non-linearity in Continuous Systems. Th. von Karman, "The Engineer
Grapples with Non-linear Problems," Bulletin Am. Math. Soc., 46 (1940). Hillel
Poritsky, "The Reduction of the Solution of Certain Partial Differential Equations to
Ordinary Differential Equations," Proceedings of the Fifth International Congress of
Applied Mechanics, John Wiley and Sons, 1939. W. Ritz, "Uber eine neue Methode
zur Losung gewisser Variationsprobleme de mathematischen Physik," Journ.f. reine
u. angew. Mathematik, 135, 1 (1908). Also "Theorie der Transversalschwingen einer
quadnitischen Platte mit freien Rander," Ann. Physik, 28, 737 (1909). A. F. Steven-
son, "On the Theoretical Determination of Earth Resistance from Surface Potential
Measurement," Physics, 5 (1934).

14. Linear Equations with Periodic Coefficients. F. R. Moulton, Periodic Orbits,
Publication 161, Carnegie Institution of Washington. F. R. Moulton, Differential
Equations, Macmillan Co., 1930. For infinite determinants see H. Poincare, Bulletin
de la Socicte de France, 14, 77. Also E. T. Whittaker and G. W. Watson, A Course in
Modern Analysis, Cambridge University Press.

15. Non-linear Circuits. B. van der Pol, "On Relaxation Oscillations," Phil.
Mag., 2, 978 (1926). B. van der Pol, "Frequency Demultiplication," Nature, Sep-
tember, 1926. B. van der Pol, Phil. Mag., 3, 65 (1927). A. Boyajian, General Electric
Review, 34 (1931). O. Martienssen, Phys. Zeitschr., 11, 448 (1910). P. H. Odessey
and E. Weber, "Critical Conditions in Ferroresonance," Trans. A.I.E.E., 57, 444
(1938). J. R. Carson, "Theory and Calculation of Variable Electrical Systems,"
Physical Review, 17 (1921). E. G. Keller, "Resonance Theory of Series Non-Linear
Control Circuits," /. Franklin Institute, 225, 561. Also "Beat Theory of Non-Linear
Circuits," 7. Franklin Institute, 228, 319.

16. Electrical Machines. T. M. Linville, "Starting Performance of Synchronous
Motors," Trans. A.I.E.E., 49, 531 (1930). W. V. Lyon and H. E. Edgerton,
"Transient Angular Oscillations of Synchronous Machines," Trans. A.I.E.E., 49,
686 (1930). H. V. Putnam, "Starting Performance of Synchronous Motors," Trans.
A.I.E.E., 46, 39 (1927).

17. Discrete Systems. E. G. Keller, "Analytical Methods of Solving Discrete
Non-Linear Problems in Electrical Engineering," Trans. A.I.E.E., 60 (1941). (Con-
tains bibliography of a hundred entries.)

INDEX

Acceleration, machines under, 190
Accelerometer, equation of motion, 22
Action, principle of least, 18
Affine connection, component parts of,

193

definition of, 195
Approximations, method of successive,

232

Arcs, admissible, 2
Automobile, equations of motion of, 21,

57

resonance in, 57, 71
spring and tire, 56
Axes, reference, 129
rotating, 180, 185
stationary, 174
Axis quantities, direct, 157
current, 158, 162
flux, 158, 162, 163, 164
voltage, 158, 162
quadrature, 157
current, 158, 162
flux, 158, 162
voltage, 158, 162, 163, 164

Braking, dynamic, 215, 298
Brushes, fictitious or real, 161, 162

Calculus, of variations, 1

n dependent variables, 13
n independent variables, 12
simplest general case, 4
operational, 171
Christoffel object, 183
Circuits, derived, definition of, 138

kinds of, 145, 146, 150
electric, 136, 137, 145, 149
elements of, 134, 135
non-linear, 274, 304
Class C prime, 2
Cofactor, 105
Coil, generalization of, 134

305

Collocation, method of, 285

for linear systems, 286

for non-linear systems, 287
Conditions, initial, 238, 246, 247, 267
Connections, scries aiding, 137

series opposing, 137

Constants, arbitrary, 61, 63, 67, 269, 276
Constraints, 26, 79, 82, 84, 86, 145

linear, 145

Convergence, of scries, 216, 219, 227, 229,
230, 2J3, 274

true radius of, 219
Coordinates, generalization of, 25

normal, 71

Cosine-amplitude function, 248
C-tensor, definition of, 125

for various systems, 143, 146, 152, 169,
173, 174, 177, 178, 179, 186

rules for deriving, 141, 169, 174, 175
Creepage, coefficient of, 73

forces of, 73

Current, generalization of, 139
Current vector, 127, 132, 139, 141
Currents, branch, 118, 128

