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FLOW-INDUCED VIBRATIONS OF SHEAR-DEFORMABLE LAMINATED PLATES EXPOSED TO AN OSCILLATING FLOW
Journal of Sound and
FLOW-INDUCED VIBRATIONS OF SHEAR-DEFORMABLE LAMINATED PLATES EXPOSED TO AN OSCILLATING FLOW L. ZHANG, H.-S. SHEN
AND
J. YANG
School of Civil Engineering and Mechanics, Shanghai Jiao ¹ong ;niversity, Shanghai 200030, People1s Republic of China. E-mail: [email protected] (Received 2 June 2000, and in ,nal form 6 November 2000) Using Reddy's higher order shear deformation plate theory, the #ow-induced vibrations of simply supported shear-deformable laminated plates exposed to an oscillating #ow are studied. The plate is assumed to be placed normal to the #ow. The #uid}structure interaction is based on Morison's model, which leads to a non-linear motion equation. The state variable approach is used in conjunction with the Spline collocation method to determine the dynamic response of the plate. Numerical illustrations concern the dynamic response of antisymmetric angle-ply and symmetric cross-ply laminated plates under oscillating #ow. The roles played by plate aspect ratio, total number of plies, "ber orientation, as well as transverse shear deformation are studied. In all cases, non-linear e!ects lead to rather complex vibrations and to essentially chaotic motion. 2001 Academic Press
1. INTRODUCTION
Vibration of plate structures induced by oscillating #ow occurs widely in practical engineering, for example, vibration of o!shore platform structures induced by ocean waves. Flow over blu! bodies causes the development of separating #ow and the vortex shedding behind the plate may be observed. The vortex patterns are somewhat di!erent from the ordinary KaH rmaH n vortex street observed behind a symmetric blu! body in a uniform #ow. Some numerical and experimental studies [1}5] on the prediction of #uid forces acting on and #ow patterns behind inclined and normal plates in oscillating #ow have already been reported. The laminated plate may vibrate in the transverse direction under the in-line #uid force, on placing the plate normal to the oscillating #ow. In spite of the practical importance of the problem, there is a need to have a better understanding of composite shear-deformable laminated plates exposed to an oscillating #ow. Many free vibration studies for composite laminated plates are available in the literature, see, e.g., references [6}16]. Numerous studies involving the application of the shear deformation plate theory to transient dynamic response of composite laminated plates can be found in references [17}20]. However, there is no literature dealing with the dynamic response of shear-deformable laminated plates exposed to an oscillating #ow. This is the problem studied in the present paper for the case when all four edges of the plate are assumed to be simply supported with no in-plane displacement. In the present study, the formulations are based on Reddy's higher order shear deformation plate theory [21]. The plate is assumed to be placed normal to the #ow. The 0022-460X/01/310029#16 $35.00/0
2001 Academic Press
30
L. ZHANG E¹ A¸.
#uid}structure interaction is based on Morison's model [22], which leads to a non-linear motion equation. The state variable approach (SVA) [18, 19] is used in conjunction with the Spline collocation method to determine the dynamic response of the plate. Numerical illustrations concern the dynamic responses of antisymmetric angle-ply and symmetric cross-ply laminated plates exposed to an oscillating #ow.
2. EQUATIONS OF MOTION
Consider a rectangular plate of length a, width b and thickness h, which consists of N plies. The plate is subjected to a transverse distributed load, q, caused by oscillating #ow. As usual, the co-ordinate system has its origin at the corner of the plate. Let u, v, w be the displacements parallel to the right-hand set of axes (x, y, z), where x is longitudinal and z is perpendicular to the plate. t and t are the rotations of transverse normals about the yV W and x-axis, respectively, at the mid-plane. Let F(x, y) be the stress function for the stress resultants, and denote di!erentiation by a comma, so that N "F N "F and V WW W VV N "!F . VW VW Attention is con"ned to the following two cases: (1) antisymmetric angle-ply laminated plates; and (2) symmetric cross-ply laminated plates, from which solutions for isotropic and orthotropic plates follow as a limiting case. The deduction of the governing equations associated with Reddy's higher order shear deformation plate theory (HSDPT) follows the same pattern in the case of its static counterpart [23, 24], so that the motion equations of shear-deformable laminated plates can be written as
*wK *wK *tG *tG V# W , # #I ¸ (w)!¸ (t )!¸ (t )#¸ (F)"q!I wK #I *x V W *x *y *y (1) *wK ¸ (w)#¸ (t )!¸ (t )#¸ (F)"I #I tG , V W *x V
(2)
*wK ¸ (w)!¸ (t )#¸ (t )#¸ (F)"I #I tG , V W *y W
(3)
¸ (F)#¸ (t )#¸ (t )!¸ (w)"0 V W
(4)
in which a superposed dot indicates di!erentiation with respect to time, and operators ¸ ( ) GH and I are de"ned in Appendix A. GH All four edges are assumed to be simply supported, the boundary conditions are x"0, a: u"w"t "0, W
F "M "P "0. VW V V
(5a)
v"w"t "0, V
F "M "P "0, VW W W
(5b)
y"0, b:
where M and M are the bending moments, and P and P are the higher order moments V W V W respectively.
31
FLOW-INDUCED VIBRATIONS OF LAMINATED PLATES
It is assumed that the plate is placed normal to the oscillating #ow and the velocity of the #uid is parallel to the z direction. As it has been shown by Morison [22], the in-line #uid force on the plate may be divided into two parts, one is drag force caused by unsteady velocity and the other is intertia force caused by acceleration or deceleration of the #uid. The #uid force per unit area is now de"ned by A A 1 q"q #q "o ;G (t)#c o (;G (t)!wK )# c o ";QM #;Q (t)!wR "(;QM #;Q (t)!wR ), ' " b ' b 2 " (6) where o is the density of #uid, c the coe$cient of added mass, c the drag coe$cient, ;Q (t) ' " the #uid velocity as a function of time, ;MQ the mean #uid velocity, and A the circular area whose diameter equals the width of the plate. For the oscillating #ow with zero mean, i.e. ;MQ "0, equation (6) becomes A A 1 ;G (t)!c o wK # c o ";Q (t)!wR "(;Q (t)!wR ) q"c o ' b K b 2 "
(7)
in which c is the inertia coe$cient and c "1#c . K K ' Substituting equation (7) into equation (1) yields A A ¸ (w)!¸ (t )!¸ (t )#¸ (F)"c o ;G (t)!c o wK V W K b ' b 1 # c o ";Q (t)!wR "(;Q (t)!wR )!I wK 2 " #I
*wK *wK *tG *tG V# W . # #I *x *y *x *y
(8)
It is noted that, if c is non-zero valued, equation (8) is a non-linear motion equation. " 3. SOLUTIONS PROCEDURE
The dynamic response of a simply supported shear-deformable laminated plate under oscillating #ow is now determined by using an analytical}numerical procedure. Firstly, the state variable approach (SVA) is used to solve equations (8), (2), (3) and (4), and we assume that w(x, y, t)" = (t) sin ax sin by, KL K L
(9a)
t (x, y, t)" X (t) cos ax sin by, V KL K L
(9b)
t (x, y, t)" > (t) sin ax cos by, W KL K L
(9c)
