Asymptotic theory of resonance in conservative non-linear systems

Int. J. Non-Lmcar

Mechomcs.

Vol 6. pp. 479-493.

ASYMPTOTIC

Pergamon

Press 1971. Printed in Great Britam

THEORY OF RESONANCE IN CONSERVATIVE NON-LINEAR SYSTEMS R. I. KOVTOLJNE?

Association Euratom-CEA. Departement de la Physique du Plasma et de la Fusion Control&e. Centre d’Etudes Nucleaires. Boite Postale n’ 6, 92 Fontenay-am-Roses. France

Abstract-~Oscillations externally excited in a conservative non-linear system are examined with the help of the asymptotic method. It is shown that phase and amplitude variations with time. in a rather’broad range of physical and techni~~al problems. might be found explicitly. and that the solutions obtained reduce smoothly to those of the linear approximation, when the absolute difference of the system eigenfrequency and the external perturbation becomes large.

1. INTRODUCTION

general problem is discussed, where a conservative non-linear system is excited by an external periodic force with a constant amplitude. A non-linearity of the system is taken into account by writing the eigenfrequency w in a form of a series expansion: A RATHER

o(a) = wg + w,a + CO242iwhere a is a system oscillation

amplitude.

.. )

The perturbation

v(u) = \?o + \‘,U + V+l2 +

frequency

is also written

as

..,

since there exist problems in physics and engineering where the perturbation frequency depends upon the excitation state of the system. Let us discuss a case where the difference o(u) - v(u) might be written in the form : o(u) - v(u).= A, + d,u + Ll*d

’ T...

(1)

with d, = 00 - “0; We shall examine

separately

A, = co, - vl;

A, = co2 - v2;....

the case of A, = 0, where

o(u) - v(u) = A, f A,u2 i A,u2 +

.. ,

(4

since expansions of the type (2) are often met in physics and engineering [l]. An external periodic perturbation a,?& sin $ = a&, sin [v(u)t + t,] is assumed to be weak enough (E G 1) to justify the application of the asymptotic method [l] to the perturbed system equation X + 02(u) x = aEO sin [v(u) t + r,], where to is an initial

phase of the perturbation,

t On leave from U.S.S.R. Institute of Radiotechnics

and x is a state variable and Electronics

479

C

(Moscow).

of the system.

R. I. KWTOUNE

480

The purpose

of the present

work is to show :

(1) that the asymptotic method permits the study of development of resonant oscillations in a system starting from the zero amplitude. even though the second term of the phase equation has a pole at u = 0 ; (2) that the time dependence of the phase and amplitude might be found explicitly for a rather broad range of problems, for which series developments (1) and (2) are valid, and (3) that the results obtained reduce smoothly to linear theory solutions when (o(u) - v(a)\ becomes large. It is known that in the linear approximation of the system equation can be written x(t) =

x0

cos cot i

If (0-V) from (3)

x0

.

~~ sm it w

+

EEO

~02 - ,2

= : / 6 1 4 co and initially

(w = const, 1’ = const) the general

solution

sin (vt + co) - sin to cos 0X - L Cos to sin wt w 1 (3) the system

was at rest (x = 0. 1 = 0) we deduce

or x(t) e Here cos(ot

-t to) is multiplied

CEO ~ w6

t cos (ot + to) + O(6) I

by a slow time function EE, 6 u(t) = ~ - sin z t, W6

(4)

which is the amplitude.

2. FIRST ORDER

Let us go over now to the solution

EQUATIONS

of an equation

X + 02(a) x = &EOsin v(a) t with the help of the asymptotic

method,

where the zero-order

(9 solution

is sought in a form [l]

x(t) = a(&t) cos [Vt + @Et)]. Here 9 is a phase difference between system oscillations and the external U(G) and 9(&t) are functions to be determined which satisfy the equations du dt

EEO -- mcos9 w + 1’

(6) perturbation.

