Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 419 Mã bài: 101 Phân tích phi tuyến động lực học tấm composite chức năng FGM đối xứng có tính chất vật liệu phụ thuộc vào nhiệt độ Nonlinear dynamic analysis of symmetrical composite FGM plates with temperature-dependent materials properties Nguyễn Đình Đức, Phạm Hồng Công Trường đại học Công nghệ - Đại học Quốc gia Hà Nội 144 Xuân Thủy, Cầu Giấy, Hà Nội - Việt Nam e-Mail: ducnd@vnu.edu.vn, congph_54@vnu.edu.vn Tóm tắt Bài báo này trình bày cách tiếp cận giải tích để nghiên cứu các đáp ứng động lực học phi tuyến của tấm composite chức năng FGM đối xứng, không hoàn hảo hình dáng ban đầu và tính chất vật liệu phụ thuộc vào nhiệt độ theo lý thuyết tấm biến dạng trượt bậc nhất của Reddy và sử dụng phương pháp giải theo các hàm ứng suất Airy. Đã xác định được 2 phương trình cơ bản để nghiên cứu động lực học của tấm FGM. Các phương trình phi tuyến được giải theo phương pháp Runger-Kutta. Bài báo đã tính toán số và thảo luận những ảnh hưởng của các tham số như tính chất vật liệu, yếu tố hình học, tính không hoàn hảo hình dáng ban đầu và nhiệt độ lên đáp ứng động học và dao động phi tuyến của tấm chức năng FGM. Abstract This paper presents an analytical approach to investigate the nonlinear dynamic response and nonlinear vibration of imperfect symmetrical composite FGM plates with temperature-dependent materials properties using the Reddy’s first order shear deformation theory using Air’s stress functions. Two basic equations are obtained to investigate the dynamic response and vibration of the FGM plate. The non-linear equations are solved by the Runger-Kutta method. Effects of material and geometrical properties, imperfection and temperature on the dynamic response and nonlinear vibration of the symmetrical FGM plates are analyzed and discussed. Keywords: Symmetrical composite functionally graded materials plates, nonlinear dynamic, temperature- dependent materials properties, first order shear deformation theory, imperfection. 1. Introduction Functionally Graded Materials (FGMs) are composite and microscopically in homogeneous with mechanical and thermal properties varying smoothly and continuously from one surface to the other. Typically, these materials are made from a mixture of metal and ceramic or a combination of different metals by gradually varying the volume fraction of the constituent metals. The properties of FGM plates and shells are assumed to vary through the thickness of the structures. Due to the high heat resistance, FGMs have many practical applications, such as reactor vessels, aircrafts, space vehicles, defense industries and other engineering structures. As a result, in recent years, many investigations have been carried out on the dynamic and vibration of FGM plates and shells. Up to date, dynamic analysis of FGM plates and shells using the higher order shear deformation theory has received great attention of the researchers. In [1], Mohammad and Singh studied static response and free vibration of symmetrical FGM plates using first order shear deformation theory with finite element method. In [2] Huang and Shen studied nonlinear vibration and dynamic response of FGM plates in thermal environments but volume fraction follows a simple power law for symmetrical FGM plate. Shariyat investigated vibration and dynamic buckling control of imperfect hybrid FGM plate subjected to thermo-electro-mechanical condition [3] and dynamic buckling of suddenly load imperfect hybrid FGM cylindrical shells [4] with temperature-dependent material properties. Kim in [5] studied