Forced vibration analysis for a FGPM cylindrical shell

Transcription

1 Shock and Vibration 0 ( DOI /SAV IOS Press Forced vibration analysis for a FGPM cylindrical shell Hong-Liang Dai a,b,c, and Hao-Jie Jiang a,c a State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan, China b Key Laboratory of Manufacture and Test Techniques for Automobile Parts, Ministry of Education, Chongqing University of Technology, Chongqing, China c Department of Engineering Mechanics, College of Mechanical and Vehicle Engineering, Hunan University, Changsha, Hunan, China Received 14 June 01 Revised 8 October 01 Accepted 7 December 01 Abstract. This article presents an analytical study for forced vibration of a cylindrical shell which is composed of a functionally graded piezoelectric material (FGPM. The cylindrical shell is assumed to have two-constituent material distributions through the thickness of the structure, and material properties of the cylindrical shell are assumed to vary according to a power-law distribution in terms of the volume fractions for constituent materials, the exact solution for the forced vibration problem is presented. Numerical results are presented to show the effect of electric excitation, thermal load, mechanical load and volume exponent on the static and force vibration of the FGPM cylindrical shell. The goal of this investigation is to optimize the FGPM cylindrical shell in engineering, also the present solution can be used in the forced vibration analysis of cylindrical smart elements. Keywords: Forced vibration, FGPM, cylindrical shell, analytical study, heat conduction 1. Introduction FGPM have experienced a remarkable increase in terms of research and development. By using the continuous change in the physical and mechanical properties of a material, it is possible to prevent fracture in composite materials, avoiding the phenomenon of stress concentration and yield in such materials. Vibration of shells is an indispensable branch of research in structural dynamics. Cylindrical shells also have vast range of applications in engineering and technology. Many studies for vibration of cylindrical structures composed of functionally graded materials are available in the literatures, Pradhan et al. [1] investigated vibration characteristics of FGM cylindrical shells under various boundary conditions. By means of the Bolotin s method, Ng et al. [] gave a stability analysis of FGM cylindrical shells under harmonic axial loading. Han et al. [3] gave a numerical method for analyzing transient waves in FGM cylindrical shells excited by impact point loads. By virtue of the introduction of a dependent variable and the separation of variables technique, the axisymmetric plane strain electroelastic dynamic problem of a special non-homogeneous piezoelectric hollow cylinder is transformed to a Volterra integral equation of the second kind about a function with respect to time, which had been solved successfully by Hou et al. [4]. According to a simple power law distribution in terms of the volume fractions of the constituents, Shen [5] presented a Corresponding author: Hong-Liang Dai, State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 41008, Hunan, China. Tel.: ; Fax: ; hldai50@sina.com. ISSN /13/$7.50 c 013 IOS Press and the authors. All rights reserved