where = (t), X (t) and > (t) are unknown functions. KL KL KL
32
L. ZHANG E¹ A¸.
Substituting equation (9) into equation (4), the solution of stress function satisfying the boundary conditions of equation (5) can be obtained as F" [g W (t)#g X (t)#g > (t)] cos ax cos by, KL KL KL K L
(10)
where g , g and g are coe$cients de"ned in Appendix B. Substituting equations (9) and (10) into equation (8) yields [K = (t)#K X (t)#K > (t)] sin ax sin by" KL KL KL A G (t)#M XG (t)#M >G (t)] sin ax sin by c o ;G ![M = KL KL KL K b 1 Q (t) sin ax sin by"(;Q != Q (t) sin ax sin by). # c o ";Q != KL KL 2 "
(11)
Making use of Galerkin's orthogonality condition, we have G (t)#M XG (t)#M >G (t)#f (t)= Q (t) M = R (t)#a = KL KL KL KL KL #K = (t)#K X (t)#K > (t)"f (t)#f (t), KL KL KL
(12a)
where f (t), f (t), f (t) and a are de"ned in Appendix C. Note that now equation (12a) is a non-linear equation. Similarly, from equations (2) and (3), we have G (t)#M XG (t)#K = (t)#K X (t)#K > (t)"0, M = KL KL KL KL KL
(12b)
G (t)#M >G (t)#K = (t)#K X (t)#K > (t)"0. M = KL KL KL KL KL
(12c)
The [M ], [K ] given in equation (12) are also de"ned in Appendix C. GH GH Next, equation (12) can be solved by an increment method and can be written as M (t)DZG #C (t)DZ #K (t)DZ "DP(t) KL KL KL
(13)
where Z "+= , X , > ,2 is the displacement vector. M (t), K (t) and C (t) are mass, KL KL KL KL sti!ness and damping matrices, respectively, DP(t) is the load increment vector, and the details can be found in Appendix D. Let the time increment be very small; then in each time increase step M (t), K (t) and C (t) can be taken as a constant, and in such a case equation (13) can be solved by the Spline collocation method numerically. De"ne the displacement vector as a 3-D B-spline function
K> t!t H , (Z ) " C X P KL PH Dt H\
(14)
where Z (r"w, x, y) correspond to the displacements =, X, > respectively, X is the third P order B-spline function, t is any Spline node in the time domain, Dt is the time increase. H Substituting equation (14) into equation (13), and letting the residuals to equal zero, one can obtain LC #L C #L C #L C "DF(t ), G> G G\ G\ G
(15)
33
FLOW-INDUCED VIBRATIONS OF LAMINATED PLATES
where L"M (t )#C (t )Dt/2#K (t )(Dt)/6, G\ G\ G\ L "!3M (t )!C (t )Dt/2!K (t )(Dt)/2, G\ G\ G\ L "3M (t )!C (t )Dt/2!K (t )(Dt)/2, G\ G\ G\ L "!M (t )#C (t )Dt/2!K (t )(Dt)/6, G\ G\ G\ DF(t )"DP(t )(Dt) G G\
(16)
from equation (15), C"+C , C , C ,2 can easily be calculated with the initial condition. U V W Resubstituting C"+C , C , C ,2 into equation (14), as a result, = (t), X (t) and > (t) U V W KL KL KL can be obtained.
4. NUMERICAL EXAMPLES AND DISCUSSION
Dynamic and chaotic responses of simply supported shear-deformable laminated plates exposed to an oscillating #ow in the time domain were investigated. A program was developed for the purpose and many examples have been solved numerically, including the following.
4.1.
COMPARISON STUDIES
Four numerical examples for free vibration and transient response of simply supported antisymmetric angle-ply and symmetric cross-ply laminated plates are presented to show the accuracy and e$ciency of the present method. Example 1. To validate the present method, the fundamental natural frequency coe$cients -"(ua/h) (o/E of (h/-h/2) antisymmetric angle-ply laminated square plates are compared in Table 1 with the "nite element method (FEM) results given by Phan and Reddy [10], using their material properties, i.e., E /E "40, G /E "G /E "0)6, G /E "0)5 and l "0)25. TABLE 1 Fudamental frequency coe.cients -"(ua/h)(o/E for (h/-h/2) square plates h"30 a/h 4 10 20 50 100
Present Phan & Present Phan & Present Phan & Present Phan & Present Phan &
Reddy [10] Reddy [10] Reddy [10] Reddy [10] Reddy [10]
h"45
N"2
N"6
N"2
N"6
9)5011 9)4456 12)927 12)873 13)869 13)849 14)177 14)174 14)223 14)223
10)5818 10)577 18)176 18)170 21)650 21)648 23)068 23)067 23)295 23)295
9)8128 9)7594 13)311 13)263 14)264 14)246 14)575 14)572 14)621 14)621
10)899 10)895 19)029 19)025 22)879 22)877 24)480 24)480 24)739 24)739
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L. ZHANG E¹ A¸.
TABLE 2 ¹he fudamental frequency coe.cients -"(ua/h)(o/E for a (0/90) square plate (a/h"5) Q with di+erent values of E /E E /E
3-D [7]
HSDPT [13]
HSDPT [10]
HSDPT [12]
Present
10 20 30 40
8)2103 9)5603 10)272 10)752
8)2940 9)5439 10)284 10)794
8)2718 9)5623 10)272 10)787
8)2718 9)5623 10)272 10)787
8)2718 9)5623 10)272 10)787
Example 2. We consider now the fundamental natural frequency coe$cients of (0/90) Q symmetric cross-ply laminated square plates, which are compared in Table 2 with HSDPT results given by Phan and Reddy [10], Khdeir [12], Khdeir and Librescu [13] and three-dimensional elasticity solutions of Noor [7]. The material properties are the same as used in Example 1. Example 3. We now turn our attention to the transient response of a (0/90/0) cross-ply laminated square plate with b/h"5 under time-dependent sinusoidal distributed load de"ned as n n q"q F(t) sin x sin y, b a
F(t)"
sin(nt/t ) 0)t)t , 0 t't
(17)
in which t "0)006 s and q "68)9476 MPa. All plies are assumed to have the same thickness and plate thickness h"0)1524 m. The material properties are: E "172)369 GPa, E "6)895 GPa, G "G "3)448 GPa, G "1)379 GPa, l "0)25 and o"1603)03 kg/m. The variations of the central de#ection as functions of time are compared in Figure 1 with HSDPT results of Khdeir and Reddy [18]. Example 4. We now compare the variations of the central de#ection as functions of time for a ($45) antisymmetric angle-ply laminated square plate subjected to a suddenly 2 applied uniformly distributed load in Figure 2 with the results of Kant et al. [17], using their computing data: E /E "25, G /E "G /E "G /E "0)5, E "21 GPa, l "0)25, a"b"0)25 m, h"0)05 m, o"800 kg/m and q"0)1 MPa. The good agreement between the present results and referenced solutions of Tables 1 and 2, and Figures 1 and 2 reveals the high accuracy of the presented method.
