General perturbational solution of a harmonically forced non-linear oscillator equation

Int. J. Non-Linear Mechanics. Vol. 8. pp. 523-538.

Pergamon Press 1973. Printed in Great Britain

G E N E R A L PERTURBATIONAL SOLUTION OF A HARMONICALLY FORCED NON-LINEAR OSCILLATOR EQUATION M. M. STANI~I(~ and J. A. EULm~ School of Aeronautics, Astronautics and Engineering Sciences, Purdue University, Lafayette, Indiana 47907, U.S.A. Abstract An extension of a general perturbational method in the theory of harmonically forced non-linear oseillati0ns has been presented, in which the interplay of two system parameters appears. The method clearly exhibits the entrainment of subharmonic, super-harmonic and other harmonic responses for various interplays of the system parameters. The mathematical insufficiency of the method to predict the behavior of the system in a limiting case of parameter interplay, which is usually attributed to the perturbational method in the Poincar6 sense, has been recognized and the method for its removal suggested.

I. I N T R O D U C T I O N

IN HIS paper [1] Kane showed that the motion of a broad class of non-linear mechanical systems is governed by the equation + ~2x + # t x 3 + #2x ~2 = N cos ~ t

(1)

where x is the responm of the system; to, fl,/~1, #2 and N are appropriate dimensional constants and a dot denotes differentiation with respect to time, t. Such a system can be illustrated by a harmonically driven oscillator as shown in Fig. 1. Here mass M 1 moves horizontally and M 2 vertically, and both are connected with a light bar of length L. The mass M t is attached to a linear spring S(k), where k is the stiffness. Static equilibrium occurs when L'is vertical, x is the displacement of mass M t from static equilibrium and h is the vertical distance of mass M2 from the horizontal position with respect to mass M1. It is assumed that x ~ L; furthermore, x = 0 implies h = L. It is worthwhile to write Equation (1) in the dimensionless form q,, + q + elq3 + e2q q,2 = COS 2Z

(2)

by setting q=--

x

(3)

x

(4)

"t" = O)t

#tH 2

•

N

~1 = --'(.02, ~2 = ]~2N2; 2 =--'(D, ~ ~-- (./)2

where the primes denote derivatives with respect to ~. Furthermore, for 2 = 1, 2 = n, 2 = l/n, n = 2, 3, 4 , . . . harmonic, and superharmonic responses, respectively. 523

(5)

we obtain the harmonic, sub-

524

M . M . STANI~I(~and J. A. EULER

I

I N~s~t

'~

\M 2

FIG. 1. The geometry of a nonlinearoscillator.

The solution of equation (2) can be obtained by means of classical methods [2, 3, 4, 5]. These methods have been devoted mainly to the analysis of periodic and almost periodic solutions and their stability criteria as the most important characteristics in the analysis of a non-linear system. Needless to say, these classical analyses are limited in their description of global characteristics of an intricate resonance problem. In this paper, the general perturbational method of Struble [6] will be extended in order to better illuminate the behavior of the system under consideration. The advantage of this method when compared with classical techniques, lies in the fact that it exhibits not only resonant and nonresonant responses, but also other characteristics of a non-linear system, including the interplay of system parameters and the entrainment of harmonic, subharmonic, superharmonic and other harmonic responses. In addition, since the method of Struble incorporates two classical techniques, i.e. a variation of parameter method and a perturbational method, it has the main advantages of both methods while avoiding the main difficulties of each. Thus a more complete picture of the behavior of a complicated non-linear system can be obtained. For example, by means of this method it is possible to better understand the phenomenological nature of turbulent motion. 2. M E T H O D O F S O L U T I O N

In this paper we examine the " h a r d " forced case, i.e. the non-autonomous system given by Equation (2), where the magnitude of the forcing term is not small. We note that in Equation (2) we have the interplay of two system parameters, ~1 and ~2- Assuming ~x < 1 and s2 < 1, the solutions are thought of as given in terms through the first power in el and e2 by expressions in the form q(T) = A cos (z - 0) + B cos Az + ~101 +