Damping, coefficient of, 32

proportional to nth power of velocity,
281

proportional to relative velocities, 35

proportional to velocity, 32
Derivative, of determinant, 110

of matrix, 109

of vector, 41

Differential analyzer, 301
Differential equations, see Equations
Dissipation function, 32

Electric motors, 169, 173, 177, 178, 179,

186
Energy, kinetic, 27, 31, 42, 84, 181

method of, 83

of rigid body, 42

306

INDEX

Energy, potential, 19, 45, 49, 59
Equations, ballistic, 201
characteristic, 60, 234
differential, analytic, 201

approximate solutions of systems,
233, 301

for minimum surface, 15

homogeneous, 60

linear, 60

non-homogeneous, 68

normal form of, 202

of type I, 219

of type II, 206

ordinary, 60, 68, 201

solution of, numerical, 242
Euler's, definition of, 5, 11

for n dependent variables, 15

for n independent variables, 13

for simplest general case, 5
holonomic systems, 26, 79, 126
Hunting, 223

invariant, of rotating machines, 190
Lagrangc's, proved by Hamilton's

principle, 26, 32
Lalesco's non-linear, 294
Lcgendrc's, 243
linear, 60
Mathieu's, 246
Maxwell's, 181
non-linear, quasi-, 304
of motion, accelcrometers, 22

automobiles, 21, 57

compound and simple pendulums,
16,25

dynamical systems, 26

electric locomotives, 77

generalization, 28, 196

gyroscopes, 50

particle in free space, 31

projectiles, 201

refrigerator units, 45

rigid body in free space, 52

seismographs, 23, 56

uniform circular motion, 34

vibration absorbers, 35

vibrating shafts, 92

wheels and shafts, 34
of performance of electric motors,
commutator, 179

compound, direct current, 1 74

Equations, of performance of electric
motors, salient-pole synchronous,
177

shaded pole, 179
single-phase induction, 173
single-phase repulsion, 169
squirrel-cage induction, 178
of performance of electric networks,

all-mesh, 142
interconnected, 149
mesh, 145

multiple transmission, 131
stationary, 124

Existence proofs, 216, 220, 229
Extremals, definition of, 6

Factors, decrement, 193
Flange forces, 75
Flux, 164, 165
Flux density, rotor, 165
Flux linkages, 160, 163, 165
Forces, generalization of, 29
Forms, linear, 106
Frequency, constrained, 86
definition of, 63, 84
fundamental, 84
of shafts, 92
critical, 92
torsional, 34

Functions, analytic, 202, 209
cosine-amplitude, 248
dissipation, 32, 33
dominant, 216, 220, 229
elliptic, 246
addition of, 258
definitions, 248
differentiation of, 249
integration of, 261
Jacobi's, 248
of complex variable, 258
of real variable, 249
expansion of, in matrices, 235

in power series, 207, 228
generating, 203, 205, 221
hyperelliptic, 264
construction of, 273
periods of, 271
implicit, 227, 228
of a matrix, 235, 236, 237
power series of, 203

INDEX

307

Galerkin's method, description of, 288

for linear systems, 289

for non-linear systems, 293

for primary and secondary boundary

conditions, 292

Geometric objects, definition of, 128
Gyroscope, description of, 50

equation of motion of, 52

Hamilton's principle, for dynamical

systems, 18
proof of Lagrange's equations by, 26,

32

statement of, 16
Hooke's law, 46

Induction motor, 173, 178
Induction-reluctance motor, 210
Inertia, curve of least moment of, 6

moments of, 44

variable moment of, 231
Instability, criterion for, 78
Integrals, elliptic, 252

hyperelliptic, 272

line, 39

Integrations, successive, method of, 232
Interpolation formula, of Lagrange, 237

Jacobi's elliptic functions, 248
Junction-pair, 135
Junction-pair theorem, 135

Kocnig's theorem, proof of, 44

statement of, 31
Kronecker deltas, 123

Lagrange's equations, 25, 26, 32, 33, 181
Lagrange's interpolation formula, 237
Lalesco's non-linear integral equation,