4.2
PARAMETRIC STUDIES
A parametric study of antisymmetric angle-ply and symmetric cross-ply laminated plates exposed to an oscillating #ow was carried out. Under the present study, we assume that the #uid velocity has the form ;Q ";Q sin ut, the material properties are: E "203)00 GPa, K E "11)20 GPa, G "G "8)40 GPa, G "4)03 GPa, l "0)32 and o"1600 kg/m. All plies are assumed to have the same thickness and the total thickness of the plate is h"0)05 m. The #uid parameters adopted here are o "1000 kg/m, ;Q "20 m/s, K u"125 rad/s, c "1)61 and c "1)12. However, the analysis is equally applicable to other K " types of #uid parameters as well. Zero initial conditions are assumed. Typical results for central de#ection as functions of time are shown in Figures 3}7. Figure 3 shows central de#ection as a function of time for a (0/90) square plate (b/h"10) Q under oscillating #ow. It can be found that very complex vibration patterns occur. Due to
FLOW-INDUCED VIBRATIONS OF LAMINATED PLATES
35
Figure 1. Comparisons of transient response of a (0/90/0) square plate under sinusoidal distributed load: **, present; - - - - -, Khdeir & Reddy [18].
Figure 2. Comparisons of transient response of a ($45) square plate subjected to a suddenly applied uniformly 2 distributed load: **, present; - - - - , Kant et al. [17].
#uid damping, the vibration "rstly dissipates and then the response amplitude only varies slightly at longer times. Figure 4 shows the e!ect of width-to-thickness ratio on the dynamic response of a ($45 ) square plate subjected to oscillating #ow and the results are compared with their 2 classical countparts (CPT). It can be found that the transverse de#ection predicted by HSDPT is larger than that predicted by the CPT. It can also be seen that the transverse de#ection of the plate with b/h"10 is much greater than that of the plate with b/h"5. Figure 5 shows the e!ect of the total number of plies N on the dynamic response of antisymmetric angle-ply laminated square plates (b/h"10) exposed to oscillating #ow. It can be seen that the transverse de#ection decreases as the total number of plies N increases, but the e!ect is very small.
36
L. ZHANG E¹ A¸.
Figure 3. Central de#ection versus time for a (0/90) square plate exposed to oscillating #ow. Q
Figure 4. E!ect of transverse shear deformation on the dynamic response of a ($45 ) plate under oscillating 2 #ow: 1, HSDPT b/h"10; 2, CPT b/h"10; 3, HSDPT b/h"5; 4, CPT b/h"5.
Figure 6 compares the variations of the central de#ection as functions of time for ($45 ) 2 and ($30 ) antisymmetric angle-ply laminated square plates (b/h"10). The results show 2 that the transverse central de#ection of the ($45 ) plate is slightly lower than that of the 2 ($30 ) plate. 2 Figure 7 shows the e!ect of plate aspect ratio on the dynamic response of ($45 ) plates 2 with either a"b"0)5 m or a"0)75 m, b"0)5 m. It can be seen that the transverse de#ection of the rectangular plate is much higher than that of the square plate. Figures 8 and 9 show the phase plane diagrams of forced vibration induced by oscillating #ow for (0/90) and ($45 ) plates with a/h"b/h"10. It can be found that Q 2 irregular motion now occurs and the phase plane trajectories for these vibrations appear chaotic.
FLOW-INDUCED VIBRATIONS OF LAMINATED PLATES
37
Figure 5. E!ect of the total number of plies N on the dynamic response of antisymmetric angle-ply laminated plates under oscillating #ow: 1, ($45 ) ; 2, ($45 ) . 2 2
Figure 6. E!ect of "ber orientation on the dynamic response of antisymmetric angle-ply laminated plates under oscillating #ow: 1, ($45 ) ; 2, ($30 ) . 2 2
Figure 7. E!ect of aspect ratio on the dynamic response of ($45 ) plates under oscillating #ow: 1, a/b"1; 2, 2 a/b"1)5.
38
L. ZHANG E¹ A¸.
Figure 8. Phase plane trajectories for (0/90) plate (a/h"b/h"10) under oscillating #ow. Q
Figure 9. Phase plane trajectories for ($45 ) plate (a/h"b/h"10) under oscillating #ow. 2
5. CONCLUDING REMARKS
Dynamic and chaotic behaviors of a simply supported, shear-deformable laminated plate exposed to an oscillating #ow have been studied by using an analytical}numerical procedure. A number of issues related to the free vibration and transient response of antisymmetric angle-ply and symmetric cross-ply laminated plates have been examined. A parametric study of shear-deformable laminated plates in an oscillating #ow has been carried out and pertinent conclusions about the in#uence played in this respect by a number of geometrical and physical parameters have been outlined. The results show that very complex vibration patterns can occur and essentially chaotic motion can develop. They also con"rm that the characteristics of dynamic behavior are signi"cantly in#uenced by transverse shear deformation, plate aspect ratio and "ber orientation. In contrast, the total number of plies has less e!ect.
FLOW-INDUCED VIBRATIONS OF LAMINATED PLATES
39
REFERENCES 1. K. KUWAHARA 1973 Journal of Physics Society of Japan 35, 1545}1553. Numerical study of #ow past an inclined #at plate by an inviscid model. 2. T. SARPKAYA 1975 Journal of Fluid Mechanics 68, 109}128. An inviscid model of two-dimensional vortex shedding for transient and asymptotically steady separated #ow over an inclined plate. 3. M. KIYA and M. ARIE 1980 Bulletin of JSME 23, 1451}1459. Discrete-vortex simulation of unsteady separated #ow behind a nearly normal plate. 4. R. CHEIN and J. N. CHUNG 1988 Computers and Fluids 16, 405}427. Discrete-vortex simulation of #ow over inclined and normal plate. 5. A. OKAJIMA, T. MATSUMOTO and S. KIMURA 1998 JSME International Journal Series B 41, 214}220. Aerodynamic characteristics of #at plates with various angles of attack in oscillatory #ow. 6. S. SRINIVAS and A. K. RAO 1970 International Journal of Solids and Structures 6, 1463}1481. Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. 7. A. K. NOOR 1973 American Institute of Aeronautics and Astronautics Journal 11, 1038}1039. Free vibration of multilayered composite plates. 8. C. W. BERT and T. L. C. CHEN 1978 International Journal of Solids and Structures 14, 465}473. E!ect of shear deformation on vibration of antisymmetric angle-ply laminated rectangular plates. 9. J. N. REDDY and N. D. PHAN 1985 Journal of Sound and
