An inverse trigonometric shear deformation theory for supersonic flutter characteristics of multilayered composite plates

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An inverse trigonometric shear deformation theory for supersonic flutter characteristics of multilayered composite plates

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Neeraj Grover , B.N. Singh

b,∗

, D.K. Maiti

b

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a

Department of Mechanical Engineering, Thapar University, Patiala, Punjab, 147 004, India b Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur, West Bengal 721 302, India

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a

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a r t i c l e

i n f o

Article history: Received 6 March 2014 Received in revised form 25 January 2016 Accepted 13 February 2016 Available online xxxx Keywords: Shear deformation theory Laminated composite Flutter Linear piston theory Finite element analysis

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a b s t r a c t

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In this study, an inverse trigonometric shear deformation theory developed by the authors is extended to assess the flutter behavior of multilayered composite plates subjected to yawed supersonic flow. The shear deformation is considered in terms of an inverse cotangent function which yields non-linear distribution of shear stresses. A generalized finite element formulation is presented to consider the shear strain function based theories. The displacement field is modified by a precise involvement of additional field variables to ensure the implementation of C 0 continuous finite element. First order piston theory is employed to consider the aerodynamic load. The applicability, validity and accuracy of the present mathematical treatment are ascertained by performing various numerical tests and comparing the present results with the existing results. The influences of various parameters such as lamination sequences, boundary conditions, material anisotropy, flow angles, etc. on the free vibration and flutter behavior are examined and significant conclusions are made. It is concluded that flow angles, lamination sequence and material anisotropy should be considered as essential design parameters for enhanced flutter boundary of supersonic vehicles. © 2016 Published by Elsevier Masson SAS.

89 90 91 92 93 94 95 96 97 98 99 100 101

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1. Introduction

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

The composite materials are increasingly being employed in the design of structures due to their enhanced mechanical properties such as specific strength, specific stiffness, ability for the tailor made designs, resistance to corrosion, and response to the environmental effects as compared to monolithic materials. The most widespread use of composites continues to be for aircraft structural components for both the airframe and the engine components. Indeed for the airframe it is anticipated that a weight saving in the range 30%–70% may be achieved through the use of composites. The increased usage of composite material in the structural applications is in the form of the layered structures. There had been significant attention in the past towards the modeling of the layered structures. The complicating effects such as shear deformation, zig-zag requirement and interlaminar continuity were dealt in the past [1]. Depending upon these factors, there exist various shear deformation theories which consider the shear deformation in a unique way. Reissner [2] and Mindlin [3] considered the linear shear deformation theory which is termed as first order shear deformation theory (FSDT). The FSDT doesn’t account

60 61 62 63 64 65 66

*

Corresponding author. Tel.: +91 3222 283026. E-mail address: [email protected] (B.N. Singh).

http://dx.doi.org/10.1016/j.ast.2016.02.017 1270-9638/© 2016 Published by Elsevier Masson SAS.

for the zero transverse shear conditions on top and bottom surfaces of the plate and therefore a shear correction factor (SCF) is required. However, it was shown that the value of SCF depends upon the lamination sequence, boundary conditions, etc. [4] and hence FSDT is less realistic. In order to eliminate the requirement of SCF and properly address the shear deformation, a number of higher order shear deformation theories (HSDT) were developed. The development of HSDTs can be observed in the form of polynomial shear deformation theories (PSDTs) and non-polynomial shear deformation theories (NPSDTs). The higher order terms in PSDTs [5–9] are expressed by means of Taylor’s expansion while in case of NPSDTs, a function of thickness co-ordinate is chosen to represent the realistic shear deformation [10–25]. The accurate mathematical modeling of the layered structures is essential since it is the prime factor which determines the accuracy of the structural kinematics of the composite and sandwich plates. The significant reviews on the development of various theories are given in Refs. [1,26–34]. Most of the aircraft and space vehicle structures, where composites have tremendous applications, are subjected to severe aerodynamic loads during their flight. These aerodynamic forces influence the static and dynamic response of the composite structures. There are instances where these structures loose their stability due to the interaction of aerodynamic, elastic and inertial forces. The

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Fig. 1. Plate kinematics in Cartesian co-ordinate system.

self excited oscillation of the external skin of the aircraft due to such interactions is called flutter. In order to ensure the safe operational conditions corresponding to flutter, it is desired to know the flutter boundary which is defined in terms of critical dynamic pressure and critical frequency parameter. In the earlier developments, the behavior of flutter was investigated by Fung [35], and Dowell [36,37]. The flutter studies belonging to isotropic and laminated composites structures were reviewed by Dowell [38], Bismark [39,40] and Mei et al. [41]. Sander et al. [42] employed a C 0 conforming quadrilateral finite element based on Kirchhoff plate theory and utilized linear piston theory to examine the supersonic flutter behavior of isotropic plates. The free vibration and flutter behavior of laminated quadrilateral plates assuming classical laminated plate theory (CLPT) and linear piston theory aerodynamics was examined by Srinivasan and Babu [43,44]. The supersonic flutter behavior of composite skew plates was studied by Chowdary et al. [45,46]. They used FSDT to consider the structural kinematics of the plate and linear piston theory for the aerodynamic forces. Olson [47] employed the finite element analysis to examine flutter modes and frequencies. Ganapathi and Patel [48] implemented a linear piston theory in conjunction with finite element to study the supersonic flutter behavior of laminated composites. Nam and Hwang [49] studied the effects of hysteresis on the supersonic flutter characteristic implementing linear piston theory. Various parametric studies [50–52] have been performed for the flutter behavior of composite and functionally graded plates. A scheme for the enhancement of flutter velocity of composites was presented by Raja [53]. They implemented direct matrix abstraction programme (DMAP) of NASTRAN and considered subsonic aerodynamics through doublet lattice method. Mahato and Maiti [54, 55] studied the aeroelastic performances of smart composites in hygro-thermal environment implementing subsonic aerodynamics. It is observed that most of the studies conducted on the flutter analysis are performed using the CLPT or FSDT. However, it should be noted that the shear deformation effects are not considered in CLPT while FSDT requires the use of shear correction factor. Moreover, no attempt has been made to study the influence of the variation of the material properties on the critical dynamic pressure. The lamination sequence and the fiber direction in the composites influence its material properties and there may be substantial effects of these variations on the flutter behavior of composite plates. In the present work, a newly developed inverse trigonometric shear deformation theory (ITSDT) in a recent work by the authors [56] is extended to model the structural kinematics of the plate and applied for the free vibration and flutter behavior of multilayered composite plates. The theory assumes the non-linear shear deformation in terms of an inverse trigonometric shear strain function and also satisfies the zero transverse shear conditions on top and bottom surfaces of the plate. Therefore, the theory considers the shear deformation in more realistic manner and SCF is no longer required. The field variables are elegantly utilized to ensure the C 0 continuity requirement. Linear piston theory is imple-

