3 subharmonic solution of elliptical sandwich plates

Applied Mathematics and Mechanics (English Edition, Vol 24, No 10, Oct 2003)

Published by Shanghai University, Shanghai, China

Article ID: 0253-4827(2003) 10-1147-11

1/3 S U B H A R M O N I C

S O L U T I O N OF ELLIPTICAL

SANDWICH PLATES * LI Yin-shan (~j~I~LLI)1,2,

ZHANG Nian-mei ( ~ z ~ )

2,

YANG Gui-tong ( ~ r ~ j ~ ) 2

( I. Institute of Engineering Mechanics, Hebei University of Technology, Tianjin 300130, P.R.China; 2. Institute of Applied Mechanics, Taiyuan University of Technology, Taiyuan 030024, P.R. China) (Contributed by YANG Gui-tong, Original Member of Editorial Committee, AMM) A b s ~ a c t : The problem of nonlinear forced oscillations for elliptical sandwich plates is dealt with. Based on the governing equations expressed in terms of five displacement components,

the nonlinear dynamic equation of an elliptical sandwich plate under a harmonic force is derived. A superpositive-iterative harmonic balance ( SIHB ) method is presented for the steady-state analysis of strongly nonlinear oscillators. In a periodic oscillation, the periodic solutions can be expressed in the form of basic harmonics and bifurcate harmonics. Thus, an oscillation system which is described as a second order ordinary differential equation, can be expressed as fundamental differential equation with fundamental harmonics and incremental differential equation with derived harmonics. The 1/3 subharmonic solution of an elliptical sandwich plate is investigated by using the methods of SIHB. The SIHB method is compared with the numerical integration method. Finally, asymptotical stability of the 1/3 subharmonic oscillations is inspected. Key words:

elliptical sandwich plate; superpositive-itemtive harmonic balance (SII-IB) method; 1/3 subharmonic solution; bifurcation

C h i n e s e Library Classification n u m b e r s : 0322; 0 3 4 3 . 9 2000 M a t h e m a t i c s Subject Classification: 741-I60; 74D10

D o c u m e n t code: A

~roducfion In recent years, with the essential advantage of light weight and high rigidity, sandwich e,ates and shells have been used as an important pattern of structural elements in aeronautical, ~stronautical and naval engineering. However, nonlinear problems for sandwich plates and shells am only investigated by a few because of the difficulties of nonlinear mathematical problems. Lin

*

R e c e i v e d date: 2002-01-29 ; R e v i s e d date: 2003-05-27

F o u n d a t i o n i t e m s : the National Natural Science Foundation of China (10172063); Shanxi Foundation of Science and Technology (20001007) ; the Key Project of Ninth Five-Year Plan of National Natural Science Foundation of China (19990510) Biography: LI Yin-shan (1961 - ), Associate Professor, Doctor (E-mail: liyinshan@ eyou. com)

1147

1148

LI Yin-shan, ZHANG Nian-mei and YANG Gui-tong

Ren-huai and Xu Jia-chu h'2] and others have made some investigations in this field. Bifurcation of nonlinear vibration for sandwich plates has not yet been investigated. In this paper, 1/3 subharmonic solution of elliptical sandwich plates is solved by superpositive-iterative harmonic balance (SIHB) method. At present a new and developing subject --chaos starts a broad prospect for analysis of nonlinear system [3-7] . The strongly nonlinear problems are difficult to solve by the classical procedures such as perturbation methods. Their main limitation may be generally due to the unreasonable assumption of the constant frequency. Recently over the past ten years, in spite of a series of findings of studies on strongly nonlinear oscillation [8~9] , quantitative studies on superharrnonic and subharmonic solution remain to be solved, which now has to be solved by numerical methods El~ . A SIHB method is presented for the steady-state analysis of strongly nonlinear oscillators. The question of steady-state solution to a strongly nonlinear dynamic system is changed into less nonlinear algebra equation groups. Program Maple can be used conveniently to solve the nonlinear algebra equation groups so as to obtain an approximately analytic representation of steady-state solution. Subharmonic or superharmonic solutions are obtained by superposition and iteration step by step. 1

