nonlinear damped vibrations of simply-supported rectangular sandwich plates

Nonlinear Dynamics 8: 417-433, 1995. © 1995 KluwerAcademic Publishers. Printed in the Netherlands.

Nonlinear Damped Vibrations of Simply-Supported Rectangular Sandwich Plates

Z. Q. XIA and S. LUKASIEWICZ Department of Mechanical Engineering, University of Calgary, Calgary, Canada T2N 1N4

(Received: 23 March 1993; accepted: 8 September 1994)

Abstract. The nonlinear, forced, damped vibrations of simply-supported rectangular sandwich plates with a viscoelastic core are studied. The general, nonlinear dynamic equations of asymmetrical sandwich plates are derived using the virtual work principle. Damping is taken into account by modelling the viscoelastic core as a Voigt-Kelvin solid. The harmonic balance method is employed for solving the equations of motion. The influence of the thickness of the layers and material properties on the nonlinear response of the plates is studied.

Key words: Damping, vibration, sandwich, plate.

Notations

i=1 , 2 , 3 ti a,b ~ti, Vi, Wi ¢,¢

Ei Gi lYi 7i #i t2 = t2/tl t3 = t3/tl E2 = E2/ E1 E3 = E3/E1 Gi = G i / E i ~1 = a / b

0~2 ~ a/~l

F2 = 72/71 F3 = "/3/'/'1

p( , , y , t )

subscript corresponding to lower facing, core and upper facing respectively thickness of the i-th layer length of plate in x and y direction respectively displacement in i-th layer in x, y, z direction respectively rotation in the core in z, y direction respectively curvature in i direction elastic modulus of i-th layer transverse shear modulus of the core in i direction Poisson's rate of i-th layer mass density of i-th layer Voigt damping parameters in the viscoelastic core core thickness ratio facing thickness ratio ratio of the Young's modulus of the core to that of the lower facing Young's modulus ratio of the facings ratio of the transverse shear modulus of the core to the Young's modulus of the lower facing aspect ratio ratio of the length of the plate in x direction to the thickness of the lower facing core density ratio facing density ratio dimensionless parameter transverse force per unit area

Introduction

The linear, flexural vibrations of damped sandwich plates have been investigated in a number of papers. The earliest work on the damping of the ordinary sandwich plate was by Plass [1] who considered that the damping was entirely due to the shear in the viscoelastic core layer. Yu

[2], who published a series of papers on sandwich plates, also studied the damping of flexural vibrations. An appropriate set of equations of motion was adopted in the structural damping analysis of the composi te plates. Ho and Lukasiewicz [3-5] studied the influence of various

418 Z Q. Xia and S. Lukasiewicz

factors on the efficiency of structural damping and determined the optimum parameters for the sandwich plates. Alam and Asnani [6, 7] devoted their works to the vibration of the resonant frequencies and the system loss factors. He and Ma [8], Bisco and Springer [9], Shin and Maurer [10] also made contributions to the study of the damped sandwich plates. They used the asymptotic solution method, finite element method and performed experiments to verify the obtained results. Thus, the linear damped vibration analysis of sandwich plates seem to be well in hand.

In the case of nonlinear vibration analysis, most researchers studied undamped vibrations. Yu [11] derived a set of nonlinear equations describing the vibrations of sandwich plates. Extending from his linear vibration study of sandwich plates with very thin faces, he ana- lyzed the geometrical nonlinearities and shear effects. However, he did not consider damping. Similarly, Ebcio~lu [12] developed a general, nonlinear theory without considering the damping. The nonlinear vibrations of undamped composite plates were studied by Chia [13]. An informative review of this work can be found in [14].

Only a few works were done on the nonlinear damped vibrations. Kovac et al. [15] gave the early paper on nonlinear damped sandwich beams. Theoretical frequency-amplitude relations were obtained using Galerkin's procedure and the method of harmonic balance. Later, Hyer et al. [16, 17] continued this study using both theoretical and experimental methods concentrating their attention on determining whether the superharmonic was a structural effect or whether it was due to some other effects in the experiment. Iu and Cheung [18, 19] were the first to consider nonlinear damped vibrations of multilayer sandwich plates using the incremental finite element technique. For the analysis of large amplitude, periodic vibrations of multilayer sandwich plates, a reduced basis composed of eigenvectors corresponding to linear, free vibrations was introduced to reduce the order of the resulting linearized equations governing the nonlinear motion.

In our previous paper [20], the nonlinear, free, damped vibrations of sandwich plates were studied. The numerical results were compared with the results presented in some references [18, 19]. The aim of this paper is to continue this study in analyzing the influence of various structural factors on the behaviour of vibrating sandwich plates. Using the virtual work principle, a set of nonlinear equations is derived. To solve these equations, Fourier expansions of the time-dependent displacements are introduced and the harmonic balance method is used. Damping is considered by modelling the behaviour of the material of the viscoelastic core as a Voigt-Kelvin solid.

