Periodic oscillations of a class of non-autonomous non-linear elastic continua

Int. J. Non-Linear Mechanics.

Vol.2, pp. 331-342. PergamonPressLtd. 1967.Printedin Great Britain

PERIODIC OSCILLATIONS

OF A CLASS OF

NON-AUTONOMOUS NON-LINEAR ELASTIC CONTINUA C. E. MCQUEARY Bell Telephone Laboratories, Incorporated, Whippany, New Jersey

and L. G. CLARK Department of Engineering Mechanics, The University o f Texas, Austin, Texas Abstraet--A modified perturbation method for obtaining periodic solutions to a class of non-autonomous non-linear partial differential equations is developed. The classic small divisor is discussed in detail and a general method for its elimination is presented. New terminology is introduced for the purpose of discussing forcing functions that produce in a system a response that is of the same form as a non-linear periodic mode for the same system. Specific examples are examined to verify the results of this work.

INTRODUCTION

TI~ purpose of this work is to develop methods for obtaining periodic solutions to a class of non-autonomous non-linear partial differential equations. The methods developed are in effect modifications of those presented in previous work [1] concerning autonomous non-linear partial differential equations. Although these methods of solution are reasonably general, the problem of forced oscillation of elastic continua on non-linear elastic foundations will be emphasized. Modifications of the general methods of solution presented herein yield a technique for successfully treating the classic small divisor difficulty [2, 3] as it arises in the general class of non-linear partial differential equations analyzed in this work. GENERAL F O R M U L A T I O N

In this section a method is developed for obtaining periodic solutions to a class of non-autonomous non-linear partial differential equations (hereafter designated NLPDE). We consider a class of problems described by an equation of motion of the type

- ~r~(~) + - ~ - + ~

+ / ~ ( ~ ) = p(~)cos~t

(1)

where ~ ~ ~(~, t) is the dependent variable. The independent variables are the spatial variable ~ (2 is defined over a finite domain) and time l. The constants ~ fl, and/~ are real and positive. The linear differential operator ~(~) is autonomous and N(~) is a nonlinear function of ~ with the property that ~V(~)/> 0 331

(2)

332

C.E. McQUEARYand U G. CLARK

for all i~. With # = 0, equation (1) is assumed to be solvable in terms of eigenfunction expansions. Since we are concerned with periodic solutions of equation (1), we find it convenient to non-dimensionalize the equation such that the period of oscillation for the non-dimensional non-linear system will be 27r. Such a non-dimensionalization can be made to yield an equation of the form - aoL(~k) + u ~

+ ~ + eN(d/) = P ( x ) c o s t

(3)

where ~k - ~k(x, t) and where all quantities in the equation are dimensionless. The constants ao, t22, and e are real and positive. The non-linear function N(~b) satisfies a condition equivalent to that of equation (2). To obtain a periodic solution of equation (3) we use a perturbation method similar to that outlined in a previous paper by the present authors. We expand the displacement function ~(x, t) into a perturbation series of the form g¢ = ~ ei~bi.

(4)

i=0

Since the fundamental frequency response of the system (3) is known a priori, nothing is to be gained at this point by expanding the known forcing frequency t2 into a perturbation series as is frequently done for systems such as the forced Duffing equation [4]. However, it will be seen that for certain values of the forcing frequency t2, it will be necessary to appropriately modify our method of solution. At this point, it is assumed that there will be no solution difficulties. We obtained a complete set {@p} of eigenfunctions and associated positive eigenvalues ~tp by solution of the homogeneous linearized form of equation (3). The differential equation relating each eigenfunction to its corresponding eigenvalue is L(~p) + ~p~p = 0.

(5)

We set 0% equal to zero and therefore it is meaningful only for those problems for which the least eigenvalue is zero. We expand the function P(x) into a series using the eigenfunctions from the set {~p}. That is, it is assumed that (6)

P(x) = ~ P.@. tl=O

where the P. are determined using the orthogonality property of the @p[@p - @p(x)]. Substituting the perturbation expansion equation (4) and the load expansion equation (6) into equation (3) and equating to zero the coefficients of like powers of e, we determine the following set of perturbation equations to be solved for a periodic solution:

E(d/o ) =- - aoL(~ko) + t2 2

02~0

+ ~bo = cos t

P.~.