~2fl

(6)

where A = A(x, ex, ~2) and 0 = 0(x, el, e2) are functions of the Kryloff-Bogoliuboff type. In addition, gt and f l depend on • only and B = 1/(1 - ,~2).If the expression (6) is substituted into (2), we obtain (A" + 2 0 A ' - AO '2 + C O cos (~ -

O) + ( A f t ' - 2 A ' + 2 A ' 0 3 sin (z - 0) + ex(g~ + gl)

+ 82(f~ + fl) + C2 cos 3 (x - 0) + C3 cos )~ + C , cos 3Ax + Cs cos [(1 + 2,~)x - 0]

525

Generalperturbational solution of a harmonicallyforced non-linear oscillator equation

+ C6 cos [(1 - 22)~ - 0] + C7 cos [(2 + 2)'c - 20] + Cs cos [(2 - 2)x - 20] = 0 (7) where C1, C2 . . . . .

Cs ar~ functions of the system p a r a m e t e r s 81 and 82 and are given by C1

~ A 8a(a 2 + 2B 2) + ¼A82(A 2 + 2B222)

C2 = ¼ A 3 (81 -

82)

C ~ - - 3 B 8 1 ( 2 A 2 + B 2) + ¼ B 82 (2A 2 + B222) C 4 - - ¼ B 3(81 - 8222)

cs

= ¼ AB ~

[381

-

C6 -~ ¼ A B 2 [381 +

8,+(22 + 2~3] 82

(22

-

22)]

C7 = ¼ A2B [381 -- 82 (22 + 1)]

(s)

Cs = ¼ A2B [381 + 82 (22 - 1)].

E q u a t i o n (7) shows that to the first order in 81 and 82 there are three resonance possibilities : A near 3, 2 near ~ a n d 2 near 0. T h e y c o r r e s p o n d to subharmonic, s u p e r h a r m o n i c a n d other h a r m o n i c resonance p h e n o m e n a , respectively. We will discuss each case separately. 3. SUBHARMONIC RESONANCE F o r 2 near 3, the nearly resonant term in coefficient Cs can be expressed in the form Ca cos [(2 - 2)~ - 20] = Ca {cos [(2 - 3)x + 30] cos (~ - 0) -

sin [(2 - 3)z + 30] sin (~ - 0)}

(9)

Substituting E q u a t i o n (9) in lieu of the last term in equation (7), a n d separating corresp o n d i n g terms of the f u n d a m e n t a l h a r m o n i c from the other h a r m o n i c s we o b t a i n : (a) Variational equations : A"+2AO'-AO '2+C1 +Cscos ~'=0 Aft' - 2A' + 2A'O' - C s sin ~ = 0

(10)

• ' = (2 - 3)~ + 30.

(11)

where

(b) Perturbational equations: g~' + gx = - ¼ A3 cos 3(~ -- 0) -- ¼B(2A 2 + B 2) cos 2~ - ¼ B 3 cos 22~ - ~ AB 2 cos [(1 + 22)~ - 0] - ¼ A B 2 cos [(1 - 22)z - 0] ¼ A2B cos [(2 + 2)z - 0] -

(12)

f ; ' + f l = ¼ A3 cos 3(z - 0) - ¼ B(2A 2 + B222) cos 2z + ¼ B322 cos 32~ + l A B 2 (22 + 22) cos [(1 + 22)z - 0] - lAB2 (22 - 22) cos [~1 - 22)z - 0] + ¼ A2B (22 + 1) cos [(2 + 2)~ - 20].

(13)

O u r interest is focussed on the variational system, equation (10). After the determination

526

M . M . STANI~I(~ a n d J. A. ELrt,ER

of A and 0 no difficulty will exist in evaluating the perturbational solutions, since the forcing function terms [-on the right side of (12) and (13)] are completely specified. The variational equation (10), to the first order in el and ~2, can be written in the following reduced form 2Aft= -C 1-Cscos 2A' = - Cs sin ~'.

(14)

Moreover, 0' = }

-

(2 -

(15)

3)].

