Non-linear resonances in the forced responses of plates, part 1: Symmetric responses of circular plates

Journal o f SowM and Vibration (1975) 41(3), 359-373

NON.LINEAR RESONANCES IN THE FORCED RESPONSES OF

PLATES, PART I: SYMMETRIC RESPONSES OF CIRCULAR PLATES

S. SRIDHAR, D. T. MOOK AND A. H. NAYFErt

Department of Engineering Science and J~lechanics, VirginiaPolytechnic lnstitute and State University, B[acksburg, Virghlia 24061, U.S.A.

(Received 19 August 1974, and in revised form 7 February 1975)

The dynamic analogue of the von Karman equations is used to study the symmetric response of a circular plate to a harmonic excitation when the frequency of the excitation is near one of the natural frequencies. The response is expressed as an expansion in terms of the linear, free-oscillation modes, and its amplitude is considered small but finite. The method of multiple scales is used to solve the non-linear equations governing the time- dependent coefficients in the expansion. It is shown that, in general, when there is no internal resonance (i.e., the natural frequencies are not commensurable), only the mode having a frequency near that of the excitation is strongly excited (i.e., is needed to represent the response in the first approximation). A clamped, circular plate is used as a numerical example to show that, when there is an internal resonance, more than one of the modes involved in this resonance can be strongly excited; moreover, when more than one mode is strongly excited, the lower modes can dominate the response, even when the frequency of the excitation is near that of the highest mode. This possibility was not revealed by any of the earlier studies which were based on the same governing equations.

1. INTRODUCTION

The dynamic analogue of the yon Karman equations has been used in several attempts to determine the effect of mid-plane stretching on the response ofplates to a harmonic excitation. For example, Yamaki [1 ] considered rectangular and circular plates having both clamped and hinged edges. He obtained an approximate solution by using a single-term expansion in conjunction with the Galerkin procedure. Eisley [2] extended the analysis for rectangular plates to include the effect of initial displacements of the edge, and he considered elliptic as well as harmonic excitations. More recently, Kung and Pao [3] also considered clamped circular plates. In conjunction with the Galerkin procedure, they assumed the same one-term expansion as Yamaki. They compared their analytical solution with their experimental data and found close agreement. Farnsworth and Evan-Iwanowski [4] considered small- amplitude oscillations about a large-amplitude static deflection. They used one- and two-term expansions in conjunction with the Galerkin procedure. Huang and Sandman [5], Sandman

r

and Huang [6], and Huang [7] considered clamped circular plates, in the form of annull having a clamped outer edge and a free inner edge, and orthotropic circular plates having an isotropic core, respectively. Assuming a simple harmonic response, they used the Kantorovich averaging procedure and solved the resulting non-linear, spatial, boundary-value problem numerically. Bennett [8] considered the response of rectangular, I.'iminated plates having simple supports. Though he used a four-term expansion in conjunction with the Galerkin procedure, he did not consider the possibility of more than one harmonic appearing in the first approximation of the response. Rehfield [9] used Hamilton's principle and perturbation theory to derive a governing integral ecluation for large-amplitud/~ motion. Then, using a single-term expansion in conjunction with the Galerkin procedure, he determined the response of a circular plate.

359

360 s. SRIDHAR, D. T. MOOK AND A. H. NAYFEH

Nayfeh, Mook, and Sridhar [I0] and Nayfeh, Mook and Lobitz [l 1] described alternatives to the procedures mentioned above. With these alternatives, the deflection is represented by an expansion in terms of the linear, free-oscillation modes, and modal interactions are considered:'The importance of considering modal interactions was illustrated with two numerical examples involving beams. The results reveal that, when two modes are involved in an internal resonance (i.e., when two of the natural frequencies are commensurable) and the amplitude of the excitation is above a certain threshold, it is possible for the lower mode involved in the internal resonance to be strongly excited (i.e., to be needed to represent the response in the

�9 first approximation) when the frequency of the excitation is near'that of the higher mode involved in the internal resonance. Consequently, in the first approximation, the steady-state response can be composed of two harmonics and actually resemble the lower mode very closely. The possibility of plates having similar behavior was not predicted in references [1 ]- [9]; it is the purpose of this paper to illustrate the presence of this phenomenon in the response of plates. In Part I, only symmetric responses of circular plates are considered. In Part II, the analysis is broadened to include asymmetric responses.

By using the dynamic analogue of the von Karman equations with forcing and damping terms included, the symmetric response of circular plates to a harmonic excitation isconsid- ered. Consideration is restricted to the resonant responses which can occurwhen the fre'quency of the excitation is near one of the natural frequencies. In contrast with references [1]-[9], the solution is obtained by using the general procedure of reference [10].

