Ordinary Non-Linear Differential Equations

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ORDINARY NON-LINEAR DIFFERENTIAL EQUATIONS IN ENGINEERING

AND PHYSICAL SCIENCES BY N. W.

McLACHLAN

D.SC. (ENGINEERING),

LONDON

OXFORD AT THE CLARENDON PRESS 1950

Oxford University Press, Amen House, London E.C,4 GLASGOW NEW YORK TORONTO MELBOURNE WELLINGTON BOMBAY CALCUTTA MADRAS CAPE TOWN Geoffrey Cwnberlege, Publisher to the University

PRINTED IN GREAT BRITAIN

PREFACE THE

purpose of this book is to provide engineers and physicists with a practical introduction to the important subject of non-linear differential equations,

and to give representative applications

in

engineering and physics. The literature, to date, exceeds 300 memoirs, some rather lengthy, and most of them dealing with applications in various branches of technology. By comparison, the theoretical side of the Subject has been neglected. Moreover, owing to the absence of a concise theoretical background, and the need to

book for economical reasons, the text is confined methods employed A wide variety of these is included, and practical details given in the hope that they will interest and help the technical reader. Accordingly, the book is not an analytical treatise with technical applications. It aims to show how certain types of non-linear problems may be solved, and how experimental results may be interpreted by aid of non -linear analysis. The reader who desires information on the justification of the methods employed, should consult the references marked with an

limit the size o^this

chiefly to the presentation of various analytical in the solution of important technical problems.

asterisk in the

Much work

list

at the

end of the book.

involving non-linear partial differential equations has

fluid mechanics, plasticity, and shock waves. The and physical analytical aspects are inseparable, and more than one treatise would be needed to do justice to these subjects. Accordingly, the present text has been confined (apart from Appendix I) to ordinary non-linear differential equations. Brief mention of work in

been done in

plasticity, etc., is will

made

in

Chapter

be found in the reference

list,

I,

while the

titles

and particularly

of

many

in [62].

papers

Appendix

has been included on account of the importance of the derived formulae in loudspeaker design. A method of using Mathieu's equation as a stability criterion of the I

solutions of non-linear equations

I

am

is

outlined in Appendix II.

particularly indebted to Mr. A. L. Meyers for his untiring checking most of the analytical work in the manuscript,

efforts in

and

for his valuable criticisms

and suggestions. Professor W. Prager

PREFACE

vi

very kindly read the manuscript, and it is to him that I owe the idea of confining the text to ordinary non-linear differential equations. I am much indebted to Professor J. Allen for reading and commenting 5.170-3 also to Mr. G. E. H. Reuter for doing likewise with upon 4.196-8, the material in which is the outcome of reading his paper on subharmonics [13 la]. My best thanks are due to Professor S. Chandrasekhar for per;

mission to use the analysis in 2.30-2 from his book [159] to Professor R. B. Lindsay for facilities in connexion with 7.22; and ;

to Sir Richard V. Southwell for permission to use the analysis in

3.180-3 from his book [206]. am much indebted to the following for either sending or obtaining papers, books, and reports: Sir Edward V. Appfeton, Professor I

W. G. Bickley, Drs. Gertrude Blanch, M. L. Cartwright, and L. J. Comrie, Mr. B. W. Connolly, the Director of Publications Massachusetts Institute of Technology, the Editor of Engineering, Professors

N. Levinson, C. A. Ludeke,

van der

J.

Marin, N. Minorsky, and Balth.

Pol.

Finally I have pleasure in acknowledging permission from the following to reproduce diagrams in the text: American Institute of

Physics (Journal of Applied Physics), M. Etienne Chiron (UOnde filectrique), the Director of Publications M.I.T., the Editors of the Philosophical

Physics

Magazine,

and the U.S.S.R. Embassy

(Technical

of the U.S.S.R.).

N.

LONDON May 1950

W. M.

CONTENTS I.

II.

III.

V.

VII. VIII.

.

.

EQUATIONS INTEGRABLE BY ELLIPTIC INTEGRALS AND FUNCTIONS .24 .

.

.

EQUATIONS HAVING PERIODIC SOLUTIONS

.

.

.

41

METHOD OF SLOWLY VARYING AMPLITUDE AND PHASE

VI.

1

EQUATIONS READILY INTEGRABLE .

IV.

.....9

GENERAL INTRODUCTION

" .

.

.

THE EQUIVALENT LINEAR EQUATION

.

.

.

.

EQUATIONS HAVING PERIODIC COEFFICIENTS

GRAPHICAL AND NUMERICAL SOLUTIONS

APPENDIX

I.

HORN APPENDIX

.103 .

113

.

145

SOUND WAVES OF FINITE AMPLITUDE IN A LOUDSPEAKER .

.

.

.

.

.

.

MATHIEU'S EQUATION AS A STABILITY CRITERION

II.

REFERENCES A. SCIENTIFIC PAPERS

.

REFERENCES B. BOOKS AND REPORTS

INDEX

.

.87

.

.

.

.

.

.

.

.

.

.

.

.

.

.180 .

188

.191

.197

.200

CHAPTER

I

GENERAL INTRODUCTION IN the so-called

classical theories of different

differential equations are

mainly

branches of science the

linear in type.

They have been the

subject of intense study, and the existence of well-known forms of solutions is now established beyond doubt. If anyone skilled in

mathematical analysis encounters a linear differential equation of standard type, the formal solution is usually not difficult to obtain. The comparatively simple nature of such equations is due to the 'characteristic' relationships of the systems, \vhich they describe symbolically, being assumed to be linear. For instance the characteristic relationship used in developing the theory of sound propagation in air is the adiabatic law pvY = a constant. Now the graphical relationship between p and v is a curve, no finite portion of which is linear. To overcome this difficulty from a mathematical viewpoint, the theory is based upon infinitesimal pressure variation, so it is

assumed that the adiabatic curve may be replaced at the workingpoint by its tangent. In practice all audible sounds necessitate finite pressure amplitude. Fortunately, however, there is no need to depart from the linear theory based on infinitesimal (and, therefore, inaudible) vibrations, until the sound is fairly intense. In modem science certain phenomena cannot be explained on the classical linear doctrine, and it is then imperative to resort to nonlinear differential equations in order to deduce the desired information. For example, consider a lowly damped vibrational system whose is represented by ay -{-by* (y the displacement), the driven we have the system being by a force fcosajt. When 6 2 co ). In linear case, and the single-valued amplitude is A //(a

restoring force

the non-linear case, where b 7^ 0, the second approximation entails a cubic equation for A^ the amplitude of the fundamental vibration of

frequency

W/ZTT.

Here a *)A l -f=0,

(1)

*JLAl+(a-
, all the roots of (1) are real. Outside this range only one root is real. By introducing a viscous damping term into the

from which amplitudePhenomena arising from the

differential equation, formulae are obtainable

frequency curves may be plotted. multi-valued nature of A l may be explained by aid of such curves. One arresting and important feature of the analysis reveals that the

motion is nearly sinusoidal. Formulae for the non-linear case, obtained merely to a second approximation, are adequate to enable a satisfactory explanation of the behaviour of the system to be given, whereas on a linear basis it could not be explained at all!

Some

thirty years ago our knowledge of non-linear differential equations might have been compared with that of linear differential

equations at the time of Newton, Leibnitz, and the elder Bernoullis (James and John), i.e. about the beginning of the eighteenth century. Until recently the subject of non-linear differential equations has been a happy (!) hunting-ground populated almost exclusively by the

Research into methods of solving these equations has been neglected by the pure mathematician. Like our resources of

technologist.

coal, those of the

hausted in a

mathematician in the

finite

Interest has been

and

in Russia,

field

of linearity

may be

ex-

time!

shown in non-linear equations,

and research has now started

chiefly in

America

in Britain, with very

promising results [26-9, 131 a]. Although the analytical difficulties to be encountered in setting the subject of non-linear differential equations

on a firm foundation are formidable,

it is

here precisely that the

pure mathematician can help the technologist. Such assistance will ultimately be mirrored in new technical devices destined to benefit the community in general and, therefore, the pure mathematician in particular.

