Free vibration of anti-symmetric angle-ply cylindrical shell walls using first-order shear deformation theory

Abstract

In the present study, free vibration of anti-symmetric angle-ply cylindrical shell walls is investigated under first-order shear deformation theory using the spline method. Assuming the displacement components in a separable form, a system of coupled equations on three displacement and two rotational functions are obtained. A generalized eigenvalue problem is obtained using the collocation procedure and applying the spline function approximation for displacement and rotational function. The resulting eigenvalue problem is solved for the frequency parameters to get as many eigenvalues as required starting from the least. The eigenvectors are the spline coefficients from which the mode shapes can be constructed. The parametric study showed the effect of length parameter, circumferential node number, ply angles, number of lay-ups, lamination materials and different boundary conditions on the shell frequency.

Keywords

Anti-symmetric angle-ply eigenvalues free vibration shear deformation splines

1. Introduction

Shell structures are a source of attraction for most engineers because of their having considerably more stiffness and strength due to the curvature. Cylindrical shell structures have wide-ranging applications in the fields of aviation, rocket, missile and sub-marine industries. The composite structures have benefits in the design of advanced lightweight, high temperature resistance, better damping and shock absorbing characteristics since they provide increased specific properties (e.g have high strength-to-weight ratio or stiffness-to-weight ratio) in comparison to their monolithic counterparts. Presently engineers are mainly focusing on structural weights while designing aerospace or automotive structures which could be achieved by selecting such composite materials which have high strength-to-weight ratio or stiffness-to-weight ratio.

Composite structures are mostly in the form of laminates consisting of multiple lamina or plies oriented in the desired direction. The natural frequencies are significantly affected by ply orientation, number of lay-ups and boundary conditions (Alibeigloo and Shakeri, 2007; Alibeigloo, 2009) . Optimal design and desired structural properties can be produced by adjusting optimal fiber orientation, geometric and material parameters. These factors also help to adjust the natural frequency of a structure (Alibeigloo and Shakeri, 2007; Alibeigloo et al., 2012).

Most of the solution techniques for composite shells are numerical methods, such as the finite element method (Toorani and Lakis, 2006), the Ritz method (Tizzi, 2006), and the differential quadrature method (Alibeigloo, 2009). Researchers have carried out work on composite cylindrical shells including shear deformation (Asadi and Qatu, 2012; Toorani and Lakis, 2006). None of the above authors used spline approximation in their analysis in the past but Viswanathan et al. (2012) used that method and produced useful results as compared to other methods. Three-dimensional free vibration of cross-ply and antisymmetric angle-ply cylindrical shells was analyzed by Soldatos and Jianqiao (1995). Further, a generalized differential quadrature method was used to investigate the vibration of antisymmetric angle-ply cylindrical panels with different boundary conditions (Loy et al., 1999). Moreover, Shao and Ma (2007) analyzed the free vibration of laminated antisymmetric angle-ply cylindrical shells using Fourier series expansion. None of the abovementioned researchers included shear deformation in their analysis. Chaudhuri and Abu-Arja (1991) investigated the static analysis of moderatly-thick antisymmtric angle-ply cylindrical panels and shells using Sander’s first-order shear deformation theory.

In this problem the authors analyzed the free vibration of antisymmetric angle-ply cylindrical shells including shear deformation and applyied spline approximation technique. The displacement and rotational functions are approximated by Bickley-type splines. Collocation with these splines yields a set of field equations which, along with the equations of boundary conditions, in return reduce to a system of homogeneous simultaneous algebraic equations on the assumed spline coefficients. Then the problem is solved using an eigensolution technique to obtain the frequency parameter. The eigenvectors are the spline coefficients from which the mode shapes can be constructed. The stability of the cylindrical structure is analyzed with respect to the ply angles, length-to-radius ratio, circumferential node number, number of lay-ups, lamination materials and two different boundary conditions. Results are presented in terms of graphs and tables.

2. Formulation

Consider a composite laminated circular cylindrical shell having length ℓ, thickness h and radius r shown in Figure 1. The x coordinate of the shell is taken along the meridional direction, θ coordinate along the circumferential direction and z along the thickness direction having the origin at the mid-surface of the shell.

