Introduction to Non-Linear Mechanics. (AM-11), Volume 11 9781400882274

Table of contents :
TABLE OF CONTENTS
Chapter I. Some Non-Linear Oscillatory Systems
Chapter II. Elementary Theory of the First Approximation
Chapter III. Refinement of the First Approximation
Chapter IV. Construction of the Higher Approximations
Chapter V. Linearization
Chapter VI. Application of Symbolic Methods To Linearization
Chapter VII. Multiply Periodic Systems
Chapter VIII. Influence of Periodic Disturbances
Chapter IX. Complements
Bibliography

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ANNALS OF MATHEMATICS STUDIES NUMBER 11

INTRODUCTION TO NON-LINEAR MECHANICS BY

N. KRYLOFF and N. BOGOLIUBOFF

A free translation by Solomon Lefschetz of excerpts from two Russian monographs

PRINCETON PRINCETON UNIVERSITY PRESS LONDON: HUMPHREY MILFORD OXFORD UNJVEF SITY PRESS

1949

Copyright 1943 PRINCETON UNIVERSITY PREss

(Reprinted 1947 with e·rrata on page 106)

INTRODUCTION During the last decade a number of Soviet scientists have investigated so-called non-linear mechanics, and among the most active are certainly to be found Kryloff and Bogoliuboff. An extensive bibliography of their contributions to the subject will be found at the end. A cursory reference to it will quickly disclose the fact that in one way or another their work is but poorly accessible to the American scientific and technical public. The present monograph is essentially a very condensed English version of their most extensive paper (No. 32 of the Bibliography, in Russi~D) except for the last chapter which is practically a small extract of their most mathematical production on the subject (No. 16 of the Bibliography, in Russian). Kryloff and Bogoliuboff consider primarily equations of the form t:.f(t, x, dx , dt

n

where t:. is a small positive quantity and f is a power series in t:., whose coefficients are polynomials in x, ~~' sin t, cos t. As a matter of fact, generally f contains neither t:. nor t. Similar equations are well known in astronomy and have been the object of systematic investigation by Linstedt, Gylden, Liapounoff and, above all by Poincare. In a general sense, one may say that the same methods are applied by Kryloff and

INTRODUCTION Bogoliuboff. However, the applicc:.tions which they have in view are quite different, being c'liefly in Engineering, Technology, or Physics, not~.~·ly electrical circuit theory. The soh;tions :J.re approximated by the first n terms of certain asymptotic representations; the first two terms usm.lly suffice u.nd yield what the ~n:thors cc:,.ll the "refined first approximation" which they discuss at length. The method of linearization descri~ed in Chapters V and VI, frew.:ently em1bles one to "Dy-pass the differenti::1l equation ~end proceed directly from the physical problem to the o.pproximate solutions. Th2t the general information obtained from the o.pproximc.tions gives importunt indications regarding the behaviour of the solution itself, is shown in the monograph, (No. 16 of the Bibliography) of which the extract given in Chapter IX will yield a few indications. Messrs. Kryloff and Bogoliuboff deserve much credit for the bold way in which they have carried out their work and for the numerous applications which they have outlined. It is believed that the present monograph will provide a fair picture of what they have accomplished. S. Lefschetz Princeton, N. J. November 20, 1942.

TABLE OF CONTENTS Chapter I. Some Non-Linear Oscillatory Systems Chapter II. Element&ry Theory of the First Approximation

30 8

ChapLer III. Refinement of the First Approximation

28

Chapter IV. Construction of the Higher Approximations

40

Chapter V.

55

Linearization

Chapter VI. Application of Symbolic Methods ·.ro Linearization Chapter VII.

Multiply Periodic Systems

30 30

Chapter VIII. Influence of Periodic Disturbances

79

Chapter IX.

87

Bibliography

Complements

100

I.

SOME NON-LINEAR OSCILLATORY SYSTEMS.

1• In the present section we will discuss a few non-linear oscillatory systems and derive the corresponding differential equ~tions. These equ2tions will serve later as illustrations for' the methods of approximation introduced in the sequel. 2. We begin with some conserv2tive (non-dissipative) systems. (2.1) Oscillating shaft. Consider a shaft composed (ideally) of two revolving masses joined by a non-linear elastic connection. Let e 1 , e 2 be the moments of inertia of the revolving masses, and e 1 , e 2 their angles of rotation. Let further M = c(e 1-e 2 ) be the angular momentum of the elastic connection represented as a function of the angle of rotation e = e.1-e 2 . The equations of motion for each of the two masses are

d 2e J--1 + c(e 1-e 2 ) 1 dt2 d 2e

J --~

2 dt2

-

c(e 1-e 2 )

o, 0.