80

mented to consider the aerodynamic load which is applicable for the flow regimes corresponding to Mach, M > 1.7. Free vibration and flutter behavior of isotropic and multilayered composite plates are investigated in terms of frequencies and dynamic pressure parameters. The ability of ITSDT to predict higher modes of vibration is ensured by computation of these modes for a few cases. The effects of boundary conditions, lamination sequence, flow angularity, and material anisotropy on the flutter behavior of isotropic and laminated composite plates are examined and the results are compared with the published results whichever available. Some new results of flutter behavior are also presented in this work.

81 82 83 84 85 86 87 88 89 90 91 92 93

2. Mathematical formulation

94

The plate under consideration is a layered plate constituted of the orthotropic layers. The schematic and dimensions (a × b × h) of a general laminated plate are shown in Fig. 1 in a Cartesian coordinate system (x– y–z).

95 96 97 98 99 100

2.1. Structural kinematics

101

The considered multilayered composite plates are modeled by utilizing an inverse trigonometric shear deformation theory [56]. The displacement field is expressed in terms of mid-plane displacements (u 0 , v 0 , and w 0 ), mid-plane rotations (θx and θ y ) and a shear strain function of thickness co-ordinate termed as g ( z) which describes the non-linear shear deformation. The displacement field for the considered plates is as follow:

 ∂ w0  u (x, y , z) = u 0 (x, y ) − z + g ( z) + Ω z θx (x, y ) ∂x  ∂ w0  v (x, y , z) = v 0 (x, y ) − z + g ( z) + Ω z θ y (x, y ) ∂y w (x, y , z) = w 0 (x, y )

102 103 104 105 106 107 108 109 110 111 112 113 114

(1)

115

Here, the shear strain function g ( z) = cot−1 (rh/ z) ensures the nonlinear shear deformation. The parameter r is the shape parameter whose value is ascertained as 0.46 by comparing the present results with the exact solution in the post processing step [56]. The value of the parameter Ω is evaluated after satisfying the zero transverse shear conditions on the top and bottom surfaces of the plate and is equal to −d/dz( g ( z))|z=± h . The finite element

116

implementation of the above mentioned displacement field requires a C 1 continuous element due to the presence of derivatives of the transverse deflection (w 0 ) in the in-plane displacements (u , v). However, in the present work, additional degrees of freedom φx = ∂∂wx0 , φ y = ∂∂wy0 are adequately imposed in order to restrict

124

the continuity requirement to C 0 . These additional degrees of freedom may contribute towards the strain energy which is taken into consideration using the variations [9] as indicated in Eq. (13). The modified displacement field is then written as:

129

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117 118 119 120 121 122 123 125 126 127 128 130 131 132

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δ=

8 

{ε } =

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46 47 48 49 50

and

ζ=

=

⎧ ⎪ ⎪ ⎪ ⎨

u ,x v,y

⎧ 0⎫ ε1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0⎪ ⎪ ⎪ ε ⎪ ⎬ ⎨ 2⎪ 0 6 0 4 0 5

ε

⎪ ⎪ ⎪ ⎪ ε ⎪ ⎪ ⎩

ε

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

+z

⎧ 0⎫ κ ⎪ ⎪ ⎪ ⎪ 1⎪ ⎪ ⎪ ⎪ ⎪ ⎨ κ20 ⎪ ⎬ 0

κ6 ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎩ 0

⎪ ⎪ ⎪ ⎪ ⎪ ⎭

55

−2Q

60 61 62 63 64 65 66



+ g ( z)

⎧ 1⎫ κ ⎪ ⎪ ⎪ ⎪ 1⎪ ⎪ ⎪ ⎪ ⎪ ⎨ κ21 ⎪ ⎬ 1

κ6 ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎩ 0

⎪ ⎪ ⎪ ⎪ ⎪ ⎭



+ g ( z)

⎪ ⎪ ⎪ κ42 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 2⎪

71 72 73 74 76 77 78

(8)

79 80 81 82 84 85

The following equation is utilized to obtain the governing equations for the flutter analysis of multilayered composite plates:

 ∂U ∂T ∂Uc ∂W + + + =0 dt ∂{˙qi } ∂{qi } ∂{qi } ∂{qi } d



1

ˆ

{ε }Tj {σ }dv =

2

1

1

1

91 92 93 94 95 96 97 99 100 101

{ε }Tj [ H ]Tj [ Q i j ][ H ] j {ε } j dv

2

89

98

{ε }Tj [ Q i j ]{ε } j dv v

ˆ

87

90

ˆ

2

86 88

(9)

where U , T , U c are the strain energy, kinetic energy, and the strain energy due to imposition of artificial constraints respectively and W is the work done due to aerodynamic forces. These terms are first expressed for the jth element and then assembled over the complete domain. Strain energy of the jth element due to linear strains is obtained by implementing equations (2) to (7) and written as follows:

=

102 103

ˆ

1

{δ}Tj [ B ]Tj [ D ] j [ B ] j {δ} j dxdy = {δ}Tj [ K j ]{δ} j

2

2

104

(10)

105 106

s

(5)

70

83

⎫ ⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ 0 ⎪ 0

69

75

sin α

v

The assembly of linear strain energies for all elements (nel) yields the total strain energy as given in Eq. (11)

107 108 109

κ5

(6) Thus, {ε } = [ H ]{¯ε }. The generalized strains and the matrix H are given in Appendix A.