The equations of equilibrium for circular cylindrical shells including shear deformation which are as follows

\frac{\partial N x}{\partial x} + \frac{1}{r} \frac{\partial N θ x}{\partial θ} = I 1 \frac{\partial 2 u}{\partial t 2} \frac{\partial N x θ}{\partial x} + \frac{1}{r} \frac{\partial N θ}{\partial θ} + \frac{1}{r} Q θ 3 = I 1 \frac{\partial 2 v}{\partial t 2} \frac{\partial Q x 3}{\partial x} + \frac{1}{r} \frac{\partial Q θ 3}{\partial θ} - \frac{1}{r} N θ = I 1 \frac{\partial 2 w}{\partial t 2} \frac{\partial M x}{\partial x} + \frac{1}{r} \frac{\partial M θ x}{\partial θ} - Q x 3 = I 3 \frac{\partial 2 ψ x}{\partial t 2} \frac{\partial M x θ}{\partial x} + \frac{1}{r} \frac{\partial M θ}{\partial θ} - Q θ 3 = I 3 \frac{\partial 2 ψ θ}{\partial t 2}

(1)

According to the first-order shear deformation theory, the displacements components $u, v, w$ can be written as

u (x, θ, z, t) = u \circ (x, θ, t) + z ψ x (x, θ, t) v (x, θ, z, t) = v \circ (x, θ, t) + z ψ θ (x, θ, t) w (x, θ, z, t) = w (x, θ, t)

(2)

where

u 0, v 0, w 0

are the mid-plane displacements, and

ψ x, ψ θ

are the shear rotations of any point on the mid-surface normal to the

xz

and

θ z

plane respectively and t is the time.

The strain-displacement relations for cylindrical shells having the radius r given as

ɛ x = \frac{\partial u 0}{\partial x} + z \frac{\partial ψ x}{\partial x}, ɛ θ = \frac{1}{r} \frac{\partial v 0}{\partial θ} + \frac{w}{r} + \frac{z}{r} \frac{\partial ψ θ}{\partial θ}, γ x θ = \frac{1}{r} \frac{\partial u 0}{\partial θ} + \frac{\partial v 0}{\partial x} + z (\frac{1}{r} \frac{\partial ψ x}{\partial θ} + \frac{\partial ψ θ}{\partial x}) γ xz = ψ x + \frac{\partial w}{\partial x} and γ θ z = ψ y + \frac{1}{r} \frac{\partial w}{\partial θ} - \frac{v 0}{r}

(3)

The stress-strain relations of the k-th layer by neglecting the transverse normal strain and stress, are of the form

{σ} = [Q (k)] {ɛ x (k) ɛ θ (k) γ x θ (k) γ xz (k) γ θ z (k)}

(4)

When the materials are oriented at an angle θ with the x-axis, the transformed stress-strain relations are

{σ} = [\bar{Q} (k)] {ɛ x (k) ɛ θ (k) γ x θ (k) γ xz (k) γ θ z (k)}

(5)

where

Q ij (k)

and

\bar{Q} ij (k)

are given in Appendix A.

The stress resultants and stress couples are given by

(N x, N θ, N x θ, Q x, Q θ) = \int z (σ x, σ θ, τ x θ, τ xz, τ θ z) dz (M x, M θ, M x θ,) = \int z (σ x, σ θ, τ x θ) zdz

(6)

Applying equation (3) into equation (5) and then substituting into equation (6), to obtain the equations of stress-resultants and moment resultant as

(N x N θ N x θ M x M θ M x θ) = (A 11 A 12 A 16 B 11 B 12 B 16 A 12 A 22 A 26 B 12 B 22 B 26 A 16 A 26 A 66 B 16 B 26 B 66 B 11 B 12 B 16 D 11 D 12 D 16 B 12 B 22 B 26 D 12 D 22 D 26 B 16 B 26 B 66 D 16 D 26 D 66) (\frac{\partial u 0}{\partial x} \frac{1}{r} \frac{\partial v 0}{\partial θ} \frac{\partial v 0}{\partial x} + \frac{1}{r} \frac{\partial u 0}{\partial θ} \frac{\partial ψ x}{\partial x} \frac{1}{r} \frac{\partial ψ θ}{\partial θ} \frac{\partial ψ θ}{\partial x} + \frac{1}{r} \frac{\partial ψ x}{\partial θ})