In a numerical example, it is found that, in addition to the solutions having the same fea- tures as those obtained in references [1 ]-[9], there are solutions in which strong modal interactions occur. For the plate, three modes are involved in the internal resonance, while for the beam only two are involved. Thus, the present example differs significantly from the one presented in reference [10]. The effects of the amplitude and the frequency of the excitation and the damping on the various solutions are illustrated in a series of graphs.

2. STATEMENT OF THE PROBLEM

The equations governing the free, undamped oscillations of non-uniform, circular plates were derived by Efstathiades [l 2]. For the purposes here, these equations need to be simplified to fit the special case of uniform plates and symmetric responses, and forcing and damping terms need to be added. The result is

0 2w 1 a [araw\ aw p h - ~ + DV4 W =r-~r ~r --ff-/) -c-~+ p(r, ,) (la)

and

1 V 4 F = - ~ r ~rLI.-~r ] j , (lb)

where p is the density, h is the thickness, D = Eh.3/[12(1 - v2)], c is the damping coefficient, p is the forcing term, Eis Young's modulus, v is Poisson's ratio, w is the deflection ofthe middle surface, F is the force function, and

v'--

The relationships between F, the in-plane displacement, u,, and the deflection are

l a., l a r v e E h " +~(~r) ] =r-~r - V'ar 2 (2a)

"!

SYMMETRIC RESPONSES OF CIRCULAR PLATES 361

and Eh ~ O" F 1 OF

= - v - - - . (2b) r Or 2 r Or

The rel~'tionships between the in-plane stress resultants, Nr and No, and the function F are

1 OF AT, = - - ~ (3a)

r ar and.

OZ F No = Or = �9 (3b)

Finally, the relationships between the moments and the transverse stress resultant, Mr, Mo and Qr, and the deflection are

[a 2w v o w I M r = - D ~ - - ~ r 2 + r ~ r ] , (4a)

Mo = - -D \ r ~ r + v O r ' ] ' (4b)

and

at 2

where t; = 12(1 - v2)h2/a 2,

0 Q, = - D ~ (172 w). (4c)

It is convenient to re-write these equations in terms of non-dimensional variables. The non-dimensional variables, which are denoted by overbars, are defined as follows:

N / / - ~ h2 h 4 r = aF, t = a z t, W = --a ~-V, u, = --a3 Ur- ,

12(1 -- v z) Dh 4 c = 2 4 ( 1 - v 2) V,ph 5 D ~, p - fi, a 4 a 7

Eh 5 _ Ell s F = - - ~ - F , (Nr, Nol=--~-(~I , , IVo) ,

Dh 2 Dh 2 _ (Mr, Mo) = 7 (Mr, Mo), Qr = --a4 Qr,

where a is the radius of the plate. Substituting these definitions into equations (1)-(4) and dropping the overbars in the final result, one obtains

a2w = l i t 3 {OwaF'~ aw ] - - + I 7` w e [ r ~r ~-~'r "~'r] - 2e ~- + p , (5a)

V 4 F = - i r ~ r L ~ r ] J '

U r 0 2 F v OF

r Or 2 r Or '

(5b)

(6a)

(6b)

362 s. SRIDHAR, D. T. MOOK AND A. It. NAYFEH

0 2 w 10w A i r = Or 2 r Or" (7a)

1 aw 0 2 w M o -- v I (7b)

r ar ar 2 '

0 Q, = - ~rr(V 2 w) (7c)

(note that the non-dimensional version of equations (3) has the same t~orm as the dimensional version). All the non-dimensional variables are assumed to be O(1) as e approaches zero. The in-plane inertial terms are higher order.

�9 The boundary conditions for plates which are clamped along a circular edge are as follows. For all t,

w = O, a w / O r = O, a n d ur = O at r = l .

Equations (6a), (b), and (8) lead to the following boundary conditions for F:

O z F vOF 0 3F 0 2F OF

Or 2 r Or - 0 and -ff~-r3 + Or 2 Or = 0 at r = I. ('~}a, b)

(8a, b, c)

The present paper is concerned with constructing an asymptotic expansion of the solution of equations (5) which satisfies equations (8) and (9). The expansion is to be uniformly valid for large time when e is much less than unity.

Note that, when e is small, the deflection is nluch smaller than the thickness. Had the deflection been the same order as the thickness (say, w = hF,), then no small parameter would have appeared in equation (5a), and the linear and non-linear terms would have the same order. Hence, the present approach must be viewed as one which provides corrections for the small- deflection theory (for which w is much smaller than h) and not as one which provides a solution for the large-deflection theory (for which w is the same order as h). This.means that some typical non-linear phenomena, such as jump phenomena, modal interactions, etc., can properly be considered part of the small-deflection theory.