Basic Equations Consider an elliptical sandwich plate under the action of an even transverse excitation

Qocoss 0 t , as shown in Fig. 1. Here a is long radius, b is short radius, h I is the thickness of the face, h 0 is the distance between middle surface of the upper and lower face. The coordinate plane xy coincides with the middle plane of the core. The two face sheets are assumed to have identical

material properties and the same thickness.

g(-,y,I) S

I_ F

Cl

T

,i

Fig. 1

Q

-1

Coordinate and geometry size of elliptical sandwich plate

The theory of isotropic sandwich plates with very thin face sheets and soft cores was first presented by Reissner. The basic assumptions are as follows: 1 ) The material is elastic and follows Hooke's law; 2) The core in the lateral direction is incompressible; 3) The core is not subjected to loads in the middle surface direction; 4) The faces are treated as membranes; 5) The line normal to the middle surface of the core remains straight during bending. Under the above assumption, using Hamilton's principle, the nonlinear dynamic equations of an elliptical sandwich plate under a harmonic force is derived [4] . The governing equations is

Subharmonic Solution of Elliptical Sandwich Plates expressed in terms of five displacement components, u, v, w, r

02~-"

+

2 o2v 8y 2 + (1

and ~by[l'2] -9

(02~ aw ) --af_+~;v2w +

+(l-v) (1

1149

v){ 32v

1

+ (8w/2 ] 3yy! j } = 0,

~

(la)

I O2v 3w ) v)(~SxZ~+ T y V 2 W +

( 1 + v){ 82u 1 3 Owl'-+(3wlZ ] a~ + 2- ~y[(~x-x! a-TY] ' } = D -+ yZ + + ~ + V-w 3x 3 3x~ 3xZay ay 3

i 7 -;'Ehl

~

+ ~~

0,%--~-

2

~j

+ 8

at"

Of

- Q0eos~t -

+ ~ :~x - 0 7 + ( 1 - ~ ) ~

+~

a-7~ , +

]072 +

Jay: + 2(1

+ ~J

(lb)

] 2"02 w

-= [ ~

~

= p~

0,

V] -~x 3~x OxOy ,

(lc)

1 + v "r Ow D a2G 1- ~a2G + _ - G + Y 7 = 0, (ld) G2h o ~ + 2 3y 2 2 D 1+ v G 1 - v 32r 02r I ~ aw G,. ho q- 3xOy + 2 3 x ,~ + 3yz:]~ G + ax 0' (le) where u, v and w are x displacement and y displacement and deflection of the middle surface of the plate, respectively; r Cy represent the rotations of a line segment, which is perpendicular to the middle plane Oxy before deformation, in the xz plane and yz plane, respectively. E and v are Young's modulus and Poisson's ratio of the face, respectively, Gz is the shear modulus of the core, D = Eho hi/(2( 1 - vz ) ) is the flexural rigidities of the Sandwich plate, p --- po + 2pf, po and pf are the mass densities per unit area of the face and the core, respectively. Let us introduce the following nondimensional parameters: a

- b"

~-

a c~9 = /77or , a4 qo = D~o Qo,

x

Y--~

~'

~y

v = b'

a = - ~oCy,

U

k -

-

au

V

h~'

D G2

ho a 2

,

by

h~' r =

w

re_~o,

& t ,

(2)

~a 2 /2 =

o,

2n -

ho CpD"

With these quantities, Eq. ( 1 ) become

2 3ZU

a~--r_ +

v)[2 zO2U

(1

-

3WIOZW

57v,_ +gg~

22 8z-W-- I

+

a~eav]]+

92V 1 0 oWl'o W e] (1+ v){2zo--~-~+ ~-~-~[(se / + 22(-~) } = O, 222-

av:

+ (1 - v)

V

OW

W

5g + ~v 5-g +

22

av~ ~ +

(3a)