Therefore, equations (14) and (15) lead to d~ =(2-3) dz dA dz

3 C1 2A ½

3 Cs cos ~, 2A

(15)

Cs sin ~'.

Hence, d~v

2A(2 - 3) - 3Ct - 3Ca cos - A C s sin

dA

(17)

The variational system, equation (17), is exact and possesses the integral A 3 c o s ~r/ _

4{[-(2 - 3)

- ~81 B 2 -

~-82B222]A2

-

T~(3et

e [ 3 e t + 82(22 -- 1 ) ]

+ 8 2 ) A 4} =

K

(18)

where K is the integration constant. Defining the Cartesian coordinates a = A cos

(19)

b = A sin ~v equation (18) can be written as ~ A * + a A 2 - flA = K

(20)

where ¢X=

3

(3el + e2)

4 BE3el + 82(22 -- 1)]

fl = 4[--(2 - 3) - ¼B2(aet + ~222)] B [ 3 e l + 82(22 - 1)]

(21)

Moreover, equation {19) leads to da dz

dA _ ~d~g cos ~' A sin dz dz

db dA ~ d~V d--~ = d-~ sin ~ + A cos dx

(22)

Generalperturbational solution of a harmonicallyforced non-linear oscillator equation

527

or, da

--

dA

=

~

cos

b d~ dz

~e -

d'c

(23)

db dA . d~ d--~-= d-~¢"sm ~' + a d z" Hence, the criteria for the critical points in the cartesian system (a, b) requires that cos

~-

~v -

d~V = ~

Lsin ~'

[:]

(24)

.

Therefore, either cos ~e

-~

sin ~'

= 0 a

i.e. tan ~v = _ b

(25)

or

d~

= 0

i.e. A = A o a n d ~P = ~ o .

(26)

F r o m equation (25) it follows that A -- 0 is one of the critical points. In addition, from equation (26), we obtain the critical points given by sin ~' = 0 and the real roots of the equation (2 - 3)

3 C~ 2A

3 Ca = 0. 2A

(27)

E q u a t i o n (27) c a n b e w r i t t e n as 4 ~ A 2 + 3 A - 2fl = 0.

(28)

The roots are 3 __. ~/(9 + 32y) Al'2

=

--

(29)

80c

with =

~fl.

(3o)

528

M.M. STANI~I(~and J. A. EULER

F r o m equation (29) it follows that for the roots A t and A2 to be real we must have y >i - 9

(31)

In addition, if these roots are to have the same sign, then y < 0. Therefore, in order that equation (28) have two real roots of the same sign we must have - 9

~< ~ < 0

(32)

Consider now the case of the integral curves, equation (20), when K = 0. Hence, equation (20) becomes A 2 ( ~ A 2 + a -- ~ = 0.

(33)

Clearly in this case we have the two curves A = 0

(34)

• A 2 + a - fl = 0. The second of equations (34) represents a circle on the a-axis with center at the point a = - 1/2~, and with radius r - x/(1 + 4y) 20~

(35)

We see that as K --, 0, the trajectories of the integral curve reduce to a finite radius if >/ - ¼

(36)

and we can clearly consider the following regions : (i) r ~ < -

9

(iO-9
- ¼

(iv) - ¼ < y < 0 (v)r = 0 (vD ~ > 0

We note that by equation (29) y = 0 implies ~ = 0 or/~ = 0; or both ~ = 0 and/~ = 0. However, from a phenomenological point of view our interest is now restricted to the case of y = 0 when ~ # 0. The case of y = 0 for 0~ = 0 will be discussed separately.

(a) Discussion of the regions under consideration Region I ? <~ - 9 . In this region, equation (27) has no real root; hence, the only critical point is a center at the origin, A = 0 (see Fig. 2). As K approaches zero, the integral curves will converge to the point solution A = 0. This means that the first "homogeneous" part of the mean solution is vanishin~ In fact, this is the case of steady vibrations of the nonlinear oscillator. No integral curve will exist for negative values of K. Region II - ~ < y < -¼. For this region we have two real roots of the same sign. One is a saddle point and the other a center. The integral curves for various values of K are shown in Fig. 3. After the curves pass through the separatrix and break into ovals about the centers, we have that, for some positive K, the curves about the center not at the origin collapse to a point while the curves remain finite about the origin. As K approaches zero, the curves about the origin also collapse to a point. No curves exist for negative values of K.