2 53 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell postbuckling analysis for a FGM cylindrical shell of finite length subjected to external pressure and in thermal environments. Utilizing the higher-order theory, Patel et al. [6] studied the free vibration characteristics of functionally graded elliptical cylindrical shells. Paulino and Silva [7] applied topology optimization to design FGM structures considering a minimum compliance criterion. Bhangale and Ganesan [8] carried out free vibration studies on functionally graded materials magneto-elastro-elastic cylindrical shells; they [9] presented linear thermal buckling and free vibration analysis for functionally graded cylindrical shells with clamped-clamped boundary conditions based on temperature-dependent material properties. By means of the separation of variables technique as well as the superposition method, Wang and Ding [10] investigated transient responses of a special non-homogeneous magnetoelectro-elastic hollow cylinder for a fully coupled axisymmetric plane strain problem. By means of the Rayleigh- Ritz method, Arshad et al. [11] performed a frequency analysis for FGM circular cylindrical shells. By means of a two-dimensional higher-order deformation theory, Matsunaga [1] investigated vibration and buckling problems of FGM cylindrical shells. Based on the three-dimensional piezoelectricity, Wu and Tsai [13] investigated cylindrical bending vibration of functionally graded piezoelectric shells using the method of perturbation. By using the elementfree Kp-Ritz method, Zhao et al. [14] gave thermoelastic and vibration analysis of functionally graded cylindrical shells. Arshad et al. [15] presented a vibration frequency analysis of a bi-layered cylindrical shell composed of two independent functionally graded layers. Shah et al. [16] investigated vibrations of functionally graded cylindrical shells based on elastic functions. Based on theory of vibrations of cylindrical shells, Sofiyev [17] presented an analytical study on the dynamic behavior of the infinitely-long FGM cylindrical shell under moving loads. By using third order shear deformation theory, Daneshjou et al. [18] gave an analytical solution for acoustic transmission through relatively thick FGM cylindrical shells. Based on Sanders thin shell theory, the vibrational behavior of functionally graded cylindrical shells with intermediate ring supports was studied by Rahimi et al. [19]. Based on the three-dimensional theory of elasticity, free vibration analysis of a functionally graded cylindrical shell embedded in piezoelectric layers was performed by Alibeigloo et al. [0]. Based on the first order shear deformation theory of shells, the free vibration analysis of rotating functionally graded cylindrical shells subjected to thermal environment is investigated by Malekzadeh and Heydarpour [1]. However, As far as we know, the exact solution presented in this paper, on forced vibration analysis for a FGPM cylindrical shell is a novel and an easily operated method in this area. In the paper, an analytical solution for forced vibration of a FGPM cylindrical shell is presented. This method is easily understood, and has been validated by comparing the results with Ma and Wang []. Numerical results are presented to show the effect of electric excitation, thermal load, mechanical load and volume exponent on the static and dynamic response of the FGPM cylindrical shell, it will be of great value when engineers design optimum cylindrical smart structures in engineering.. Theoretical analysis Consider a FGPM cylindrical shell subjected to electric excitation φ(z, mechanical load q and thermal load T (z. Its inner radius a, length L and thickness h are shown in Fig. 1. The Cartesian coordinate system (x, y, z is set on the mid-plane (z =0of the FGPM cylindrical shell, where x and y denotes the axial and circumferential directions of the mid-plane of the FGPM cylindrical shell, respectively..1. FGPM material properties The FGPM cylindrical shell is composed of metal and ceramic materials and the material constituents of the shell varies from the outer surface to the inner surface, i.e. the outer surface (z = h/ of the cylindrical shell is metal-rich and the inner surface (z = h/ is ceramic-rich. In such a way, material properties P (e.g., modulus of elasticity E, elastic constants c ij (i, j =1,, thermal expansion coefficient α i (i =1,, thermal conductivity K, piezoelectric constants e i and pyroelectric coefficients p(z of the FGPM cylindrical shell are assumed to vary through the thickness of the shell. Here, FGPM s material properties P are related not only to material properties of the material constituent, but also to their volume fractions V 1 and V, therefore, one has P (z =P 1 V 1 + P V = P +(P 1 P V 1 (1 where subscript 1 and denotes, respectively, metal material and ceramic material.

3 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell 533 Fig. 1. (a The geometry of a FGPM cylindrical shell; (b Fixed of both ends; (c Simply supported of both ends. Assuming V 1 follows a simple power law distribution [3 3]: ( h z V 1 = h n ( where n (0 <n< is the exponent of volume, and n represents the inhomogeneity of FGPMs, and it degenerates into metal material at n =0,whenn, it becomes ceramic material... Heat conduction equation Assuming that the rise of temperature occurs only in the thickness direction of the FGPM cylindrical shell, the one-dimensional steady heat conduction equations is [ ] d dt (z K(z =0 (3 dz dz The corresponding thermal boundary conditions of the FGPM cylindrical shell are T ( h = T 1, T ( h = T (4 where T 1 and T are, respectively, the inner and outer surface temperature of the FGPM cylindrical shell. Solving Eq. (3, solution of the temperature can be written as T (z =T 1 1+ z ( T 1 T 1 h/ h/ h/ 1 K(z dz 1 K(z dz (5

4 534 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell.3. Governing equations According to the geometric symmetry of the FGPM cylindrical shell, the strain-displacement relations are where ε x = ε 0 x + zk 0 x, ε y = ε 0 y + zk 0 y, ε z =0 (6 ε 0 x = u x, ε0 y = w a, k0 x = w x, k0 y = w a (7 where u and w are the displacements along x and z axes, respectively, ε 0 x and ε 0 y are strain components along x and y axes on the middle surface of the shell s structure, respectively, kx 0 and k0 y are the curvature change of x and y directions on the middle surface of the shell s structure, respectively. The electrothermoelastic constitutive relations of the FGPM cylindrical shell are σ x = c 1 (zε x + c (zε y + λ(zt (z e 1 (z φ z (8a σ y = c (zε x + c 1 (zε y + λ(zt (z e (z φ z (8b D z = e 1 (zε x + e (zε y + p(zt (z g(z φ z (8c where σ i (i = x, y, D z, c i (i =1,, e i (i =1,, p(z, λ(z and φ are the components of stresses, radial electric displacement, elastic constants, piezoelectric constants, pyroelectric coefficients, thermal expansion coefficients and electric potential, respectively, and c 1 (z = E(z 1 v, c (z = ve(z E(z, λ(z = 1 v 1 v α(z, e 1 (z =e 1 +(e 11 e 1 V 1, e (z =e +(e 1 e V 1 (9 where v denotes Poisson s ratio, and it is assumed to be a constant in this paper. In absence of free charge density, the charge equation of electrostatics (Heyliger [33] is expressed as D z z + D z z a =0 Solving the Eq. (10, it can be obtained as D z = k d a z where k d is determined by the boundary conditions, substituting Eq. (11 into Eq. (8c, one gets φ z = e 1(zε x + e (zε y k d p(zt (z + g(z g(z(a z g(z Define the electric potential boundary conditions as: φ z= h = φ h, φ z=+ h = φ + h Substituting Eq. (1 into Eqs (8a and (8b, yields σ x =Θ 11 ε x +Θ 1 ε y +Θ 13 +Θ 14 +Θ 15 σ y =Θ 1 ε x +Θ ε y +Θ 3 +Θ 4 +Θ 5 (10 (11 (1 (13 (14a (14b