U=

nel 

U ( j) =

j =1

1 2

Uc =

nel 

( j)

Uc

=

j =1

( j)

=

2.3. Constitutive relations

γ

ˆ 

2



The stress–strain relation of a kth orthotropic layer are expressed as

where {σ } is the vector containing stress components, [ Q i j ](k) is the transformed reduced stiffness matrix and obtained in terms of material properties (E 1 , E 2 , G 12 , G 23 , G 13 , ν12 ) of the composite material and fiber orientation (β ) of the kth layer.

(11)

γ 2

+

1 2

114 115 116

(12)

118 119

∂ w0 − φx ∂x

∂ w0 − φy ∂y

113

117

{δ}T [ K c ]{δ} T 

T  j

j

∂ w0 − φx ∂x

∂ w0 − φy ∂y

120



121 122

j

123

  dV ( j )

124

(13)

˙ T [ M ]{δ} ˙ where [ M ] = {δ}

z nl ˆ k  k=1 zk−1

125 126

j

In the similar fashion, the expressions for kinetic energy, T is obtained as follows:

T=

111 112

V

(7)

110

{δ}T [ K ]{δ}

The strain energy due to imposition of artificial constraints is expressed by means of a penalty parameter (γ = 1 × 106 ) as follows [9]:

Uc

{σ }5×1 = [ Q i j ] {ε }5×1



where M, Q , V and α are Mach number, dynamic pressure, free stream velocity and yaw angle respectively.

=

{¯ε }13×1

T = ε10 ε20 ε60 k01 k02 k06 k11 k12 k16 ε40 ε50 k24 k25

(k)



∂w ∂w cos α + 2 ∂x ∂y M −1    1 M2 − 2 ∂ w + V M2 − 1 ∂t

58 59



v

56 57



p= √

U ( j) =

where ε10 , ε20 , etc. are the generalized strains and are represented by vector

52 54

(4)

⎫ ⎪ ⎪ ⎪ ⎬

γxy = u , y + v ,x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ v ,z + w , y ⎪ ⎪ ⎪ ⎩ γ yz ⎪ ⎭ ⎪ ⎭ ⎩ γzx u , z + w ,x

51 53

N i ζi

i =1

⎧ ⎫ εxx ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ εyy ⎪ ⎬

The aeroelastic analysis of plate structures requires an additional relation providing the information of aerodynamic load. Aerodynamic load is calculated using first order piston theory which is applicable to supersonic flow with Mach number, M > 1.7. The aerodynamic pressure due to supersonic flow and inclined at yaw angle, α is given as [42]:

2.5. Governing equations

In this study, it is assumed that displacements and strains are very small relative to thickness of the plate. Therefore, the linear strain displacement relations are implemented and they are given in Eq. (5)

38

45

N i δi

8 

2.2. Strain displacement relations

37

44

(3)

where N i are the shape functions corresponding to eight nodded element [17], δ is the generalized field variable with δi as the value of field variable at node i and ζ is the generalized geometrical co-ordinate, ζi is the value of corresponding coordinate at node i.

36

43

θx θ y ϕx ϕ y ] T

w0

i =1

34

42

v0

In the finite element analysis, the solution for a complicated problem is approximated by subdividing the region of interest and representing the solution within each subdivision by a relatively simple function. In the present work, we consider an eight nodded quadrilateral finite element with seven degrees of freedom at each node. Since the element chosen is an isoparametric element, the same functions are used to represent the geometrical co-ordinates and physical degrees of freedom i.e.,

33

41

(2)

The displacement vector now comprises of seven degrees of freedom given by

{δ}7×1 = [u 0

67 68

w (x, y , z) = w 0 (x, y )

32

35

2.4. Aerodynamic load

 v (x, y , z) = v 0 (x, y ) − zφ y + g ( z) + Ω z θ y (x, y ) 

18 19





u (x, y , z) = u 0 (x, y ) − zφx + g ( z) + Ω z θx (x, y )

3

127 128 129 130

[ N ]T ρ (k) [ N ]dz

(14)

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Table 1 Non-dimensional frequencies of simply supported isotropic plate (a/h = 10, v = 0.3).

2 3 4 5 6 7 8 9 10

68

Mode

Present (ITSDT)

Exact [57]

HSDT [58]

Nayak et al. [59]

CLPT [58]

1 2 3 4 5 6 7 8

0.0931 0.2221 0.3409 0.4156 0.5215 0.6543 0.6850 0.7472

0.0932 0.2226 0.3421 0.4171 0.5239

0.0931 0.2222 0.3411 0.4158 0.5221 0.6545 0.6862 0.7481

0.0934 0.2253 0.3463 0.4299 0.5368 0.694 0.7081 0.7864

0.0955 0.236 0.3732 0.4629 0.5951 0.7668 0.809 0.8926

0.6889 0.7511

11 12 13 14 15

18

W =

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

(15)

¨

[ As] =



 [ N ]T [ N ,x ] + [ N ]T [ N , y ] dxdy

51 52 53 54 55 56 57 58 59 60 61 62 63

66

76

83 84 85 86 87

¨ [ Ad ] =



 [ N ] [ N ] dxdy T

88 89

(17)

90 91

The implementation of Eqs. (11), (12), (14), and (15) in Eq. (9) yields the following algebraic equation which can be solved to obtain the desired response.

