A Rayleigh-Ritz approach for the estimation of the dynamic properties of symmetric composite plates with general boundary conditions

Composites Science and Technology 53 (1995) 289-299 0 1995 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0266-3538/95/$09.50

0266-3538(95)00002-X

ELSEVIER

A RAYLEIGH-RITZ APPROACH FOR THE ESTIMATION OF THE DYNAMIC PROPERTIES OF SYMMETRIC COMPOSITE PLATES WITH GENERAL BOUNDARY CONDITIONS A. Al-Obeid Scientific Studies and Research Centre, Damascus, Syria

J. E. Cooper* School of Engineering,

University of Manchester,

Oxford Road, Manchester,

UK, Ml3 9PL

(Received 30 July 1993; accepted 25 November 1994) Abstract A Ritz method using polynomial functions has been developed to analyse the vibration properties of composite laminate plates for boundary conditions that have not been analysed previously by means of such an approach. An investigation is made into the effect of ply orientation, orthotropic ratio and stacking sequence on the natural frequencies and mode shapes for diflerent boundary conditions. The results are compared with finite element (FE) analysis and other analytical techniques. The results show a good agreement with the FE analysis and better convergence characteristics than other analytical methods.

employ different versions of the latter technique and these have dealt with a variety of boundary conditions.4’5 However, not all possible boundary conditions have been considered and there has also been the problem of obtaining convergence of the solution when the number of terms in the series solution is increased. The classical beam solution has often been used, but requires fairly complicated techniques if it is to be employed so as to ensure convergence and it is not reliable for all boundary conditions.6 The Rayleigh-Ritz approaches are suitable for the symmetric balanced and unbalanced cases. However, the amount of computation required becomes prohibitive for the asymmetric case where the finite element technique should be used. In this paper a more general shape function is introduced to the Rayleigh-Ritz technique that enables some previously unsolved composite plate boundary conditions to be computed. It is demonstrated that the solution possesses good numerical characteristics and convergence properties. The technique is compared for a number of boundary conditions and ply orientations with results from other continuous series approaches and also finite element studies. The effects of different orthotropic properties and stacking sequence are also considered.

Keywords: composite

plates, free vibration, RayleighRitz method, mode shapes 1 INTRODUCTION

The use of composite materials in place of more traditional isotropic materials has increased dramatically over the past decade in areas such as the aerospace industry. It has therefore been of interest to determine the vibration properties of structures such as plates made up of varying orientations of anisotropic materials. Although a large number of publications exist which investigate this problem, they are restricted at present to certain boundary conditions. ‘-’ It is usual to approach the problem by using energy principles either with a discretisation (finite element) or continuous series (Rayleigh-Ritz) type of technique. A number of studies have been made which

2 MATHEMATICAL

Considering the composite plate shown in Fig. 1, it is well known for the symmetric and isotropic case that the kinetic and strain energies can be written as: T = +pw2

* TO whom correspondence

MODEL

should be addressed.

II 4

289

W*(x, y> dx dy

(1)

A, Al-Obeid, J. E. Cooper

290 t

4

the middle surfaces of the plate are written as:

I

2

I

K,=Middle surface /

t

Fig. 1. Geometry

a2w ay2

-2 a2w K.r_v = ___ ax ay

t

Layer number

ax2

KY = -

and

? N

-a2w

(3)

(4)

(5)

It is assumed that the displacement in the axial direction of any point on the plate is a sinusoidal function of time, such that w(x, y, t) = W(x, y) sin wt. A summation of polynomial products’ will be used rather than the Ashton approach’ as it is arguably simpler. With the non-dimensional coordinate system shown in Fig. 2, the displacement function becomes:

of an N-layered laminate.

where p is the mass per unit area and

A

.2 2 +ZD,,*‘” ax* ay2

+4

!

D,$+.,,,

+4D,---

a2w a2w 1 ax ay

a2W

ax ay I

(2)

h dy

The D;j terms are the conventional laminate stiffness coefficients7 which allow for changes in the orientation of each laminate. The bending curvature changes of

where &&‘) and q,(v) are polynomial functions depending upon the boundary conditions of the plate and 5 = xla - t&, r] = y/b - v. as defined in Fig. 2. The A,, terms are amplitude coefficients that need to be found. One advantage of using such a coordinate conditioning system is that possible numerical problems do not occur. Table 1 shows the combinations of polynomial displacement functions for boundary conditions that have not been considered elsewhere in the literature. A similar approach can be used for other boundary conditions by using the same type of displacement functions. The shape functions described by eqn (6) are then substituted into the kinetic energy and strain energy expressions. Minimisation of the difference between the maximum kinetic energy and maximum strain energy with respect to each of the A,,, coefficients leads to: (7)

where m = m,,, m,, + 1, . . . , A4 and II = n,,, 1, . . . , N. These equations take the form (K - PM)c = 0 Fig. 2.