3. SOLUTION

Following a straightforward procedure, one can assume expansions in powers of~ for it' and /7, substitute these expansions into equations (5), (8) and (9), balance powers of e, and obtain a sequence of problems which can be solved for the coefficients. However, the expansions which are generated for large time by such a procedure are not uniformly valid. One modi- fication of this straightforward procedure which can be used to generate uniformly valid expansions is the method of multiple scales [13].

Following the derivative-expansion version of this method, one replaces the single time scale (variable) t, which appears naturally, by a multitude of time scales. These new scales are defined as follows:

Tj = eJt for j = 0, 1, 2 . . . . . (10a)

Introducing this multitude of scales restdts in the derivatives with respect to the original scale being transformed into expansions in terms of the derivatives with resoect to the new scales as follows:

a at Do + eDl + ,2 D2 + . . . (10b)


and 92

- Do 2 + e2D o D~ + e2(D~ + 2Do Dz) + . . . . 9t 2

Where Dj-~9/gTj. The expansions for w and F a r e given the following forms:

(IOc)

and

w(r, t; e) = ~ d w~(r, To, TI,...) (1 la) J=O

ec F(r, t; e) = ~ eJFj (r, To, T1,...). (1 lb)

J=O

Substituting equations (10) and (11) into equations (5) and balancing powers of e, one obtains

D~wo + V4wo = 0, (12)

, o 174Fo=--~r~r[~'~-r ] j ' (13)

%,

1 9 [OFo 9WoX

etc. From equations (8), it follows that

W j = O,

and from equations (9), it follows that

92 F~ 9Fj 9 r 2 - - v - ~ - r = O,

(14)

9w/gr = 0 at r = 1, (15a, b)

o 3FJ 9"Fj 9Fj + Or ' ' - T 9r = 0 at r = 1. (16a, b)

In addition, one must require the solution to be bounded at r = 0. Using the method of separation of variables, one can solve the problem defined by equations

(12) and (15) and the requirement that w be finite at r = 0. The result is

where

!

to. is chosen so that

the tl. are the roots of

Wo = ~. cb.(r) [A.( TI, T2 .. . . ) exp(iog. To) + cc]

Jo( , . ) 1 qS. = h'. Jo(q. r) - ~ Io(q. r)] ;

(17)

(18a)

1

frqb~.dr = 1; (18b) O

Io(,t.) J~(n.) - ~ , l . ) Jo(qn) = 0; (18C)

o9. = ll.z; (18d)

the A. are unknown complex functions ofall T. for n > 1 at this point, but they are to be determined from the solvability conditions at higher levels of approximations; and the cc represents the complex conjugate ofthe preceding terms. The q~. are the linear, free-oscillation modes, and the 09. are the (linear) natural freouencies.

364 S. S R I D H A R , D. T. MOOK A N D A. H . NAYFEII

i~quations (13) and (14) suggest that it may be more convenient to solve for aFo/~r instead o f Fo. Consequently, we put

~k = aFo/ar (19a)

and , after integrating and using equation (16b), obtain f rom equation (13) a2~ a@ 1 1 (a,','o~ 2

r-~r2 + ~r - - r O = - - g i ~ r ] (19b)

and f rom equation (16a) a~, - - - v ~ = 0 a t r - - I . (19c) ar

The solution o f the problem defined by equations (19) and the requirement that ~, be finite at r = 0 can be obtained as an expansion in terms of any convenient or thonormal , complete set of functions. Because [rd2/dr 2 + d /dr - l/r]Jt(~mr) =-~rJ~(~'mr), it is convenient to choose

= ~. Am(To, T~ .. . . )Jl(~mr), (20) m . 1

where the ~,, are the roots of ~m Jo(~',,) - (1 + v) J,(~r~) = 0.

For v = 1/3, the first twelve roots are as follows:

/~1 = 1.545, ~2 = 5.266, ~'3 = 8.497, ~4 = 11.68,

~'5 = 14.84, ~6= 18.00, ~7 = 21-15, ~s = 24.30,

~9 = 30"59, ~1o = 33.74, ~lt = 36.88, G2 = 40.03.

Substituting equations (17) and (20) into equation (19b), multiplying the result by J~(~,r) and integrating f rom r = 0 to r = 1, one obtains

A,, = ~. ~ S,.,o{A, Aoexp[i(co, + ~ov) To] + An-~v exp[i(co, - ~ov) To] +ce} , n - I p - I

where 1

f ~ ' ( r ) ~ ( r ) Jl(~,, r) dr

Stun, = o ( ~ - I + V 2) J i(~, . )

One can now rewrite equation (20) as follows:

~,= ~. ~. ~. SmnoJl(~_mr){AnAoexp[i(oa~ +c%)To + AnAoexp[i(~176 (211 m . l n . l p . !