40
L. ZHANG E¹ A¸.
APPENDIX A
In equations (1)}(4)
* * * 4 F* #(F * #F * #4F * ) #F * , ¸ ( )" *x*y *y 3h *x
4 * 4 * ¸ ( )" D* ! F* # (D* #2D* )! (F * #2F * ) , 3h *x *x*y 3h
4 * 4 * ¸ ( )" (D* #2D* )! (F * #2F * ) # D* ! F* , *x*y 3h *y 3h * * #(2B* !B* ) , ¸ ( )"(2B* !B* ) *x*y *x*y
* 8 16 ¸ ( )" A ! D # F h h *x 4 # 3h
* * 4 4 F *! H* # (F * #2F * )! (H* #2H* ) , 3h *x 3h *x*y
* 8 16 8 16 ¸ ( )" A ! D # F ! D* ! F* # H* h h 3h 9h *x
8 16 * ! D* ! F* # H* , 3h 9h *y
4 16 * (F * #F * #2F * )# (H* #H* ) , ¸ ( )"¸ ( )" (D* #D* )! 3h *x*y 9h
4 4 * * ¸ ( )"¸ ( )" (B* !B* )! (E * !E * ) # B* ! E* , 3h *x*y 3h *y
8 16 * ¸ ( )" A ! D # F h h *y 4 # 3h
4 * 4 * (F *#2F * )! (H * #2H * ) # F*! H* , 3h *x*y 3h *y
8 16 8 16 * ¸ ( )" A ! D # F ! D* ! F* # H* h h 3h 9h *x
8 16 * ! D* ! F* # H* , 3h 9h *y
4 * 4 * ¸ ( )"¸ ( )" B* ! E* # (B* !B* )! (E * !E * ) , 3h *x 3h *x*y
FLOW-INDUCED VIBRATIONS OF LAMINATED PLATES
41
* * * ¸ ( )"A* #(2A* #A* ) #A* , *x *x*y *y
* * 4 (2E* !E* ) #(2E* !E* ) , ¸ ( )" *x*y *x*y 3h
(A1)
where [A*], [B*], [D*], [E*], [F *] and [H*] (i, j"1, 2, 6) are reduced sti!ness matrices, GH GH GH GH GH GH de"ned as A*"A\,
B*"!A\B,
F*"F!EA\B,
D*"D!BA\B,
E*"!A\E,
H*"H!EA\E,
(A2)
where A , B , etc., are the plate sti!nesses, de"ned by GH GH FI (QM ) (1, z, z, z, z, z) dz (i, j"1, 2, 6), (A , B , D , E , F , H )" GH GH GH GH GH GH GH I I FI\
FI (A , D , F )" (QM ) (1, z, z) dz (i, j"4, 5), GH GH GH GH I I FI\
where QM are the transformed elastic constants, de"ned by GH
QM c 2cs s 4cs QM cs c#s cs !4cs QM s 2cs c 4cs " QM cs cs!cs !cs !2cs(c!s) QM cs cs!cs !cs 2cs(c!s) QM cs !2cs cs (c!s)
and
where
QM
QM
QM
E Q " , (1!l l )
c
" !cs s
s cs
c
Q Q
E Q " , (1!l l )
Q "G ,
Q Q Q Q
Q "G ,
,
(A3a)
(A3b)
(A4a)
(A4b)
l E Q " , (1!l l ) Q "G
(A4c)
in which E , E , G , G , G , l and l have their usual meanings, and c"cos h, s"sin h,
(A4d)
42
L. ZHANG E¹ A¸.
where h"lamination angle with respect to the plate x-axis. Also I "I ,
I "
4 4 4 4 I , I # IM ! I #IM 3h 3h 3h 3h I
I "
4 IM 4 ! I #IM IM #IM 3h I 3h
4 4 IM I , I " IM ! 3h 3h I I "I ,
I "
,
(IM ) !IM , I
I "I ,
(A5a)
where 4 I , IM "I ! 3h
4 IM "I ! I , 3h
8 16 IM "I ! I # I . 3h 9h
(A5b)
The inertias I (i"1}5, 7) are de"ned by G , FI oI (1, z, z, z, z, z) dz, (I , I , I , I , I , I )" I FI\
(A6)
where oI is the mass density of the kth layer.
APPENDIX B
In equation (10) (!4/3h) [(2E * !E * )ab#(2E * !E * ) ab] g " , A * a#(2A * #A * )ab#A * b [(B * !B * )!(4/3h) (E * !E * )]ab#[B * !(4/3h) E * )] b g " , A * a#(2A * #A * )ab#A * b [B * !(4/3h) E * ] a#[(B * !B * )!(4/3h) (E * !E * )] ab g " . A * a#(2A * #A * )ab#A * b
(B1)
APPENDIX C
In equation (12), for two cases of (;Q !wR )*0 and (;Q !wR )(0 f (t)"sgn o c ;Q , " 32oc ", a "!sgn 9mnn
16 f (t)" o hc ;G , mnn K
8 f (t)"sgn o c ;Q , mnn " (C1)
43
FLOW-INDUCED VIBRATIONS OF LAMINATED PLATES
where sgn is the signal function of (;Q !wR ), and 4 [F * a#(F * #F * #4F * )ab#F * b]![2B* !B* ]g ab K " 3h !(2B* !B* ) g ab,
4 4 D* ! F * a# (D* #2D* )! (F * #2F * ) ab 3h 3h
K "!
#(2B* !B* )abg #(2B* !B* )abg ,
4 (D* #2D* )! (F * #2F * ) ab 3h
K "!
4 # D* ! F * b#(2B* !B* )abg #(2B* !B* )abg , 3h
8 16 K " A ! D # F a h h 4 ! 3h
4 4 F* ! H * a# (F* #2F* )! (H * #2H * ) ab 3h 3h
4 4 # (B* !B* )! (E* !E* ) abg # B* ! E* bg , 3h 3h
8 16 8 16 K " A ! D # F # D* ! F* # H* a h h 3h 9h
8 16 4 # D* ! F* # H* b# (B* !B* )! (E* !E* ) abg 3h 9h 3h
4 E* bg , # B* ! 3h
4 16 K " (D* #D* )! (F * #F * #2F * )# (H* #H* ) ab 3h 9h
4 4 # (B* !B* )! (E* !E* ) abg # B* ! E* bg , 3h 3h
8 16 K " A ! D # F b h h 4 ! 3h
4 4 F * #2F * )! (H* #2H* ) ab# F * ! H* b 3h 3h
4 4 # B* ! E* ag # (B* !B* )! (E* !E* ) abg , 3h 3h
44
L. ZHANG E¹ A¸.
4 16 K " (D* #D* )! (F * #F * #2F * )# (H* #H* ) ab 3h 9h
4 4 # B* ! E* ag # (B* !B* )! (E* !E* ) abg , 3h 3h
8 16 8 16 K " A ! D # F # D* ! F* # H* a h h 3h 9h
8 16 4 # D* ! F* # H* b# B* ! E* ag 3h 9h 3h 4 # (B* !B* )! (E* !E* ) abg , 3h
M "I #
4 4 4 4 I (a#b)#o hc , I # IM ! I #IM ' 3h 3h 3h 3h I
M "
4 IM 4 ! I #IM IM #IM 3h I 3h
a,
M "
4 IM 4 ! I #IM IM #IM 3h I 3h
b,
4 IM I 4 ! M " IM a, 3h 3h I (IM ) M " IM ! , I
4 IM I 4 ! M " IM b, 3h I 3h M "M "0,
M "M .