5 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell 535 where the expressions of Θ ij (i =1, ; j =1, 5 are shown in Appendix A. According to the axial-symmetry of the FGPM cylindrical shell, when investigating on the force vibration of this structure, the equilibrium equations of the shell are a Q x x N y + aq i = aq + aq t (15a a N x x =0 (15b a M x x + aq x =0 (15c where Q x, N i (i = x, y and M x denote, respectively, the transversely shear force, axial force of membrane and bending moment, and q, q t and q i denote the static load, dynamic load and inertia force, respectively. When investigating the static problem or the free vibration of the cylindrical shell, the first equilibrium equation of Eq. (15 degenerates into the following form, respectively, a Q x x N y = aq a Q x x N y + aq i = aq The corresponding expressions are shown as follows: h/ ( N x = σ x 1 z dz a and h/ h/ N y = σ y dz h/ h/ ( M x = σ x z 1 z dz a q i = h/ h/ h/ ( f z 1 z dz a (15a,1 (15a, (16a (16b (16c (17 where q i is defined as the inertia force resultant along unit length of the circumferential central line, and f z = ρ(z w t (18 The negative sign indicates the opposite direction between the direction of acceleration and that of inertial force. Substituting Eq. (18 into Eq. (17, yields q i = h/ h/ [ ρ(z w (1 t z ] dz = ρ w a t (19 where ρ = h/ h/ ( ρ(z 1 z dz a (0 Substituting Eqs (15c and (19 into Eqs (15a and (15b, yields a M x x + N y + a ρ w t = aq + aq t (1a

6 536 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell a N x x =0 Then, substituting Eq. (14 into Eq. (16, yields u N x = A 11 x + A w 1 x + A 13w + A 14 + A 15 + A 16 u N y = A 1 x + A w x + A 3w + A 4 + A 5 + A 6 u M x = A 31 x + A w 3 x + A 33w + A 34 + A 35 + A 36 where A ij (i =1,, 3; j =1, 6 are shown in Appendix B. (1b (a (b (c 3. Solution of the electrothermoelastic deformation problem Substituting Eq. ( to Eq. (1, yields 4 w aa 3 x 4 +(aa 33 + A w x + A 3 u 3w + aa 31 x 3 u +A 1 x + A 4 + A 5 + A 6 + a ρ w t + aq + aq t =0 u A 11 x + A 3 w 1 x 3 + A w 13 x =0 Integrating both sides of the Eq. (3b with respect to x, yields (3a (3b u x = A 1 A 11 w x A 13 A 11 w k 1 A 11 (4 where k 1 is an integral constant to be determined by the boundary conditions. Differentiating on both sides of Eq. (3b, yields 3 u x 3 = A 1 A 11 4 w x 4 A 13 A 11 w x (5 Substituting Eqs (4 and (5 into Eq. (3a, yields 4 w G 1 x 4 + G w x + G 3w + G 4 + a ρ w t + aq t =0 (6 where G 1 = aa 3 aa 31A 1, G = aa 33 + A aa 31A 13 A 1A 1, A 11 A 11 A 11 G 3 = A 3 A 1A 13, G 4 = A 4 + A 5 + A 6 A 1 k 1 + aq (7 A 11 A 11 where G 1, G, G 3 and ρ are constants, and G 4 is a term related to the boundary conditions at the two ends and the electric field boundary conditions. First, assuming that the cylindrical shell is only subjected to static mechanical and temperature loads,subsequently the cylindrical shell reaches the state of equilibrium, it can be written as G 1 4 w e x 4 + G w e x + G 3w e + G 4 + a ρ w e t =0 (8

