nonstationary stochastic response analysis of composite laminated plates with aerodynamic and thermal loads

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Stationary/Nonstationary Stochastic Response Analysis of Composite Laminated Plates with Aerodynamic and Thermal Loads Kai Zhou , Zhen Ni , Xiuchang Huang , Hongxing Hua PII: DOI: Reference:

S0020-7403(19)33883-4 https://doi.org/10.1016/j.ijmecsci.2020.105461 MS 105461

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

14 October 2019 13 January 2020 19 January 2020

Please cite this article as: Kai Zhou , Zhen Ni , Xiuchang Huang , Hongxing Hua , Stationary/Nonstationary Stochastic Response Analysis of Composite Laminated Plates with Aerodynamic and Thermal Loads, International Journal of Mechanical Sciences (2020), doi: https://doi.org/10.1016/j.ijmecsci.2020.105461

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Highlights 

A unified solution for the stationary/nonstationary stochastic vibration problems of the composite laminated plate with thermal and aerodynamic loads is proposed.



The modified Fourier method combined with pseudo excitation method (PEM) is adopted to obtain the solutions for the cases with classical and elastic boundary conditions.



An experiment study concerning random vibration of the composite plate under acoustic loads is performed.



The effects of the boundary condition, thermal load, fiber orientation and aerodynamic pressure on the random vibration behaviors of the composite laminated plate in supersonic airflow are clearly figured out.

1

Stationary/Nonstationary Stochastic Response Analysis of Composite Laminated Plates with Aerodynamic and Thermal Loads Kai Zhou1,*, Zhen Ni1, Xiuchang Huang1 and Hongxing Hua1 1. Institute of Vibration, Shock and Noise, State Key Laboratory of Mechanical System and Vibration, Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai Jiao Tong University, Dongchuan Road 800, Shanghai, 200240, China Email: [email protected]

Abstract： This paper investigates the stationary/nonstationary stochastic responses of composite laminated plates under thermal and aerodynamic loads, where the point random excitation, distributed random excitation and base acceleration random excitation can be considered. The effects of the thermal stress and aerodynamic pressure are taken into account by employing the thermo-elastic theory and supersonic piston theory, respectively. The Hamilton’s principle is used to formulate the governing equations of the system, and the solutions for the dynamic problems of cases having classical and non-classical (elastic) boundary conditions are obtained by the modified Fourier method combined with pseudo excitation method (PEM). To validate the proposed formulation, a sufficient number of numerical and experimental studies are conducted for the free vibration, flutter and stochastic response analysis of composite laminated plates with various boundary conditions. Satisfactory agreements are shown between the computed results and those from the finite element method (FEM), published literature and experiments. Finally, the effect of boundary 2

condition, thermal load, fiber orientation and aerodynamic pressure on the random vibration behaviors of composite laminated plates is also presented. Keywords: Composite laminated plate; Stationary/Nonstationary response; Random excitation; Thermal environment; Aerodynamic pressure.

1. Introduction Composite laminated plates have been extensively used in aeronautic, naval, civil and other engineering applications due to their excellent performance and reliable properties. It is well known that aerospace structures made by composite materials are always exposed to the severe environment during the cruise. The combination of mechanical, thermal and aerodynamic loads will have a significant effect on the dynamic behaviors of these composite material structures, and the thermal buckling and flutter phenomena may occur. Meanwhile, these structures are always subjected to the random excitations induced by turbulent boundary and jet noise. Therefore, investigations on stochastic responses of composite laminated plates considering thermal and aerodynamic loads are of great significance in the optimal design of these structures. In recent years, many researchers have investigated the random vibration behaviors of the thin-walled structures under stochastic excitations. Several numerical cases can be analyzed by the analytical methods. Cederbaum et al. [1] proposed an analytical method to investigate the dynamic response of the simply supported composite plate under nonstationary excitations. Hosseinloo et al. [2] performed the random vibration analysis of a thin plate subjected to boundary excitations by an 3

analytical method. The effects of boundary conditions, modal damping and excitation frequency range on stochastic responses of the plate were also discussed in detail. Meanwhile, some approximate methods, such as finite element method (FEM) and Galerkin method also have been widely employed in engineering applications. Chang et al. [3] used the FEM to investigate the random response of the shell structures excited by nonstationary stochastic loadings. The obtained results can be used for estimating the safety and reliability of structures. Guo et al. [4] applied the FEM to study random vibration behaviors of the composite shallow shell in the thermal environment. The linear and nonlinear stochastic responses of shells can be predicted by the proposed model. De Rosa and Franco [5] obtained the exact and numerical solutions for the stochastic response of the plate excited by turbulent boundary layer by using the analytical method and FEM, respectively. Franco et al. [6] used the FEM and other numerical methods to study the stochastic responses of the plate excited by random and convective loads. Dogan [7] adopted the Galerkin method to investigate the random vibration of the FGM plate under thermal environment. The influence of sound pressure levels, temperature and material mixture parameters on the dynamic behaviors of the plate was investigated. By using the Galerkin method, the random vibration of a plate resting on elastic foundation and excited by stochastic excitations was investigated by Hosseinkhani et al. [8]. Meanwhile, the influence of the foundation stiffness and damping on the random vibration behaviors of the plate was discussed. Besides, quite a number of other numerical methods have been adopted to study the random vibration problems of structures. Maury et al. [9,10] presented a 4

wavenumber method to study the stochastic response of the plate excited by turbulent boundary layer. Sadri and Younesian [11] adopted the Laplace transform inversion algorithm to analyze the vibro-acoustic responses of a coach platform subjected to stochastic loadings. The influence of the rail irregularity and vehicle speed on the stochastic responses was also analyzed. Gao and Kessissoglou [12] used the random factor method to study the stochastic response of truss structures under nonstationary random excitation. Meanwhile, the effect of the randomness of the structural parameters on the dynamic response was also investigated. In order to enhance the efficiency and accuracy, Lin et al. [13] proposed an efficient method termed as pseudo excitation method (PEM). Chen et al. [14] used the PEM to analyze the sensitivity characteristics of coupled structural acoustic systems excited by random excitation. Si et al. [15] carried out the random vibration analysis of a beam-soil structure under a moving stochastic loading based on the PEM. The effects of the viscous damping and elastic modulus on the critical velocity were also analyzed. Chen et al. [16, 17] obtained the exact solutions for the random vibration of the thin plate excited by stationary and nonstationary stochastic loadings based on the PEM, respectively. Besides, the PEM has been widely used in other engineering applications and showed the excellent performances [18-22]. Since the aerospace vehicles with high speed are always subjected to aerodynamic pressure, considerable research efforts have been devoted to investigating the dynamic characteristics of the plate structures in airflow. Thus, abundant computational methods have been developed, such as the analytical method 5

[23,24], Rayleigh-Ritz method [25-27], Galerkin method [28-30], FEM [31-35] and other numerical methods [36-38]. In these studies, the flutter problems of the isotropic, composite and functionally graded plates with high speed have been sufficiently investigated. It is worth noting that almost all of these publications focus on the cases with classical boundary conditions. Actually, it is more realistic to use the elastic constraints to simulate the boundaries of these plate structures in some engineering cases [39]. Recently, the flutter behaviors of plates having classical and elastic boundaries in supersonic airflow have been substantially studied by Zhou et al. [39-41]. As can be seen from the literature review, although many researchers have studied the random vibration of plate structures, few publications concerning the stochastic response of composite laminated plates with aerodynamic and thermal loads considered can be found. Although the dynamic problems of composite plates under aerodynamic, thermal and random acoustic loads have been studied in Ref. [42], the acoustic load is assumed as the normal Gaussian distribution and its spectral density functions are uniformly distributed in the frequency domain. Actually, the random excitation is very complicated in practical engineering cases, and its spectral density functions are often unevenly distributed in the frequency domain. Motivated by such reality consideration, the present work aims to develop a unified solution for the stationary/nonstationary stochastic responses of heated composite laminated plates excited by random excitations in supersonic airflow. It is worth noting that the stochastic excitation can be one or any combination of the stationary/nonstationary 6

point excitation, distributed excitation and base acceleration excitation. The proposed model is suitable for the cases subjected to classical and elastic constraints. The first-order shear deformation theory (FSDT) together with supersonic piston theory is adopted to derive the formulations of the system and the solutions for the random responses of the composite laminated plate are achieved by using the modified Fourier method combined with PEM. The remainder of the present study is arranged as follows. The governing equations of the heated composite laminated plate excited by random excitations in supersonic airflow and corresponding solutions for the flutter and stochastic response problems are presented in Section 2. Numerical comparisons and discussion are carried out in Section 3, and conclusions are drawn in Section 4.