(7)

and

(Q θ Q x) = K (A 44 A 45 A 45 A 55) (ψ θ + \frac{1}{r} \frac{\partial w}{\partial θ} - \frac{v 0}{r} ψ x + \frac{\partial w}{\partial x})

(8)

in which

A ij

B ij

and

D ij

are respectively, extensional rigidities, the bending-stretching coupling rigidities and the bending rigidities defined as

A ij = \sum_{k} Q ij (k) (z k - z k - 1), B ij = \frac{1}{2} \sum_{k} Q ij (k) (z k 2 - z k - 1 2) D ij = \frac{1}{3} \sum_{k} Q ij (k) (z k 3 - z k - 1 3) (i, j = 1, 2, 6) and A ij = K \sum Q ij (k) (z k - z k - 1) (i, j = 4, 5)

(9)

where K is the shear correction factor depending on lamina properties and lamination scheme (Pai, 1995; Pai and Schulz, 1999) and calculated by various static and dynamic methods.

z k - 1

and z_k are the boundaries of the k-th layer.

Substituting for case of anti-symmetric angle-ply lamination the coefficients $A 16$ , $A 26$ , $A 45$ , $B 11$ , $B 12$ , $B 22$ , $B 66$ , $D 16$ , $D 26$ (Gibson, 1994; Staab, 1999) equal to zero in equation (7) and equation (8) and then applying to equilibrium equation (1) will get differential equations as follows.

(A 11 \frac{\partial 2}{\partial x 2} + A 66 \frac{1}{r 2} \frac{\partial 2}{\partial θ 2}) u + ((A 12 + A 66) \frac{1}{r} \frac{\partial 2}{\partial x \partial θ}) v + A 12 \frac{1}{r} \frac{\partial w}{\partial x} + (2 B 16 \frac{1}{r} \frac{\partial 2}{\partial x \partial θ}) β x + (B 16 \frac{\partial 2}{\partial x 2} + B 26 \frac{1}{r 2} \frac{\partial 2}{\partial θ 2}) β θ = I 1 \frac{\partial 2 u}{\partial t 2} ((A 66 + A 12) \frac{1}{r} \frac{\partial 2}{\partial x \partial θ}) u + (A 66 \frac{\partial 2}{\partial x 2} + A 22 \frac{1}{r 2} \frac{\partial 2}{\partial θ 2} + KA 44 \frac{1}{r 2}) v + (A 22 + KA 44) \frac{1}{r 2} \frac{\partial w}{\partial θ} + (B 16 \frac{\partial 2}{\partial x 2} + B 26 \frac{1}{r 2} \frac{\partial 2}{\partial θ 2}) β x + (2 B 26 \frac{1}{r} \frac{\partial 2}{\partial x \partial θ} + KA 44 \frac{1}{r}) β θ = I 1 \frac{\partial 2 v}{\partial t 2} - A 12 \frac{1}{r} \frac{\partial u}{\partial x} - (KA 44 + A 22) \frac{1}{r 2} \frac{\partial v}{\partial θ} + (KA 55 \frac{\partial 2}{\partial x 2} + KA 44 \frac{1}{r 2} \frac{\partial 2}{\partial θ 2} - A 22 \frac{1}{r 2}) w + (KA 55 \frac{\partial}{\partial x} - B 26 \frac{1}{r 2} \frac{\partial}{\partial θ}) β x - (B 26 \frac{1}{r} \frac{\partial}{\partial x} - KA 44 \frac{1}{r} \frac{\partial}{\partial θ}) β θ = I 1 \frac{\partial 2 w}{\partial t 2} (2 B 16 \frac{1}{r} \frac{\partial 2}{\partial x \partial θ}) u + (B 16 \frac{\partial 2}{\partial x 2} + B 26 \frac{1}{r 2} \frac{\partial 2}{\partial θ 2}) v - (KA 55 \frac{\partial}{\partial x} - B 26 \frac{1}{r 2} \frac{\partial}{\partial θ}) w + (D 11 \frac{\partial 2}{\partial x 2} + D 66 \frac{1}{r 2} \frac{\partial 2}{\partial θ 2} - KA 55) β x + (D 12 + D 66) \frac{1}{r} \frac{\partial 2 β θ}{\partial x \partial θ} = I 3 \frac{\partial 2 β x}{\partial t 2} + (B 16 \frac{\partial 2}{\partial x 2} + B 26 \frac{1}{r 2} \frac{\partial 2}{\partial θ 2}) u + (2 B 26 \frac{1}{r} \frac{\partial 2}{\partial x \partial θ} + KA 44 \frac{1}{r}) v + (B 26 \frac{1}{r} \frac{\partial}{\partial x} - KA 44 \frac{1}{r} \frac{\partial}{\partial θ}) w + ((D 12 + D 66) \frac{1}{r} \frac{\partial 2}{\partial x \partial θ}) β x + (D 66 \frac{\partial 2}{\partial x 2} + D 22 \frac{1}{r 2} \frac{\partial 2}{\partial θ 2} - KA 44) β θ = I 3 \frac{\partial 2 β θ}{\partial t 2}