Because wl and ~% satisfy the same boundary conditions, it also is convenient to express w, as an expansion in terms of the qSn:

wl = ~. H,.(To, Tx . . . . ) ~ m ( r ) . ( 2 2 ) m - 1

Then substituting equations (21) and (22) into equation (14), multiplying by r~bk, and integrating f rom r = 0 to r = 1 leads to

Do 2 Hk + w~, Hk = -- 2koRD, Ak + C~ A~) exp(iwk To) + �89 exp(i).To)

+ ~. ~. ~_ Fkmp~{Am Ao Aq exp[i(COm + co v + 094) To] m - I p - I 4 -1

-a t- A m A p ~Zlq exp[i(COm + coo - o~q)To] + Ar~ A v Aq exp[i(co o + co~ -- corn)To]

+ 4,. A o Aq exp [i(c% -- to,, - COd)To]} + CC, (231


where the'constants r'k,,pq are produced by the integration. Harmonic excitation and modal damping have been assumed. �9 Equations (23) are a set of linear, uncoupled equations which can be solved for the functions

tlk. However, unless certain conditions are imposed on the non-homogeneous terms in these equations, the l i t will contain terms having the factors Toexp(itokTo) and Toexp(-ioJ~To). Because equations (23) are linear and uncoupled, these secular terms can be eliminated simply by requiring the coefficients of exp(kok To) and exp(--i~k To) to vanish. This leads to a set of non-linear coupled equations--the so-called solvability conditions--which can be solved for the d,,.

Note that the combinations of natural frequencies in the cubic terms have essentially three forms: tom + m~, + toq, tom- top- a~q, tom- a~, + to~. It is always possible for the last of these to equal ~o~. This occurs, for example, when p = q and m = k. Thus, the last combin- ation is always responsible for a series of non-linear terms coupling the A,. If either of the other combinations equals, or nearly equals, to~ (i.e., if there exists a commensurable relationship between the natural frequencies), internal resonance is said to exist. Next we consider a numerical example which illustrates the far-reaching effects of the internal resonance on the response.

4. NUMERICAL EXAMPLE

For a clamped plate, the first five natural frequencies (see, e.g., reference [14]) are to, = 10.2158, to2=39.7710, r t o ,= 158.1830, and to5=247.0050. Note that an internal resohance exists because to~ + 2to2 = 89.7578 ~ 0)3. To express the nearness of this approximation quantitatively, one can introduce a detuning parameter, th, defined as follows:

�9 . tot + 2co2 - ton = 0"6538 = ~o1. (24)

Secular terms are eliminated from the expansion by requiring the coefficient of exp(i~k To) to vanish in each of equations (23). This solvability condition leads to the following set of coupled, non-linear equations: f o r k = l,

-2ko~(Dl A, + Ct A,) + �89 exp(itr2 7",) + [Qt ,~2 A3 exp(-itrl 7",) r

+vttA21,~t+vt2AtA2,~2+vt3AtA3,43+At ~ v.jAjAj = 0 ; (25a)

f o r k = 2 ,

f o r k = 3,

f o r k > 3,

--2ito2(Dl A2 + C2 A2) + �89 exp(itr2 7"1) + [Q2 A3 Ax A2 exp(-iat T~)

+ )'2t A2AI,41 +~22A~A2+)'23A2A3,43+A2 .i-* ~ )'2.t Aj ,4j] = 0; (25b)

- 2ir A3 + C3 As) + �89 exp(i~r2 T,) + [Q3 A x A~ exp(iax 7',)

- ] +~3~A3A,2~+~,32A3A2A2+v33A]A3+A3 ~. V3jAjAj = 0 ; (25c) J-4

--i2tok(D, Ak + Ck Ak) + 6k~cPk exp(ia2 T,) + Ak ~. 7kj Aj,4j = 0. (25d) J - ,

36 6 s. SRIDHAR, D. T. MOOK AND A. tL NAYFEtl

Hc'r'e (5~ is the Kronecker delta and 0" 2 is a second detuning paramete r defined as follows:

2 = eo,v + ea2. (26)

Th6~,~j and Q~ are combinat ions of the Fk,.p~ appear ing in equat ion (23). The latter are given by [: ] ~ ; ~ J l ( ( . r ) d r f ~ q ~ ' J l ( ( " r ) d , (27)

n=l 0

which leads to the results shown in Table 1.