Generalperturbational solution of a harmonicallyforced non-linear oscillator equation

^

~

529

r--o.a

FIG. 2. Subharmonic region for 7 ~ -~92-.

Region III

7 = -¼. In this region the trajectories for various values of K are illustrated in Fig. 4. The integral curves in this case differ from those in Region II in that the curves about both centers converge to points simultaneously as K tends to zero. Region IV-¼ < 7 < 0. The graph of the curves for this region is presented in Fig. 5. After the curves pass through the separatrix and break into two ovals, the curve about the origin collapses to a point as K approaches zero. However, the curves around the second center converge to a circle with radius r = x/(1 + 4y)/2~.

----Y/ I I /

J

/¢

/

- - 1-0565

,8-0"2514 y

FIO. 3. Subharmonic region for _ 9 < 7 < - ~ .

=-0-2656

530

M . M . STANr~It~and J. A. EULER 0.02

6b

\

FIG. 4. Subharmonic region tor 7 = - ~ .

F o r negative K the curves break away from the circle and collapse to a point for some negative value of K. Region V ~ = 0 for fl = 0. For ), = 0 one of the roots of equation (27) is equal to zero and a double root exists at the origin, i.e. a saddle point which cancels the center. The only critical point that remains is the center away from the origin, (see Fig. 6). As K approaches zero the family of integral curves will converge to a circle passing through the origin of radius 1/2~. The curves for negative values of K are similar to those of Region IV. Region V/~ > 0. Figure 7 illustrates the integral curves of this region for various values of K. The two real roots in this region will have opposite signs so that the saddle point is no longer between the two centers. Unlike the previous regions, as K approaches zero the

6b

I

I

143 7" =-0.125

1~6. 5. Subharmonic region for - ~ < ~, < 0.

General perturbational solution of a harmonically forced non-linear oscillator equation

531

P~

K~O

a=-1-1249 FiG. 6. S u b h a r m o n i c r e # o n for y = O,/7 = O.

integral curves will now collapse to a circle that encloses all three critical points. F o r K negative we have two sets of curves occurring simultaneously. One set consists of the curves that break away from the circle corresponding to K equal zero and converge toward the separatrix. The other set are curves that expand away from the origin to the separatrix. When these curves meet at the separatrix, they combine into one curve that encloses the center away from the origin. As K becomes more negative these curves converge to this center. (b) Special case y = 0 f o r ~ = 0

Figure 8 represents the integral curves in this case. These curves can be considered to be the limiting case of the curves of Fig. 5 as 0t approaches zero. In order to understand this,

'>o / t ~ o

i

::-o0:.o:

\'\\ FIo. 7. S u b h a r m o n i c r e g i o n for y > O.

532

M . M . STANI~I~ a n d J. A. EULER

K>O

1 /t 3

60 4

FIG. 8. S u b h a r m o n i c r e g i o n tor ~ = 0, ¢ t = 0 .

it should be noticed that as a tends to zero one of the roots of equation (27) will converge to ¼/~, while the other goes to infinity. Therefore, Fig. 8 represents the integral curves for the case where one of the centers is at infinity. It then follows that the curve for K = 0 is the straight line, a = fl, while the curves for K negative and positive are to the right and the left of this line, respectively. Note that in all other previously discussed cases the integral curves were closed. However, in this case we see that all curves are open with db/da approaching o0, (A ~ oo and ~' ~ ~/2) except for the curves about the center at the origin. Hence, as a ~ 0, i.e. along the line az = - 3~1, we obtain a "pseudo" resonance independent of 2, which cannot be interpreted physically. Mathematically speaking, this implies the absence of convergence of the solution, equation (6), along the line ~2 = - 3~1. Physically, however, the theory fails to predict amplitude and phase for long period perturbations uniform in time for a specific interplay of the parameters ~ and ~zThe hidden difficulty, as indicated by Lighthill [7], is a direct consequence of the perturbational method of Poincar6 attempting the solution uniform in time by means of a power series expansion, even in the case of a perturbation parameter. However, the Lighthill method is designed to eliminate such difficulties by means of a power series expansion in 8~ and ~2 not only of the dependent variable q, equation (6), but also the independent variable ~. Such a procedure, as shown by Tsien [8], will extend the range of the convergent method, but requires further investigation and will not be presented in this paper.