When a problem involving a new kind of non-linear equation arises, or

new

non-linear boundary conditions are encountered, the technologist may be puzzled as to the proper form of solution to be assumed. Usually these equations cannot be integrated explicitly in terms of

known

functions. Thus the solution must be adequate to account for

will

be an approximate one, but

the salient physical features of the problem. Moreover, theoretical knowledge formulated by the pure mathematician in the guise of suitable forms of sol/tion, existence

it

all

theorems, and methods of solution would prove invaluable. So far as questions of periodicity, stability,

and

instability are concerned,

ORDINARY NON-LINEAR DIFFERENTIAL EQUATIONS IN ENGINEERING AND PHYSICAL SCIENCES

GENERAL INTRODUCTION

CH.I

we may turn profitably, but not exclusively, [14],

Liapounoff

[87],

and Poincare

A

to the

3

work of Bendixon

[116].

knowledge of the experimental aspect of a problem may give guidance as regards an appropriate form of solution, e.g. 7.230.

But it will be realized that the accumulation of experimental data

may

in certain cases

be either too costly or impracticable, so that other must then be invoked. Graphical or numerical methods, procedure although tedious, are sometimes useful. Better still is the employment of a differential analyser, or an electronic computing machine. From a purely utilitarian point of view, difficult and troublesome non-linear differential equations involving much numerical work are best solved by a machine. Just as the expert craftsman in various branches of industry has been replaced largely so in the

by machines,

course of time the technical mathematician

may be replaced by other calculating machines. Then the intrinsic interest in mathematical problems will have ceased to exist. At the moment, however, these machines are so rare and the differential

analysers

or

price so high, that they are beyond the reach of the average technologist,

and from

his standpoint

may, therefore, be counted out. a concrete idea of the growing importance of non-linear give differential equations, the following typical (but not exhaustive) branches of science where they occur may be cited: acoustics, aero-

To

dynamics, astronomy, cable telegraphy, circuits,

elasticity, electrical

power

electrical

machinery, electronics, engine governors, fluid jets, hydraulics, hydrodynamics, naval architecture (stabilization of ships), plasticity, wave motion of finite amplitude on fluids and in solids.

S.

One of the earliest non-linear equations of acoustics was given by Earnshaw in 1860 [32], and pertained to the propagation of plane

sound waves of finite amplitude in air. A general equation for expanding sound waves of finite amplitude, of which the above is a given by the author in reference [183]. Solutions of and exponential loudspeaker horns have been obtained by S. Goldstein, N. W. McLachlan, and A. L. Meyers [41, 100, 101]. Non-linear equations were encountered by Lord particular case,

is

this equation for conical

Rayleigh in connexion with an electrically maintained tuning-fork [131], and by C. V. Raman in his experimental work on vibrating strings [127, 128].

An

important non-linear equation which occurs in astronomy

GENERAL INTRODUCTION

4

CH.

i

concerns the gravitational equilibrium of gaseous configurations (stars ) It originated with J. Homer Lane in 1870 [78], and has been discussed .

some length by S. Chandrasekhar [159], V. R. Emden [165], R. H. Fowler [36, 37], and Lord Kelvin [68]. Of other non-linear equations

at

in celestial mechanics, those pertaining to the white

and the pulsation of cepheid

may

variables, treated

dwarf starsj [159]

by A.

S.

Eddington,

be mentioned [164].

About twenty-six years ago the speed of signalling on long submarine telegraph cables was increased some fivefold by 'loading* the inner copper conductor with thin nickel-iron tape or wire. This alloy -> 0) of the order 4,000, and has an initial permeability (dB/dll as

H

with normal sending voltages (40 to 60) it becomes saturated magnetically, thereby introducing non-linearity and consequent distortion of the signals. Using a method of G. Riemann for the propagation of impulses in a gas, H. Salinger showed that a wave front, vertical at the sending end, tilts backwards (on a time basis) with increase in

distance along the cable. If the sending voltage exceeds a critical value, the wave front along the cable has both vertical and nonvertical parts [135].

There are a variety of non-linear problems in the theory of elasticity

An

.

5

early problem

is

that of the 'elastica or flexible rod bent in one

plane so that the two ends approach each other. It attracted the attention of the Bernoullis, Euler, and Lagrange. R. V. Southwell

has given the solution for a uniform bar in terms of elliptic integrals [206], while W. G. Bickley has studied the problem in which fabrics

bend under their own weight [16]. In connexion with large deflexions of beams, solutions have been obtained by H. L. Cox, K. 0. Friedrichs, Th. von Karman, J. J. Stoker, J. S. Way [62], and others. Finite deformation of solids has been discussed analytically by M. Biot, J. Boussinesq, G. Kirchhoff, and F. D. Murnaghan [62]. The author has studied the deformation of steel istics,

due to impulsive forces

(

shells,

with non-linear character-

8.20-8.23).

In the realm of vibra-

tional mechanical systems having non-linear restoring forces, analytical and experimental work has been done by E. V. Appleton [11],

G. Duffing [162], J. P. Den Hartog [47-9], and C. A. Ludeke [91, 92, 92 a]. In certain cases the control stiffness decreases with increase in The radius of such a star is much smaller than that of one of th/main stars. Thus same luminosity the former will have a much higher effective temperature than the latter. Hence the smaller star will be much 'whiter' than the larger one. This is t

for the

the origin of the

name

'white dwarf.

OH.

GENERAL INTRODUCTION

i

5

amplitude, resulting in instability which is evinced by a 'jump' or discontinuity. Appleton obtained this effect with a 'magnetic' vibration galvanometer [11], and solved the appropriate non-linear equation. The decreasing stiffness characteristic occurs also in con-

nexion with a simple pendulum and a synchronous electrical motor. Non-linearity arising from iron-cored apparatus in electrical power circuits may introduce oscillations, whose frequency is a sub-multiple of the supply frequency,

when the

circuit switch is closed.

frequencies must not be confused with sub -harmonics circuits

These sub-

in electrical

under different conditions or excited parametrically as in J. D. McCrumm and are

The former have been discussed by initiated by 'shock' [99]. 7.12.

An

experimental investigation into resonance effects in LCR having iron-cored inductances has been made by C. G. Suits

circuits

A variable potential difference (50 c.p.s.) was applied in series with the LCR combination. Provided R < a certain value R Q when [140].

,

the potential difference reaches a certain magnitude, the reactance vanishes and the current suddenly jumps to many times its former

The matter has been investigated analytically by E. G. Keller [64, 65], R. J. Duffin has discussed the behaviour of electrical networks having positive non-linear resistors [31]. value.

In connexion with transient effects in suddenly loaded synchronous have been

electrical motors, solutions of the non-linear equations

given by H. E. Edgerton, P. Fourmarier, W. V. Lyon, and F. J. Zak [33, 34, 93]. The solutions were obtained in graphical form by means of a differential analyser. In the field of electronics the triode oscillator

the outstanding The characteristic is curved, the is

example of a non-linear device. curvature changing from positive to negative, there being a point of inflexion. During a period of the oscillation the damping of the associated electrical circuit is sometimes negative (maintenance or growth) and at others positive (limitation and loss). The non-linear differential equation of the circuit,

namely,

was first studied by E. V. Appleton and B. van der Pol [8], Later van der Pol obtained solutions by the isocline method (see 8.12) with f(y)

==

e (l

2

y ),foT

=

0-1, 1-0, 10.

In the latter case he found that

GENERAL INTRODUCTION

6

under certain conditions a triode

oscillator

OH.

I

executed relaxation

being a limiting case when the parameter is large also showed that relaxation phenomena occur in a number

oscillations, this

[119].