temperature dependent vibration analysis of functionally graded rectangular plates by finite-elements method. Notice that in all the publication mentioned above [1-5], all authors use the displacement functions and volume fraction follows a simple power law. This paper presents an analytical approach to investigate the nonlinear dynamic response and 420 Nguyễn Đình Đức, Phạm Hồng Công VCM2012 nonlinear vibration of imperfect symmetrical FGM plates with temperature-dependent properties using the Reddy’s first order shear deformation theory [6]. Moreover, other than [1-5], the paper uses Air’s stress functions for solutions and volume fraction follows a Sigmoid distribution. Numerical results for dynamic response of the FGM plate are obtained by Runger-Kutta method. 2. Theoretical formulation In the modern engineering and technology, there are many structures usually working in a very high heat resistance environment. To increase the ability to adjust to a high temperature, structures with the top and bottom surfaces are made of ceramic and the core of the structure is made of metal. The symmetrical FGM plate considered in this paper is the one example of these structures [7]. Consider a symmetrical rectangular plate that consists of two layers made of functionally graded ceramic and metal materials and is midplane- symmetric. The outer surface layers of the plate are ceramic-rich, but the midplane layer is purely metallic. The plate is referred to a Cartesian coordinate system , , x y z , where xy is the midplane of the plate and z is the thickness coordinator, / 2 / 2 h z h    .The length, width , and total thickness of the plate are a , b and h , respectively (Fig.1). Fig. 1. Symmetrical FGM plate For symmetrical FGM plates, applying a Sigmoid distribution, the volume fractions of metal and ceramic, m V and c V , are assumed as [8]: 2 , /2 0 ( ) 2 , 0 /2 N z h h z h V z m N z h z h h                                   ( ) 1 ( ) c m V z V z   (2.1) where the volume fraction index N is a nonnegative number that defines the material distribution and can be chosen to optimize the structural response. The effective Young’s modulus E , thermal expansion coefficient  and the mass density  are independent to the temperature, vary in the thickness direction z and can be written as follows:         ( , ), , , , ( ), ( ), ( ) 2 ; / 2 0 ( ), ( ), ( ) 2 ;0 / 2 c c c N mc mc mc N E T z T z T z E T T T z h h z h E T T T z h z h h                                     (2.2) where       ( ) ( ) ( ) ( ) ( ) ( ) mc m c mc m c mc m c E T E T E T T T T T T T             (2.3) subscripts m and c stand for the metal and ceramic constituents, respectively, and the Poisson ratio  depends weakly on temperature change and is assumed to be a constant ( )z    . 2.1. Nonlinear dynamic of imperfect symmetrical FGM plate using first order shear deformation theory Suppose that the FGM plate is subjected to a transverse load of intensity 0 q . In the present study, the Reddy’s first-order shear deformation theory is used to obtain the motion and compatibility equations, as well as expression for determining the dynamic response of the FGM plate. The train-displacement relations taking into account the Von Karman nonlinear terms and the first - order shear deformation theory are [6,8]: 0 0 0 , , ε ε χ x x x ε = ε + z χ y y y γ γ χ xy xy xy w xz x x w yz y y                                                            (2.4) with 0 2 , , 0 2 , , 0 , , , , / 2 / 2 x x x y y y y x x y xy u w v w u v w w                                 ; , , , , x x x y y y