1150

LI Yin-shan, ZHANG Nian-mei and YANG Gui-tong (1 + ~),[ 2n 3 W

3 ~-W

a2U

@y 33 (~Sy _ • ~3 3 7]

33 (~Sx _ A2 33 (~Sx

aV l a 2 W l _ 2 [

(I

-~-J

awl'- 1

awl 2

1 ~

aW

_

2

w

.I ar]2 -

4(1 - ; ) ; t 2 a W a W

a2W 3~ 87] 3~3r1 - q0cos~v = 0,

(3~,[

k

.~.28'-#~

;~ ~

+

(3c)

1+ ,. 3'r

2

3~ 2 +

Oy _

~ r + '~

= 0.

(3e)

The boundary conditions for an elliptical sandwich plate is rigidly clzmped edge. Eq. (3) will be solved by the following nondimensional boundary conditions usually encountered in engineering : U = V = W = ~ = ~y = 0, at ~- + 7/2 = 1. (4) To obtain an exact solution of Eq. ( 3 ) with constraint equations ( 4 ) is not a task. A modified Galerkin's technique appears most suitable for an approximate solution. It is assumed that i=O j=0

V = ~_~___jbii(r)(1- $ 2 _ ?]2)~2i712j+1 ,

(5b)

i=O j=O

W = 9(v)[1

2 + + 16k. ,

+

+

1 + 1 6k-( C-

+

r/2 )2],

(5c)

T]2)~:21+1 v2j,

(5d)

~_a~__adij(v)( 1 _ ~2 _ y]2)~2iT]2j+l ,

(5e)

C~) = G ~ d r

i=0

1

~2

rf- )

j=O oo

ff)y

/=0 j=0 where 9 ( r )

is an unknown function of r , the maximum is 9.,= = Wm = wm/ho, Wm is the

centric deflection of an elliptical sandwich plate. Substituting (5) into (3) and applying Galerkin' s procedure, we obtain a nonlinear ordinary differential equation for the time function 9 + 2n~ + p2 9 +,up 3 = q c o s ~ r , (6) in which p is the natural frequency of an elliptical sandwich plate, ( ~ _~) 1/2 ~2 ~4 p = , /z - a 3 ' q = ~3 q ~ where

(7)

Subhannonic Solution of Elliptical Sandwich Plates fl

f 1%/1-~ ~ [ 8 3 ~

,~2 8 3 ~ x

83~Y

,t3 83~y] 2 + 16k 1 ~ - 3 1 [1 - 1 + 16k (t2 + ?]2) + 1 6+~

f

0/2=-

l

("l'v/~-~2 l" [

8U

+ ~2(8r

4(1 - v)), 2

1

1 + 16k

f

3?]

$z

+

?]z)2Jdr/d~ '

v))'z_.,

l f~--r/2 -lJ-~

(8a)

~-10---~J+

+

v( 8~/'- + ~(8e/2 ]sZW 3 7?2

2 r + 2;-[

- -

2 + 16k(~z

[1 _ 2_ +_ l_6k 1 + 16k

2 + 16k [1

_1 _+( ~16k z

~2

?]2 )~2i+ 1 9j

+

+ ?]2)-I-

1+1~"

3-g~Jt 1

(~2 + 712)2]d?]d~ '

0/3 = jfl_ l jf_4~~ - v " 0/4 =

-

OWSW 8ZW~[ 3~

(

8~ r 8 2

) z ( 2 ~ _~+3~' v_ff~_)8?]28~-/82~-+ (1

3~12

1151

(8b) + ?]2) + ~ ( 1

+ ?]z)2

1

+ 7/2) + 16------k 1+ (

~2

dr/de,

+ r/2)Z]d?]d~,

(8c) (8d)

where U = ~ 2 ( 1 i=0 j=0

= 1 ~

= iffi0 j=0

2 + 16k ( ~z ?]z ~ 1 + 16k + ) + 1

= 22(1

i~0 j=0

(~2 +

- ~ z _ ?]2)~2~+1@i,

?]2

)2,

~, = 2 2 ( 1 _ i=0

~ 2 _ ?]2)~zi~zj+,.