Hence the equation governing the oscillations is ( 2. 2)

Fig. 1

Fig. 2

2

INTRODUCTION TO NON-LINEAR MECHANICS

In this relation the function c(e) is usually given graphically and may have the most diverse form. (2.3) Electrical circuit without resistance. Consider an electrical oscillating circuit (Fig. 1 ) containing an iron core. Let ~ denote the magnetic flux, i the line current, C the capacity. We then have ~ it (2.4) dt + c Sidt = o. The relation between~ and i is shown in Fig. 2. With sufficient accuracy and within certain limits, one may represent this relation analytically, for instance as: (2.5)

i =A~ + ~ 3

We have then for ( 2. 6)

p the differential equation o.

3. In the examples of oscillating systems that we have examined so far, we have not taken into consideration friction which causes dissipation of the oscillations of the system. Generally speaking the laws of mecnanical friction have been but little investigated. In practice, one chiefly assumes one of the following three: a) The force of friction is proportional to the velocity (oscillations in the atmosphere). b) The force of friction is proportional to the square of the velocity (for oscillations in a liquid). c) Coulomb's law: The force of friction is constant in magnitude but depends upon the velocity and its direction is opposite the velocity (for example in slipping of surfaces upon one another). (3.1) Pendulum freely oscillating in tne.atmosphere. r.t· we assume that friction is proportional to the velocity, the equation of oscillations will·be

I.

SOME NON-LINEAR OSCIUATORY SYSTEMS.

( 3. 2)

0

where A is a proportionality coefficient, called friction coefficient. (3.3) ElectricCJ.l circ1Jit with resistance. We suppose that the circuit contains an iron core, an ohmic resistance and a capacity (Fig. 3). Let ip be the flux, i the current, R the ohmic resiste:.nce, C the ce,naci tv. We will have this time

Fig. 3

~ dt

t

+

Ri +

c S idt 1

0

and hence assuming that (2.5) holds: ( 3. 4)

o.

4. 'Jp to the present, we 11ave considered oscillating systems with or without dissipation (friction). Since in practice dissipation is always present in some form in oscillating systems, the oscillations will fail to die down only if the system contains some source of energy whicL may compensate for the loss of energy due to dissipation. This condition may be fulfilled in two ways. First, the force acting upon the oscillating body (due to its connection with the source of energy), may possess a definite periodicity. The simplest example of oscillations of this type, said to be forced, is found in the vibrations of linear

3

INTRODUCTION TO NON-LINEAR MECHANICS

4

systems subjected to a harmonic disturbance: ( 4.1)

2

~ + 11 dx + kx dt 2 dt

=

F sintzt

where m is the mass, x the displacement, A the dissipation coefficient, k the sprin>~; constant, F the amplitude of the exterior force, ~the frequency of the disturbance. Second, the source of energy itself may have no specific periodicity but its ac'tion upon the oscillating body ~ppears to introduce into the system a negative dissipation which may compensate for the normal positive dissipation caused by the dissipative forces. Oscillations of this last type, called auto-oscillations, are quite wide-spread and have great importance in Physics and Technology. To obtain some idea of the manner in which autooscillations arise, we will examine a system with one degree of freedom. If the oscillations are of rather small amplitude we may write down the customary linear equation: (4.2)

m d 2x

dx

dt2 + i\dt + kx

=0.

As is well known the general solution will be -6t x ae cos(wt+~) where a,

~

6

are arbitrary constants, ..1_

2m

w

2

k m

( .1)

2

m.

Hence if A)o, then the amplitude of the small oscillations ae- 6 t will die down according to an exponential law. If on the contrary 11(0 then the small OSn (:.~)

I~.

is

eq' : .~i'

•!l " •

··~·.rdly ~!ecess.r·r·~1

for forc13d ~o

osc::.~l~_. ti,y~·1s.

Thus we

[:. van dcr PoJ system ::,n

Plr~y

::.ave in relc..tion

cq:lnt.ic~11

( ... 2)

F sin c:tt,

:....nd likewise fc•r t::e r·t:·,er systems.

II.

ELEMENTARY THEORY OF THE FIRST APPROXIMATION

6. All tlJe exc.mples discussed in the preceding chapter lead to equations of tlle form d 2x dt 2

( 6. 1 )

+ F(x,dx,t) dt

o.

