- Email: [email protected]
Stability of parametric vibrations of laminated composite plates
Applied Mathematics and Computation 223 (2013) 127–138
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Stability of parametric vibrations of laminated composite plates Wei-Ren Chen a,⇑, Chun-Sheng Chen b, Jenq-Huey Shyu c a b c
Department of Mechanical Engineering, Chinese Culture University, Taipei 11114, Taiwan Department of Mechanical Engineering, Lunghwa University of Science and Technology, Guishan Shiang 33306, Taiwan Department of Mechanical Engineering, National Taipei University of Technology, Taipei 10608, Taiwan
a r t i c l e
i n f o
Keywords: Dynamic stability Bolotin’s method Dynamic instability index
a b s t r a c t The dynamic stability of laminated composite plates subjected to arbitrary periodic loads is studied based on the first-order shear deformation plate theory. The in-plane load is taken to be a combination of periodic biaxial and bending stress. A set of second-order ordinary differential equations with periodic coefficients of Mathieu–Hill type is formed to determine the regions of dynamic instability based on Bolotin’s method. Numerical results reveal that the dynamic instability is significantly affected by the modulus ratio, number of layer, static and dynamic load parameters. The effects of various important parameters on the instability region and dynamic instability index are investigated. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction The lightweight high-strength composite plates have been used successfully in mechanical, aircraft industry and other engineering applications. When a plate is subjected to periodic loads, it is well known that under some circumstances the ordinary forced response will become dynamically unstable, leading to a violent vibration which is called the dynamic instability phenomenon of parametric resonance. To use them efficiently, a good understanding of dynamic behavior under various load conditions for laminated composite plates are needed. Thus, the problem of dynamic instability has become of ever increasing importance in modern composite structures design. It is necessary to accurately determine dynamic stability regions for suitable applications. Numerous references pertaining to the parametric resonance of plates can be found in the books by Bolotin [1] and Evan-Ivanowski [2]. The dynamic stability of plates subjected to a periodic in-plane loading had received a great deal of attention in the past years. Petry and Fahlbusch [3] investigated the dynamic stability behavior of imperfect plates subjected to in-plane pulse loadings. The governing equations were solved by the Galerkin method and Navier’s double Fourier series. Gilat and Aboudi [4] studied the dynamic instability of elastic composite plates subjected to a periodic in-plane loading. The effect of the matrix non-linearity on the overall response of the composite plate was predicted by the micromechanical method of cells. The dynamic stability of plates under periodic in-plane forces was analyzed and the corresponding stability regions of the first and second order were calculated by Baldinger et al. [5]. The dynamic stability analysis of a plate under a follower force was performed through the finite element method based on the Kirchhoff–Love plate theory by Kim and Kim [6]. The effects of shear deformation and rotary inertia were investigated. A dynamic stability of an undeflected state of isotropic elastic circular plate was analyzed. A dynamic instability index is devised by Wang and Dawe [7] to measure the dynamic instability of composite laminated plates under periodic dynamic loads. The boundary frequencies were determined by using Bolotin’s method, Sturm sequence method and multi-level substructuring technique. Ye et al. [8] investigated the parametric excitation of a simply supported cross-ply laminated composite rectangular plate. The relations between the steady state nonlinear ⇑ Corresponding author. E-mail address: [email protected] (W.-R. Chen). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.07.095
128
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
responses and the amplitude and frequency of parametric excitation were obtained. Yang et al. [9] conducted a dynamic stability analysis of FGM plates subjected to a uniaxial periodic in-plane load and under a uniform temperature change. A Galerkin-differential quadrature approach was employed to convert the governing equations into a linear system of Mathieu–Hill equations from which the boundary points on the unstable regions were determined by Bolotin’s method. The nonlinear dynamical equations of three-dimensional frame in the orthogonal coordinates system were obtained by Wang et al. [10]. Then the equations were transformed into the axial symmetry nonlinear dynamical equations in the polar coordinates system. Wei and Yang [11] investigated the nonlinear dynamic stability of composite laminated plates. The influence of the large deflection, initial imperfection, support conditions and ply-angle of the fibers was considered. The dynamic stability of laminated plates with four free edges subjected to pulsating follower forces was discussed by Choo and Kim [12]. The results were obtained using the first-order shear deformation theory. The influences of aspect ratios, lamination angles, and the number of layers were also studied. The stability analysis of laminated plates under in-plane compressive loading was studied by Iyengar and Chakraborty [13]. The change in initial buckling response of thick laminates with respect to the fiber orientation angle was investigated. The dynamic stability characteristics of laminated composite skew plates subjected to a periodic in-plane load were investigated by Dey and Sinqha [14] based on the finite element approach. The principal and second instability regions were identified for different skew angles, thickness-to-span ratios, fiber orientations and static in-plane loads. Udar and Datta [15] predicted a combination resonance of parametrically excited laminated composite plates. The modal transformation was applied to transform the equilibrium equation into a suitable form for the application of the multiple scales method. Goren [16] analyzed the dynamic stability of cantilever symmetric laminated beams based on the classical plate theory. The effects of the ply orientation, static and dynamic load parameters on the stability were examined. The dynamic instability of imperfect composite laminates was presented by Chakrabarti [17] by using a finite element plate model. The excitation frequency was found to decrease rapidly with the increase in the imperfection parameter. Patel et al. [18] studied the dynamic instability behavior of laminated composite stiffened plates using Hill’s infinite determinant method. The type and the width of loadings had remarkable effects on the dynamic instability characteristics of the stiffened plates. The static and dynamic transitions between the stable states for laminated composite plates were considered by Diaconu et al. [19]. The dynamic analysis model of the snap-through phenomena was proposed for the plates based on the strain field approximation. The model was used to investigate the static and dynamic transitions from one stable state to another. The finite element dynamic stability of laminated skew plates with cutouts and subjected to a distributed periodic in-plane load was analyzed by Lee [20] based on the higher-order shear deformation theory. The influences of skew angles, cutout sizes, layup sequences and thickness-to-length ratios on the onset of the load frequency, width of instability regions, and dynamic instability indexes were presented. Pradyumna and Bandyopadhyay [21] investigated the dynamic instability behavior of laminated hypar and conoid shells using a higher-order shear deformation theory and a finite element approach. The Bolotin’s method is applied to find the boundaries of the instability regions. The effect of number of layers, ply orientation, thickness, curvature, static load component and boundary conditions on the dynamic instability regions was studied. Ramachandra and Panda [22] studied the dynamic instability of a shear deformable composite plate under periodic nonuniform in-plane loads with either parabolic or linear distributions. Based on Bolotin’s method, the first-order and second-order instability regions were obtained for various dynamic in-plane loads. The effects of the span to thickness ratio, shear deformation, aspect ratio, boundary conditions and static load factor on the instability regions were also investigated. The parametric instability of laminated longitudinally stiffened panels with internal cutouts under dynamic in-plane loadings was investigated by Fazilati and Ovesy [23] based on the finite strip method. The effects of cutouts and stiffeners on the instability regions were investigated using the Bolotin’s first order approximation. The results revealed that the finite strip method is capable of predicting dynamic instability regions of laminated panels with complex geometries with fair accuracy. Based on the Lindstedt–Poincare perturbation technique, Khalili et al. [24] studied the buckling stability of a non-ideal simply supported laminated plate resting on elastic foundations. One of the edges of the plate is assumed to allow a small non-zero deflection and moment. The effects of non-ideal boundary conditions, aspect ratio, stiffness of foundation, shear modulus, in-plane pre-load on the buckling load of laminated plates were discussed. The vibration behaviors of composite plates under arbitrary initially stresses had been analyzed previously by the second author [25–27]. In this paper, the problem to be considered is the dynamic oscillations of laminated plates subjected to arbitrary periodic loads. The dynamic load is taken to be a combination of a periodic bending stress and a periodic biaxial stress. Based on the first-order shear deformation theory, the governing equations of motion of the laminate plates are derived by a perturbation technique. The Galerkin method is then applied to the governing partial differential equations to yield a set of second-order ordinary differential equations with periodic coefficients of Mathieu–Hill type. Based on Bolotin’s method, a set of homogeneous linear algebraic equations is formulated and used to obtain the boundaries of the dynamic instability regions of laminated composite plates. An eigenvalue problem is solved to determine the boundary frequencies for the boundaries of instability regimes. The effects of various parameters on the region of dynamic stability and dynamic instability index are studied. 2. Problem formulation In order to approximate the three-dimensional problem, Cartesian coordinates are used to specify the plate geometry in two-dimensional model. The xy plane is coincident with the middle plane of the plate, and the origin of the coordinate
