Large amplitude free exural vibrations of laminated composite skew plates

Transcription

1 International Journal of Non-Linear Mechanics 39 (2004) Large amplitude free exural vibrations of laminated composite skew plates Maloy K. Singha, M. Ganapathi Engineering Analysis Center of Excellence, General Electric (Aircraft Engines), Bangalore , India Accepted 6 April 2004 Abstract Here, the large amplitude free exural vibration behaviors of thin laminated composite skew plates are investigated using nite element approach. The formulation includes the eects of shear deformation, in-plane and rotary inertia. The geometric non-linearity based on von Karman s assumptions is introduced. The non-linear governing equations obtained employing Lagrange s equations of motion are solved using the direct iteration technique. The variation of non-linear frequency ratios with amplitudes is brought out considering dierent parameters such as skew angle, number of layers, ber orientation, boundary condition and aspect ratio. The inuence of higher vibration modes on the non-linear dynamic behavior of laminated skew plates is also highlighted. The present study reveals the redistribution of vibrating mode shape at certain amplitude of vibration depending on geometric and lamination parameters of the plate. Also, the degree of hardening behavior increases with the skew angle and its rate of change depends on the level of amplitude of vibration.? 2004 Elsevier Ltd. All rights reserved. Keywords: Composite skew plate; Non-linear vibration; Finite element; Frequency ratio; Amplitude 1. Introduction The increased utilization of thin-walled structural components with relatively low exural rigidity in the design of space vehicles and their vibration characteristics at large amplitudes in response to the conditions they are subjected to, have attracted the attention of many researchers in recent years. These studies are reviewed and are well documented by Leissa [1,2], Bert [3], Sathyamoorthy [4], Chia [5], and Kapania and Raciti [6]. It is observed from the existing literature that the large amplitude exural vibration Corresponding author. address: mganapathi@redimail.com (M. Ganapathi). of rectangular and circular plates has received considerable attention of the researchers. However, limited work has been focused on the geometrically non-linear free vibration analysis of plates other than rectangular/circular plates. The plates with non-rectangular plan-forms like skew plates nd wide application in the aerospace industry. Though a huge body of literature exists on the linear free vibration of isotropic and single layer orthotropic skew plates, such analysis concerning the laminated composite skew plates has received relatively little attention, despite the increasing use of such components in aircraft [7]. The notable recent contributions pertaining to linear vibration behavior of laminated composite skew plates are dealt with /$ - see front matter? 2004 Elsevier Ltd. All rights reserved. doi: /j.ijnonlinmec