We propose to investigate more particularly the so called quasi-harmonic case, where there are oscillations near the sinusoidal: x

a sin (v

t+~),

that is to say when we may write

II. ( 6. 2)

ELEMENTARY THEORY OF THE FIRST APPROXIMATION dx

F(x,dt't) =

.~2 y

9

dx

x+tf(x,dt't),

where t is c.. pa:mmeter ch2ro.cterizing the smallness of the deviation of F from 11 2 x. Thltil further notice we assume F, and hence also f, free from the explicit vari:1ble t. The basic differential equation will thus be ( 6. 3)

d 2x

dx + Y 2 x + tf(x,-) = dt

dt 2

O,·

and this is the equation which we shall investigate. If we endeavor to solve this equation by the usual methods of approximation, notably by the method of Poisson, we encounter a classical difficulty which baffled the astronomers of the eighteenth century, namely the presence of so-called secular terms, or terms of the form t~' trigonometric functio~. In the same spirit as the 0stronomers did in their day, we sh2"ll endenvor to find methods of approximation which yield results free from secular terms. In the present chapter we shall describe a very intuitive method enabling us to construct an approximate solution which will be free from secular terms. 7. We first observe that for t = o, (6.3) has the solution ( 7.1)

x =a sin (.,lt+.P),

(7.2)

dt = av cos (..Jt+~),

dx

where the amplitude a and the phase ~ are constant. For convenience the term "frequency" will designate v rather than the customary 2 rrv. Consider a, ~ , as new unknown functions of the time which are to be determined so that (7.1) becomes a solution of (6.3). We must have first

INTRODUCTION TO NON-LINEAR MECHANICS

10

Hence if we wish to preserve (7.2), or we may say if we impose (7.2), then

~~ sin ('lit+~) + a~: cos (vt+ 4»)

( 7. 4)

o.

From these relations we deduce: 2

(7.5)