jffiO

Equation (6) is a Duff'rag's equation for a hardening spring. Undimensional parameters are introduced as foUows: ~ s , e = pr, g, q = qff-fi p3 , p , P P and then standard Duffing' s equation can be obtained. While doing that, ~, ~ , ~ and ~ must be changed respectively over to t, s q and n :e +2n.r + X + x 3 = qcosY2t. (9) x

2

=

A Superpositive-Iterative Harmonic

Balance Method

Instead of special example ( 6 ) , we treat general systems. Thus we consider the forced oscillations of strongly nonlinearity systems having a single degree of freedom: + 2n~ + f(x) = qcosg2t. (10) The procedure of the SIHB method for seeking periodic solution is divided into two main steps. The first step is Newton-Raphson procedure. Make a hypothesis:

1152

LI Yin-shan, ZHANG Nian-mei and YANG Gui-tong

x ( t ) + Xo(t) + y ( t ) ,

(11) in which X o ( t ) is a basic

which is a periodic solution obtained by solving equation ( 1 0 ) , solution, and y ( t )

is a derived solution. The fundamental solution meets the fundamental

equation : ~0 + 2n~0 + f ( x 0 )

= qcost2t.

(12)

The derived solutions meet the incremental equation: y

+ 2rty +

21f(k)(x) k= 1

n"

yk

i

(13)

= O.

.'g = x o

The second step of the SIHB method is Ritz's mean course. According to form ( 1 2 ) , basic harmonic solution can be obtained by Ritz' s mean method: x = a 0 + alcosOt + blsinOt.

(14)

According to form ( 1 3 ) , the derived solution y ( t ) can be solved Ritz' s mean method step by step. 3 3.1

Analytical .Solution Basic harmonics solution A hypothesis of basic harmonic solution of Eq. (6) formed as follows q~( t ) = al ( t ) c o s / T t

+ bl ( t ) s i n O t .

(15)

If the basic harmonic solution ( 1 5 ) is taken into Eq. ( 6 ) ,

by means of Ritz' s mean

method, a group of nonlinear differential equations concerning amplitude parameters can be obtained 3 al + 2hal + 2Ob I 4" ( p 2 _ f22)al + 2nOb 1 + _ ~ t t a l ( a i + b~) - q = 0, 3 + 2nD1 - 2nF2al + (P'- - /72)bl + - ~ t t b l ( a 7 + b l ) = 0.

bl - 2s

(16a) (16b)

Steady-state basic harmonic solution is ~(t)

= glcosOt + blsinOt.

(17)

Whose coefficients are singularities of differential equation group ( 1 6 ) , which can be obtained by solving the following group of nonlinear algebraic equations: 3

_z

(p2 _ y22)g1 + 2rtObl + ~ - t t g l ( a 7 + b~) - q = 0,

(18a)

3 - 2ng2~ + (pZ _ ~ 2 ) ~ 1 + ~_t~bl(~2 4- ~2) = 0.

(18b)

The stability of steady-state basic harmonic solution (17) can be judged by the characteristic roots of the following linearized coefficient matrix of differential equation group (16) : 3albl

4/7 A 1 =

3.2

1 - /72 2/7

n

3 ( 3 a l2 + b)) 8/7

1 - g'22 3 ( a ~ + 3b~) + 20 80

-

3 -

It

-

~-~a

1

b1

1/3 s u b h a r m o n i c s o l u t i o n A hypothesis of 1/3 subharmonic solution of Eq. (6) formed as follows :

(19)

Subharmonic Solution of Elliptical Sandwich Plates ~(t)

= ~l(t)

1153

+ ~xn(t),

(20)

in which fundamental solution q~l ( t ) is

qol(t) = a l ( t ) c o s ~ t

+ bl(t)sin~t;

(21a)

Derived solution 91/3 ( t ) is Ot ~t = a m(t)cos-~- + bm(t)sin-~-.