He

of branches of science, e.g. physiology, the heart-beat being a relaxation oscillation. B. van der Pol and S. van der Mark constructed an electrical

model working on the same principle as the heart and

exhibiting similar characteristics [121]. V. Volterra has investigated the variation in the numbers of individuals in animal species which

time

FIG.

1.

in

live together,

The ordinates in represent the number of soles, and the number of sharks (different vertical scales).

(2)

one feeding on the other [147, 209]. The curve of varia-

a relaxation type. The problem may be illustrated by the simple case of two species offish in the sea, e.g. sharks and soles. The sharks eat soles, and the latter live on food in the sea, of which an ever

tion

is

If the soles existed alone, they would an ever increasing rate. The sharks, however, multiply in number at take care that this does not happen, by devouring large quantities of soles, so the latter diminish rapidly in number. Ultimately there is an inadequate supply of soles to sustain the sharks which commence

present supply

is

assumed.

to die off at a high rate. Thereafter the soles again begin to increase in number, so the sharks now have more food, thus entailing a growth in the shark population, which reaches a maximum. Then the cycle is

repeated indefinitely. The relationship between the two populations is depicted in Fig. 1, being in the nature of a relaxation

and time

although the changes from maxima to minima and vice versa are less precipitous than those in an electronic relaxation oscillation,

oscillation (Fig. 63).

parasites

show

The recurrence of epidemics and the problem of when a time base is used [182].

similar characteristics

Other aspects of non-linearity in valve

circuits

have been treated by

W. M. Greaves, and B. van der Pol [7-10], e.g. the 'silent interval' when a valve oscillator is driven by an external source.

E. V. Appleton,

The problem of parametric excitation and of oscillations in electriand other systems having non-linear elements has been studied

cal

by A. Andronow,

S.

Chaikin, N. Kaidanowsky, L. Mandelstam,

CH.

GENERAL INTRODUCTION

i

7

A. Melikian, N. Papalexi, H. Sekerska, S. Strelkow, K. Theodortschik, and A. Witt [94, 95, 139, 142]. Methods of solving the types of non-

above kind have been N. developed by Kryloff and N. Bogoliuboff The procedure is one of successive approximation, being based on rational assumptions relating to such applications. The method may be classified with linear equations occurring in researches of the .

Lagrange's variation of parameters. It is described in [175] by S. Lefschetz, and in [187] by N. Minorsky. The Poincare perturbation

method, developed primarily for astronomical problems,

may

be

applied (with limitations) to non-linear equations for various types of oscillatory system.

Mandelstam and Papalexi have extended the

procedure to cover the equations for a self-oscillatory system when acted upon by an external source. The extended analysis enables certain resonance

explained, e.g.

phenomena

peculiar to non-linear circuits to be

sub-harmonics in a thermionic valve circuit into \vhich

an e.m.f. is injected from an external source [94]. Problems in hydrodynamics involving non-linear equations occur in connexion with rivers, artificial channels, and hydro-electric

They have been studied by J. Boussinesq, R. Forchheimer, and others [30, 62, 166]. systems.

S. Cole, P.

Non-linear problems in the science of plasticity have received attention by various authors, of whom we cite J. Boussinesq, Th. von

Karman, A. Nadai, W. Prager, and G.

I. Taylor [62, 125, 126, 189]. Extensive researches in connexion with viscous and with compressible

fluids

have been conducted by many authors during the past century.

Of these W. G. Bickley, A. Busemann, S. Goldstein, D. R. Hartree, Th. von Karman, C. W. Oseen, L. Prandtl, 0. Reynolds, and G. I. Taylor

may

be mentioned

[62].

The theory of ship stabilization by means of anti -rolling tanks and auxiliary mechanism involves non-linear differential equations. This subject has been studied by N. Minorsky [108-11]. Wave motion of finite amplitude on fluid surfaces has been treated

by T. H. Havelock, T. Levi-Civita, Lord Rayleigh, G. G. Stokes, D. J. Struik, and others [62]. For additional information on the subject of non-linear equations

analytically

in general, the reader may consult the references at the end of the book, and also those in [62], which contains a bibliography of 178

The importance of the purely mentioned already. During the past few

items classified under nine heads. theoretical aspect has been

GENERAL INTRODUCTION

8

CH.

i

years the theory of non-linear equations for (a) mechanical vibrators, (b) triode oscillators (including relaxation oscillations), has been studied by M. L. Cartwright and J. E. Littlewood [26-9], G. E. H. Reuter [131 a], H. J. Eckweiler, D. A. Flanders, K. O. Friedrichs, J. J. Stoker, F. John [163], N. Levinson and O. K. Smith [82-6]. Although solutions of a number of the problems mentioned above were obtained either by graphical means or by a differential analyser, the majority of cases to date have been solved approximately by analytical or numerical processes. In some problems the amount of computation is considerable, calculating machines being needed. The methods of solution used in this book are summarized below:

Chap.

II.

Integrable exactly, using suitable transformations.

(with some exceptions) in terms of either Jacobian or Weierstrassian elliptic functions. IV. Approximate periodic solutions by (a) successive approxi-

III. Integrable exactly

mation

(iteration), (b)

perturbation method,

(c)

assuming a

Fourier series, and determining early coefficients therein. V. Approximate periodic and non-periodic solutions by method of slowly varying amplitude and phase. VI. Method in V applied to derive equivalent linear equations. VII. Approximate periodic solutions, assuming Fourier series, as based upon theory of Mathieu functions. VIII.

Appendix

(a) Isocline

graphical construction,

construction, (c) Maclaurin series, As at IV (a).

I.

Lienard graphical numerical methods.

(b)

(d)

CHAPTER

II

EQUATIONS READILY INTEGRABLE 2.10. Definition. If in

dent variable y and

an ordinary

differential

equation the depen-

derivatives are of the first degree only, there 2 3 no like being products yy' y'y'', y y the equation is said to be linear. But when the degree of y and/or its derivatives differs from unity, or its

,

',

if

,

they occur as products, the equation

said to be non-linear in y.

is

For example

is

a linear equation of the second order. The presence of # 3 x 5 does not ,

constitute non-linearity in y.

But

dx 2 are non-linear equations of the second order, y dy/dx, y 2 being non-linear terms.

Examples. In

2.11.

this chapter

we

shall deal

which are integrable exactly without recourse to

We

commence with those

of the

(dy/dx)*

with equations

elliptic functions.

order.

= o, ^+? dx y

1. Solve

the initial condition being y t/ The equation may be written

x so the second

first

,

term

is

(1)

when x

dx+y dy

=

=X

Q.

0,

(2)

we have

non-linear in y. Integrating,

= A, a constant. The initial condition gives A = x%-\-y% = a say. is #

2

-f 7/2

2

,

(3)

Hence the

solution

2

(4)

,

the equation of a family of concentric circles solution may also be written in the form

radius a

y=(a-s)*, so y

is

double-valued, and

integration.

is

also a function of a

(variable).

The

(5) 2 ,

the constant of

10

EQUATIONS READILY INTEGRABLE Comparison may be made with the linear equation

whose solution

A ^

where

is is

l

y

=

CH.

AJx,

n

(7)

an arbitrary constant multiplier dependent on the

initial condition.

=

2. Solve dx the

initial

condition being x

the form

(x+y) dx

the term

=

y

=

y dy

=

1.

(x

If the equation

is

written in

_ y) ^

(2)

Let

is non-linear in y.

x

rcos6,

rsind,

(3)

dO,

(4)

dr+r cos 6 dd.

(5)

y

being variable. Then

r,

dx

and

dy

Substituting from (cos

= =

cos 9 dr sin 6

(5) into

(4),

(2),

r sin

we

get

drr sin 9 dd) = (cos0 sin 0)(sin0 dr+r cos rsin dO = sin rfr+rcos ^ d0, = r. rfr/d0 logr = 0+A,

0+sin 0)(cos d cos 2

so

2

2

dr

giving

Thus J.