2 1710 M.K. Singha, M. Ganapathi / International Journal of Non-Linear Mechanics 39 (2004) Refs. [8 16]. Iyengar and Umaretiya [8] have studied a hybrid laminated skew plate using Galerkin s method. Malhotra et al. [9] have carried out the vibration of rhombic orthotropic plates using parallelogrammic orthotropic plate nite element whereas, Krishnan and Deshpande [10] have examined the free vibration of thin cantilever skew laminates employing nite element method, based on discrete Kirchho theory. Kapania and Singhvi [11], and Anlas and Goker [12] have analytically solved the problem using Ritz method. Wang [13] has studied using B-spline Rayleigh Ritz method, while Han and Dickinson [14] have applied a Ritz Hierarchical nite element in analyzing the laminated skew plates. A numerical approach with assumed solution considering Green function is introduced in the work of Hosokawa et al. [15]. Reddy and Palaninathan [16] have dealt with the problem using high precision plate bending nite element. An attempt is also made to study the vibration characteristics of skew sandwich plates with laminated facings by Makhecha et al. [17] and Wang et al. [18]. It may be concluded that most of these work are carried out based on classical plate theory. Further, the non-linear free vibration of skew plates is treated sparsely in the literature. Nowinski [19] and most recently Ray et al. [20] have analytically studied the non-linear vibration behavior of isotropic skew plates. Sathyamoorthy and Pandalai [21] have analyzed single layer orthotropic skew plates for movable in-plane edges and Berger approximation for immovable in-plane edge conditions. Prathapand Varadan [22], and Sathyamoorthy and Chia [23] have investigated large amplitude vibration of anisotropic skew plates using the Galerkin method on the basis of a single term assumed vibration mode. All these investigations introduce assumed mode while solving the problem. It is known that one-term approximation for the free vibration mode is insucient for rectangular as well as skew plates, and such results lose accuracy with increase in aspect ratio and skew angles. It may be further observed that the geometrically non-linear free vibration of higher modes of structures including rectangular seems to be scarce in the literature [24]. Also, to the best of the authors knowledge, the work on the non-linear free exural vibrations of laminated anisotropic composite skew plates is not commonly yet available in the literature. For accurate solution, the assumed mode shape in the analytical approach should have more terms that lead to unwieldy algebraic and more numerical work. Numerical methods such as nite element procedure is preferable over the analytical methods, while studying the large amplitude dynamic analysis of structures, as there is no need for an a priori assumption of the mode shapes, and the solution itself predict the mode shapes [25]. Such analysis for skew plates appears to be lacking in the literature. In the present paper, a four-noded shear exible quadrilateral high precision plate bending element developed recently [26,27] is extended to analyze the large amplitude vibrations of thin laminated composite skew plates. As the element is free from locking phenomenon, all the energy terms are evaluated using full numerical integration scheme. The formulation includes in-plane and rotary inertia eects. Furthermore, the present study makes use of incremental matrices to represent the geometrical non-linearity. The non-linear governing equations obtained here are solved using direct iteration technique where the linear mode shape is taken as the starting vector. The amplitude frequency relationships are predicted. While calculating the results, necessary convergence criteria [28] are satised for the displacement vector and frequency value. The formulation developed here is validated with the available analytical results. A detailed parametric study, to bring out the inuences of the skew angle, lay-up, aspect ratio, and boundary conditions on the non-linear free vibration behavior of laminated skew plates has been carried out. Further, the signicance of non-linear free vibration of higher modes is investigated in the present analysis. 2. Formulation Fig. 1 shows the rectangular Cartesian co-ordinate system along with the associated covariant base vectors (g 1 ; g 2 ; g 3 ) and contravariant base vectors ( 1 g; 2 g; 3 g) for the skew plate having a and b as the length and width, and as the skew angle. (g 1 ; g 2 ; g 3 ) and ( 1 g; 2 g; 3 g) are related to the rectangular Cartesian unit base vectors (e 1 ; e 2 ; e 3 )by g 1 = e 1 ; g 2 = (sin )e 1 + (cos )e 2 ; g 3 = e 3

3 M.K. Singha, M. Ganapathi / International Journal of Non-Linear Mechanics 39 (2004) and the covariant components of the strain tensor ij ( = i ij g j g) are related to their counterparts c mn & c mn and in the rectangular Cartesian co-ordinate system by c mn = ij (g i e m )(g j e n ) (4) and c mn = ij ( i g e m )( j g e n ): (5) and Fig. 1. The oblique co-ordinate system of skew plate. 1 g = e 1 (tan )e 2 ; 2 g = (sec )e 2 ; 3 g = e 3 : (1) The covariant components of the displacement vector u (u = u 1 1 g + u 2 2 g + u 3 3 g) of a shear deformable plate can be expressed in terms of the contravariant components of the position vector r (r= 1 rg rg rg 3 )as[27] u 1 ( 1 r; 2 r; 3 r)=u 0 1( 1 r; 2 r)+ 3 r 1 ( 1 r; 2 r); u 2 ( 1 r; 2 r; 3 r)=u 0 2( 1 r; 2 r)+ 3 r 2 ( 1 r; 2 r); u 3 ( 1 r; 2 r; 3 r)=u 3 ( 1 r; 2 r); (2) where 1 ( 1 r; 2 r)={u 3;1 + 1 ( 1 r; 2 r)}; 2 ( 1 r; 2 r)={u 3;2 + 2 ( 1 r; 2 r)}: Here, 1 ( 1 r; 2 r) and 2 ( 1 r; 2 r) are the total rotations; 1 ( 1 r; 2 r) and 2 ( 1 r; 2 r) are the rotations due to shear deformation of the normal to the plate middle surface around 2 g axis and 1 g axis respectively; ( ) ;i represents the partial dierentiation of the variable preceding it with respect i r. Now the covariant components of strain tensor ( = i ij g j g) can be written in terms of displacement components u (u = u i i g = k ug k )as[27,29] ij = 1 2 (u i;j + u j;i + k u ;i u k;j ): (3) The normal strain component 33 is zero. The contravariant components of stress tensor ij (= ij g i g j ) For a composite laminate of thickness h, comprising of N layers with stacking angles i (i=1; 2;:::;N) and layer thicknesses h i (i =1; 2;:::;N), the necessary expression to compute the stiness coecients are available in the literature [30]. The stress strain relation in the rectangular Cartesian co-ordinate system can be written as c { i } =[Q ij ] k c { j }; (6) where, the stress c {} = { } T and strain c {} = { } T components. [ Q ij ] k represents the reduced stiness matrix of the kth laminate of the plate. The strain energy of the plate can be written as U = 1 ij ij dv; (7) 2 v where dv =d 1 r d 2 r d 3 r cos. The kinetic energy of the plate is given by T()= 1 [ u 2 2 i +( u 3;j + j ) 3 r 2 ]dv i=1; 3; V j =1; 2; (8) where, is the mass density. Substituting Eqs. (7) and (8) in Lagrange s equations of motion, one obtains the governing equations as ([K L ]+[K NL ()]){} +[M]{ } = {0}; (9) where [K L ] and [K NL ] are the linear and non-linear stiness matrices, respectively. The governing Eq. (9) is solved using nite element approach based on C 1 continuous element developed recently [26,27]. Here, a four-noded quadrilateral plate element with ten degrees of freedom per node, namely u1 0, u0 2, u 3;1, u 3;2, u 3;11, u 3;12, u 3;22, 1 and 2 are used. The linear polynomial shape functions are employed to describe the eld variables corresponding to in-plane displacements (u1 0;u0 2 ) and rotations due to shear of the middle