~m(t)

(21b)

Substituting (20) into ( 6 ) , we obtain an incremental equation qgl/3 + 2/~113 +

p2

2

3/zpl @2/3

~01/3 + 3/tq91 ~91/3 +

(22)

"1- ~t@3/3 ----- 0.

Substituting (20) into ( 6 ) , a group of nonlinear differential equations concerning amplitude parameters can be obtained by Ritz' s mean method:

al + 2net1 + 2Obl + (p2 _ O 2 ) a l + 2ng-2bI _ q + ,u ~ a l ( a ~

+ b21 + 2ai/3 + 2b~/3) +

"bl + 2nbl - 2 0 a l

+ (P'- - OZ)ba - 2 n O a i

+ 3bT13) = 0,

+

~ 7 b a ( a i " + bi" + 2a~,3 + 2b~3) + - ~ b m ( 3 a ~ , 3 al/3 + 2 n d l / 3 + ~ ' ~ b l / 3

+

-

(23a)

f2 2 ai/3 + ~ n , Q b l / 3

- b~/3

= O,

(23b)

+

{ 3 ~ 2 tz.~al/,3(ay/3 + ba/3 + 2a 2 + 2b~) + 3 , ~al(aT/3

,

3

- b])3) + ~ b l a l / 3 b l / 3 .

bl/3 + 2nDt13 - ~-~dl/3 +

p2 _

~z

}

= 0,

(23c)

bm _ ~_ns

m +

/z { 3 bl/3 ( a~/3 + b12/3 + 2ay,, + 2b~) - ~3a l a t l 3 b i l 3 +

3

o - b~/3) }

~bl(a~/3

= 0.

(23d)

Steady-state 1/3 subharmonic solution is ~(t)

= ~l(t)

+ ~1/3(t),

(24)

in which steady-state fundamental solution is ~l(t)

= aieos~t

+ blsinl2t,

(25a)

steady-state derived solution is ~I/3 ( t )

~t

~t

= (~i/3eos ~ - + bi/3sin ~ - .

(25b)

Whose coefficients are singularities of differential equation group (23), which can be obtained by solving the following group of nonlinear algebraic equations: -~ (p'- - 0 2 ) ~ 1 + 2nY2D 1 - q + ,u -3- ~_a i ( a_ ,~ + D2 + 291z/3 + 2bl/3 ) +

1 _, -2 } ~a113(a~/3 - 3 b y 3 ) . = 0, 3 (p2 _ O~_)~a _2mQgz + ,u ~-ba(a~-" + ~2 + 2~2/3 + 2/~z3 ) +

(26a)

1154

LI Yin-shan, ZHANG Nian-mei and YANG Gui-tong 1 -

--'~

-2

~ b 2 1 3 ( 3 a ' ~ l 3 - bl/3)

(p~ _

1

~-22-

,)

3_

2

}

-

I3

51/3 - I - - ~ r k O b 2 1 3

_,

~ a 2 ( a~/3 --

b12/3) q-

(26b)

= 0, + /z[-~-

1/3

3-

(_,

a~13 + b12/3 + 291 + 2D}) +

}

(26c)

~b15113~)113 = O, 3-

p2 _

2 n22al13 _ ~ 222 ) b213 _ _.~ + /z {-~- b 1/3 ( a-'~i/3 + /~/3 + 2g~ + 2b}) 3 3} ~ a l S l 1 3 b l l 3 + -~-b1(52/3 - b}/3) = 0.

(26d)

The stability of steady-state 1/3 subharmonic solution (24) can be judged by the characteristic roots of the following linearized coefficient matrix of differential equation group (23). The nonlinear algebraic equations (26) can be deduced and calculated by Maple programme and iterative method. Iterative steps are First step

Solving equation group (18), initial values d~0) ,

obtained; Second step Third step

Let ga

d~~

=

,

b2

=

b~~

b~0) , t,~-(~

solving(26c), (26d)

,

,

-- 0 ,

= 0 are

ui'(~

to obtain ~213 ~(1)

(1) ; , ~01/3

Let al/3 = 5}13) , bl/3 = ~(t)1/3, solving ( 2 6 a ) , ( 2 6 b ) , to obtain 5}1) ,/~2) ;

...