7 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell 537 Noticing that the static deflection is time-independent, it gets w e t =0 (9 Then, consider the FGPM cylindrical shell subjected to dynamic load q t and assuming that the transient perturbation at any point is w t,then 4 w t G 1 x 4 + G w t x + G 3w t + G 4 + aq t + a ρ w t =0 (30 t Equation (30 minus Eq. (8, yields 4 (w t w e (w t w e G 1 x 4 + G x + G 3 (w t w e +aq t + a ρ (w t w e t =0 (31 where (w t w e is the displacement measured from the static equilibrium position, for the sake of analysis, we mark it as w a = w t w e. Then the vibration equation from the static equilibrium position can be obtained 4 w a G 1 x 4 + G w a x + G 3w a + a ρ w a t + aq t =0 (3 The perturbation is assumed to have the form as follows w a (x, t = = w am (x, t = Sin L x H m (t Sin L x [A m Sin(ω m t+b m Cos(ω m t+τ m (t] (33 where ω m is the mth order of natural frequency. Substituting Eq. (33 into Eq. (3, yields G 1 L For free vibration, G 1 L 4 G + G3 a ρωm + aq t =0 (34 L 4 G + G3 a ρωm =0 (35 L Then, the natural frequency of the FGPM cylindrical shell is obtained as ω m = 1 a ρ G 1 L Assume the dynamic load as 4 G + G3 (36 L q t = q 0 (xcos (ω F t (37 where ω F is the loading frequency, expanding Eq. (37 into the form of mode shape, yields q t = q t,m = F m Sin L x Cos (ω F t (38

8 538 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell where F m = L q 0 (xsin L 0 L x dx (39 The orthogonality of mode shapes leads to the uncoupled form of equation for motion, in other words, it allows for the equations of motion to be written in an uncoupled form. Just take the mth mode shape into discussion, then w a = w am =Sin L x H m (t (40 q t = q t,m = F m Sin L x Cos (ω F t (41 Substituting Eqs (40 and (41 into Eq. (34, yields [ 4 ] G 1 G + G3 H m (t+a ρ d H m (t L L dt + af m Cos (ω F t=0 (4 From Eq. (35, one knows G 1 L 4 G + G3 = a ρωm (43 L Substituting Eq. (43 into Eq. (4, yields a ρωm H m(t+a ρ d H m (t dt + af m Cos (ω F t=0 (44 Then d H m (t dt + ω m H m(t = F m ρ Cos (ω F t (45 One particular solution of Eq. (45 is h m (t = F mcos (ω F t ρ (ω m ω F (46 Express it as τ m (t, gives τ m (t =h m (t = F mcos (ω F t ρ (ω m ω F (47 Thus, the deflection may be written as w a (x, t = = and the velocity may be expressed as Sin L x [A m Sin (ω m t+b m Cos (ω m t+τ m (t] [ Sin L x A m Sin (ω m t+b m Cos (ω m t F ] (48 mcos (ω F t ρ (ωm ωf w (x, t v (x, t = = w a (x, t t t [ ] (49 = Sin L x F m Sin (ω F t [A m Cos (ω m t B m Sin (ω m t] ω m + ω F ρ (ωm ω F

9 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell 539 Assume the initial condition as w(x, t t=0 = w 0 (x v(x, t t=0 = w a (x, t t = v 0 (x t=0 The Eqs (50a and (50b can be changed into the following series form w 0 (x = C m Sin L x v 0 (x = D m Sin L x Applying the orthogonality of trigonometric series results in C m = L w 0 (xsin L 0 L x D m = L v 0 (xsin L 0 L x Substituting the initial condition Eq. (50a into the deflection expression, yields w a (x, t t=0 = = Sin [ Sin L x A m Sin(ω m t+b m Cos (ω m t F ] mcos (ω F t ρ (ωm ωf t=0 L x [ B m From Eqs (51a and (53, it can be obtained as (50a (50b (51a (51b (5a (5b ] (53 F m ρ (ωm ω F B m = C m + F m ρ (ω m ω F (54 Substituting the initial condition Eq. (50b into the velocity expression, yields w a (x, t t = t=0 = From Eqs (51b and (55, gives [ ] Sin L x F m Sin(ω F t [A m Cos(t B m Sin(ω m t] ω m + ω F ρ (ωm ω F t=0 (55 Sin L x A m ω m A m = D m ω m (56 Substituting the expressions of A m and B m into Eq. (48, gives w a (x, t = Sin L x [( Dm ω m Sin(ω m t+ ( C m + F m ρ (ωm ωf Cos (ω m t F ] (57 mcos (ω F t ρ (ωm ωf