d x dt 2

=

da dt

cos (vt+f>) - va.9! sin (vt+~) dt

and so finally from ( 6. 3) :

~~~·cos (vt+~) - va1t- sin (vt++)

( 7. 6)

- tf(a sin

(vt+~),a~cos

(vt+~)).

By combining with (7.4) there comes (7.7) ~

=-

d4> (7.8) dt

=

~f(a sin (Yt+4»), avcos

f. avf(a sin (vt+.,), av cos

(vt++)) cos (vt+f>),

(Yt+~))

sin ("t+4>).

Thus instead of the single differential equation of the second order (6.3) in the unknown x, we have two differential equations of the first order in the two unknowns a, cj> • Notice now that the right hand sides of (7.7), (7.8) admit with respect to t the period T = ~· Moreover ~~ ~~ are proportional to the small parameter l, so that a, +will be slowly varying functions of the time during the period T, and as a first approximation we may, therefore, consider them as constant. On the strength of this observation, we will indicate at once a simple intuitive method for constructing an approximate solution of (7.7l, (7.8). For this purpose consider the expressions

B.

II.

ELEMENTARY THEORY OF THE FIRST APPROXIMATION

f(a sin

cos

6,a~ cos~)

~,f(a

sin

11

~,av cos~) sin~.

o.nd let us expand them in Fourier series.

We find

f(a sin~, a)i cos~) cos~= K0 (a)+J(K_n(a) rl)o cos n~+~(a) sinn«P), ( 8. 1 )

f(a sin lb,av cos ~) sin ill= P0 (a)+) (Pn(a)

Dyo

cos nlb+Qn(a) sin nib. The coefficients Pn(a) • • . . , are calculated in the usual way. It will be sufficient to give the explicit expressions:

( 8. 2)

i

K 0 (a) =

~-rr

P 0 (a) =

~rr

S2~f(a

sin· ill, al" cos ill ) cos lbdlb,

S f(a

sin~.

9n: 0

aY cos ill ) sin ibdQ>.

Taking advantage of (8.1) we can represent (7.7), (7.8) in the following expanded forms:

~~ (8.3)

dcj,

=-

dt =

~K0 (a)

-

~L(K_n(a)cos

n)o sin n(Yt+Q>) ),

t t > raP 0 (a)+y8nTo(Pn(a)

n(vt-td>

)+~(a)

cos n(..,t+Q>)+Qn(a)

sin n(-Jt+Q>)). Let us integrate these espressions in the interval t, t + T, within which we consider a, ~. as constant and equal to the values a(t), ~(t).

12

INTRODUCTION TO NON-LINEAR MECHANICS We thus obtain:

( 8. 4)

1

a(t+TJ-a(t)

= _

p(t+T)-~(t) T

=

~Ko(a(t)),

~P (a(t)) va o "

Since T and the increments a(t+T)-a(t), ~(t+T-~(t) are small, we replace in (8.4) the left sides by ~~· ~t, and thus arrive at the equations of the first approximation:

( 8. 5)

l

~~ = - tK d~

dt

0

(a)

t

=

TPo(a).

If we compare with the exact relations (8.3) we find that the equations of the first approximation are obtained from the exact equations by averaging the right hand sides with respect to the time. This process duly generalized in the obvious way will be described as the averaging principle. It need not be said that the preceding reasoning cannot pretend to any sort of mathematical rigor. For this reason we shall examine in the next chapters the questions of the mathematical foundations of the averaging principle and likewise the question of forming the higher approximations. 9. Returning to (8.5), if we have a solution in a and~ and substitute it in (7.1 ), we obtain an approximate expression for x. If we choose in place of ~ the unlmown ljl = Yt+ •, then (8.5) yields ( 9. 1 )

Substituting in (8.5) and (9.1), in place of K0 ,

ELEMEm'ARY THEORY OF THE FIRST APPROXIMATION

II.

P0 their expressions from (8.2) we obtain explicitly da dt

(9. 2 )

~

(9.3)

=

-r.

2ITY

21r

S f( 2-rl

Y+ 2 ;a¥~

=

,a¥

sin~

cos~)

cos

~d~,

0

sin~.

f(a

0

aY cos

~)

sin

~dq>.

Thus the"first approximation to the solution of (6.3) will be of the form x

( 9. 4)

a sin ljl,

=

where the amplitude a and the full phase ljJ are to be determined from (9.2), (9.3). lo. Suppose that Fin (6.1) does not contain~' in which case f will likewise be free from it. Thus we will have

f(x;~)

( 1 0. 1 )

f(x),

=

and hence instead of (9.2), (9.3): da

-

( 1 0. 2)

dt

dljJ

( 1 0. 3)

dt

=

2 rr

t

5 f(a sin¢>) cos

=--

211v 0

l..l(a)

=

t

~d~,

2rr

v+ 2rrav5

f(a sin ~) sin~ d4>.

0

If we set

+(x)

X

=

S

f(x)dx

=

~srrd~(ad~in ~)

0

then s-rrr(a sin

~)

cos

~d~

0

and

hence da

dt = o.

o,

J3

INTRODUCTION TO NON-LINEAR MECHANICS

14

Thus the amplitude of the oscillations is now constant a= a 0 , and so instead of (10.3) we have ljJ

= w(a)t+e,

where the phase e is constant and equal to the initial value of ljJ. An approximate solution of (6.3) is then x =a sin (w(a)t+e).

( 1 o. 4)

We may say that here the nonlinear character of the equation has no other effect in the first approximation than to make the frequency depend upon the amplitude. If we square both sides of (10.3) and retain only terms in E. we obtain ( 1 o. 5)

w

2

2 t

(a) = Y +arr

Since F(x) =

v2x+E.f(x)

(10.6)

2 (a )

w

2 1(

S0 f(a

sin ci>) sin ci>dcl>.

we have finally:

1 2ft

= rr aS F (a sin

cl> )

0