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
129
system is at the left corner of the middle plane. The displacements components u, v and w at any position in the plate space are expanded with the thickness coordinate. Based on the Mindlin plate theory, the displacement fields have the following expressions:
uðx; y; z; tÞ ¼ ux ðx; y; tÞ þ zux ðx; y; tÞ
v ðx; y; z; tÞ ¼ v y ðx; y; tÞ þ zuy ðx; y; tÞ
ð1Þ
wðx; y; z; tÞ ¼ wz ðx; y; tÞ; where u, v and w are the displacement function of the x, y and z axes, respectively. ux and vy are in-plane displacements; wz is transverse displacement; ux and uy are angles of rotation of the normal to the middle plane about y and x axes. An initially stressed plate, which is in static equilibrium and subjected to a time-varying incremental deformation, is considered in this study. Following a similar technique described by Brunelle and Robertson [28] and Chen et al. [27], the governing equations of motion of the laminate plate including the effects of rotary inertia and transverse shear are established using a perturbation technique. The governing equation is expressed as
€ s is;i þ F s þ DF s ¼ qu s;j Þ;i þ r ðrij u
ð2Þ
is , u s , Fs and DFs denote the initial stress, perturbing stress, displacement, perturbing body force and body force, where rij, r respectively. In this paper, a rectangular plate is considered so the equations will be rephrased in xy coordinates. The constitutive relations for a kth lamina are given by:
3 2 r xx ðkÞ C 11 7 6r 6 6 yy 7 6 C 12 7 6 6 7 6r 6 6 yz 7 ¼ 6 0 6 7 6 4 rzx 5 4 0 r xy C 16 2
C 12
0
0
C 22 0
0 C 44
0 C 45
0
C 45
C 55
C 26
0
0
C 16
3ðkÞ 2
7 C 26 7 7 0 7 7 7 0 5 C 66
3 exx ðkÞ 6 e 7 6 yy 7 7 6 6 eyz 7 7 6 6 7 4 ezx 5 exy
ð3Þ
Consider a rectangular laminated plate of uniform thickness h subjected to arbitrary time-dependent initial stresses. The periodic initial stress system is assumed to have the form: Dm rij ¼ rnij þ 2zrmij =h ¼ ðrSij þ rDij cos xtÞ þ 2zðrSm ði; j ¼ x; y; zÞ ij þ rij cos xtÞ
ð4Þ
which consists of the spatially uniform longitudinal, transverse, shear, bending and twisting stress. Here rSij and rDij are the Dm respective static and dynamic component of the periodic normal or shear stress rnij ; rSm represent the respective ij and rij static and dynamic component of the periodic pure bending or torsion stress rm ; x is the angular frequency of excitation. ij Substitute displacement field (1) and Eqs. (3) and (4) into Eq. (2), perform all necessary partial integrations and group terms together by the displacements variation to yield the dynamic equations of rectangular laminated plates as
½A11 ux;x þ A16 ðux;y þ uy;x Þ þ A12 uy;y þ B11 ux;x þ B16 ðux;y þ uy;x Þ þ B12 uy;y þ Nxx ux;x þ M xx ux;x þ Nxy ux;y þ M xy ux;y þ N xz uz;x ;x þ ½A16 ux;x þ A26 uy;y þ A66 ðux;y þ uy;x Þ þ B16 ux;x þ B66 ðux;y þ uy;x Þ þ B26 uy;y þ Nyy ux;y þ M yy ux;y þ Nxy ux;x þ Mxy ux;x €x þ Nyz uz;x ;y þ fx ¼ qhu h
ð5Þ
i A16 ux;x þ A66 ðux;y þ uy;x Þ þ A26 uy;y þ B16 ux;x þ B66 ðux;y þ uy;x Þ þ B26 uy;y þ N xx uy;x þ Mxx uy;x þ Nxy uy;y þ M xy uy;y þ Nxz uz;y ;x h i þ A12 ux;x þ A26 ðux;y þ uy;x Þ þ A22 uy;y þ B12 ux;x þ B26 ux;y þ uy;x Þ þ B22 uy;y þ Nyy uy;y þ M yy uy;y þ Nxy uy;x þ M xy uy;x þ Nxz uz;y
;y
€y þ fy ¼ qhu € ½A55 ðw;x þ ux Þ þ A45 ðw;y þ uy Þ þ Nxx w;x þ N xy w;y ;x þ ½A45 ðw;x þ ux Þ þ A44 ðw;y þ uy Þ þ Nxy w;x þ Nyy w;y ;y þ fz ¼ qhw
ð6Þ ð7Þ
½B11 ux;x þ B16 ðux;y þ uy;x Þ þ B12 uy;y þ D11 ux;x þ D16 ðux;y þ uy;x Þ þ D12 uy;y þ M xx ux;x þ M xy ux;x þ M xy ux;y þ M xy ux;y þ M xz uz;x ;x þ ½B16 ux;x þ B66 ðux;y þ uy;x Þ þ B26 uy;y þ D16 ux;x þ D66 ðux;y þ uy;x Þ þ D26 uy;y þ Myy ux;y þ M yy ux;y þ Mxy ux;x þ M xy ux;y 3 € x =12 þ Myz uz;x ;y A55 ðw;x þ ux Þ A45 ðw;y þ uy Þ ðNxz ux;x þ M xz ux;x þ Nzz ux þ Nzy ux;y þ M zy ux;y Þ þ mx ¼ qh u
ð8Þ
½B16 ux;x þ B66 ðux;y þ uy;x Þ þ B26 uy;y þ D16 ux;x þ D66 ðux;y þ uy;x Þ þ D26 uy;y þ Mxx uy;x þ Mxx uy;x þ Mxy uy;y þ Mxy uy;y þ M xz uz;y ;x þ ½B26 ðux;y þ uy;x Þ þ B12 ux;x þ B22 uy;y þ D12 ux;x þ D26 ðux;y þ uy;x Þ þ D22 uy;y þ Myy uy;y þ M yy uy;y þ Mxy uy;x þ M xy uy;x 3 € y =12 þ Mxz uz;y ;y A45 ðw;x þ ux Þ A44 ðw;y þ uy Þ ðNxz uy;x þ M xz uy;x þ Nzz uyx þ N zy uy;y þ Mzy uy;y Þ þ my ¼ qh u
where
ð9Þ
130
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
ðAij ; Bij ; Dij Þ ¼ ðAij ; Bij ; Dij Þ ¼
Z Z
ðNij ; Mij ; M ij Þ ¼
C ij ð1; z; z2 Þdz ði; j ¼ 1; 2; 6Þ
jC ij ð1; z; z2 Þdz ði; j ¼ 4; 5Þ Z
ð10Þ
h=2
rij ð1; z; z2 Þdz ði; j ¼ x; y; zÞ
h=2
Here Cij’s are the elastic constant of the stiffness matrix; q is the mass density; Aij, Bij and Dij are the laminate stiffness coefficients; j is the shear correction factor; Nij, Mij and M ij are arbitrary initial stress resultants; fx, fy, fz, mx and my are the lateral loadings. All the integrations are carried out through the thickness of the plate from h/2 to h/2. 3. Dynamic instability analyses The dynamic behaviors of laminated plate governed by Eqs. (5)–(9) are so complicated that it would be difficult to present results for all cases. The case to be investigated is the dynamics of the simply supported rectangular laminated plate subjected to the periodic spatially uniform biaxial in-plane stress system consisted of a pulsating longitudinal normal stress and a pulsating pure bending stress. With all other stresses assumed to be zero, the stress system Eq. (4) is reduced to
rxx ¼ rn þ 2zrm =h S
D
ð11Þ Sm
Dm
where rn = r + r cos xt = ðr þ r þ ðr þ r xt and rm = r + r cos xt = ðr þ r þ ðr þ r xt. All Sm Dm Dm the static and dynamic stress components rSxx , r r rDyy , rSm xx , ryy , rxx and ryy are taken to be constants. b = rm/rn is the ratio of bending stress to normal stress. The rectangular plate considered has boundary sides along x = 0 and a, y = 0 and b, all of which are simply supported. To satisfy above geometric boundary conditions, the following shape modes of displacement fields are used
ux ¼ uy ¼
XX XX XX
S xx
S yy Þ
D xx
D yy Þ cos S D yy , xx ,
Sm yy Þ
Dm xx
Dm yy Þ cos
hU mn ðtÞcosðmpx=aÞsinðnpy=bÞ hV mn ðtÞsinðmpx=aÞcosðnpy=bÞ
hW mn ðtÞsinðmpx=aÞsinðnpy=bÞ XX ux ¼ Wxmn ðtÞcosðmpx=aÞsinðnpy=bÞ XX Wymn ðtÞsinðmpx=aÞcosðnpy=bÞ uy ¼ w¼
Sm xx
ð12Þ
Assume that the D = [Umn, Vmn, Wmn, Wxmn, Wymn]T which can be expressed as
DðtÞ ¼ Df ðtÞ
ð13Þ
Here D denotes time independent displacement vector. Substituting Eqs. (12) and (13) into governing Eqs. (5)–(9) and applying the Galerkin method, we obtain the following set of ordinary differential equations as 2
fð½K ðNxx þ Nyy Þ½GÞf ðtÞ þ ½Mðd f ðtÞ=dtÞgD ¼ 0
ð14Þ
where [K] is the elastic stiffness matrix, [G] is the geometric stiffness matrix and [M] is the consistent mass matrix. The system Eq. (14) is related to the eigenvalue problems of the static buckling, free vibration and dynamic instability. The static buckling problem is formed from Eq. (14) by setting f(t) = 1, which leads to
f½K ðNxx þ Nyy Þ½GgD ¼ 0
ð15Þ
Eq. (15) can be used to determine the smallest applied loads Nxx and Nyy, which cause the buckling of the laminated plate. The eigenvalue equation of the free vibration of the laminated plate is obtained from Eq. (14) by setting Nxx + Nyy = 0 and f(t) = eixt
f½K x2 ½MgD ¼ 0
ð16Þ
The roots of the determinant of the coefficients of Eq. (16) are the natural frequencies of the plate. The non-zero initial load and moment resultants in Eq. (11) can be written as:
Nxx þ Nyy ¼ ½ðaS þ aYS ÞPcr þ ðaD þ aYD ÞPcr cos -t
ð17Þ
where aS ¼ hr a ¼ hr aD ¼ hr and a ¼ hr P cr is the buckling load of the plate subjected to uniaxial in-plane load. aS ðaYS Þ and aD ðaYD Þ are static and dynamic load parameters, respectively. Substituting Eq. (17) into Eq. (14) gives S xx =P cr ;
Y S
S yy =P cr ;
D xx =P cr
Y D
D yy =P cr
2
fð½K þ ðaS þ aYS ÞP cr ½G þ ðaD þ aYD ÞPcr ½G cos -tÞf ðtÞ þ ½Mðd f ðtÞ=dtÞgD ¼ 0
ð18Þ
131
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
Eq. (18) represents the dynamic stability problem of a parametrically excited system subjected to a periodic biaxial in-plane load. According to Bolotin’s method [1], the boundaries of the dynamic instability regions can be constructed by the periodic solutions with period 2T and T respectively in Fourier series as
f ðtÞD ¼
f ðtÞD ¼
1 X k-t k-t ak sin þ bk cos 2 2 k¼1;3;5... 1 X k¼0;2;4...
ak sin
k-t k-t þ bk cos 2 2
ð19Þ
ð20Þ
where ak and bk are arbitrary time invariant constants. Substituting the series expansions (19) and (20) into Eq. (18) and separating the sine and cosine parts, two sets of linear algebraic equations in ak and bk are obtained for each solution. Then, the boundaries between stable and unstable regions could be obtained from the condition for the set of equations to have nontrivial solution. The eigenvalue system of the dynamic stability boundaries with period 2T is given as
1 ½K þ ðaS þ aY ÞPcr ½G 1 ðaD þ aY ÞPcr ½G 1 -2 ½M ðaD þ aYD ÞPcr ½G . . . S D 2 4 2 1 ðaD þ aYD ÞPcr ½G ½K þ ðaS þ aYS ÞPcr ½G 94 -2 ½M . . . ¼ 0 2 ...