Non-dimensional coordinates for the plate.

Boundary conditions [“(& - 1)‘~“~

CFCF CSCF CCCF

[‘(5 - l)‘[“-’ [‘(LJ- l)‘[“-’ (‘(5 - l)*em-’

(8)

where K and M are stiffness and mass matrices and c is the vector of unknown coefficients.

Table 1. Values of m,, n,,b and qa aud the corresponding function in the x aud y directions for various boundary conditions

cccc

no +

7j2(r/- 1)2n”~’ V 77” 17”

1

1

0

0

1

1

0 1

0 0

0.5 0

1

2

0

0

Dynamic properties of symmetric composite plates

291

Table 2. Material properties for E-glass/epoxy and graphite/epoxy

Material E-glass/epoxy

Graphite/epoxy

E,(GPa)

WGPa) 60.7

24.8

138

8.96

The (m - m,, + l)(n - n,, + 1) simultaneous linear can be solved as an homogeneous equations eigenproblem, where the eigenvalues, A, are the non-dimensional frequency parameter defined as: A = wa’(p/D,$

(9)

with D,, = Elh3/12(1 - Y,~Y~,)

(10)

Hence it is possible to estimate the natural frequencies and mode shapes of the composite plates for any combination of layers and orientations. 3 APPLICATION

OF THE METHOD

The approach described above has been applied to a number of different composite plates. The material properties used in this study are given in Table 2 and are taken from previous work.’ The convergence characteristics of the solution have been investigated by calculating frequencies and mode shapes for a varying number of terms in the polynomial shape functions. The investigation can be categorised into two different sections: clamped (i) composite plates with completely boundary conditions that have been analysed using other Rayleigh-Ritz type techniques and finite element analysis; (ii) composite plates with boundary conditions that have not been analysed before using the Table 3. Convergence

Boundary condition

&(GPa)

VI2

E,IE2

12.0

0.23

7.1

0.30

2.45

15,4

Rayleigh-Ritz techniques. Results for the CSCF, CCCF, and CFCF conditions are compared with finite element calculations (where F, S and C represent the free, and clamped conditions, simply-supported respectively). In both cases a variety orientations are examined.

of

different

laminate

4 RESULTS A convergence study has been made for symmetrically laminated square plates of four layers with stacking sequence (0, - 8, - 0, 0). Two materials were considered, namely graphite/epoxy (G/E) and Eglass/epoxy (E/E). Convergence studies of the lowest eight frequency parameters for graphite/epoxy with four different boundary conditions and a lay up of (30, -30, -30,30) are given in Table 3 and for E-glass/epoxy in Table 4. This study indicates that the results are reasonably accurate for engineering applications. It was seen that the maximum difference between the 49-term and 64-term solutions is less than 3% for all boundary conditions. The poorest convergence is observed for the plate with three clamped edges (i.e. CCCF plates) for the G/E material. The fastest convergence for both materials was observed for the plate with fully clamped edges (i.e. CCCC plates). The