1,

xy

being a constant of integration, so in terms of $log(x

Since y

=

1

when #

implicit form

For

initial

i

I,

+y*)t&n-

A=

l

=

y/x

|(loge 2

77),

2

1

#,

XQ y

?/

,

,

(8) (9)

y we have (10)

and the solution takes the

-

the solution

0.

(11)

is

=

2

+y )/(*+^

0.

Here again we

see that the solution of a non-linear equation

function of the

initial condition.

+? = ^ dx y

3. Solve which is non-linear and (1) becomes

Write y

in y. 7

,

(6) (7)

A.

og |(a: +2/ 2)_2[tan- (^)-|7r]

condition x 2

log[(*

=

2

dO),

2

is

a

(1)

2,

vx,

(12)

then dy/dx

= &(dv/dx)-{-v

EQUATIONS READILY INTEGRABLE

2.11

fj?)

xv

or

(tx

-

r

Thus

y

-

J (V 1

(y>

-f 2 .

+

-$=!?

C

=

2

1)

=

0.

Cdx

dv

+ (^11)5 ,

J

(3)

log A, a constant,

X

J

L)

Cd(v-l)

2 J

/*

fti)

+ (v

11

^

J

(4)

==

and, therefore,

=

Iog(t;-l)+loga;-log4

Hence

log[(t;

=

1)/-4]

l/(v-l).

l/(v

(6)

1),

(7)

which yields the implicit form

(yx)

A

=

Ae*l v- x \ (

dy/dx+x/y

=

(8)

The equation

being the constant of integration. 2/c

> 0)

(ic

(9)

should be solved by the reader, using the substitution y 2.12.

Bernoulli's equation. This

may

=

vx.

be written

C

tjc

+f(x)y==g(x)y,

(i)

^

where/, g are continuous functions of x, but not of y, and /x l which entails non-linearity in y. Write u y ~^ and we get

=

=- +00. This equation occurs hydrodynamics

The

first

=

CH. (1),

is

(D

0.

when #

=

0,

w is

or

,

we

so

write x

=

2 )

3u,

(3) (3)

may

be

*

=

~- (w"+2ww')

du

so

t^"

=

2ww'-{-A

0,

(4)

a constant.

)

> 0,

virtue of the above conditions, with when ^ -> +00, so 0, and w

A

1

w"+2ww'

=

= --K+w; du

0.

(5)

w = w"

Thus, from

when (5),

0,

(6)

0,

(7)

2

or

)

w'

so

B=

=

0,

the accents denoting differentiation with respect to u.

=

when

a thrice differentiate function of u. Then

w"'+2(u>w"+w'

written

By

=

y'

in the theory of the plane jet in

remove the constant

to

%w(u), where y(x) (1) becomes

u

the general

[17].

step

n

= a -^ 2

2

(8)

,

a 2 being a non-zero constant of integration. Hence j

dw/(a

w+(7,

w*)

-tanh~ 1 (^/a)

giving

Since

2

d

w

when ^

=

0, it

satisfies

= u-{-C.

follows that

w= which

a constant,

C

=

atanhaw,

(9)

(10) 0,

so (11)

the three conditions above.

Restoring the original

variables leads to the solution

y

which

=

2eatanh(a#/3),

B=a

(12) 2

is a function of the constant Since t^e differential equation is of the third order, there are three constants of integration, C 0, and B. namely, A .

EQUATIONS READILY INTEGRABLE

2.27

Solution of

2.27.

with

2.26

(1),

when A,

B,

C

21

^ 0. We commence

2.26, then

(5),

^(w'+^^A, Writing

z

__

dw

dz __ ~~

dz

du

du so (2)

(2)

= Au+B, we get dw

This

= Au+B.

w'+w 2

so

(1)

,

dw '

dz

= o. *?+^!_l dz A A

becomes

(4)

= =

the Riccati type of equation at (4), 2.130, with a b I/A, 2.130. In the present 1, so mutatis mutandis the solution is (12), \L case a 0, jp 1/2, v 1/3, q 3/2, Z 2/34, which gives is

=

=

=

=

=

C being a constant of integration. Finally by f

=

z

in (5) above,

is

2.30.

Some

[4(a;/3)+5]

we obtain y

which

2.26 with

=

2tw[A(x/3)

+ B],

(6)

a function of the three constants of integration A, B, and C.

astronomical equations.

[159].

These arise in con-

nexion with the gravitational equilibrium of a gaseous configuration

Here the total pressure is due to the usual gas in virtue of radiation. It is given by the formula that pressure plus

in stellar structure.

p

= &T*+RT/v,

(I)

where p = pressure, T = absolute temperature, a = radiation constant, v = volume of unit mass, R = gas constant. The pressure p the relationship and density p = l/v both vary with the radius r,

between the two former being

p

m

-

Kp l +

l

IP,

(2)

is the mass of matter within a sphere of K, ^ being constants. If radius r, and G the gravitational constant, namely, 6-67 x 10~ 8 dynes cm. 2 gm." 2 the equations of equilibrium for the configuration are ,

EQUATIONS READILY INTEGRABLE

22

_4wr = 2

and

/>

aT

From

(3)

p ar

and by aid of

+ 0w =

CH.

n

0.

(4)

o

(5)

f

becomes

(4), this

O.

(6)

p dr)

Write p = A0J1 where A is, for the time being, an arbitrary constant. Then from this and (2) we get ,

p Substituting from

= KW+^6^.

(7)

into (6) yields

(7)

?

=

with & 2

4arGX l -WI(ii+l)K. Putting

=

fer

in (9) leads to

+ This

is

known

as the

at r = sity

,

0,

Lane-Emden

equation. Unless

and we have 9

is

=

I,andd0/d

/*

=

or

1, it is

by making 6=1 = the central denOwhen^ = 0. A solution which

arbitrary, but we now fix the centre of the sphere, which gives A

non-linear. So far A

it

/o c ,

these boundary conditions is termed a Lane-Emden function of index //,. Tabular values of some of the functions are given in

satisfies

reference [105].

2.31. | =

Transformation of

i/x,

d/dg

=

(10),

2.30 [159]. The substitution

x 2d/dx transforms the equation to

*g + * = We

=

shall consider the integrable case

0.

(1)

where

/x

=

5.

Substituting f

(J)*aty into (1) gives

'-ly(i-y*)

=

o,

(2)

EQUATIONS READILY INTEGRABLE

2.31

while with x =

eu

,

(2)

23

becomes

-O, which

is

2.32.

Solution of

non-linear

(3)

by virtue of the term %y 5

Write v

2.31 [159].

(3),

and we get

.

-

dy/du, du

=

+00, r -> 0. From (4) we have

u Let sin

2

^

\y*,

= -2 J

and

(5)

u

(dyly)l(l-W+Bi.

(5)

becomes

=

(

dw/smw-\-B 1

(6)

(7)

tan^w =

so

.#e- M

jB^.

(8)

to/(l+tan \w^ = so (9) y = [12JS2fa/(i + J82^)2]t. = = But = (\) x*y, and I' so ^ (i)ty~*> ^ n(i on substituting for from (9), we get Q = 3jB + B 2|a)2p (10) ~ 5, we take 0=1, For the Lane-Emden function of index = = the function of 5 which Hence at 0, 1/3. gives d0/dg

Now

sin 2

= l

4 tan 2

4

2

J?/

a;

,

1

,

[

2/( 1

t

JJL

2

index 5

is

-

(11)

CHAPTER

III

EQUATIONS INTEGRABLE BY ELLIPTIC INTEGRALS AND FUNCTIONS 3.10. Solve y+ay+by* = 0, (1) = = a initial 6 and the are conditions where > 0, ^ 0, 0, y y y when = 0. This equation refers to a mass-spring system of the type ,

'

but the characteristic' of the spring is i2/+s 3 ?/ 3 no damping. The differential equation of the system is

illustrated in Fig. 2,

and there

is

,

my+s^+Szy*

= sjm, and b = s

=

0,

(2)

being the mass. A spring control of the form Siy+s^y* may be obtained by using a flat bar and suitably 0. The effective shaped blocks, as shown in Fig. 4 A, provided s 3

so a

3 /ra, ra

>

length of the spring decreases with increase in the amplitude of vibration. The stiffness is defined to be the derivative of the restoring force, so

s

Thus the

= 5l +3^

2

3 ?/

stiffness increases or decreases

(3)

.

with increase in the displace-

s 3 > or < 0. A case where s 3 > is illustrated in We shall see later that in the case of a simple pendulum

ment, according as Fig. 12 B, c. 53




0,

so in (1) b

>

0.