(10)

The displacement components u₀, v₀, w and shear rotations $ψ x$ , $ψ θ$ are assumed in the form of

u 0 (x, θ, t) = U (x) e n θ e i ω t v 0 (x, θ, t) = V (x) e n θ e i ω t w (x, θ, t) = W (x) e n θ e i ω t ψ x (x, θ, t) = Ψ x (x) e n θ e i ω t ψ θ (x, θ, t) = Ψ θ (x) e n θ e i ω t

(11)

where ω is the angular frequency of vibration, t is the time and n is the circumferential node number.

Using the above equation (11) into equation (10), the resulting equation becomes in the matrix form as

[L 11 L 12 L 13 L 14 L 15 L 21 L 22 L 23 L 24 L 25 L 31 L 32 L 33 L 34 L 35 L 41 L 42 L 43 L 44 L 45 L 51 L 52 L 53 L 54 L 55] {U V W Ψ x Ψ θ} = {00000}

(12)

where

L 11 = A 11 \frac{d 2}{dx 2} + A 66 \frac{n 2}{r 2} + I 1 ω 2, L 12 = (A 12 + A 66) \frac{n}{r} \frac{d}{dx}, L 13 = A 12 \frac{1}{r} \frac{d}{dx} L 14 = 2 B 16 \frac{n}{r} \frac{d}{dx}, L 15 = B 16 \frac{d 2}{dx 2} + B 26 \frac{n 2}{r 2} L 21 = (A 66 + A 12) \frac{n}{r} \frac{d}{dx}, L 22 = A 66 \frac{d 2}{dx 2} + A 22 \frac{n 2}{r 2} - KA 44 \frac{1}{r 2} + I 1 ω 2 L 23 = (A 22 + KA 44) \frac{n}{r 2}, L 24 = B 16 \frac{d 2}{dx 2} + B 26 \frac{n 2}{r 2}, L 25 = 2 B 26 \frac{n}{r} \frac{d}{dx} + KA 44 \frac{1}{r} L 31 = - A 12 \frac{1}{r} \frac{d}{dx}, L 32 = - (KA 44 + A 22) \frac{n}{r 2}, L 33 = KA 55 \frac{d 2}{dx 2} + KA 44 \frac{n 2}{r 2} - A 22 \frac{1}{r 2} + I 1 ω 2 L 34 = KA 55 \frac{d}{dx} - B 26 \frac{n}{r 2}, L 35 = - B 26 \frac{1}{r} \frac{d}{dx} + KA 44 \frac{n}{r} L 41 = 2 B 16 \frac{n}{r} \frac{d}{dx}, L 42 = B 16 \frac{d 2}{dx 2} + B 26 \frac{n 2}{r 2}, L 43 = - KA 55 \frac{d}{dx} + B 26 \frac{n}{r 2} L 44 = D 11 \frac{d 2}{dx 2} + D 66 \frac{n 2}{r 2} - KA 55 + I 3 ω 2, L 45 = (D 12 + D 66) \frac{n}{r} \frac{d}{dx} L 51 = B 16 \frac{d 2}{dx 2} + B 26 \frac{n 2}{r 2}, L 52 = 2 B 26 \frac{n}{r} \frac{d}{dx} + KA 44 \frac{1}{r} L 53 = B 26 \frac{1}{r} \frac{d}{dx} - KA 44 \frac{n}{r}, L 54 = (D 12 + D 66) \frac{n}{r} \frac{d}{dx} L 55 = D 66 \frac{d 2}{dx 2} + D 22 \frac{n 2}{r 2} - KA 44 + I 3 ω 2