of the frequency parameter, A, for graphite/epoxy (3% -30, -WW

No. of terms

square plates with stacking sequence

h Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Mode 7

Mode 8

cccc

36 49 64

22.07 22.07 22.07

34.86 34.85 34.85

53.04 53.03 53.03

54.87 53.83 53.83

68.01 67.95 67.95

77.53 77.39 77.25

92.29 92.15 91.80

98.82 98.62 98.59

CFCF

36 49 64

16.43 16.43 16.42

16.32 18.27 18.26

26.42 26.15 26.10

40-03 39.83 38-85

46.25 46.21 46.07

48.06 47.74 47.65

59.67 59.42 59.05

70.86 62.14 60.98

CSCF

36 49 64

17.05 17.05 17.04

22.89 22.83 22.76

34.36 34.11 34.09

46.38 46.28 46.20

54.35 52.82 51.88

55.05 54.82 54.79

70.03 69.62 69.17

84.86 76.62 76.29

CCCF

36 49 64

17.14 17.14 17.14

24.12 24.10 24.10

36.99 36.98 36.96

46.52 46.47 46.45

55.93 55.83 55.81

58.08 56.19 56.13

73.11 71.88 71.80

85.03 82.11 79.76

292

A. Al-Obeid,

Table 4. Convergence Boundary condition

of the frequency

J. E. Cooper

parameter, A, for E-@ass/epoxy (30, -30, -3&W

No. of terms

square

plates

with

stacking

sequence

A Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Mode 7

Mode 8

cccc

36 49 64

28.70 28.70 28.70

53-69 53.69 53.69

62.81 62.81 62.81

85.20 85.21 85.19

95.37 95.24 95.24

114.58 114.39 114.39

123.69 122.97 122.96

139.40 139.22 139.18

CFCF

36 49 64

18.63 18.62 18.62

21.96 21.96 21.94

34.45 34.06 34.06

51.41 51.38 51.36

56.12 56.09 56.06

59.48 59.34 57.72

7244 71.72 71.58

97.97 96.19 93.95

CSCF

36 49 64

19.57 19.56 19.56

28.26 28-26 28.26

48.99 48.62 48.61

52.82 52.81 52.80

63.81 63.77 63.77

83.23 81.85 80.81

90.10 88.69 87.63

102.63 102.46 102.43

CCCF

36 49 64

19.90 19.90 19.89

30.99 30.99 30.99

52.93 52.92 52.91

54.79 54.78 54.77

65.93 65.92 65.91

8844 87.72 87.69

99.22 95.27 95.27

102.88 102.66 102.65

Table 5. Frequency

parameter,

A, for graphite/epoxy

square four-layer plates with stacking sequence

Boundary condition

Present study cccc

Ref. 9 cccc

Present study CFCF

Present study CSCF

Present study CCCF

(0, -0, -8, @)

A Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Mode 7

Mode 8

0 15 30 45

23.69 23.13 22.07 21.59

29.59 30.78 34.85 3844

4144 44.67 53.03 47.75

59.82 59.68 53.83 59.71

62.54 64.68 67.95 74.82

67.01 66.75 77.25 83.83

75.94 79.63 92.80 84.76

84.47 89.44 98.59 102.62

0 15 30 45

23.86 23.29 22.24 21.75

29.71 30.99 35.09 39.20

41.73 44.98 53.40 48.07

60.24 60.09 54.21 60.12

62.93 65.23 68.42 75.33

67.45 67.18 77.91 84.43

76.46 80.14 92.76 85.20

85.06 90.11 99.29 104.20

0 1.5 30 45 60 75 90

22.21 20.53 16.42 11.64 7.84 6.02 5.66

22.85 21.34 18.26 15.14 12.02 9.15 7.81

25.47 25.50 26.10 26.19 21.87 1664 15.59

32.52 34.83 39.85 32.56 24.33 20.80 18.77

46.17 50.61 46.07 37.89 30.17 26.30 25.43

61.17 56.80 47.65 44.33 41.29 32.72 30.43

62.09 58.13 59.09 56.27 43.42 35.35 34.17

65.01 62.90 60.98 63.66 51.31 40.62 34.66

0 15 30 45 60 75 90

22.37 20.79 17.04 12.81 9.22 6.98 6.27

24.03 23.58 22.76 21.66 19.70 17.43 1644

29.08 30.51 34.09 34.28 24.54 19.49 18.33

40.19 43.73 46.20 37.45 35.07 29.62 27.22

60.73 57-06 51.88 48.65 44.97 34.71 31.50

61.45 6048 54.79 58.47 48.85 45.67 41.62

63.35 66.89 69.17 66.81 54.31 50.29 51.75

67.92 68.24 76.29 74.09 67.22 56.33 51.83

0 15 30 45 60 75 90

22.43 20.83 17.14 13.12 9.90 7.99 7.40

24.58 24,27 24.10 24.01 22-43 18.47 17.10

30.76 32.43 36.96 34.68 26.26 24.51 24.35

42.73 46.29 46.45 41.28 38.28 32.99 31.97

61.48 57.81 55.81 51.71 46-63 36.25 32.14

63.70 60.99 56.13 62.38 55.66 49.31 45.55

66.16 69.26 71.80 67.60 58.11 57.06 52.18

68.93 71.23 79.76 78.11 71.59 60.70 63.14

-

Dynamic properties of symmetric composite plates

plates with CSCF and CCCF boundaries converge faster with E/E than G/E material. The difference between the 49-term and 64-term solutions for the CCCC condition is 0.03% for G/E and 0.028% for E/E. For the CFCF condition the difference between the 49-term and 64-term solutions is 1.9% for G/E and 2.38% for E/E materials which shows that the solution of the CFCF condition converges faster for G/E than E/E material. From these findings it was decided to use a 64-term solution in the subsequent results. Tables 5 and 6 give the lowest eight frequency parameters for G/E and E/E square plates with a four-layer laminate with a stacking sequence [e, - 8, - 8, 01. The orientation angle 8 is varied between 0” and 90” with an increment of 15” for the CFCF, CSCF and CCCF conditions. Only orientations between 0” and 45” for the CCCC condition are considered because the frequency parameters with

293

fibre angles 60”, 75” and 90” are the same with fibre angles 0 = 30”, 15” and O”, respectively. The results shown in Tables 5 and 6 indicate that increasing 8 from 0” to 45” for CCCC plates with G/E material, the maximum frequency parameters of the second, fifth, sixth and eight modes occur at 0 = 45”. However for the first and fourth modes, and the third and seventh modes, the maximum frequency parameters occur at 0” and 30” respectively. For the E/E material, the maximum frequency parameters of the second and fifth modes occur at 8 = 45”, whereas the maximum frequency parameters of the first, third, fourth and seventh modes, and the sixth and eighth modes occurred at 8 = 0” and 15”, respectively. For CFCF G/E square plates, the maximum frequency parameters of the first, second, sixth, seventh and eighth modes occurred at f? = 0”. For the third and fifth, and the fourth modes, the maximum frequency parameters occurred at 0 = 0” and 30

Table 6. Frequency parameter, A, for E-glass/epoxy square four-layer plates with stacking sequence (0, -8, -0,O)