294

Legendre's equation, 243
Locomotives, creepage forces for, 73

differential equations of motion of, 77

electric, 72

oscillations of, 72

potential energy of, 49

stability of, 78

Machines, balancing, 94
derived, 168, 169, 173, 174, 177, 179
definition of, 155

Machines, primitive, 156, 180

rotating, electric, 153
generalization of, 157
invariant equation of, 190

synchronous, dynamic braking of, 215,
298

under acceleration. 190

waves in, 165

Maclaurin's expansion, 207
Magnets, permanent, 196
Mathieu's equation, 246
Matrices, addition of, 105, 112, 151

division of, 109

three, 110
Matrix, adjoint of, 104

derivative of, 236

diagonal, 106

functions of, 235, 236, 237

high power of, 237

integral of, 236

inverse of, 105

rank of, 104

scalar, 104

skew-symmetric, 106

transpose of, 105

zero, 104

Matrix determinant, 109
Matrix unit, 104
Matri/ant, 238
Maxwell's equations for voltage and

torque, 181
Modulus, Young's, 92
Modulus transformation, reciprocal, 257
Motor, compound direct-current, 174

electric, 169, 173, 177, 178, 179, 186
equations, see Equations, for electric
motors

induction, 173, 178

induction-reluctance, 210

squirrel-cage, 178

synchronous, below synchronous speed,

210
Muffler, equation for pressure in, 225

Networks, all-mesh, 137

equation of performance, 142
rules for C-tensor of, 139

component parts of, 134

definition of, 99

derived, 138

308

INDEX

Networks, interconnection of, 149

junction, 101

mesh, 145

primitive, 100

stationary, 99, 124

sub-, 134
Nosing of electric locomotives, 72

Orthogonality condition, 89
Oscillation, torsional, 25, 34

Parameters, method of variation of, 221

of dynamical systems, 131

of networks, 139

of rotating machines, 163

power series in, 206, 229
Postulates, generalization of, 130, 132

of Kron, 130, 132
Power, equation of, 149, 161
Product, inner, 122
Products, of inertia, 44

of matrices, 107

of tensors, 122

of vectors, 37, 38, 39

scalar, 37, 38

Projectiles, equations of motion, 201
Pulling-into-step, 223
Pulsatance, 84

Quotient rule, 123

Rayleigh's dissipation function, 33
Rayleigh's principle, critical speeds, 92

orthogonality condition, 89

statement of, 85
Resistance tensor, 163
Resonance, definition of, 71

in linear systems, 71

in non-linear systems, 274
Reversion of series, 227
Revolution, minimum surface of, 7
Roots, latent, 234
Rotor, generalization of, 157
Rotor flux density, 165

Seismograph, 23, 56

Sequences, 232

Series, dominant functions for, 216, 220,

229,
reversion of, 227

Shafts, critical speeds of, 92, 93

frequencies of, critical, 92

rotating, 34

torsional, 34
Solutions, estimates to, 233, 301

generating, 203, 205, 221
Spring constant, 46
Spring-mounted systems, 46
Spring stiffness, variable, 246
Stator, generalization of, 157
Springs, linear, 45

non-linear, 268

with shock-absorber and tire, 56
Summation convention, 114
Superposition theorem, 296
Systems, conservative, 19

holonomic, 26
equations of, 26, 79, 126

non-holonomic, 82

non-linear, 200

of equations, 61

oscillating, 58

primitive, 138, 154, 156, 180

transmission, 131

Taylor's expansion, 207

Tensor unit, 123

Tensors, addition of, 122, 151, 166

admittance, 117, 121, 132

associated, 193

C t see C-tensor

contraction of, 122

contra variant, 115

covariant, 115

historical note on, 96

impedance, 166 -

inductance, 163

in mechanical engineering, 198

metric, 191, 192

mixed, 120

of resistance, 163

torque, 165
Transformation formulas, for i, c, z t 149

for interconnected networks, 152

for rotating machines, 168, 1 72, 183, 185
Transmission systems, equation of, 131

Unit, matrix, 104
tensor, 123
vector, 157

INDEX

309

Valence, definition of, 115
Vector unit, 157
Vectors, addition of, 36, 151
definitions of, 36, 115
differentiation of, 41
line integral of, 39
of current, 127, 132, 139, 141
of displacement, 45
products of, 37, 38, 39
Vibration absorber, dynamic, 35, 36
Vibrations, by means of matrices, 239
free, damped, 60
definition of, 59
general theory of, 58
non-linear, 268, 275
forced, damped, 68
definition of, 68

Vibrations, forced, general theory of, 68
non-linear, 275

linear, 58

normal modes of, 83

resonance in, 70
Voltage, generalization of, 139

generated, 160, 164, 176

induced, 160

Waves in machines, 165
Winding, moving, 159
Winding diagrams, 169, 174

Young's modulus, 92

Zero matrix, 104
Zero-phase-scquence quantities, 167