¨ + [ K + γ K C ]{δ} + g [ A d ]{δ} ˙ + λ[ A s ]{δ} = {0} [ M ]{δ}

92 93 94

(18)

95

where λ = 2Q / M 2 − √1 is the aerodynamic pressure parameter and g = λ( M 2 − 2)/ v M 2 − 1 is the aerodynamic damping parameter. Eq. (18) corresponds to an eigen value problem. Physically, the eigen values relate with the frequencies while the eigen vectors represent the mode shapes. The problem corresponds to that of a free vibration for zero dynamic pressure. The changes in the eigen values are observed due to increment in the dynamic pressure parameter. It should be noted that the aerodynamic stiffness matrix is a non-symmetric matrix due to the fact that the aerodynamic forces are non-conservative in nature. The influence of the presence of non-symmetric matrix is to yield some complex eigen values. The further increment of dynamic pressure may create the instability in the plate due to coalescence of the complex modes. The lowest value of dynamic pressure for which the first coalescence occurs corresponds to the flutter phenomenon and termed as critical dynamic pressure.

97

√

3. Results and discussion The governing equations for free vibration and flutter behavior are solved to get the desired response characteristics. The solution methodology discussed in Section 2 is coded in MATLAB environment. Various numerical experiments are performed in order to ascertain the validity of ITSDT for isotropic, laminated composite and sandwich plates for the free vibration and flutter analysis. The nondimensional results are obtained and compared with the existing results. The influences of various parameters such as lamination sequence, boundary conditions, span-thickness ratio, and material anisotropy ratio are examined on free vibration and aeroelastic behavior for multilayered composites. The simply supported, clamped and their combination are considered as boundary conditions in the analysis. These conditions are specified at the boundary in the following manner:

64 65

75

81

(16)

49 50

74

82

47 48

73

80

pdxdy

which yields aerodynamic stiffness ( A s ) and aerodynamic damping ( A d ) matrices given as follows:

22 24

72

79

ˆ

21 23

71

78

Work done due to aerodynamic forces is given by

19 20

70

77

16 17

69

• Clamped condition (C): u 0 = v 0 = w 0 = θx = θ y = φx = φ y = 0 • Simply supported condition (S):

96

Fig. 2. Coalescence of frequencies of a simply supported isotropic plate (a/h = 100, v = 0.3) with increment in dynamic pressure parameter.

98 99 100

– Edge parallel to x-axis: u 0 = w 0 = θx = φx = 0 – Edge parallel to y-axis: v 0 = w 0 = θ y = φ y = 0

101 102 103

3.1. Isotropic plate: free vibration and flutter behavior

104 105

A simply supported isotropic plate is considered in this section to ensure the applicability of ITSDT for free vibration responses. The span-thickness ratio (a/h) of the plate is 10. The frequencies corresponding to first eight √ modes are obtained in the nondimensional forms ωcr = ωh ρ /G and the results are compared in Table 1 along with the exact solution [57], analytical solution of Reddy’s HSDT [58], finite element solution of Reddy’s theory presented by Nayak et al. [59], and CLPT [58]. In order to ensure the accuracy, the percentage differences of various methodologies are obtained with respect to exact solution [57] and are given in parenthesis in Table 1. It is observed that the present ITSDT finite element predicts the results in the error range of 0.16–0.514% while the error range of the HSDT based finite element [49] is 0.215–4.7%. Overall, the percentage difference of the present finite element solution based on ITSDT is 0.38% as compared to 0.29% of analytical solution of Reddy’s theory [58], 2.24% of finite element solution [59] and 11.2% of CLPT [58]. Further, the square isotropic plate is considered to examine the flutter behavior. The dynamic pressure parameter is increased subsequently and mode coalescence is observed. This mode coalescence, indicated in Fig. 2, decides the flutter boundary in terms of critical dynamic pressure λcr = λa3 / Eh3 (12(1 − ν 2 )) and critical flutter frequency parameters. The convergence of the present ITSDT based finite element and the validation of the converged solution is presented in Table 2. The mesh size is varied from 6 × 6 to 14 × 14 to understand the convergence behavior and it is observed that a good convergence is achieved. On the basis of this conver-

106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

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Table 2 Flutter behavior of simply supported isotropic plate (a/h = 100, v = 0.3).

3

Source

Critical dynamic pressure

Flutter frequency

4

Present (6 × 6) Present (8 × 8) Present (10 × 10) Present (12 × 12) Present (14 × 14) Chowdary et al. [45] Sander et al. [42] Sander et al. (CQ) [42] Singha and Ganapathi [51] Liao and Sun [60] Valizadeh et al. [61]

511.23 511.55 511.82 511.94 511.99 512.6 509.7 503 513.35 498.73 511.11

42.9587 43.0122 43.0306 43.0377 43.0407 42.84 42.77

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

5

First six modes of vibration are investigated and the non-dimensional frequencies ω = ω(ρ / E 2 )1/2 (a2 /h) corresponding to these modes are listed in Table 3 along with existing 3D, higher-order zigzag theory (HOZT), and HSDT results [62] for various span-thickness ratio and different boundary conditions. The comparison of the results indicates the validity, accuracy and applicability of the present ITSDT. The present results are more accurate than HSDT [62]; however, less accurate than HOZT [62] when compared with 3D results. It should be noted that ITSDT possesses the similar computational efforts as that of HSDT and provides more accurate results. It is observed that the nondimensional frequencies corresponding to each mode increase with the increment in span-thickness ratio (a/h = 5, 10, and 20) for both the boundary conditions. Moreover, the clamped plate (CCCC) possesses higher frequencies relative to simply supported (SSSS) plate for each span-thickness ratio due to increment in overall stiffness of the plate for the clamped conditions. In order to study the anti-symmetric plates, we consider a simply supported cross ply plate with anti-symmetric layup sequence [0/90/0/90]. The free vibration behavior is investigated in terms of first eight eigen frequencies and corresponding mode shapes and the results are shown in Fig. 4. Further, a seven layered [0/90/0/C /0/90/0] symmetric sandwich plate with all its edges simply supported is considered for different span-thickness ratio and different aspect ratio (a/b). The ratio of core thickness to total thickness (hc /h) is taken as 0.88. The considered sandwich plate possesses the isotropic core (E = 0.10363 GPa, G = 0.05 GPa, ρ = 130 kg/m3 ) and orthotropic face sheets of equal thickness with following material properties [63]:

30 32

Fig. 3. Effect of flow angle on critical dynamic pressure parameter of an isotropic plate (a/b = 1, a/h = 100, v = 0.3).