Another example of a non-linear restoring force is that of a mass m at the centre of a taut uniform vertical wire, as illustrated in Fig. 4 B. It is left to the reader to show that if y/l

2/

,

(1)

(2)

displacement y Q

Thus equating the

/ y dy+s 3 j

we have

y*

dy

=

K2/g+i*32/o

=

9^+yo)-

=y

Q,

the motion

3.13, 3,14, so the y-t curve

is

the

is

identical with that discussed in

elliptic cosine

en t. Since damping to y Q is obtained by

absent, the initial part of the curve from y reflecting the first quadrant of cut in the y-axis, so that

is

behind the

origin.

In

=A Thus if we commence at = for

a>t,

it

comes

we assumed

3.14

y

\TT)

(3)

(3) yields

After reaching y

(cot

may be

strain energy

I/O

1/0

si

0,

l

cos wt+As cos

3^.

(5)

t when the impact occurs, we must write which gives the approximate solution

y

=

^4 1

sin^

A l and A^ are calculated as in

.4 3

sin3co.

(6)

3.14, the value of y Q being that in

(4)

above. 3.160.

The simple pendulum.

of motion

Referring to Fig. 6 A the equation

is

mass X acceleration -f- restoring force or so

m(ld 0/dt*)+mgsm6 =

=

0,

(1)

2

d 2 9/dt 2

+ -sin0 =

0.

0,

(2)

(3)

EQUATIONS INTEGRABLE BY

32

Since

sin0=:0

03 3!

05

OH. in

-)----..., the degree of the equation differs

from

5!

unity, so it is non-linear. Owing to the shape of a sine curve, the control 'stiffness' decreases with increase in (Fig. 6 B).

mq sm0 (restoring)

(8)

placement)

V-mg

cos0

(stiffness)

\

FIG. 6 A. Schematic diagram for simple pendulum. * B. Restoring force and stiffness* curves for simple

pendulum, showing non-linearity.

Write v

dd/dt y g/l

= a, and (3) becomes v dv ... + asm0 = 0, ,

do

f

so

giving If the

v dv v

maximum

= 2

a

sin0 d9-\-%A,

f

2a cos 0+-4

swing (amplitude) t;

=

0,

and

is

6

A=

(4)

(5) (6)

.

= ^, 2acosi/r.

Thus Put

AND FUNCTIONS

ELLIPTIC INTEGRALS

3.160

v

=

(2a)*(cos0 k,

sin|i/r

cos^)*

sin|0 = ksiny,

and

(7)

2

^--sin becomes

2a*(sin

33

2

(7)

|0)*.

the negative sign bei^g chosen, since decreases with increase in as reckoned from the instant of maximum deflexion. Also i cos

W d0 = k cos 9 d^, =

2&COS9 dq>/(l~k sm y)*. the for d0 from (8), (9) gives Equating expressions d0

or

t

=

0,

v

2

-

2

=

Accordingly on integrating

between the

(10)

(9)

2

2

-d9/a*(l-& sin 9)*. and 9 = |TT, and when Z = JTO dt

When

tt

limits

(10)

= =

,

9

0,

9

=

0.

we

(|TT, 0),

obtain the quarter period, so a~*

|T O

\

fc

dy/(I

2

sin 2 9)*

(11)

(l/g)*F(k,%7r),

o

where F(k, k

=

sin

\\fj.

a complete elliptic integral of the first kind, modulus Hence the time for a complete period is given by

|TT) is

From a table of elliptic integrals we find that F increases with increase in k, and, therefore, the greater the amplitude the longer the periodic

Consequently the motion is not isochronous, i.e. the time of swing a function of the amplitude, which is to be expected, since (3)

time.

is

is

non-linear in

3.161.

0.

Approximate solution of

write sin0

^

3

J0

.

3.160. If

(3),

Then the non-linear equation

0+a0+60

3

=

|0|




0,

=

but small enough, and

(2) is satisfied,

the equation

to be solved takes the approximate equivalent linear form

Sf+2fcy+a>fy

For the initial conditions y = V = y

T/ O ,

an approximate solution of

is

Q

y e-

=

= Kt

0.

at

0, it

follows that

cosa)t,

Since

(1).

t

(5)

=

(6)

A ^ yz~

Kl

l

>

from

(3)

we

obtain (7)

so

co

-> a* as

t

-> +00.

qualitatively when to motion is oscillatory at hold. that the (6) Suppose the start with by% :> ay Q When the amplitude decreases to a value y,

The behaviour of the system may be considered

K

is

too large for

.

inonosay, the subsequent motion will be non-oscillatory, and y -> a as t -> -{-co. if the control For still /c, tonically (ay -{-by*) is larger

monotonically from the start.

inadequate to promote oscillation, y -> If

we change

the signs of K and

y

and

o>

^

= yQe

b, (6), (7) Kt

become, respectively,

cosa>t,

(a-3fo/ge

(8)

2

(9)

'/4)*.

Hence as

t increases from zero, the amplitude builds up, but the rate of oscillation decreases by virtue of the decrease in stiffness with increase in amplitude. The function ay by 3 has a zero value when K* y (a/6)*, so to avoid instability, y Q e (a/b)* in (8), (9).

=




2

=

0,

A

non-periodic term in the particular so the complete solution of (7) is

A

2

integral,

we

2

(8)

.

a>Q

With the above

A =^ Vl

-

(^

2

2

/a>

find that

2

2

l

so

we

initial conditions, /3co

J?!

,

=

0,

(9)

)[-^H|cosa; ^+icos2a;

(10)

^.

l 2/o+ 2 2/o 2/i)

A* 3oj [A*

J3

1

(12)

To avoid a

non-periodic term in the particular integral of (12), the coefficient of cos o> 1 must vanish, so o>|

Then the complete 2/2

=

=

5A 2 l6ajl

solution of (12)

is

(13) f

1

n 9

A3

1

cog

48

A* co

OS3c 0^ (14)

EQUATIONS HAVING PERIODIC SOLUTIONS conditions above, we obtain

4.131

Using the

A2 = Hence by of

(!),

y

-

51

initial

(1), (5),

4.130,

29,4 3 / 144w o>

and

JB 2

=

0.

(15)

(10), (14), (15) to the second order in 6, the solution

is

-[

o

^

(16)

bA /6a)%, i.e. bA/6a

2

cog

If a

>

56 2 4 2 /6 Q decreases with increase in A. For a spring control ay+by*, the stiffness is (a+36?/ 2 ) and increases with increase in y, whether i.e. it is an even function of y (see ,

3 Fig. 12 c). The graph ofay-\-by is anti-symmetrical, so cu increases with increase in amplitude. The minimum ordinate in Fig. 1 1 A occurs when y a/26, and for stability the negative swing must be

less

than

a/6, i.e. |ymin

|

< a/6.

apposite to remark in connexion with the foregoing analysis that a tone of double frequency (the octave) is audible when a tuningIt

is

EQUATIONS HAVING PERIODIC SOLUTIONS

52

CH. iv

fork vibrates with a large amplitude. As the latter decreases, the pitch of the fundamental tone rises. This is in accord with the analysis.

EXAMPLE. The reader should solve (1), 3.10, by the perturbation method using the initial conditions y = A, y = 0. The result to order three in 6

is:

bA 3

-- ---^^* ,

4-

IbA*

\

,

1

1

,

cos 3o)

.

-.