(7)

(8)

‘“2 -+ 1 (27)~6q (9) put? (s) u! 1 (D),I xeldal uoy3q.mvad aql alaqM am v u! paufvlqo ale

waga

uo!lypuoD

.waug-uou aq4 /IIuo q3y

pue ‘0 t

wnu

aM MON

aq ue3 qnsal

ZI uaqm uoyqos

1vql aas aM snyL

‘(t) u!wqo

.wauq

‘O> aseyd

~ag!u!

alues aql leyl aql 0~ uogye~l

aM %yx%alu!

olaz-uou

MOU ~oqs lDallo3

pgyt

ql!~ -

olu!

6 u!s JO anleA syl

e say

sn $27 ~a8 aM (6)

put2 (L) o~u! 8 sg

Buylnd

: lad &peal

uo!lcnba OB3 ‘0 - - c?m

amq am (6) suoypuo3

.“JE

alnlpsqns

aM

aM JI

= fj U!S

poylaur

u@ue.&?~

aql

dq uogenba

syl

%!AIOS

=~u~sq+sg;-~

!?“Z S’I! uall!lM

la] asodInd

&sea

aq UBD llnsal

s!ql .IO~ ‘(6) suo!~~puo3 9 =

(ISUOD =

‘(p) ynsaJ

ll’?qs a&

,z -

Jaauq

ayL ‘atug awugla

ayl (~HL)

01 (L) dq (8) uoyenba

[k?!l!u! ayl auInSSe aM J as??3 aql s! s!ql leyl

aql (33)x) .ro~ aA$! plnoqs m) lueisuo~

walsds

(&(L)

a.w (t pur! 0 J! .alowayl.In~

~pxus JoJ J! (L) ur0.q LI13a.up pauyqo

uoynlos

aq pIno

qflsal

aql

pur! leauq

*I-

awvs

= (o)fl

ayl

leyl

ap!Arp sn MOU ~oqs

s! (g) uo!lcnba

so3 aAeq aM 3 aug

uaas aq Qsva

uw

11

uroq

urelqo plnoys ah4 ‘sl3aga lgauq-uou vaI8au utx au0 pw aql uaqM ‘uoyel!Dxa malsils ayl30 a%ls Ai-rea ue le paapu1

~pms /ZlaA II~S SFD apnlqdm

(6)

‘0 =

2.t = (oh 1 0 =

6 UIS

anvy SAEMI~ aM ‘(9) w103 aqi 30 uoyrqos I? yaas akt 3! leyi ‘JaAaMoq uMoqs aq uw $1 .aIqmqddeu! aq 01 maas poy$auI ~!wduIAse aql pm ‘snonuguomp s! (8) uo!pnba 30 ap!s putq-l@y aql ‘[o = (())D] @p!u! ISa1 lx?SBM LualSiCS aql ‘.IaAaMoq JI .~pxus am I?p&p 1 pue I~pIvp 1 iE!yi 8uywnsst? [I]

PI u! pampap an (~HL) suoywnba ‘(g) Jo uoynlos laplo-olaz aqi la8 a.kt

wd pue (8) pur! (L) WOJJ (13)~ pug (13)~ pug aM 31 ‘[I] u! (p’s~) uownba

(9) oiu! myi

aas

R I

482

KDVTOUNE

However it is obvious that we can still seek a solution in the form (6) assumxg 9(O) = x -c <. Applying now the asymptotic method to the equation

only lhat

jl $ w’(u) x = i:E, cos &, sin \(a) t + eE, sin 5 cos ~(a) r, we readily

obtain du

a& ~ cos (9 - &J 0 i 1’

dt

It can be eaaly seen that by inneducing the inittai conditions (9).