Nonlinear Equations of Motion

The cross-section of the sandwich plate in one direction is shown in Figure 1. The x, y axes are in the middle plane of the core of the undeformed plate defined as the reference plane. The positive direction of the axis z is normal to and down from the reference plane. The corresponding displacement components are ui, vi, w with i used to define the i-th layer respectively. To derive the governing equations, the assumptions are made as follows:

1. Each layer is bonded together perfectly and there is no slide between interfaces.

2. The first and third layer are considered as perfect elastic facings and the second layer as the viscoelastic core. The first and third layer have relatively higher moduli of elasticities as compared with those of the core, and can withstand both bending and membrane stress. Layer 2 is able to carry the transverse shear stress.

Nonlinear Damped Vibrations of Simply-Supported Rectangular Sandwich Plates 419

7

×

t , i

! t ,

Fig. 1. Configuration of a undeformed and deformed sandwich plate.

3. A straight line which is normal to the undeformed middle plane of the core remains straight after deformation but not necessarily normal to the middle plane of the core. The Kirchhoff hypothesis is only used for the facings.

4. Compression effects are neglected. The normal strains in the thickness direction are ignored. The transverse displacement is assumed to be the same for each layer.

The displacement components ul, v~ and w, that include the effect of transverse shear deformation, may be described by the following expressions:

t2 ~ = u ° - g (¢ - m,~)- z ~ , ~ ,

t2 u3 = u ° + ~ ( ¢ - w , ~ ) - zw,~,

u2 = u ° - z¢, ~2 = ~ - ~¢,

W i = W~

t2 vt = ~o _ ~ ( ¢ _ ~ , ~ ) _ ZWty~

t2 v3 = v° + -~ ( ~ - W'y)- ZW'y,

(1)

where u°, v2 ° are the in-plane displacements in the reference plane of the plate; ¢ and ¢ are the rotatory angles of the core in x and y directions, respectively.

Considering the nonlinear strains due to large transverse displacements, the following strain-displacement relations are obtained from equation (1). For elastic facings

~ixx : ~ixx + Z T + nixx~ ~iyy -" ~iyy + Z T + niyy,

420 Z. Q. Xia and S. Lukasiewicz

o ci~,., = q~:y+ q: + ~,:y, (2)

where

0 ~ixx =

0 ~ixy --

t2 ti 1 w2, 0 = v0 t2 ti 1 w,2y '

?z 0 V0 t2 2,~ + 2,~ q: 7 (¢'~ + ¢'2 T t~,x~ + w,~,~,

- - W , x x ~ I~iyy .-~ - -Wlyy , I~ixy = - - 2 W ' x y .

C2xx -- COxx ~ Zl~2xx,

C2xy = C O 2xy ~- Zl~2xy,

where

soxx = uO 1 :,~+ ~ ¢2,

the facings can be written as:

El (ei~ + uiciu), ai,: - 1 7 v ~

El aixY -- 2(1 - ui) cixy,

and for the core they become

o o 1 c2yy = v2,y + ~ ¢2,

COzy = U 0 2,y + v°,~ + ¢¢, c°= = - ¢ + ~,~,

C o 2yz = - ¢ + w,y, n2~ = -¢ ,~ ,

1~2yy -- - - ¢ ' y , g2xy = - - ( ¢ ' y -~ ¢ ' x ) " (5)

e ° are the nonlinear strains in the middle plane of the core. In these equations, a comma denotes partial differential with respect to the corresponding coordinates. The transverse shear strains in facings have been neglected.

After neglecting the influence of transverse normal stress, the stress-strain relations for

tY2x - -

~72y --

Ei (eiy+ uiei:), air - 1 7"t.'~

E2 1 ~- v2 2 ((e2x + # J 2 ~ ) + v2(c2y + #yg2y)),

E2 1 =~,2 ((eay + #yg2y) + u2(e2o: + #xeax)),

(6)

nixz = (3)

In equations (2) and (3), i = 1 and i = 3 indicate the corresponding components in the o first and third layer, e i are the nonlinear strains in the middle plane of the i-th layer, nj are

the curvatures in j - th directions of the facings. The curvatures in the first and third layer are assumed to have the same value expressed in terms of the derivatives of the transverse displacement w. The upper and lower overlapped algebraic signs are corresponding to i = 1 and i = 3 respectively.

The strain-displacement relations for the core have the similar form to those of the facings

0 C2yy = C2yy "Jr" Zl~2yy,

= C 0 C2xz = COxz, C2yz 2yz" (4)


E2 az~y - 2(1 ' v2) (E2~y +#~vg2~y),

ff2xz = G2xz(~xz + #xzgxz),

a2y~ = G2y~(Syz + #wgv~). (7)

The Voigt-Kelvin model is used for the stress-strain relation (7) of the viscoelastic material core #ij are the damping parameters determined from experiments [21]. For simplification, the Poisson's ratio ui for the first and third layers are assumed to have the same value Ul.