(C2)
APPENDIX D
In equation (13)
M
M M " M M M 0 , M 0 M
K K K K " K K K , K K K
Q (t) 0 0 f (t)#2a = KL C (t)" 0 0 0 , 0
0 0
Q (t) fQ (t)Dt fQ (t)Dt#fQ (t)Dt!= KL DP(t)" 0 . 0
(D1)
FLOW-INDUCED VIBRATIONS OF SHEAR-DEFORMABLE LAMINATED PLATES EXPOSED TO AN OSCILLATING FLOW L. ZHANG, H.-S. SHEN
AND
J. YANG
School of Civil Engineering and Mechanics, Shanghai Jiao ¹ong ;niversity, Shanghai 200030, People1s Republic of China. E-mail: [email protected] (Received 2 June 2000, and in ,nal form 6 November 2000) Using Reddy's higher order shear deformation plate theory, the #ow-induced vibrations of simply supported shear-deformable laminated plates exposed to an oscillating #ow are studied. The plate is assumed to be placed normal to the #ow. The #uid}structure interaction is based on Morison's model, which leads to a non-linear motion equation. The state variable approach is used in conjunction with the Spline collocation method to determine the dynamic response of the plate. Numerical illustrations concern the dynamic response of antisymmetric angle-ply and symmetric cross-ply laminated plates under oscillating #ow. The roles played by plate aspect ratio, total number of plies, "ber orientation, as well as transverse shear deformation are studied. In all cases, non-linear e!ects lead to rather complex vibrations and to essentially chaotic motion. 2001 Academic Press
1. INTRODUCTION
Vibration of plate structures induced by oscillating #ow occurs widely in practical engineering, for example, vibration of o!shore platform structures induced by ocean waves. Flow over blu! bodies causes the development of separating #ow and the vortex shedding behind the plate may be observed. The vortex patterns are somewhat di!erent from the ordinary KaH rmaH n vortex street observed behind a symmetric blu! body in a uniform #ow. Some numerical and experimental studies [1}5] on the prediction of #uid forces acting on and #ow patterns behind inclined and normal plates in oscillating #ow have already been reported. The laminated plate may vibrate in the transverse direction under the in-line #uid force, on placing the plate normal to the oscillating #ow. In spite of the practical importance of the problem, there is a need to have a better understanding of composite shear-deformable laminated plates exposed to an oscillating #ow. Many free vibration studies for composite laminated plates are available in the literature, see, e.g., references [6}16]. Numerous studies involving the application of the shear deformation plate theory to transient dynamic response of composite laminated plates can be found in references [17}20]. However, there is no literature dealing with the dynamic response of shear-deformable laminated plates exposed to an oscillating #ow. This is the problem studied in the present paper for the case when all four edges of the plate are assumed to be simply supported with no in-plane displacement. In the present study, the formulations are based on Reddy's higher order shear deformation plate theory [21]. The plate is assumed to be placed normal to the #ow. The 0022-460X/01/310029#16 $35.00/0
2001 Academic Press
30
L. ZHANG E¹ A¸.
#uid}structure interaction is based on Morison's model [22], which leads to a non-linear motion equation. The state variable approach (SVA) [18, 19] is used in conjunction with the Spline collocation method to determine the dynamic response of the plate. Numerical illustrations concern the dynamic responses of antisymmetric angle-ply and symmetric cross-ply laminated plates exposed to an oscillating #ow.
2. EQUATIONS OF MOTION
Consider a rectangular plate of length a, width b and thickness h, which consists of N plies. The plate is subjected to a transverse distributed load, q, caused by oscillating #ow. As usual, the co-ordinate system has its origin at the corner of the plate. Let u, v, w be the displacements parallel to the right-hand set of axes (x, y, z), where x is longitudinal and z is perpendicular to the plate. t and t are the rotations of transverse normals about the yV W and x-axis, respectively, at the mid-plane. Let F(x, y) be the stress function for the stress resultants, and denote di!erentiation by a comma, so that N "F N "F and V WW W VV N "!F . VW VW Attention is con"ned to the following two cases: (1) antisymmetric angle-ply laminated plates; and (2) symmetric cross-ply laminated plates, from which solutions for isotropic and orthotropic plates follow as a limiting case. The deduction of the governing equations associated with Reddy's higher order shear deformation plate theory (HSDPT) follows the same pattern in the case of its static counterpart [23, 24], so that the motion equations of shear-deformable laminated plates can be written as
*wK *wK *tG *tG V# W , # #I ¸ (w)!¸ (t )!¸ (t )#¸ (F)"q!I wK #I *x V W *x *y *y (1) *wK ¸ (w)#¸ (t )!¸ (t )#¸ (F)"I #I tG , V W *x V
(2)
*wK ¸ (w)!¸ (t )#¸ (t )#¸ (F)"I #I tG , V W *y W
(3)
¸ (F)#¸ (t )#¸ (t )!¸ (w)"0 V W
(4)
in which a superposed dot indicates di!erentiation with respect to time, and operators ¸ ( ) GH and I are de"ned in Appendix A. GH All four edges are assumed to be simply supported, the boundary conditions are x"0, a: u"w"t "0, W
F "M "P "0. VW V V
(5a)
v"w"t "0, V
F "M "P "0, VW W W
(5b)
y"0, b:
where M and M are the bending moments, and P and P are the higher order moments V W V W respectively.
31
FLOW-INDUCED VIBRATIONS OF LAMINATED PLATES
It is assumed that the plate is placed normal to the oscillating #ow and the velocity of the #uid is parallel to the z direction. As it has been shown by Morison [22], the in-line #uid force on the plate may be divided into two parts, one is drag force caused by unsteady velocity and the other is intertia force caused by acceleration or deceleration of the #uid. The #uid force per unit area is now de"ned by A A 1 q"q #q "o ;G (t)#c o (;G (t)!wK )# c o ";QM #;Q (t)!wR "(;QM #;Q (t)!wR ), ' " b ' b 2 " (6) where o is the density of #uid, c the coe$cient of added mass, c the drag coe$cient, ;Q (t) ' " the #uid velocity as a function of time, ;MQ the mean #uid velocity, and A the circular area whose diameter equals the width of the plate. For the oscillating #ow with zero mean, i.e. ;MQ "0, equation (6) becomes A A 1 ;G (t)!c o wK # c o ";Q (t)!wR "(;Q (t)!wR ) q"c o ' b K b 2 "
(7)
in which c is the inertia coe$cient and c "1#c . K K ' Substituting equation (7) into equation (1) yields A A ¸ (w)!¸ (t )!¸ (t )#¸ (F)"c o ;G (t)!c o wK V W K b ' b 1 # c o ";Q (t)!wR "(;Q (t)!wR )!I wK 2 " #I
*wK *wK *tG *tG V# W . # #I *x *y *x *y
(8)
It is noted that, if c is non-zero valued, equation (8) is a non-linear motion equation. " 3. SOLUTIONS PROCEDURE
The dynamic response of a simply supported shear-deformable laminated plate under oscillating #ow is now determined by using an analytical}numerical procedure. Firstly, the state variable approach (SVA) is used to solve equations (8), (2), (3) and (4), and we assume that w(x, y, t)" = (t) sin ax sin by, KL K L
(9a)
t (x, y, t)" X (t) cos ax sin by, V KL K L
(9b)
t (x, y, t)" > (t) sin ax cos by, W KL K L
(9c)
