Nonlinear free vibrations analysis of truncated conical shells made of bidirectional functionally graded materials

Abstract

For the first time, present paper addresses the nonlinear vibrations analysis of a truncated conical shell made of bidirectional functionally graded (BDFG) materials whose mechanical properties varied continuously through the thickness and the length directions of the conical shell based on the power law distribution. Accordingly, the set of nonlinear equation of motions of the conical shell are derived based on first-order shear deformation theory (FSDT) and von Karman strain-displacement relations using Hamilton’s principle. System of partial differential equations of the BDFG conical shell are transformed into time-dependent ordinary differential equations using Galerkin approach. Afterward, the nonlinear equations are analytically solved by means of modified Poincaré–Lindstedt method, and the nonlinear frequency of the BDFG conical shell is obtained. For verification purpose, the present outcomes are compared with those available in previous researches. Eventually, the effect of longitudinal and transverse gradient indexes, vibration amplitude, and geometrical parameters on the nonlinear frequency of the BDFG conical shell is investigated. The present research may give benchmark solutions and some guidelines for designing BDFG truncated conical shells.

Keywords

nonlinear vibrations truncated conical shells bidirectional functionally graded first-order shear deformation theory

1. Introduction

Shell is a three-dimensional structure whose thickness is very small compared to other dimensions. Among them, cylindrical and conical shells have more applications in engineering structures; some examples are pressure tanks, submarine and ship hulls, plane wings and hulls, pipes, outer surfaces of rockets, vehicle tires, reinforced concrete roofs, and liquid tanks (Sobhani et al., 2021). The mentioned components and structures are subjected to extreme dynamical loads during the operation most of the time. When the excitation frequency of the external load approaches the natural frequency of the structure, a resonance phenomenon occurs, and consequently, the structure immensely vibrates and is finally damaged. Therefore, vibrations behavior analysis of such structures has become a noteworthy issue among researchers in recent years, which some of them will be explored in the next paragraph.

Asymmetric transverse vibrations analysis of truncated conical shells with consideration of flexural and membrane rigidity as well as inertia terms resulting from transverse and in-plane motion was investigated by Lindholm and Hu (1966). Weingarten (1966) determined the internal and external pressure on the free vibrations of fully simply supported conical shells using Donnel assumptions and Galerkin method. Qin et al. (2017) developed a comparison study on the free vibrations investigation of cylindrical shells with various boundary supports based on Sanders shell theory and using Rayleigh–Ritz method. Dong et al. (2022) studied nonlinear vibrations of a cylindrical shell coupled with multi-mode under different boundary supports by considering in-plane inertia terms. Lian et al. (2022) investigated the free vibration analysis of semi-closed shells of revolution with variable curvature, with a focus on the analytical solution to the natural frequency of such shells. Sobhani et al. (2022) studied the vibrational behavior of coupled hemispherical-conical-conical shells structures made composite materials reinforced with nanofillers using Donnell’s shell theory.

Conventional composite materials experience a rapid change in their mechanical properties at the interface where different phases meet. This creates elevated levels of stress at the boundary, which frequently results in failure (Kazemzadeh-Parsi et al., 2023). With the advances in technology, new materials with unique properties have been replaced by traditional ones. Functionally graded materials (FGMs) are inhomogeneous composite materials that occur of two or more materials with different properties (such as ceramic and metal) which are changed gradually and continuously throughout one or more directions (Civalek and Avcar, 2022). Nowadays, FGMs have attracted the attention of many researchers due to their impressive mechanical and thermal characteristics. Hence, numerous studies on the vibrations analysis of functionally graded (FG) shells have been reported recently, which some of them will be reported in the following paragraph.

Xue et al. (2023) studied free vibrations analysis of FG cylindrical panels and shells with porosity that has been symmetrically and non-symmetrically distributed in thickness direction with two novel dispersal types. Kim et al. (2023) investigated free and forced vibration behavior of relatively thick FG doubly curved shell of revolution according to first-order shear deformation theory (FSDT). Sofıyev and Kuruoglu (2015) studied vibration characteristics of the geometrically imperfect FG truncated conical shell with nonlinear von Karman–Donnell strain-displacement relations. Esfahani et al. (2022) examined the free vibrations analysis of sandwich conical shell with a saturated FG porous core via FSDT using differential quadrature method (DQM). Hashemi and Jafari (2020a) investigated nonlinear free vibrations of FG rectangular plates based on FSDT using modified Lindstedt–Poincaré method. Hashemi and Jafari (2021) determined the effect of fluid on the nonlinear fundamental frequency of the FG rectangular plates based on FSDT using modified Lindstedt–Poincaré method. Hashemi et al., 2023 analyzed the free vibrations of FG annular plate with elastic edge supports resting on Winkler foundation. Dastjerdi et al. (2020) investigated the nonlinear dynamic analysis of FG torus-shaped and cylindrical shell-like structures using semi-analytical method. Civalek et al. (2022) analyzed the buckling and free vibrations of FG carbon nanotube reinforced cross-ply laminated composite plates based on FSDT.