The dynamic virtual work principle requires

~((~x~slx + ~@~@ + ~ x y ~ y ) - 71(~ui + ~v~ + ~w)) "i=1

+ + + : o,

where 7i are the mass density of i-th layer per unit volume and p(x, y, t) is the transverse load per unit area. A dot denotes differentiation with respect to the time.

With the application of usual integration and variational procedures to the dynamic virtual work equation (8), a system of nonlinear equations in terms of five variables u °, v °, ¢, ¢, w can be obtained as follows

Kl(2U,xx + (1 - Vl)U,yy) + (1 + Ul)UlV, xy + K4(2u,xx + (1 - t]2)U,yy )

+ (1 + Y2)IC4V,xy + g2(2(9,xx + (1 - Pl)¢'yy)

+ (1 + ,~1)1c2¢,xy + g3(~,xxx + ~,y.y) + SC4(2~x~,~ + (1 - .2)Vx.~,.~)

+ Ksi~,xy + 2K4(¢¢,x + ¢¢'x) + (1 - u2)K4(¢,y ¢ + ¢¢,y)

+ 2Kl(w,xW,xx + ulW,yW,xy ) + (1 - ul)Kl(w,xyW,y + w,xW,yy )

+ I(6(O~'x + (90'm) -}- /]2/(6(~¢'x + ¢~'x) + 1(7;~ + I(8~ + l(9w'x -- O,

(1 + ul)Klu,xy + IQ(2v, vv + (1 - ul)v,xx) + (1 + u2)K4u,xv

+ K4(2V,yy + (1 - u2)v,zz) + (1 + ul)K2¢,xy + K2(2¢,yy + (1 - Ul)¢'xx)

+ K3(w,yyy + w,xxy) + Kllit'xy + K4(2#yiS'vy + (1 - v2)#xyi;'xx)

+ 2I;4(¢,y¢ + ¢¢,y) + (1 -- u2)I(4(~b,xO + ¢¢'x)

+ 2K~(~,~w,~y + ~,~,y~) + (1 - ~)IC~(~,~,~ + ~ , ~ , = )

+ ,Yll(¢¢,y + ¢¢,y) + u2IQl(¢¢,y + ¢¢,y) + K7~ + Zs¢ + g9ib,v = O,

K2(2U,xx + (1 - ui)U,yy) + (1 + ul)K2v,xy + gl2(2¢,xx + (1 - Ul)¢,yy) + K13¢

+ (1 + ul)g12¢,xy + I(15(2¢,xx + (1 - u2)¢,yy) + (1 + u2)gls¢,xy

+ K14(w,zx~ + w,xv . ) - I(13w,x -}- K15(2#~¢,x~ + ( 1 - u2)#~.q$'u.)

+ K16q$ + ICl7~,xy - ICl6(V,x - K4(Z(u,x¢ + u2v,v¢) + (1 - u2)(U,y¢ + v,x¢))


+ 2K2((w,xw,zx + l,'lW,yW'xy) 4- ( l - ul)/2(W, zyW, y + w,zW,yy))

- K6,~,~¢ - ,,2K~+,,,¢ + K18(% + ,~,~)¢ - K4(~2¢ + ¢3)

- K6(¢2~ + z,2¢¢4) + K18(~2¢ + ¢~¢) + Ks~Z + K19~; + K20~,~ = 0,

(1 + pl)lr4f2Utxy -k K2(2v,y v + (1 - ul)v, zx) + (1 + pl)K12fO,xy

+ K12(21~,yy+ ( 1 - l.'l)l~,xx)-+- K21/~+ (1 + t/2)K'lS¢,xyq- Kls(21~,yyq- ( 1 - u2)l~,xx)

+ < 4 ( ~ , , ~ + ~, ,~) - K2~.~ + K23¢,~ + K15(2~4,,~ + (1 - . 2 )#~ , .~ )

-Jr- K 2 2 ~ - K221J3,y - K 4 ( 2 ( V , y ¢ -I-- p211,,x~)) ...I- (1 - u 2 ) ( v , ~ ¢ -I- u,u¢)) + 2K2((w,yW,yy + UlW,~W,~) + (1 - Ul)/2(w, xW,~y + W, yW,~=))

-- Kll+,yff3 - pzK6~,x~) -~- K18(qS?), x --}- qS~,y) - K 4 ( q 5 2 ~ + 2/33)

- _r;l~ (¢% + ,,2¢¢~;) + -<8(¢% + ¢¢~) + K~V + K~945 + K=o,~,,, = o,

-K3(u'xxx + ?~'xyy) -- K3(V'yyy q- v'xxy) -- I(14(¢'xzz + ¢'xyy) -- K14(¢'yyy "-k ~3'xxy)