where = (t), X (t) and > (t) are unknown functions. KL KL KL
32
L. ZHANG E¹ A¸.
Substituting equation (9) into equation (4), the solution of stress function satisfying the boundary conditions of equation (5) can be obtained as F" [g W (t)#g X (t)#g > (t)] cos ax cos by, KL KL KL K L
(10)
where g , g and g are coe$cients de"ned in Appendix B. Substituting equations (9) and (10) into equation (8) yields [K = (t)#K X (t)#K > (t)] sin ax sin by" KL KL KL A G (t)#M XG (t)#M >G (t)] sin ax sin by c o ;G ![M = KL KL KL K b 1 Q (t) sin ax sin by"(;Q != Q (t) sin ax sin by). # c o ";Q != KL KL 2 "
(11)
Making use of Galerkin's orthogonality condition, we have G (t)#M XG (t)#M >G (t)#f (t)= Q (t) M = R (t)#a = KL KL KL KL KL #K = (t)#K X (t)#K > (t)"f (t)#f (t), KL KL KL
(12a)
where f (t), f (t), f (t) and a are de"ned in Appendix C. Note that now equation (12a) is a non-linear equation. Similarly, from equations (2) and (3), we have G (t)#M XG (t)#K = (t)#K X (t)#K > (t)"0, M = KL KL KL KL KL
(12b)
G (t)#M >G (t)#K = (t)#K X (t)#K > (t)"0. M = KL KL KL KL KL
(12c)
The [M ], [K ] given in equation (12) are also de"ned in Appendix C. GH GH Next, equation (12) can be solved by an increment method and can be written as M (t)DZG #C (t)DZ #K (t)DZ "DP(t) KL KL KL
(13)
where Z "+= , X , > ,2 is the displacement vector. M (t), K (t) and C (t) are mass, KL KL KL KL sti!ness and damping matrices, respectively, DP(t) is the load increment vector, and the details can be found in Appendix D. Let the time increment be very small; then in each time increase step M (t), K (t) and C (t) can be taken as a constant, and in such a case equation (13) can be solved by the Spline collocation method numerically. De"ne the displacement vector as a 3-D B-spline function
K> t!t H , (Z ) " C X P KL PH Dt H\
(14)
where Z (r"w, x, y) correspond to the displacements =, X, > respectively, X is the third P order B-spline function, t is any Spline node in the time domain, Dt is the time increase. H Substituting equation (14) into equation (13), and letting the residuals to equal zero, one can obtain LC #L C #L C #L C "DF(t ), G> G G\ G\ G
(15)
33
FLOW-INDUCED VIBRATIONS OF LAMINATED PLATES
where L"M (t )#C (t )Dt/2#K (t )(Dt)/6, G\ G\ G\ L "!3M (t )!C (t )Dt/2!K (t )(Dt)/2, G\ G\ G\ L "3M (t )!C (t )Dt/2!K (t )(Dt)/2, G\ G\ G\ L "!M (t )#C (t )Dt/2!K (t )(Dt)/6, G\ G\ G\ DF(t )"DP(t )(Dt) G G\
(16)
from equation (15), C"+C , C , C ,2 can easily be calculated with the initial condition. U V W Resubstituting C"+C , C , C ,2 into equation (14), as a result, = (t), X (t) and > (t) U V W KL KL KL can be obtained.
4. NUMERICAL EXAMPLES AND DISCUSSION
Dynamic and chaotic responses of simply supported shear-deformable laminated plates exposed to an oscillating #ow in the time domain were investigated. A program was developed for the purpose and many examples have been solved numerically, including the following.
4.1.
COMPARISON STUDIES
Four numerical examples for free vibration and transient response of simply supported antisymmetric angle-ply and symmetric cross-ply laminated plates are presented to show the accuracy and e$ciency of the present method. Example 1. To validate the present method, the fundamental natural frequency coe$cients -"(ua/h) (o/E of (h/-h/2) antisymmetric angle-ply laminated square plates are compared in Table 1 with the "nite element method (FEM) results given by Phan and Reddy [10], using their material properties, i.e., E /E "40, G /E "G /E "0)6, G /E "0)5 and l "0)25. TABLE 1 Fudamental frequency coe.cients -"(ua/h)(o/E for (h/-h/2) square plates h"30 a/h 4 10 20 50 100
Present Phan & Present Phan & Present Phan & Present Phan & Present Phan &
Reddy [10] Reddy [10] Reddy [10] Reddy [10] Reddy [10]
h"45
N"2
N"6
N"2
N"6
9)5011 9)4456 12)927 12)873 13)869 13)849 14)177 14)174 14)223 14)223
10)5818 10)577 18)176 18)170 21)650 21)648 23)068 23)067 23)295 23)295
9)8128 9)7594 13)311 13)263 14)264 14)246 14)575 14)572 14)621 14)621
10)899 10)895 19)029 19)025 22)879 22)877 24)480 24)480 24)739 24)739
34
L. ZHANG E¹ A¸.
TABLE 2 ¹he fudamental frequency coe.cients -"(ua/h)(o/E for a (0/90) square plate (a/h"5) Q with di+erent values of E /E E /E
3-D [7]
HSDPT [13]
HSDPT [10]
HSDPT [12]
Present
10 20 30 40
8)2103 9)5603 10)272 10)752
8)2940 9)5439 10)284 10)794
8)2718 9)5623 10)272 10)787
8)2718 9)5623 10)272 10)787
8)2718 9)5623 10)272 10)787
Example 2. We consider now the fundamental natural frequency coe$cients of (0/90) Q symmetric cross-ply laminated square plates, which are compared in Table 2 with HSDPT results given by Phan and Reddy [10], Khdeir [12], Khdeir and Librescu [13] and three-dimensional elasticity solutions of Noor [7]. The material properties are the same as used in Example 1. Example 3. We now turn our attention to the transient response of a (0/90/0) cross-ply laminated square plate with b/h"5 under time-dependent sinusoidal distributed load de"ned as n n q"q F(t) sin x sin y, b a
F(t)"
sin(nt/t ) 0)t)t , 0 t't
(17)
in which t "0)006 s and q "68)9476 MPa. All plies are assumed to have the same thickness and plate thickness h"0)1524 m. The material properties are: E "172)369 GPa, E "6)895 GPa, G "G "3)448 GPa, G "1)379 GPa, l "0)25 and o"1603)03 kg/m. The variations of the central de#ection as functions of time are compared in Figure 1 with HSDPT results of Khdeir and Reddy [18]. Example 4. We now compare the variations of the central de#ection as functions of time for a ($45) antisymmetric angle-ply laminated square plate subjected to a suddenly 2 applied uniformly distributed load in Figure 2 with the results of Kant et al. [17], using their computing data: E /E "25, G /E "G /E "G /E "0)5, E "21 GPa, l "0)25, a"b"0)25 m, h"0)05 m, o"800 kg/m and q"0)1 MPa. The good agreement between the present results and referenced solutions of Tables 1 and 2, and Figures 1 and 2 reveals the high accuracy of the presented method.