4 1712 M.K. Singha, M. Ganapathi / International Journal of Non-Linear Mechanics 39 (2004) surfaces ( 1 ; 2 ), whereas quantic polynomial function is considered for the lateral displacement (u 3 ) and are expressed as follows: u 0 1 = c k ( 1 r) i ( 2 r) j ; i;j =0; 1 and k =1; 4; u 0 2 = c k ( 1 r) i ( 2 r) j ; i;j =0; 1 and k =5; 8; u 3 = c k ( 1 r) i ( 2 r) j + c k ( 1 r) m ( 2 r) n + c k ( 1 r) n ( 2 r) m ; i; j =0; 1; 2; 3; m=0; 1; n=4; 5; and k =9; 32; 1 = c k ( 1 r) i ( 2 r) j ; i;j =0; 1 and k =33; 36; 2 = c k ( 1 r) i ( 2 r) j ; i;j =0; 1 and k =37; 40; (10) where c k are constants and are expressed in terms of nodal displacements in the nite element discretization. The full integration scheme with 6 6 Gaussian integration rule is adopted here for computing the element stiness and mass matrices. This element is free from locking syndrome. It has good convergence properties and has no spurious rigid modes. For the sake of brevity, these are not presented here, as they are available in the literature [26,27]. Substituting characteristics of the time function at the point of reversal of motion { } max =! 2 {} max (11) in Eq. (9), will lead to the following non-linear algebraic equation of the form [31], ([K L ]+[K NL ]){}! 2 [M]{} = {0}; (12) where,! is the natural frequency. The frequency amplitude relation is obtained by solving Eq. (12) iteratively. 3. Solution procedure The vibration problem is solved using eigenvalue formulation. To solve the non-linear eigenvalue problems, an iterative procedure is used. Firstly, the eigenvector (mode shape) is obtained from the linear vibration analysis, neglecting the non-linear stiness matrix in Eq. (12) and then normalized. Next, the normalized vector is amplied/scaled up so that the maximum displacement is equal to the desired amplitude, say w=h =0:4 (w is the maximum lateral displacement, h is the thickness of the plate). This gives the initial vector, denoted by. The iterative solution procedure for the non-linear analysis starts with initial vector,. Based on this initial mode shape ( ), the non-linear stiness matrix, that depends on displacement, is formed and subsequently, the updated eignvalue and its corresponding eigenvector are obtained. This eigenvector is further normalized, and scaled upby the same amplitude (w=h), and the iterative procedure adopted here continues till the frequency values and mode shapes evaluated from the subsequent two iterations satisfy the prescribed convergence criteria [28] as N (! k r! r 1 k )=! k r 6 0:0001; (13a) k N i ( r i r 1 i )= r i 6 0:0001; for kth mode; (13b) where, k, i, N and r represents the mode number, degree of freedom of the nite element model, total degree of freedom and iteration number, respectively. 4. Results and discussion The study, here, has been focused on the large amplitude free exural vibration characteristics of thin laminated composite skew plates. The material properties, unless specied otherwise, used in the present analysis are E L =E T =40:0; G LT =E T =0:6; G TT =E T =0:5; LT =0:25; where E, G and are Young s modulus, shear modulus and Poisson s ratio. Subscripts L and T represent the longitudinal and transverse directions respectively with respect to the bers. All the layers are of equal thickness. The boundary conditions considered here are: simply supported on all sides (SS) : u 1 = u 2 = u 3 = 0 clamped on all edges (CC) : u 1 = u 2 = u 3 =0;u 3;1 =0 u 1 = u 2 = u 3 =0;u 3;2 =0 along the boundary nodes at 1 r =0;a; at 2 r =0;b:

5 M.K. Singha, M. Ganapathi / International Journal of Non-Linear Mechanics 39 (2004) Table 1 Convergence study of free vibration ($ =!a 2 = 2 h =E T ) of 5-layered [45 = 45 =45 = 45 =45 ] simply supported skew laminates (a=b =1; a=h = 1000:0) Skew angle Mesh size Modes Present study Ref. [13] Present study Ref. [13] Before proceeding for the detailed study for the non-linear free vibration behavior, the formulation developed herein is validated against the linear free vibration of laminated composite skew plate. The non-dimensional natural frequencies ($=!a 2 = 2 h =E T ; a and h are length and thickness of the plate) obtained for simply supported (SS) and clamped (CC) angle-ply [45 = 45 =45 = 45 =45 ] skew plate are presented in Tables 1 and 2, respectively along with the analytical solutions [13] and they match very well with the available results. It is also observed from Table 1 that the element employed here has a good convergence property and thus, a mesh is found to be adequate to model the full plate. Further, the ecacy of the present formulation is tested studying the non-linear free vibrations of thin isotropic and orthotropic square plates (a=h = 1000) with simply supported boundary condition for which the analytical/numerical results are available in the work of Chu and Herrmann [32] and Ganapathi et al. [31], respectively. The variation of non-dimensional non-linear frequency ratio (! NL =! L ; where subscripts NL and L correspond to the non-linear and linear, respectively) with respect to non-dimensional maximum amplitude (w=h; w is the exural amplitude of the plate) evaluated is shown in Table 3. These results are found to be in excellent agreement with the existing solutions. It is furthermore observed from Table 3 that there is a dropin the value of frequency ratio at certain amplitude and then it increases with amplitude exhibiting hardening behavior. This trend for isotropic case is noticed at amplitude higher than that of orthotropic plates. This is possibly due to change in stiness values and thus leading to the redistribution of mode shape at higher amplitudes of vibration. The redistributed mode shapes for the orthotropic case, loosing the symmetry and shifting the maximum displacement towards one side of the plate, are presented in Fig. 2. This phenomenon may be termed as mode redistribution. This mode redistribution or sudden decrease of non-linear stiness are reported in the literature by Singha et al. [26,27] and Ganapathi et al. [33] while investigating the thermal post-buckling of composite plates. Next, the non-linear free vibration behavior of clamped isotropic/orthotropic and single layer anisotropic skew plate (30 ) are examined and the results (! NL =! L at w=h =1:0) obtained using the present formulation are compared in Table 4 with the available solutions in the literature [22]. In Ref. [22], the non-linear frequency ratio is expressed in terms of hardening parameter as! NL =! L =(1+(w=h) 2 ) 1=2. From Table 4, it may be observed that, the results are fairly in good agreement. However, a little dierence observed in the results is due to the assumption of one-term approximation in the solution and enforcing the immovable boundary condition in an averaged manner [22] whereas there