TABLE I Vahtes of )'tj and Ql

Coefficient Values [Number of terms in equation (27): 2 I0 I I ]

)'11 = 3Fllt l --162"22 --166"22 --166-22 7J2 = 2(2/-'1212 + /-"1122) = 72J --848"34 --883"80 --883"80 7Z3 = 2(2/-'131~ + F1133) = 7r --1428"6 --1644"8 --1644"8 722 = 3F2222 -4991 "2 -5552"1 -5552"1 )'23 = 2(2F3232 + ]-'2233) = 732 -10333 --14220 --14220 ~33 = 3Fas33 -27914 -34401 -34401 QI = 2F1223 -I- / ~ 1 3 2 2 = Q3 -331.04 -556.77 -556.77 Qz=2QI -662.08 -1113.5 -1113.5

In the present numerical example, considerat ion is restricted to the si tuation when 2 is near one of the first three natural frequencies; hence, the values o f the remaining.hLar e irrelevant, as will be shown.

Because the calculation is to be s topped after obtaining one te rm in the expansion, all T, for n/> I are considered constants, and in the following, a pr ime denotes differentiation with respect to 7"1. For convenience, polar nota t ion is int roduced as follows:

A. = �89 exp(i:t.), (28)

where a,(T1) and ~,(TI) are real. Substi tuting equat ion (28) into equations (25) and separat ing the results into real and imaginary parts , one obtains the following equat ions: f o r k = 1,

col(a~ + C1 al) + ~QI a2 a3 sin fl - ~61~. P1 s inlq = 0, (29a)

~'O1 a l ~ -{-~ ( ) ' I t a~ -']'-)'12 al a 2 + ],'13 ax a 2 + Q la22 a3 c o s / / + o1 ~ ' Y I j a.]] +�89 PI cos /q ~ 0 ~ \ j=4 /

(29b) f o r k = 2 ,

oJ2(al + C2 a2) + ~tQ2 al a2 a3 sin fl - �89 sin ll2 = O, (30a)

OJ 2 a 2 r i -k- ~ ()'22 a23 + )'21 a2 a 2 + )'23 a2 a32 + Q2 al a2 a3 cos/3 \

~

+ a2 ~ Y2j + �89 P2 cos P2 = 0; (30b)


for k = 3,'

f o r k > 3,

where

033(a~ + 6'3 a3) - ~'Q3 al a, 2 sin fl - ~(~3NP 3 sin lt3 = 0,

coa a.3 ~ + "~ 0'33 a 3 + )'31 a2 a3 + )'a2 a2 aa + Q3 at a 2 cos fl

q- a3 j~ff4"Yaj. -}- �89 COSII3 = 0 ;

(31a)

(31 b)

03k( a~, + Ck ak) -- "~ 6tu Pk sin ltk = O,

03kak~, + ~a~ ~ vkja~ + ~6kNPkCoSltk = O, J - t

(32a)

(32b)

I~, = a2 Tt - c~,, (33a)

fl = cq + 2ct2 - ct3 + al 7"1. (33b)

For the steady-state solution, all the a,, fl, and the appropriate/~, are constants. Hence, it follows immediately from equation (32a) that ,..

ak = 0 for k > 3 (34)

when 2 is not near co~. The terms containing QI, Q2, and Q3 appear as a result of the internal resonance; because

of these terms one cannot accept the conclusion indicated by equation (34) for k = 1, 2, and 3 without further study. Next consider the following possibilities: 2 is not near col, to2, or co3; 2 is near'col; 2 is near to2; and 2 is near to3.

4.1.2 Is NOT NEAR tot, (02, OR 033 ( N > 3)

In this case one can establish that in the first approximation none of the first three modes is excited. To do this, suppose a, and a3 are not zero. Then it follows from equations (30a) and (31a) that

a2 = C3033Q2 \a3/ C2 032 Qa" (35)

This is an impossible relationship because Q2 and Q3 have the same sign. Consequently, a2 = a3 = 0. Then it follows from equation (29a) that at = 0. Thus, if2 is not near the frequency of one of the modes involved in the internal resonance, then none of these modes are excited in the first approximation.

The amplitude, ak, the frequency shift, ct~, and the phase, cq, can be found from equations (32) and (33) as functions of the amplitude, Pk, and the detuning, tr2, of the excitation and the damping coefficient, Ck. It follows from equation (33a) that

tXk = 0"2 Z l - - ,ttk,

and hence

Ak exp(i03, To) = �89 exp[i(03k + ecr2) t --/tk].

Thus, in the first approximation the steady-solution has the form

Wo = ak qS~ cos (2t - irk). (36)

Note that equations (32) yield the steady-state solution of Duffing's equation for a hardening spring. Hence, there is a range of the frequency of the excitation for which two stabler

1" The m a n n e r in which the stabili ty of the various branches of the solut ion was studied is briefly described in section 5: �9

368 s. SRIDHAR, D. T. MOOK AND A. H. NAYFEH

solutions and one unstable solution exist. In this frequency range, the initial conditions determine which branch represents the actual response.