2. Theoretical formulations 2.1. Kinematic relations and stresses Consider a heated composite laminated plate with multiple layers with length L1 and width L2 in supersonic airflow, and h is the total thickness of the laminated plate, as shown in Fig. 1. The Cartesian coordinate system established on the referenced middle plane of the laminated plate is adopted to account for the geometry and deformation information of the structure. Zk and Zk+1 are the distances from the bottom surface and top surface of the k-th layer to the referenced plane.  air represents the yawed flow angle between the airflow and x axis. In order to simulate the possible constraints, it is assumed that translational degrees of freedom of the composite laminated plate for each edge are restrained by three types of translational 7

springs ku, kv and kw. Similarly, rotational degrees of freedom for each edge are restrained by two types of rotational springs kx and ky. For example, kux0, kvx0, kwx0, kxx0 and kyx0 are used to represent the uniformly distributed springs for the edge x=0 of the composite laminated plate [43]. It is noteworthy that the present study focuses on the dynamic behaviors of the composite plate before the thermal buckling and flutter, thus the linear plate theory is adopted according to the Ref. [26]. Besides, since the thermomechanical coupling effect is very weak in the free vibration analysis [44-46], it will be ignored in the following formulations. As shown in Fig. 1(b), the external random excitations can be one or any combination of the stationary/nonstationary point excitation, distributed excitation and base acceleration excitation.

z n

y


k+1 k k-1

O

x


1

(a) Composite laminated rectangular plate

8

zn+1 zn zk+2 zk+1 zk x zk-1 z2 z1

y z Distributed Excitation

 air

Base Acceleration Excitation

Point Excitation

x Thermal environment (b) Various random excitations Fig. 1. Geometry and coordinate systems of the heated composite laminated plate in supersonic airflow.

By using the FSDT, each displacement of the composite laminated plate is expressed as [47] U ( x, y, z, t )  u ( x, y, t )  z x ( x, y, t ) V ( x, y, z, t )  v( x, y, t )  z y ( x, y, t )

(1)

W ( x, y, z, t )  w( x, y, t )

where u, v and w denote the middle plane displacements of the laminated plate in the x, y and z directions, respectively.  x and  y are the shear rotations around the y and x axes and t is the time variable. The linear strains-displacement relations of the composite plate are obtained based on the elasticity theory [47]  u z x x x  v y   z y y y

x 

  u v  )  z( x  y ) y x y x w  xz   x x w  yz   y y

 xy  (

9

(2)

For the k-th composite layer, the following relation concerning the Poisson's ratio and elasticity modulus can be obtained [47]

 12k E1k



 21k

(3)

E2k

With the thermal effects considered, the normal and shear stresses of the composite laminated plate are expressed by the generalized Hooke’s law [40]

 xk   Q11k Q12k  k  k k  y  Q12 Q22  k   0  yz    0  k   0  xz   0  k   k  xy  Q16 Q26k

0

0

0

0

Q44k

Q45k

Q45k

Q55k

0

0

k k Q16k   x  11T    k k Q26k   y   22 T     k  0   yz   k 0    xz   k  k k Q66   xy  12 T 

(4)

where  xk and  yk denote the normal stresses in the x and y directions of the k-th layer, respectively.  yzk ,  xzk and  xyk are the corresponding shear stresses. 11k ,  22k and 12k represent the coefficients of linear thermal expansion. The temperature variation is defined as T  T1  T0 , where T0 is the reference temperature and T1 is the current temperature, respectively. The elastic stiffness coefficients Qijk (i, j=1, 2, k 4, 5 and 6) and thermal expansion coefficients  pq (p, q=1 and 2) of the k-th layer

are given by [43] Q11k Q12k  k Q12k Q22  0  0  0  0  k k Q16 Q26

0

0

0

0

k Q44

k Q45

k Q45

Q55k

0

0

Q16k  Q11k Q12k  k  k k  Q26 Q12 Q22  0 0   Tk  0   0  0 0   0 0 k  Q66 

10

0 0 k Q44 0 0

0 0   0 0  T 0 0  Tk  Q55k 0  k  0 Q66 

(5)

11k  11k    k 1 T  k   22   (Tk )  22   k  k   12  12 

(6)

k where superscript T denotes the transposition. Qijk (i, j=1, 2, 4, 5 and 6) and  pq (p,

q=1 and 2) are the elastic stiffness coefficients and linear thermal expansion coefficients along the principal axes of the k-th composite layer, where the material constants Qijk are given by [43]

Q11k 

E1k

1   12k  21k

, Q12k 

 21k E1k E2k k k k k k , , Q44 , Q55 Q   G23  G12k (7)  G13k , Q66 22 k k k k 1   12 21 1   12 21

It is worth emphasizing that Tk denotes the transformation matrix, which is employed to describe the relation of the principal material and global coordinate systems of the k-th layer [43]

 cos2  k  2  sin  k Tk   0  0  sin  k cos  k

sin 2  k cos2  k 0 0  sin  k cos  k

0 0 cos  k  sin  k 0

0 0 sin  k cos  k 0

2sin  k cos  k   2sin  k cos  k   0  0  2 2 cos  k  sin  k 

(8)

where  k denotes the angle between the principal direction and the x-axis for the k-th composite layer. The lamina stiffness coefficients Qij (i, j=1, 2, 4, 5 and 6) and thermal expansion k coefficients  pq (p, q=1 and 2) can be obtained by substituting Eqs. (7) and (8) into

Eqs. (5) and (6), and the detailed expressions of them are listed in Appendix A. The thermal stresses induced in the k-th layer of the composite plate can be expressed by [40] 11

Q11k Q12k Q16k  11k T   xTk      Tk   k k k k    y    Q12 Q22 Q26  22 T   k  k  Tk   k k  Q Q Q  16 26 66  12 T   xy    

(9)

The strains caused by temperature variation can be written as [40]

U 2 V 2 W 2 ) ( ) ( ) x x x U 2 V 2 W 2 d yy  ( ) ( ) ( ) y y y U U V V W W d xy  ( )( )  ( )( )  ( )( ) x y x y x y d xx  (

(10)

According to the Refs. [34, 48], since the aerodynamic damping always stabilizes the structure, the aerodynamic pressure induced by supersonic airflow with aerodynamic damping effect ignored is given by p  

 U 2

w w cos air  sin air ) y M 2  1 x

(11)

(

where   , U  and M  represent the air density, velocity and Mach number of the free stream, respectively. For the sake of convenience in the following studies, the parameter of 2 3 1 2 non-dimensional aerodynamic pressure    U  L1 / ( D1 M   1) is adopted. In the

expression,

D11

is

the

reference

flexural

rigidity

defined

as

1 1 1 1 and  21 are the elastic parameters of D11  E11h3 / [12(1   12  21 )] , where E11 ,  12

the 1-th composite layer.

2.2. Energy expressions The strain energies of the composite laminated plate are expressed as [40]

12

k k k k k k k k k 1  n zk 1  x ( x  11T )   y ( y   22 T )   xy ( xy  12 T )     dz  dxdy U e     Tk 2   k 1 zk   K s xzk  xzk  K s yzk  yzk   xTk d xx   Tk   y d yy  2 xy d xy  

(12) where Ks denotes the shear correction factor and is chosen as 5/6 in this study [47]. By substituting Eqs. (1)-(10) into Eq. (12), the strain energies Ue of the heated composite laminated plate are obtained, where the detailed expressions are listed in Appendix B. The elastic energies stored in distributed springs of the composite laminated plate for four edges can be given as [43]

U BC 

1 L1  kuy 0 (u ) 2  kvy 0 (v ) 2  k wy 0 ( w) 2  k xy 0 ( x ) 2  k yy 0 ( y ) 2  dx  0 y 0 2

1 L1  kuyL2 (u ) 2  kvyL2 (v ) 2  k wyL2 ( w) 2  k xyL2 ( x ) 2  k yyL2 ( y ) 2  dx  0 y  L2 2 L 1 2 +   kux 0 (u ) 2  kvx 0 (v ) 2  k wx 0 ( w) 2  k xx 0 ( x ) 2  k yx 0 ( y ) 2  dy x 0 2 0 L 1 2 +   kuxL1 (u ) 2  kvxL1 (v ) 2  k wxL1 ( w) 2  k xxL1 ( x ) 2  k yxL1 ( y ) 2  dy x  L1 2 0 

(13)

The kinetic energies of the composite laminated plate can be given as [43]

  u 2 v 2 w 2  u  x   I 0 ( t )  ( t )  ( t )   2 I1 ( t )( t )   1 L1 L2    Tk     dxdy 2 0 0    x 2  y 2  v  y  2 I ( )( )  I 2 ( ) ( )   1 t t  t   t n

where I 0    k ( zk 1  zk ) , I1  k 1

(14)

1 n 1 n 2 2  ( z  z ) I   k ( zk31  zk3 ) . and 2   k k 1 k 2 k 1 3 k 1

In the present study, the stationary/nonstationary stochastic excitations

p( x, y, t ) perpendicularly exerted on the composite plate surface are considered. Thus, the work done by the airflow and stochastic excitations are respectively expressed as 13

Wa   pwdxdy

(15)

Ws   pwdxdy

(16)





2.3. Solutions In order to obtain sufficiently accurate solutions for the cases with various boundary conditions, different trial functions for the plate need to be sought in traditional numerical methods, such as Ritz method and Garlekin method. Usually, seeking the admissible functions for the plate with different boundary conditions is tedious. Meanwhile, the convergence problem always occurs due to the discontinuities at edges of the plate by employing the traditional Fourier series method [43]. In order to satisfy general restraints and avoid convergence problems, the displacement fields of the composite laminated plate are expressed in the form of cosine Fourier series together with auxiliary functions [43]