ð21Þ
The governing equations of dynamic stability boundaries with period T is
1 ½K þ ðaS þ aY ÞPcr ½G -2 ½M ðaD þ aYD ÞPcr ½G . . . S 2 1 ðaD þ aYD ÞPcr ½G ½K þ ðaS þ aYS ÞPcr ½G 4-2 ½M . . . ¼ 0 2 ...
ð22Þ
½K þ ðaS þ aY ÞPcr ½G ðaD þ aYD ÞPcr ½G . . . S ðaD þ aYD ÞPcr ½G ½K þ ðaS þ aYS ÞPcr ½G -2 ½M . . . ¼ 0 ...
ð23Þ
and
Generally, it is impossible to solve the determinant equations (21)–(23) of infinite order to find the stability boundaries. Hence, different orders of approximation can be used to obtain approximate results which are sufficiently close approximations of the infinite eigenvalue problem. The boundaries of primary instability region with period 2T are usually much larger than those of secondary instability region with period T, and are therefore of greater practical importance. Furthermore, useful results can be obtained by considering only the first-order approximation of the major instability boundaries, which is capable of obtaining solutions with sufficient accuracy [29]. Hence, only the first-order approximation of the principal stability boundaries will be given in the present study. From element (1, 1) of Eq. (21), the first-order solution of primary instability regions can be obtained in the following form
1 1 ½K þ ðaS þ aYS ÞP cr ½G ðaD þ aYD ÞPcr ½G -2 ½M ¼ 0 2 4
ð24Þ
The eigenvalues of the above equation are just the frequencies of excitation for the boundaries of instability regimes.
Table 1 Excitation frequencies for a simply supported symmetrically four-layer cross-ply laminate plate with various static and dynamic loads.
aS
aD
Wang’s results [7] U
0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8
0.0 0.3 0.6 0.9 1.2 1.5 0.06 0.12 0.18 0.24
Present results L
x
x
xU
xL
144.57 155.03 164.83 174.08 182.87 191.25 131.71 117.45 101.20 81.78
144.57 133.29 120.95 107.21 91.43 72.28 126.86 106.24 80.49 40.89
144.36 155.64 165.12 174.43 183.21 191.75 132.12 117.96 101.84 82.31
144.36 133.79 121.45 107.63 91.86 72.62 127.26 106.82 81.10 41.32
132
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
Fig. 1. Effect of dynamic load parameter on the instability regions of laminated plates with various layer numbers (a/b = 1, a/h = 10, Ex/Ey = 10, aS = 0.1, q: N = 2, h: N = 4, s: N = 6, N: N = 8).
Fig. 2. Effect of static compressive load parameter on the instability regions of laminated plates with various layer numbers (a/b = 1, a/h = 10, Ex/Ey = 10, aD /aS = 0.3, q: N = 2, h: N = 4, s: N = 6, 4: N = 8). Table 2 Effect of the static and dynamic loads on the instability regions DX of laminated plates under various layer numbers N (a/b = 1, a/ h = 10, Ex/Ey = 10, b = 0).
aS 0.1 0.1 0.1 0.1 0.1 0 0.2 0.4 0.6 0.8
aD 0 0.4 0.8 1.2 1.6 0 0.06 0.12 0.18 0.24
N 2
4
6
8
0 4.2051 8.5478 13.2302 18.7104 0 0.7018 1.6224 2.9961 5.9166
0 5.9862 12.1984 18.8341 26.6355 0 0.9990 2.3096 4.2651 8.4229
0 6.2180 12.6378 19.5234 27.6668 0 1.0377 2.3990 4.4302 8.7490
0 6.4235 12.7950 19.8040 28.0071 0 1.0504 2.4285 4.4848 8.8567
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
133
Fig. 3. Effect of number of layer N on dynamic instability index of laminated plates under various dynamic loads. (a/b = 1, a/h = 10, Ex/Ey = 10, aS = 0.1, q: aD = 0.4, h: aD = 0.8, s: aD = 1.2, 4: aD = 1.6).
Fig. 4. Effect of number of layer N on dynamic instability index of laminated plates under various static loads. (a/b = 1, a/h = 10, Ex/Ey = 10, aD /aS = 0.3, q: aS = 0.2, h: aS = 0.4, s: aS = 0.6, 4: aS = 0.8).
4. Result and discussion To check the accuracy of the present model, numerical results of the excitation frequency for a simply supported symmetric four-layer cross-ply laminate plate under various static and dynamic loads are computed and compared with the results by Wang [7]. The upper and lower excitation frequencies of the primary instability region are presented in Table 1. It can be observed that the presented excitation frequencies are in good agreement with those obtained by Wang. In the next, the influence of various parameters of the cross-ply laminate plate on its dynamic instability behaviors will be examined qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
and discussed. The non-dimensional coefficients of excitation frequency X ¼ -b
q=h2 Ey , the instability region
DX ¼ XU XL and the dynamic instability index XDI ¼ DX=ðxnf K cr Þ are defined and used throughout the dynamic instability studies. Here XU and XL are the upper and lower boundary excitation frequency, respectively; xnf is the non-dimensional
134
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
Fig. 5. Effect of modulus ratio on dynamic instability index of laminated plates under various dynamic loads. (a/b = 1, a/h = 10, N = 2, aS = 0.1, q: aD = 0.4, h: aD = 0.8, s: aD = 1.2, 4: aD = 1.6).
Fig. 6. Effect of modulus ratio on dynamic instability index of laminated plates under various static and dynamic loads. (a/b = 1, a/h = 10, N = 2, aD /aS = 0.3, q: aS = 0.2, h: aS = 0.4, s: aS = 0.6, 4: aS = 0.8).
2
fundamental natural frequency given by xnf ¼ xb 2
4
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q=h2 Ey ;Kcr is the dimensionless critical buckling load given by
Kcr = ðN xx Þcr b =Ey h . The effects of static and dynamic load parameters on the principal unstable regions of laminated plates with various layer numbers are shown in Figs. 1 and 2 and Table 2. In Fig. 1, the static load parameter aS = 0.1 and the dynamic load parameter increases from 0 to 1.6. As can be seen, the upper excitation frequency increases and lower excitation frequency decreases with the increase of dynamic load parameter. The instability regions tend to shift to higher frequencies of excitation as the layer number is increased. The onset of dynamic instability occurs much later with the increasing layer number. In Fig. 2, the static load parameter aS varies from 0 to 0.8 and the ratio of aD =aS is kept as 0.3. It can be observed that both the upper and lower excitation frequency decrease with the increasing static load parameter. This is due to the fact that the compressive static load tends to reduce the plate stiffness. Similarly, when the layer number is increased, the instability region is found to move to the higher excitation frequency. As shown in Table 2, the primary instability region DX for the laminated plate
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
135
Fig. 7. Effect of dynamic load parameter on instability region of laminated plates under various static loads. (a/b = 1, a/h = 10, N = 2, E1/E2 = 40, s: aS = -0.4, 4: aS = 0.2, w; aS = 0, N: aS = 0.2, d: aS = 0.4).
Fig. 8. Effect of dynamic load parameter on instability region of laminated plates under various lateral static loads. (a/b = 1, a/h = 10, N = 2, E1/E2 = 40, aS = 0.2, s: aYS = 0.3, w: aYS = 0, d: aYS = 0.3).
increases with the increasing layer number, static and dynamic load parameter, and the width of the unstable zone is becoming much wider at the higher dynamic load parameter. Meanwhile, the instability region is largely controlled by the magnitude of dynamic load parameter rather than the static load parameter. The effects of the packing layer number on the dynamic instability index of laminated plates under different dynamic and static loads are showed in Figs. 3 and 4. The dynamic instability index is decreased with the increasing layer number, but increased with the increasing static and dynamic load parameter. However, the effect of layer number N on the dynamic instability of the plates is diminished as N is greater than 4. Meanwhile, the dynamic load parameter has an apparent influence than the static load parameter on the dynamic instability index. Thus, the lower layer number and the higher dynamic load parameter are the more dynamically unstable is the laminate composite plate. Figs. 5 and 6 present the effect of the modulus ratio on the dynamic instability index for the laminated plates under various dynamic and static load parameters. With the increase in the static or dynamic load parameter, the dynamic instability index increases regardless of the
136
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
Fig. 9. Effect of the dynamic load parameter on dynamic instability index of laminated plates under various lateral static loads. (a/b = 1, a/h = 10, N = 2, E1/E2 = 40, aS = 0.2, s: aYS = 0.3, w: aYS = 0, d: aYS = 0.3). Table 3 Effect of the bending stress parameter on the principle unstable region and dynamic instability index of two-layer cross-ply laminated plate under various dynamic loads (a/b = 1, a/h = 10, aS = 0.1, Ex/Ey = 10).
aD
b
U
0
10
20
30
40
X XL XDI XU XL XDI XU XL XDI XU XL XDI XU XL XDI
0
0.4
0.8
1.2
1.6
14.7038 14.7038 0 14.7074 14.7074 0 14.7178 14.7178 0 14.7340 14.7340 0 14.7546 14.7546 0
16.2556 12.9675 6.9729 16.2516 12.9770 6.9785 16.2401 13.0047 6.9949 16.2221 13.0475 7.0215 16.1993 13.1016 7.0573
17.6717 10.9595 14.2342 17.6587 10.9730 14.2480 17.6209 11.0119 14.2889 17.5622 11.0722 14.3547 17.4876 11.1482 14.4428
18.9825 8.4892 22.2525 18.9591 8.5038 22.2816 18.8914 8.5460 22.3671 18.7860 8.6113 22.5046 18.6517 8.6935 22.6873
20.2084 4.9012 32.4610 20.1736 4.9119 32.5243 20.0726 4.9435 32.7096 19.9149 4.9919 33.0069 19.7135 5.0527 33.4013
Table 4 Effect of the bending stress parameter on the principle unstable region and dynamic instability index of two-layer cross-ply laminated plate under various static compressive loads (a/b = 1, a/h = 10, aD/aS = 0.3, Ex/Ey = 10).