Fourth step Fifth step

Let51

=

g~ k) ,

h I ----

--(k+l) , Let if1/3 = au3

b~k ) , solving(26c)

D 1 1 3 = -1/3 ~(k+l) .

,

( 2 6 d ) , t o o b t a i n o,x(k+l) 1/3

solving . (26a), (26b) .

~ i(k+l) t) 1/3 ;

to. obtain ff~k+2) ~k+2)

if [ ff~k+l) _ if{k) [ < e = 10-6 , then going on to the next step, else, going to the Fourth step; Sixth step

At last, to obtain coefficient of 1/3 subharmonic solution (24), 51 = 5~ k+2) ,

b2 = b~ k + l ) , ~ / 3 4

Numerical

= ~~ (mk + 2 ) , a l l 3

Calculation

= 19 (.3k + l ) , judging the stability of those coefficients (End)

Results

Investigating Duffing's equation (9) to the situation when n = 0.05, 22 = 6 . 3 , q = 40, by means of Maple program, Eq. (9) can be solved and determine the stability with characteristic roots. By means of Eq. (18), coefficients A, B, C of steady-state basic harmonic solution (17) are given in Fig. 2 ( a ) ,

in which B is a saddle point, seven 1/3 subharmonic solutions are

bifurcated to come from A, superharmonic solutions will be bifurcated from C ( I t will be discussed in another paper). f = -- 1.056 434 372 A : I al b2 0.017 582 713 07 {~i = 7.593 367 164 C:

/~a = 0.921 507 309 5

(focus) ,

f52 = - 6 . 5 6 3 3 9 2 7 9 1 B: l b2 0.685 889 977 4

(saddle point)

(focus).

The coefficients of steady-state 1/3 harmonic solutions (24) are given by means of Eq. (26), in which coefficients A o , A I , A

2 of fundamental solution ( 2 5 a )

are given in Fig.2(b)

coefficients O ~ (7) of derived solution (25b) are given in F i g . 2 ( c ) . { A~

= 51 bl

-0.017 1.056582 434713 37207

(focus) '

A1 -"

{ ~l = - 1 " 2 0 3 9 4 2 2 1 0 0 bl = 0.029 939 240 54

(focus),

and

Subharmonic Solution of Elliptical Sandwich Plates

A~_:

{~1 = - 1 . 1 0 9 5 6 4 257 bl 0 . 0 2 1 158 117 17

(~)

0"1/3_

= 2 . 0 1 2 720 533 00

b 1/3

0 . 0 6 0 974 947 77

{

( saddle p o i n t ) .

rdl/3 = -

(focus)

'

~) ~ bl/3

1.773 554 586 0

fly3

= - 0.998 901 094 9 0.091 263 497 09

~

0

(focus)

1 . 0 5 9 166 12

(focus),

1.712 579 638

t-a1/3 = 0 . 5 7 8 487 054 3 (fi) t b u 3 0 . 8 1 9 441 975 5

= - 0 . 9 5 3 554 412 7 t b 1/3

=

[bl~

1155

(focus), (saddle point),

(saddle p o i n t ) ,

I~a/3 = 0.420 414 040 50 (~) [bl/3 705 0.910 472 6

(saddle point).

Substituting those coefficients into ( 2 4 ) , seven 1/3 subharmonic solutions (27) are obtained, in which 1/3 subharmonic solution ( 2 7 b ) is given in F i g . 3 . 8

0.04

4

0.02

:..3"

B

-\

G

0

bl

A

.............

\

0

-4

- 0.02

-8

-0.04

-1 .......

-4

0

5

-i--O

4

1

2

4

3 -3

-1

3

1

14I

(a) Basic harmonics (b) Fundamental harmonics (c) 1/3 subharmonics n = 0.05, /2 = 6.3, q = 40, O - - f o c u s , O - - s a d d l e Fig.2 Amplitude distribution of 1/3 subharmonic solution on Van der Pol' s plane 15

t t

-1 II !~9149 i~

-5

II -9165 ..... .~.... -3

84 ,.i

0

....