+ K13¢,~ + Kzl¢,y + K 2 4 ( w , ~ + w,vyv~ + 2w,~yy) - Ki3w,.~

- K21w,yy + K16¢,~ + K22¢,~ - K16~,~ - Kz2~,yy

+ 2K1((~,=~,~ + u , ~ , ~ ) + , 1 ( ~ , ~ + ~,~w,~)

+ (1 - ,1 ) /a (~ ,~ ,~ + ~.~,~,~ + ~ ,~ ,~ ) )

+ 2K1 ((V, yyW, y q- V, yW, yy + lel(V'xyW' x + VyW'xx)

+ (1 - , ,~ ) /2( , , ,~ ,~ + 2 , , , ~ , ~ + ~ , , ~ , ~ ) )

+ 2K2((¢,~W,x + ¢,~,~,=) + ~i(¢,~yw,y + ¢,~,yy)

+ (1 - pl) /2(¢,yyW,x --~ 2¢,yW,xy .-{- ¢,xyW, y))

+ 2/~2((¢,yy~,y + ¢ ,y~ ,~) + ,~ (¢,~w,~ + ¢,~w,~)

+ (1 - Pl)/E(¢,xyW,x Jr- 2¢,xW, xy q-¢,xxW, y))

- 2K~(ulW2.y + (1 - Ul)W,~w,uy + (1 - Ul/2)w, uw,~.y + (1 - Ul/2)w,~W,~yy)

~,~w,~) - 8Z-'l K2w,xxW,yy + K1 (3w2zw,zx + 3W2,yW, yy + 4w, xW,zyW,y + W,yW,zz +

- Kggt,x - K9~),y - K20¢,x - K2O~,y - K25(v),xx + ib,yy) + K7v) = 0 (9)

where the subscripts 2 and 0 of u and v respectively in equations (9) have been omitted for simplicity. The coefficients Kj can be found in the Appendix.

A valid check of equations (9) was performed by comparing the undamped nonlinear frequencies with those of [18, 19] for a sandwich plate with immovable edges. This comparison is rewritten in Table 1 in the present paper. A good agreement is noticed for a wide range of amplitude A.


Table 1. Frequency ratio w/wn (nonlinear frequency/natural frequency) of three-layer rectangular plate with immovable edges.

Frequency ratio w/wn Amplitude at centre Iu and Cheung [18] Present results

0.1 1.0021 1.0020 0.2 1.0083 1.0081 0.5 1.0507 1.0499 1.0 1.1886 1.1858 1.5 1.3862 1.3789 2.0 1.6214 1.6072 2.5 1.8802 1.8489 3.0 2.1542 2.1489

a = 72 in, b = 0.016 in, t2 = 0.25 in, I/1 = 1/3 = / ]2 = 0.33, E1 = E3 = 107 lb/in 2, G:z = Guz = 13.5.103 lb/in 2, 71 = 3+3 = 259 * 10 -6 lb sec2/in 4, 7z = 11.4 * 10 -6 lb sec2/in 4

Solution Methodology

Consider a simply-supported rectangular sandwich plate of the dimensions a , b, the boundary condi t ions for the plate can be wri t ten as fo l lows

v = w = ¢ = 0 ,

1 1 Nx - (1 - u9 ~f) ( E l t l + E3t3)(u,x + UlV,y) + (1 - u~)-9~ E2t2(u,x + l/2V,y)

M : =

+ 1 2(1 - l/2) ( -E l t l t2 + E3t3t2)(¢,x + l / l ¢ ' y )

+ 1 (_Elt2 + E3t23)(w,: x + l/'w'uY ) 2(1 - l/12)

+ 1 2(1 - l/~) (E lq + Elt3)(W2x q" tqW2y) +

1 2(1 - E2t (¢2 + l/2¢2)

2(1 - 1/2) (Eltl(t2 -1- t l ) - E3t3(t3 q- t 2 ) ) (u , z q- l/lV,y)

1 + 4(1 - 1/21) (Eltlt2(t2 q- t l ) --}- E3t3tz(t3 q- t 2 ) ) (~ 'x + l / l~ 'y )

1 + 12(1 - u2t3(¢' + l/2¢,y)

(1 --- l/~)

1 (Eltl(tz+tl)+E3t3(t3+tz))(wZ,~+l/twZ, y ) a t x = 0 , a + 4(1 - l/~)


u = w = ¢ = 0 ,

My 1 (1 - Vl 2) ( E l t ! + E3t3)(V'Y + VlU'x) + - -

1 (1 - v2 2) E2tz(v,v + u2u,=)

+ 1 vg'i') ( -E l t l t2 + E3t3t2)(¢'Y + vl¢'x) 2(1

1 (-Elt~ + E3t~)(W,y u + VlW'xx) + 2(1 - v 2)

+ 1 (Eltl + E3t3)(w2y + l]lW2x) + 2(1 - 4 )

1 2(1 - u2z) " E2t2(¢2 + u2¢2)