Nguyen et al. (2021) investigated nonlinear vibration analysis of full-filled fluid corrugated sandwich FG cylindrical shells using the fourth-order Runge–Kutta method.

In the above-aforementioned articles, the vibrations behavior of conical and cylindrical shells is limited to one-directional FGMs. However, conventional FGM may also not be very effective in some engineering components such as rocket warheads and propulsion systems. Therefore, vibrations characteristics of bidirectional functionally graded (BDFG) (Hashemi et al., 2022; Hashemi and Jafari, 2020b) conical and cylindrical shells is an important issue that has been worked on less, which will be discussed in the following.

Amirabadi et al. (2021) worked on linear free vibrations of the BDFG graphene nanoplatelets reinforced rotating conical shells resting on elastic foundation whose material properties varied along the length and thickness of the shell. Allahkarami et al., 2020b analyzed the dynamical buckling of BDFG porous conical and cylindrical shells on elastic foundation with various boundary supports based on FSDT and using generalized differential quadrature method (GDQM). Chen et al. (2018) examined the vibrations analysis of BDFG porous sector cylindrical shells with elastic boundary conditions using FSDT and Hamilton’s principle. Tahouneh and Naei (2016) investigated the effects of multi-directional nanocomposites on the linear vibrations of thick cylindrical shell panels on elastic foundation by means of two-dimensional DQM. Bahadori and Najafizadeh (2015) studied linear vibrations of BDFG porous cylindrical shells resting on Winkler elastic foundation on the basis of FSDT. Shariyat and Asgari (2013) worked on nonlinear buckling and postbuckling temperature-dependent BDFG truncated cylindrical shells with variable thickness according to the higher-order shear deformation theory (HSDT) and Hamilton’s principle. Mohammadzadeh et al. (2013) examined buckling analysis of BDFG cylindrical shells subjected to combined axial and external pressure using classical shells theory and DQM. Jafari Mehrabadi and Aragh (2013) investigated thermal analysis of temperature-dependent BDFG cylindrical shells based on HSDT using DQM. Pilafkan et al. (2013) reported a three-dimensional frequency analysis for thick BDFG cylindrical shells. Wu and Li (2021) presented a semi-analytical method based on finite element in order to analyze three-dimensional bending of BDFG toroidal shells exposed to uniform compression load. Khorsand et al. (2019) utilized multi-directional FG to increase the durability and longevity of the shell structures whose mechanical properties changed along the radial and axial direction and were subjected to mechanical loads. Wu and Huang (2020) studied stress and deformation analysis of BDFG truncated conical shells subjected to sinusoidal compression load using a semi-analytical method.

To the best knowledge of the authors, up to now, the nonlinear free vibrations analysis of BDFG truncated conical shell based on FSDT has not been reported in open literatures. Therefore, the present study introduces a novel investigation into the bidirectional effect of FGMs on the nonlinear vibrational behavior of truncated conical shells. The mechanical properties of the graded materials vary continuously through both the thickness and length directions of the shell based on a power law distribution, which has not been previously explored in the literature. The main innovation of this paper is to examine the bidirectional nature of the graded materials and its impact on the nonlinear vibrations analysis of the conical shell for the first time. For this purpose, in the first section, the equations of motion of the system have been derived using Hamilton’s principle and nonlinear von Karman strain-displacement relations based on FSDT. In the second section, in order to obtain nonlinear natural frequency of the system, the nonlinear equations of the BDFG truncated conical shell have been analytically solved by modified Poincaré–Lindstedt method. Eventually, the effect of some significant parameters on the nonlinear frequency of the shell has been deeply investigated.

2. Model description

Figure 1 depicts the schematic view of a BDFG truncated conical shell with length of $L$ , small radius of $R_{1}$ , large radius of $R_{2}$ , thickness of $h$ , and semi-vertex angle of α which is simply supported on both circular edges. The origin of the cylindrical coordinate system (r,θ,z) is located in the mid-plane of the shell at point O. As indicated in Figure 1, (x) and (z) stand for directions of the longitudinal distance and outwards normal of mid-surface from the origin of the truncated conical shell, respectively.