Boundary condition Present study cccc

Ref. 9 cccc

Present study CFCF

Present study CSCF

Present study CCCF

h cd&,

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Mode 7

Mode 8

30 45

29.13 28.99 28.70 28.55

50.88 51.62 53.69 55.63

67.36 66.04 62.81 60.29

85.77 85.19 85.19 85.34

87.23 89-67 95.24 102.68

118.69 120.05 114.39 105.34

126.27 122,93 122.97 124.69

137.74 140.93 139.22 137.75

0 15 30 45

29.13 28.98 28.69 28.53

50.82 51.56 53.62 55.56

67.29 65.97 62.74 60.22

85.67 85-11 85.09 85.25

87.14 89.57 95.09 102.6

118-6 119.9 114.3 105.2

126.2 122.8 122.9 124.5

0 15 30 45 60 75 90

21.49 20.67 18.62 16.30 1464 13.89 13.73

23.78 23.25 21.96 20.40 18.85 17.59 17.09

33.09 33.42 34.06 34.36 34.00 33.35 33.04

55.92 56.18 51.38 45.06 40.43 38.32 37.85

59.26 57.45 56.09 50.99 46.63 43.70 4264

62.41 60.56 59.34 61.90 62.65 60.83 59.59

73.05 72.67 71.72 70.37 69.45 70.06 70.57

93.65 94.38 96.19 88.40 79.37 75.18 79-59

0 15 30 45 60 75 90

22.09 21.37 19.56 17.48 15.84 14.92 14.65

28.36 28.36 28.26 27.84 27.02 26.12 25.72

45.82 46.65 48.62 46.75 42.16 39.77 39.12

60.08 58.00 52.81 50.84 52.28 51.76 50.89

67.26 66.28 63.77 60.53 57.42 56.25 56.39

78.70 79.75 81.85 82.72 81.19 76.79 7564

83.95 85.12 8864 90.47 81.93 80.26 7964

114.37 112.64 102.46 94.59 96.99 90.55 87.87

0 15 30 45 60 75 90

22.36 21.65 19.89 17.93 16.43 15.59 15.33

30.66 30.77 30.99 31.13 31.01 30.73 30.59

51.51 52.41 52.92 47.07 42.56 40.20 39.55

60.25 58.19 54.78 57.24 5744 55.49 54.47

68.82 63.54 65.92 63.82 63.89 65.79 66.69

86.08 67.97 87.72 88.46 81.60 77.10 75.95

88.07 86.45 95.27 90.88 88.39 87.82 88.19

117.23 90.11 102.66 102.75 lOO@I 93.80 9064

0 15

137.5 140.1 139.1 137.6

A. Al-Obeid,

294

J. E. Cooper

~~~::~~~~~~~:~~~~~~ f----L 0

0.5

1.0 0

22.21

0.5

1.0 0

22.85

0.5 25.47

1.0 0

0.5

1.0 0

0.5

1.0 0

46.17

32.52

0.5

1.00

0.5

1.0 0

05

61.17

62.09

65.01

50.61

0.5 56.80

58.13

62.90

e=o

~~~‘:~~~~~~~ IO

0.5 21.34

1.00

05 25.41

1.0 0

0.5

1.0 0

34.83

0.5

1.0 0

0=15

16.42

18.26

26.10

39.85

46.07

47.65

59.09

60.98

11.64

15.14

26.19

32.56

37.89

44.33

56.21

63.66

30.17

41.29

43.42

51.31

e=45

0 7.84

12.05

21.87

24.33

O=bO

6.02

9.15

16.64

20.80

26.30

32.72

35.72

40.62

5.66

7.81

15.59

18.77

25.43

30.57

34.17

34.66

0=90

Fig. 3. Mode shapes and natural frequencies

(Hz) for CFCF square plate of four-layer G/E.

1.0

295

Dynamic properties of symmetric composite plates

~~1::~:‘~“~.‘~~~~~~~ 0

0.5

1.0 0

20.79

0.5

1.00

23.58

0.5

1.0 0

30.51

0.5

100

43.73

0.5

1.0 0

57.06

0.5

1.0 0

60.48

0.5

1.0 0

0.5 68.24

1.0

10 0

0.5 76.29

1.0

66.48

6=15

“‘i~~~~~~~~~~~~~:n 0

1.. o 0.5 17.04

1.0 0

0.5 22.76

1.00

0.5 34.09

1.0 0

0.5

1.0 0

46.20

51.88

: ... .-. . ., 0.5 54.79

48.65

58.47

66.81

44.97

48.85

54.31

67.22

34.71

45.67

50.29

56.33

31.50

41.62

51.75

51.83

0.5

1.00

0.5 -’ 69.17

0=30

34.28

9.22

19.70

24.54

35.07 0=60

6.98

17.43

19.49

29.62 9=75

6.27

16.44

18.33

27.22 e=90

Fig. 4. Mode shapes and natural frequencies

(Hz) for CSCF square plate of four-layer G/E.

296

A. Al-Obeid, J. E. Cooper

22.43

24.58

30.76

42.73

61.48

63.70

66.16

68.93

57.15

60.99

69.26

71.23

0=0

20.83

U

0.5

17.14

24.27

1.00

0.5 24.10

32.43

1.00

0.5 36.96

46.29

10 0

0.5 46.45

1.0 0

55.81

0.5 56.13

0.5

1.0 0

1.00

71.80

0.5 79.76

78.11

0.5

1.0 0

I.0

e=30

13.12

24.01

34.68

41.28

51.71

62.38

67.60

9.90

22.43

26.26

38.28

46.63

55.66

71.59

00

0.5

1.0

58.11

0=60

“‘I~~~l~~l.‘:;~~~~~~~~~ U

0.5

7.99

1.00

0.5

18.47

1.0 0

.c::.:;~F.Yy..

I

0.5

IO

24.5 1

u

1.0 0

0.5

32.99

0.5

1.0 u

0.5

1.00

0.5

36.25

49.31

57.06

32.14

45.55

52.18

1.0 0

0.5

1.0

60.70

0=75

0.5 7.40

17.10

24.35

31.97

1.0 0

0=90

Fig. 5. Mode shapes and natural frequencies

(Hz) for CCCF square plate of four-layer G/E.