36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

gence study, a mesh size of 12 × 12 is chosen to examine further analysis. However, higher mesh size can be adopted for more accurate solutions. The obtained results in terms of critical dynamic pressure, (λcr ) and critical frequency parameter (Ω cr ) are also validated with some of the existing results such as Chowdary et al. [45], Sander et al. [42], Singha and Ganapathi [51], Liao and Sun [60] and Valizadeh et al. [61]. It should be noted that Sander et al. [61] employed CLPT while Chowdary et al. [45], Singha and Ganapathi [51], Liao and Sun [60] and Valizadeh et al. [61] employed FSDT. The comparison of the results affirms the applicability of ITSDT for flutter behavior. The effect of yaw angle on the supersonic flutter characteristics is also examined for the same plate and indicated in Fig. 3 along with the existing results. It is observed that the flutter behavior is qualitatively similar to that predicted by Sander et al. [42] though it is quantitatively different. Moreover, for the flow angle 0◦ and 90◦ , the present results are more accurate than Sander et al. [42] when compared with the analytical solution.

55 56

E 2 = 7.77 GPa,

G 12 = G 13 = 3.34 GPa,

33 35

59 60 61

3.2. Free vibration of composite and sandwich plates

64 65 66

70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95

ν12 = 0.246,

G 23 = 1.34 GPa,

ρ = 1800 kg/m3 .

101 102 103 104 105 106 107 108 109 110 111

3.3. Flutter behavior of composite plates

112

The flutter bounds along with first six natural frequencies are obtained for an eight layered clamped composite plate with symmetrical lay up as [0/90/0/90]s . All the orthotropic layers are of equal thickness and possess the following material properties [44]:

G 12 = 0.277 × E 2 ,

99 100

The non-dimensional frequencies ω = ω(ρc / E 2c )1/2 (a2 /h) corresponding to first three modes-of-vibration are evaluated and the results are compared with Xiang et al. [63] and Nayak et al. [59]. The results are given in Table 4. The effect of span-thickness ratio is similar to that of laminated plates i.e., increasing the spanthickness ratio increases the non-dimensional frequencies. Also, for a particular span-thickness ratio, the non-dimensional frequency corresponding to each mode increases with increase in aspect ratio of the plate.

E 1 / E 2 = 11.481481,

98

6

E 2 = 2.7 × 10 psi,

ν12 = 0.28,

ρ = 0.192 × 10−3 lb sec2 /inch4 .

113 114 115 116 117 118 119 120 121 122 123

A four layered symmetric square plate with cross ply lamination sequence [0/90/90/0] is considered to study the free vibration behavior. Each ply is of equal thickness and possesses the following orthotropic material properties.

62 63

69

97

57 58

68

96

E 1 = 24.51 GPa,

31

34

67

E 1 = 181 GPa,

E 2 = 10.3 GPa,

G 12 = G 13 = 7.17 GPa,

ν12 = 0.28,

G 23 = 2.87 GPa,

ρ = 1578 kg/m3 .

Non-dimensional frequency parameters ω = ω(a2 /h)(ρ / E )1/2 and flutter bounds in terms of critical dynamic pressure, λcr = λa3 / E 2 h3 and critical flutter frequency parameter, Ω cr obtained using present finite element based on ITSDT are listed in Table 5 along with existing results obtained using various techniques. The results are compared with integral equation (IE) and series solution presented by Srinivasan and Babu [44], Mindlin theory based finite element solution presented by Lee and Cho [64], and Rayleigh– Ritz (RR), FEM and experimental results by Thornton and Clary

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1 2 3

a/h

Mode

SSSS

4 5 6

5

7 8 9 10 11

10

12 13 14 15 16 17

67

Table 3 First six frequencies of symmetric laminates subjected to simply supported and clamped boundary conditions (type b).

20

18 19 20 21

68 69

CCCC

Present

3D [62]

HOZT [62]

HSDT [62]

Present

3D [62]

HOZT [62]

HSDT [62]

1 2 3 4 5 6

8.6226 15.9192 17.7168 22.4038 25.688 27.8064

8.5611 15.8799 17.5087 22.2319 25.6164 –

8.5692 16.1903 17.6861 22.482 26.5115 –

8.7079 16.0482 17.8355 22.4856 25.9036 –

11.5948 18.3978 19.5906 24.4875 27.5081 29.3704

11.486 18.2956 19.3319 24.2519 27.3411 28.9117

12.0965 19.3528 21.1266 26.2179 29.0997 33.12

11.9516 19.0208 20.3969 25.4967 28.519 30.9182

1 2 3 4 5 6

11.347 21.2829 28.6437 34.4953 37.6314 46.9217

11.2981 21.2529 28.3362 34.2444 37.5751 –

11.2857 21.3772 28.3239 34.1788 38.2535 –

11.4121 21.3602 28.9828 34.7299 37.8238 –

17.8985 28.3239 32.8477 40.007 43.913 51.1732

17.7502 28.2032 32.4505 39.6697 43.7521 50.457

18.1118 29.0729 33.5629 41.0151 45.6649 52.7698

18.2744 28.9047 33.8184 41.0769 44.9767 52.739

1 2 3 4 5 6

12.7407 23.9967 38.9219 45.1568 45.3978 60.9355

12.721 23.9803 38.6947 45.0944 45.1926 –

12.7066 23.9855 38.6446 45.3214 45.0057 –

12.7576 23.9939 39.1669 45.1583 45.5095 –

23.8401 36.4154 50.0723 58.0855 58.4416 74.8679

23.7212 36.3153 49.6061 57.6707 58.2579 74.4577

23.8689 36.6889 50.2908 58.4074 59.3371 75.6414

24.113 36.7473 51.1651 59.1384 59.0253 75.9905

70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87

22

88

23

89

24

90

25 26 27 28

a/h

Mode

10

31

a/b = 0.5

92

a/b = 1

93

a/b = 1.5

Present

Xiang et al. [63]