aS
b
0
10
20
30
40
XU XL XDI XU XL XDI XU XL XDI XU XL XDI XU XL XDI
0
0.2
0.4
0.6
0.8
15.4991 15.4991 0 15.4991 15.4991 0 15.4991 15.4991 0 15.4991 15.4991 0 15.4991 15.4991 0
14.1204 13.6004 1.1026 14.1263 13.6081 1.1043 14.1433 13.6304 1.1091 14.1698 13.6648 1.1169 14.2033 13.7084 1.1275
12.5915 11.3895 2.5492 12.6021 11.4023 2.5567 12.6325 11.4396 2.5791 12.6796 11.4972 2.6153 12.7391 11.5699 2.6638
10.8494 8.6295 4.7075 10.8630 8.6441 4.7286 10.9023 8.6865 4.7907 10.9631 8.7518 4.8909 11.0399 8.8343 5.0249
8.7676 4.3838 9.2966 8.7822 4.3936 9.3528 8.8246 4.4225 9.5176 8.8901 4.4667 9.7837 8.9726 4.5222 10.1392
137
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
Table 5 Effect of the bending stress parameter on the principle unstable region and dynamic instability index of two-layer cross-ply laminated plate under various static tensile loads (a/b = 1, a/h = 10, aD/aS = 0.3, Ex/Ey = 10).
aS
b
U
0
10
20
30
40
X XL XDI XU XL XDI XU XL XDI XU XL XDI XU XL XDI
0
0.2
0.4
0.6
0.8
15.4991 15.4991 0 15.4991 15.4991 0 15.4991 15.4991 0 15.4991 15.4991 0 15.4991 15.4991 0
17.1894 16.7649 0.9002 17.1796 16.7579 0.8989 17.1515 16.7376 0.8949 17.1077 16.7061 0.8884 17.0522 16.6661 0.8797
18.7277 17.9415 1.6671 18.7065 17.9265 1.6622 18.6451 17.8831 1.6474 18.5494 17.8155 1.6234 18.4276 17.7295 1.5905
20.1489 19.0457 2.3395 20.1147 19.0218 2.3291 20.0154 18.9525 2.2981 19.8605 18.8446 2.2470 19.6626 18.7070 2.1769
21.4762 20.0892 2.9415 21.4276 20.0556 2.9240 21.2863 19.9580 2.8718 21.0652 19.8059 2.7855 20.7817 19.6114 2.6662
modulus ratio. As can be observed, the dynamic instability index is reduced when the modulus ratio is increased. Hence, the laminated plate is becoming more dynamically unstable as its modulus ratio decreases. The effect of the static loading type on the instability region of the laminated plate is depicted in Fig. 7. It shows that when the static load varies from the tensile (aS < 0) to the compressive type (aS > 0), the instability regions tend to shift to lower excitation frequencies and become wider. Hence, the tensile static load has a strengthen effect on the dynamic stability but the compressive static load has the opposite influence. The laminated plate under the compressive static load is more dynamically unstable than that under the tensile one. Fig. 8 shows the effect of the biaxial static load on the instability regions. As can be seen, the onset of the dynamic instability shifts to a lower excitation frequency as the lateral static load varies from the tensile (aYS < 0) to the compressive load (aYS > 0). The width of the instability regions are also enlarged significantly with the increasing compressive lateral static load. The effect of various lateral static load types on the dynamic instability index is presented in Fig. 9. It is observed that the dynamic instability index is significantly affected by the compressive lateral static load. The tensile lateral static load has little effect on the dynamic instability index. Therefore, the compressive lateral static load shows a destabilizing effect on the dynamic stability behavior of the laminate plate. The effects of the bending stress ratio on the excitation frequency and dynamic instability index of laminate plates under different dynamic loads are presented in Table 3. It can be found that the increasing bending stress produces a slightly reduction in the upper boundary frequency and increase in the lower boundary frequency and dynamic instability index. Table 4 shows the effects of the bending stress ratio on the dynamic stability for the plates under various compressive static loads. As the bending stress ratio increases, the excitation frequency and dynamic instability index increase. Hence, the higher are the bending stress ratio and compressive static or dynamic load parameter, the more dynamically unstable is the laminate plate. Table 5 presents the influence of the bending stress and tensile static load. As can be observed, the values of the excitation frequency and dynamic instability index reduce slightly when the bending stress ratio arises. Therefore, the laminate plate under the higher tensile load and bending stress ratio is more dynamically stable. 5. Concluding remarks The dynamic stability of laminate plate subjected to periodic spatially uniform biaxial loads has been described and discussed in this paper. Following the above discussions, some conclusions are addressed as follows: (1) The excitation frequency, instability region and dynamic instability index are significantly affected by the static load, dynamic load, layer number and modulus ratio. They are slightly affected by the bending stress. (2) The upper excitation frequency increases with the increasing of the tensile static load, dynamic load and layer number. The lower excitation frequency increases with the increase of the tensile static load and layer number, and the decrease of the dynamic load. As the static, dynamic load parameter and layer number is increased, the instability region is increased. (3) The dynamic instability index increases with the increasing compressive static loading and dynamic loading and reduces with the increase of the layer number, modulus ratio and tensile static loading.
References [1] V.V. Bolotin, The Dynamic Stability of Elastic Systems, Holden-Day, San Francisco, 1964. [2] R.M. Evan-Ivanowski, Resonance Oscillations in Mechanical Systems, Elsevier, Amsterdm, 1976.
138 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138 D. Petry, G. Fahlbusch, Dynamic buckling of thin isotropic plates subjected to in-plane impact, Thin-Walled Struct. 38 (2000) 267–283. R. Gilat, J. Aboudi, Parametric stability of non-linearly elastic composite plates by Lyapunov exponents, J. Sound Vib. 235 (2000) 627–637. M. Baldinger, A.K. Belyaev, H. Irschik, Principal and second instability regions of shear-deformable polygonal plates, Comput. Mech. 26 (2000) 288–294. J.H. Kim, H.S. Kim, Study on the dynamic stability of plates under a follower force, Comput. Struct. 74 (2000) 351–363. S. Wang, D.J. Dawe, Dynamic instability of composite laminated rectangular plates and prismatic plate structures, Comput. Methods Appl. Mech. Eng. 191 (2002) 1791–1826. M. Ye, Q. Ding, Y.H. Sun, W. Zhang, Nonlinear oscillations of an antisymmetric cross-ply laminated composite thin rectangular plate under parametric excitation, ASME Des. Eng. Div. 116 (2003) 621–630. J. Yang, K.M. Liew, S. Kitipornchai, Dynamic stability of laminated FGM plates based on higher-order shear deformation theory, Comput. Mech. 33 (2004) 305–315. X.Z. Wang, G. Wang, Y.Y. Zhao, K.Y. Yeh, Nonlinear dynamical stability analysis of the circular three-dimensional frame, Appl. Math. Mech. 25 (2004) 367–372. D.M. Wei, G.T. Yang, Nonlinear dynamic stability of composite laminated plates, Appl. Math. Mech. 25 (2004) 1215–1219. Y.S. Choo, J.H. Kim, Dynamic stability of rectangular plates subjected to pulsating follower forces, AIAA J. 38 (2004) 353–361. N.G.R. Iyengar, A. Chakraborty, Interaction curves for composite laminates subjected to biaxial in-plane loads, Int. J. Struct. Stab. Dyn. 5 (2005) 539– 555. P. Dey, M.K. Sinqha, Dynamic stability analysis of composite skew plates subjected to periodic in-plane load, Thin-Walled Struct. 44 (2006) 937–942. R.S. Udar, P.K. Datta, Combination resonance characteristics of laminated composite plates subjected to non-uniform harmonic edge loading, Air Eng. Aero. Tech. 78 (2006) 107–119. K.A. Goren, Static and dynamic stability analyses of the symmetric laminated cantilever beams, Adv. Comput. Lett. 17 (2008) 155–160. A. Chakrabarti, An efficient FE model for dynamic instability analysis of imperfect composite laminates, Struct. Eng. Mech. 30 (2008) 383–386. S.N. Patel, P.K. Datta, A.H. Sheikh, Parametric study on the dynamic instability behavior of laminated composite stiffened plate, J. Eng. Mech. 135 (2009) 1331–1341. G.G. Diaconu, P.M. Weaver, A.F. Arrieta, Dynamic analysis of bi-stable composite plates, J. Sound Vib. 322 (2009) 987–1004. S.Y. Lee, Finite element dynamic stability analysis of laminated composite skew plates containing cutouts based on HSDT, Comput. Sci. Technol. 70 (2010) 1249–1257. S. Pradyumna, J.N. Bandyopadhyay, Dynamic instability behavior of laminated hypar and conoid shells using a higher-order shear deformation theory, Thin-Walled Struct. 49 (2011) 77–84. L.S. Ramachandra, S.K. Panda, Dynamic instability of composite plates subjected to non-uniform in-plane loads, J. Sound Vib. 331 (2012) 53–65. J. Fazilati, H.R. Ovesy, Parametric instability of laminated longitudinally stiffened curved panels with cutout using higher order FSM, Comput. Struct. 95 (2013) 691–696. S.M.R. Khalili, P. Abbaspour, K. Malekzadeh Fard, Buckling of non-ideal simply supported laminated plate on Pasternak foundation, Appl. Math. Comput. 219 (2013) 6420–6430. C.S. Chen, Analysis of nonlinear vibration of composite laminated plate, Comput. Part B Eng. 38 (2007) 437–447. C.S. Chen, C.P. Fung, R.D. Chien, Nonlinear vibration of an initially stressed laminated plate according a higher order theory, Comput. Struct. 77 (2007) 521–532. C.S. Chen, C.P. Fung, J.G. Yang, Assessment of plate theories for initially stressed hybrid laminated plates, Comput. Struct. 88 (2009) 195–201. E.J. Brunell, S.R. Robertson, Initially stressed Mindlin plates, AIAA J. 12 (1974) 1036–1045. L.W. Chen, J.Y. Yang, Dynamic stability of laminated composite plates by the finite element method, Comput. Struct. 36 (1990) 845–851.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Stability of parametric vibrations of laminated composite plates Wei-Ren Chen a,⇑, Chun-Sheng Chen b, Jenq-Huey Shyu c a b c