4.2
PARAMETRIC STUDIES
A parametric study of antisymmetric angle-ply and symmetric cross-ply laminated plates exposed to an oscillating #ow was carried out. Under the present study, we assume that the #uid velocity has the form ;Q ";Q sin ut, the material properties are: E "203)00 GPa, K E "11)20 GPa, G "G "8)40 GPa, G "4)03 GPa, l "0)32 and o"1600 kg/m. All plies are assumed to have the same thickness and the total thickness of the plate is h"0)05 m. The #uid parameters adopted here are o "1000 kg/m, ;Q "20 m/s, K u"125 rad/s, c "1)61 and c "1)12. However, the analysis is equally applicable to other K " types of #uid parameters as well. Zero initial conditions are assumed. Typical results for central de#ection as functions of time are shown in Figures 3}7. Figure 3 shows central de#ection as a function of time for a (0/90) square plate (b/h"10) Q under oscillating #ow. It can be found that very complex vibration patterns occur. Due to
FLOW-INDUCED VIBRATIONS OF LAMINATED PLATES
35
Figure 1. Comparisons of transient response of a (0/90/0) square plate under sinusoidal distributed load: **, present; - - - - -, Khdeir & Reddy [18].
Figure 2. Comparisons of transient response of a ($45) square plate subjected to a suddenly applied uniformly 2 distributed load: **, present; - - - - , Kant et al. [17].
#uid damping, the vibration "rstly dissipates and then the response amplitude only varies slightly at longer times. Figure 4 shows the e!ect of width-to-thickness ratio on the dynamic response of a ($45 ) square plate subjected to oscillating #ow and the results are compared with their 2 classical countparts (CPT). It can be found that the transverse de#ection predicted by HSDPT is larger than that predicted by the CPT. It can also be seen that the transverse de#ection of the plate with b/h"10 is much greater than that of the plate with b/h"5. Figure 5 shows the e!ect of the total number of plies N on the dynamic response of antisymmetric angle-ply laminated square plates (b/h"10) exposed to oscillating #ow. It can be seen that the transverse de#ection decreases as the total number of plies N increases, but the e!ect is very small.
36
L. ZHANG E¹ A¸.
Figure 3. Central de#ection versus time for a (0/90) square plate exposed to oscillating #ow. Q
Figure 4. E!ect of transverse shear deformation on the dynamic response of a ($45 ) plate under oscillating 2 #ow: 1, HSDPT b/h"10; 2, CPT b/h"10; 3, HSDPT b/h"5; 4, CPT b/h"5.
Figure 6 compares the variations of the central de#ection as functions of time for ($45 ) 2 and ($30 ) antisymmetric angle-ply laminated square plates (b/h"10). The results show 2 that the transverse central de#ection of the ($45 ) plate is slightly lower than that of the 2 ($30 ) plate. 2 Figure 7 shows the e!ect of plate aspect ratio on the dynamic response of ($45 ) plates 2 with either a"b"0)5 m or a"0)75 m, b"0)5 m. It can be seen that the transverse de#ection of the rectangular plate is much higher than that of the square plate. Figures 8 and 9 show the phase plane diagrams of forced vibration induced by oscillating #ow for (0/90) and ($45 ) plates with a/h"b/h"10. It can be found that Q 2 irregular motion now occurs and the phase plane trajectories for these vibrations appear chaotic.
FLOW-INDUCED VIBRATIONS OF LAMINATED PLATES
37
Figure 5. E!ect of the total number of plies N on the dynamic response of antisymmetric angle-ply laminated plates under oscillating #ow: 1, ($45 ) ; 2, ($45 ) . 2 2
Figure 6. E!ect of "ber orientation on the dynamic response of antisymmetric angle-ply laminated plates under oscillating #ow: 1, ($45 ) ; 2, ($30 ) . 2 2
Figure 7. E!ect of aspect ratio on the dynamic response of ($45 ) plates under oscillating #ow: 1, a/b"1; 2, 2 a/b"1)5.
38
L. ZHANG E¹ A¸.
Figure 8. Phase plane trajectories for (0/90) plate (a/h"b/h"10) under oscillating #ow. Q
Figure 9. Phase plane trajectories for ($45 ) plate (a/h"b/h"10) under oscillating #ow. 2
5. CONCLUDING REMARKS
Dynamic and chaotic behaviors of a simply supported, shear-deformable laminated plate exposed to an oscillating #ow have been studied by using an analytical}numerical procedure. A number of issues related to the free vibration and transient response of antisymmetric angle-ply and symmetric cross-ply laminated plates have been examined. A parametric study of shear-deformable laminated plates in an oscillating #ow has been carried out and pertinent conclusions about the in#uence played in this respect by a number of geometrical and physical parameters have been outlined. The results show that very complex vibration patterns can occur and essentially chaotic motion can develop. They also con"rm that the characteristics of dynamic behavior are signi"cantly in#uenced by transverse shear deformation, plate aspect ratio and "ber orientation. In contrast, the total number of plies has less e!ect.
FLOW-INDUCED VIBRATIONS OF LAMINATED PLATES
39
REFERENCES 1. K. KUWAHARA 1973 Journal of Physics Society of Japan 35, 1545}1553. Numerical study of #ow past an inclined #at plate by an inviscid model. 2. T. SARPKAYA 1975 Journal of Fluid Mechanics 68, 109}128. An inviscid model of two-dimensional vortex shedding for transient and asymptotically steady separated #ow over an inclined plate. 3. M. KIYA and M. ARIE 1980 Bulletin of JSME 23, 1451}1459. Discrete-vortex simulation of unsteady separated #ow behind a nearly normal plate. 4. R. CHEIN and J. N. CHUNG 1988 Computers and Fluids 16, 405}427. Discrete-vortex simulation of #ow over inclined and normal plate. 5. A. OKAJIMA, T. MATSUMOTO and S. KIMURA 1998 JSME International Journal Series B 41, 214}220. Aerodynamic characteristics of #at plates with various angles of attack in oscillatory #ow. 6. S. SRINIVAS and A. K. RAO 1970 International Journal of Solids and Structures 6, 1463}1481. Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. 7. A. K. NOOR 1973 American Institute of Aeronautics and Astronautics Journal 11, 1038}1039. Free vibration of multilayered composite plates. 8. C. W. BERT and T. L. C. CHEN 1978 International Journal of Solids and Structures 14, 465}473. E!ect of shear deformation on vibration of antisymmetric angle-ply laminated rectangular plates. 9. J. N. REDDY and N. D. PHAN 1985 Journal of Sound and