(20)

4.132. Effect of viscous

Write y

= e~'

c/

-u(^),

damping. The equation

y+Zxy+ay+by* and we get 2

Q

for

(a/c 2 ),

(1) is

-

16^ 2e-^\

bA 2e-' ,

+^3

2

cog

6

/

cog

(3)

When

=

0, y =

A,y =

KA which is small enough to be neglected.

A better approximate solution of (1) may be obtained using the perturbation method. This

is left

as an exercise for the reader.

4.140. Solve

y+ay+by* =fcosa)t

(a

>

0,

b

%

0).

(1)

the equation for a loss-free mass-spring system, with control proportional to ay+by*, driven by a force/cos a>t as shown schemati-

This

is

cally in Fig. 12 A.

The

of the driving force at t

=

inevitable transient follows the application 0, but we may suppose that there is sufficient

damping to extinguish this and enable the periodic gftate to be attained after a short time interval. The damping is supposed to be small enough to have negligible influence on the amplitude of the motion. When a, 6, co, / have appropriate values, a subharmonic of

4.140

EQUATIONS HAVING PERIODIC SOLUTIONS

Fcoswt

53

(A)

m

A. Schematic diagram of mass and non-linear driven by an alternating force /cosorf. B. Antisymmetrical curve showing combination of linear and

FIG. 12. spring

s,

cubical springs ay, 6i/ 8 , respectively. The restoring force is an odd function of y, o. Stiffness-displacement curve for case B: a-f36t/ a is an even function of y.

frequency CO/GTT occurs. This aspect is considered in 4.190 et seq. For the time being we shall assume that the conditions for the existence of a subharmonic are not satisfied. In analytical work herein,

it is

tacitly

presumed that the reaction of the system on the

EQUATIONS HAVING PERIODIC SOLUTIONS

54

CH. iv

driving agent may be neglected. Thus the amplitude and functional form of the applied force is invariable.

For a periodic solution we may assume a Fourier series. Since ay -{-by* is an odd function of y, the corresponding force-displacement graph in Fig. 12 B is anti-symmetrical about the force axis. It follows that the solution may take the form

where

=

ifj

ay

=

We shall restrict ourselves

a)t.

stituting into

(1),

we

to

two terms

in

(2).

Sub-

get

a(A l cosi/j-{-A 3 cos3ifj)

by* J == (3)

Equating the have

or

coefficients of cos

on each side of

(1),

*

or*

=

by

aid of

=/,

(a+lbAl-f/A l )+lbAl

+ 2Yl

(3),

we (4)

(5)

This may be regarded as an approximation to the amplitude-frequency relation.

For the

coefficient of cos

3*)Az+lbAl+lbA\Az+lbAl so

In

.4 3

(5)

=

assume \A^A^\

0,

lbAH($a>*-a-lbAl-lbAl).


\A*IAi\ 0, b < 0. Hence if (13)

so

if

1

|

1,

(13)

is satisfied,

the forced motion of the system is nearly cosinusoidal, the displacement of the fundamental being in phase with the driving force. In the foregoing procedure, the non-linear equation is solved directly, whereas

the methods of iteration and perturbation entail the solution of linear differential equations.

Energy considerations. Writing

4.141.

takes the form so

v dv/dy +ay+b v dv-\-a

I

By

4.

(2),

j

y dy-\-b

dy/dt, (1),

^ = /cos ^ =/

3

\

=

v

y dy

J

cos cut dy.

4.140, (1) (2)

140,

dy Substituting

(3)

=

into

=

oj(A l sin

(3) ^+3^4 3 sin ty+...)dt. the right-hand side of (2), and integrating over a

(0, ^TT/OJ), the left-hand side vanishes by virtue of periodiand the right-hand side by virtue of orthogonality of the circular city, functions. Hence during steady motion, no energy is supplied to the system from the driving mechanism, as we should expect, since there is no dissipation.

period

t

4.142. Equivalent linear differential equation. By virtue of the motion being almost simple harmonic, the original equation may be replaced by what tion.

Then by

may be regarded as an equivalent linear approxima-

(8)

4.140, if f/A l

is

we have

negligible,

y+(a+I64|)y=/cosarf,

(1)

the particular integral being

y =/cosa>^/(a+|6^4f

co

2 )

A

l

cos wt.

(2)

Experimental illustration of analysis in 4.140 [92]. The apparatus shown schematically in Fig. 13 A, B has been used to obtain

4.15.

EQUATIONS HAVING PERIODIC SOLUTIONS

56

wave forms (1),

4.140.

for

CH. iv

a system akin to one represented symbolically by

A beam is mounted on a fulcrum so that oscillation occurs

about the latter in a vertical plane. The mechanical construction is such as to reduce friction to a small amount. One end of the beam is constrained by a non-linear type of spring while a mass m rotates at Rotating Unbalance

Fulcrum Timing motor Slotted time disc

Parabolic mirror fastened to ax le v

Axle fastened to beam

-s

s

\

Point sources ,

of

light

;yxNs.

Springs and supports rfot

Shown

Base/ Beam Concrete pier (approximate weight 1-5 ton)

Revolving

drum Slotted isc which rotates with unbalance

Fio. 13. A. Schematic diagram of apparatus for investigating system with non-linear restoring force. B. Plan view of apparatus.

radius r with angular velocity o> about the other end of the beam, thereby causing unbalance. If I is the moment of inertia of the oscillating parts about the fulcrum, f(0) the spring-control torque,

distance of the centre of rotation of

of motion

I

the

m from the fulcrum, the equation

is

2 where m(a) r cos ojt) is the accelerational force due to the rotation of m, and the right-hand side of (1) the corresponding driving torque.

The motion of the lever was recorded photographically, and a record is 14. The torque-deflexion curve /(#) for the spring is depicted in Fig. 15, being such that f(0) = 9), i.e. it is an odd /( reproduced in Fig.

4.15

EQUATIONS HAVING PERIODIC SOLUTIONS

57

1y

3o sec.

FIG. 14. Record of wave form obtained from apparatus in Fig. 13 using load-deflexion curve of Fig. 15. 60

h

50

O

0_

-30 &>

3

cr

10

-02

01

Deflection

-03 in

-04

radians

FIG. 15. Load-deflexion curve pertaining to Fig. 14.

function of

0,

and the curve

is

anti-symmetrical about the torque

Despite appreciable departure from linearity everywhere in Fig. 15, the wave form in Fig. 14 is almost a simple harmonic type. axis.

4.16.

Amplitude-frequency relation for

y+2 Ky+ay-\-by* = fcos(ajt+y). This equation is for a dynamical system of the type considered in 4.140, but with a term 2/cy representing viscous damping. The

68

driving

EQUATIONS HAVING PERIODIC SOLUTIONS force will now be out of phase with the displacement

CH. iv

corre-

sponding to the fundamental vibration, so to simplify the analysis have introduced the constant phase angle 9. may assume as

We

adequate approximation that with

=A

y

= 2/cy

Then while

an

a)t

iff

cost/j+A 3 cos3i/j.

(I)

2a)K(A l sin *fj+3A 3 sin

/cos(o)+9)

.

l

we

= /(cos 9 cos

(2)

3 0, 6 0, and 6 a*, \A^\ 0, is

>

=


0, and

1^1

It is evident

from

=

(- a, the amplitude-frequency curves for b are asymptotic to the straight line I-4J b

0,

to

=

>

(B)

FIG. 16.

Amplitude-frequency curves for driven mass-spring system, with spring characteristic of the

form

ay-}- by

3 .

For

6




increased gradually. The operating point will travel to Q, where \A\ drops suddenly to R and continues to move along to S with increase in

From S suppose the frequency is decreased gradually On reaching T \A!\ jumps suddenly to U and then follows the curve down to P.

a>.

.

y

Thus a form of hysteresis

when b

is

exhibited.


/ >/

The point there are three amplitude curves for forces /3 x 2 a If is lies on such w an increase constant, f'2 P! that/3 2 curve/g

>

>/

.

.