Nayak et al. [59]

Present

Xiang et al. [63]

Nayak et al. [59]

Present

Xiang et al. [63]

Nayak et al. [59]

1 2 3

11.4058 15.4592 20.4133

11.53 15.55 21.74

11.2 15.05 20.41

15.4583 29.0432 29.7289

15.55 28.93 30.22

15.04 28.1 29.2

21.7409 33.664 40.8267

21.74 34.05 44.203

21.08 32.86 40.82

1 2 3

14.1359 19.8235 29.8748

14.2 19.88 29.87

13.85 19.24 29.16

19.8215 42.8626 45.631

19.88 42.79 46.14

19.23 41.7 44.88

29.8509 52.2816 73.2108

29.87 52.74 72.83

28.97 51.12 71.17

1 2 3

14.907 21.1476 32.7257

14.94 21.18 32.71

14.6 20.5 31.95

21.1452 48.5906 52.9656

21.18 48.55 53.33

20.49 47.36 52.03

32.6959 61.1414 88.8165

32.71 61.49 88.53

31.73 59.66 86.73

1 2 3

15.2098 21.6811 33.9434

15.23 21.7 33.92

14.9 21.01 33.14

21.6785 51.2362 56.556

21.7 51.2 56.81

20.99 49.99 55.53

33.9102 65.5609 97.2917

33.92 65.81 97.08

32.9 63.9 95.26

29 30

91

Table 4 First three frequencies of seven layered simply supported symmetric sandwich plate [0/90/0/C /090/0].

32 33 35

38 40 41

96 97 99 100 101

30

37 39

95

98

20

34 36

94

102 103 104

40

105 106 107

42

108

43

109

44

110

45

111

46

112

47

113

48

114

49

115

50

116

51

117

52

118

53

119

54

120

55

121

56

122

57

123

58

124

59

125

60

126

61

127

62

128

63

129

64

130

65 66

131

Fig. 4. First eight mode shapes of vibration for simply supported anti-symmetric plate [0/90/90/0] (a/h = 10, a/b = 1).

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1 2 3

6 7 8 9 10

67

Table 5 Natural frequencies and flutter boundaries of 8 layered Clamped symmetric laminate [0/90/0/90]s . Source

68

Frequency

4 5

7

Present (ITSDT) Srinivasan and Babu (IE) [44] Srinivasan and Babu (Series) [44] Lee and Cho [64] Thornton and Clary (RR) [65] Thornton and Clary (FEM) [65] Thornton and Clary [65] (Experimental)

69

Flutter bounds

1st

2nd

3rd

4th

5th

6th

λcr

Ω cr

23.2041 23.33 23.63 23.34 22.71 23.25 17.44

41.8898 42.36 42.28 42.3 44.14 45.23 35.61

52.8777 53.77 53.76 53.62 49.59 50.68 43.42

64.5176 65.56 65.42 65.4 63.58 64.31 54.68

74.7374 76.52 75.89 76.25 80.84 83.2 64.85

90.7076 92.16 92.19 92.46 94.1 95.19 79.75

441.84 446.36 474.6 471.16 – – –

45.8576 46.09 47.19 46.89 – – –

11

14 15

18 19

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61

Source

66

75 76

Present (ITSDT) Srinivasan & Babu (IE) [44] Srinivasan & Babu (Series) [44]

Flutter boundary

81 82

1st

2nd

3rd

4th

5th

6th

λcr

Ω cr

16.2066 16.22 16.71

33.3765 33.82 33.71

45.7325 46.35 46.14

61.5964 63.84 62.22

70.5638 65.39 62.36

70.5638 73.96 71.19

167.43 163.23 173.31

29.2388 28.99 29.79

83 84 85 86

[65]. It is observed that the natural frequencies agree well with the existing computational results; however, there is a significant difference from the existing experimental results. In case of the flutter bounds, the present results are closer to IE solution [44] as compared to other results. This is due to consideration of different shear deformation theories and different approximate solution techniques. Further, we consider an anti-symmetric laminate [0/90] to examine the free vibration and supersonic flutter behavior. The material properties of each layer are same as that of the symmetrical plate considered above. All the edges of the plate are clamped and the flow is assumed parallel to the plate axis. The numerical results in terms of non-dimensional natural frequencies, non-dimensional dynamic pressure and flutter frequencies are obtained and compared with the IE and series solution presented by Srinivasan and Babu [44] in Table 6. Though the frequencies corresponding to free vibration match well for different approaches, the deviation in the flutter bounds is substantial. It should be noted that Srinivasan and Babu [44] employed FSDT for modeling the laminated quadrilateral plates. Thus, the choice of shear deformation has more influence on the flutter behavior as compared to free vibration response. In order to investigate the influence of lamination sequence and boundary conditions on the flutter behavior, five layered symmetric laminated plate is considered. The cross ply [0/90/0/90/0] and angle ply [45/−45/45/−45/45] plates are considered, each with simply supported and clamped conditions. The laminates constitute equal-thickness plies possessing the following orthotropic properties:

E 1 / E 2 = 40, G 23 = 0.5,

E 2 = 1,

ν12 = 0.25,

G 12 = G 13 = 0.6,

ρ = 1.