(c) Effect of ~2 on the solution The effect of 8z is illustrated in Fig. 9, which represents a plot of ~, vs t$ = e2/a~. F o r positive values of 6z the domain to the right of the ~-axis represents a hard spring and that to the left a soft spring. When ~2 is zero, ~, will be the 7o, which corresponds to the Dulling equation. The regions discussed previously for various values of ~ are also indicated on this figure. Clearly, ~ is highly dependent on t$ for small values. For t~ positive y quickly approaches the region ~ < - 7~r until at 6 ~ 1 its dependence on b becomes small. For

Generalperturbational solution of a harmonicallyforced non'linearoscillator equation

533

6 = [ - 3/(22 - 1)], y possesses a singularity due to the fact that the coefficient Cs becomes zero at this point. H o w e v e r , this region can be excluded from consideration since as a p p r o a c h e s - 13/(22 - 1)], C8 b e c o m e s of second order, a n d therefore, would not be included in the first order equations. M o r e o v e r , 6 = - 3 corresponds to • = 0, which was

r,o

¢ -32 a ~ 2" _ I -4 <_O_

FIG. 9. Effect of the parameters on the solution.

discussed previously (see Fig. 8) while/~ = 0 is represented by the zero of 7 near the origin. F o r large 6, either positive br negative, y will a p p r o a c h the value of - 43-as can be seen as a limiting case of equation (30). Thus, the effect of 52, except for small values with respect to 51, will be to shift the phase plane curves to the region ~ < - 9 . 4.

S U P E R H A R M O N I C RESONANCE

When the input frequency 2 is nearly ~ the t e r m C , cos 32, in equation (7) becomes nearly resonant. H o w e v e r . this t e r m can be expressed in the following f o r m : C , cos 32, = C,{cos [(32 - 1) • + (9] cos (, - (9) -

sin [(32 - 1) ~ + 0] sin (, - 0)}

(37)

Therefore, the variational equations reduce to A" + 2AO' - AO '2 + Ct + C4 cos ~' = O

(38)

A0" - 2A' + 2A'0' - C , sin ~ = 0 where ~, = (32 - 1 ) , + 0

(39)

T h e p e r t u r b a t i o n a l equations are given by g't + 01 = -- ¼A 3 cos 3(, -- 0) - ¼B(2A 2 + B 2) cos 2, -

-~AB 2 cos [(1 + 2 2 ) , - 0] - ¼AB 2 cos [(1 - 2 2 ) , - (9] -

i A 2 B cos [(2 + 2) • - 20] - -~A2B cos [(2 - 2) • - 20]

(40)

534

M . M . STANI~I(~ a n d J. A. EULeR

f ~ + f l = 1A3 cos 3(x - 0) - 1B(2A 2 + + ¼ A B 2 ( 2 2 + 22) c o s [(1 + 22) ~ -

B222)cos 2z

0] - ¼ A e 2 ( 2 2

- 2 ~) c o s [(1 - 22) • -

0]

+ 1AZB(22 + 1) cos [(2 + 2) z - 20] - ¼AZB(22 - 1) cos [(2 - 2) • - 20]

(41)

T h e variational system to the first order in 51 and 52 leads to the reduced equations d9 ~ = 2A(32 - 1) - C1 - C4 cos ~P d'c'

dA -dz -

2A

(42) =

- ½C4 sin 97.