0.5 63.14

1.0

297

Dynamic properties of symmetric composite plates

decrease as the fibre orientation increases from 0” to 90”. This effect is due to the fact that increasing the fibre orientation decreases the stiffness in the x direction which affects primarily the first mode. The estimated mode shapes for four layers with stacking sequence (0, - 8, - 8, 0) of square plates with the ply angle varying between 8 = 0” and 90” and different boundary conditions are shown in Figs 3-6. The solid lines are nodal lines whereas the dashed lines are contour lines. The displacement between the contour lines is one-fifth of the maximum displacement. A comparison has been carried out between the present results and the analytical results obtained previously.’ The predicted results reveals excellent agreement for all cases of the CCCC condition but the present method uses fewer terms. Finite element results were produced for all the boundary conditions using the PAFEC package. The finite element frequency results are presented in Table 7. These results are for G/E square plates with the form [e,, 01, with 8 = 0, 15, 30, 45, 60, 75 and 90”. Due to the fragile nature of unidirectional laminates, the inner plies of the laminate with 8 = 0” were rotated to 90” to

respectively. For E/E material, maximum frequency parameters of the first, second, fifth, sixth and seventh modes were found at 8 = 45”, 15”, 30”, respectively. For CSCF G/E square plates, the maximum frequency parameters of the first, second, fifth and sixth modes occurred at 8 = 0”. The maximum frequency parameters for the third and fourth, seventh and eighth modes occurred at 8 = 45”, and 30”, respectively. For E/E material, maximum frequency parameters of the first, second, fourth, fifth and eighth modes were found at 8 = 0”. Meanwhile for the third and the sixth and seventh modes, the maximum frequency parameters occur at 30” and 45”. For CCCF G/E square plates, the maximum frequency parameters of the first, second, fifth, and the sixth modes occurred at 8 = 0”. For the third, fourth, seventh and eighth modes, the maximum frequency parameters were at 8 = 30”. For E/E material the maximum frequency parameters of the first, fourth, fifth and eighth, modes occur at 8 = 0”. For the second and sixth, and the third and seventh modes, the maximum frequency parameters occur at 45” and 30”, respectively. Also it can be seen that for all boundary conditions the first frequency parameters

0.5 23.69

1.0 (I

0.5 29.52

1.00

0.5 41.44

1.0 0

0.5 59.82

1.0 0

0.5 62.54

1.0 0

0.5 67.01

100

0.5 75.94

10 0

0.5 84.47

1.0

0=0

0

0.5

10 0

0.5

100

0.5

1.0 0

0.5

100

23.13

30.78

44.61

59.68

64.68

66.75

79.63

89.44

22.07

34.85

53.03

53.83

67.95

77.25

92.80

98.59

tk30

IO

0.5 38.94

1.0 0

0.5 47.75

1.0 0

0.5 59.71

1.0 0

0.5 74.82

1 .o 0

0.5 83.83

1.00

0.5

1.0 0

84.76

Fig. 6. Mode shapes and natural frequencies (Hz) for CCCC square plate of four-layer G/E.

0.5 103.62

1.0

Table 7. Comparison of analytical and finite element natural frequencies (Hz) for laminated plates with stacking sequence of the laminate [9,, 01,; E, = 98 GPa, E,=7*9GPa, GU=56GPa, v U = O-28, a = 30444mm, b = 3044 mm, h/ply = O-134mm, po = 1520 kg/m’ Boundary condition Present study cccc