EQUATIONS HAVING PERIODIC SOLUTIONS

60

M

(B)

f = O.K =

V

b>o

CH. IV

b.

A when A =

=

A when A ^ 0. When

(4) is

9a+27(6/

2

(18)

/4)*,

then

(1) is

y=(4//6)*costarf.

.

(19)

These results enable us to obtain an equation which has a subharmonic solution A cos at. Writing 3o> for w, 4//6 = A* in (18), gives a = ( so (1) becomes ,

3

= /cos 3^.

(20)

The other subharmonic solutions are A cos(a>+277/3),4 cos(a>^+ 477/3).

A

similar equation is given in [27]. a subharmonic solution of 2

2/+(4o> /3.4

3

2 )i/

=

If 3/

= 4o>

2 ,

then

A Goscot

is

(21)

EQUATIONS HAVING PERIODIC SOLUTIONS

4.191

4.191.

a

=

2, b

Example. To illustrate the analysis in 4.190, we take 0-05, f = 2. When the subharmonic starts

A l ~ -f/8a = Using

(9),

4.190, co

2

=

given in Table

2.

A = -\A^ =

and

-0-125,

we

Assigning values to

A^

and using

4-24

~

(1)

4.190,

(5), (6),

3a*.

we

(2)

obtain the data

These are exhibited graphically in Fig.

A =

Data

is

relationship between tion of the case when b


g+6cof+6 o;l+...

being determinable constants. Here we expand in ascending powers of 6 (see 4.110, 4.131).

the

y'(Q)

o>

2

(not a)

EQUATIONS HAVING PERIODIC SOLUTIONS

68

Substituting from

(2)

CH. iv

into (1) gives

(co8+&co|+&i+..0(y;;+^^

+b(y*+2by Q yl +...) Equating the

coefficients of 6

6:

r ,

r =

"g^+a^^O,

For period

27r

s

in

if/,

or

so the complete solution of (4) 2/

2

y"Q +(a/a> )y Q

=

(3)

side of (3) yields: 0.

(4)

=^

= a,

(5)

is

sini/f+JS

(6)

cosi/r.

4.110, the initial conditions entail

in

y

on each

bFcos

f

i

,

n

n v/OS OyJ

(17)

Hence

to the second order in b

(18)' v

2

\2

Also to order two in 6

In virtue of the constant term in 'centre of oscillation'

is

(18),

on the negative

what may be regarded

as the

side of the origin of the force-

2 displacement curve ay-^-by (Fig. 11 A). We showed in Chapter II that the solution of a non-linear differen-

equation depends upon the initial conditions. If (1) is solved such that the coefficient of cos ^ is A for all t, the coefficients of the various

tial

terms will differ from those in y(Q)

= A.

(18),

and the initial condition will not be

Also the last term in (19) will not appear. The reader should

verify these remarks as an exercise.

4.195. Amplitude-frequency relation. (18), at t A, y' subject to the conditions y

=

must be

satisfied also,

=

which means that

=

4.194, 0.

But

was derived (19),

for fixed values of a, 6,

4.194,

and/,

EQUATIONS HAVING PERIODIC SOLUTIONS

70 co

2

CH. iv

complies with this equation. Written in the form

a cubic equation for A, which

it is

The consequences of such a

may be compared with

relation are

examined

4.140.

(9),

4.16 et seq.

in

From the analysis therein it appears that a physical system operating in accordance^ with the equation

y+2 K y+ay+by 2 = /cos wt

(2)

should exhibit the jump phenomenon described in 4. 1 70. Consideration of the signs of the terms in (1), where b 0, shows that it is of the

>

same type K

>

4.140, with b


0, 6, K > turbation. Let wt = z, 2/c small,

for

and

becomes

(1)

oj

2

2

y"+eb y'+ay+by y = y

Assume that

=

2

bF cos 2z.

(2)

2

Q

+by 1 +b y 2 +...,

(3)

the yr being periodic twice differentiate functions of

where the

o> r

are to be determined. Substituting from

z,

and that

(3), (4)

into (2)

gives

^2) + ..-K + %I + 6

2

2/2+...)

+ (5)

Equating

coefficients of b

r

6:

,

r = 0, uW+ay -

1, 2,...,

so for a solution with period

yQ

A,

13,

shall

we must have co

b, /, K, to;

yields (6)

=

a,

and, therefore,

-4sinz4-7? cos z-

(7)

to effect simplification of the

keep them the same throughout.

stage) differ slightly b:

0,

(5),

the equation non-linear, the ultimate initial conditions (unknown at the present

algebra is

=

are functions of a,

we

2?7,

on each side of

from

Sftice

= J5, y'(Q) ~ A. = - 2^^ yl~yl+F cos 2z,

j/(0)

wgyl+ay 1

(8)

EQUATIONS HAVING PERIODIC SOLUTIONS

4.196

71

so

-(l/a){(A*+B*)/2+ABsm2z-[(A*-B*)/2+F]cos2z}. (9) To avoid non-periodic terms in y l9 we must have a) l = 0. Thus with g

- (A*+B

2

)/2,

-

h

F+(A*-B*)/2, we

get

sin 2z

"2 cos z

.

J3 sin

2)+ 2a>

a> 2 (^4

(A/3a)cos 20.

(10) (11)

2/o2/i

sin

2+ JB cos 2)

so

(12)

(13)

where P, Q are given below. To avoid non-periodic terms and Q must vanish, and with A, B, non-zero we have

= 2(o> K) + 2(^/a )-(A/3a Q/B = 2(^> + 2((7/a + (A/3a 2

P/A

2

Subtracting

w2

=

)--(^ /3a

)

2

+ (^^

)~M/co B)

(14),

=

= -

P

0,

(14)

0.

(15)

a^/2, gives

-5(A +B*)/$a+(Ka> Q /b*)(A/B-BJA). (15),

obtain

(16)

and multiplying throughout by

=

(A/B+B/A)

Solving (17) yields

A/B

=

6 2 co /2/c,

=

6//3/ccog

- 0.

(17)

{)8(j3-4)t}/2,

K Mo (A/J3-~BfA)

so

2

2

j/ 2 ,

2

o> /e

we

2

)-(J5 /3a

and multiplying throughout by a/2

(14), (15) 2co Q

2

)

)

Adding

2

2

in

=

(18)

2 {(6//3a) -4*c a}!. 2

(19)

a. = a>2+26 WoWl +& K+2a> a> )+..., By (4) = 0, by aid of (16), (19), we get and since w^ = a, w = a-56 (4 +^ )/6a{(6//3a) -4/c a}i. 2

2

2

(20)

){(6//3a) -4/c a}*]}i

(22)

which gives the amplitudes. of the subharmonics. To order one in 6 (small) the solution is y = yQ Jrbyv so

y

= .4sin2+Bcosz+(6^5/3a)sin22 = -(6/2a)F2 +rcos(z-6' )-(6/6a)r 1

(6^/3a)cos 2 2

(bg/a)

(23)

cos(2z+6 2 )-(//3a)cos22, (24)

EQUATIONS HAVING PERIODIC SOLUTIONS

72

= AB/{(A -B )/2}, = where tan = A/B, tan when there are tion is valid only for Y real > 0, 2

2

2

2

l

i.e.

The

CH. iv

20^ This solusubharmonics.

term represents a unidirectional displacement, and as in 4.195 the 'centre of oscillation' is on the left of the origin. The second and third terms represent, respectively, the subharmonic and its first overtone, while the last term gives the forced oscillation which first

has the same period as the driving force.

4.197.

Subharmonics and

The motion

stability.

defined to be

is

stable (unstable), if after being subjected to a small disturbance

returns to (moves

away from)

its

former

In [131

state.

a],

by

it

aid of

stability criteria given in [85], it is shown that the upper (lower) internal and the two external signs in (22), 4. 196 correspond to a pair

of stable (unstable) subharmonics. Thus there are four subharmonics, two stable and two unstable. By virtue of the two external signs in 4.196, the

(22),

components of each pair

differ in

phase by

TT

radians

Under

certain conditions specified in [131 a], the forced of oscillation driving frequency becomes unstable (see remarks in

or

7T/o) sec.