14







u ( x, y )   Amn cos(m x) cos(n y )   ( d11m1b ( y )  d 21m 2b ( y )) cos(m x) m0 n0

m0



  ( f11n1a ( x)  f 21n 2 a ( x)) cos(n y ) n 0







v( x, y )   Bmn cos(m x) cos(n y )   ( d12m1b ( y )  d 22m 2b ( y )) cos(m x) m0 n0

m0



  ( f12n1a ( x)  f 22n 2 a ( x)) cos(n y )

(17)

n 0







w( x, y )   Cmn cos(m x) cos(n y )   ( d13m1b ( y )  d 23m 2b ( y )) cos(m x) m0 n0

m0



  ( f13n1a ( x)  f 23n 2 a ( x)) cos(n y ) n 0







 x ( x, y )   Dmn cos(m x) cos(n y )   ( d14m1b ( y )  d 24m 2b ( y )) cos(m x) m0 n0

m0



  ( f14n1a ( x)  f 24n 2 a ( x)) cos(n y ) n 0







 y ( x, y )   Emn cos(m x) cos(n y )   ( d15m1b ( y )  d 25m 2b ( y )) cos(m x) m0 n0

m0



  ( f15n1a ( x)  f 25n 2 a ( x)) cos(n y ) n 0

where m  m / L1 , n  n / L2 (m=0,1,…,∞, n=0,1,…,∞). Amn, Bmn, Cmn, Dmn and Emn represent the coefficients of Fourier series expansion. d1lm , d 2l m , f1ln and f 2ln (l=1,2,…,5) are the coefficients of supplement functions. The supplement functions are given by [43]

x x x x 1a ( x )  L1 ( )(  1) 2 , 2 a ( x)  L1 ( ) 2 (  1) L1 L1 L1 L1 y y y y 1b ( y )  L2 ( )(  1)2 , 2 a ( y )  L2 ( ) 2 (  1) L2 L2 L2 L2

(18)

It can be found that,

1a (0)=1a ( L1 )=1a ( L1 )=0, 1a (0)=1  2 a (0)= 2 a ( L1 )= 2 a (0)=0,  2 a ( L1 )=1 1b (0)=1b ( L2 )=1b ( L2 )=0, 1b (0)=1  2b (0)= 2b ( L2 )= 2b (0)=0,  2b ( L2 )=1 15

(19)

which demonstrates the first-order partial derivatives of the proposed admissible expressions are continuous at any location of the composite laminated plate. It is assumed that infinite series terms in Eq. (17) are uniformly truncated to M and N in the actual numerical calculations. Then, substituting Eqs. (12)-(17) into the Hamilton’s principle [47] t1

  (Tk  U e  U BC  Wa  Ws ) dt  0

(20)

t0

the obtained linear algebraic equations can be further expressed as the following matrix form

MQ  KQ  F

(21)

where M, K and F denote the mass matrix, stiffness matrix and external force vector of the system. The vector

Q  [q1; q2 ; q3 ; q4 ; q5 ]

represents the unknown

coefficients. Then, the governing equations of system can be further expressed as  K uu K T  uv  0  T  K ux  K Tuy

K uv K vv 0 K Tvx K Tvy

0 0 K ww K Twx K Twy

K ux K vx K wx K xx K Txy

K uy  q1   M uu K vy  q 2   0      K wy  q 3    0   T K xy  q 4   M ux   K yy   q 5    0

0 M vv 0 0 M Tvy

0 0 M ww 0 0

M ux 0 0 M xx 0

0  q1   0  M vy  q 2   0          0  q 3   Fw   0  q 4   0      M yy   0   q 5   

(22) where Fw  

L1

0



L2

0

HT p( x, y, t )dxdy and the detailed submatrices in Eq. (22) are

given in Appendix C. The nonstationary random process p(t) is expressed as the following form based on the Priestley evolutionary power spectral theory [17] 

p(t )   A(, t ) exp(it )d () 

16

(23)

where

A(, t ) represents the time-frequency modulated function and  ( )

satisfies the following relations [17] x( t ) 





e x pi ( t d)  ( )

E[d * (1 )d  (2 )]  S xx (1 ) (2  1 )d 1 d 2

(24) (25)

where x(t ) is a stationary random process with zero-mean-value and its power spectral density (PSD) function is defined as S xx ( ) ; E[] represents the mathematical expectation.  is the Dirac delta function and superscript * denotes the complex conjugate operator. Thus, the PSD function of the random process p(t) can be further written as

S p (, t )  A(, t ) S xx () 2

(26)

The uniform modulated nonstationary PSD function can be obtained by changing the time-frequency modulated function A(, t ) to the frequency independent modulated function g(t) 

p(t )  g (t )  exp(it )d ()  g (t ) x(t ) 

(27)

Assume the PSD function of the stochastic excitations p( x, y, t ) exerted on the plate structure is expressed by S p ( x, y, , t )  2 ( x, y ) g 2 (t ) S xx [, (t )] . Based on the PEM, the following harmonic excitation is used to replace the Gaussian random excitation p( x, y, t ) [13] p( x, y, t )  ( x, y ) g (t ) S xx [, (t )]eit

(28)

Here, the stochastic excitation p( x, y, t ) can be one of the following three loads or any combination: (1) the concentrated excitation exerted on the point (x0, y0); (2) the distributed excitation over q( x, y ) ; (3) the base acceleration excitation with 17

acceleration wb (t ) . Thus, the detailed pseudo harmonic excitations for these three kinds of nonstationary random loads can be respectively given by [13] p( x, y, t )   ( x  x0 , y  y0 ) g (t ) S xx [, (t )]eit

(29)

p( x, y, t )  q( x, y ) g (t ) S xx [, (t )]eit

(30)

p( x, y, t )   I 0 g (t ) Swb [, (t )]eit

(31)

It's worth noting that the pseudo harmonic excitations for nonstationary excitations listed in Eqs. (29)-(31) can readily degenerate to the pseudo harmonic excitations for stationary excitations. For example, the following harmonic excitation is used to replace the stationary random excitation p( x, y, t )

p( x, y, t )  S p ( x, y, )eit

(32)

Then, the detailed pseudo harmonic excitations for these three kinds of stationary random loads can be respectively given as

p( x, y, t )   ( x  x0 , y  y0 ) S f ()eit

(33)

p( x, y, t )  q( x, y ) S f ()eit

(34)

p( x, y, t )   I 0 Sw ()eit

(35)

b

With the eigenvalue problem of Eq. (22) solved, the natural frequencies and mode shapes of the composite laminated plate with aerodynamic and thermal loads can be obtained. As described in Ref. [26], the flutter phenomenon of structures can be predicted by the first coalescence of two consecutive natural frequencies of the composite laminated plate and the non-dimensional aerodynamic pressure at the occurrence of the flutter is called as the critical flutter aerodynamic pressure (CFAP). 18

In this study, the CFAP is denoted as cr . In the stochastic response analysis, the pseudo response of the composite laminated plate with pseudo excitation can be calculated firstly and the auto-PSD functions for the stationary/nonstationary random transverse responses of the composite plate can be further obtained as [13] Sww ( x, y,  )  w( x, y,  )* w( x, y, )

(36)

Sww ( x, y,  )  w( x, y, )* w( x, y, )   2 Sww ( x, y, ) Sww ( x, y,  )  w( x, y, )* w( x, y, )   4 Sww ( x, y, )

(37) (38)

where * denotes the complex conjugate. The root mean square (RMS) of the stochastic response of the composite laminated plate can be expressed by [17]

 r ( x, y ) 

1



Gr ( x, y,  )d  

0

N

 G ( x, y,  ) i 1

r

i

(39)

where 0 and 1 are the lower cutoff frequency and upper cutoff frequency of the random excitation, respectively. Gr ( x, y, ) is the one-side PSD function of the response of the composite laminated plate and the number of intervals is defined as

N  (1  0 ) /  .

3. Results and discussion 3.1. Convergence study Based on the proposed formulations, a corresponding computational code is developed in MATLAB. Firstly, the convergence study concerning the influence of the distributed springs on the modal characteristics of the composite laminated plate is conducted. Since a composite laminated plate with four edges clamped should be 19

restrained by all types of boundary distributed springs, a clamped composite laminated plate with thermal and aerodynamic effects ignored is chosen as the computational model. A five-layer rectangular laminated plate with the same composite material for each layer is considered, whose length and width are both defined as 1m. The detailed material property parameters of each layer are listed in Table 1. The orientations of five layers from top to bottom are [0/0/0/0/0] and the thickness for each layer is uniformly chosen as 0.002m. The changes of the first two order natural frequencies of the composite laminated plate are plotted in Fig. 2. It is clearly observed that these two natural frequencies are almost unchanged when the stiffness of distributed boundary springs exceeds 5*107N/m2, which means the clamped constraint of the composite plate can be simulated by giving these boundary springs greater than 5*107N/m2. Thus, for the following listed three types of classical constraints, the stiffness values of boundary springs are assigned by: (1) Clamped constraint (C): ku=kv=kw=kx=ky=5*1010N/m2 ; (2) Simply supported constraint (S): kx=0N/m2

and

ku=kv=kw=ky=5*1010N/m2

(x=0,

L1);

ky=0N/m2

and

ku=kv=kw=kx=5*1010N/m2 (y=0, L2); (3) Free constraint (F): ku=kv=kw=kx=ky=0N/m2. Besides, the non-classical (elastic) constrains can be easily achieved by giving these boundary springs with actual stiffness values. Table 1. The material parameters for each single layer [40]. E1(GPa) E2(GPa) G12(GPa) G13(GPa) 175

32

12

12

3 G23(GPa)  (Kg/m ) v12

5.7

20

1760

0.25

11 (K-1)  22 (K-1) 12 (K-1) 1.2e-6

2.3e-6

0

Natural frequency(Hz)

200 1st mode 2nd mode

150

100

50

0

1

2

3

4

5

6

7

8

9

10

10 10 10 10 10 10 10 10 10 10 10 Stiffness for boundary springs(N/m2)

11

Fig. 2. The changes of the first two natural frequencies of the composite laminated plate with various distributed springs.