4.2..). IS NEAR 0)1 (N = 1)

In this case one can establish that in the first approximation only the first mode is excited. To do this, suppose that a 2 and aa are not zero. Then it follows from equations (30a) and (31a) that (_/2

a2 (37) \a3/ C2 0)2 Q3

This is an impossible relationship because Q2 and Q3 have the same sign. Consequently, a2 = aa = 0. Hence, in the first approximation the solution has the form given in equation (36). The comments concerning the multiple branches which were made in section 4.1 apply here as well.

Note that, for the vibrating beams considered in references [10] and [I 1 ], only two modes are involved in the internal resonance (the frequencies are in the ratio three to one). Also it was found that, when the frequency of the excitation is near the fundamental, both modes are always excited.

4.3. 2 Is NEAR 0) 2 (N = 2)

In this case, one can establish that in the first approximation only the second mode is excited. To do this, suppose a, and aa are not zero. Then it follows from equations (29a) and (~31a) that

a.-!i2=Ca0)3Q* (38) a3] Ci col Q3 "

This is an impossible relationship because Q, and Qa have the same sign. Consequently, al = a3 = 0, and in the first approximation, the solution has the form given in equation (36). The comments concerning the multiple branches which were made in section 4.1 apply here as well.

4.4. 2 Is NEAR 0)3 (N = 3)

In this case, one can proceed as above and suppose that a, and a2 are not zero. Then it follows from equations (29a) and (30a) that

a_2/2 C2 0)2 Q,

a2/= c-S 0)--7 Q-S" In contrast with equations (35), (37), and (38), this relationship is possible. Thus, there are two possibilities: either both at and a2 are zero or neither one is zero. The two possibilities are discussed separately next.

In the first approximation when a, and a2 are zero, the solution has the following form: Wo = aaq~3cos(2t -/ t3). The amplitude, a3, is plotted as a function of the detuning of the excitation, ca2, in Figure l(a) and as a function of the amplitude of the excitation, eP3, in Figure l(b). The jump phenomenon in Figure l(a) is perhaps more familiar than the one in Figure l(b). Recently, the latter wag observed in an experiment involving strings by Eller [15], and both are discussed by Stoker [16], for example. This solution is similar to the solutions for the cases discussed in sections 4.1, 4.2, and 4.3. Those solutions are not presented.

In the first approximation when a, and a2 are not zero, the solution has the following form:

Wo = a, q~, cos[(ah + ecrU) t + .~l] + a2 t~2 cos[(w2 + e0~) t + T2] + a3 qb3 cos0.t --/q). (39) !

SYMblETRIC RESPONSES OF CIRCULAR PLATES 369

12

I0

8

6

4

2

- 2

i I I I I I i I l (o)

. / s / .

f d I I F ~ l I t r

- I 0 2 3 4 5 6

~o" 2

i i i i . = . _ _ . _ _ _ = . _ i

61 (b)

\

2

I

I 0 100 Z(X) 300 4 0 0 ~

~P3 Figure i . The amplitude of the steady-state response aj, (a) as a function of the detuning of the excitation,

ta,, a n d (b) as a func t ion o f the ampl i tude o f the exci tat ion, tPa, for the s ing le -mode solu t ion . (a ) ) . near co3; eP3 = 1 5 0 : (b ) ) . nea r m3(ea2 = 1"5). - - , Stable; - - - , uns tab le ; t = 1-057 x 10-3; t C = 0.01 ; a~ -- a , = O.

In this case, a, , a2, a3, ct~, ~ and Pa are obtained as fol lows. One finds from equat ion (33a) that

~ = a2 (40a) and hence, f rom equat ion (33b), that

�9 t a2 - al = cq + 2~z. (40b)

;then one can combine equations (29)-(31) to obtain

Cl al + 8-~ a2,as sin fl O, (41a)

Q~ C2 as + ~ al a2 as sin fl = O, (41 b)

Ca as - = - - al a2 sln fl - ~ sin P3 = O, (41 c) ~ ( , o 3

1 P3 a2as +.~m3(?3taza~ + ~,32asa~ + rs3al + Osala~cosfl) + ~ c o s p s = O , (41d)

370

and

S. SRIDHAR, D. T. MOOK AND A. YI. NAYFEt:[

a 2 - ~rl + ~ [(~'11 + 2~'2t / a~ + ('~12 + 2~22/a~ + (7,._]3 + 2 ~ a _ ) a ~ +

Q'a2a3 +2Q2ata3)cos~] =0 . (41e) tOl a l to 2

One can solve these equations numerically for aa, a2, a3, fl and It3. After this is done, one can �9 obtain ctl and ct~ from equations (29b) and (30b); cq and ct2, and hence the constants rx and r2 appearing in equations (39), cannot be determined from this analysis. This phasing depends on the initial conditions. Note that, if the plate were forced at two frequencies, each near a different natural frequency, then the phasing would be determined as in the cases when only a single mode is excited in the first approximation. And if the plate were forced at three or more frequencies, each near a different natural frequency, no steady-state solution would exist in general.