(7a)

n=O

E(~I) = -- Nl(~bo)

(7b)

E(g¢2) = - N2(~bo, ~1)

(7c)

Periodic oscillations of a class of non-autonomous non-linear elastic continua

333

For simplicity and in order to demonstrate certain critical points we choose N(~b) = ~bs and study the resulting equations. The periodic solution of period 21t for fro is ~0o = cos t .=oZ [~oE:

(8)

b2]

where 2 (/)nO

= 1 + a . a o.

2 (DmO

_ 95 = O(e)

(9)

In equation (8) we note that if

(10)

(m being a specific integer) with the corresponding Pm being non-zero, then we have the classic small divisor difficulty [-2, 3]. If this difficulty is encountered, then we must modify our method of solution. Such a modification will be shown in the next section. Assuming that there is no small divisor difficulty, we proceed to obtain the solution for ~h1. Let the function ~ho 3 be written as ~ks

= cosSt ~, Q.¢I,.

(11)

n=0

where (12) .=o

n=O

[~o

-

~]J

"

Then the differential equation for ~t may be written as

E(¢,0 = - ~ cos t ~ Q.~. - ¼ cos 3t ~. Q.~.. n=0

(13)

n=O

The periodic solution for ffl is o0

~,1 = - ~ c o s t

BO~o- b ~2

¼cos 3t

g

Qn~

[o~ o Z 9122]"

(14)

n=0

n----O

Here we see that we must require that 9~22 :~ COmo 2 (m being a particular integer and the corresponding Qm being non-zero) to eliminate the possibility of unbounded terms in the solution. Noting the pattern established for elimination of certain forcing frequencies, we see that if we desire a qth-order correction term ~hq,then we must demand that 2 (2m + 1)29 2 ~ CO.o

n = 0, 1, 2 . . . .

(15)

m = 0,1,2 . . . . q if the corresponding coefficient of the term is non-zero. The general form of solution could be carried to higher order to yield better approximate solutions to ~k in equation (3). However, the qualitative nature of the periodic solution has U

334

C . E . McQuEARY and L. G. CLARK

been established for the class of problems under consideration. In a later section we consider specific solutions of a more detailed nature. SMALL D I V I S O R S

In previous work [1] the implications of small divisors in perturbation solutions to autonomous N L P D E has been thoroughly discussed. As indicated in the earlier work the presence of a small divisor in a perturbation solution does not mean that the perturbation method is invalid; the small divisor is simply an indication that the method of solution must be modified. We present the necessary modification to the theory such that nonautonomous N L P D E of the class considered can be adequately treated. Consider the following functional form for a non-autonomous N L P D E : M(~k) + eN(~k) = F(x, t)

(16)

where M(g0 is a linear differential operator and N(¢) is a non-linear function of ~,. The forcing term F(x, t) is time periodic, i.e. F(x, t) = F(x, t + T), T being the period. The symbol x is used to designate functional dependence in three spatial variables Ix = (x l,

x~, x3)]. We assume that the autonomous (F(x, t) = 0) linearized (~ = 0) form of equation (16) is solvable in terms of eigenfunction expansions. Separation of variables of this linearized form of equation (16) yields a set of eigenfunction solutions of the form {Xp(x) gp(t)}.

(17)

Any homogeneous solution of the linearized form of (16) is composed of elements from the set (17). It is assumeff that F(x, t) can be written as F(x, t) = ~ &X~(x) fk(t)

(18)

k

where

fk(t) =A(t + T)

(19)

and Ak is a constant. If a perturbation solution to equation (16) is desired, we may expand the function ~b into an ascending power series of the perturbation parameter e. Then by equating to zero the coefficients of like powers of s, we obtain a set of linear partial differential equations of the form: M(~bo) = ~ AkXk(x) fk(t) k

M ( ~ 0 = Nl(~'o)

(20)

M(tP2) = N2(g¢o, ~1) J~/(~j) = Nj(ffo, ~1 . . . . ~kj_1)Each of the above equations is linear with the forcing terms determined from preceding equations (except for @o)- It is necessary to expand the non-linear forcing terms into linear combinations of forcing terms before the solution for a given qJj can be obtained.