As mentioned above, the infinite terms in trial functions should be truncated in the actual calculations. In order to obtain the suitable truncation number for the modified Fourier series solution, the computed first ten order natural frequencies of the composite laminated plate established in the previous part with different truncation numbers and the corresponding computation time are compared with those from FEM, as listed in Table 2. It should be worth emphasizing that the FEM results for the composite plate are computed using quadratic S8R elements with 50×50 finite element mesh in the ABAQUS. Obviously, the computed results exhibit excellent accuracy and the proposed method shows a great convergence property. Therefore, all the series will be truncated into M=N=16 in the following cases. Table 2. The natural frequencies of the composite laminated plate with different truncation numbers. Mode 1 2 3

Natural frequencies (Hz) M=N=10 M=N=11 M=N=12 M=N=13 M=N=14 M=N=15 M=N=16 116.76 172.02 276.14

116.75 171.79 276.13

116.72 171.77 275.50

116.70 171.66 275.53 21

116.68 171.67 275.24

116.67 171.61 275.27

116.67 171.64 275.10

FEM 116.68 171.55 274.95

4 5 6 7 8 9 10 Time(s)

292.95 332.14 412.79 427.02 543.22 560.25 592.78 4.46

292.66 331.62 412.49 424.26 540.30 560.14 592.42 5.65

292.60 331.58 411.87 424.25 540.24 559.39 591.68 6.96

292.47 331.37 411.76 423.17 539.08 559.33 591.50 8.82

292.47 331.35 411.49 423.18 539.06 559.02 591.21 12.32

292.42 331.26 411.44 422.69 538.53 558.98 591.11 15.88

292.40 331.25 411.29 422.71 538.51 558.86 590.99 20.76

292.35 331.13 411.07 422.12 537.85 558.71 590.77 58.32

3.2. Numerical validation In this subsection, the solutions obtained from the present method are verified by comparing the computed results with those from FEM and published references. Firstly, the results for the free vibration of the heated composite laminated plate are verified. The computational model is the same as that used in the convergence study and three types of fiber orientations are considered here. The temperature change is chosen as T  20K . The comparisons of the obtained results with those from FEM are listed in Tables 3-5. It can be clearly seen that the present results agree well with the FEM results, which reveals the present method is reliable to predict the free vibration of the composite laminated plate in the thermal environment. Table 3. The natural frequencies of the heated composite laminated plate computed by present method and FEM ( T  20K ). Natural frequencies (Hz) ([0/0/0/0/0]) Num 1 2 3 4 5 6 7 8

S-S-S-S

S-C-S-C

C-C-C-C

FEM

Present

FEM

Present

FEM

Present

47.909 97.323 182.688 190.790 217.268 291.809 324.403 407.542

47.991 97.616 182.858 191.132 217.816 292.725 324.961 407.724

63.030 135.729 188.374 239.621 250.043 336.576 402.990 410.776

63.062 135.799 188.409 239.743 250.237 336.859 403.597 410.862

111.850 165.906 269.125 286.607 324.869 404.484 416.265 531.206

111.846 165.972 269.284 286.661 324.980 404.714 416.838 531.870

Table 4. The natural frequencies of the heated composite laminated plate computed by present 22

method and FEM ( T  20K ). Natural frequencies (Hz) ([45/-45/45/-45/45]) Num 1 2 3 4 5 6 7 8

S-S-S-S

S-C-S-C

C-C-C-C

FEM

Present

FEM

Present

FEM

Present

60.531 139.797 162.362 244.670 290.750 309.749 375.681 432.374

60.821 139.898 163.371 245.045 291.062 311.886 376.203 433.774

86.283 163.970 208.567 280.249 319.493 380.486 418.806 483.478

86.368 164.222 209.019 280.721 320.389 381.679 419.726 484.846

106.456 205.390 230.457 324.688 380.914 401.114 471.779 532.949

106.583 205.762 230.963 325.539 382.009 402.480 473.349 535.286

Table 5. The natural frequencies of the heated composite laminated plate computed by present method and FEM ( T  20K ). Natural frequencies (Hz) ([0/90/0/90/0]) Num 1 2 3 4 5 6 7 8

S-S-S-S

S-C-S-C

C-C-C-C

FEM

Present

FEM

Present

FEM

Present

47.897 119.174 169.096 217.092 250.537 326.532 373.155 410.638

48.071 119.517 169.394 217.841 251.133 327.797 373.855 411.905

73.311 175.933 179.297 255.305 334.528 378.733 397.126 434.252

73.365 176.245 179.455 255.733 335.496 379.293 398.265 435.105

111.784 196.527 266.064 324.417 346.899 447.057 505.791 550.175

111.877 196.846 266.525 325.093 347.819 448.340 507.251 551.889

The obtained solution for the free vibration of the heated plate is also compared with that from Ref. [49], as shown in Table 6. The clamped composite plate with dimensions of 0.5×0.4×0.01m3 is chosen as the computational model, and the material parameters

are

E1=37.78GPa,

E2=10.9GPa,

G12=G13=4.91GPa,

v12=0.3,

 =1870Kg/m3 , 11 =7  10-6 /℃ and 22 =2.3  10-5 /℃ . As shown in Table 6, the obtained results agree well with those from the Ref. [49].

23

Table 6. The natural frequencies of the heated composite plate computed by present method and literature.

T (℃)

Method

Natural frequency (Hz) 1

2

3

4

5

0

Ref. [49] Present Error

264 264 0.03%

510 509 0.25%

573 566 1.15%

770 766 0.46%

912 905 0.79%

30

Ref. [49] Present Error

234 233 0.50%

471 469 0.53%

543 535 1.39%

732 728 0.54%

871 861 1.10%

60

Ref. [49] Present Error

197 196 0.44%

426 424 0.42%

510 502 1.48%

691 687 0.54%

825 816 1.14%

90

Ref. [49] Present Error

151 150 0.74%

377 374 0.70%

475 467 1.71%

648 644 0.66%

777 767 1.27%

100

Ref. [49] Present Error

132 131 1.05%

359 356 0.80%

462 454 1.66%

633 629 0.71%

760 750 1.29%

Meanwhile, the thermal buckling temperatures for these aforementioned cases are calculated and compared with the FEM results, as listed in Table 7. Obviously, excellent agreements are demonstrated. Table 7. The thermal buckling temperature of the composite laminated plate computed by present method and FEM. The thermal buckling temperature variation (K) Method FEM Present

[0/0/0/0/0]

[45/-45/45/-45/45]

[0/90/0/90/0]

S-S-S-S S-C-S-C C-C-C-C S-S-S-S S-C-S-C C-C-C-C S-S-S-S S-C-S-C C-C-C-C 71.28 71.62

103.29 103.30

239.07 239.02

101.66 102.50

167.99 168.38

222.45 222.81

71.25 71.58

127.29 127.42

238.17 238.42

Subsequently, the solutions for the aero-thermo-elastic analysis of the plate are validated. It is noteworthy that Tcr denotes the critical thermal buckling temperature of the plate. The geometric parameters and material properties of the plate are listed in Table 8. The obtained results of a simply supported plate are calculated and compared with those from Ref. [26], as shown in Fig. 3. Here, the non-dimensional natural

24

frequency is defined as    L12 1h / D11 . From Fig. 3, excellent agreements can be observed between the obtained results and those from literature, which means the proposed method is accurate to evaluate the flutter behaviors of the heated plate in supersonic airflow. Table 8. Material properties and geometric parameters of the plate [26]. L1(m)

L2(m)

 (Kg/m3)

h(m)

E(GPa)

v

 (K-1)

0.305

0.305

2764

0.0013

68.9

0.3

2.25e-5

45

T=0.8Tcr

40 Non-dimensional natural frequency

35 30 25 20 15 Present Method Ref.[26]

10 5

0

100

200

300

400

500



Fig. 3. The variations of the non-dimensional natural frequencies of the S-S-S-S plate under thermal environment with different aerodynamic pressures.

Then, the obtained results for the stochastic vibration of the plate excited by stationary random excitations are compared with those from the published literature. The material properties and geometric parameters of the plate employed in Ref. [16] are listed in Table 9. The plate is assumed to be subjected to a band-limited base acceleration with PSD function S0  0.5g 2 / Hz within [20, 2000] Hz. In the calculations, the frequency step is chosen as 5Hz and modal damping ratio is equal to 0.05. The obtained acceleration response PSD curves measured at center of the plate are compared with those from Ref. [16], as shown in Fig. 4. Obviously, excellent agreements can be observed. Meanwhile, the obtained RMS results of displacement, 25

velocity and acceleration responses measured at center of the plate are also compared with those from Ref. [16], as shown in Table 10. It can be found that the present results are in good agreements with those from literature. From these cases, it can be concluded that the present model is accurate to predict stationary random vibration of the plate.