Mechanical properties $(P)$ of the BDFG truncated conical shell have been assumed to continuously change along the length (x-axis) and thickness (z-axis) of the shell, and can be expressed as

P (x, z) = P_{c} V_{c} (x, z) + P_{m} V_{m} (x, z)

(1)

where

P_{c}

and

P_{m}

are material properties for ceramic and metal. If the volume fraction of the ceramic part is 𝑉_𝑐 and the metallic part is 𝑉_𝑚, the sum of all the volume fractions must be one and is written as

V_{c} + V_{m} = 1

(2)

According to power law distribution, the volume fraction of a BDFG truncated conical shell is written as the following form (Allahkarami et al., 2020a):

V_{m} (x, z) = 1 - V_{c},

(3)

V_{c} (x, z) = {(\frac{z}{h} + \frac{1}{2})}^{N_{z}} {(\frac{x - L_{1}}{L})}^{N_{x}}, N_{x}, N_{z} \geq 0

(4)

where

N_{x}

and

N_{z}

designate the power indexes in the

x -

and

z -

axes, respectively.

Young’s modulus

E

and the density

ρ

of the BDFG conical shell have the forms (Allahkarami et al., 2020a):

E (x, z) = E_{m} + (E_{c} - E_{m}) {(\frac{z}{h} + \frac{1}{2})}^{N_{z}} {(\frac{x - L_{1}}{L})}^{N_{x}}

(5)

ρ (x, z) = ρ_{m} + (ρ_{c} - ρ_{m}) {(\frac{z}{h} + \frac{1}{2})}^{N_{z}} {(\frac{x - L_{1}}{L})}^{N_{x}}

(6)

Poisson’s ratio

ν

is considered to be constant. The subscripts c and m are related to ceramic and metallic phases, respectively. Table 1 shows the material properties of pure ceramic and pure metal.

Table 1.

Mechanical properties of pure ceramic and pure metal (Hashemi et al., 2023).

Properties	Ceramic $(A l_{2} O_{3})$	Metal $(A l)$
Young’s modulus	380 GPa	70 GPa
Density	3800 $\frac{k g}{m^{3}}$	2702 $\frac{k g}{m^{3}}$
Poisson’s ratio	0.3	0.3

3. Theory and equations of motion

Longitudinal (U), circumferential (V), and transverse (W) displacement of any point on a truncated conical shell based on the FSDT is written as follows (Allahkarami et al., 2020a):

U (x, θ, z, t) = u (x, θ, t) + z ϕ_{x x} (x, θ, t)

(7)

V (x, θ, z, t) = v (x, θ, t) + z ϕ_{θ θ} (x, θ, t)

(8)

W (x, θ, z, t) = w (x, θ, t)

(9)

where $u$ , $v$ , and $w$ are the displacement of an arbitrary point on middle surface of the BDFG truncated conical shell in the $x$ , $θ$ , and $z$ directions, respectively, and $ϕ_{x x} a n d ϕ_{θ θ}$ are rotations about the $x$ and $θ$ axes, respectively, also t is the time. The strain components according to the von Karman strain-displacement relations are written in the following form (Mohammadrezazadeh and Jafari, 2021a):

ε_{x x} = \frac{\partial u}{\partial x} + \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2} + z \frac{\partial ϕ_{x x}}{\partial x}

(10)

ε_{θ θ} = \frac{u + z ϕ_{x x}}{x} + \frac{\csc (α) \frac{\partial v}{\partial θ}}{x} + \frac{\csc^{2} (α) {(\frac{\partial w}{\partial θ})}^{2}}{2 x^{2}} + \frac{\cot (α) w}{x} + \frac{z c s c (α) \frac{\partial ϕ_{θ θ}}{\partial θ}}{x}

(11)

γ_{x θ} = \frac{\csc (α) \frac{\partial u}{\partial θ}}{x} + \frac{\partial v}{\partial x} - \frac{v + z ϕ_{θ θ}}{x} + \frac{\csc (α) \frac{\partial w}{\partial x} \frac{\partial w}{\partial θ}}{x} + \frac{z c s c (α) \frac{\partial ϕ_{x x}}{\partial θ}}{x} + z \frac{\partial ϕ_{θ θ}}{\partial x}

(12)

γ_{θ z} = - \frac{\cot (α) v}{x} + \frac{\csc (α) \frac{\partial w}{\partial θ}}{x} + ϕ_{θ θ}

(13)

γ_{x z} = \frac{\partial w}{\partial x} + ϕ_{x x}

(14)

In which $ε$ and $γ$ is normal and shear strain, respectively, and $α$ is semi-vertex angle. The stress-strain relations for an isotropic material according to the Hook’s law are defined as follows (Allahkarami et al., 2020a):

(\begin{array}{c} σ_{x x} \\ σ_{θ θ} \\ τ_{θ z} \\ τ_{x z} \\ τ_{x θ} \end{array}) = [\begin{array}{c} Q_{11} & Q_{12} & 0 & 0 & 0 \\ Q_{12} & Q_{22} & 0 & 0 & 0 \\ 0 & 0 & Q_{44} & 0 & 0 \\ 0 & 0 & 0 & Q_{55} & 0 \\ 0 & 0 & 0 & 0 & Q_{66} \end{array}] (\begin{array}{c} ϵ_{x x} \\ ϵ_{θ θ} \\ γ_{θ z} \\ γ_{x z} \\ γ_{x θ} \end{array})