3. SOLUTION

9, = JJ - 4” we return

OF THE FIRST

ORDER

SYSTEM

10 the system (7t(8)

with

(7We)

It will be seen below that in the system (7)-(S) one can put w = wg 1’ = I’~. in all the terms multiplied by E. If we now introduce dimensionless quantities s = o,t,

we have from (7)(S)

95 = -ECOSQ

(11)

d>

d9 ds

=:A -+ d,an + c12a2 +

+ 5 sin 9 %I

(12)

with A,&,) = A + dlun + d,a,2 T..... Drvrding

again (12) by (11) to eliminate

and by taking account

(13)

time s, we obtain

of (9) WC have sing=

-pa,-q+ru,3-,..

with

Putting (14) into (11) we obtain an equation with separate variables from which u(t) can be found. Substituting this a(t) IWO (14) we readily obtain s(t). In a great number of physics and engmeering problems d, in (13) IS assumed to be zero [I], [Z]. Th ere fore we shall discuss separately the case of d, # 0 and that of d, = (1

Asymptotic theory of resonunce in conservutive non-lineur systems

483

(1) The first case (d, # 0) According to (15) we have now q # 0. Assuming that )d, ) and 1d2 1 in (13) are of order of unity, let us examine the maximum values of a, as functions of the difference (w,, - vO). then we have p = (d/Zs) 9 1 and from the condition If IAl I(% - V,)/% I -1 pa, + qa.2 + . 1= 1sin 9 1 d 1 it follows immediately that uM = Max (a,) is of the order OflPV- s.Thus, the first term in (14) is of the order of unity (Ipa,1- 1) and all the others are -E, E2,. . ) and can be therefore neglected in the zero order approximation So we have, from equation (15) : A sin 9 = - pa, = - s a, = -

00

-

2E00

“(J

a,.

We see here that the amplitude dependence of w and v is neglected and sin 9 = const. a,, which is equivalent to the linear approximation discussed in the preceding paragraph. It is obviously a non-resonant case, when the amplitude is small (la,, - E) and non-linear effects are negligible. Resonance in the system takes place when Ip I is of the order of E+ or less (IpJ 5 s3). In this case we see from (14) and (15) that Ipa, 1 - I qa,f I - 1. However, the term I ru,”1 is of the order of E+ and can be neglected. So we have sin9

= -pa,

- qa,2 + O(E+).

(16)

We see that in the resonant case the amplitude a, is roughly l/J& times larger than the case far from resonance. Substituting (16) in (11) and separating the variables we have da,, Jr1

-

- (~a, +

W321

-sds.

(17)

Let us introduce f(p, q, a,) = 1 - (pa, 4 qajy

= (1 - pa, - qa,f) (1 i- pa, i qa,f).

(18)

It is easily seen that we can impose here a condition q 3 0 and discuss all the possible values of p, positive and negative. To integrate (17) first we must find roots ai of the equation f(p, q, a,) = 0 which are a1 = - &P

+ J(P2

c(2 = - Jm [p + J(p2 2q

+ Jq)],

- 4q)],

a3

a4

=

=

-

-

;;

;;

IIP -

CP -

J(P2

J(P2

-

+

4qf-J.

4dl

(19) I-

with J(p2 f 4q) = I J(p2 + 4q)l in all a,. It can be easily verified that ai < a2 < (Yj <

Cf.4

a1 > c12 > cI3 > CC4

P>O p < 0.

Cw

This is naturally true only if p2 > 4q, otherwise CI~and cls are complex values. Let us first discuss the large positive values of p(p2 $ 4q). Next, reducing p values, we shall examine the character of all the solutions possible.

484

R. I. KOVTOUNE

(19) that in this case M,IX~and c(~ are (a) p % J2q 3 0. It follows directly from equation negative and only ~1~> 0. Here CI,,varies between a3 and Q (or l/p and - 1‘p). Coming back to the dimensional values [see equation (lo)] and putting 1,;~ = 28/A, we have

which is in full agreement with (4) and which is quite natural, since for very large lp( we have sin 9 z -pa,, + O(E),i.e. the non-resonant regime. (b) p2 2 49. That is, p2 is of the order of 4q but remains still greater than it We easily see that here again clj d a, d ma. Now it is convenient to write (18) as follows f = - Y2(% - ai) (a, - Q) ((1, - Q) (a, - K$) and substitute

the c[~‘sfrom (19) we get

or

(21) f(P, 4, a,) = q2(cY2- x’) (x2 - j?),

where x = a, +

-c &7’