My 1 tfi ~1) (Eltl(t2 + t l ) - E3t3(t3 + t2))(V,y + UlU,x) 2(1

+ 1 tf'i') (Eltlt2(t2 + tl) + E3t3t2(t3 + t2))(~b'y -t- Vl¢'x)

4(1

+ 1 12(1- vx x) E2t3(¢'v + v2qS,~)

l (Elt2(~-ff-~)-q-E3t2(~q-~))(w,yy-l-lllWtxx) (1 - ,,~)

+ 1 (Eltl(t2 + t l ) + E3t3(t3 + t2))(w2v + VlW2,z) 4(1 - v~)

at y = O, b. (10)

If the response functions expressed in equations (11) are chosen, the boundary conditions can be approximately satisfied. It can be observed that only the nonlinear terms in the expressions (10) for the forces N~ and Ny and moments M~ and My become not identically zero at the edges. These forces result from the elongations in the directions parallel to the plane of the plate due to the transverse deflection.

(7) u = ~ u l j ( t ) c o s - - sin j

v = ~ ~v i j ( t ) s in ( ~ - ~ ) c o s ( ~ - ~ ) , i j

i j

,= r r sin ( )cos i j

w = ~ ~-~wij(t)sin ( ~ - ~ ) s i n ( ~ - ~ ) . i j

(11)


Here, the numbers i, j = 1, 2, . . . express the number of half-waves contained in the directions x and y, respectively. For simplicity, we took only i, j = 1 to study the fundamental motion. Substituting equations (11) into equations (9) and multiplying them by cos(iTrx/a) sin(jTry/b), sin(iTrx/a) cos(jTry/b), cos(iTrx/a) sin(jTry/b), sin(iTrx/a) cos(jTry/b), sin(iTrx/a) sin(jTry/b) respectively, then integrating from 0 to a for x and from 0 to b for y, a set of five nonlinear ordinary differential equations is obtained for the time-dependent function u(t), v(t), ¢(t), ¢(t), w(t). The general representation of this set of equations with quadratic and cubic nonlinear terms can be written as

5 5 5 5 5

E mijgzj(t) + E cijitj(t) + E kijaj(t) + E E dijkaj(t)ak(t) j= l j= l j=l j=t k=l

5 5 5 5 5

+ E E eijkaj(t)itk(t) + E E E fijkzaj(t)ak(t)al(t) j= l j= l j=l k=l l=l

5 5 5

+ ~ ~ ~-~gijklaj(t)ak(t)hl(t) + Pi(t) = O, i , j ,k, l= 1,. . .5. (12) j= l k=l /=1

Here, ai(t) are the generalized coordinates which represent U(t), V(t), ~(t), ~(t), W(t), respectively, mij are the inertia coefficients, cij the linear viscous damping coefficients, kij the linear stiffness coefficients, dijk and fljkt are the quadratic and cubic nonlinear stiffness coefficients, eijk and gijkt are quadratic and cubic nonlinear damping coefficients, respectively. These coefficients are not presented here. The dimensionless parameters used here are

a a w u v = 0~1' tl 0~2' 0~2¢ ¢ ' 0~2¢ ~ ' tl - W, tl U, tl - V,

7A = r2, 72 = r3, 71 t2 _ fl, Gz = Gz, 71 71 E1 E1

a3 / / ( ~ - ) ( ~ - ) Elt~ p(x, y, t) sin sin dx dy = /5 . (13)

This time-dependent equation (12) was solved by using the implicit harmonic balance method [22]. The unknowns U(t), V(t), ~(t), ~(t), W(t) are expressed as Fourier series including up to the third harmonic term

U(t) = Uo + gl cos(wt) + U2 sin(wt) + U3 cos(2wt) + U4 sin(2wt)

+ U5 cos(3wt) + U6 sin(3wt),

V(t) = Vo + V1 cos(wt) + V2 sin(wt) + V3 cos(2wt) + V4 sin(2wt)

+ V5 cos(3wt ) + V6 sin(3wt ),

¢(t) = '~0 + ~1 cos(wt) + ¢9_ sin(wt) + ~3 cos(2wt) + ff,~ sin(2wt)

+ ¢'5 cos(3wt) + ~6 sin(3wt),

g/(t) = k~0 + g21 cos(wt) + g22 sin(wt) + ~3 cos(2wt) + ~4 sin(2wt)

+ ~5 cos(3~zt) + ff~6 sin(3~t),


W(t ) "- Wo q- W1 cos(o)t) q- W2 sin@t) + W 3 COS(2O3t) -t- W4 sin(2wt)

+ W5 cos(3~vt)+ W6sin(3wt), (14)

where Ui, ~, q~i, q!i, and Wi are constant Fourier coefficients for the K-th harmonic ampli- tudes of U(t), V(t), q~(t), q~(t) and W(t) respectively. The transverse force/5 can also be approximated in a Fourier form

/5 = P0 + Pl cos@t) + P2 sin(wt) + P3 cos(2~t) + P4 sin(2wt)

+ P5 cos(3wt) + P6 sin(3wt). (15)

Substituting (14) and (15) into (12), equating the produced coefficients at the similar terms of cosines and sines functions, a system of 35 nonlinear algebraic equations has been obtained. These equations are very complicated and are not shown here. To solve this set of nonlinear algebraic equations, a subroutine named NEQNF in IMSL Fortran Library [23] was employed.