(15)

where variable elastic coefficients Q_ij are

Q_{11} (x, z) = Q_{22} (x, z) = \frac{E (x, z)}{1 - ν^{2}}

(16)

Q_{12} (x, z) = \frac{E (x, z) ν}{1 - ν^{2}}

(17)

Q_{44} (x, z) = Q_{55} (x, z) = Q_{66} (x, z) = \frac{E (x, z)}{2 (1 + ν)}

(18)

The strain energy (VE) and kinetic energy (KE) of the BDFG truncated conical shell is stated as follows (Allahkarami et al., 2020a):

V E = \frac{1}{2} \int_{L_{1}}^{L_{2}} \int_{0}^{2 π} \int_{- \frac{h}{2}}^{\frac{h}{2}} (σ_{x x} ϵ_{x x} + σ_{θ θ} ϵ_{θ θ} + τ_{θ z} γ_{θ z} + τ_{x z} γ_{x z} + τ_{x θ} γ_{x θ}) R (x) d z d θ d x

(19)

K E = \frac{1}{2} \int_{L_{1}}^{L_{2}} \int_{0}^{2 π} \int_{- \frac{h}{2}}^{\frac{h}{2}} ρ (x, z) ({(\frac{\partial U}{\partial t})}^{2} + {(\frac{\partial V}{\partial t})}^{2} + {(\frac{\partial W}{\partial t})}^{2}) R (x) d z d θ d x

(20)

where

R (x) = x \sin (α)

. The motion equations of the BDFG conical shells are derived using Hamilton’s principle:

\int_{0}^{T} (δ V E - δ K E) d t = 0

(21)

where T is the time period. By substituting equations (7)–(18) into equations (19) and (20) and replacing the results into equation (21) using integration by parts and equaling the coefficients of

(δ u_{0}, δ v_{0}, δ w_{0}, δ ϕ_{x x}, δ ϕ_{θ θ})

to zero, the motion equations of BDFG conical shell can be written in the following types:

- \frac{\sin (α) N_{θ θ}}{R (x)} + \frac{\frac{\partial N_{x θ}}{\partial θ}}{R (x)} + \frac{\sin (α) N_{x x}}{R (x)} + \frac{\partial N_{x x}}{\partial x} = I_{0} \ddot{u} + I_{1} {\ddot{ϕ}}_{x x}

(22)

\frac{\frac{\partial N_{θ θ}}{\partial θ}}{R (x)} + \frac{2 \sin (α) N_{x θ}}{R (x)} + \frac{\partial N_{x θ}}{\partial x} + \frac{\cos (α) Q_{θ θ}}{R (x)} = I_{0} \ddot{v} + I_{1} {\ddot{ϕ}}_{θ θ}

(23)

\frac{2 N_{x θ} \frac{\partial^{2} w}{\partial x \partial θ}}{R (x)} + \frac{N_{θ θ} \frac{\partial^{2} w}{\partial θ^{2}}}{{R (x)}^{2}} + \frac{\frac{\partial w}{\partial θ} \frac{\partial N_{θ θ}}{\partial θ}}{{R (x)}^{2}} + \frac{\frac{\partial w}{\partial θ} \frac{\partial N_{x θ}}{\partial x}}{R (x)} + \frac{\frac{\partial w}{\partial x} \frac{\partial N_{x θ}}{\partial θ}}{R (x)} + \frac{\sin (α) N_{x x} \frac{\partial w}{\partial x}}{R (x)} - \frac{\cos (α) N_{θ θ}}{R (x)} + N_{x x} \frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial N_{x x}}{\partial x} + \frac{\frac{\partial Q_{θ θ}}{\partial θ}}{R (x)} + \frac{\sin (α) Q_{x x}}{R (x)} + \frac{\partial Q_{x x}}{\partial x} = I_{0} \frac{\partial^{2} w}{\partial t^{2}}

(24)

\frac{\frac{\partial M_{x θ}}{\partial θ}}{R (x)} + \frac{M_{x x} - M_{θ θ}}{x} + \frac{\partial M_{x x}}{\partial x} - Q_{x x} = I_{1} \ddot{u} + I_{2} {\ddot{ϕ}}_{x x}

(25)

\frac{\frac{\partial M_{θ θ}}{\partial θ}}{R (x)} + \frac{2 \sin (α) M_{x θ}}{R (x)} + \frac{\partial M_{x θ}}{\partial x} - Q_{θ θ} = I_{1} \ddot{v} + I_{2} {\ddot{ϕ}}_{θ θ}

(26)

In which force resultants (N), moments resultants (M), transverse force resultants (Q), and inertia related terms (I) of the BDFG truncated conical shell are defined as follows:

{\begin{array}{c} N_{x x} \\ N_{θ θ} \\ N_{x θ} \end{array}} = \int_{- \frac{h}{2}}^{+ \frac{h}{2}} {\begin{array}{c} σ_{x x} \\ σ_{θ θ} \\ τ_{x θ} \end{array}} d z

(27)

{\begin{array}{c} M_{x x} \\ M_{θ θ} \\ M_{x θ} \end{array}} = \int_{- \frac{h}{2}}^{+ \frac{h}{2}} {\begin{array}{c} σ_{x x} \\ σ_{θ θ} \\ τ_{x θ} \end{array}} z d z

(28)

{\begin{array}{c} Q_{x x} \\ Q_{θ θ} \end{array}} = K \int_{- \frac{h}{2}}^{+ \frac{h}{2}} {\begin{array}{c} τ_{x z} \\ τ_{θ z} \end{array}} d z

(29)

{\begin{array}{c} I_{0} \\ I_{1} \\ I_{2} \end{array}} = \int_{- \frac{h}{2}}^{+ \frac{h}{2}} {\begin{array}{c} 1 \\ z \\ z^{2} \end{array}} ρ (x, z) d z

(30)

where $K$ is shear correction factor that is considered as 5/6 in this study. By substituting equations (6) and (15) into equations (27)–(30) and replacing the results into equations (22)–(26), the motion equations of the system can be expanded as

\frac{\csc α A_{12} \frac{\partial^{2} v}{\partial x \partial θ}}{x} + \frac{\csc α A_{66} \frac{\partial^{2} v}{\partial x \partial θ}}{x} + \frac{\csc^{2} α A_{12} \frac{\partial w}{\partial θ} \frac{\partial^{2} w}{\partial x \partial θ}}{x^{2}} + \frac{\csc^{2} α A_{66} \frac{\partial w}{\partial θ} \frac{\partial^{2} w}{\partial x \partial θ}}{x^{2}} + \frac{\csc α B_{12} \frac{\partial^{2} ϕ_{θ θ}}{\partial x \partial θ}}{x} + \frac{\csc α B_{66} \frac{\partial^{2} ϕ_{θ θ}}{\partial x \partial θ}}{x} + \frac{\csc^{2} α A_{66} \frac{\partial^{2} u}{\partial θ^{2}}}{x^{2}} - \frac{A_{22} u}{x^{2}} + A_{11} \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial A_{11}}{\partial x} \frac{\partial u}{\partial x} + \frac{\frac{\partial A_{12}}{\partial x} u}{x} + \frac{A_{11} \frac{\partial u}{\partial x}}{x} - \frac{\csc α A_{22} \frac{\partial v}{\partial θ}}{x^{2}} - \frac{\csc α A_{66} \frac{\partial v}{\partial θ}}{x^{2}} + \frac{\csc α \frac{\partial A_{12}}{\partial x} \frac{\partial v}{\partial θ}}{x} - \frac{\csc^{2} α {A_{12} (\frac{\partial w}{\partial θ})}^{2}}{2 x^{3}} - \frac{\csc^{2} α {A_{22} (\frac{\partial w}{\partial θ})}^{2}}{2 x^{3}} + \frac{\csc^{2} α A_{66} \frac{\partial w}{\partial x} \frac{\partial^{2} w}{\partial θ^{2}}}{x^{2}} - \frac{\cot α A_{22} w}{x^{2}} + \frac{\csc^{2} α \frac{\partial A_{12}}{\partial x} {(\frac{\partial w}{\partial θ})}^{2}}{2 x^{2}} + A_{11} \frac{\partial w}{\partial x} \frac{\partial^{2} w}{\partial x^{2}} + \frac{\cot α \frac{\partial A_{12}}{\partial x} w}{x} + \frac{\cot α A_{12} \frac{\partial w}{\partial x}}{x} + \frac{1}{2} \frac{\partial A_{11}}{\partial x} {(\frac{\partial w}{\partial x})}^{2} + \frac{{A_{11} (\frac{\partial w}{\partial x})}^{2}}{2 x} - \frac{{A_{12} (\frac{\partial w}{\partial x})}^{2}}{2 x} + \frac{\csc^{2} α B_{66} \frac{\partial^{2} ϕ_{x x}}{\partial θ^{2}}}{x^{2}} - \frac{B_{22} ϕ_{x x}}{x^{2}} + B_{11} \frac{\partial^{2} ϕ_{x x}}{\partial x^{2}} - \frac{\csc α B_{22} \frac{\partial ϕ_{θ θ}}{\partial θ}}{x^{2}} - \frac{\csc α B_{66} \frac{\partial ϕ_{θ θ}}{\partial θ}}{x^{2}} + \frac{\partial B_{11}}{\partial x} \frac{\partial ϕ_{x x}}{\partial x} + \frac{B_{11} \frac{\partial ϕ_{x x}}{\partial x}}{x} + \frac{\frac{\partial B_{12}}{\partial x} ϕ_{x x}}{x} + \frac{\csc α \frac{\partial B_{12}}{\partial x} \frac{\partial ϕ_{θ θ}}{\partial θ}}{x} = I_{0} \ddot{u} + I_{1} {\ddot{ϕ}}_{x x}