L1(m)

L2(m)

 (Kg/m3)

h

E (GPa)

v

0.4

0.2

2700

0.002

70

0.33

10

5 4

10

3

10

2

10

1

10

0

10

6

10

5

10

4

10

3

10

2

10

1

10

0

2

10

Present Method Ref. [16]

3

6

Acceleration PSD (m /s )

10

10

-1

10

-2

0

500

1000 1500 Frequency(Hz)

2000

10

-1

10

-2

10

-3

Present Method Ref. [16]

0

500

1000 1500 Frequency(Hz)

(a) S-S-S-S 10

5

10

4

10

3

10

2

10

1

10

0

2000

(b) S-C-S-C Present Method Ref. [16]

2

3

Acceleration PSD (m /s )

2

3

Acceleration PSD (m /s )

Table 9. The material and geometric information of the plate [16].

10

-1

10

-2

0

500

1000 1500 Frequency(Hz)

2000

(c) S-C-S-F Fig. 4. The comparisons of the obtained acceleration response PSD function with those from the literature: (a) S-S-S-S; (b) S-C-S-C; (c) S-C-S-F.

26

Table 10. The comparisons of obtained RMS responses of the plate with those from literature. BC

S-S-S-S

S-C-S-C

S-C-S-F

RMS of response

Method Present Ref. [16] Error Present Ref. [16] Error Present Ref. [16] Error

Dis. (m)

Vel. (m/s)

Acc. (m/s2)

6.021e-4 5.966e-4 0.919% 2.339e-4 2.320e-4 0.802% 8.710e-4 8.629e-4 0.940%

0.599 0.591 1.269% 0.451 0.447 0.828% 0.409 0.405 1.037%

933.33 929.13 0.452% 1222.60 1220.66 0.159% 696.06 702.84 0.965%

Finally, the solutions for the nonstationary response of the plate excited by random excitation are validated. In this study, the uniform modulated nonstationary excitation and fully nonstationary excitation are both considered, which have been employed in Ref. [17]. The following modulated filtered Gaussian white-noise process is adopted to study the uniform modulated nonstationary response of plate structures [17] t

x(t )   h(t   , )W ( )d

(40)



where h(t , ) is the response function under the unit impulse and   [g ,  g ] denotes the filter parameters. g represents the filtered frequency parameter of the random excitation and  g is the corresponding damping ratio. W ( ) is the Gaussian white-noise random process and its PSD function is denoted by S0 [17]. The acceleration PSD function of the filtered white-noise random excitation x(t) can be expressed as [17] S xx (, ) 

1  4 2 g2 / g2 (1   2 / g2 )2  4 2 g2 / g2

S0

(41)

For the fully nonstationary stochastic excitation, with the temporal and spectral 27

nonstationarity separated, the fully nonstationary model of the filtered Gaussian white-noise random excitation x(t) can be expressed by [17] t

x(t )   h[t   , (t ))]W ( )d

(42)



Similarly, the acceleration PSD function of the filtered white-noise random excitation x(t) in this case can be expressed as [17]

S xx (, (t )) 

1  4 2 g2 (t ) / g2 (t ) (1   2 / g2 (t ))2  4 2 g2 (t ) / g2 (t )

(43)

S0

Since the spectral densities of PSD functions at zero frequency as shown in Eq. (41) and Eq. (43) are not equal to zero, the modified PSD functions are expressed with a Butterworth high-pass filter used [17]

S xx (, )  H h ( ) S xx (, (t ))  H h ( )

1  4 2 g2 / g2

2

(1   2 / g2 )2  4 2 g2 / g2 1  4 2 g2 (t ) / g2 (t )

2

(44)

S0

(1   2 / g2 (t ))2  4 2 g2 (t ) / g2 (t )

S0

(45)

where H h ()   2 N / ( 2 N  h2 N ) . H h ( ) represents the frequency response 2

function of the filter. h and N are the high-pass filtered frequency and filter order number of the Butterworth high-pass filter, respectively [17]. The obtained nonstationary random responses of the plate are compared with those from Ref. [17]. The length, width and thickness of the plate are defined as 6m, 6m and 0.12m, respectively. The Young’s modulus, Poisson’s ratio and mass density of the plate are E=36.2GPa,  =0.2 and   2400Kg / m3 . Besides, the damping ratio is defined as 0.05. For the temporally uniform time modulated nonstationary (TNON) acceleration 28

PSD function listed in Eq. (44), the filtered frequency, filtered damping ratio, high-pass filtered frequency and filter order number are respectively defined as

g  15.6rad / s ,  g  0.64 , h  2rad / s and N=2. Meanwhile, the intensity of the white-noise is S0  1m2 / s3 . For the fully nonstationary (FNON) acceleration PSD function described in Eq. (45), the filtered frequency is further expressed as

g (t )  0  (0  T )

t T

(46)

where T is the total duration time of the nonstationary excitation. In the following numerical cases, the total duration time is chosen as T=40s, 0  39.4rad / s ,

T  4.86rad / s and  g (t )  0.64 [17]. Besides, in this study, three kinds of time modulated functions g(t) are chosen, where the detailed information of them is listed in Table 11 [17]. Meanwhile, the shapes of these time modulated functions are also plotted in Fig. 5. Table 11. Three kinds of time modulated functions [17]. Type 1

Parameter

B4   0.0995   2

Type 2

t0  0.0004s, t1  t2  12.2s

  0.2 tb  1.5s, tc  15.0s

g (t )  B(et  e  t )

B  0.0744 g ,   0.413,   0.552

t  t0 0  t t (t / tb ) 0  t  tb 0 2  B(  t  t ) t0  t  t1  g (t )  1 tb  t  t c g ( t )   1 0   ( t tc ) B t1  t  t2 t  tc e    ( t  t2 )  t  t2  Be 2

g(t)

Type 3

29

1.2

1.2

Type 1

Type 2 0.8

g (t)

g (t)

0.8

0.4

0.4

0.0

0

10

20 Time (s)

30

0.0

40

0

10

(a) Type 1

20 Time (s)

30

40

(b) Type 2

1.2

Type 3

g (t)

0.8

0.4

0.0

0

10

20 Time (s)

30

40

(c) Type 3 Fig. 5. Shapes of three kinds of time modulated functions.

It is assumed that the frequency range of the band-limited base acceleration is [0, 160] rad/s. In the following calculations, the frequency step is chosen as 1rad/s. The variations of the obtained maximum displacement RMS (L1/2, L2) of the S-C-S-F plate with different nonstationary excitations and time modulated functions are plotted and compared with the results from Ref. [17], as shown in Fig. 6 and Fig. 7. It can be found that the present results and those from literature are in good agreements.

30

16 Present Method Ref. [17]

Displacement RMS (mm)

Displacement RMS (mm)

16

12

8

4

0

0

10

20 Time (s)

30

Present Method Ref. [17]

12

8

4

0

40

0

10

(a) Type 1

20 Time (s)

30

40

(b) Type 2

Displacement RMS (mm)

12 Present Method Ref. [17]

8

4

0

0

10

20 Time (s)

30

40

(c) Type 3 Fig. 6. The variation of the maximum displacement RMS of S-C-S-F subject to three types of TNON excitation. 25 Present Method Ref. [17]

20

Displacement RMS (mm)

Displacement RMS (mm)

25

15 10 5 0

0

10

20 Time (s)

30

40

(a) Type 1

Present Method Ref. [17]

20 15 10 5 0

0

10

20 Time (s)

(b) Type 2

31

30

40

Displacement RMS (mm)

20 Present Method Ref. [17]

15

10

5

0

0

10

20 Time (s)

30

40

(c) Type 3 Fig. 7. The variation of the maximum displacement RMS of S-C-S-F subject to three types of FNON excitation.

Meanwhile, the displacement RMS, velocity RMS and acceleration RMS measured at the center of the plate with different boundary conditions are also compared with those from Ref. [17], as listed in Table 12 (Type 2 time modulated function of FNON). It is worth noting that the obtained first natural frequency is also compared with that from Ref. [17] in Table 12. Obviously, excellent agreements can be observed, which means the proposed method is accurate to predict the nonstationary response of the plate under random excitation.