Finite element cccc

Present study CFCF

Finite element CFCF

Present study CSCF

Finite element CSCF

Present study CCCF

Finite element CCCF

Natural frequencies WI 0 15 30 45

77.21 7464 69.38 66.75

104-84 100.38 106.19 114.55

159.66 148.27 160.75 153.51

199.47 192.44 169.89 16864

218.99 214.43 212.78 231.98

241.39 215.89 225.15 232-78

0 15 30 45

77.06 74-50 69.76 67-47

103.18 99.45 106.56 114.32

164.79 152.69 166.96 156.10

195.09 187.65 168.86 184.53

203.80 208.43 221.68 228.80

239.55 221.60 234.33 235.39

0 15 30 45 60 75 90

70.40 65.78 52.19 38.34 29.04 25.06 24.24

72.73 6744 54.86 45-06 38.46 32.90 30.42

83.58 80.42 76.25 73.62 73-80 69.13 66.83

115.11 111.49 114.16 104.96 8064 74.50 75.69

176.56 16484 145.39 117.73 99.71 88.08 83.46

194.07 182.27 148.26 123.11 121.53 120.62 121,77

0 15 30 45 60 75 90

69.88 6560 52.66 38.69 29.08 25.05 24.23

71.80 67.43 55.85 45.90 38.97 33.08 30.41

82.49 80.32 76.74 74.30 73.59 69.17 66.82

109.24 106.09 111.53 107.15 81.37 74.83 75.72

16944 154.99 150.27 121.81 102.32 89.18 83.80

192.12 181-87 150.84 122.06 123.37 122.67 122.26

0 15 30 45 60 75 90

70.99 66.35 53.20 40.74 32.05 27.35 25.93

77.74 75.05 68.95 63.42 60.18 60-10 61.31

100.66 97-78 100.17 102.46 86.61 71.00 69.13

151.59 143.15 142.48 113.31 105.61 101.80 99.42

194.86 182.57 158.32 151.06 147.01 140-01 133.56

202.15 193.64 171.04 158.87 158.72 159.83 161.33

0 15 30 45 60 75 90

70.43 66.25 53.89 41.25 32.21 27.37 25.92

77.26 75.08 69.37 64.03 60.93 60.56 61.50

99.72 97.11 100.36 103.86 87.90 73.68 69.12

151.43 141.43 147.33 115,96 107.69 103.18 99.51

192.70 182.62 157.68 153.66 147.90 140.65 133.91

197.46 187.43 170-62 167.65 165.38 160.69 165.50

0 15 30 2; 75 90

71,25 66.42 53.32 41.37 33.73 29.93 28.84

80.45 77-39 72-74 70.38 7064 7144 70.81

108.93 104.65 109.08 109.75 89.56 80-44 79.76

163,61 152.13 144.95 118.56 116.70 114.25 113.54

195.03 183.03 166.46 160.77 161.35 141.83 118.54

203.80 195.15 174.07 170.61 172.79 173.62 134.87

0 15 30 45 60 75 90

70.63 66.34 54.04 41.95 33.96 29.99 28.86

79.79 77.32 73.30 71.16 71.28 71.60 70.78

106.36 102.98 109.24 112.59 90.94 80.78 79.82

167.12 153-00 150,05 119-21 118.28 114.97 112.75

192.76 182.84 169.63 163.45 161.63 142.81 118.84

198.01 189.42 173.50 187-13 180.83 179.45 135.46

Dynamic properties of symmetric composite plates

the form [&, 901s. Comparing the calculated frequencies with those calculated by the finite element method, it can be seen that the maximum difference for all cases is 7.9% and generally there is a very good agreement.

5 CONCLUSIONS A new Rayleigh-Ritz shape function has been introduced which produces good estimates of natural frequencies and mode shapes for symmetric composite plates with any boundary conditions. When compared with previous Rayleigh-Ritz approaches, improved convergence properties are obtained and less degrees of freedom are required than the finite element approach to obtain the same degree of accuracy.

ACKNOWLEDGEMENT The authors are grateful for the financial support of the Scientific Studies and Research Centre, Damascus, Syria.

299

REFERENCES 1. Kapania, R. K. & Raciti, S., Recent advances in analysis of laminated beams and plates. Part II: Vibration and wave propagation. AIAA Journal, 27 (1989) 935-46. 2. Leissa, A. W. & Narita, Y., Vibration studies for simply supported symmetrically laminated rectangular plates. Comp. Struct., 12 (1989) 113-32. 3. Sivakumaran, K. S., Natural frequencies of symmetrically laminated rectangular plates with free edges. Comp. Struct., 7 (1987) 191-204. 4. Qatu, M. S., Free vibration of laminated composite rectangular plates. Int. J. Solids Stntct., 28 (1991) 941-54. 5. Dickinson, S. M. & Blasio. A. D., On the use of orthogonal polynomials in the Rayleigh-Ritz method for the study of the flexural vibration and buckling of isotropic and orthotropic rectangular plates. J. Sound Vibration, 108 (1986) 51-62.

6. Jenson, D. W. & Crawley, E. F., Frequency determination techniques for cantilevered plates with bendingtorsion coupling. AIAA Journal, 22 (1984) 415-20. 7. Vinson. J. R. & Sierakowski. R. L., 7% Behauiour of Structures Composed of Composite Materials. Martinus Nijhoff, Dordrecht, 1986. 8. Whitney, J. M., Structural Analysis of Laminated Plates. Technomic Publishing, Lancaster, PA, 1987. 9. Chow, S. T., Liew, K. M. & Lam, K. Y.. Transverse vibration of symmetrically laminated rectangular composite plates. Comp. Struct., 20 (1992) 213-26.