(31)

\frac{\csc α A_{12} \frac{\partial^{2} u}{\partial x \partial θ}}{x} + \frac{\csc α A_{66} \frac{\partial^{2} u}{\partial x \partial θ}}{x} + \frac{\csc α A_{12} \frac{\partial w}{\partial x} \frac{\partial^{2} w}{\partial x \partial θ}}{x} + \frac{\csc α A_{66} \frac{\partial w}{\partial x} \frac{\partial^{2} w}{\partial x \partial θ}}{x} + \frac{\csc α B_{12} \frac{\partial^{2} ϕ_{x x}}{\partial x \partial θ}}{x} + \frac{\csc α B_{66} \frac{\partial^{2} ϕ_{x x}}{\partial x \partial θ}}{x} - \frac{\cot^{2} α A_{44} v}{x^{2}} + \frac{\cot α \csc α A_{44} \frac{\partial w}{\partial θ}}{x^{2}} + \frac{\cot α A_{44} ϕ_{θ θ}}{x} + \frac{\csc α A_{22} \frac{\partial u}{\partial θ}}{x^{2}} + \frac{\csc α A_{66} \frac{\partial u}{\partial θ}}{x^{2}} + \frac{\csc α \frac{\partial A_{66}}{\partial x} \frac{\partial u}{\partial θ}}{x} + \frac{\csc^{2} α A_{22} \frac{\partial^{2} v}{\partial θ^{2}}}{x^{2}} - \frac{A_{66} v}{x^{2}} + A_{66} \frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial A_{66}}{\partial x} \frac{\partial v}{\partial x} - \frac{\frac{\partial A_{66}}{\partial x} v}{x} + \frac{A_{66} \frac{\partial v}{\partial x}}{x} + \frac{\csc^{3} α A_{22} \frac{\partial w}{\partial θ} \frac{\partial^{2} w}{\partial θ^{2}}}{x^{3}} + \frac{\csc α A_{66} \frac{\partial w}{\partial θ} \frac{\partial^{2} w}{\partial x^{2}}}{x} + \frac{\csc α A_{66} \frac{\partial w}{\partial x} \frac{\partial w}{\partial θ}}{x^{2}} + \frac{\cot α \csc α A_{22} \frac{\partial w}{\partial θ}}{x^{2}} + \frac{\csc α \frac{\partial A_{66}}{\partial x} \frac{\partial w}{\partial x} \frac{\partial w}{\partial θ}}{x} + \frac{\csc α B_{22} \frac{\partial ϕ_{x x}}{\partial θ}}{x^{2}} + \frac{\csc α B_{66} \frac{\partial ϕ_{x x}}{\partial θ}}{x^{2}} + \frac{\csc^{2} α B_{22} \frac{\partial^{2} ϕ_{θ θ}}{\partial θ^{2}}}{x^{2}} - \frac{B_{66} ϕ_{θ θ}}{x^{2}} + B_{66} \frac{\partial^{2} ϕ_{θ θ}}{\partial x^{2}} + \frac{\csc α \frac{\partial B_{66}}{\partial x} \frac{\partial ϕ_{x x}}{\partial θ}}{x} + \frac{\partial B_{66}}{\partial x} \frac{\partial ϕ_{θ θ}}{\partial x} + \frac{B_{66} \frac{\partial ϕ_{θ θ}}{\partial x}}{x} - \frac{\frac{\partial B_{66}}{\partial x} ϕ_{θ θ}}{x} = I_{0} \ddot{v} + I_{1} {\ddot{ϕ}}_{θ θ}

(32)