xy x y y x                                  (2.5) Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 421 Mã bài: 101 where 0 x  and 0 y  are the normal strains, 0 xy  is the shear strain on the midplane of the plate, and xz  and yz  are the transverse shear strains; , u v , and w are the midplane displacement components along the , x y , and z axes; x  and y  are the rotation angles in the xz and yz planes, respectively; (,) indicates a partial derivative. The strains are related in the compatibility equation 2 2 0 2 0 2 0 2 2 2 2 2 2 2 w w w y xy x x y x y y x x y                              (2.6) Hooke law for an FGM plate is defined as         2 , ( , ) , (1 ) (1,1) 1 , , , , 2(1 ) x y x y y x xy xz yz xy xz yz E T E                              (2.7) In this study, it is assumed that the temperature is uniformly raised and T  is the constant. The force and moment resultants of the plate can be expressed in terms of stress components across the plate thickness as     /2 /2 /2 /2 , 1, ; , , ; , h i i i h h i iz h N M z dz i x y xy Q dz i x y           (2.8) Inserting Eqs. (2.4), (2.5) and (2.7) into Eq. (2.8) gives the constitutive relations as                                         0 0 1 2 2 2 3 1 2 0 0 1 2 2 2 3 1 2 0 1 2 2 3 1 1 , [ , 1 , 1 , ] 1 , [ , 1 , 1 , ] 1 , [ , , ] 2 1 , , 2 1 x x x y x y y y y x y x xy xy xy xy x y xz yz N M E E E E N M E E E E N M E E E E E Q Q                                          (2.9) where   1 2 3 3 3 /2 1 2 /2 ; 0 1 12 2( 1)( 2)( 3) , ( , ) ( , ) (1, ) mc c c mc h h E h E E h E N E h E h E N N N E T z T z T z dz                 1 [ 1 1 ] 2 1 c c c mc c mc mc mc h h E h E E N N h E T N              (2.10) For using late, the reverse relations are obtained from Eq. (2.9) 0 0 1 1 1 1 0 1 1 1 ; 2(1 ) x x y y y x xy xy N N N N E E N E                         (2.11) According to Love’s theory the equations of motion are 2 2 1 1 2 2 2 0 1 2 ; 0; 0 w w w w w xy xy y x xy xy y x x y y x x xy xy y N N N N u v x y x y t t M M M M Q Q x y x y Q Q N N x y x x y N N q y x y t                                                                      (2.12) where /2 1 /2 ( ) 1 h m c c h z dz h N                   (2.13) With the assumption w, w u v   , in (2.12) the inertia 2 1 2 0 u t     and 2 1 2 0 v t     . Within the framework of the first order shear deformation theory, the nonlinear motion equations for a perfect plate can be written in terms of deflection w and force resultants as 0; 0 xy xy y x N N N N x y x y             (2.14) 4 2 , , , 1 2 1 , , , 0 2 2 1 2 2(1 ) ( 2 w ) ( 2 w ) 0 x xx xy xy y yy x xx xy xy y yy D D w N w N w N w E N w N w N w q t t                     (2.15) 422 Nguyễn Đình Đức, Phạm Hồng Công VCM2012 where 2 2 2 2 2 x y        and 3 2 1 E D    For solving Esq. (2.14) and (2.15) we introduce Air’s stress function ( , ) x y    so that 2 2 2 2 2 ; ; x y xy N N N x y y x               (2.16) Inserting Eq. (2.16) into the Eq. (2.15) for prefect plate leads to 2 2 2 2 4 2 2 2 1 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 0 1 2 2 2 2(1 ) w w w+ [ 2 w w w w ] [ 2 w w ] 0 D D E x y x y y x x y x y x y t y x q x y t                                                     (2.17) Equation (2.17) includes two dependent unknowns w and  . To obtain a second equation, relating the unknowns, the geometrical compatibility 0 0 0 2 , , , , , , x yy y xx xy xy xy xx yy w w w        (2.18) Setting Eqs. (2.11), (2.16) into Eq. (2.18) gives the compatibility equation of an perfect FGM plate as 2 4 4 4 2 2 2 4 2 2 4 2 2 1 1 w w w 2 E x y x x y y x y                                         (2.19) For an imperfect plate, following to the Volmir’s approach [9] for an imperfect FGM plate, Eqs. (2.19) and (2.17) are modified into form as 2 4 4 4 2 2 2 4 2 2 4 2 2 1 2 2 * 2 * 2 * 2 2 1 w w w 2 w w w E x y x x y y x y x y x y                                                               (2.20)   2 2 2 2 4 * 2 2 2 1 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 0 1 2 2 2 2(1 ) w w w w + [ 2 w w w w ] [ 2 w w ] 0 D D E x y x y y x x y x yx y t y x q x y t                                                      (2.21) in which * w is a known function representing initial small imperfection of the FGM plate. Equations (2.20) and (2.21) are the basic relations used to investigate the dynamic response and vibration of imperfect FGM plate with temperature-dependent material properties. They are nonlinear in the dependent unknowns w and  . 2.2. Nonlinear vibration of imperfect symmetrical FGM plate Suppose that the imperfect symmetrical FGM plate is simply supported at its edges and subjected to q transverse loads 0 ( ) q t . The boundary conditions can be expressed as 0 w 0, 0, , 0 x x x xy M N N N     at 0, x x a   0 w 0, 0, , 0 y y y xy M N N N     at 0, y y b   (2.22) Taking into account temperature-dependent material properties, the mentioned conditions (2.22) can be satisfied if the deflection * w, w and the stress function  are represented by [7]: 1 2 3 2 2 4 0 0 w ( )sin sin 2 2 os os sin sin 1 1 os os 2 2 x y m x n y f t a b m x n y m x n y Ac A c A a b a b m x n y A c c N y N x a b                 (2.23) * 0 ( , ) sin sin m x n y w x y f a b    (2.24) in which , 1,2, m n  are natural numbers representing the number of half waves in the x and y directions respectively; f is the deflection amplitude; 0 f const  , varying between 0 and 1 , represents the size of the imperfections. Introduce Eqs. (2.23), (2.24) into Eq. (2.20), we can obtain the coefficients ( 1 4) i A i   as follow :     2 2 2 2 1 1 0 2 2 2 2 2 2 1 2 0 3 4 2 2 ( ) 32 ( ) ; 0 32 E n a A f t f m b E m b A f t f A A n a       (2.25) The condition expressing the immovability of edges (2.22) is fulfilled on the average [32, 34]: 0 0 0 0 0; 0 b a a b u v dxdy dydx x y         (2.26) From Eqs. (2.5) and (2.9), taking into account Eq. (2.16) , we can obtain the following relations: Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 423 Mã bài: 101     2 1 , , , 1 1 2 1 , , , 1 1 1 w / 2 1 w / 2 yy xx x xx yy y u x E E v y E E                          (2.27) Introduce (2.23) into (2.27) and then integrate the results as in (2.26), we have: 2 2 2 2 1 0 1 2 2 2 1 ( ) 1 8(1 ) x m n N E f t a b                 (2.28) 2 2 2 2 1 0 1 2 2 2 1 ( ) 1 8(1 ) y m n N E f t a b                 (2.29) Introduce (2.23), (2.24) into (2.21) and apply the Bubnov-Galerkin method for the resulting equation, we obtain:       2 2 2 4 0 2 2 2 2 4 2 2 2 2 1 2 2 2 1 2 2 2 4 0 2 2 2 2 2 2 2 4 2 0 1 2 2 2 2 2 2 4 2 2 1 2 ( ) 4 2(1 ) 4 [ ( ) 8 8 ( ) 4 ( ) ( )] 4 4 [ ( ) 2 4 x y m n ab D f t f a b D m n m b n a A A f t E a ba b m n m b N f t aa b m n n a m n ab N f t f t ba b a b m n m b A A f t ab a                                                          0 2 2 0 0 1 2 ( ) 4 ( ) ( )] 0 4 4 x y N f t n a ab ab N f t q f t b mn        (2.30) Eq. (2.30) can be represented as:            5 10 1 0 2 6 1 2 3 7 0 4 8 0 9 0 ( ) ( ) ( ) ( ) ( ) x y B B f t B f t f B B A A f t B B N f t B B N f t B q            (2.31) Where 2 2 2 4 1 2 2 2 2 4 2 2 2 2 2 2 2 1 2 2 2 4 3 2 2 1 2 2 2 4 4 2 2 1 2 2 2 5 1 2 2 1 2 2 4 2 2 2 6 7 8 4 2(1 ) 4 8 8 2(1 ) 4 2(1 ) 4 2(1 ) 4 ; ; 2 4 m n ab B D a b