Department of Mechanical Engineering, Chinese Culture University, Taipei 11114, Taiwan Department of Mechanical Engineering, Lunghwa University of Science and Technology, Guishan Shiang 33306, Taiwan Department of Mechanical Engineering, National Taipei University of Technology, Taipei 10608, Taiwan
a r t i c l e
i n f o
Keywords: Dynamic stability Bolotin’s method Dynamic instability index
a b s t r a c t The dynamic stability of laminated composite plates subjected to arbitrary periodic loads is studied based on the first-order shear deformation plate theory. The in-plane load is taken to be a combination of periodic biaxial and bending stress. A set of second-order ordinary differential equations with periodic coefficients of Mathieu–Hill type is formed to determine the regions of dynamic instability based on Bolotin’s method. Numerical results reveal that the dynamic instability is significantly affected by the modulus ratio, number of layer, static and dynamic load parameters. The effects of various important parameters on the instability region and dynamic instability index are investigated. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction The lightweight high-strength composite plates have been used successfully in mechanical, aircraft industry and other engineering applications. When a plate is subjected to periodic loads, it is well known that under some circumstances the ordinary forced response will become dynamically unstable, leading to a violent vibration which is called the dynamic instability phenomenon of parametric resonance. To use them efficiently, a good understanding of dynamic behavior under various load conditions for laminated composite plates are needed. Thus, the problem of dynamic instability has become of ever increasing importance in modern composite structures design. It is necessary to accurately determine dynamic stability regions for suitable applications. Numerous references pertaining to the parametric resonance of plates can be found in the books by Bolotin [1] and Evan-Ivanowski [2]. The dynamic stability of plates subjected to a periodic in-plane loading had received a great deal of attention in the past years. Petry and Fahlbusch [3] investigated the dynamic stability behavior of imperfect plates subjected to in-plane pulse loadings. The governing equations were solved by the Galerkin method and Navier’s double Fourier series. Gilat and Aboudi [4] studied the dynamic instability of elastic composite plates subjected to a periodic in-plane loading. The effect of the matrix non-linearity on the overall response of the composite plate was predicted by the micromechanical method of cells. The dynamic stability of plates under periodic in-plane forces was analyzed and the corresponding stability regions of the first and second order were calculated by Baldinger et al. [5]. The dynamic stability analysis of a plate under a follower force was performed through the finite element method based on the Kirchhoff–Love plate theory by Kim and Kim [6]. The effects of shear deformation and rotary inertia were investigated. A dynamic stability of an undeflected state of isotropic elastic circular plate was analyzed. A dynamic instability index is devised by Wang and Dawe [7] to measure the dynamic instability of composite laminated plates under periodic dynamic loads. The boundary frequencies were determined by using Bolotin’s method, Sturm sequence method and multi-level substructuring technique. Ye et al. [8] investigated the parametric excitation of a simply supported cross-ply laminated composite rectangular plate. The relations between the steady state nonlinear ⇑ Corresponding author. E-mail address: [email protected] (W.-R. Chen). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.07.095
128
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
responses and the amplitude and frequency of parametric excitation were obtained. Yang et al. [9] conducted a dynamic stability analysis of FGM plates subjected to a uniaxial periodic in-plane load and under a uniform temperature change. A Galerkin-differential quadrature approach was employed to convert the governing equations into a linear system of Mathieu–Hill equations from which the boundary points on the unstable regions were determined by Bolotin’s method. The nonlinear dynamical equations of three-dimensional frame in the orthogonal coordinates system were obtained by Wang et al. [10]. Then the equations were transformed into the axial symmetry nonlinear dynamical equations in the polar coordinates system. Wei and Yang [11] investigated the nonlinear dynamic stability of composite laminated plates. The influence of the large deflection, initial imperfection, support conditions and ply-angle of the fibers was considered. The dynamic stability of laminated plates with four free edges subjected to pulsating follower forces was discussed by Choo and Kim [12]. The results were obtained using the first-order shear deformation theory. The influences of aspect ratios, lamination angles, and the number of layers were also studied. The stability analysis of laminated plates under in-plane compressive loading was studied by Iyengar and Chakraborty [13]. The change in initial buckling response of thick laminates with respect to the fiber orientation angle was investigated. The dynamic stability characteristics of laminated composite skew plates subjected to a periodic in-plane load were investigated by Dey and Sinqha [14] based on the finite element approach. The principal and second instability regions were identified for different skew angles, thickness-to-span ratios, fiber orientations and static in-plane loads. Udar and Datta [15] predicted a combination resonance of parametrically excited laminated composite plates. The modal transformation was applied to transform the equilibrium equation into a suitable form for the application of the multiple scales method. Goren [16] analyzed the dynamic stability of cantilever symmetric laminated beams based on the classical plate theory. The effects of the ply orientation, static and dynamic load parameters on the stability were examined. The dynamic instability of imperfect composite laminates was presented by Chakrabarti [17] by using a finite element plate model. The excitation frequency was found to decrease rapidly with the increase in the imperfection parameter. Patel et al. [18] studied the dynamic instability behavior of laminated composite stiffened plates using Hill’s infinite determinant method. The type and the width of loadings had remarkable effects on the dynamic instability characteristics of the stiffened plates. The static and dynamic transitions between the stable states for laminated composite plates were considered by Diaconu et al. [19]. The dynamic analysis model of the snap-through phenomena was proposed for the plates based on the strain field approximation. The model was used to investigate the static and dynamic transitions from one stable state to another. The finite element dynamic stability of laminated skew plates with cutouts and subjected to a distributed periodic in-plane load was analyzed by Lee [20] based on the higher-order shear deformation theory. The influences of skew angles, cutout sizes, layup sequences and thickness-to-length ratios on the onset of the load frequency, width of instability regions, and dynamic instability indexes were presented. Pradyumna and Bandyopadhyay [21] investigated the dynamic instability behavior of laminated hypar and conoid shells using a higher-order shear deformation theory and a finite element approach. The Bolotin’s method is applied to find the boundaries of the instability regions. The effect of number of layers, ply orientation, thickness, curvature, static load component and boundary conditions on the dynamic instability regions was studied. Ramachandra and Panda [22] studied the dynamic instability of a shear deformable composite plate under periodic nonuniform in-plane loads with either parabolic or linear distributions. Based on Bolotin’s method, the first-order and second-order instability regions were obtained for various dynamic in-plane loads. The effects of the span to thickness ratio, shear deformation, aspect ratio, boundary conditions and static load factor on the instability regions were also investigated. The parametric instability of laminated longitudinally stiffened panels with internal cutouts under dynamic in-plane loadings was investigated by Fazilati and Ovesy [23] based on the finite strip method. The effects of cutouts and stiffeners on the instability regions were investigated using the Bolotin’s first order approximation. The results revealed that the finite strip method is capable of predicting dynamic instability regions of laminated panels with complex geometries with fair accuracy. Based on the Lindstedt–Poincare perturbation technique, Khalili et al. [24] studied the buckling stability of a non-ideal simply supported laminated plate resting on elastic foundations. One of the edges of the plate is assumed to allow a small non-zero deflection and moment. The effects of non-ideal boundary conditions, aspect ratio, stiffness of foundation, shear modulus, in-plane pre-load on the buckling load of laminated plates were discussed. The vibration behaviors of composite plates under arbitrary initially stresses had been analyzed previously by the second author [25–27]. In this paper, the problem to be considered is the dynamic oscillations of laminated plates subjected to arbitrary periodic loads. The dynamic load is taken to be a combination of a periodic bending stress and a periodic biaxial stress. Based on the first-order shear deformation theory, the governing equations of motion of the laminate plates are derived by a perturbation technique. The Galerkin method is then applied to the governing partial differential equations to yield a set of second-order ordinary differential equations with periodic coefficients of Mathieu–Hill type. Based on Bolotin’s method, a set of homogeneous linear algebraic equations is formulated and used to obtain the boundaries of the dynamic instability regions of laminated composite plates. An eigenvalue problem is solved to determine the boundary frequencies for the boundaries of instability regimes. The effects of various parameters on the region of dynamic stability and dynamic instability index are studied. 2. Problem formulation In order to approximate the three-dimensional problem, Cartesian coordinates are used to specify the plate geometry in two-dimensional model. The xy plane is coincident with the middle plane of the plate, and the origin of the coordinate