sin cl> del> •

Formula (10.6) has the considerable advantage that the function F enters into it directly and not merely through its nonlinear part as it does in (10.5 ). 11. We will now examine a certain number of examples. (11 .1) Example 1. Consider the equation of the pendulum reduced for small oscillations (say not exceeding 30°) to the form (11.2)

d 2x sz., x3 dt 2 + l'x- 6)

We have at once from (10.6):

o.

II.

ELEMENTARY THEORY OF THE FIRST APPROXIMATION

and so approximately 2

(11.3)

Zl

w(a)=T'1-

a2

8 ).

As the amplitude increases the frequency decreases and hence the period increases also. This is likewise shown by the approximate formula (11.4)

T

=

211: w

=

To take a concrete example for a = 30° we find T = 1 • 01 4 X

2tr~.

(11 .5) Example 2. Consider the differential equation (2.2) for the oscillations of a shaft. Here: (11.6)

2

w (a)=

211:

J1 +J2 1 (" ~1TaJ 1 2

0

c(a

sin~) sin~d+.

To take a concrete case, suppose that M = c(e) is represented by the graph of Fig. 6, or more explicitly that

Fig. 6

c(e)

We find here

Jh+ke, e)o }h+ke, e(o.

15

16

INTRODUCTION TO NON-LINEAR MECHANICS 21f

~ c ( a s in

4> )

s in ~ d ~

4h + nka

0

and so by (11 .6): J1+J2 4h J J k( 1+nak).

( 11.7)

1

2

In order that this formula be applicable it will be clearly necessary that ~ be small. This quantity measures in a sense the deviation of M from linearity. (11 .8) Example 3. Take the case of the electrical2 circuit of (2.3) and rel2ted equation (2.5). Assuming B~ small, we find by (10.6): ( 11.9)

from which follows approximately: ( 11 .1 0) 12.

dx

We will now examine some cases where F con-

tains dt. (12.1) Example 4. Consider the equation of van der Pol (4.4) where the p~rameter t is assumed small. Comparing with the basic equation (6.3) we have here: 2dx

-(1-x )dt' As a consequence we find f(a sin~' av cos~)= -a(1-a 2 sin 2 ~)cos~ a2 a3 a( 4 -1 ) cos ~ - 4 cos 3 ~,

II.

ELEMENTARY THEORY OF THE FIRST APPROXIMATION

and therefore •

21l

rr

o

;' v S f(a sin, av ccs ¢)cos 1

2

2-rt

rrv S

f(a sin~'

0

av

d~

cos ) sin¢ d + =. Thus an arbitrary oscillation will tend to the stationary oscillation (12.7). The systems of the van der Pol type differ essentially from those of the conservative type with eq~ations: d

2

dt

~ + Y2 x +

t f ( x) = o.

Indeed in the conservative systems as we have seen there may occur steady oscillations of arbitrary amplitude whereas in the van der Pol system steady amplitudes are possible only for special values. Physically this is evident from the following considerations: since a conservative system neither dissipates nor creates

II.

ELEMENTARY THEORY OF THE FIRST APPROXIMATION

19

energy, oscillations once started have no reason to die down or to grow and so their amplitudes remain fixed. On the contrary in a "self-exciting" system there is creation as well as dissipation of energy and so the amplitude may increase if the source of energy provides more energy that there is dissipated or conversely. There will thus arise a fixed amplitude only if the two processes compensate. 13. (13.1) Example 5. As our next example we will take Rayleigh's equation (4.3). Here the function f of (6.3) will be ( 13.2)

Hence we have ( 13.3)

x

a sin (vt+e),

=

as our first approximation, with

.Y

=~ ,

e = canst. ,

and (13.4)

da

dt

=

-1

2nmv

2

S1tf(a" 0

cos t) cos~ d~.

However we find from (13.2)

and so from (13.4): (13.5)

da

dt

_2 2.12 = a2m(A-4Ba Y ).

20

INTRODUCTION TO NON-LINEAR MECHANICS

It follows from (13.5) that the trivial solution a = o will be unstable, since A)o, and so we have here a self-excited oscillation. The stationary amplitude satisfies

which yields ( 1 3. 6)

Whatever the initial amplitude a 0 (fo) we have from ( 1 3 • 5) : a(t)----.. 1\14A t--->+) +

X

32 3 (1
)

cos S(wt+d>),

where a, ill are arbitr&ry constants and w is given by ( 28. 7)

The, same method may be applied in an obvious wccy to

(28.8)

d 2x

dt 2 +

,,2 v

x + tf(x) +

c2 f 1 (x) + . . .

0.

Consider again the system (26.1) with f(x) power series in x: 29.

0,

( 29.1)

Here there is no small parnmeter c. However if we merely wish to consider small oscillations then clearly f(x) will be small with respect to v 2 and furthermore it consists of a series of terms of increasing orders of small magnitude. This justifies to a certain extent the following procedure. Replace (26.1) by (29.2)

2 2 + p2b x 3 + . • . = -d x + ,,2 v X + pb X 3 dt2 2

0

IV.

CONSTRUCTION OF THE HIGHER APPROXIMATIONS

where we will consider p

~,s

~c"

smo.ll p.",rameter.

47

This

eouc"tion is now solved as 'cefore CiS a power series in p , after which the parameter p is m~:.de equal to 1.mi ty, thc:s yielding Gn co.pproximate solc.