Results and Discussion

The material constants and aspect ratios used in the following examples are given below:

E3 _ E2 73 72 E--~ - 1, - - = 0 . 0 0 2 , - - = 1 , - - = 0 . 0 5 , V l = V 2 = V 3 = 0 . 3 3 3 ,

E1 71 71

a a = 50, 71t2 - 0 . 4 . 1 0 -13, P1 = P2 = 0.08. (16) = 1, tl E1

Other parameters such as t2/tl, t3/tl, G~/E1 will be given for different problems and will be studied as the factors in following examples. Here for simplicity the shear moduli are assumed to have the value G2xz = G2yz = Gz.

It was found that presenting the force P by means of the two terms corresponding to the harmonic motion, P = P1 cos@t) + P2 sin@t), only the terms of the first and third order harmonics responded to the excitation. The coefficients for those of constant and second order terms in (14) were equal to zero. Figure 2 through Figure 7 give a comparison of r.m.s, of the first and third order harmonic terms of U(t), q~(t), and W(t). For the example in Figure 2 to Figure 4, the parameters have the values in (16) and t2/tl = 5, ta/tl = 1, Gz/EI = 0.04, the sandwich plate with such parameters has the frequencies Wl = 0.943753. 105 and ~3 = 0.445342, 10 6. For the example in Figure 5 to Figure 7, all parameters are the same except the facing thickness ratio t3/tl changes to the value 2 which gives the frequencies wl = 0.944148 • 105 and w3 = 0.453355.106. By studying the obtained results corresponding to the first and third harmonic terms, it was found that the terms corresponding to the third order harmonic motion were much smaller than that of the first order terms in general. When the value ofw/wn is about 1, the third order harmonic is excited and has a great increase while the first harmonic response decreases at this area. Another sudden change was found in the response curves at the value ofw/w~ -- 4.72 in the case for t3/tl = 1 (Figure 2 to Figure 4) and at w/a~n = 4.8 for t3/tl = 2 (Figure 5 to Figure 7). These values are of the same ratio of w/w3 = 1, where w3 are the third linear frequencies. It is noticed that the internal resonance due to the third linear frequency does not change the transverse deflections Wi very much. In the symmetrical case t3/tl = 1, in Figure 2, the change of the internal resonant response of w


60.0

40,0

20.0

0.0 0.0

- - r - - T I ' I

A ~

. . . . . . . . . . . ' " ' " , 1 . , . . . . . . . . . . . . .

- - . . . . . . . . . . . .

2,0 4,0 6,0 8,0

Fig. 2. Comparison of the transverse response of the first and third order harmonic terms, ta/tl = 1. The transverse displacement has the form W = ~ + W2 z cos(wt + 01) + V / ~ + W g cos(3wt + 02).

100.0

50,0

0.0 0.0

- - - - r ' - ~ ' r • I

2,0 4,0 6.0 8,0

co/ca n

Fig. 3. Comparison of the slope response of the first and third order harmonic terms. 13/t~ = 1. The slope has the form (b = ~ ¢2 cos(wt + 0~') + V~52 + ¢2 cos(3wt + 0;).


2 . ( 3 i J I l i f[ I

1.5

1.0

0.5

i i j /

.." ',.. / " . , . . . . .

UI + U[

V t J5 ,6

0 , 0 . . . . . . . . . . . ' - . . . . . . . . . . . . . . . . . . . . . . . , , , , . . . . . . , . . . . . . . . . . . . . . . 77z,.

0,0 2.0 4,0 6.0 8.0

cJ/cJ

Fig. 4. Comparison of the in-plane response of the first and third order harmonic terms. 13/tl = 1. The in-plane displacement has the form U = V/-~(+ U~ cos(wt + 0[* + ~ cos(3wt + 0~*).

80.0 I / I

20.0

40,0

0.0 0.0

, , . ""~, , . , . , . . . . . . ~ . . . .

I , , , , , . . . . . I . . . . . . . . . . . . . . . "~ . . . . . . . . I ,

2,0 4.0 6,0 /3.0

Fig. 5. Comparison of the transverse response of the first and third order harmonic terms, t3/t 1 = 2. The transverse

displacement has the form W = V/-~12 + W~ cos(wt + Ol) + V/'~-~2 + We ~ cos(3wt + 02).

at o~/w3 = 1 cannot even be seen. The response curves of slopes ffi and ~ i also show that the unsymmetrical case t3 / t l has a more considerable response at the third resonance. In Figure 4 and Figure 7, the in-plane displacements are studied. It is interesting to find that the in-plane


100,0

80.0

60,0

40,0

20,0

0.0 0.0 8,0

' I ' I ' I '

2.0 4.0 6,0

~ / ~

Fig. 6. Comparison of the slope response of the first and third order harmonic terms. ¢3/tl = 2. The slope has the form @ = ~ @~ cos(wt + 0;) + V/@~ + ~ cos(3wt + 0;).