6 1714 M.K. Singha, M. Ganapathi / International Journal of Non-Linear Mechanics 39 (2004) Table 2 Comparison of non-dimensional linear natural frequency ($ =!a 2 = 2 h =E T ) of a clamped 5-layered [45 = 45 =45 = 45 =45 ] skew laminates (a=b = 1; a=h = 1000:0) B.C. Skew angle Modes Present Ref. [13] CC 30 Present Ref. [13] Present Ref. [13] Table 3 Comparison of non-linear frequency ratio (! NL =! L ) of simply supported square plate (a=b =1, E L =E T =3:0, G LT =E T =0:5, LT =0:25) w=h Isotropic plate Orthotropic plate Present Ref. [32] Ref. [31] Present Ref. [31] : : is no such approximation involved in the present work. Thereafter, a detailed study is carried out considering thin multi-layered cross-ply and angle-ply skew laminates (a=h = 1000) for which results are not available in the literature. The variation of non-linear free vibration frequency ratio (! NL =! L ) of a simply supported four-layered cross-ply [0 =90 =90 =0 ] skew plate (a=b =1:0) is presented in Table 5 for dierent values of skew angle. From this table, it is observed that the non-linear frequency ratio increases with the increase in amplitude of vibration, indicating hardening type of non-linear behavior. It is also seen that, as noticed for the non-linear free vibration of square isotropic, and orthotropic cases, the non-linear frequency value drops down at certain amplitude and then it increases with the further increase of amplitude of vibration of skew plate. Similar trend is viewed, irrespective of skew angle, at the amplitude of vibration (w=h) of 0.8 for the case considered here. Furthermore, for the chosen amplitude, it can be noted that, with the increase in skew angle, the degree of non-linearity is high before the occurrence of mode redistribution. For a velayered angle-ply [45 = 45 =45 = 45 =45 ] skew composite plate (a=b = 1), the non-linear frequency ratio highlighted in Table 6 brings out hard spring type of non-linear characteristics. Unlike cross-ply case, it is, in general, opined for the angle-ply laminates that the dropin the frequency ratio occurs at low amplitudes with the increase in skew angle. It is further concluded that, for the chosen amplitude, the degree of non-linearity with skew angle, before the occurrence of mode-redistribution phenomenon, is less in comparison with those of cross-ply case. For the given amplitude, the change in the degree of non-linearity depends on skew angle. Fig. 3 shows the mode shapes associated with

7 M.K. Singha, M. Ganapathi / International Journal of Non-Linear Mechanics 39 (2004) Fig. 2. The redistribution of normalized non-linear mode shape contours of simply supported orthotropic plate (a) w=h =1:0, (b) w=h =1:4, (c) mode shapes along y = b=2, (d) mode shapes along x = a=2. Table 4 Comparison of non-linear frequency ratio (! NL =! L ) of single layer anisotropic/isotropic skew plate (a=b =1, w=h =1:0, E L =E T =10:0, G LT =E T =0:5, LT =0:333) Sl. No. Skew angle Ply-angle (single layer) Ref. [22] Present! NL =! L! NL =! L 1 0 Isotropic Isotropic From graph. the cross- and angle-ply square plates, and skew laminates, for certain amplitudes wherein a sudden decrease in non-linear frequency value occurs. The mode shape after the occurrence of mode redistribution reveals the change in the spatial distribution of mode shape and shifting the maximum displacement of the plate. The inuence of clamped boundary condition on the non-linear vibration behavior of skew laminates is studied assuming a ve-layered angle-ply [45 = 45 =45 = 45 =45 ] plate (a=b=1) and the results are tabulated in Table 7. The behavior, in general, is qualitatively similar to those of cross-ply case. It can be further concluded that although the type of non-linear

8 1716 M.K. Singha, M. Ganapathi / International Journal of Non-Linear Mechanics 39 (2004) Table 5 The non-linear frequency ratios (! NL =! L ) of a simply supported cross-ply [0 =90 =90 =0 ] skew plate (a=b =1) w=h Skew angle :3127 1:3252 1:3622 1:4108 1: Table 6 The non-linear frequency ratios (! NL =! L ) of a simply supported angle-ply [45 = 45 =45 = 45 =45 ] skew plate (a=b =1) w=h Skew angle : :4627 1: :4094 1: behavior is hardening, the degree of non-linearity is less compared to those of simply supported skew laminates. Furthermore, it is seen that the dropin the frequency ratio occurs at higher amplitudes in comparison with that of simply supported case. The eect of aspect ratio is also investigated considering a simply supported ve-layered angle-ply [45 = 45 =45 = 45 =45 ] skew plate and the variation of non-linear frequency ratio (! NL =! L ) with the non-dimensional amplitude of vibration (w=h) is reported in Table 8. In general, it is observed that the sudden dropin frequency occurs at low amplitude with the increase in the aspect ratio. Further, for the given amplitude, the change in the degree of non-linearity highly depends on the aspect ratio. Lastly, the non-linear free vibration of higher modes of laminated plates is investigated considering a simply supported ve-layered angle-ply [45 = 45 =45 = 45 =45 ] square, and skew plates and the non-linear frequency ratios evaluated are presented in Tables 9 and 10 for rst three modes by exciting the individual modes, respectively. It