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
129
system is at the left corner of the middle plane. The displacements components u, v and w at any position in the plate space are expanded with the thickness coordinate. Based on the Mindlin plate theory, the displacement fields have the following expressions:
uðx; y; z; tÞ ¼ ux ðx; y; tÞ þ zux ðx; y; tÞ
v ðx; y; z; tÞ ¼ v y ðx; y; tÞ þ zuy ðx; y; tÞ
ð1Þ
wðx; y; z; tÞ ¼ wz ðx; y; tÞ; where u, v and w are the displacement function of the x, y and z axes, respectively. ux and vy are in-plane displacements; wz is transverse displacement; ux and uy are angles of rotation of the normal to the middle plane about y and x axes. An initially stressed plate, which is in static equilibrium and subjected to a time-varying incremental deformation, is considered in this study. Following a similar technique described by Brunelle and Robertson [28] and Chen et al. [27], the governing equations of motion of the laminate plate including the effects of rotary inertia and transverse shear are established using a perturbation technique. The governing equation is expressed as
€ s is;i þ F s þ DF s ¼ qu s;j Þ;i þ r ðrij u
ð2Þ
is , u s , Fs and DFs denote the initial stress, perturbing stress, displacement, perturbing body force and body force, where rij, r respectively. In this paper, a rectangular plate is considered so the equations will be rephrased in xy coordinates. The constitutive relations for a kth lamina are given by:
3 2 r xx ðkÞ C 11 7 6r 6 6 yy 7 6 C 12 7 6 6 7 6r 6 6 yz 7 ¼ 6 0 6 7 6 4 rzx 5 4 0 r xy C 16 2
C 12
0
0
C 22 0
0 C 44
0 C 45
0
C 45
C 55
C 26
0
0
C 16
3ðkÞ 2
7 C 26 7 7 0 7 7 7 0 5 C 66
3 exx ðkÞ 6 e 7 6 yy 7 7 6 6 eyz 7 7 6 6 7 4 ezx 5 exy
ð3Þ
Consider a rectangular laminated plate of uniform thickness h subjected to arbitrary time-dependent initial stresses. The periodic initial stress system is assumed to have the form: Dm rij ¼ rnij þ 2zrmij =h ¼ ðrSij þ rDij cos xtÞ þ 2zðrSm ði; j ¼ x; y; zÞ ij þ rij cos xtÞ
ð4Þ
which consists of the spatially uniform longitudinal, transverse, shear, bending and twisting stress. Here rSij and rDij are the Dm respective static and dynamic component of the periodic normal or shear stress rnij ; rSm represent the respective ij and rij static and dynamic component of the periodic pure bending or torsion stress rm ; x is the angular frequency of excitation. ij Substitute displacement field (1) and Eqs. (3) and (4) into Eq. (2), perform all necessary partial integrations and group terms together by the displacements variation to yield the dynamic equations of rectangular laminated plates as
½A11 ux;x þ A16 ðux;y þ uy;x Þ þ A12 uy;y þ B11 ux;x þ B16 ðux;y þ uy;x Þ þ B12 uy;y þ Nxx ux;x þ M xx ux;x þ Nxy ux;y þ M xy ux;y þ N xz uz;x ;x þ ½A16 ux;x þ A26 uy;y þ A66 ðux;y þ uy;x Þ þ B16 ux;x þ B66 ðux;y þ uy;x Þ þ B26 uy;y þ Nyy ux;y þ M yy ux;y þ Nxy ux;x þ Mxy ux;x €x þ Nyz uz;x ;y þ fx ¼ qhu h
ð5Þ
i A16 ux;x þ A66 ðux;y þ uy;x Þ þ A26 uy;y þ B16 ux;x þ B66 ðux;y þ uy;x Þ þ B26 uy;y þ N xx uy;x þ Mxx uy;x þ Nxy uy;y þ M xy uy;y þ Nxz uz;y ;x h i þ A12 ux;x þ A26 ðux;y þ uy;x Þ þ A22 uy;y þ B12 ux;x þ B26 ux;y þ uy;x Þ þ B22 uy;y þ Nyy uy;y þ M yy uy;y þ Nxy uy;x þ M xy uy;x þ Nxz uz;y
;y
€y þ fy ¼ qhu € ½A55 ðw;x þ ux Þ þ A45 ðw;y þ uy Þ þ Nxx w;x þ N xy w;y ;x þ ½A45 ðw;x þ ux Þ þ A44 ðw;y þ uy Þ þ Nxy w;x þ Nyy w;y ;y þ fz ¼ qhw
ð6Þ ð7Þ
½B11 ux;x þ B16 ðux;y þ uy;x Þ þ B12 uy;y þ D11 ux;x þ D16 ðux;y þ uy;x Þ þ D12 uy;y þ M xx ux;x þ M xy ux;x þ M xy ux;y þ M xy ux;y þ M xz uz;x ;x þ ½B16 ux;x þ B66 ðux;y þ uy;x Þ þ B26 uy;y þ D16 ux;x þ D66 ðux;y þ uy;x Þ þ D26 uy;y þ Myy ux;y þ M yy ux;y þ Mxy ux;x þ M xy ux;y 3 € x =12 þ Myz uz;x ;y A55 ðw;x þ ux Þ A45 ðw;y þ uy Þ ðNxz ux;x þ M xz ux;x þ Nzz ux þ Nzy ux;y þ M zy ux;y Þ þ mx ¼ qh u
ð8Þ
½B16 ux;x þ B66 ðux;y þ uy;x Þ þ B26 uy;y þ D16 ux;x þ D66 ðux;y þ uy;x Þ þ D26 uy;y þ Mxx uy;x þ Mxx uy;x þ Mxy uy;y þ Mxy uy;y þ M xz uz;y ;x þ ½B26 ðux;y þ uy;x Þ þ B12 ux;x þ B22 uy;y þ D12 ux;x þ D26 ðux;y þ uy;x Þ þ D22 uy;y þ Myy uy;y þ M yy uy;y þ Mxy uy;x þ M xy uy;x 3 € y =12 þ Mxz uz;y ;y A45 ðw;x þ ux Þ A44 ðw;y þ uy Þ ðNxz uy;x þ M xz uy;x þ Nzz uyx þ N zy uy;y þ Mzy uy;y Þ þ my ¼ qh u
where
ð9Þ
130
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
ðAij ; Bij ; Dij Þ ¼ ðAij ; Bij ; Dij Þ ¼
Z Z
ðNij ; Mij ; M ij Þ ¼
C ij ð1; z; z2 Þdz ði; j ¼ 1; 2; 6Þ
jC ij ð1; z; z2 Þdz ði; j ¼ 4; 5Þ Z
ð10Þ
h=2
rij ð1; z; z2 Þdz ði; j ¼ x; y; zÞ
h=2
Here Cij’s are the elastic constant of the stiffness matrix; q is the mass density; Aij, Bij and Dij are the laminate stiffness coefficients; j is the shear correction factor; Nij, Mij and M ij are arbitrary initial stress resultants; fx, fy, fz, mx and my are the lateral loadings. All the integrations are carried out through the thickness of the plate from h/2 to h/2. 3. Dynamic instability analyses The dynamic behaviors of laminated plate governed by Eqs. (5)–(9) are so complicated that it would be difficult to present results for all cases. The case to be investigated is the dynamics of the simply supported rectangular laminated plate subjected to the periodic spatially uniform biaxial in-plane stress system consisted of a pulsating longitudinal normal stress and a pulsating pure bending stress. With all other stresses assumed to be zero, the stress system Eq. (4) is reduced to
rxx ¼ rn þ 2zrm =h S
D
ð11Þ Sm
Dm
where rn = r + r cos xt = ðr þ r þ ðr þ r xt and rm = r + r cos xt = ðr þ r þ ðr þ r xt. All Sm Dm Dm the static and dynamic stress components rSxx , r r rDyy , rSm xx , ryy , rxx and ryy are taken to be constants. b = rm/rn is the ratio of bending stress to normal stress. The rectangular plate considered has boundary sides along x = 0 and a, y = 0 and b, all of which are simply supported. To satisfy above geometric boundary conditions, the following shape modes of displacement fields are used
ux ¼ uy ¼
XX XX XX
S xx
S yy Þ
D xx
D yy Þ cos S D yy , xx ,
Sm yy Þ
Dm xx
Dm yy Þ cos
hU mn ðtÞcosðmpx=aÞsinðnpy=bÞ hV mn ðtÞsinðmpx=aÞcosðnpy=bÞ
hW mn ðtÞsinðmpx=aÞsinðnpy=bÞ XX ux ¼ Wxmn ðtÞcosðmpx=aÞsinðnpy=bÞ XX Wymn ðtÞsinðmpx=aÞcosðnpy=bÞ uy ¼ w¼
Sm xx
ð12Þ
Assume that the D = [Umn, Vmn, Wmn, Wxmn, Wymn]T which can be expressed as
DðtÞ ¼ Df ðtÞ
ð13Þ
Here D denotes time independent displacement vector. Substituting Eqs. (12) and (13) into governing Eqs. (5)–(9) and applying the Galerkin method, we obtain the following set of ordinary differential equations as 2
fð½K ðNxx þ Nyy Þ½GÞf ðtÞ þ ½Mðd f ðtÞ=dtÞgD ¼ 0
ð14Þ
where [K] is the elastic stiffness matrix, [G] is the geometric stiffness matrix and [M] is the consistent mass matrix. The system Eq. (14) is related to the eigenvalue problems of the static buckling, free vibration and dynamic instability. The static buckling problem is formed from Eq. (14) by setting f(t) = 1, which leads to
f½K ðNxx þ Nyy Þ½GgD ¼ 0
ð15Þ
Eq. (15) can be used to determine the smallest applied loads Nxx and Nyy, which cause the buckling of the laminated plate. The eigenvalue equation of the free vibration of the laminated plate is obtained from Eq. (14) by setting Nxx + Nyy = 0 and f(t) = eixt
f½K x2 ½MgD ¼ 0
ð16Þ
The roots of the determinant of the coefficients of Eq. (16) are the natural frequencies of the plate. The non-zero initial load and moment resultants in Eq. (11) can be written as:
Nxx þ Nyy ¼ ½ðaS þ aYS ÞPcr þ ðaD þ aYD ÞPcr cos -t
ð17Þ
where aS ¼ hr a ¼ hr aD ¼ hr and a ¼ hr P cr is the buckling load of the plate subjected to uniaxial in-plane load. aS ðaYS Þ and aD ðaYD Þ are static and dynamic load parameters, respectively. Substituting Eq. (17) into Eq. (14) gives S xx =P cr ;
Y S
S yy =P cr ;
D xx =P cr
Y D
D yy =P cr
2
fð½K þ ðaS þ aYS ÞP cr ½G þ ðaD þ aYD ÞPcr ½G cos -tÞf ðtÞ þ ½Mðd f ðtÞ=dtÞgD ¼ 0
ð18Þ
131
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
Eq. (18) represents the dynamic stability problem of a parametrically excited system subjected to a periodic biaxial in-plane load. According to Bolotin’s method [1], the boundaries of the dynamic instability regions can be constructed by the periodic solutions with period 2T and T respectively in Fourier series as
f ðtÞD ¼
f ðtÞD ¼
1 X k-t k-t ak sin þ bk cos 2 2 k¼1;3;5... 1 X k¼0;2;4...
ak sin
k-t k-t þ bk cos 2 2
ð19Þ
ð20Þ
where ak and bk are arbitrary time invariant constants. Substituting the series expansions (19) and (20) into Eq. (18) and separating the sine and cosine parts, two sets of linear algebraic equations in ak and bk are obtained for each solution. Then, the boundaries between stable and unstable regions could be obtained from the condition for the set of equations to have nontrivial solution. The eigenvalue system of the dynamic stability boundaries with period 2T is given as
1 ½K þ ðaS þ aY ÞPcr ½G 1 ðaD þ aY ÞPcr ½G 1 -2 ½M ðaD þ aYD ÞPcr ½G . . . S D 2 4 2 1 ðaD þ aYD ÞPcr ½G ½K þ ðaS þ aYS ÞPcr ½G 94 -2 ½M . . . ¼ 0 2 ...