The non-dimensional parameters are introduced as follows:

X = \frac{x}{ℓ}, adistancecoordinate, and X \in [0, 1]; λ = ω ℓ \sqrt{\frac{I 1}{A 11}}, afrequencyparameter; L = \frac{ℓ}{r}, alengthparameter; δ k = \frac{h k}{h}, relativelayerthicknessofthe k -thlayer

(13)

The resulting equation in the matrix form is obtained as follows.

[L 11 * L 12 * L 13 * L 14 * L 15 * L 21 * L 22 * L 23 * L 24 * L 25 * L 31 * L 32 * L 33 * L 34 * L 35 * L 41 * L 42 * L 43 * L 44 * L 45 * L 51 * L 52 * L 53 * L 54 * L 55 *] {\bar{U} \bar{V} \bar{W} \bar{Ψ} X \bar{Ψ} Θ} = {00000}

(14)

The differential operators $L ij *$ of the matrix are given as

L 11 * = \frac{d 2}{d X 2} + s 10 n 2 \frac{1}{R 2} + λ 2, L 12 * = (s 2 + s 10) n \frac{1}{R} \frac{d}{d X}, L 14 * = 2 s 15 n \frac{1}{R} \frac{d}{d X}, L 15 * = s 15 \frac{d 2}{d X 2} + s 16 n 2 \frac{1}{R 2} L 21 * = (s 10 + s 2) n \frac{1}{R} \frac{d}{d X}, L 22 * = s 10 \frac{d 2}{d X 2} + (s 3 n 2 - Ks 13) \frac{1}{R 2} + λ 2 L 23 * = (s 3 + Ks 13) n \frac{1}{R 2}, L 24 * = s 15 \frac{d 2}{d X 2} + s 16 n 2 \frac{1}{R 2}, L 25 * = 2 s 16 n \frac{1}{R} \frac{d}{d X} + Ks 13 \frac{1}{R} L 31 * = - s 2 \frac{1}{R} \frac{d}{d X}, L 32 * = - (Ks 13 + s 3) n \frac{1}{R 2}, L 33 * = Ks 14 \frac{d 2}{d X 2} + (Ks 13 n 2 - s 3) \frac{1}{R 2} + λ 2 L 34 * = Ks 14 \frac{d}{d X} - s 16 n \frac{1}{R 2}, L 35 * = - s 16 \frac{1}{R} \frac{d}{d X} + Ks 13 n \frac{1}{R} L 41 * = 2 s 15 n \frac{1}{R} \frac{d}{d X}, L 42 * = s 15 \frac{d 2}{d X 2} + s 16 n 2 \frac{1}{R 2}, L 43 * = - Ks 14 \frac{d}{d X} + s 16 n \frac{1}{R 2} L 44 * = s 7 \frac{d 2}{d X 2} + s 12 n 2 \frac{1}{R 2} - Ks 14 + λ 2 p 1, L 45 * = (s 8 + s 12) n \frac{1}{R} \frac{d}{d X} L 51 * = s 15 \frac{d 2}{d X 2} + s 16 n 2 \frac{1}{R 2}, L 52 * = 2 s 16 n \frac{1}{R} \frac{d}{d X} + Ks 13 \frac{1}{R} L 53 * = s 16 \frac{1}{R} \frac{d}{d X} - Ks 13 n \frac{1}{R}, L 54 * = (s 8 + s 12) n \frac{1}{R} \frac{d}{d X} L 55 * = s 12 \frac{d 2}{d X 2} + s 9 n 2 \frac{1}{R 2} - Ks 13 + λ 2 p 1

where

p 1 = \frac{I 3}{I 1 ℓ 2} and S ij aredefinedinAppendixB

2.1. Spline collocation procedure

The displacement functions $\bar{U}$ , $\bar{V}$ , $\bar{W}$ and rotational functions $\bar{Ψ} X$ , $\bar{Ψ} Θ$ are approximated by cubic spline functions in the range of $X \in [0, 1]$ as