9 M.K. Singha, M. Ganapathi / International Journal of Non-Linear Mechanics 39 (2004) Fig. 3. The redistribution of normalized non-linear mode shape of a simply supported plate: (a) cross-ply [0 =90 =90 =0 ] square plate, (b) angle-ply [45 = 45 =45 = 45 =45 ] square plate, (c) angle-ply [45 = 45 =45 = 45 =45 ] skew (a=b =1, =30 ) plate. is revealed from these tables that the vibration of the lowest modes can signicantly inuence the frequency ratio of higher modes whereas the higher mode vibration, to some extent, aects the values of non-linear frequency ratio pertaining to the lower modes. 5. Conclusions Large amplitude free vibration of composite skew plate has been investigated using a four-noded shear exible quadrilateral high precision plate bending element. The present formulation accounts for moderately nite deformation of the plate. Numerical studies are conducted here to examine the eect of skew angle, ber angle orientation, aspect ratio, and boundary conditions on the large-amplitude frequency and the mode shapes of composite skew plate. Some general observations are made as follows: (i) The non-linear frequency increases with the amplitude of vibration up to a certain value of amplitudes, where the non-linear mode shape suddenly changes leading to decrease in the frequency value.

10 1718 M.K. Singha, M. Ganapathi / International Journal of Non-Linear Mechanics 39 (2004) Table 7 The non-linear frequency ratios (! NL =! L ) of a clamped angle-ply [45 = 45 =45 = 45 =45 ] skew plate (a=b =1) w=h Skew angle : : : :3046 1: Table 8 The non-linear frequency ratios (! NL =! L ) of simply supported angle-ply [45 = 45 =45 = 45 =45 ] skew plate for various aspect ratios w=h Skew angle = 0 Skew angle = 30 b=a =1:0 b=a =1:5 b=a =2:0 b=a =1:0 b=a =1:5 b=a =2: : : :3343 1: : : (ii) The symmetric fundamental mode shapes become unsymmetric at the amplitude corresponding to sudden change in mode shape and the maximum displacement shifts towards one side for cross-ply laminates and towards one corner for angle-ply laminates. (iii) With increase in skew angle and aspect ratio, the mode redistribution phenomenon occurs at low amplitude of vibration for angle-ply laminates. (iv) The degree of non-linearity, in general, increases with skew angles and its rate of change depends on the occurrence of mode redistribution phenomenon. (v) The degree of hardening type of non-linearity is less for clamped boundary condition compared to those of simply supported boundary condition. (vi) In general, the degree of non-linearity highly depends on ply-angle and aspect ratio. (vii) The increase in non-linear frequency value pertaining to the vibrating mode is signicant compared to those of non-exciting modes.

11 M.K. Singha, M. Ganapathi / International Journal of Non-Linear Mechanics 39 (2004) Table 9 First three non-linear frequency ratios (! NL1 =! L1,! NL2 =! L2,! NL3 =! L3 ) of a simply supported angle-ply [45 = 45 =45 = 45 =45 ] square plate (a=b =1) w=h Vibration in rst mode Vibration in second mode Vibration in third mode! NL1 =! L1! NL2 =! L2! NL3 =! L3! NL1 =! L1! NL2 =! L2! NL3 =! L3! NL1 =! L1! NL2 =! L2! NL3 =! L Table 10 First three non-linear frequency ratios (! NL1 =! L1,! NL2 =! L2,! NL3 =! L3 ) of a simply supported angle-ply [45 = 45 =45 = 45 =45 ] skew plate (a=b =1, =30 ) w=h Vibration in rst mode Vibration in second mode Vibration in third mode! NL1 =! L1! NL2 =! L2! NL3 =! L3! NL1 =! L1! NL2 =! L2! NL3 =! L3! NL1 =! L1! NL2 =! L2! NL3 =! L References [1] A.W. Leissa, Plate vibration research, : classical theory, Shock Vib. Digest. 13 (1981) [2] A.W. Leissa, Plate vibration research, : complicating factors, Shock Vib. Digest. 13 (1981) [3] C.W. Bert, Research on dynamic behavior of composite and sandwich plates IV, Shock Vib. Digest. 17 (1985) [4] M. Sathyamoorthy, Nonlinear vibration analysis of plates: a review and survey of current developments, ASME, Appl. Mech. Rev. 40 (1987) [5] C.Y. Chia, Geometrically nonlinear behavior of composite plates: a review, ASME, Appl. Mech. Rev. 41 (1988) [6] R.K. Kapania, S. Raciti, Recent advances in analysis of laminated beams and plates, Part II: vibration and wave propagation, Am. Inst. Aeronaut. Astronaut. J. 27 (1989)

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