ð21Þ
The governing equations of dynamic stability boundaries with period T is
1 ½K þ ðaS þ aY ÞPcr ½G -2 ½M ðaD þ aYD ÞPcr ½G . . . S 2 1 ðaD þ aYD ÞPcr ½G ½K þ ðaS þ aYS ÞPcr ½G 4-2 ½M . . . ¼ 0 2 ...
ð22Þ
½K þ ðaS þ aY ÞPcr ½G ðaD þ aYD ÞPcr ½G . . . S ðaD þ aYD ÞPcr ½G ½K þ ðaS þ aYS ÞPcr ½G -2 ½M . . . ¼ 0 ...
ð23Þ
and
Generally, it is impossible to solve the determinant equations (21)–(23) of infinite order to find the stability boundaries. Hence, different orders of approximation can be used to obtain approximate results which are sufficiently close approximations of the infinite eigenvalue problem. The boundaries of primary instability region with period 2T are usually much larger than those of secondary instability region with period T, and are therefore of greater practical importance. Furthermore, useful results can be obtained by considering only the first-order approximation of the major instability boundaries, which is capable of obtaining solutions with sufficient accuracy [29]. Hence, only the first-order approximation of the principal stability boundaries will be given in the present study. From element (1, 1) of Eq. (21), the first-order solution of primary instability regions can be obtained in the following form
1 1 ½K þ ðaS þ aYS ÞP cr ½G ðaD þ aYD ÞPcr ½G -2 ½M ¼ 0 2 4
ð24Þ
The eigenvalues of the above equation are just the frequencies of excitation for the boundaries of instability regimes.
Table 1 Excitation frequencies for a simply supported symmetrically four-layer cross-ply laminate plate with various static and dynamic loads.
aS
aD
Wang’s results [7] U
0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8
0.0 0.3 0.6 0.9 1.2 1.5 0.06 0.12 0.18 0.24
Present results L
x
x
xU
xL
144.57 155.03 164.83 174.08 182.87 191.25 131.71 117.45 101.20 81.78
144.57 133.29 120.95 107.21 91.43 72.28 126.86 106.24 80.49 40.89
144.36 155.64 165.12 174.43 183.21 191.75 132.12 117.96 101.84 82.31
144.36 133.79 121.45 107.63 91.86 72.62 127.26 106.82 81.10 41.32
132
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
Fig. 1. Effect of dynamic load parameter on the instability regions of laminated plates with various layer numbers (a/b = 1, a/h = 10, Ex/Ey = 10, aS = 0.1, q: N = 2, h: N = 4, s: N = 6, N: N = 8).
Fig. 2. Effect of static compressive load parameter on the instability regions of laminated plates with various layer numbers (a/b = 1, a/h = 10, Ex/Ey = 10, aD /aS = 0.3, q: N = 2, h: N = 4, s: N = 6, 4: N = 8). Table 2 Effect of the static and dynamic loads on the instability regions DX of laminated plates under various layer numbers N (a/b = 1, a/ h = 10, Ex/Ey = 10, b = 0).
aS 0.1 0.1 0.1 0.1 0.1 0 0.2 0.4 0.6 0.8
aD 0 0.4 0.8 1.2 1.6 0 0.06 0.12 0.18 0.24
N 2
4
6
8
0 4.2051 8.5478 13.2302 18.7104 0 0.7018 1.6224 2.9961 5.9166
0 5.9862 12.1984 18.8341 26.6355 0 0.9990 2.3096 4.2651 8.4229
0 6.2180 12.6378 19.5234 27.6668 0 1.0377 2.3990 4.4302 8.7490
0 6.4235 12.7950 19.8040 28.0071 0 1.0504 2.4285 4.4848 8.8567
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
133
Fig. 3. Effect of number of layer N on dynamic instability index of laminated plates under various dynamic loads. (a/b = 1, a/h = 10, Ex/Ey = 10, aS = 0.1, q: aD = 0.4, h: aD = 0.8, s: aD = 1.2, 4: aD = 1.6).
Fig. 4. Effect of number of layer N on dynamic instability index of laminated plates under various static loads. (a/b = 1, a/h = 10, Ex/Ey = 10, aD /aS = 0.3, q: aS = 0.2, h: aS = 0.4, s: aS = 0.6, 4: aS = 0.8).
4. Result and discussion To check the accuracy of the present model, numerical results of the excitation frequency for a simply supported symmetric four-layer cross-ply laminate plate under various static and dynamic loads are computed and compared with the results by Wang [7]. The upper and lower excitation frequencies of the primary instability region are presented in Table 1. It can be observed that the presented excitation frequencies are in good agreement with those obtained by Wang. In the next, the influence of various parameters of the cross-ply laminate plate on its dynamic instability behaviors will be examined qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
and discussed. The non-dimensional coefficients of excitation frequency X ¼ -b
q=h2 Ey , the instability region
DX ¼ XU XL and the dynamic instability index XDI ¼ DX=ðxnf K cr Þ are defined and used throughout the dynamic instability studies. Here XU and XL are the upper and lower boundary excitation frequency, respectively; xnf is the non-dimensional
134
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
Fig. 5. Effect of modulus ratio on dynamic instability index of laminated plates under various dynamic loads. (a/b = 1, a/h = 10, N = 2, aS = 0.1, q: aD = 0.4, h: aD = 0.8, s: aD = 1.2, 4: aD = 1.6).
Fig. 6. Effect of modulus ratio on dynamic instability index of laminated plates under various static and dynamic loads. (a/b = 1, a/h = 10, N = 2, aD /aS = 0.3, q: aS = 0.2, h: aS = 0.4, s: aS = 0.6, 4: aS = 0.8).
2
fundamental natural frequency given by xnf ¼ xb 2
4
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q=h2 Ey ;Kcr is the dimensionless critical buckling load given by
Kcr = ðN xx Þcr b =Ey h . The effects of static and dynamic load parameters on the principal unstable regions of laminated plates with various layer numbers are shown in Figs. 1 and 2 and Table 2. In Fig. 1, the static load parameter aS = 0.1 and the dynamic load parameter increases from 0 to 1.6. As can be seen, the upper excitation frequency increases and lower excitation frequency decreases with the increase of dynamic load parameter. The instability regions tend to shift to higher frequencies of excitation as the layer number is increased. The onset of dynamic instability occurs much later with the increasing layer number. In Fig. 2, the static load parameter aS varies from 0 to 0.8 and the ratio of aD =aS is kept as 0.3. It can be observed that both the upper and lower excitation frequency decrease with the increasing static load parameter. This is due to the fact that the compressive static load tends to reduce the plate stiffness. Similarly, when the layer number is increased, the instability region is found to move to the higher excitation frequency. As shown in Table 2, the primary instability region DX for the laminated plate
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
135
Fig. 7. Effect of dynamic load parameter on instability region of laminated plates under various static loads. (a/b = 1, a/h = 10, N = 2, E1/E2 = 40, s: aS = -0.4, 4: aS = 0.2, w; aS = 0, N: aS = 0.2, d: aS = 0.4).
Fig. 8. Effect of dynamic load parameter on instability region of laminated plates under various lateral static loads. (a/b = 1, a/h = 10, N = 2, E1/E2 = 40, aS = 0.2, s: aYS = 0.3, w: aYS = 0, d: aYS = 0.3).
increases with the increasing layer number, static and dynamic load parameter, and the width of the unstable zone is becoming much wider at the higher dynamic load parameter. Meanwhile, the instability region is largely controlled by the magnitude of dynamic load parameter rather than the static load parameter. The effects of the packing layer number on the dynamic instability index of laminated plates under different dynamic and static loads are showed in Figs. 3 and 4. The dynamic instability index is decreased with the increasing layer number, but increased with the increasing static and dynamic load parameter. However, the effect of layer number N on the dynamic instability of the plates is diminished as N is greater than 4. Meanwhile, the dynamic load parameter has an apparent influence than the static load parameter on the dynamic instability index. Thus, the lower layer number and the higher dynamic load parameter are the more dynamically unstable is the laminate composite plate. Figs. 5 and 6 present the effect of the modulus ratio on the dynamic instability index for the laminated plates under various dynamic and static load parameters. With the increase in the static or dynamic load parameter, the dynamic instability index increases regardless of the
136
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
Fig. 9. Effect of the dynamic load parameter on dynamic instability index of laminated plates under various lateral static loads. (a/b = 1, a/h = 10, N = 2, E1/E2 = 40, aS = 0.2, s: aYS = 0.3, w: aYS = 0, d: aYS = 0.3). Table 3 Effect of the bending stress parameter on the principle unstable region and dynamic instability index of two-layer cross-ply laminated plate under various dynamic loads (a/b = 1, a/h = 10, aS = 0.1, Ex/Ey = 10).
aD
b
U
0
10
20
30
40
X XL XDI XU XL XDI XU XL XDI XU XL XDI XU XL XDI
0
0.4
0.8
1.2
1.6
14.7038 14.7038 0 14.7074 14.7074 0 14.7178 14.7178 0 14.7340 14.7340 0 14.7546 14.7546 0
16.2556 12.9675 6.9729 16.2516 12.9770 6.9785 16.2401 13.0047 6.9949 16.2221 13.0475 7.0215 16.1993 13.1016 7.0573
17.6717 10.9595 14.2342 17.6587 10.9730 14.2480 17.6209 11.0119 14.2889 17.5622 11.0722 14.3547 17.4876 11.1482 14.4428
18.9825 8.4892 22.2525 18.9591 8.5038 22.2816 18.8914 8.5460 22.3671 18.7860 8.6113 22.5046 18.6517 8.6935 22.6873
20.2084 4.9012 32.4610 20.1736 4.9119 32.5243 20.0726 4.9435 32.7096 19.9149 4.9919 33.0069 19.7135 5.0527 33.4013
Table 4 Effect of the bending stress parameter on the principle unstable region and dynamic instability index of two-layer cross-ply laminated plate under various static compressive loads (a/b = 1, a/h = 10, aD/aS = 0.3, Ex/Ey = 10).