Evidently, d~' 2A(32 - 1) - C1 - C4 cos 'P = dA - C4 sin ~P

--

(43)

T h e exact form of the integral curve is A cos ~P --

-1 B3(51

{1A4(351 + 52) + A2[B2(351

+

5222) -- 4(32

--

1)]} + K *

(44)

5222 )

where K * is the integration constant. E q u a t i o n (44) can be written in Cartesian coordinates (a, b), as ot*A 4 - f l * A 2 + a = K *

(45)

where 0C~

1 35x + 52 4 B3(51 - 5222) B2(351 + 5222) - 4(32 - 1)

(46)

B3(51 -- ,B222) M o r e o v e r , the critical points, by virtue or (26). E q u a t i o n (25) leads to a crtical of e q u a t i o n (42) we have that d ~V/d~ and point. F r o m e q u a t i o n (26) we have that the of the e q u a t i o n

of equation (42), are given either by equation (25) point at A = 0. However, if A = 0 then by m e a n s d A / d z tend to infinity. Hence, A = 0 is not a critical critical points are for sin 97 = 0 and the real roots

A3 - - ~p* A

+ ~ =10

(47)

E q u a t i o n (47) possesses either one real root and two conjugate complex roots, three real roots of which at least two are equal, or three different real roots, depending on whether the quantity

is positive, equal to zero, or negative, respectively.

Generalperturbational solution of a harraonicallyforced non-linear oscillator equation

\

\

I,, I

\

I \

\ \ \~ ,..

~

535

,YJJ "

.,///

..///

,,*=o.68o7

e.oo

FIG. 10. Superharmonieregion for I fl~**)3 + ( 1~ ), 2 > 0.

The form of the integral curves in the phase plane for one real root, i.e. (fl*l~*) ~ + (1/4~*) 2 > 0 are illustrated in Fig. 10. Fig. 11 shows the integral curves in the case

(rS~ (~.)2= o \~l +

i.e. three real roots of which two are equal. The curves enclose the non-equal singular point. The two other roots which are equal, correspond to a saddle point and a center, which cancel each other.

/6/

/

~° "°'

o.I ~

\\

~([////~/" ///// \X,, (~r~ ~+ FIG. 11. Supcrharmonicregion for ~6~]

o--,,~ (~-,)' = 0.

M. M. STANI~Idand J. A. EULmt

536

Finally, +

02


corresponds to three different real roots two of which are centers and the other a saddle point as indicated in Fig. 12.

4 I'!

K~

5a

Fro. 12. Superharmonic region for (fl*~ 2 (_~ 2 \6~*) + \4~*} <0.

N o t e that all phase curves illustrated in this paper represent only a qualitative behaviour of the system; however, for those interested, the values of the parameters used are indicated o n the curves.

5. O T H E R

RESONANCE

PHENOMENA

W h e n the input frequency 2 is near 0, then the terms at Cs and C 6 b e c o m e nearly resonant. In this case we have to express both terms as a p r o d u c t of fundamental harmonics. It is easy to show that Cs cos[(1 + 22) • - 0] + C6 cos [(1 - 22) ~ - 0] = ½AB2(3el - e222) cos 22~ cos (~ - 0) + e22AB ~ sin 22x sin (z - 0)

(48)

Hence, the variational equations b e c o m e : A" + 2A0' - A0 '2 + C1 + ½AB2(3el - e222) cos 22~ = 0 AO" - 2A' + 2A'0' + ~2AB22 sin 22~ = 0

(49)

T h e p e r t u r b a t i o n a l equations are g~ + g l = - ¼A3 cos 3(~ - 0) - ~B(2A 2 + B 2) cos ;t~ --

¼Ba cos 32~ -- ~ A 2 B cos [(2 + 2) z - 20] - ~ A 2 B cos [(2 -- 2) z -- 20]

(50)

General perturbational solution o f a harmonically forced non-linear oscillator equation

537

f ~ + f~ = ¼Aa cos 3(x - 0) - ¼B(2A 2 + B222) cos 2z + ¼Ba22 cos 32x + ¼A2B(22 + 1) cos [(2 + 2) x - 20] --