\bar{U} (X) = \sum_{i = 0}^{2} a i X i + \sum_{j = 0}^{N - 1} b j (X - X j) 3 H (X - X j) \bar{V} (X) = \sum_{i = 0}^{2} c i X i + \sum_{j = 0}^{N - 1} d j (X - X j) 3 H (X - X j) \bar{W} (X) = \sum_{i = 0}^{2} e i X i + \sum_{j = 0}^{N - 1} f j (X - X j) 3 H (X - X j) \bar{Ψ} X (X) = \sum_{i = 0}^{2} g i X i + \sum_{j = 0}^{N - 1} p j (X - X j) 3 H (X - X j) \bar{Ψ} Θ (X) = \sum_{i = 0}^{2} l i X i + \sum_{j = 0}^{N - 1} q j (X - X j) 3 H (X - X j)

(15)

The assumed spline functions given in equation (15) are approximated at the nodes (coincide with the knots) and these splines satisfy the differential equations given in equation (14), at all X_s and resulting into the homogeneous system of $(5 N + 5)$ equations in the $(5 N + 15)$ unknown spline coefficients.

The following boundary conditions are considered to analyze the problem.

Clamped-Clamped (C-C) (both the ends are clamped)

Simply-Supported (S-S) (both the ends are simply-supported)

By applying each of these boundary conditions separately, 10 more equations can be obtained on spline coefficients. Combining these 10 equations with the earlier $(5 N + 5)$ equations, there are $(5 N + 15)$ homogeneous equations in the same number unknowns. Thus, there is a generalized eigenvalue problem in the form

[M] {q} = λ 2 [P] {q}

(16)

where

[M]

and

[P]

are the square matrices,

{q}

is the column matrix of the spline coefficients and λ is the eigenfrequency parameter.

3. Results and discussion

In this problem, the frequencies are analyzed for two and four layered anti-symmetric angle-ply cylindrical shells under C-C and S-S boundary conditions using Kevler-49 Epoxy and (KE) Graphite Epoxy (AS4/3501-6) (GE) materials arranging them in different orders. A convergence study has been made for the frequency parameters of two and four layered shells under various boundary conditions with fixing the length ratio, thickness ratio and circumferential node number. The program is performed for N (number of knots) = 2 onwards and finally it is seen that N=14 would be enough to achieve the change in percentage is 3.7.

Table 1 depicts the variation of fundamental frequency parameter λ with respect to the circumferential node number n for four layered shells having ply orientation as

45 \circ / - 30 \circ / 30 \circ / - 45 \circ

and different arrangements of layered materials. It can be seen that the layers having the material combination of KE-GE-GE-KE show the least frequency for most of the values of the fundamental frequency parameter. For this reason it was decided to choose this combination for future analysis. Moreover, as the circumferential node number increases the fundamental frequency parameter value randomly decreases. In addition, for

n = 6, 7

there is a slight increase in the values of the fundamental frequency parameter followed by a decrease.

Table 1.

Variation of fundamental frequency parameter λ with circumferential node number n using different materials for four layered plate $45 \circ / - 30 \circ / 30 \circ / - 45 \circ$ under clamped-clamped boundary conditions.

n	$45 \circ / - 30 \circ / 30 \circ / - 45 \circ$
	GE-GE-GE-GE	KE-KE-KE-KE	GE-KE-KE-GE	KE-GE-GE-KE
	λ	λ	λ	λ
1	0.640523	0.611779	0.667822	0.587476
2	0.616632	0.586263	0.632524	0.569530
3	0.550910	0.517846	0.543890	0.518493
4	0.405938	0.365696	0.333405	0.410264
5	0.186592	0.240101	0.374737	0.134405
6	0.484699	0.500956	0.608363	0.384702
7	0.539151	0.551161	0.602045	0.544835
8	0.501526	0.437969	0.480915	0.473725
9	0.366931	0.310052	0.348993	0.351967
10	0.270866	0.257304	0.282986	0.252614

GE: graphite epoxy; KE: Kevler-49 epoxy.

Variation of fundamental frequency parameter λ with respect to circumferential node number n of three different ply orientations is shown in Table 2 under C-C boundary conditions. It can be clearly seen that as the circumferential node number increases the fundamental frequency parameter value decreases in general except for few values where there is an increase. Moreover, the value of the fundamental frequency parameter increases with the ply angles.