Periodic oscillations o f a class o f non-autonomous non-linear elastic continua

335

It now becomes clear how a small divisor may be encountered. If the expanded form of a non-linear forcing term contains a term that is near a homogeneous solution of the corresponding differential equation, then a term of large magnitude in the particular integral will result. In other words, an apparent resonance condition exists for the particular perturbation equation being studied. This condition necessitates a modification in the method of solution if it is known that such a condition cannot exist for the system under consideration. We now indicate a method for eliminating the small-divisor difficulty. Let us assume that in the solution of thejth (j > 0) perturbation equation, we encounter a small divisor. As noted earlier, the small divisor simply means that the functional form of the large term is "near" a homogeneous solution for ~Oj.Since all of the homogeneous differential equations of equations (20) are of the same form, then the functional form of the term with apparent resonance in the solution of ~j is also "near" a homogeneous solution of ~bj_ 1. Thus we may eliminate the difficult term in ~kj simply by including the term as an arbitrary homogeneous portion of the solution for ~Oj_1 and then we may eliminate the arbitrariness of the term by choosing its magnitude from the requirement that there be no unbounded (or very large) terms in the solution for ~Jj. (It should be noted that Struble [5] has previously developed a method for elimination of small divisors in perturbation solutions to non-linear ordinary differential equations. However, his method is formally different from the one presented herein.) The above idea works very well unless the apparent resonance condition exists in the non-homogeneous solution for ~bo. If the apparent resonance Condition exists for a term in the solution for qJo, then we obviously cannot move the term to a lower order in the perturbation solution since the zeroth order is the lowest order. We may effectively eliminate the apparent resonance condition in the solution for qJo by pragmatically assuming that the part of the forcing function that gives rise to the apparent resonance condition is of order e. This moves that part of the forcing function to a higher order in the perturbation equation. We then include in ~o a homogeneous solution of the same form as the apparent resonance term and alleviate the arbitrariness of the term from the requirement that there be no large terms in the solution for ¢'lIn the solution for ~o, equation (8), assume that the condition of equation (10) is satisfied with the corresponding Pm in equation (8) being non-zero. Therefore, let (21)

P m = eP*.

Additionally, we assume that co

O2

- - (DmO ~

V

i

2

(22)

~-~ F, (~Omi i=1

where the value of the infinite summation will be precisely known. We now substitute the above forms into the previously obtained perturbation equations (with N(~J) = ~ba) and regroup terms according to powers of e to obtain the following set of perturbation equations: E(~ko) ~ - aoL(~o) + o92 mO 02~k° at---T- + ~9o = cos t ~ p ~ / i

(23a)

= 02~o E(~kx) = - ogre1~ -- ~9S0+ P*q~,, COSt.

(23b)

n=0 n:Cm

336

C . E . McQUEARV and L. G. CLARK

The desired periodic solution for ~0 can be shown to be Ipo = cost

,.1~,. +

ao(0t. _ ~,,).j n=0 n:~m

where A,., is arbitrary since 4,. cos t is a homogeneous solution to the differential equation (23a). To obtain the particular integral for ~q, we first must expand the function ~Oo a. It can be readily shown that

~p3 =

COS3t

[A

ml~rn +

~ P"" 13

(25a)

ao(Ot. - ~=)3

n=O n:#m

or

~,3 = (¼ cos t + ¼ cos 3t) ~ R.~..

(25b)

n=0

Now the differential equation (23b) for 01 may be written as E(~,I) = [ m m ~ J t , ~ -- ~ - g . + P.*]~.. c o s t

+ cos t E (°921S. - lR.)q~. -- ¼ cos 3t ~, R . e . n=O n~m

(26)

n=O

where S, -

Pit

(27)

ao(0t. - ~.,)

To preserve the periodicity of solution we must demand that 2 ~o,.xA,.1 - ¼R, + P~ = 0.