5

2 i ....

10 t

i

....

15

(a) Time h~story Fig. 3

20

-1: -3

.i.i.~.i -1

1

3

iiiiiii

iii i

0f

i.

0

2

X

4

6

10

8

09

(b) Phase plane portrait (c) Frequency spectra 1/3 subharmonic solution (27b) by SIHB method

= 1 . 0 5 6 6 e o s ( 6 . 3 t - 3 . 1 2 5 0)

(27a)

(stability),

= 1 . 2 0 4 3 e o s ( 6 . 3 t - 3 . 1 1 6 7) + 2 . 0 1 3 6 c o s ( 2 . 1 t - 0 . 0 3 0 286)

(stability), (27b)

= 1 . 2 0 4 3 e o s ( 6 . 3 t - 3 . 1 1 6 7) + 2 . 0 1 3 6 e o s ( 2 . 1 t - 2 . 1 2 4 7)

(stability), (27c)

1156

LI Yin-shan, ZHANG Nian-mei and YANG Gui-tong = 1.2043 cos(6.3t-

3.116 7) + 2 . 0 1 3 6 e o s ( 2 . 1 t -

4.219 1)

(stability), (27d)

= 1.109 8 c o s ( 6 . 3 t - 3. 122 5) + 1.003 1 e o s ( 2 . 1 t - 0.956 09)

(unstability), (27e)

= 1.109 8 e o s ( 6 . 3 t - 3. 122 5) + 1.003 1 e o s ( 2 . 1 t - 3.050 5)

(unstability), (27f)

= 1.109 8 e o s ( 6 . 3 t -

3.122 5) + 1.003 1 c o s ( 2 . 1 t - 5.144 9)

(unstability) . (27g)

The numerical integration solution of 1/3 subharmonic solution is given by Runge-Kutta ( R K ) in Fig. 4, bifurcation diagram for forced amplitude is given in Fig. 5, in which time step is At

= T/35,

3f

T = 27t/g], initial conditions are x0 = 3, ~0 = 1.

~

15

!

!

!

!

14

!

1

-3 520

540

i i

. H

iiiiiiiiiiiiiiiilliii iii

2 0 iJ 2 0

- 15

530

i.

....

.

.

.

.

6 7-! ....

-5

-1

10 ....

-3

-1

1

3

4

6

8

k

10

t

(a) Time history Fig .4

(c) Frequency spectra (b) Phase plane portrait 1/3 subhannonic solution by RK method

.0 ,d o.o

i

~

!

j

0

9,

0

Fig. 5

5

9 ...

................... ......~, ........................ .~,

"~.': ....

|[

ir 25 q

30

Bifurcation diagram for forced amplitude ( n = 0.05, O = 6.3)

Conclusions

1) The 1/3 subharmonic oscillations for elliptical sandwich plates can be described by solving the hypothesis ( 2 4 ) . In t h e subharmonic frequency scope, the steady-state 1/3 subharmonic solution obtained by SII-IB method considerably tallies with that by numerical integration. 2) By means of SIHB, the numbers of 1/3 subharmonic solution, the stability of the solutions the analytic representations of the Solutions and bifurcate situation of the solutions can be obtained.

Subharmonic Solution of Elliptical Sandwich Plates 3) There are four stable 1/3 subharmonic solutions in

Fig . 2 ( c )

1157

in which one solution is 1/1

harmonic fundamental response yet, there is the same fundamental solution in the other three solutions. The 1/3 harmonics are symmetric, their phase difference is 27r/3. Focus 1 is a zero solution. Three foci ( 2 , 3 , 4 )

constitute a regular triangle. Three saddle points ( 5 , 6 , 7 )

constitute

a regular triangle too.

References: [ 1] [ 2 ] [ 3 ]

[4 ]

[ 5 ]

[ 6 ]

[ 7 ] [ 8 ] [ 9 ] E10 ]

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