aS
b
0
10
20
30
40
XU XL XDI XU XL XDI XU XL XDI XU XL XDI XU XL XDI
0
0.2
0.4
0.6
0.8
15.4991 15.4991 0 15.4991 15.4991 0 15.4991 15.4991 0 15.4991 15.4991 0 15.4991 15.4991 0
14.1204 13.6004 1.1026 14.1263 13.6081 1.1043 14.1433 13.6304 1.1091 14.1698 13.6648 1.1169 14.2033 13.7084 1.1275
12.5915 11.3895 2.5492 12.6021 11.4023 2.5567 12.6325 11.4396 2.5791 12.6796 11.4972 2.6153 12.7391 11.5699 2.6638
10.8494 8.6295 4.7075 10.8630 8.6441 4.7286 10.9023 8.6865 4.7907 10.9631 8.7518 4.8909 11.0399 8.8343 5.0249
8.7676 4.3838 9.2966 8.7822 4.3936 9.3528 8.8246 4.4225 9.5176 8.8901 4.4667 9.7837 8.9726 4.5222 10.1392
137
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138
Table 5 Effect of the bending stress parameter on the principle unstable region and dynamic instability index of two-layer cross-ply laminated plate under various static tensile loads (a/b = 1, a/h = 10, aD/aS = 0.3, Ex/Ey = 10).
aS
b
U
0
10
20
30
40
X XL XDI XU XL XDI XU XL XDI XU XL XDI XU XL XDI
0
0.2
0.4
0.6
0.8
15.4991 15.4991 0 15.4991 15.4991 0 15.4991 15.4991 0 15.4991 15.4991 0 15.4991 15.4991 0
17.1894 16.7649 0.9002 17.1796 16.7579 0.8989 17.1515 16.7376 0.8949 17.1077 16.7061 0.8884 17.0522 16.6661 0.8797
18.7277 17.9415 1.6671 18.7065 17.9265 1.6622 18.6451 17.8831 1.6474 18.5494 17.8155 1.6234 18.4276 17.7295 1.5905
20.1489 19.0457 2.3395 20.1147 19.0218 2.3291 20.0154 18.9525 2.2981 19.8605 18.8446 2.2470 19.6626 18.7070 2.1769
21.4762 20.0892 2.9415 21.4276 20.0556 2.9240 21.2863 19.9580 2.8718 21.0652 19.8059 2.7855 20.7817 19.6114 2.6662
modulus ratio. As can be observed, the dynamic instability index is reduced when the modulus ratio is increased. Hence, the laminated plate is becoming more dynamically unstable as its modulus ratio decreases. The effect of the static loading type on the instability region of the laminated plate is depicted in Fig. 7. It shows that when the static load varies from the tensile (aS < 0) to the compressive type (aS > 0), the instability regions tend to shift to lower excitation frequencies and become wider. Hence, the tensile static load has a strengthen effect on the dynamic stability but the compressive static load has the opposite influence. The laminated plate under the compressive static load is more dynamically unstable than that under the tensile one. Fig. 8 shows the effect of the biaxial static load on the instability regions. As can be seen, the onset of the dynamic instability shifts to a lower excitation frequency as the lateral static load varies from the tensile (aYS < 0) to the compressive load (aYS > 0). The width of the instability regions are also enlarged significantly with the increasing compressive lateral static load. The effect of various lateral static load types on the dynamic instability index is presented in Fig. 9. It is observed that the dynamic instability index is significantly affected by the compressive lateral static load. The tensile lateral static load has little effect on the dynamic instability index. Therefore, the compressive lateral static load shows a destabilizing effect on the dynamic stability behavior of the laminate plate. The effects of the bending stress ratio on the excitation frequency and dynamic instability index of laminate plates under different dynamic loads are presented in Table 3. It can be found that the increasing bending stress produces a slightly reduction in the upper boundary frequency and increase in the lower boundary frequency and dynamic instability index. Table 4 shows the effects of the bending stress ratio on the dynamic stability for the plates under various compressive static loads. As the bending stress ratio increases, the excitation frequency and dynamic instability index increase. Hence, the higher are the bending stress ratio and compressive static or dynamic load parameter, the more dynamically unstable is the laminate plate. Table 5 presents the influence of the bending stress and tensile static load. As can be observed, the values of the excitation frequency and dynamic instability index reduce slightly when the bending stress ratio arises. Therefore, the laminate plate under the higher tensile load and bending stress ratio is more dynamically stable. 5. Concluding remarks The dynamic stability of laminate plate subjected to periodic spatially uniform biaxial loads has been described and discussed in this paper. Following the above discussions, some conclusions are addressed as follows: (1) The excitation frequency, instability region and dynamic instability index are significantly affected by the static load, dynamic load, layer number and modulus ratio. They are slightly affected by the bending stress. (2) The upper excitation frequency increases with the increasing of the tensile static load, dynamic load and layer number. The lower excitation frequency increases with the increase of the tensile static load and layer number, and the decrease of the dynamic load. As the static, dynamic load parameter and layer number is increased, the instability region is increased. (3) The dynamic instability index increases with the increasing compressive static loading and dynamic loading and reduces with the increase of the layer number, modulus ratio and tensile static loading.
References [1] V.V. Bolotin, The Dynamic Stability of Elastic Systems, Holden-Day, San Francisco, 1964. [2] R.M. Evan-Ivanowski, Resonance Oscillations in Mechanical Systems, Elsevier, Amsterdm, 1976.
138 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
W.-R. Chen et al. / Applied Mathematics and Computation 223 (2013) 127–138 D. Petry, G. Fahlbusch, Dynamic buckling of thin isotropic plates subjected to in-plane impact, Thin-Walled Struct. 38 (2000) 267–283. R. Gilat, J. Aboudi, Parametric stability of non-linearly elastic composite plates by Lyapunov exponents, J. Sound Vib. 235 (2000) 627–637. M. Baldinger, A.K. Belyaev, H. Irschik, Principal and second instability regions of shear-deformable polygonal plates, Comput. Mech. 26 (2000) 288–294. J.H. Kim, H.S. Kim, Study on the dynamic stability of plates under a follower force, Comput. Struct. 74 (2000) 351–363. S. Wang, D.J. Dawe, Dynamic instability of composite laminated rectangular plates and prismatic plate structures, Comput. Methods Appl. Mech. Eng. 191 (2002) 1791–1826. M. Ye, Q. Ding, Y.H. Sun, W. Zhang, Nonlinear oscillations of an antisymmetric cross-ply laminated composite thin rectangular plate under parametric excitation, ASME Des. Eng. Div. 116 (2003) 621–630. J. Yang, K.M. Liew, S. Kitipornchai, Dynamic stability of laminated FGM plates based on higher-order shear deformation theory, Comput. Mech. 33 (2004) 305–315. X.Z. Wang, G. Wang, Y.Y. Zhao, K.Y. Yeh, Nonlinear dynamical stability analysis of the circular three-dimensional frame, Appl. Math. Mech. 25 (2004) 367–372. D.M. Wei, G.T. Yang, Nonlinear dynamic stability of composite laminated plates, Appl. Math. Mech. 25 (2004) 1215–1219. Y.S. Choo, J.H. Kim, Dynamic stability of rectangular plates subjected to pulsating follower forces, AIAA J. 38 (2004) 353–361. N.G.R. Iyengar, A. Chakraborty, Interaction curves for composite laminates subjected to biaxial in-plane loads, Int. J. Struct. Stab. Dyn. 5 (2005) 539– 555. P. Dey, M.K. Sinqha, Dynamic stability analysis of composite skew plates subjected to periodic in-plane load, Thin-Walled Struct. 44 (2006) 937–942. R.S. Udar, P.K. Datta, Combination resonance characteristics of laminated composite plates subjected to non-uniform harmonic edge loading, Air Eng. Aero. Tech. 78 (2006) 107–119. K.A. Goren, Static and dynamic stability analyses of the symmetric laminated cantilever beams, Adv. Comput. Lett. 17 (2008) 155–160. A. Chakrabarti, An efficient FE model for dynamic instability analysis of imperfect composite laminates, Struct. Eng. Mech. 30 (2008) 383–386. S.N. Patel, P.K. Datta, A.H. Sheikh, Parametric study on the dynamic instability behavior of laminated composite stiffened plate, J. Eng. Mech. 135 (2009) 1331–1341. G.G. Diaconu, P.M. Weaver, A.F. Arrieta, Dynamic analysis of bi-stable composite plates, J. Sound Vib. 322 (2009) 987–1004. S.Y. Lee, Finite element dynamic stability analysis of laminated composite skew plates containing cutouts based on HSDT, Comput. Sci. Technol. 70 (2010) 1249–1257. S. Pradyumna, J.N. Bandyopadhyay, Dynamic instability behavior of laminated hypar and conoid shells using a higher-order shear deformation theory, Thin-Walled Struct. 49 (2011) 77–84. L.S. Ramachandra, S.K. Panda, Dynamic instability of composite plates subjected to non-uniform in-plane loads, J. Sound Vib. 331 (2012) 53–65. J. Fazilati, H.R. Ovesy, Parametric instability of laminated longitudinally stiffened curved panels with cutout using higher order FSM, Comput. Struct. 95 (2013) 691–696. S.M.R. Khalili, P. Abbaspour, K. Malekzadeh Fard, Buckling of non-ideal simply supported laminated plate on Pasternak foundation, Appl. Math. Comput. 219 (2013) 6420–6430. C.S. Chen, Analysis of nonlinear vibration of composite laminated plate, Comput. Part B Eng. 38 (2007) 437–447. C.S. Chen, C.P. Fung, R.D. Chien, Nonlinear vibration of an initially stressed laminated plate according a higher order theory, Comput. Struct. 77 (2007) 521–532. C.S. Chen, C.P. Fung, J.G. Yang, Assessment of plate theories for initially stressed hybrid laminated plates, Comput. Struct. 88 (2009) 195–201. E.J. Brunell, S.R. Robertson, Initially stressed Mindlin plates, AIAA J. 12 (1974) 1036–1045. L.W. Chen, J.Y. Yang, Dynamic stability of laminated composite plates by the finite element method, Comput. Struct. 36 (1990) 845–851.