(2-q

It can easily be seen that 0 < p < x < M. Substituting (21) into (17) and integrating the equation

(23) in the limits ((Y,x) we have, according F[arcsin.

to the formula

/&I$),

J(I

- !g

(3.152.10)b of [3] : = -caq(s

- so),

(24)

where s0 = wt, should be determined from the condition a,(t) = 0 at t = 0. Here F(. .) is the elliptic integral of the first kind, with the modulus k = J[l - (P,a)‘]. Let us introduce for brevity z = mqs. It is known that the inversion of the equation is a Jacobi function (elliptic sine). Applying cx2 - x2 = (a2 - fl”) sn2 (z - zO)

F(cp, k) = x gives sin cp = sn x where sn x this transformation to (24) we obtain

or x = [cx’ - (a2 - /I’) sn* (z - rO)]+ = cc[l - k2 sn* (z - z,)]+,

(25)

and finally x = Mdn

(7 -

z,,),

i.e. 2 J(l

- k’) < x < a.

(26)

Asymptotic

theory ofresonc~nce in conservative

485

non-linerrr systems

Let us determine now z,, = axqs using the condition a,(O) = 0, or according to equation (22), x(O) = p!2q. Putting this x into equation (25) we find sn2 4,, = i. Then cn2 to = 1 -k2 sn2 to = 1 - )k2. Therefore, applying the formula (8.156) in [3] for dn ( u+ 18)we have from equation (26) x = ~ JU

- ik’)++--k2

sn 7 cn 7

(27)

1 -ik2snZT It is easy to show that this result reduces to (4) when p becomes means the transition to a linear case. Indeed we have then k2 =

1

large or q + 0, which

_!!!= _!t:-4~+!!+() 1

P2+ 4q

P2

sn 7 + sin z, cn z + cos 2, and equation

(27) is rewritten - $k2)+(l - k2 sn2 r.)+ + +k2 sn z cn z 1 - 3k2 sn2 t

I

+ )k2 sn t cn T + 0 Taking

account

of (28) we finally have 2 2 a,(t)+-snzcnz---smzcost P P

2 A = E-sin-s A 2

since ps A --t-=-_-s 2 2 according to (15) and (22). Thus CI, really tends to (4) when 1coo - v. 1becomes (c) p2 = 4~. Now B’ = 0 and integrating (23) we get

large.

CI

x

(29)

Ch (z - zO)

(d) p2 < 4q. In that case, /I2 < 0 and we put p: Integrating (23) again we obtain F [arcsin.J(l and after inversion

-~~).~~~~~j]=

= -fi’

> 0.

-ECtq(s

-So)

486

R I. KOVTOUNE

Thus in all cases we have considered we have found explicit expressions for x(t) = u,(t) + (p/2(1) = a,(t) _t (3&4d,) showmg that a, and x are periodic functions. In the case of p2 > 44 we see from equation (26) that the period is equal to 2K(k) = 2K{J[l - (fl”:~‘)]}. where K(k) IS a complete elliptic integral of the first kind of the modulus k = [ 1 - (/j2)/cr2] *. If p2 < 4q the corresponding solution equation (30) has a period 4K(k) = 4k[a *J( x2 -b”)]. But since t = ccxqs = ; J(p2 the same periods

+ 4qs) = 7

(T1 and T,) in dimenslonal

J(LP

+- y Ed&).