The non-dimensional critical dynamic pressure (λcr = λa3 / E 2 h3 ) is evaluated for these test cases and the results are compared with Singha and Ganapathi [51] as provided in Fig. 5. The influence of lamination sequence is clearly observable from the figure. It is observed that the present results agree well with the existing results. Moreover, it should be noted that the flutter boundary for the clamped plate is higher than that of a simply supported plate for both the lamination sequences. 3.4. Influence of flow angle on flutter boundary

64 65

74

80

Frequency

62 63

73

79

20 21

72

78

Table 6 Flutter behavior of 2 layered anti-symmetric laminate [0/90] with clamped conditions.

16 17

71

77

12 13

70

It should be noted that all the test cases on composite plates in the previous section are subjected to supersonic flow parallel to

87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106

Fig. 5. Flutter behavior of symmetric laminated plates subjected to different boundary conditions.

the plate axis. However, there are instances when the panel structure experiences the yawed flow instead of a parallel flow during the operating flight. In order to investigate the influence of flow angularity, the yaw angle is varied from 0◦ to 90◦ and the effect is studied for the composite plates with varying lamination sequence and boundary conditions. We consider five layered symmetric laminated plates with cross ply [0/90/0/90/0] and angle ply [45/−45/45/−45/45] lamination sequences. All edges simply supported (SSSS) and all edges clamped (CCCC) boundary constraints are considered to investigate the influence of boundary conditions. The orthotropic properties of equal-thickness plies are as follows:

107 108 109 110 111 112 113 114 115 116 117 118 119 120 121

E 1 / E 2 = 40, G 23 = 0.5,

E 2 = 1,

ν12 = 0.25,

G 12 = G 13 = 0.6,

ρ = 1.

Figs. 6 and 7 show the effect of flow angle, lamination sequence and boundary conditions on the non-dimensional critical dynamic pressure and non-dimensional flutter frequency respectively. It is observed that lamination sequence, boundary conditions and flow angle significantly influence the flutter bounds. The effect of clamped boundary conditions is to enhance the flutter boundary (λcr and Ω cr ) for both the lamination sequences. This can be accounted towards the stiffness increment of the plate due

122 123 124 125 126 127 128 129 130 131 132

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67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14

80

15

81

16

82

17

83

18

84

19

85

20 21 22

Fig. 6. Effect of flow angle on critical dynamic pressure of symmetric laminated composite plate (a/h = 100).

86

Fig. 8. Effect of material anisotropy on critical dynamic pressure of a simply supported symmetric cross ply laminated composite plate (a/h = 100) [0/90/0/90/0].

87 88

23

89

24

90

25

91

26

92

27

93

28

94

29

95

30

96

31

97

32

98

33

99

34

100

35

101

36

102

37

103

38

104

39

105

40

106

41

107

42 43 44

108

Fig. 7. Effect of flow angle on flutter frequency of symmetric laminated composite plate (a/h = 100).

Fig. 9. Effect of material anisotropy on critical dynamic pressure of a clamped symmetric cross ply laminated composite plate (a/h = 100) [0/90/0/90/0].

to clamped edges. A slight variation in flow angle significantly influences the flutter boundaries. The critical dynamic pressure for cross ply plate is maximum when the flow is parallel to the plate (α = 0◦ ) and gradually decreases to its minimum value when the flow angle is 90◦ . However, for the angle ply plate, maximum critical dynamic pressure is observed for the flow aligned at 45◦ while λcr is minimum for the flow aligned at 0◦ or 90◦ . Moreover, it should be noted that this behavior is independent of the boundary conditions as shown in Fig. 6. The similar effects of the flow angle, boundary conditions and lamination sequences are observed on the flutter frequency parameter as indicated in Fig. 7. This information is significant for the structural design of a supersonic aircraft since desired composite plates’ configuration may be implemented for the design purpose in order to avoid the flutter or at least delay the flutter phenomenon.

material anisotropy play a significant role to enhance the flutter characteristics and hence delay the flutter which is always desirable. Five layered symmetric plates are considered constituted of equal thickness layers and possessing the following material properties: E 1 / E 2 = variable, E 2 = 1, G 12 = G 13 = 0.6, G 23 = 0.5, ν12 = 0.25, ρ = 1. The material anisotropy (E 1 / E 2 ) ratio is varied from 3 to 40 for four different cases. Figs. 8–9 show the influence of material anisotropy on supersonic flutter boundary for symmetric cross ply plate [0/90/0/90/0] with simply supported and clamped boundary conditions respectively. The influence of material anisotropy on the critical dynamic pressure for the symmetric angle ply [45/−45/45/−45/45] plate is shown in Figs. 10–11 for simply supported and clamped conditions respectively. The variation in flow angle is also considered in the above analyses with yaw angle as 0◦ , 30◦ , and 45◦ . It is observed that flutter boundary (critical dynamic pressure) increases with increase in material anisotropy ratio for all the cases (see Figs. 8–11). This can be accounted towards the fact that an increase in material anisotropy leads to increased overall stiffness thereby increasing the resistance of the plate against aerodynamic loads. However, the rate of increment in flutter boundary with respect to change in ma-

45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

3.5. Influence of material anisotropy on flutter boundary

63 64 65 66

110 111

61 62

109

In this section, we examine the influence of material anisotropy (E 1 / E 2 ) on the supersonic flutter characteristics of laminated composite plates and establish, with the help of numerical tests, that

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1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14

80

15

81

16

82

17

83

18

84

19

85

20 21 22 23

86

Fig. 10. Effect of material anisotropy on critical dynamic pressure of a simply supported symmetric angle ply laminated composite plate (a/h = 100) [45/−45/45/−45/45].

24

Fig. 12. Effect of span-thickness ratio on the critical dynamic pressure of simply supported laminated composite plate.