32

Table 12. The comparisons of RMS responses of plates subjected to type 2 time modulated function with different boundary conditions [17]. BC

Method

SS SS SS SC SC SC SF SF SS SF SC SF

Ref. [17] Present Ref. [17] Present Ref. [17] Present Ref. [17] Present Ref. [17] Present Ref. [17] Present

Natural frequency (rad/s) 75.29 75.31 90.19 90.08 110.43 110.07 37.27 37.24 45.52 45.48 49.64 49.55

Time-varying response RMS Displacement(mm)

Velocity(m/s)

Acceleration(m/s2)

5s

15s

25s

5s

15s

25s

5s

15s

25s

6.870 6.833 4.349 4.346 2.769 2.781 26.413 26.333 14.824 14.747 10.082 10.059

5.259 5.236 3.349 3.350 2.149 2.161 22.192 22.051 11.769 11.697 7.896 7.877

0.503 0.501 0.322 0.322 0.209 0.210 2.183 2.149 1.130 1.126 0.758 0.741

0.472 0.470 0.349 0.349 0.264 0.265 0.959 0.956 0.645 0.642 0.485 0.474

0.348 0.347 0.258 0.258 0.196 0.197 0.788 0.783 0.498 0.495 0.361 0.359

0.031 0.031 0.023 0.023 0.018 0.018 0.075 0.073 0.046 0.045 0.033 0.032

35.322 35.272 31.106 31.125 28.581 28.658 36.684 36.658 30.595 30.535 25.189 25.206

25.987 25.940 22.953 22.961 21.188 21.241 29.710 29.565 23.325 23.227 18.920 18.903

2.341 2.331 2.071 2.053 1.916 1.910 2.798 2.743 2.126 2.120 1.717 1.672

3.3. Experimental validation To further verify the developed method, an experimental study concerning the dynamic response of the composite plate is performed in the progressive wave tube [50]. The schematic diagram of the testing system is shown in Fig. 8, where a rectangle composite plate subjected to random acoustic loads is considered. The high-intensity acoustic load is generated by air modulators and two microphones P1 and P2 are set symmetrically in the upstream and downstream of the test specimen to record the pressure. Two acceleration transducers pasted at center point (0.5L1, 0.5L2) and 1/4 point (0.75L1, 0.5L2) are adopted to achieve the acceleration responses of the plate. 33

The C/SiC composite plate is chosen as the test specimen and its dimensions are 0.38×0.26×0.0015 m3. The width of 0.015m on all edges of the composite plate are fastened with bolted joints to achieve the clamped boundary condition. Thus, the acoustic load exerts at the central region of the composite plate with an area of 0.35×0.23m2. The material parameters of the test plate are given as: E=116GPa,

 =2100 Kg/ m3 and v=0.25. The input PSD function of the acoustic loads with 150dB sound pressure is shown in Fig. 9 and the dynamic responses of the composite plate are obtained from the experimental study and proposed method, respectively. The comparisons of the theoretical and experimental results of the dynamic response measured at the center and 1/4 point (0.75L1, 0.5L2) of the composite plate are shown in Fig. 10. Meanwhile, the natural frequencies and peak values of acceleration PSD functions of the composite plate from the proposed model and experiment are listed in Table 13. From these results, satisfactory agreements are observed. Thus, it can be concluded that the proposed model is accurate to evaluate the dynamic response of the plate structures under random excitations.

Test section

Exponential horn P  P0 cos(0t  k0 x )

Termination section

Pi

Pr

P1

P2

Composite plate z y x

Temperature ( x1, y1)

Acoustic pressure ( x2 , y2 )

34

Fig. 8. The schematic diagram of the experimental setup [50]. 3000

2

PSD(Pa /Hz)

2500 2000 1500 1000 500 0

0

200

400 600 Frequency(Hz)

800

1000

2

10

1

10

0

10

3

10

2

10

1

10

0

2

10

Acceleration PSD (g /Hz)

10

2

Acceleration PSD (g /Hz)

Fig. 9. The PSD function of the acoustic load with 150dB sound pressure. 3

10

-1

10

-2

10

-3

Present Method Experiment

50 100 150 200 250 300 350 400 450 500 Frequency(Hz)

10

-1

10

-2

10

-3

Present Method Experiment

50 100 150 200 250 300 350 400 450 500 Frequency(Hz)

(a) Center point (b) 1/4 point Fig. 10. The comparisons of the theoretical PSD function and experimental PSD function. Table 13. The comparisons of the theoretical results and experimental results.

Point

Center 1/4 Point 1/4 Point

Natural frequency (Error= (Theo.-Exp.)/ Exp.)

PSD (Error=20log10(Theo. /Exp.))

Experiment (Hz)

Theoretical (Hz)

Error

Experiment (g2/Hz)

Theoretical (g2/Hz)

Error

254 254 398

266 266 409

4.7% 4.7% 2.8%

415.8 50.91 166.1

482.1 60.2 99.1

1.285 1.456 -4.486

Note: The results of 1/4 point listed in Table 13 denote two peaks information of dynamic response measured at 1/4 point of the composite plate.

3.4. Parametric study With the developed model validated above, the parametric study concerning the effects of the boundary condition, thermal loads, fiber orientation and aerodynamic pressure on the stochastic vibration behaviors of the composite laminated plate is 35

carried out in this part. It is noteworthy that the computational model used in this subsection follows the same configuration as those in the subsection concerning the convergence study, and the random excitation is defined as a band-limited base acceleration with PSD function S0  0.5g 2 / Hz within [2, 1000] Hz. Besides, the structural damping of the composite plate is set as 0.01 and the frequency step is chosen as 2Hz in the following cases. Firstly, the effect of the boundary condition on the random response of the composite laminated plate is analyzed. Three types of boundary conditions, namely, C-C-C-C, S-S-C-C and S-S-S-S are chosen. The auto-PSD curves of displacement and acceleration responses measured at the center of the [0/0/0/0/0] composite laminated plate with different boundary conditions are shown in Fig. 11. As is known to all, the C-C-C-C plate possesses the largest structural stiffness and the S-S-S-S plate possesses the smallest structural stiffness among these constraints. It can be found that the resonance peaks of these curves move to lower frequency range with the structural stiffness decreased. The amplitude of the first resonance peak of the displacement response PSD increases with the structural stiffness decreased, while the amplitude of the first resonance peak of the acceleration response PSD decreases with the structural stiffness decreased. It can be readily explained that the frequency has a more significant effect on the acceleration response PSD values of the plate than the displacement response PSD values. Moreover, the effect of elastically constrained boundaries on the auto-PSD curves of displacement and acceleration responses of the plate is also discussed in Fig. 12. Three kinds of cases are chosen as: 36

ku=kv=kw=kx=ky=5*1010N/m2 (Case 1); ku=kv=5*1010N/m2 and kw=kx=ky=1*107N/m2 (Case 2); ku=kv=5*1010N/m2 and kw=kx=ky=5*106N/m2 (Case 3). As expected, the resonance peaks of these curves shift to lower frequency range with the stiffness of

-5

-7

10

-9

10

-11

10

-13

10

-15

0

200

400 600 Frequency (Hz)

800

3

10

7

10

5

10

3

10

1

CCCC SSCC SSSS

2

10

CCCC SSCC SSSS

Acceleration PSD (m /s )

10

2

Displacement PSD (m *s)

elastically restrained boundaries decreased.

1000

10

-1

10

-3

10

-5

0

200

400 600 Frequency (Hz)

800

1000

Case1 Case2 Case3

10

-7

10

-9

10

-11

10

-13

10

-15

0

200

400 600 Frequency (Hz)

800

3

10

7

10

5

10

3

10

1

Case1 Case2 Case3

2

-5

Acceleration PSD (m /s )

10

2

Displacement PSD (m *s)

(a) Displacement response (b) Acceleration response Fig. 11. The stochastic responses of the composite laminated plate with different boundary conditions.

1000

10

-1

10

-3

10

-5

0

200

400 600 Frequency (Hz)

800

1000

(a) Displacement response (b) Acceleration response Fig. 12. The stochastic responses of the composite laminated plate with different elastically restrained boundaries.

The variations of auto-PSD curves of displacement and acceleration responses measured at the center of the [0/0/0/0/0] clamped composite laminated plate with different thermal loads are plotted in Fig. 13. Tcr denotes the critical thermal buckling temperature of the composite plate, which has been calculated in Table 7. Since the thermal loads will significantly soften the structural stiffness of the plate, the similar 37

phenomena can be observed in these figures, which can be described as the resonance peaks of these curves move to lower frequency range with temperature elevated. Also, it is found that the amplitude of the first resonance peak of the displacement response PSD increases with thermal loads increased, while the amplitude of the first resonance peak of the acceleration response PSD decreases with thermal loads increased. The reason can be summarized that the acceleration response PSD values are more

-7

10

-9

10

-11

10

-13

10

-15

0

200

400 600 Frequency (Hz)

800

10

7

10

5

10

3

10

1

deltaT=0 deltaT=0.5Tcr deltaT=0.8Tcr

3

10

deltaT=0 deltaT=0.5Tcr deltaT=0.8Tcr

2

-5

Acceleration PSD (m /s )

10

2

Displacement PSD (m *s)

sensitive to frequency than the displacement response PSD values.

1000

10

-1

10

-3

10

-5

0

200

400 600 Frequency (Hz)

800

1000

(a) Displacement response (b) Acceleration response Fig. 13. The stochastic responses of the composite laminated plate with different thermal loads.

The influence of the fiber orientation of the composite laminated plate on the stochastic response of the composite laminated plate is further discussed. The auto-PSD curves of displacement and acceleration responses measured at the center of the [  /  /  /  /  ] clamped composite laminated plate are calculated and illustrated in Fig. 14. Three kinds of fiber orientations, namely 0º, 30º and 45º, are taken into consideration here. Since the structural stiffness of the composite laminated plate decreases with  increased when  ranges from 0º to 45º in this case, the resonance peaks of these curves move to lower frequency range with the fiber orientation  elevated. 38

10

-8

10

-10

10

-12

10

-14

10

-16

0deg 30deg 45deg

0

200

400 600 Frequency (Hz)

800

10

7

10

5

10

3

10

1

0deg 30deg 45deg

3

10

-6

2

-4

Acceleration PSD (m /s )

2

Displacement PSD (m *s)

10

1000

10

-1

10

-3

10

-5

0

200

400 600 Frequency (Hz)

800

1000

(a) Displacement response (b) Acceleration response Fig. 14. The stochastic responses of the composite laminated plate with different fiber orientations.