10 540 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell From Eqs (40 and (57, it can be obtained as [( ] Dm H m (t = Sin(ω m t+c m Cos (ω m t ω m [ + F m ρ (ω m ω F [Cos (ω mt Cos (ω F t] ] (58 There are two terms included in Eq. (58. First consider the first term and simplify it, yields ( ( Dm Sin(ω m t+c m Cos (ω m t= (C m Dm + Sin(ω m t + ϕ 0 (59 ω m ω m where ( ϕ 0 =Tan 1 Cm ω m D m (60 It is easily noticed that the amplitude in Eq. (59 doesn t vary with time, it is a term of free vibration. Next, discussing the second term of Eq. (58, noticing that ω F, which is the frequency of the dynamic load q t, approaches to one of the natural frequency ω m, terms like 0/0 appears. In such case, measures could be taken as follows: F m ρ (ωm ω F [Cos (ω mt Cos (ω F t] [ F m = ρ (ωm ωf Sin (ω F + ω m t Sin (ω ] (61 F ω m t when ω F ω m, the term with the corresponding infinitesimal need to be replaced as Sin (ω F ω m t = (ω F ω m t Substituting Eq. (6 into Eq. (61, and take the limit ω F ω m, yields [ [ F m lim ω F ω m ρ (ωm ω F Sin (ω F + ω m t Sin (ω ]] F ω m t [ [ ( ]] F m (ωf + ω m t (ωf ω m t = lim Sin ω F ω m ρ (ω F ω m (ω F + ω m = F mt ρω m Sin (ω m t Substituting Eq. (63 into Eq. (58, yields ( H m (t ωf ω m = (C m Dm + Sin (ω m t + ϕ 0 + ω m ω m [ ] Fm t Sin (ω m t ρω m Finally substituting Eq. (64 into Eq. (57, yields ( w (x, t ωf ω m = (C m Dm + Sin (ω m t + ϕ 0 Sin L x + [ ] Fm t Sin (ω m t Sin ρω m L x (6 (63 (64 (65

11 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell 541 Fig.. Validation of the present method. Fig. 3. Variation law of the fraction of ceramic along z-axis with different k. Consider the initial state of rest, then w 0 (x =0, v 0 (x =0 (66 Substituting Eq. (66 into Eq. (5, yields C m = L w 0 (xsin L 0 L x =0 (67a D m = L v 0 (xsin L 0 L x =0 (67b Therefore, Eq. (65 can be turned into [ ] Fm t w a (x, t ωf ω m = Sin (ω m t Sin ρω m L x (68 4. Numerical examples and discussion 4.1. Simple validation of the present method In order to validate the analytical method presented in the paper, it was applied to solve the bending and postbuckling problem of a FGM circular plate under mechanical and thermal loadings taken the same condition and parameters as a reference (Ma []. Variations of the maximal deflection of the mid-plane for the fixed and simply supported FGM structures are depicted in Fig., which reveals that the results are nearly the same. 4.. Static deflection of the FGPM cylindrical shell Consider an FGPM cylindrical shell with length L =10m, inner radius a =m, the ratio of radius to thickness is 0. In all the following numerical calculations, the following material constants for the FGPM cylindrical shell are shown in Table 1. (Dunn and Taya [34]; Dai and Wang [35]. The variation law of the fraction of ceramic along z-axis with different k is shown in Fig. 3. In this section, assuming the two ends of the FGPM cylindrical shell is fixed, then the boundary conditions may be expressed as dw w x=0,l =0; u x=0,l =0; dx =0 (69 x=0,l

12 54 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell Table 1 Material parameters of FGPM cylindrical shell Material properties Symbol Unit Al min um/al (i =1 Zironia/ZrO (i = Young s modulus E i GPa Coefficient of thermal expansion α i /k Thermal conductivity K i W/mk 04.9 Piezoelectric constants e i1 C/m e i C/m Dielectric constant g i C /Nm Pyroelectric coefficient p i N/m k Density ρ i kg/m Fig. 4. Deflection of the mid-plane for the clamped supported FGPM cylindrical shell imposed by a uniform mechanical load, with different volume fraction index n (q =1MPa, T 1 =0, T =0, a =1m, L =4m, a/h =0, φ 1 =0, φ =0. Fig. 5. Deflection of the mid-plane for the clamped supported FGPM cylindrical shell imposed by a thermal load, with different volume fraction index n (q = 0 MPa, T 1 = 00, T = 0, a = 1 m, L =4m, a/h =0, φ 1 =0, φ =0. Figures 4 and 5 show the distribution of the radial displacement w along x-axis with different volume fraction index n when the clamped-clamped FGPM cylindrical shell is only loaded by mechanic load or thermal load, respectively. As can be seen from Fig. 4 that w increases with the decrease of n. It depends mostly on the variation of the stiffness of the material. Figure 5 reveals that comparing with the mechanic load, the thermal load exerts more remarkable influence on w. With the increase of n, the hot deformation decreases when n, but increases slowly to a stable value when n>. That depends mostly on the change of thermal effect of the FGPM. Both the Figs 4 and 5 indicate that when the FGPM cylindrical shell is under either the mechanic load or the thermal load, the maximum value of w along the x-axis happens not in the center but in the places where, respectively, is 0.4 m away from each end. Figure 6 shows the effect of the thermal condition on the radial displacement w along x-axis of the FGPM cylindrical shell when q =1MPa, n =1. It is clearly revealed that the larger the thermal load is, the larger the radial displacement is. For the FGPM cylindrical shell, the effect of the electric potential on its response is worthy of concern. Figure 7 shows the distribution of the radial displacement w along x-axis of the FGPM cylindrical shell with n =1,whenit is located in different electric field. It is clearly shown that the influence of the electric potential boundary conditions over w is not obvious. Figure 8 reveals that the distribution of the electric potential does not vary linearly along z- axis, and the maximum value of the electric potential occurs in the points deviated from the neutral surface. Figures 9 and 10 show the relationship of the mechanical load or the thermal load with the electric potential distribution, respectively. They indicate that with the increases of the mechanical load or the thermal load, the value of the electric potential in the same place along the radial direction increases, and the maximum value is obtained in the points deviated from the neutral surface.