The non-linearity adjusts the frequencies so that the third mode is tuned to the forcing frequency exactly and the frequencies of the three modes involved in the internal resonance

( 0 ) i I I ' i I I i y I

I ;~ 0 i

IO

8

6

4

0 2 3 4 5 6 7

~o- z

(b) I l i

t I I I I 0 I00 200 ~00 400 500 600

Figure 2. The amplitudes of the strongly excited modes, a~, a,, and a~, (a) as functions of the detuning of the excitation, c~r,, and (b) as functions of the amplitude of the excitation, eP3, for the three-mode solution. (q)'2 near to3; eP3 = 150: (b) 2 near to3(eo', = 1.5). - - , Stable; - - - , unstable; e = 1.067 x 10-3; ,'C= 0.01.

SYMMETRIC RESPONSES OF C I R C U L A R PLATES 3 7 1

are exactly commensurable. To explain this, one can recall equations (40a) and (40b) and note the following:

c01 + e ~ + 2(co2 + e~t~) = col + 2co2 + e(r - al) = 2.

The amplitudes at, a2, and a3 are plotted as functions of err2 in Figure 2(a) and as functions 0feP3 in Figure 2(b). Generally, when a~ and a2 are not zero, a~ is larger than a 2 and a2 is much larger than a3. Consequently, the solution is dominated by the first mode. Figures l (b) and 2(b) illustrate that, when the amplitude of the excitation is small, there is only one branch tO the solution and it is practically indistinguishable from the solution of the linear problem. For convenience, all the calculations were made with C1 = (72 = C3 = C.

2 0 - - I I i i I i

-

1 6 - -

,4

I , I , I t't I , 0 2 4 6 8 I0 12 14

~o" 2

Figure 3. The amplitudes of the strongly excited modes, a,, a,, and aj, as functions of the detuning of the excitation, ctr~, for two values of the damping coefficient, eC. 2 near ~ ; e = 1"067 x 10-3; cP3 = 460.

Figure 3 illustrates that decreasing the damping coefficients has the effect of extending the range of frequency for which at and a2 can be different from zero. Figure 4 illustrates that increasing the amplitude of the excitation has the same effect as decreasing the damping coefficients. In these two figures, only the stable branches of the solution are plotted.

I I I I l I -- 20

8 , ~Pa" I SO

4

�9 ~ , . . . . . . . . . ~ / 0 ~ (P3=150 03 , i : P 3 . 4 6 0

" - - ' r - - , - ~ - ' - - - ~ , i i/ i 0 2 4 6 - 8 I0 12 14

,Eo" z

Figure 4. The amplitudes of the strongly cxcited modes, az, a2, and a3, as/unctions of the detuning of the excitation, r for two values of the amplitude of the excitation, r near c~; c = 1.067 x 10-3; ~C= 0"01.

' Finally Figure 5 is a log-log plot of the amplitude of the excitation as a function of the amplitude o f the response. For this graph, 2 is exactly to3; consequently, it is not possible for

372 S. S R I D H A R , D. T. M O O K A N D A. H . N A Y F E H

I 0 0 0 I I 1 I I

I00

I0

I

0.1

0.01 t ~ O'OI 0-1 I I0

Q3

Figure 5. Log-log plot of the amplitude of the excitation, cP~, as a function of the amplitude of the response, a3, for the single-mode solution. 2 = toj; c = 1-067 • 10-3; eC= 0-01 ; al = a, = 0.

ax and a 2 to differ from zero. This graph illustrates that the present results are qualitatively in agreement with the experimental data obtained by Jacobson and van der Heyde [17], though they experimented with honeycomb panels.

5. STABILITY

The stability of the various branches was determined by adding an infinitesimal disturbance to the steady-state solution. From equations (29)-(31), one can obtain a system of linear, homogeneous equations, having constant coefficients, which govern the disturbance. Conse- quently, the disturbance will be of the form exp(MTl) where M is an eigenvalue of the coefficient matrix. I f the real parts ofall the eigenvalues are negative, the branch is said to be stable; otherwise, it is said to be unstable. More details can be found in reference [10].