D m n m b n a B E a b a b D m n m b B E aa b D m n n a B E ba b D m n ab B E a b m n m b n B B B ab a                                                              2 9 10 1 2 4 4 ; 4 a b ab ab B B mn      (2.32) Introducing i A and 0 0 , x y N N at Eqs. (2.25), (2.28) and (2.29) into (2.31) gives 3 1 2 3 0 4 0 ( ) ( ) ( ) f t m f t m f t m f m q     (2.33) where             2 2 2 2 2 1 1 5 10 2 1 2 6 0 2 2 2 2 1 1 3 7 4 8 3 1 2 2 2 2 1 4 2 6 2 2 2 2 2 2 2 3 7 1 2 2 2 2 2 2 4 8 1 2 2 2 2 4 1 2 3 1 1 32 1 1 32 1 8(1 ) 1 8(1 ) ; ; E n a m b C B B C B B B f m b n a B B B B C B E n a m b C B B m b n a m n B B E a b m n B B E a b C C m m m C C                                                               3 9 4 1 1 ; C B m C C  (2.34) and ( ) f t - deflection of middle point of the plate   /2 /2 ( ) w x a y b f t    . For linear free vibration for FGM plate the equation (2.33) gets form: 1 ( ) ( ) 0 f t m f t   (2.35) one can determine the fundamental frequency of natural vibration of the FGM plate : 1 L m   (2.36) The equation (2.33) for obtaining the nonlinear dynamic response the initial conditions are assumed as . 0 (0) , (0) 0 f f f   . The applied loads 424 Nguyễn Đình Đức, Phạm Hồng Công VCM2012 are varying as function of time. The nonlinear equation (2.33) can be solved by the Newmark’s numerical integration method or by the Runger- Kutta method. 3. Numerical results and Discussion The imperfect symmetrical FGM plate considered here a square plate: 1 a b m   , 0.1 h m  . Here, several numerical examples will be presented for perfect and imperfect simply supported midplane-symmetric FGM plates. The silicon nitride and stainless steel are regarded as constituents of the composite FGM plates. A material property Pr , such as the elastic modulus and thermal expansion coefficient, can be expressed as a nonlinear function of temperature [1-5].   1 1 2 3 0 1 1 2 3 Pr 1 P P T PT PT PT        (3.1) in which 0 T T T    and 0 300 T K  (room temperature); P 0 , P -1 , P 1 , P 2 and P 3 are temperature-dependent coefficients characterizing the constituent materials. The typical values of the coefficients of the materials mentioned are listed in Table1 [10]. The Poisson ratio is chosen to be 0.28 for simplicity. Table 1. Material properties of the constituent materials of the considered FGM shells Material Property P 0 P - 1 P 1 P 2 P 3 Si 3 N 4 (Ceramic) E(Pa) 348.43e9 0 -3.70e-4 2.160e-7 -8.946e-11  (kg/m 3 ) 2370 0 0 0 0 1 ( ) K   5.8723e-6 0 9.095e-4 0 0 SUS304 (Metal) E(Pa) 201.04e9 0 3.079e-4 -6.534e-7 0  (kg/m 3 ) 8166 0 0 0 0 1 ( ) K   12.330e-6 0 8.086e-4 0 0 The equation of non-linear free vibration of a perfect plate can be obtained from (2.33) 3 1 2 ( ) ( ) ( ) 0 f t m f t m f t    (3.2) Seeking solution as ( ) os( ) f t c t    and applying procedure like Galerkin method to Eq.(3.2), the frequency-amplitude relation of non- linear free vibration is obtained 1 2 2 2 2 3 1 4 NL L L m             (3.3) Where NL  is the non-linear vibration frequency and  is the amplitude of non-linear vibration. Fig. 2: Frequency- amplitude relation ( NL  -  ) The plate subjected by an uniformly distributed excited transverse load 0 ( ) sin q t p t   . From Eq. (2.36), natural frequencies of FGM plate L  are shown the Table 2. Obviously the natural frequencies of the FGM plate are observed to dependent on the constituent volume fraction, they increase when the power law index N increases. Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 425 Mã bài: 101 Table 2. Effect of power law index N and temperature (with m=n=1) on natural frequencies L  for perfect and imperfect ( 0 0.002 f  ) symmetrical FGM plates T  (K) N=1 N=2 N=3 Perfect Imperfect Perfect Imperfect Perfect Imperfect 0 4.2875e3 4.2872e3 4.8770e3 4.8767e3 5.2236e3 5.2232e3 300 3.4524e3 3.4521e3 4.0143e3 4.0139e3 4.3377e3 4.3373e3 500 2.7085e3 2.7081e3 3.2735e3 3.2731e3 3.5852e3 3.5847e3 600 2.2611e3 2.2607e3 2.8365e3 2.8360e3 3.1415e3 3.1411e3 The nonlinear dynamic response of the FGM plate acted on by the harmonic uniformly excited transverse load 0 ( ) sin q t p t   are obtained by solving Eq. (2.33) combined with the initial conditions and by use of the Runge-Kutta method. Fig. 3 shows dynamic responses (relations deflection of middle point of the plate   /2 /2 ( ) w x a y b f t    and the time ) of the symmetrical FGM plate subjected (without temperature and with temperature change) to the harmonic transverse load 0 ( ) 75000sin(4000 ) q t t  . We can see that temperature change has serious effect on dynamic response of the FGM plates. 0 0.01 0.02 0.03 0.04 -4 -3 -2 -1 0 1 2 3 4 x 10 -4 t(s) w(m) Fig.3. Linear and nonlinear transient response of the FGM plate with temperature change 50 T K   ( N=1, : nonlinear; - - -: linear) Relation of maximum deflection and velocity of maximum deflection when (N=1) and 0 ( ) 75.000sin(2500 ) q t t  is presented in Fig.4a and Fig.4b -2 -1 0 1 2 x 10 -4 -0.5 0 0.5 w dw/dt Fig.4a. Deflection velocity relation, 0 T K   -5 0 5 x 10 -4 -1.5 -1 -0.5 0 0.5 1 1.5 w dw/dt Fig.4b. Deflection velocity relation, 500 T K   Fig.5a and Fig.5b show nonlinear response of the FGM plate of long period with different intensity of loads : 2 75000 / p N m  and 2 95000 / p N m  ; 4000   426 Nguyễn Đình Đức, Phạm Hồng Công VCM2012 0 0.02 0.04 0.06 0.08 0.1 -3 -2 -1 0 1 2 3 4 x 10 -4 t(s) w(m) p=75000 p=95000 Fig. 5a. Dynamic response with different intensity of loads, 0 T K   0 0.02 0.04 0.06 0.08 0.1 -4 -3 -2 -1 0 1 2 3 4 x 10 -4 t(s) w(m) p=75000 p=95000 Fig. 5b. Dynamic response with different intensity of loads, 200 T K   Fig.6a and Fig.6b show the effect of the imperfection ( 0 0 (0,001;0,003); ( ) 750.000sin(3500 ) 1 f q t t N    ) on nonlinear dynamic responses of the FGM plate. Fig.6a. Influence of imperfection on nonlinear dynamic response of FGM plate, 0 T K   Fig.6b. Influence of imperfection on nonlinear dynamic response of FGM plate, 300 T K   Fig. 7a, Fig.7b show the effect of the geometrical parameters ( / ) a b on dynamic response of the FGM plate with 0 ( ) 75.000sin(2200 ) q t t  and 1 N  . Fig. 8a, Fig. 8b show the effect of the ( / ) a h on dynamic response of the FGM plate with 0 ( ) 75.000sin(1700 ) q t t  , 1 N  and a b  . Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 427 Mã bài: 101 0 0.02 0.04 0.06 0.08 0.1 -3 -2 -1 0 1 2 3 x 10 -4 t(s) w(m) a/b=1 a/b=2 Fig.7a. Effect of dimension ration / a b on dynamic response, 0 T K   0 0.02 0.04 0.06 0.08 0.1 -4 -3 -2 -1 0 1 2 3 4 x 10 -4 t(s) w(m) a/b=1 a/b=2 Fig.7b. Effect of dimension ration / a b on dynamic response, 100 T K   0 0.01 0.02 0.03 0.04 0.05 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x 10 -4 t(s) w(m) a/h=10 a/h=15 Fig. 8a. Effect of dimension / a h on dynamic response of the square plate   a b  , 0 T K   0 0.01 0.02 0.03 0.04 0.05 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 x 10 -4 t(s) w(m) a/h=10 a/h=15 Fig. 8b. Effect of dimension / a h on dynamic response of the square plate   a b  , 100 T K   Fig. 9 shows effect of temperature T  on dynamic response of the FGM plate with 0 ( ) 75000sin(4000 ) q t t  . 