Table 2.

Variation of fundamental frequency parameter λ with circumferential node number of four layered cylindrical shells $30 \circ / - 30 \circ / 30 \circ / - 30 \circ$ , $45 \circ / - 45 \circ / 45 \circ / - 45 \circ$ and $60 \circ / - 60 \circ / 60 \circ / - 60 \circ$ under clamped-clamped boundary conditions.

n	$30 \circ / - 30 \circ /$ $30 \circ / - 30 \circ$	$45 \circ / - 45 \circ /$ $45 \circ / - 45 \circ$	$60 \circ / - 60 \circ /$ $60 \circ / - 60 \circ$
	KE-GE-GE-KE	KE-GE-GE-KE	KE-GE-GE-KE
	λ	λ	λ
1	0.457351	0.702846	1.422302
2	0.486073	0.598322	0.897975
3	0.474921	0.497906	1.138034
4	0.426517	0.629605	1.326433
5	0.345306	0.689929	1.025334
6	0.199376	0.557799	0.772561
7	0.199824	0.411085	0.638277
8	0.313359	0.399209	0.652484
9	0.301410	0.353173	0.715397
10	0.220564	0.241314	0.403776

GE: graphite epoxy; KE: Kevler-49 epoxy.

Table 3 shows the variation of fundamental frequency parameter value for a four layer anti-symmetric angle-ply cylindrical shell having different ply orientations under C-C boundary conditions. It is observed that the ply orientation

45 \circ / - 30 \circ / 30 \circ / - 45 \circ

resulting in the least fundamental frequency parameter value for

1 < n < 6

and the ply orientation

60 \circ / - 30 \circ / 30 \circ / - 60 \circ

shows the least fundamental frequency parameter value for

7 < n < 10

Table 3.

Variation of fundamental frequency parameter λ with circumferential node number of four layered cylindrical shell $45 \circ / - 30 \circ / 30 \circ / - 45 \circ$ , $60 \circ / - 30 \circ / 30 \circ / - 60 \circ$ and $60 \circ / - 45 \circ / 45 \circ / - 60 \circ$ under clamped-clamped boundary conditions.

n	$45 \circ / - 30 \circ /$ $30 \circ / - 45 \circ$	$60 \circ / - 30 \circ /$ $30 \circ / - 60 \circ$	$60 \circ / - 45 \circ /$ $45 \circ / - 60 \circ$
	KE-GE-GE-KE	KE-GE-GE-KE	KE-GE-GE-KE
	λ	λ	λ
1	0.587476	0.791230	0.959125
2	0.569530	0.806238	0.811491
3	0.518493	0.763367	0.248207
4	0.410264	0.610199	0.902966
5	0.134405	0.254889	0.893721
6	0.384702	0.393622	0.750689
7	0.544835	0.421250	0.632106
8	0.473725	0.304303	0.597727
9	0.351967	0.175489	0.489552
10	0.252614	0.104099	0.324157

GE: graphite epoxy; KE: Kevler-49 epoxy.

Figure 2 depicts the variation of the angular frequency $ω m (m = 1, 2, 3)$ with respect to the length parameter L of two layered anti-symmetric cylindrical shells under C-C boundary conditions. The ply orientation for Figure 2(a) is $30 \circ / - 30 \circ$ , for Figure 2(b) is $45 \circ / - 45 \circ$ and for Figure 2(c) is $60 \circ / - 60 \circ$ and the KE material is used here. The circumferential node number $n = 2$ and thickness ratio $H = 0.02$ are fixed for Figure 2((a)–(c)). It can be seen that the value of the angular frequency $ω m (m = 1, 2, 3)$ generally decreases with the increase of length parameter L. The decrease is fast for short shells $0.5 < L < 1$ further, there is a minimal decrease for $L > 1$ onwards. In addition, as the lamination angle increases the angular frequency $ω m (m = 1, 2, 3)$ also increases. Moreover the angular frequency value is higher for higher modes. The percentage changes in $ω 1$ over the range of $0.5 < L < 2$ for different ply angles, are (a) 54.711% (b) 68.141% (c) 65.357%

The variation of the angular frequency $ω m (m = 1, 2, 3)$ with length parameter L of two layered anti-symmetric cylindrical shells under S-S boundary conditions is shown in Figure 3. The same trend for the slope of curvature can be seen in Figure 2 the only difference being that the angular frequency $ω m (m = 1, 2, 3)$ values are significantly lower for S-S boundary conditions as compared to C-C boundary conditions. The angular frequency $ω m (m = 1, 2, 3)$ increases with the increase of the lamination angle. The percentage changes in $ω 1$ over the range of $0.5 < L < 2$ for different ply angles are (a) 40.357% (b) 60.236% (c) 74.589%.