2,5 I I I III I

2.0

1.5

1.o L _ J

0.5 j

, / ' - .

0.0 ......... ' : , ' " " l . . . . . . . . . . . . . . . . v . .

0.0 2.0 4,0 ~.0 fl.O

Fig. 7. Comparison of the in-plane response of the first and third order harmonic terms, t3/tl = 2. The in-plane displacement has the form U = ~ l 2 + U2 2 eos(w~ + 0~'*) + ~ + U6 2 cos(3wt + 0~*).


800 I I

75.0

70.0

0

65,0

60.0

55.0

50.0

t2 / t I =3

t z / t 1=5

"'"""-..,.. . . . .

4 5 . 0 , ¢ , I

0 . 0 1 . 0 2 . 0 3 , 0

%/q

Fig. 8. Effect of the core thickness ratio (t2/tl) on the resonant response of the sandwich plates. G~/EI = G~z/El = 0.04, tl is fixed.

displacements U and V are strongly excited at both resonant frequencies especially at the third one. It means that even the sandwich plates bear only the exciting force in transverse direction, the strongly coupled in-plane displacements can also be excited greatly.

In Figures 8 through 10 the influence of geometric and material parameters on the resonant response, which is defined as the maximum response of response curves for forced vibration in transverse direction (point A at Figure 2), is presented. Figures 8 and 9 show that by increasing the thickness of the facings and of the core, a smaller resonant response is obtained. When the facing thickness ratio t3 / t l is small, the change of the core thickness ratio t z / q does not affect the resonant response very much (Figure 8). The reason for this phenomenon is due to the fact that the core does not carry much bending. When the third layer is very thin, only the first layer contributes to carry the bending force. In the second part, the resonant response calculated for the thicker core is obviously smaller than that of a thinner core. A small change of the facing thickness results in a big difference of the resonant response for a plate with a thicker core. But after a drastic decrease of the resonant response, the slopes of the curves become mild. The conclusion from Figure 8 is that the resonant response of a sandwich plate with a thicker core is more sensitive to the facing thickness. This observation can be useful for the designers of sandwich plates. In Figure 9, all the curves have almost the same shape. It is noticed that the resonant response does not change very much in the initial and final parts of the curves. The increase of the thickness of the core makes the resonant response decrease. This phenomenon is more evident in the middle part of the curves.

The curves in Figure 10 demonstrate the effect of the changes of the material constants on the resonant response. If a different material is used for the core, even if the geometric parameters are constant, the resonant response changes. In Figure 10, the resonant response goes up with the increase of shear moduli of the core after an initial decrease. This means that the damping of the sandwich plates does not always decrease with the increase of the shear


75.0

70.0

65.0

60.0

o

55.0

50.0

o

45.0

40.0

I } I I I I ' I I

35.0

30.0 q I I I I i I I

1.0 2.0 3.0 4 .0 5.0 6.0 7 .0 8.0 9.0 10.0

tJtl

Fig. 9. Effect of the face thickness ratio (t3/tl) on the resonant response of the sandwich G~z/El = Gyz/El = 0,04, tl is fixed,

plates.

80,0 I I ' I

0

70.0

60.0

50,0

4 0 0 0.00

- tJtl=3, %/ti=2

.. _---

_ _ / ~ t2/t l=5, t~/tl= 2

, I I , I , I

0.02 0.04 0,06 0.08 O. 10

Gx~/EI=G~/EI

Fig. 10. Resonant response of the sandwich plates changes with the shear modulus of the core.

moduli. For a certain value of G / E the resonant response has a minimum. Then an optimum

shear moduli can be found for each sandwich plate.


Appendix

K's for equation (9)

1 Nl - 2 ( 1 - .2) (El t l n L E3t3)

1 K2 - 4(1 - u 2) tz (El t l - E3t3)

1 ( E I t ~ - E3t~) K 3 - 2 ( 1 -

1 K4 - 2 ( l _ v 2 ) E2t2

K5 = (2n#x q- (1 - u2)#xy)l( 4

1 K6 - (1 - u z) E2t2tzz

K7 = (71tl q- 72t2 -t- 73t3)

t2 K8 = ~ ( -71t l + 73t3)

1 1(9 = ~ (-71t~ + 73t 2)

KlO = (2U2#u q- (1 - u2)tzxuK4 1

g l l - (1 - / ]2 ) E2t2#y

1 t2(Elt 1 q- E3t3 ) 1(12 = - 8 ( i - u 2)