6. CONCLUDING REMARKS

Here the same governing equations have been used as by several previous investigators, and all the single mode solutions obtained in the present paper are in essence the same as the solutions obtained by them. However, all the earlier studies apparently failed to reveal the additional, multi-mode solutions which can exist when the natural frequencies are commensurable (i.e., when an internal resonance exists).

For the example problem considered in the present paper, the first three modes are involved in the internal resonance. When the frequency of the excitation is near that of either the fundamental or the second mode, it is only possible to excite one mode, essentially in agreement with the earlier solutions. However, when the frequency of the excitation is near that of the third mode, it is possible to excite either the third mode only or all three modes; the latter possibility apparently was overlooked before. When the lower two modes are also excited, the fundamental mode dominates the response. Consequently, the response can


strongly resemble the response which occurs when the frequency of the excitation is near that of the fundamental mode.

For the examples involving beams (references [10] and [11]), only the two lowest modes are involved itt, the internal resonance (3co~ ~ o92). When the frequency of the excitation is near that of the fundamental mode, both modes must be excited, in contrast with the results for plates. However, the amplitude of the second mode generally is much smaller than that of the fundamental. When the frequency of the excitation is near that of the second mode, either the second mode alone is excited or both modes are excited. I f both modes are excited, the fundamental mode dominates the response, similar to the results for plates. Consequently, in the examples involving beams and plates, there can be a significant transfer of energy down from the highest mode to the lower modes, but a significant transfer of energy in the opposite direct!on cannot occur.

ACKNOWLEDGMENT

This research was supported by the NASA Langley Research Center under Contract No. NAS 1-10646-13.

REFERENCES

1. N. YAMAKI 1961 Zeitschriftfiir angewandte Mathematik und2~fechanik 41, 501-510. Influence of large amplitudes on flexural vibrations of elastic plates.

2. J. G. EtSLEY 1964 Zeitschriftfiir angewandte Mathematik undPhysik 15, 167-175. Non-linear vibration' of beams and rectangular plates.

3. G. C. KUNG and Y.-H. PAO 1972JournalofAppliedMechanics39, 1050-1054. Non-linear flexural vibrations of a clamped circular plate.

4. C. E. FARNSWORTH and R. M. EVAN-IWANOWSKI 1970 Journal of Applied Mechanics 37, 1043- 1049. Resonance response of non-linear circular plates subjected to uniform static loads.

5. C.-L. HUANG and B. E. SANDMAN 1971 blternationalJournalofNon-linear Mechanics 6, 451--468. Large amplitude vibrations of a rigidly clamped circular plate.

6. B. E. SANDMAN and C.-L. HUANG 1971 Developments in Mechanics 6 (Proceedings of the 12th Midwestern Mechanics Conference), 921-934. Finite amplitude oscillations of a thin elastic annulus.

7. C.-L. HtrANG 1973 International Journal of Non-linear Mechanics 8, 445-457. Finite amplitude vibrations of an orthotropic circular.plate with an isotropic core.

8. J. A. BENNETT 1971 American blstitute of Aeronautics and Astronautics Journal 9, 1997-2003. Non-linear vibration of simply supported angle ply laminated plates.

9. L. W. REHFIELD 1974 American blstitute of Aeronautics and Astronautics Journal 12, 388-390. Large amplitude forced vibration of elastic structures.

10. A. H. NAYFEH, D. T. MOOK and S. SRmHAR 1974 Journal of the AcousticalSociety of America 55, 281-291. Non-linear analysis of the forced response of structural elements.

11. A.H. NAVFEH, D. T. Moor: and D. W. LoBrrz 1974 American blstitute of Aeronautics and Astro- ' nautics Journal 12, 1222-1228. A numerical-perturbation method for the non-linear analysis of

structural vibrations. 12. G.J. EFSTA'rmADES 1971 Journal ofSoundand Vibration 16, 231-253. A new approach to the large-

deflection vibrations of imperfect circular disks using Galerkin's procedure. 13. A. H. NAVFEH 1973 Perturbation Methods. New York: John Wiley and Sons. See Chapter 6. 14. A. W. LEISSA 1969 NASA-SP-160. Vibration of plates. 15. A. I. ELLER 1972 Journal of the Acoustical Society of America 51, 960-966. Driven non-linear

oscillations of a string. 16. J. J. STOKER 1950 Non-linear Vibrations. New York: John Wiley and Sons. See p. 94. 17. M.Y. JACOBSON and R. C. W. VAN DER HEYOE 1972 Journal of Aircraft 9, 31-42. Acoustic fatigue

design information for honeycomb panels with fiber-reinforced facings.

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Non-linear resonances in the forced responses of plates, part 1: Symmetric responses of circular plates