Since the present study focuses on the stochastic dynamic behaviors of the plate before the flutter, the CFAP of the fully clamped composite laminated plate is calculated firstly. Based on the developed model, the variations of the first six order modes of the [0/0/0/0/0] clamped composite plate with aerodynamic pressure are shown in Fig. 15. Obviously, the non-dimensional CFAP cr of the clamped composite plate is 669. Meanwhile, the first four mode shapes of the composite plate without aerodynamic pressure and at flutter boundary are compared and shown in Fig. 16. Obviously, the sequence of modes of the composite laminated plate will be changed with the aerodynamic pressure elevated, which also can be found in Fig. 15. An interesting phenomenon is that the 2nd mode and 3rd mode will coincide when the aerodynamic pressure is equal to the CFAP, and these two modes evolve from the 1st mode and 4th mode of the plate without aerodynamic pressure. Besides, the mode shape is also significantly changed at the flutter boundary.

39

450

Natural frequency(Hz)

400 350 300 250 200 (1,1) (1,3) (2,2)

150 100

0

200

400



600

(1,2) (2,1) (2,3)

800

1000

Fig. 15. The variations of natural frequencies of the composite laminated plate with aerodynamic pressure.

1st mode

1st mode

2nd mode

2nd mode

3rd mode

3rd mode

4th mode

4th mode

 0

 cr

Fig. 16. The comparisons of the mode shapes of the composite laminated plate without and with aerodynamic pressure.

The effect of aerodynamic pressure on the random vibration of the composite laminated plate is further analyzed. Fig. 17 shows the RMS of the displacement response of the [0/0/0/0/0] clamped composite laminated plate with different aerodynamic pressures. Unlike the aforementioned cases, it can be found that the 40

location of the maximum RMS of the displacement response of the composite plate with aerodynamic pressure considered usually deviates from the center of the plate. Thus, it can be concluded that the aerodynamic pressure significantly affects the dynamic behaviors of the composite laminated plate. The auto-PSD curves of displacement and acceleration responses of the [0/0/0/0/0] clamped composite laminated plate at the point where the maximum RMS of displacement responses occurs are plotted in Fig. 18. From these curves, it can be found that some resonance peaks move to higher frequency range while others shift to lower frequency range with aerodynamic pressure increased. This phenomenon can be readily explained by the rise up of the natural frequencies of some modes and the drop down of those of other modes with the increase of aerodynamic pressure before flutter. Especially, it can be found that the amplitude of the first resonance peak of displacement and acceleration PSD curves significantly increases with the aerodynamic pressure increased.

(b)  =300

(a)  =0

41

-7

10

-9

10

-11

10

-13

10

-15

0

200

400 600 Frequency (Hz)

800

10

7

10

5

10

3

10

1

 =0  =300  =600

3

10

 =0  =300  =600

2

-5

Acceleration PSD (m /s )

10

2

Displacement PSD (m *s)

(c)  =600 Fig. 17. The RMS of displacement responses of the composite laminated plate with different aerodynamic pressures.

1000

10

-1

10

-3

10

-5

0

200

400 600 Frequency (Hz)

800

1000

(a) Displacement response (b) Acceleration response Fig. 18. The stochastic responses of the composite laminated plate with different aerodynamic pressures.

4. Conclusions This paper investigates the stationary/nonstationary stochastic responses of the heated composite laminated plate excited by random excitations in supersonic airflow, in which the random excitations can be one of concentrated excitation, distributed excitation and base acceleration excitation or any combination of them. The FSDT together with supersonic piston theory is used to derive the formulations and the governing equations of the system are obtained by using the Hamilton’s principle. The solutions for stochastic responses of the composite laminated plate with general 42

restraints are computed based on the modified Fourier method and PEM. A great number of computational and experimental cases are tested to validate the accuracy, efficiency and reliability of the proposed model. Finally, the influence of some key parameters such as boundary condition, thermal load, fiber orientation and aerodynamic pressure on the random vibration of the composite laminated plate is analyzed. It is demonstrated that the present model provides a high-efficiency and accurate way to investigate the stationary/nonstationary stochastic vibration problems of the heated composite laminated plate having general constraints in supersonic airflow.

Author Statement Kai Zhou: Conceptualization, Methodology, Formal analysis, Writing - Original Draft Zhen Ni: Software, Validation, Investigation Xiuchang Huang: Writing - Review & Editing, Resources, Data Curation Hongxing Hua: Supervision, Project administration, Funding acquisition

Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Conflict of interest statement There is no conflict of interest. 43

Acknowledgements The authors express their gratitude for the grants provided by the Open Fund of Defense Key Disciplines Laboratory of Ship Equipment Noise and Vibration Control Technology (VSN201801).

Appendix A The lamina stiffness coefficients Qij (i, j=1, 2, 4, 5 and 6) and thermal expansion k coefficients  pq (p, q=1 and 2) are obtained as 4 2 Q1k 1  Q k1 cos  k  (2Q k 124Q k ) cos  k sin 2 k  Q k sin2 2 4 k 1 66 2 4 4 Q1k 2  (Q k1 1 Q k 224Q k ) cos  k sin 2 k  Q k (cos 66 1 2  k  sin  k )

Q16k  (Q11k  Q12k  2Q66k ) cos3  k sin  k  (Q12k  Q22k  2Q66k ) cos  k sin 3  k Q22k  Q11k sin 4  k  Q22k cos 4  k  (2Q12k  4Q66k ) cos 2  k sin 2  k Q26k  (Q11k  Q12k  2Q66k ) cos  k sin 3  k  (Q12k  Q22k  2Q66k ) cos3  k sin  k (A-1) Q66k  Q66k (cos 4  k  sin 4  k )  (Q11k  2Q12k  Q22k  2Q66k ) cos 2  k sin 2  k Q44k  Q44k cos 2  k  Q55k sin 2  k Q45k  (Q55k  Q44k ) cos  k sin  k Q55k  Q55k cos 2  k  Q44k sin 2  k

11k  11k cos 2  k   22k sin 2  k  12k sin  k cos  k  22k  11k sin 2  k   22k cos 2  k  12k sin  k cos  k

(A-2)

12k  2(11k   22k )sin  k cos  k  12k (cos 2  k  sin 2  k )

Appendix B The strain energies Ue of the heated composite laminated plate are obtained as

44

u 2 v 2 u v u v 2    A11 ( x )  A22 ( y )  2 A12 ( x )( y )  A66 ( y  x )     2 A ( u  v )( u )  2 A ( u  v )( v )  K A ( w   ) 2  16 26 s 44 y   y x x y x y y     K A ( w   ) 2  2 K A ( w   )( w   )  x s 45 x y  s 55 x  x y    y 2  x 2  x  y  x  y 2     D11 ( x )  D22 ( y )  2 D12 ( x )( y )  D66 ( y  x )     x  y  x  x  y  y    2 D16 ( y  x )( x )  2 D26 ( y  x )( y )      u  x u  y v  x  2 B11 ( x )( x )  2 B12 ( x )( y )  2 B12 ( y )( x )      v  y u v  x  y )  2 B66 (  )(  )  2 B22 ( )(  y y y x y x       u v  u v 2 B16 (  )( x )  2 B26 (  )( y )   1 L1 L2  y x x y x y  Ue     dxdy 2 0 0   x  y u  x  y v  2 B16 (  )( )  2 B26 (  )( )   y x x y x y      v  w    T u 2  y 2 T 2 T 2 T T x 2   N x ( x )  N x ( x )  N x ( x )  M x ( x )  M x ( x )     2 B T ( u )(  x )  2 B T ( v )(  y )  x x   x x x x     N T ( u ) 2  N T ( v ) 2  N T ( w ) 2  M T (  x ) 2  M T (  y ) 2  y y y y  y y  y y y y    2 B T ( u )(  x )  2 B T ( v )(  y )  y y   y y y y    2 N T ( u )( u )  2 N T ( v )( v )  2 N T ( w )( w )  xy xy xy   x y x y x y    2 M T (  x )(  x )  2 M T (  y )(  y )  xy xy   x y x y    2 B T ( u )(  x )  2 B T ( u )(  x )  2 B T ( v )(  y )  2 B T ( v )(  y )  xy xy xy xy  x y y x x y y x  (B-1) where the stiffness coefficients Aij, Bij and Dij listed in Eq. (B-1) are defined as follows: 45

n

Aij   Qij ( zk 1  zk ), (i, j  1, 2, 4,5,6) , Bij  k 1

Dij 

1 n Qij ( zk21  zk2 ), (i, j  1, 2, 4,5,6) ,  2 k 1

1 n Qij ( zk31  zk3 ), (i, j  1, 2, 4,5,6)  3 k 1

and the stress resultants read n

( N xT , BxT , M xT )    k 1

n

zk 1

zk

(N , B , M )    T xy

T xy

T xy

k 1

n

 xTk (1, z, z 2 )dz , ( N Ty , BTy , M Ty )    k 1

zk 1

zk

zk 1

zk

2  Tk y (1, z, z )dz

and

 xyTk (1, z, z 2 )dz .