(28)

Thus to the first order in ~, the frequency-amplitude relation is given by 8 2 = 1 + ~.ao - ~

+ ~\4a,.,f

(29)

From the form of equation (25a) we note that R,. will contain at least a cubic term in A,.I. Thus equation (29) yields a backbone curve similar to that of the forced Duffing equation [4]. The solution for ~b1 is given by qq = cost E ~l=O n~m

(a~'S" ao(Ct _- - -'R") - ~) ~. + ¼cos3t S

R.q~. 0t.)]. [8 + ao(90t/_

(30)

n=0

The general solution is carried no further since the qualitative nature of the solution has been established. We now examine the first order approximate solution for ~ with all P. = 0 except for n = m (a particular integer). Using equations (24), (25), (29), and (30) we determine that

Periodic oscillations of a class of non-autonomous non-linear elastic continua

~k=A,.ld.,cost+eAax[

337

3 - 4ao7 cos t ~'-~ b"(~a)~"-ot. --- --oc.. n=O n~m

+ ¼cos 3t

[8 + ~

2_ ~t.)]

(31a)

n=O

and Pm

3

3

2

122 = i + ot,nao ~4ma+ e[~bm(~m)Ama]

(31b)

where we have used ~3 = ~ b.(~)~..

(32)

n=O

Equation (31a) is identically of the form given for the mth non-linear periodic mode [1]. Therefore, we give specific terminology to the class of forcing functions which produce such a response as this in our system. We define the forcing function, Pm~m cos t, (with 2 ) as a "modal forcing function." f22 near OgmO DEFI~TION. A modal forcin# function for a non-linear system is one that is of the same functional form as any one of the eioenfunction solutions of the linearized equation of motion for the system and has a forcino frequency "near" the correspondino linear mode frequency [6]. We see for a given continuous system of the type studied that there are an infinite number of modal forcing functions.

EXAMPLES

Example 1 In order to verify the previously obtained results for cases where there are no small divisors, we consider, as a specific example, the periodic small-amplitude transverse oscillations of a taut uniform string attached to a non-linear Duffing-type foundation. Appropriately non-dimensionalizing the physical equation of motion such that the nondimensional string has a length of Ttand a period of oscillation of 2n, we obtain the following dimensionless equation of motion: 02~ ~22 02~ -- ao ~ + - ~ - + ~k + eq/3 = P(x) cos t

(33)

[Compare equation (33) with equation (3) where N(~b)---if3]. Now we must expand P(x) into a series equivalent to that of equation (6) where the set of spatial eigenfunctions for this example is {~p} - {sin px},

p = 1, 2, 3 . . .

(34)

Therefore, we have

P(x) = ~ P, sin nx. n=l

(35)

C . E . McQUEARY and L. G. CLARK

338

F o r simplicity, we consider an example w h e r e

P(x) =

P : sin x

(36)

and present the results for this type of load distribution. C o r r e s p o n d i n g to the a p p r o x i m a t e solution obtained from equations (8), (9) and (14), we have the following first-order approximate solution for this e x a m p l e :

P1

e(

Pt

2)3[

~b(x, t) = 0)20 _ t22 sin x cos t - ~ \ 0 9 2 0 _ t2

9 -0920 _ 02 sin x cos t

+ O9~o -3 9f22 sin x cos 3t - O9~o 3- 02 sin 3x cos t

O)lo _1 9 0 2 sin 3x cos 3 t]

(37)

where 2 O.)nO

= 1 + n2ao,

n = 1,3.

(38)

In order that equation (37) be a reasonable periodic approximate solution to ~k(x, t), ANALYTICAL SOLUTION AT t "-2n'ff AND NUMERICAL SOLUTION AT t=O ANALYTICAL SOLUTION AT t = (2n + 4-~--)1T --" - - ANALYTICAL SOLUTION AT t = (2n + -J~--)'n" e NUMERICAL SOLUTION AT t = 12w • NUMERICAL SOLUTION AT t = -~=Tr 0"0":F~ o NUMERICAL~SOLUTIONAT t = (12--+ 2-~-)'n"

0-2

0'4 0'6 0'8 (a) DISPLACEMENT

I'0

11"

9/.--f°" %

05 .~

•

_." 0.3 "~" 0.2

ol

o

I

c

i ~

(b) VELOCITY

Io

~II-~

X~ "IT

FIG. I. C o m p a r i s o n of analytical and numerical solutions for forced vibrations. P ( x ) = - sin x, ~ = 1.0, ao = 1.0, I2 = 2.0.