40
L. ZHANG E¹ A¸.
APPENDIX A
In equations (1)}(4)
* * * 4 F* #(F * #F * #4F * ) #F * , ¸ ( )" *x*y *y 3h *x
4 * 4 * ¸ ( )" D* ! F* # (D* #2D* )! (F * #2F * ) , 3h *x *x*y 3h
4 * 4 * ¸ ( )" (D* #2D* )! (F * #2F * ) # D* ! F* , *x*y 3h *y 3h * * #(2B* !B* ) , ¸ ( )"(2B* !B* ) *x*y *x*y
* 8 16 ¸ ( )" A ! D # F h h *x 4 # 3h
* * 4 4 F *! H* # (F * #2F * )! (H* #2H* ) , 3h *x 3h *x*y
* 8 16 8 16 ¸ ( )" A ! D # F ! D* ! F* # H* h h 3h 9h *x
8 16 * ! D* ! F* # H* , 3h 9h *y
4 16 * (F * #F * #2F * )# (H* #H* ) , ¸ ( )"¸ ( )" (D* #D* )! 3h *x*y 9h
4 4 * * ¸ ( )"¸ ( )" (B* !B* )! (E * !E * ) # B* ! E* , 3h *x*y 3h *y
8 16 * ¸ ( )" A ! D # F h h *y 4 # 3h
4 * 4 * (F *#2F * )! (H * #2H * ) # F*! H* , 3h *x*y 3h *y
8 16 8 16 * ¸ ( )" A ! D # F ! D* ! F* # H* h h 3h 9h *x
8 16 * ! D* ! F* # H* , 3h 9h *y
4 * 4 * ¸ ( )"¸ ( )" B* ! E* # (B* !B* )! (E * !E * ) , 3h *x 3h *x*y
FLOW-INDUCED VIBRATIONS OF LAMINATED PLATES
41
* * * ¸ ( )"A* #(2A* #A* ) #A* , *x *x*y *y
* * 4 (2E* !E* ) #(2E* !E* ) , ¸ ( )" *x*y *x*y 3h
(A1)
where [A*], [B*], [D*], [E*], [F *] and [H*] (i, j"1, 2, 6) are reduced sti!ness matrices, GH GH GH GH GH GH de"ned as A*"A\,
B*"!A\B,
F*"F!EA\B,
D*"D!BA\B,
E*"!A\E,
H*"H!EA\E,
(A2)
where A , B , etc., are the plate sti!nesses, de"ned by GH GH FI (QM ) (1, z, z, z, z, z) dz (i, j"1, 2, 6), (A , B , D , E , F , H )" GH GH GH GH GH GH GH I I FI\
FI (A , D , F )" (QM ) (1, z, z) dz (i, j"4, 5), GH GH GH GH I I FI\
where QM are the transformed elastic constants, de"ned by GH
QM c 2cs s 4cs QM cs c#s cs !4cs QM s 2cs c 4cs " QM cs cs!cs !cs !2cs(c!s) QM cs cs!cs !cs 2cs(c!s) QM cs !2cs cs (c!s)
and
where
QM
QM
QM
E Q " , (1!l l )
c
" !cs s
s cs
c
Q Q
E Q " , (1!l l )
Q "G ,
Q Q Q Q
Q "G ,
,
(A3a)
(A3b)
(A4a)
(A4b)
l E Q " , (1!l l ) Q "G
(A4c)
in which E , E , G , G , G , l and l have their usual meanings, and c"cos h, s"sin h,
(A4d)
42
L. ZHANG E¹ A¸.
where h"lamination angle with respect to the plate x-axis. Also I "I ,
I "
4 4 4 4 I , I # IM ! I #IM 3h 3h 3h 3h I
I "
4 IM 4 ! I #IM IM #IM 3h I 3h
4 4 IM I , I " IM ! 3h 3h I I "I ,
I "
,
(IM ) !IM , I
I "I ,
(A5a)
where 4 I , IM "I ! 3h
4 IM "I ! I , 3h
8 16 IM "I ! I # I . 3h 9h
(A5b)
The inertias I (i"1}5, 7) are de"ned by G , FI oI (1, z, z, z, z, z) dz, (I , I , I , I , I , I )" I FI\
(A6)
where oI is the mass density of the kth layer.
APPENDIX B
In equation (10) (!4/3h) [(2E * !E * )ab#(2E * !E * ) ab] g " , A * a#(2A * #A * )ab#A * b [(B * !B * )!(4/3h) (E * !E * )]ab#[B * !(4/3h) E * )] b g " , A * a#(2A * #A * )ab#A * b [B * !(4/3h) E * ] a#[(B * !B * )!(4/3h) (E * !E * )] ab g " . A * a#(2A * #A * )ab#A * b
(B1)
APPENDIX C
In equation (12), for two cases of (;Q !wR )*0 and (;Q !wR )(0 f (t)"sgn o c ;Q , " 32oc ", a "!sgn 9mnn
16 f (t)" o hc ;G , mnn K
8 f (t)"sgn o c ;Q , mnn " (C1)
43
FLOW-INDUCED VIBRATIONS OF LAMINATED PLATES
where sgn is the signal function of (;Q !wR ), and 4 [F * a#(F * #F * #4F * )ab#F * b]![2B* !B* ]g ab K " 3h !(2B* !B* ) g ab,
4 4 D* ! F * a# (D* #2D* )! (F * #2F * ) ab 3h 3h
K "!
#(2B* !B* )abg #(2B* !B* )abg ,
4 (D* #2D* )! (F * #2F * ) ab 3h
K "!
4 # D* ! F * b#(2B* !B* )abg #(2B* !B* )abg , 3h
8 16 K " A ! D # F a h h 4 ! 3h
4 4 F* ! H * a# (F* #2F* )! (H * #2H * ) ab 3h 3h
4 4 # (B* !B* )! (E* !E* ) abg # B* ! E* bg , 3h 3h
8 16 8 16 K " A ! D # F # D* ! F* # H* a h h 3h 9h
8 16 4 # D* ! F* # H* b# (B* !B* )! (E* !E* ) abg 3h 9h 3h
4 E* bg , # B* ! 3h
4 16 K " (D* #D* )! (F * #F * #2F * )# (H* #H* ) ab 3h 9h
4 4 # (B* !B* )! (E* !E* ) abg # B* ! E* bg , 3h 3h
8 16 K " A ! D # F b h h 4 ! 3h
4 4 F * #2F * )! (H* #2H* ) ab# F * ! H* b 3h 3h
4 4 # B* ! E* ag # (B* !B* )! (E* !E* ) abg , 3h 3h
44
L. ZHANG E¹ A¸.
4 16 K " (D* #D* )! (F * #F * #2F * )# (H* #H* ) ab 3h 9h
4 4 # B* ! E* ag # (B* !B* )! (E* !E* ) abg , 3h 3h
8 16 8 16 K " A ! D # F # D* ! F* # H* a h h 3h 9h
8 16 4 # D* ! F* # H* b# B* ! E* ag 3h 9h 3h 4 # (B* !B* )! (E* !E* ) abg , 3h
M "I #
4 4 4 4 I (a#b)#o hc , I # IM ! I #IM ' 3h 3h 3h 3h I
M "
4 IM 4 ! I #IM IM #IM 3h I 3h
a,
M "
4 IM 4 ! I #IM IM #IM 3h I 3h
b,
4 IM I 4 ! M " IM a, 3h 3h I (IM ) M " IM ! , I
4 IM I 4 ! M " IM b, 3h I 3h M "M "0,
M "M .
(C2)
APPENDIX D
In equation (13)
M
M M " M M M 0 , M 0 M
K K K K " K K K , K K K
Q (t) 0 0 f (t)#2a = KL C (t)" 0 0 0 , 0
0 0
Q (t) fQ (t)Dt fQ (t)Dt#fQ (t)Dt!= KL DP(t)" 0 . 0
(D1)