13 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell 543 Fig. 6. Deflection of the mid-plane for the clamped supported FGPM cylindrical shell imposed by different thermal loads (q =1MPa, a =1m, L =4m, a/h =0, φ 1 = 1000, φ =0, n =1. Fig. 7. Deflection of the mid-plane for the clamped supported FGPM cylindrical shell under different electric fields (q =1MPa, T 1 = 00, T =0, a =1m, L =4m, a/h =0, n =1. Fig. 8. Distribution of the electric potential φ along z-axis for the clamped supported FGPM cylindrical shell with different n (q =1MPa, T 1 = 00, T =0, a =1m, L =4m, a/h =0, φ 1 = 1000, φ =0. Fig. 9. Distribution of the electric potential φ along z-axis for the clamped supported FGPM cylindrical shell imposed by different mechanical loads (T 1 = 00, T = 0, a = 1 m, L = 4 m, a/h =0, φ 1 = 1000, φ = Free vibration of the FGPM cylindrical shell In this section, the effect of volume fraction index n on the free vibration of the simply supported FGPM cylindrical shell is studied. All geometric parameters of the FGPM cylindrical shell and material parameters are the same as the Section 4.. Figure 11 gives the effect of volume fraction index n on each order natural frequency ω for the simply supported FGPM cylindrical shell. Seen from Fig. 11, the effect of volume fraction index n on natural frequency is small when the lateral half wave number m<4. After that the high order natural frequency increase rapidly, and the effect of volume fraction index n on high order naturalfrequencygets larger. Figure 1 shows the effect of volume fraction index n on the low order natural frequency ω for the simply supported FGPM cylindrical shell. It can be easily seen that the 1th order, th order and 3th order natural frequency fall abruptly with the increase of volume fraction index n. The slippage reach to the maximum value near n =5, after that, the curves decrease gently. Figure 13 presents the effect of volume fraction index n on the high order natural frequency ω for the simply supported FGPM cylindrical shell. From Fig. 13, the effect of volume fraction index n on the high order natural frequency is obviously larger than that the low order natural frequency is. The high order natural frequency ω

14 544 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell Fig. 10. Distribution of the electric potential φ along z-axis for the clamped supported FGPM cylindrical shell imposed by different thermal loads (q = 1 MPa, a = 1 m, L = 4 m, a/h = 0, φ 1 = 1000, φ =0. Fig. 11. The effect of volume fraction index n on each order natural frequency ω for the simply supported FGPM cylindrical shell. Fig. 1. The effect of volume fraction index n on the low order natural frequency ω for the simply supported FGPM cylindrical shell. Fig. 13. The effect of volume fraction index n on the high order natural frequency ω for the simply supported FGPM cylindrical shell. decrease with the increase of volume fraction index when n<5, and reach to a minimal value at this location. After that, the natural frequency increase with the increase of volume fraction index, which can t be found in low order frequency. In other words, the constituents of material reach to optimum in high order frequency Force vibration of the FGPM cylindrical shell In this section, the force vibration of the simply supported FGPM cylindrical shell is investigated. As calculated in the former Section 4., the influence of the electric potential boundary conditions over w is not obvious, and the thermal boundary conditions have also been discussed in detail. Then in this section the temperatures of inner and outer surfaces are, respectively, taken as T 1 =0 CandT =0 C, and the corresponding electric potential boundary conditions are taken as φ 1 =0, φ =0.