Periodic oscillations of a class o f non-autonomous non-linear elastic continua

339

we see that we must impose the condition of equation (15) with m = 0, 1 and n = 1, 3. In other words, we must require that there be no small divisor difficulty. It is interesting to note from equation (37) that, in contrast to the linear problem (5 = 0), the displacement geometry is not maintained during oscillation, but is repeated with period 2rr. To verify that equation (37) is indeed a reasonable approximate solution for the case where there is no small divisor difficulty, equation (33) [with P(x) given by equation (36)] was numerically integrated using initial conditions determined from equation (37). The resulting digital computer solution indicated periodicity of solution, as predicted analytically, for all cases examined where there were no small divisors in the approximate solution for $(x, t) [equation (37)]. Figure 1 shows the results of numerical integration for a case where Pt = - 1"0, ao = 1.0, = 1.0, and f2 = 2-0. We note that the numerical results differ only slightly from the analytical results even though the nonlinear term in equation (33) is of the same order of magnitude as the linear term in the equation. We note that both the maximum displacement and the maximum velocity are slightly increased from the value of one-half predicted for each of them by the linearized solution. These increases are indicated by the correction terms in equation (37). To improve the agreement between numerical and analytical results, we would need to obtain a higher-order approximate solution.

Example 2 In this section we investigate the response of the system described by equation (33) when the system is subjected to a modal forcing function. A modal forcing function for this system is given by P1 sin x cos t with the requirement that I22 ,~ CO2o= 1 + a o

(39)

in equation (33). We have previously obtained the response of a general system to a modal forcing function [see equations (31)]. In equation (31a) we let m = 1 and 41 -- sin x to obtain the response (through the first order) of the system (33) when it is subjected to a modal forcing function. Thus ~A~I [ 3 ~(x, t) = A,1 s i n x c o s t + ~ sin3xcos t 3 + 1 + a 0 sin x cos 3t -- sin 3x cos 3t]

(40a)

and

..Q" = 1 + ao

PI 9 2 A1, + e[1-~A11].

(40b)

Equation (40a) is of the same form as the first non-linear periodic mode solution obtained in a previous paper [1]; hence, we derive the name "modal forcing function". Equation (40b) was solved to determine the amplitude A11 vs. flz the square of the driving frequency. The results of this solution are presented in Fig. 2 for ao = 1.0, e = H), and PI = 0.5. The results of this curve show that the response of the system (33) is bounded

C. E, MCQUEARYand L. G. CLARK

340

E =0<) I

5"0

II

.~ 2.o

I I I I

E = 1.0

I

F

1

o

,

/

I

2

ONLINEAR THEORY (MODAL FORCING FUNCTION)

~),/ ~/v"

---LINEAR T.EORY

i

3

i

4

5

i

e

~

I

;

9~'

DRIVING FREQUENCY (SQUARED) FIG. 2. Amplitude vs. square of driving frequency for modal forcing function.

ANALYTICAL SOLUTION AT t • 2nw AND NUMERICAL SOLUTION AT t = 0 ANALYTICAL SOLUTION AT t = (2n +~-)w

--~ --'-

ANALYTICAL SOLUTION AT t = (2n + --~ )'~ NUMERICAL SOLUTION AT t = 12w NUMERICAL SOLUTION AT t • -~--w NUMERICAL SOLUTIONAT t = (12 ÷ # ) w

~

_

I-0 08 ,,O6 ). 04 0.2 0.2

.

.

0'4 0-6 0.8 (o) DISPLACEMENT

1.0

x_. 11"

I I°r 0.8

...._.o......

/.o~

~-

i •D- 08[-

/ ~"/

.~

,¢.,--" - ~ . ~ \

0"4L .I j i / //"

,,No, \.,

\\'.

o.2~C [,

\~, o,

o,o ,o (b) VELOCITY

,o

__~

x__ "IT

FIG. 3. Comparison of analytical and numerical solutions for modal forcing function. P(x) = 0.5 sin x, e = 1.0, ao = 1-0, g2 = 1.414, A l l = 0.964.