time units are

When k + 1 both periods 17;tend to infinity, and both solutions (27) and (30) tend to (29). It is easy to see that for negative p’s we get the same solutions for a,(t). A lack of space does not permit discussion here of the numerous particular examples which could be found for different combinations of E. d and d values. It is worthwhile. however, to discuss once more the limit of the validity of Ihe linear approximation. We know already that the linear theory can be applied if we consider only the linear term IJI a, on the right-hand side of equalIon (14), i.e. if 1 9 Isa; 1. In lhat case a, is of the order UT F and the preceding inequality means that p2 9 ~1or jd/ 9 ,:(~qj if we take I~IIO account (1.5). However, ifs and 1A 1are sufficiently small there exists a range of A values where

pa,,]

and in that range we are equally justified to apply both the asymptotic method (1 9 1A () and a linear approximation [I A ( B J(.zql)]. This fact explains a continuous transition from non-linear solutions to linear ones when 1A 1 - 1coo - 11~1 becomes large. (2) The second case (d, = 0) Now we have A,(a) = A + d,a* t- d,a3 + . . , and according sins= As in the above

case we immediately

-pa,

-ru,”

to (14)

-.,..

deduce from (31) the validity

011

conditmn

for the linear

approximation IP( 9 I+$

(AI + Jk2 (4 1).

Far from resonance, a,, is naturally of the order of E, but in the resonant region, where (pa, ( _ I ra,”1 - 1, we easily get Ia,\ rr E+((p I = ( A/‘~E/ - F*). We see that the resonant effect in case of (31) is stronger than is the previous case when we have had [anI - I-:). Let us substitute (31) into equation (11) and separate the variables :

(32)

Asymptotic theory of’resonmce

in conservutive non-linear systems

487

Introducing x = r*a,2, into equation

R = (p/r+), = $$

(33)

(32) we have

J{x[l

_

;‘;R

+

X)2]}

=

2EytdS:

or

(34) __~ J[x(x

- B1) (XE b2) (x - &)I

= 2Erf ds,

where /Ii’s are roots of an equation x(R + x)’ = 1. When the last equation has two complex to rewrite equation (34) in the form

Using the Cardan

method

X3

roots /12, 3 = m k in, it is convenient

conjugate

to determine

(35)

the roots of equation

- 2Rx2 + R’x - I = 0,

(35) rewritten

as (37)

we put into (37) x=$R+y

(38)

and obtain y3 + 3py + 2a = 0,

(39)

with p=

-$R2

a=&R3

-;.

(40)

Then for the roots of (39) we have y,=u+u Y2

=

-

+(u

+

v)

+

i 2J3(u + u)

Y3

=

-

$(u

+

u)

-

i 2J3 (u - ~1).

(41)

where

u = ;/ 0 =

[-u + J(u’

J I]-cJ

+

+ /I?)]

(42)

JCU’+ $)I.

(43)

R. I. KOVTOUNE

488

It is known discriminant

that the number of real roots of equation (37) depends 9 = 0’ + p3 which can be rewritten, using (40) as

9=aLet us separately root). It is convenient

According

discuss the solutions to introduce

0

R 3

3

-

for positive

(CJ+

the sign of the

$,.

and negative

(44) 9’s:

GS > 0 (a single real

here

to (44) and (45) 0 = - (a + 5’2)

and putting

=

upon

R = 3
- (2),

(46)

(46) into (42) and (43) we have 11 =

(5 +

+,,

L‘ =

(< -

5)‘.

(47)

To simplify the final results it is convenient to express all the variables of u and c. For example, by substituting u and zi from (47) into

of interest

in terms

X, = +R + u $ c we have immediately (48) In the same way we get

or. with the help of (38) and (41) we obtain x2.3

Comparing

=

-x~

-

$(ui- V) +

i$

13 (U - v) = - &(n f v) - 2 J(m)) +_ i $_ (U _ u).

this result to (36) we see that 171= - &A -t c) - 2 ,/(ua),

Let us now integrate

equation

(36) using the formula

I? = $(U

- t‘).

(49)

(3.145.2) in [3]. We obtain

(51)

Asymptotic theory

489

ofresomw~e in conservative non-lineur systems

and p2 = (in - xi)’ + n2, Inversing

cr2 = m2 + n2.