428 Nguyễn Đình Đức, Phạm Hồng Công VCM2012 Fig. 9. Effect of temperature on dynamic response of the square symmetrical FGM plate From the obtained results, we observe these interesting conclusions: Temperature change has serious effect on vibration and dynamic response of FGM plates. Frequencies of nonlinear vibration of composite FGM plate increase when amplitudes, the intensity of dynamic loads, volume fraction index increase and frequencies of vibration decrease when temperature increase. 4. Conclusions This paper presents an analytical approach to investigate the nonlinear dynamic analysis of imperfect symmetrical composite FGM plates with temperature-dependent material properties. The strong point in this article is the achievement of analytical fundamental equations for FGM plates using Air’s stress function and the Reddy’s first order shear deformation theory. Numerical results for dynamic response of the FGM plate are obtained by Runger-Kutta method. Thus it is obvious that dynamic response of the considered plate depends on many parameters significantly: temperature, imperfection   0 f and geometrical parameters   / ; / a b a h of the imperfect symmetrical composite FGM plate. Therefore, when we change these parameters, we can control the dynamic response and vibration of the FGM plate actively. References [1] Mohammad Talha, Singh B.N, Static response and free vibration analysis of FGM plates using higher order shear deformation theory. Applied Mathematical Modeling, 2010, Vol.34 (12), pp.3991- 4011. [2] Xiao-Lin Huang, Hui-Shen Shen, Nonlinear vibration and dynamic response of functionally graded plates in thermal environments. International Journal off Solid and Structures, 2004, Vol 41, pp.2403-2427 [3] Shariyat M., Vibration and dynamic buckling control of imperfect hybrid FGM plates with temperature-dependent material properties subjected to thermo-electro-mechanical loading conditions. J. Composite Structures, 2009, Vol 88, pp. 240-252 [4] Shariyat M., Dynamic buckling of suddenly loaded imperfect hybrid FGM cylindrical shells with temperature – dependent material properties under thermo-electro-mechanical loads. International Journal of mechanics Sciences, 2008, Vol.50, pp.1561-1571. [5] Young-Wann Kim, Temperature dependent vibration analysis of functionally graded rectangular plates. Journal of Sound and Vibration, 2005, 284: 531-549. [6] Reddy J.N. Mechanics of laminated composite plates and shells: Theory and analysis. Boca Raton: CRC Press; 2004. [7] Duc N.D, Tung H.V, Mechanical and thermal post-buckling of shear-deformable FGM plates with temperature-dependent properties. J. Mechanics of Composite Materials, 2010, 46 (5), pp. 461-476. [8] Hui-Shen Shen, Functionally Graded materials, Non linear Analysis of plates and shells, CRC Press, Taylor & Francis Group, London, Newyork, 2009 [9] Volmir A.S, Non-linear dynamics of plates and shells, Science Edition, M. 1972 [10] Reddy JN, Chin CD. Thermoelastical analysis of functionally graded cylinders and plates. Journal of thermal Stresses 1998; 21:593-626 . Mã bài: 101 Phân tích phi tuyến động lực học tấm composite chức năng FGM đối xứng có tính chất vật liệu phụ thuộc vào nhiệt độ Nonlinear dynamic analysis of symmetrical composite FGM plates. cứu các đáp ứng động lực học phi tuyến của tấm composite chức năng FGM đối xứng, không hoàn hảo hình dáng ban đầu và tính chất vật liệu phụ thuộc vào nhiệt độ theo lý thuyết tấm biến dạng trượt. số như tính chất vật liệu, yếu tố hình học, tính không hoàn hảo hình dáng ban đầu và nhiệt độ lên đáp ứng động học và dao động phi tuyến của tấm chức năng FGM. Abstract This paper presents