87 88 89

3.6. Influence of span-thickness ratio on flutter boundary

25

90 91

In this section, a numerical example is considered to investigate the influence of span thickness ratio on the flutter boundary of laminated composite plate. A simply supported square laminated composite plate with all layers having fibers aligned with x-axis ([0]4s ) is considered. The material properties of equal thickness layers of the laminate are as follows:

26 27 28 29 30 31 32

E 1 = 155 Gpa,

33

E 2 = 8.07 Gpa,

93 94 95 96 97

G 12 = 4.55 Gpa,

98

ν12 = 0.22,

100

99

34

G 23 = 3.03 Gpa,

35

ρ = 1550 kg/m3

101

The flutter boundary is examined in terms of non-dimensional critical dynamic pressure for a variety of span thickness ratio ranging from 10 to 100. Also, the analysis is performed for the flow aligned at 0◦ and 30◦ . The obtained results are indicated in Fig. 12 along with the results presented by Xia and Ni using HSDT [66]. It is observed that non-dimensional critical dynamic pressure increases with the increase in span-thickness ratio which implies that increment in span-thickness ratio increases the flutter boundary. However, the rate of increment in flutter boundary with a/h ratio reduces with increase in a/h and the dynamic pressure becomes almost constant beyond a/h > 40.

103

36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

Fig. 11. Effect of material anisotropy on critical dynamic pressure of a clamped symmetric angle ply laminated composite plate (a/h = 100) [45/−45/45/−45/45].

terial anisotropy ratio is maximum for the flow aligned at 0◦ for cross ply plates for both the boundary conditions (see Figs. 8–9) while it is minimum for the flow aligned at 45◦ . It should also be noted that this rate of increment of flutter boundary is higher for the clamped plate as compared to simply supported plate for a particular flow angle. On the contrary, the angle ply plates exhibit quite different behavior. The maximum rate of increment of flutter boundary is achieved for the flow aligned at 45◦ while this rate is least for the flow aligned at 0◦ to the plate (see Figs. 10–11). However, the effect of boundary conditions is similar to that of the cross ply plates i.e., the rate of increment of flutter boundary with respect to material anisotropy ratio is higher for the clamped plate as compared to simply supported plate. A comparison of Figs. 8–9 and Figs. 10–11 also reveal that clamped plates (both cross ply and angle ply) possess higher flutter boundary than a simply supported plate for a particular material anisotropy ratio and a particular flow angle. This observation is similar to that established in the discussions made in earlier sections.

G 13 = 4.55 Gpa,

92

102 104 105 106 107 108 109 110 111 112 113 114

4. Conclusion

115 116

An inverse trigonometric shear deformation theory (ITSDT) is implemented to examine the free vibration and flutter behavior of isotropic and multilayered composite plates. Shear deformation of ITSDT is chosen in terms of an inverse cotangent function which yields non-linear distribution of shear stresses. The displacement field is modified by a precise involvement of additional field variables to ensure the applicability of C 0 continuous finite element. The discretization of the plate is done using an isoparametric quadrilateral element with eight nodes and 56 degrees of freedom. First order piston theory which is applicable for the supersonic flow is implemented to consider the aerodynamic loads. The methodology is validated for the free vibration and flutter behavior and it is concluded, on the basis of various numerical experiments and their comparison with the established results, that the present approach is quite accurate and efficient for evaluating free vibration and flutter behavior. The effects of span-thickness

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1 2 3 4 5 6 7

ratio, boundary conditions, lamination sequences, etc. on the free vibration behavior are investigated. Few analysis are performed for investigating higher modes in terms of frequencies and mode shapes of vibration and the present results are compared with the existing results. The effect of boundary conditions, lamination sequence and flow angles on the flutter behavior are examined and various conclusions are made.

8 9

• Boundary conditions, lamination sequence, flow angle, and

10

material anisotropy play significant roles on the flutter boundaries (dynamic pressure and flutter frequency). The effect of the clamped conditions is to increase the flutter boundary for both cross ply and angle ply plates due to increment in overall stiffness of the plate. The cross ply plates possess highest dynamic pressure for the flow aligned at 0◦ and lowest dynamic pressure for the flow aligned at 45◦ . The considered angle ply plate possesses highest dynamic pressure for the flow aligned at 45◦ and lowest dynamic pressure for the flow aligned at 0◦ . The effect of increment in material anisotropy is to increase the flutter boundary irrespective of the boundary conditions, lamination sequence and flow angle. However, the rate of increment significantly depends upon the flow angle, lamination sequence and boundary conditions.

11 12

•

13 14 15

•

16 17 18

•

19 20 21 22 23 24 25

•

26 27

Conflict of interest statement

28 29

None declared.

30 31

Appendix A

32 33 34 35 36 37 38 39

[H ] ⎡

1 ⎢0 ⎢ =⎢0 ⎣0 0

0 0 z 1 0 0 0 1 0 0 0 0 0 0 0

0 0 z 0 0 z 0 0 0 0

g ( z) 0 0 0 0

0 g ( z) 0 0 0

0 0 0 0 0 0 g ( z) 0 0 0 1 0 0 0 1

0 0 0 g  ( z) 0

⎤

0 0 ⎥ ⎥ 0 ⎥ ⎦ 0  g ( z)

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

∂ u0 ∂ v0 ∂ u0 ∂ v 0 ; ε20 = ; ε60 = + ; ∂x ∂y ∂y ∂x ∂ w0 ∂ w0 ε40 = − φy; ε50 = − φx ; ∂y ∂x ∂θ y ∂φ y ∂θx ∂φx k01 = Ω − ; k02 = Ω − ; ∂x ∂x ∂y ∂y     ∂φx ∂φ y ∂θx ∂θ y − ; k06 = Ω + + ∂y ∂x ∂y ∂x ∂θ y ∂θx k11 = g ( z) ; k12 = g ( z) ; ∂x ∂y   ∂θx ∂θ y ; k16 = g ( z) + k24 = θ y ; k25 = θx ∂y ∂x

ε10 =

56 57

References

58 59 60 61 62 63 64 65 66

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