Periodic oscillations of a class o f non-autonomous non-linear elastic cont&ua

341

when subjected to a modal forcing function whereas the general solution of equation (37) predicts unbounded amplitudes. The coefficient of sin x cos t as determined from the linearized form (5 = 0) of equation (37) is also graphed on Fig. 2 so that the transition in order of magnitude of the coefficient may be noted. This transition indicates a necessity for modification of the general procedure used in obtaining an approximate solution. Figure 3 shows the results of numerical integration of equation (37) ]with P(x) = P1 sin x] with initial conditions determined by equation (40a). The results are for f2z = 2.0, P1 = 0.5, and ~ = 1.0. As determined from Fig. 2, the value of A H is 0.964 for t22 = 2.0. Figure 3 shows that the initial conditions used for numerical integration of equation (33) are proper for periodic motion. One last comment is made regarding the response curve shown in Fig. 2. The crossed portions of the curve indicate unstable-solution values for the amplitude Art. This fact has not been established mathematically, but the instability has been indicated by numerical solutions obtained by the authors. By analogy with the results of the forced Duffing equation, we would indeed expect the crossed portions of the frequency response curve to correspond to unstable solutions. CONCLUDING REMARKS

In this work we have developed a method for obtaining steady-state solutions to a rather broad class of non-autonomous non-linear partial differential equations. We have shown a logical method for recognition and elimination of the difficulties of small divisors. Additionally, we have introduced the concept of a modal forcing function. The information contained herein should lead to a better understanding of oscillations of non-linear continuous systems. In the interest of ease of understanding, only simple load distributions were used for the examples considered. The authors have investigated examples with more complex loading distributions. The results in all instances were as equally rewarding as those presented herein. A problem that is of interest, in conjunction with those considered herein, is subharmonic response of a non-linear distributed parameter system. This problem is presently being investigated by the authors.

REFERENCES [1] C. E. McQuEARY and L. G. CLARK,Nonlinear periodic modes of oscillation of elastic continua. Proc. lOth Midwestern Mech. Conf., 1967. To be published. [2] R. M. ROSENa~G, Nonlinear oscillations. Appl. Mech. Rev. 14, 837 (1961). [3] J. J. STOKER,Nonlinear Vibrations. Interscienee (1950). [4] N. MINORSKY,Nonlinear Oscillations. Van Nostrand (1962). [5] R. A. STRtraLE, Nonlinear Differential Equations. McGraw-Hill (1962). [6] C. E. McQt~ARY, On Periodic Solutions o f Autonomous and Nonautonomous Nonlinear Partial Differential Equations o f Motion of Certain Elastic Continua. Ph.D. Dissertation, The University of Texas (Aug. 1966). (Received 31 October 1966)

342

Periodic oscillations of a class of non-autonomous non-linear elastic continua

R&um6---On d6veloppe une m6thode modifi6e des perturbations pour obtenir des solutions p6riodiques d'une classe d'6quations aux d6riv6es partiel!es non lin6aircs et non autonomes. On discute en d&ail le classique petit diviseur et on montre une m6thodc g6n6rale pour son 61imination. On pr6scnte une nouvelle terminologie dans le but de discuter les fonctions forc6es qui produisent sur un syst6me une r6ponse qui est de la m&ne forme qu!un mode p6riodique non lin6aire pour ce m~me syst~me. On &udie des exemples particuliers pour v6rifier les r6sultats de ce travail. Zusammeafa~ung--Es wird eine modifizierte Perturbationstheorie entwickelt, um periodische LSsungen f'tir eine Katergorie von nieht-autonomen niehtlinearen, partiellen Differentialgleiehungen zu erhalten. Der Hassische Heine Divisor wird ausffihrlich besprochen, mad eine allgemeine Methode fur dessen Eliminierung wird aufgezeigt. Eine neue Terminologie wild eingeffuhrt, um zwingende Funktionen zu besprechen, die in einem System eine Antwortfunktion erzeugen, die die gleiche Form besitzt wie ein nicht-linearer periodischer Modus ffir dasselbe system, Spezifische Beispiele werden untersucht, um die Ergebnisse dieser Arbeit zu best~tigen.

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