The effect of length parameter L on the variation of the angular frequency $ω m (m = 1, 2, 3)$ of four layered anti-symmetric cylindrical shells under C-C boundary conditions is shown in Figure 4. The ply orientation considered is $30 \circ / - 30 \circ / 30 \circ / - 30 \circ$ and materials arranged in the order of KE-GE-GE-KE, the circumferential node number fixed for Figure 4(a) as $n = 2$ and for Figure 4(b) as $n = 4$ respectively. The thickness ratio $H = 0.02$ is fixed. It can be seen that the value of the angular frequency $ω m (m = 1, 2, 3)$ generally decreases with the increase of length parameter L. Moreover the angular frequency value is higher for higher modes. In addition there is a slight increase in the fundamental angular frequency value as the circumferential node number increases. The percentage changes in $ω 1$ over the range of $0.5 < L < 2$ for different ply angles are (a) 57.745% (b) 72.422%.

The effect of length parameter L on the angular frequency $ω m (m = 1, 2, 3)$ of a four layered anti-symmetric cylinder is studied in Figure 5 under S-S boundary conditions. It can be seen that the value of the angular frequency $ω m (m = 1, 2, 3)$ decreases with the increase of length parameter L. It is concluded from Figures 4 and 5 that the angular frequency value is lower for S-S boundary conditions than C-C boundary conditions. The percentage changes in $ω 1$ over the range of $0.5 < L < 2$ for different ply angles are (a) 40.031% (b) 71.033%.

Variation of angular frequency $ω m (m = 1, 2, 3)$ with respect to length parameter L is shown in Figures 6 and 7. The ply orientation for both figures considered are $45 \circ / - 30 \circ / 30 \circ / - 45 \circ$ and $60 \circ / - 45 \circ / 45 \circ / - 60 \circ$ respectively, and materials are arranged in the order of KE-GE-GE-KE by fixing the thickness ratio $H = 0.02$ . The circumferential node number $n = 2$ and $n = 4$ is fixed for Figures 6, 7(a) and Figures 6, 7(b) respectively. It can be seen that fundamental angular frequency ω value is highest for ply orientation $60 \circ / - 45 \circ / 45 \circ / - 60 \circ$ followed by $45 \circ / - 30 \circ / 30 \circ / - 45 \circ$ . The percentage changes in $ω 1$ over the range of $0.5 < L < 2$ for different ply angles, are Figure 6(a) 42.479%, Figure 6(b) 72.75% for Figure 7(a) 61.011%, Figure 7(b) 77.759%.

The effect of circumferential node number n on the fundamental frequency λ for two layered shells having ply orientation as $30 \circ / - 30 \circ$ and order of the material as KE-KE and four layered shells having ply orientation as $30 \circ / - 30 \circ / 30 \circ / - 30 \circ$ and order of the material as KE-GE-GE-KE is depicted in Figure 8. The length parameter $L = 1$ and the thickness parameter $H = 0.02$ is fixed. It can be seen that as the number of layers increases the fundamental frequency value decreases.

4. Conclusion

The influence on the frequencies of the two and four layered anti-symmetric angle-ply cylindrical shells is analyzed. The vibrational response of the shell is significantly affected by layer material, ply orientation, length parameter and boundary conditions. A frequency can be tailored by a proper choice of the length parameter, material and arrangement of ply-angles. The C-C boundary conditions give rise to higher frequencies in comparison with the simply supported boundary conditions. The value of the frequency parameter increases as the mode shapes increase. The effect of increasing the length of the cylinder is to decrease in frequencies, for the different number of layers and boundary conditions considered.

Footnotes

Funding

This work was supported by the MOHE, flagship research grant vote (grant number 01G40) Research management Centre (RMC), Universiti Teknologi Malaysia.

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