¼A2B(22 -- 1) cos [(2 - 2) x - 20]

(51)

Evidently, the variational equations to the first order in e, and e2 become: 2AO' = -

C1 - ½AB2(3el - e222) cos 22z

2 A ' = ~2AB22 sin 22z

(52)

F r o m the second of equations (52) it follows that A = A o e x p ( - ¼~2B2 cos 22x)

(53)

where A0 is an integration constant. Expanding equation (53) in Taylor series we obtain the value of A to the first order as A = Ao(1 - ¼82B2 cos 220

(54)

Evidently, the first of equations (52) leads to 0

=

0o

-

3 2o + 2B 2) + ~g2(Ao 1 2 + 2B222)] t - ~-~ B2 (3et - e222) sin 22x []el(A

(55)

Further investigation for other resonance phenomena could be considered along the same lines as those for subharmonic and superharmonic responses~ However, for the sake of brevity, this part of the analysis will not be presented in this paper. Acknowledgements--The authors of this paper are indebted to Dr. R. N. Iyengar and Dr.M.A. Zoeller, from Purdue University, for their helpful criticism during the progress of this work. 7. REFERENCES [1] T. R. Kxr,m, Free and forced oscillations of a class of mechanical system. Int. J. non-linear Mech. 1, 157-167 (1966). [2] N. IOtYLOrr and N. BOC_,OLIO~O~,Introduction to Non-Linear Mechanics Princeton University Press (1949). [3] W. RITZ. Uber eine neue Methode zur Losung gewisser Variationsprobleme der Mathematische Physik, J. f reine u. angew. Math. 135 (1908). [4] M. M. STAI~V_~lt~.Free vibration of a rectangular plate with damping considered; Q. Appl. Math. 7, No. 4 (1955). [5] N. N. BoGoutraorF and Y. A. MrntoPotsKY. Asymptotic Methods in the Theory o f Non-linear Oscillations; Hindustan Publishing Corp. (India)(1961 ). [6] R. A. STRtraU~. The geometry of the orbits of artificial satellites, Arch. rational Mech. Anal. 7 87-104 (1961). [7] M. J. LxGrrrmLL. A technique for rendering approximate solutions to the physical problems uniformly valid; Phil. Maa. 7, 40, 1179 (1940). [8] H. S. TsmN. The Poincar6-Ligbthill-Kuo Method ; Advances in Applied Mechanics, Vol. IV, 281-349, Acad. Press, N.Y. (1956).

(Received 13 April 1972)

538

M . M . STANI~I~and J. A. EULER

R6sum~-On pr6sente tree extension de la m6thode g6n6rale de perturbation darts la th6orie des oscillations no~ lin6aires fore6es par tree fonetion harmonique; dans cette extension apparait l'interaction de deux param6tres dl syst~me. La m6thode montre elairement rentrainement des r~ponses sous-harmoniques, sur-harmoniques e autres harmoniques pour diverses interactions des param~tres du syst6me. On reeonnalt l'insuffisanee math6matique de la m~thode pour pr&lire ie comportement du syst~me dans m eas limite d'interaetion des param~tres qui est d'habitude attribu6/t la m~thode de perturbation au sens de Poinear6 on sugg~re une m6thode pour la lever. Zusammeafassang--Eine Erweiterung einer allgemeinen Perturbationsmethode in der Theory harmonisc] erzwungener nichtlinearer Schwingungen wird dargestellt, in welcher die Wechselwirkung zweier System parameter auftritt. Die Methode verdeutlicht das Mitschleppen yon unter- und fiberharmonischer und andere harmonischer Reaktionen f'tir verschiedene Wechselwirkungen der Systemparameter. Die mathematische Unf'fihigkeit der Methode das Systemverhalten f'tir einen Grenzfall in der Parameterwechsel wirkung zu beschreiben, was gewfhnlich der Perturbationsmethode im Sinne Poincares zugeschrieben wird wurde erkannt und ein Verfahren f'tir deren Beseitigung vorgeschlagen.

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