K13 = Gxzt2

1 t2(Elt 2 + E3t2 ) K 1 4 - 4(1 _ u2 )

1 E2t~ tQ5 = 24(1 - u2)

K16 = #xzK13

g17 = (2vz u~ + (1 - u2)#~v)K15

1 K18 -- 2(1 +/]2) E2t2#xy

l t z z ( 7 , t l + l ) K19 = ~ ~ 72t2 + 73t3

1 K20 = ~t2(71 t 2 + 7 3 t2)


K21 = Gyzt2

K22 = #yzK21

K23 = (2/,'2#y + (1 - l.'2)#xy)Kl5

1 (Elt~ + E3t~) / ' ( 2 4 - 3 ( 1 _ _ . 2 )

1 1(25 = ~ (71t~ + 73t ] ) .

References

1. Plass, H. J., 'Damping of vibration in elastic rods and sandwich structures by incorporation of additional viscoelastic material', Proceedingsofthe ThirdMidwestern Conferenceon Solid Mechanics, 1957, pp. 48-71.

2. Yu, Y. Y., 'Damping of flexural vibration of sandwich plates', Journal of Aerospace Science 29, 1962, 790-803.

3. Ho, T. T. and Lukasiewicz, S., 'Some aspects of structural damping in sandwich plates', Archiwum Budowy Maszyn 24(1), 1977, 25-39.

4. Ho, T. T. and Lukasiewicz, S., 'Compression effects in structural damping in sandwich plates', Rozprawy In~ynierskie. Engineering Transactions 26(1), 1978, 27-40.

5. Ho, T. T. andLukasiewicz, S., 'Structural damping in sandwich plates',Archiwum Budowy Maszyn 22, 1975, 145-161.

6. Alam, N. and Asnani, N. T., 'Vibration and damping analysis of fibre reinforced composite material plates', Journal of Composite Materials 20, 1986 1-18.

7. Alam, N. and Asnani, N. T., 'Refined vibration and damping analysis of multilayered rectangular plates', Journal of Sound and Vibration 119(2), 1987, 347-362.

8. He, J. E and Ma, B. A., 'Analysis of flexural vibration of viscoelastically damped sandwich plates', Journal of Sound and Vibration 126(1), 1988, 37-47.

9. Bisco, A. S. and Springer, G. S., 'Analysis of free damped vibration of laminated composite plates and shells', International Journal of Solids and Structures 25(2), 1989, 129-149.

10. Shin, Y. S. and Maurer, G. J., 'Vibration resonance of constrained viscoelastically damped plates', Finite Element in Analysis and Design 7, 1991, 291-297.

11. Yu, Y. Y., 'Nonlinear flexural vibration of sandwich plates', The Journal of the Acoustical Society of America 34(9), 1962, 1176-1183.

12. Ebcio~lu, I. K., 'A general nonlinear theory of sandwich panels', International Journal of Engineering Science 27(8), 1989, 865-878.

13. Chia, C. Y., Nonlinear Analysis of Plates, McGraw-Hill, Inc., 1980. 14. Chia, C. Y., 'Geometrically nonlinear behaviour of composite plates: A review', Applied Mechanics Review

41(2), 1988, 439-451. 15. Kovac, E. J., Anderson, W. J., and Scott, R. A., 'Forced nonlinear vibrations of a damped sandwich beam',

Journal of Sound and Vibration 17(1), 1971,25-39. 16. Hyer, M. W., Anderson, W. J., and Scott, R. A., 'Nonlinear vibration of three-layer beams with viscoelastic

core. I. Theory', Journal of Sound and Vibration 46(1), 1976, 121-136. 17. Hyer, M. W., Anderson, W. J., and Scott, R. A., 'Nonlinear vibration of three-layer beams with viscoelastic

core. II. Experiment', Journal of Sound and Vibration 61(1), 1978, 25-30. 18. Iu, V. P. and Cheung, Y. K., 'Non-linear vibration analysis of multilayer sandwich plates by incremental finite

elements: 1. Theoretical development', Engineering Computers 3, 1986, 36-42. 19. Iu, V. R and Cheung, Y. K., 'Non-linear vibration analysis of multilayer sandwich plates by incremental finite

elements: 2. Solution techniques and examples', Engineering Computers 3 1986, 43-52. 20. Xia, Z. Q. and Lukasiewicz, 'Nonlinear, free, damped vibrations of sandwich plates', Journal of Sound and

Vibration 175(2), 1994, 219-232. 21. Lockell, E J., Nonlinear Viscoelastic Solid, William Clowes & Sons Ltd., 1972. 22. Cheung, Y. K. and Iu, V. E, 'An implicit implementation of harmonic balance method for non-linear dynamic

system', Engineering Computations 5, 1987, 134-140. 23. IMSL Library Reference Manual, 1 lth ed., Intemational Mathematical and Statistical Libraries, Houston,

Texas, 1991.

nonlinear damped vibrations of simply-supported rectangular sandwich plates

Documents