15 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell 545 Fig. 14. Deflection of the mid-plane for the simply supported FGPM cylindrical shell imposed by a uniform mechanical load, when n =1. Fig. 15. Deflection of the mid-plane for the simply supported FGPM cylindrical shell imposed by a uniform mechanical load, when t = The simply supported boundary conditions may be expressed as follows w x=0,l =0, u x=0,l =0, M x x=0,l =0 (70 In the following numerical calculations, three kinds of loading are considered respectively. The driving forces are supposed to be harmonic forces with the cosine form as Eq. (37. Case 1 Consider a simply supported FGPM cylindrical shell which is imposed by a uniform distributed load q 0 (x =10 5 Pa (71 Figure 14 shows that the instantaneous deflections of the simply supported FGPM cylindrical shell with n =1at different time points. From Fig. 14, it is seen that the amplitude value of deflection is equal to zero at two ends of the shell, which satisfies the given boundary conditions. It is also seen from the curves that the amplitude of vibration for the simply supported FGPM cylindrical shell increases as the time increases, and the amplitude peak appears near two ends of the shell, and the place of the amplitude peak appears further away two ends of the shell as the time gets longer. Figures 15 and 16 show that the instantaneous deflections of the simply supported FGPM cylindrical shell with various gradient indexes n at t = sandt =0.00 s, respectively. From these figures, it is seen that amplitude value of deflection decreases except two ends of the shell as the volume exponent increases. Comparing Fig. 15 with Fig. 16, the change of the amplitude value of deflection decreases as the volume exponent increases, and the change trend of deflection is contrary as the volume exponent increases. Figure 17 illustrates the change trend of the instantaneous deflections at four different points with the time goes. From Fig. 17, it shows that the pace of the vibration of this four points different from each other, not on the same rhythm. It is also found that the slight variations of the deflection happened at the start-up time, and the form of vibration changes and the amplitude increases significantly as time goes. Case Consider a simply supported FGPM cylindrical shell imposed by a sinusoidal distributed load ( x q 0 (x =10 5 Sin L π Pa (7 Figure 18 shows that variation of the instantaneous deflections of the simply supported FGPM cylindrical shell with n =1at different time points. From Fig. 18, it is found that the sharp of deflection for different time points all are sinusoidal form, which is due to the same form of the driving load and the mode sharp of the cylindrical shell. Figures 19 and 0 show the instantaneous deflections of the FGPM cylindrical shell imposed a sinusoidal distributed load with various volume exponents n at t = sandt =0.001 s, respectively. From Fig. 19, it is

16 546 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell Fig. 16. Deflection of the mid-plane for the simply supported FGPM cylindrical shell imposed by a uniform mechanical load, when t =0.00. Fig. 17. Deflection of various points for the simply supported FGPM cylindrical shell imposed by a uniform mechanical load as the time increasing, when n =1. Fig. 18. Deflection of the mid-plane for the simply supported FGPM cylindrical shell imposed by a sinusoidal distributed load, when n =1. Fig. 19. Deflection of the mid-plane for the simply supported FGPM cylindrical shell imposed by a sinusoidal distributed load, when t = noticed clearly that the change trend of deflection is similar with various volume exponents, and the deflection of the simply supported FGPM cylindrical shell decreases as the volume exponent increases at the same radial point. Comparing Fig. 17 with Fig. 0, one knows the amplitude of deflection of the simply supported FGPM cylindrical shell increases significantly as the time increases. Figure 1 shows the change trend of various points deflection with sinusoidal distributed loads as the time increases. From Fig. 1, it is seen easily that the vibrations at different points are in the same step, it means that the deflections reach the maximum or minimum values at the same time, and the amplitude increases linearly with the increase of time. Case 3 Considering a simply supported FGPM cylindrical shell imposed a cosine distributed load. ( x q 0 (x =10 5 Cos L π Pa (73 Figure shows deflection of the mid-plane for the simply supported FGPM cylindrical shell imposed a cosine distributed load with n =1at different time points. From Fig., it is found that there exists big difference of

17 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell 547 Fig. 0. Deflection of the mid-plane for the simply supported FGPM cylindrical shell imposed by a sinusoidal distributed load, when t = Fig. 1. Deflection of various points for the simply supported FGPM cylindrical shell imposed by a sinusoidal distributed load as the time increasing, when n =1. Fig.. Deflection of the mid-plane for the simply supported FGPM cylindrical shell imposed by a cosine distributed load, when n =1. Fig. 3. Deflection of the mid-plane for the simply supported FGPM cylindrical shell imposed by a cosine distributed load, when t =0.00. vibration sharp at different time point. At the beginning, the amplitude is small, and changes gently. However, as the time goes by, the amplitude turns to be quite large. Figures 3 and 4 show deflections of the simply supported FGPM cylindrical shell imposed a cosine distributed load with various volume exponents n at t =0.00 sandt =0.005 s, respectively. From Fig. 3, it is seen easily that the amplitude of deflection for the shell decreases from two ends to middle place of the shell, and it decreases as the volume exponent increases at the same point of the x axis. Comparing Figs 3 and 4, one knows, the change trend of deflection for the shell is similar. Figure 5 gives effect of the loading frequency on the mid-plane deflection of the simply supported FGPM cylindrical shell imposed by a cosine distributed load at x =3. From Fig. 5, The deflection reach to maximum when loading frequency ω F equal to ω m (achieved resonance, the greater between the natural frequency ω m and loading frequency ω F, the smaller the deflection is by and large.