(50) we find sin [arc ctg j~(~~.~~x))]

where sn is an elliptic

=

2

_Jlpo(x,IX)l= po+o(x,

-x)

sn

(p

(53)

’

sine with an argument (s = w(g).

4 = 2&Y+&OS), Solving

(52)

(54)

(53) for x we get 2p(l - cn 4) - (p - g) sn’ 4 x = Xi~_~__ -~~~ ~I -2~--~, 4~0 + (p - .a) sn 4

where cn 4 is an elliptic cosine. We see thus, that x = r 3a,’ is a periodic period 4&C(k). The same period (T) in the dimensional time units is obtained from (54)

To simplify

the result (55) it is convenient y=

(55) function

with

to introduce

J(!)=(s&

Since in the present case 0 < < < m, we have for ‘1: - 1 d y < + 1. It will be shown that all the parameters of the solution (55) and those in (51) (52) and (54) can be easily expressed in terms of n(g). First we have r = ,:- _ 1. 2>

substituting

this [ into (56) and solving

the result for u we easily get

u = (1 - $-4.

Now directly

(57)

from (48) (49) and (52) we obtain Xi = u(1 - q)2. pz = 3u2(1 + 112+ q‘+), fJ2

=

u2

(58)

[(l + y)2 (1 i- V2) -t r’J

where u must be taken from (57). For example, (59) Let us now introduce

the functions

g(q) and H(q) (60)

R J KOVTOUNE

490

(61) Substituting

g(q) and H(Y) into (51) and (55) we get 2H( 1 - cn 4) - (1 - H) sn2 4 X=X1--d4H-(1 _f#sn2~

(62) (63)

The functions Y(q), H(q), k*(q), x,(q) and u-‘(q) are easily calculated. decrease in the following from - I to t 1 all these functions continually 0 < 9(v) s 4;

0 < u-yq-) d 1.59;

0 < k2(v) ,< 1 ;

When ‘1 changes ranges :

1 d H(vl)b4; 0 d x,(q) d 2.52.

Let us consider now a limiting case ( R 1 -+ co, when 4 + CC and q + I [see (45) and (56)l. In thus case the oscillation regime moves away from the resonant one. It can be shown that with g -+ CC the solution (62) reduces to the linear solution (4). Let us first consider the limiting value of x1 when q -+ 1. According to (59) and (56) we have in this case

(64)

Therefore.

(65) But (45) gives in the case of /R 1 + cc

(66) Putting

this < into (65) we get *3-‘-l Xl

3:” ’ (IjE

I)

1 =R’

The same limiting value for the “small” root can be obtained directly from (35). From (60) and (61) we see also that Y(q) -+ 0, H(q) -+ 1 when n + 1. According we then have k2 + 0, cn 4 -+ cos 4, sn 4 -+ sin C$and from (62) it follows that x-+x,.+(1

-cos~$)=,,sin

to (63)

1 .24

z

Let us determine now the limiting value of the phase 4 = 2cn*,lpas. we see thal p + CT+ 314when q -+ 1 and

see (54). From

(58)

Asymptotic

theory of resonance in consrrvutive

non-lineur systems

491

On the other hand, from (57) it follows that U = (1 - q)-+(l and if we substitute

+ II + +-+

account

- r/)-f

(64) and (66) we obtain u + 3-‘(35)*

Taking

-+ 3-+(1

= 4f = +R.

of (33) we finally have for 4 4

--t

:

2~r+Rs

=

As.

X and R by the corresponding

If we put now this 4 into (67) and replace (33) we obtain

expressions

in

i.e. the result of the linear theory. Let us go now to a case where CS < 0 (three real roots) Again we apply formula (3&(43) to solve equation (37). but now SP = and instead of (44) it is more convenient to introduce ;“=_Q+R3_$ and

Then we have from (42) and (43) U= Moreover

(5 + ii)‘,

u =

(5 - ii)+.