4.198). Nevertheless, there is still the oscillation due to the overtone of the subharmonic. The analysis pertaining to stability is beyond our present purview, so we shall merely state what results may be

expected on the basis of reference [131 4.198.

Forms

of a and

to,

of oscillation. For

a].

7 to be real,

whatever the values

we must have /

>

(I)

6i

we must have

(w*-a),

(2)

~4* 2a}*.

(3)

for the unstable pair,

(a-aj

2 )

>

2

{(6//3a)

Variation in f.

For

(i)

co

2


a,

as

/

is

forced oscillation of amplitude (ii)

>

For a

6/ca f

>w

2 ,

when /

=

increased from zero, there

X C2 //(4co

6/ca*/6

2

a),

is

a stable

but no subharmonics.

the threshold

is

reached, and for

a pair of stable (unstable) subharmonics having amplitude given by (22), 4.196 with the upper (lower) internal sign.

/

When

/6 there is

2

a

[a-co -{(6//3o) -4ic^}]

=

0,

the unstable subharmonics vanish while the stable ones remain.

(4)

At

4.198

EQUATIONS HAVING PERIODIC SOLUTIONS

this point, however, the forced oscillation

73

becomes unstable and the

amplitude jumpsf to that for one of the stable subharmonics, as given by (22), 4.196 with the upper internal sign. Increase in /is accompanied by that of

Y

and

also in the amplitude of the

subharmonic

now

decreased, the stable subharmonic persists but with decreasing F, until the threshold/ 6*a*/& is reached. It then

overtone. If/

is

=

stable forced oscillation; CA = unstable subharmonics; CB A. OA subharmonics. At A, B, /8 = (3a/&){(a-o> 2 ) a -f 4* 2 a}i; at C, ^ = 6*a*/6. For / increasing from O, the forced oscillation becomes unstable at A, and there is a jump to one of the stable subharmonics at B. B. OA = forced oscillation AB = stable forced oscillation which is unstable between B and subharmomcs. c. ABDF ; BC = stable subharmonics DE unstable subharmonics.

FIG. 21.

A

;

D

;

vanishes and the amplitude of the motion jumpsf down to that for the 2 forced oscillation (now stable), namely, a). The cycle c//(4o>

X

is portrayed in Fig. 21 A, and there is a type of 'hysteresis' as in Fig. 16 A. In the first case two frequencies are involved, but only one in the second.

of changes

system were started suddenly by applying a force / (/!

2

a,

7

4.196 cannot be real with the lower

in (22),

internal sign, so there are no unstable subharmonics. The stable pair cannot occur unless (1), (2) are satisfied. As /in (2) increases from zero, there is the forced oscillation

f=/

the stable subharmonics

now

with

X ^//(4o>

is

tone. Increase in

/ is accompanied by that

Variation in If (1)

(i)

There

a),

and when

X

commence with amplitude

unstable, but there

forced oscillation

illustrated in Fig. 21 B.

2

is

is

.

The

the subharmonic over-

These remarks are

in Y.

no hysteresis

effect [131 a].

CD.

not

is

satisfied, there are

no subharmonics, but there

is

a

stable forced oscillation. (ii)

If (1)

is

and w 2 > a+{(bf/3a) 2 4* 2a}*, there is a stable but no subharmonics, since Y is imaginary. But

satisfied

forced oscillation

when \aaj 2




2 If a-{(bf/3a) 2 -4:K 2 a}* to there is a stable forced also two of the oscillation, subharmonics, provided a is not too pairs

to

co

2

-{

nearf to

}*.

co

2

+{

}*.

The trend of these remarks

is

illustrated in Fig.

21 c. Subharmonic resonance does not occur.

In general the forced oscillation is unstable when accompanied by a single pair of subharmonics, but remains stable if another pair of subharmonics is present to 'counteract the instability. This remark 5

is

exemplified in Figs. 21 A, B, o [131

a].

In Fig. 11 A the slope of the force-displacement curve changes sign to the left of the minimum, and if

Amplitude

limitation.

Iftninl

>ab l

>

the displacement increases continually with increase in time. Usually the characteristic for a physical system has no minimum, but it may

have a point of inflexion. In the case considered restoring force occurs.

pneumatic, neither a

Example. Imagine a hollow

4.199. length

is

I,

closed

in

4.199,

where the

minimum nor an

circular cylinder, of

inflexion

working

by a rigid disk of mass J m which is driven axially by a

The approximate analysis does not hold if a is too near to these values, j This includes 'accession to inertia' due to the external air [183].

t

EQUATIONS HAVING PERIODIC SOLUTIONS

4.199

75

such that the displacement | is 'finite'. If p Q is the atmospheric pressure, and v the corresponding undisturbed volume,

force /cos

2a)t,

we have

for adiabatic operation

pv7

p Q v%

=

a constant.

If the displacement during expansion

p(Z+f )? so

i>

Thus the

=

jp

IY,

(2)

(i+^)- y ^i>o[i-y^+y(y+i)^/2P-...].

'spring control' per unit cross-sectional area,

within the cylinder,

Accordingly disk

then

is -)-

= PO

(1)

if g/l

is

is

^

air

given by

1,

the approximate equation of motion of the

= /cos 2co,

2

fw"+2*ffaf +6

=

(3)

due to the

=

(5)

yp Q Ajl, b y(y+l)p ^4/2Z ^ being the cross-sectional area of the cylinder. Here the slope of the p-v curve is negative, and what may be regarded as the 'centre of oscillation' moves down this with a

curve (see

minimum large

(18),

,

4.194 with b

value, so

would

2


+00. As example of the latter we cite the familiar case of the free oscillation

of a linear electrical

LCR /

Here A(t)

=

Ce~ K* ->

circuit in

=

which the current

is

given by

(7e-*<sin(a>+9).

(2)

-{-co, while 9 is a constant phase angle. Consider a differential equation of the form

as

t

->

2

y+*g(y,y)+< y

=

o,

(3)

which the non-linear term eg(y,y) is relatively small, and g is a function J of both the displacement y (or its equivalent) and its first time derivative y. Neglecting this term, the equation reduces to the in

linear one 2y

of which the complete solution

y

=A

l

=

Q?

(4)

is

coQa>t-{-B 1 sma)t

(5)

i

A!, B! being arbitrary constants. In a sense this may be regarded as a first approximation, although it possesses no non-linear characteristics. (5) is expressible in the form (1), with A (A\ ^i)*, and

=

=

+

1

tsm- (A l /B 1 ) these being constants. 9 Writing (a)t+q>) == x> the solution takes the form )

= Asmx; y = AOJ cos xy

so

(6) (7)

t Strictly 'phase' pertains to sine waves, but it is convenient to use it here. 3 J If terms of the form 6y occur, they are to be included in g, i.e. g comprises damping and non-linear control terms.

METHOD OF SLOWLY VARYING

88

We now tiating (6)

A

suppose that

we

and 9 vary slowly with

CH. t,

so

by

v

differen-

get [77, 175]

y

=

Substituting for y from

Asmx+A(a)+^)GOs x (7)

0.

(9)

we have

(7),

AOJ cos x~ ^4co(co+9)sin ^.

y Substituting from

(8)

into (8) leads to

A sin x+^9 cos X = Differentiating

^

(6), (7), (10)

(10)

into (3) yields

A GOSX~~ ^4

0.

From

increases monotonically with increase in value 2 as t -> +00. If we suppose that

decay and -> 2 as

t

-> +00.

t,

(11) it follows that

and tends

AQ >

2,

A

to the ultimate

the amplitude will

METHOD OF SLOWLY VARYING

90

By

(2)

above, with

x and

for

v

5.10,

(15),

,

OH.

27T

-

9 == Since

where (11),

o>

cp

t

(l-^ 2 sin 20)cos