Appendix C The displacement components of the composite laminated plate are written as follows, where H is the Fourier cosine series combined with supplementary functions and ql (l=1,2,…, 5) represents the unknown Fourier coefficient eigenvector u ( x, y )  Hq1 v( x, y )  Hq 2 w( x, y )  Hq 3

(C-1)

 x ( x, y )  Hq 4  y ( x, y )  Hq 5 The Fourier cosine series and supplementary functions are given by

H  [H1 , H2 , H3 ] H1  [cos 0 x cos 0 y,…， cos m x cos n y,…， cos M x cos N y]

(C-2) (C-3)

H2  [1b ( y) cos 0 x,…, 1b ( y) cos M x, 2b ( y) cos 0 x，…, 2b ( y) cos M x] (C-4) H3  [1a ( x) cos 0 y,…，1a ( x) cos N y, 2a ( x) cos 0 y, …，2 a ( x) cos N y] (C-5) The unknown Fourier coefficient eigenvectors are expressed as q1  [q11 , q12 , q13 ]T , q 2  [q 21 , q 22 , q 23 ]T , q3  [q31 , q32 , q33 ]T q 4  [q 41 , q 42 , q 43 ]T , q5  [q51 , q52 , q53 ]T 46

(C-6)

q11  [ A00 ,…，Amn ,…，AMN ] 1 q12  [d101 ,…，d11m ,…，d11M , d 20 , …，d 21m , …，d 21M ]

(C-7)

q13  [ f ,…，f ,…，f , f , …，f , …，f ] 1 10

1 1n

1 1N

1 20

1 2n

1 2N

q 21  [ B00 ,…，Bmn ,…，BMN ] q 22  [d102 ,…，d12m , …，d12M , d 202 , …，d 22m , …，d 22M ]

(C-8)

q 23  [ f102 ,…，f12n ,…，f12N , f 202 , …，f 22n , …，f 22N ] q31  [C00 ,…，Cmn ,…，CMN ] 3 q32  [d103 ,…，d13m , …，d13M , d 20 , …，d 23m , …，d 23M ]

(C-9)

q33  [ f103 ,…，f13n ,…，f13N , f 203 , …，f 23n , …，f 23N ] q 41  [ D00 ,…，Dmn ,…，DMN ] q 42  [d104 ,…，d14m , …，d14M , d 204 , …，d 24m , …，d 24M ]

(C-10)

q 43  [ f ,…，f ,…，f , f , …，f , …，f ] 4 10

4 1n

4 1N

4 20

4 2n

4 2N

q51  [ E00 ,…，Emn ,…，EMN ] 5 q52  [d105 ,…，d15m , …，d15M , d 20 , …，d 25m , …，d 25M ]

(C-11)

q53  [ f105 ,…，f15n ,…，f15N , f 205 , …，f 25n , …，f 25N ] The detailed expressions in the stiffness and mass matrices of the heated composite laminated plate in supersonic airflow are listed as follows K uu K T  uv K 0  T  K ux T  K uy 

K uv K vv 0 T K vx T K vy

0 0 K ww K Twx K Twy

K ux K vx K wx K xx K Txy

K uy  K vy  K wy   K xy  K yy 

(C-12)

 H T H H T H H T H H T H   A11 ( x ) ( x )  A66 ( y ) ( y )  A16 ( y ) ( x )  A16 ( x ) ( y ) dxdy   L1 L2   H H H H H H H H       N xT ( )T ( )  N Ty ( )T ( )  N xyT ( ) T ( )  ( ) T ( )  dxdy 0 0 x x y y y y x    x 

K uu  

L1

0



L2

0

L1

  (kuy 0HT H 0

y 0

 kuyL2 HT H

L2

y  L2

)dx   ( kux 0HT H 0

x 0

 kuxL1 HT H

x  L1

)dy (C-13)

K uv  

L1

0



L2

0

 H T H H T H H T H H T H   A12 ( x ) ( y )  A66 ( y ) ( x )  A16 ( x ) ( x )  A26 ( y ) ( y ) dxdy   (C-14) 47

H H H H H H H H   B11 ( )T ( )  B66 ( )T ( )  B16 ( )T ( )  B16 ( )T ( )   L1 L2 x x y y y x x y dxdy K ux     0 0   BT ( H )T ( H )  BT ( H )T ( H )  BT ( H )T ( H )  BT ( H ) T ( H )  y xy xy  x x x y y x y y x   (C-15)

K uy  

L1

0



L2

0

 H T H H T H H T H H T H   B12 ( x ) ( y )  B66 ( y ) ( x )  B16 ( x ) ( x )  B26 ( y ) ( y ) dxdy   (C-16)

 H H H H H H H H  A22 ( )T ( )  A66 ( ) T ( )  A26 ( ) T ( )  A26 ( ) T ( ) dxdy  0 0 y y x x x y y x   L1 L2   H H H H H H H H       N xT ( )T ( )  N Ty ( )T ( )  N xyT ( ) T ( )  ( ) T ( )  dxdy 0 0 x x y y y y x    x 

K vv  

L1



L2

L1

  (kvy 0HT H 0

y 0

 kvyL2 HT H

L2

y  L2

)dx   ( kvx 0HT H 0

x 0

 kvxL1 HT H

x  L1

)dy (C-17)

K vx  

L1

0



L2

0

 H T H H T H H T H H T H   B12 ( y ) ( x )  B66 ( x ) ( y )  B16 ( x ) ( x )  B26 ( y ) ( y ) dxdy   (C-18)

H T H H T H H T H H T H   B ( ) ( )  B ( ) ( )  B ( ) ( )  B ( ) ( )  22 66 26 26 L1 L2  y y x x x y y x dxdy K vy     0 0   BT ( H )T ( H )  BT ( H )T ( H )  BT ( H )T ( H )  BT ( H ) T ( H )  y xy xy  x x x y y x y y x   (C-19)

48

 H H H H H H H H  K s A44 ( )T ( )  K s A55 ( )T ( )  K s A45 ( ) T ( )  K s A45 ( ) T ( ) dxdy  0 0 y y x x x y y x   L1 L2   H H H H H H H H       N xT ( )T ( )  N Ty ( )T ( )  N xyT ( ) T ( )  ( ) T ( )  dxdy 0 0 x x y y y y x    x 

K ww  

L1



L2

L1

  ( k wy 0HT H 0



L1

0



L2

0

y 0

 k wyL2 HT H

L2

y  L2

)dx   (k wx 0HT H 0

x 0

 k wxL1 H T H

x  L1

)dy

  D11 T H  T H  L3 ( H x cos air  H y sin air ) dxdy  1  (C-20)

K wx  

L1

K wy  

L1

0

0



L2



L2

0

0

  H T H T  K s A55 ( x ) (H)  K s A45 ( y ) (H) dxdy  

(C-21)

  H T H T  K s A44 ( y ) (H)  K s A45 ( x ) (H) dxdy  

(C-22)

H H H H   K s A55 ( H)T ( H)  D11 ( ) T ( )  D66 ( ) T ( )   L1 L2 x x y y dxdy K xx     0 0   D ( H )T ( H )  D ( H ) T ( H )  16  16 x   y  y  x   L1 L2   H H H H H H H H       M xT ( )T ( )  M Ty ( )T ( )  M xyT ( ) T ( )  ( ) T ( )  dxdy 0 0 x x y y y y x    x  L1

  ( k xy 0HT H 0

y 0

 k xyL2 HT H

L2

y  L2

)dx   (k xx 0HT H 0

x 0

 k xxL1 H T H

x  L1

)dy (C-23)

H H H H   K s A45 (H)T ( H)  D12 ( )T ( )  D66 ( )T ( )   L1 L2 x y y x dxdy K xy     0 0   D ( H )T ( H )  D ( H )T ( H )  26  16 x  x y y  

(C-24)

H H H H   K s A44 ( H)T ( H)  D22 ( )T ( )  D66 ( ) T ( )   L1 L2 y y x x dxdy K yy     0 0   D ( H )T ( H )  D ( H ) T ( H )  26  26 x  y y x   L1 L2   H H H H H H H H       M xT ( )T ( )  M Ty ( )T ( )  M xyT ( ) T ( )  ( ) T ( )  dxdy 0 0 x x y y y y x    x  L1

  ( k yy 0HT H 0

y 0

 k yyL2 HT H

L2

y  L2

)dx   ( k yx 0HT H 0

49

x 0

 k yxL1 H T H

x  L1

)dy

(C-25) M uu  0  M 0  T  M ux  0 

0 M vv 0 0 M Tvy

Muu  

L1

Mux  

L1

Mvv  

L1

Mvy  

L1

0



L2



L2



L2

0

0

0

L2

0

0

L1



0

0

M ww  

0 0 M ww 0 0

0



L2

0

M xx  

L1

M yy  

L1

0

0



L2



L2

0

0

M ux 0 0 M xx 0

0  M vy  0   0  M yy 

(C-26)

I 0 (H)T (H)dxdy

(C-27)

I1 (H)T (H)dxdy

(C-28)

I 0 (H)T (H)dxdy

(C-29)

I1 (H)T (H)dxdy

(C-30)

I 0 (H)T (H)dxdy

(C-31)

I 2 (H)T (H)dxdy

(C-32)

I 2 (H)T (H)dxdy

(C-33)

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Graphical Abstract

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