(6%

from (68) it follows 3 R = 2 $($ + i2)

and from (38) we have xi = 2 $(a + i2 + _VJ. If we introduce

yi from (41) and take account xi = & $2 X2 =m3[ x3 = s-,1 (cos cp)#

after some transformations

of (69) we obtain

[l - cos (60” - 5 cp)]

(70)

1 - cos (60” -i- f cp)]

(71)

+ cos$cp],

where cp is determined

(72)

from the formula (73)

492

R. I. KOVTOUNE

To find some particular solution one has to first determine R from (33) and put it into (68) to find i. and then one can easily determine from (73) the angle cp, which immediately gives all the roots X, by virtue of (70)-(72). Let us integrate now (34) using formula (3.147.2) in [3]. This gives (74) where (75) Inversion

of (74) gives, after some transformations, (76)

with

Thus, x = r*ai is again a periodic

function

with a period 2K(k) in 4 or

Wk) -

T=-,

-~~-

E’WO \J$2(P3

-

m

in time t. Let us consider now the limiting case. R + ,x! when < 4 z and cp + z 2. [see (68) and (73)]. If we introduce x = 71.2 - cp cz--f 0 when i: + x. then we have directly from (73) 1 1 cos2 cp = i-t. _+ .__ 5 + c2 (21?J2

and 1 cos (90” - cp) = sin ‘z --f IX2 2, (77) Substituting

cp = 90

- x into (70) and taking

account

of (77) we have

However, according to (68) we have for large R’s: I; r (R’3)‘: therefore /j’, > 1 R2. as should be expected directly from (35). In the same way we obtain for the “large” roots B2 and P3

Therefore.

far from the resonant

frequency

we have

Asymptotic theory of resonance in conservutive non-lineur systems

493

and from (75) it follows that k + 0. Then from (76)

(78) where 4 = U+

JIIa2tfl2

Putting this phase in (78) and replacing solution. AcknowledgementsPI should like to thank work and for their encouraging remarks.

- /c?~)s]+ wtRs

= 3A.s.

X by rtu,, [see (33)] we finally

Drs. Blaquiere

and Cotsaftis

for the interest

obtain

the linear

they have shown in this

REFERENCES [I] N. N. BOGOLIUBOV and Y. A. MITROPOLSKI, Asymptotic Methods in rhr 7J~ory of Non-linear Oscillations. Fizmatgiz, Moscow (1958); English translation, Gordon and Breach, New York (1961). [2] Y. A. MITROPOLSKI, Problems of the Asymptotic 7heory of Non-linerrr Oscillations. Fizmatgiz, Moscow (1964); French translation, Gauthier and Vlllars. Editeur. Paris (1966). [3] I. S. GRADSHTEYN and Y. M. RYZHIK. Table of‘Inteyrals, Series und Products. Academic Press. New York and London (1965).

(Received 13 August 1970)

R&umLOn examine B I’aide de la mCthode asymptotique les oscillations excitkes extirieurement de systbmes entretenus non lintaires. On montre que les variations de la phase et de l’amplitude avec le temps pour une gamme assez &endue de problkmes physiques et techniques peuvent &tre trouvCes explicitement et que les solutions obtenues se rtduisent de mani&re continue g celles de l’approximation lintaire lorsque la diffkrence. en valeur absolue. de la frequence propre du systkme et de la perturbation externe devient tlevCe.

Zusammenfassung-Von aussen angeregte Schwingungen in einem konservativen nichtlinearen System werden mit Hilfe der asymptotischen Methode untersucht. Es wird gezeigt. dass zeitliche Phasen- und Amplitudenschwankungen in einem ziemlich grossen Bereich physikalischer und technischer Probleme explizit gefunden werden k(innen und dass die erhaltenen LGsungen sich ohne weiteres auf diejenigen der linearen NBherung reduzieren lassen, wenn der absolute Unterschied zwischen der Systemeigenfrequenz und der Busseren Stiirung gross wird.

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