18 548 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell Fig. 4. Deflection of the mid-plane for the simply supported FGPM cylindrical shell imposed by a cosine distributed load, when t = Fig. 5. Effect of the loading frequency on the mid-plane deflection of the simply supported FGPM cylindrical shell imposed by a cosine distributed load at x =3. 5. Concluding remarks The paper presents an analytic study for forced vibration of a FGPM cylindrical shell. To investigate the interaction among the effects of electric excitation, thermal load, mechanical load and volume exponent, static deflection of the FGPM cylindrical shell is first discussed in the numeral examples. As depicted in the relevant section, it is known that the influence of the electric excitation over the static deflection is not obvious, while with the increase of the mechanical load or thermal load, the static deflection increases monotonously. The effect of volume fraction index on high order frequency is larger than that is in low order frequency. In the second example, the force vibration of this FGPM cylindrical shell is investigated in detail, the influences of vibration amplitude and volume exponent have been examined. The greater between the natural frequency and loading frequency, the smaller the deflection is by and large. It is confirmed that the characteristics of deflection are significantly influenced by the volume exponent and various kinds of loads. Therefore, FGPM cylindrical shell structures deserve special attention in order to optimize their mechanical response, it is possible to optimize and design the FGPM cylindrical shell by selecting a proper volume exponent and suitable loads. Acknowledgments The authors wish to thank reviewers for their valuable comments and the funded by the National Natural Science Foundation of China (Grant No , Key Laboratory of Manufacture and Test Techniques for Automobile Parts, Ministry of Education, Hunan Provincial Natural Science Foundation for Creative Research Groups of China (Grant No.1JJ7001, State key Laboratory of Advanced Design and Manufacturing for Vehicle Body (Grant No , and the central colleges of basic scientific research and operational costs (funded by the Hunan University. Appendix A Θ 11 = c 1 (z [e 1(z] g(z, Θ 1 = c (z e 1(ze (z, Θ 13 = λ(zt (z, g(z

19 Θ 14 = e 1(zk d g(z(a z, H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell 549 Θ 15 = e 1(zp(zT (z g(z, Θ 1 = c (z [e (z], g(z Θ = c 1 (z e 1(ze (z, Θ 3 =Θ 13, Θ 4 = e (zk d g(z g(z(a z, Θ 5 = e (zp(zt (z. g(z Appendix B h/ ( A 11 = Θ 11 1 z h/ ( dz, A 1 = Θ 11 z 1 z dz, h/ a h/ a h/ ( A 13 = Θ 1 1 z h/ a h/ A 15 = Θ 14 dz, A 16 = h/ h/ A = Θ 1 zdz, A 3 = h/ h/ A 5 = Θ 4 dz, A 6 = h/ A 3 = h/ h/ ( 1 a + 1 a dz, A 14 = h/ h/ h/ h/ h/ h/ ( Θ 11 1 z z dz, a h/ ( A 34 = Θ 13 1 z h/ a h/ ( A 36 = Θ 15 1 z h/ a zdz, zdz. Θ 15 dz, A 1 = h/ h/ h/ h/ Θ 13 dz, Θ 1 dz, Θ ( 1 a + 1 a dz, A 4 = Θ 5 dz, h/ h/ h/ ( A 31 = Θ 11 1 z h/ a h/ ( A 33 = Θ 1 1 z h/ a h/ ( A 35 = Θ 14 1 z h/ a Θ 3 dz, zdz, ( 1 a + 1 a zdz, zdz, References [1] S.C. Pradhan, C.T. Loy, K.Y. Lam et al., Vibration characteristics of functionally graded cylindrical shells under various boundary conditions, Applied Acoustics 61(1 (000, [] T.Y. Ng, K.Y. Lam, K.M. Liew et al., Dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading, International Journal of Solids and Structures 38(8 (001, [3] X. Han, D. Xu and G.R. Liu, Transient responses in a functionally graded cylindrical shell to a point load, Journal of Sound and Vibration 51(5 (00, [4] P.F. Hou, H.M. Wang and H.J. Ding, Analytical solution for the axisymmetric plane strain electroelastic dynamics of a special nonhomogeneous piezoelectric hollow cylinder, International Journal of Engineering Science 41 (00, [5] H.S. Shen, Postbuckling analysis of pressure-loaded functionally graded cylindrical shells in thermal environments, Engineering Structures 5(4 (003, [6] B.P. Patel, S.S. Gupta, M.S. Loknath et al., Free vibration analysis of functionally graded elliptical cylindrical shells using higher-order theory, Composite Structures 69(3 (005, [7] P.H. Glaucio and C.N.S. Emilio, Design of functionally graded structures using topology optimization, Materials Science Forum (005, , (005, [8] R.K. Bhangale and N. Ganesan, Free vibration studies of simply supported non-homogeneous functionally graded magneto-electro-elastic finite cylindrical shell, Journal of Sound and Vibration 88(1 (005, [9] R.K. Bhangale and N. Ganesan, Buckling and free vibration analysis of functionally graded cylindrical shells subjected to a temperature specified boundary condition, Journal of Sound and Vibration 89(3 (006,

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