\frac{3 {(\frac{\partial w}{\partial θ})}^{2} \frac{\partial^{2} w}{\partial θ^{2}} A_{22} \csc^{4} α}{2 x^{4}} + \frac{\frac{\partial^{2} v}{\partial θ^{2}} \frac{\partial w}{\partial θ} A_{22} \csc^{3} α}{x^{3}} + \frac{\frac{\partial v}{\partial θ} \frac{\partial^{2} w}{\partial θ^{2}} A_{22} \csc^{3} α}{x^{3}} + \frac{\frac{\partial^{2} w}{\partial θ^{2}} \frac{\partial ϕ_{θ θ}}{\partial θ} B_{22} \csc^{3} α}{x^{3}} + \frac{\frac{\partial w}{\partial θ} \frac{\partial^{2} ϕ_{θ θ}}{\partial θ^{2}} B_{22} \csc^{3} α}{x^{3}} + \frac{\frac{\partial w}{\partial x} {(\frac{\partial w}{\partial θ})}^{2} \frac{\partial A_{12}}{\partial x} \csc^{2} α}{2 x^{2}} + \frac{\frac{\partial w}{\partial x} {(\frac{\partial w}{\partial θ})}^{2} \frac{\partial A_{66}}{\partial x} \csc^{2} α}{x^{2}} + \frac{\frac{\partial u}{\partial θ} \frac{\partial w}{\partial θ} \frac{\partial A_{66}}{\partial x} \csc^{2} α}{x^{2}} + \frac{\frac{\partial w}{\partial θ} \frac{\partial B_{66}}{\partial x} \frac{\partial ϕ_{x x}}{\partial θ} \csc^{2} α}{x^{2}} - \frac{\frac{\partial w}{\partial x} {(\frac{\partial w}{\partial θ})}^{2} A_{12} \csc^{2} α}{2 x^{3}} + \frac{{(\frac{\partial w}{\partial θ})}^{2} \frac{\partial^{2} w}{\partial x^{2}} A_{12} \csc^{2} α}{2 x^{2}} + \frac{{(\frac{\partial w}{\partial x})}^{2} \frac{\partial^{2} w}{\partial θ^{2}} A_{12} \csc^{2} α}{2 x^{2}} + \frac{\frac{\partial u}{\partial x} \frac{\partial^{2} w}{\partial θ^{2}} A_{12} \csc^{2} α}{x^{2}} + \frac{\frac{\partial^{2} w}{\partial θ^{2}} \frac{\partial ϕ_{x x}}{\partial x} B_{12} \csc^{2} α}{x^{2}} + \frac{\frac{\partial w}{\partial θ} \frac{\partial^{2} u}{\partial x \partial θ} A_{12} \csc^{2} α}{x^{2}} + \frac{2 \frac{\partial w}{\partial x} \frac{\partial w}{\partial θ} \frac{\partial^{2} w}{\partial x \partial θ} A_{12} \csc^{2} α}{x^{2}} + \frac{\frac{\partial w}{\partial θ} \frac{\partial^{2} ϕ_{x x}}{\partial x \partial θ} B_{12} \csc^{2} α}{x^{2}} + \frac{\cot α {(\frac{\partial w}{\partial θ})}^{2} A_{22} \csc^{2} α}{2 x^{3}} + \frac{\frac{\partial u}{\partial θ} \frac{\partial w}{\partial θ} A_{22} \csc^{2} α}{x^{3}} + \frac{\frac{\partial w}{\partial θ} \frac{\partial ϕ_{x x}}{\partial θ} B_{22} \csc^{2} α}{x^{3}} + \frac{\frac{\partial^{2} w}{\partial θ^{2}} u A_{22} \csc^{2} α}{x^{3}} + \frac{\cot α \frac{\partial^{2} w}{\partial θ^{2}} w A_{22} \csc^{2} α}{x^{3}} + \frac{\frac{\partial^{2} w}{\partial θ^{2}} A_{44} \csc^{2} α}{x^{2}} - \frac{\frac{\partial w}{\partial x} {(\frac{\partial w}{\partial θ})}^{2} A_{66} \csc^{2} α}{x^{3}} + \frac{\frac{\partial^{2} u}{\partial θ^{2}} \frac{\partial w}{\partial x} A_{66} \csc^{2} α}{x^{2}} - \frac{\frac{\partial u}{\partial θ} \frac{\partial w}{\partial θ} A_{66} \csc^{2} α}{x^{3}} + \frac{{(\frac{\partial w}{\partial θ})}^{2} \frac{\partial^{2} w}{\partial x^{2}} A_{66} \csc^{2} α}{x^{2}} + \frac{{(\frac{\partial w}{\partial x})}^{2} \frac{\partial^{2} w}{\partial θ^{2}} A_{66} \csc^{2} α}{x^{2}} - \frac{\frac{\partial w}{\partial θ} \frac{\partial ϕ_{x x}}{\partial θ} B_{66} \csc^{2} α}{x^{3}} + \frac{\frac{\partial w}{\partial x} \frac{\partial^{2} ϕ_{x x}}{\partial θ^{2}} B_{66} \csc^{2} α}{x^{2}} + \frac{\frac{\partial w}{\partial θ} \frac{\partial^{2} u}{\partial x \partial θ} A_{66} \csc^{2} α}{x^{2}} + \frac{2 \frac{\partial u}{\partial θ} \frac{\partial^{2} w}{\partial x \partial θ} A_{66} \csc^{2} α}{x^{2}} + \frac{4 \frac{\partial w}{\partial x} \frac{\partial w}{\partial θ} \frac{\partial^{2} w}{\partial x \partial θ} A_{66} \csc^{2} α}{x^{2}} + \frac{2 \frac{\partial ϕ_{x x}}{\partial θ} \frac{\partial^{2} w}{\partial x \partial θ} B_{66} \csc^{2} α}{x^{2}} + \frac{\frac{\partial w}{\partial θ} \frac{\partial^{2} ϕ_{x x}}{\partial x \partial θ} B_{66} \csc^{2} α}{x^{2}} + \frac{\frac{\partial^{2} w}{\partial θ^{2}} B_{22} ϕ_{x x} \csc^{2} α}{x^{3}} + \frac{\frac{\partial v}{\partial θ} \frac{\partial w}{\partial x} \frac{\partial A_{12}}{\partial x} \csc α}{x} + \frac{\frac{\partial v}{\partial x} \frac{\partial w}{\partial θ} \frac{\partial A_{66}}{\partial x} \csc α}{x} + \frac{\frac{\partial w}{\partial θ} \frac{\partial B_{66}}{\partial x} \frac{\partial ϕ_{θ θ}}{\partial x} \csc α}{x} + \frac{\frac{\partial w}{\partial x} \frac{\partial B_{12}}{\partial x} \frac{\partial ϕ_{θ θ}}{\partial θ} \csc α}{x} - \frac{\frac{\partial w}{\partial θ} \frac{\partial A_{66}}{\partial x} v c s c α}{x^{2}} + \frac{\frac{\partial v}{\partial θ} \frac{\partial^{2} w}{\partial x^{2}} A_{12} \csc α}{x} + \frac{\frac{\partial^{2} w}{\partial x^{2}} \frac{\partial ϕ_{θ θ}}{\partial θ} B_{12} \csc α}{x} + \frac{\frac{\partial w}{\partial x} \frac{\partial^{2} v}{\partial x \partial θ} A_{12} \csc α}{x} + \frac{\frac{\partial w}{\partial x} \frac{\partial^{2} ϕ_{θ θ}}{\partial x \partial θ} B_{12} \csc α}{x} - \frac{\cot α \frac{\partial v}{\partial θ} A_{22} \csc α}{x^{2}} - \frac{\cot α \frac{\partial ϕ_{θ θ}}{\partial θ} B_{22} \csc α}{x^{2}} - \frac{\cot α \frac{\partial v}{\partial θ} A_{44} \csc α}{x^{2}} + \frac{\frac{\partial ϕ_{θ θ}}{\partial θ} A_{44} \csc α}{x} - \frac{\frac{\partial v}{\partial θ} \frac{\partial w}{\partial x} A_{66} \csc α}{x^{2}} - \frac{\frac{\partial v}{\partial x} \frac{\partial w}{\partial θ} A_{66} \csc α}{x^{2}} + \frac{\frac{\partial^{2} v}{\partial x^{2}} \frac{\partial w}{\partial θ} A_{66} \csc α}{x} - \frac{\frac{\partial w}{\partial θ} \frac{\partial ϕ_{θ θ}}{\partial x} B_{66} \csc α}{x^{2}} - \frac{\frac{\partial w}{\partial x} \frac{\partial ϕ_{θ θ}}{\partial θ} B_{66} \csc α}{x^{2}} + \frac{\frac{\partial w}{\partial θ} \frac{\partial^{2} ϕ_{θ θ}}{\partial x^{2}} B_{66} \csc α}{x} + \frac{\frac{\partial w}{\partial x} \frac{\partial^{2} v}{\partial x \partial θ} A_{66} \csc α}{x} + \frac{2 \frac{\partial v}{\partial x} \frac{\partial^{2} w}{\partial x \partial θ} A_{66} \csc α}{x} + \frac{2 \frac{\partial ϕ_{θ θ}}{\partial x} \frac{\partial^{2} w}{\partial x \partial θ} B_{66} \csc α}{x} + \frac{\frac{\partial w}{\partial x} \frac{\partial^{2} ϕ_{θ θ}}{\partial x \partial θ} B_{66} \csc α}{x} + \frac{\frac{\partial w}{\partial θ} v A_{66} \csc α}{x^{3}} - \frac{2 \frac{\partial^{2} w}{\partial x \partial θ} v A_{66} \csc α}{x^{2}} - \frac{\frac{\partial w}{\partial θ} \frac{\partial B_{66}}{\partial x} ϕ_{θ θ} \csc α}{x^{2}} + \frac{\frac{\partial w}{\partial θ} B_{66} ϕ_{θ θ} \csc α}{x^{3}} - \frac{2 \frac{\partial^{2} w}{\partial x \partial θ} B_{66} ϕ_{θ θ} \csc α}{x^{2}} + \frac{1}{2} {(\frac{\partial w}{\partial x})}^{3} \frac{\partial A_{11}}{\partial x} + \frac{\partial u}{\partial x} \frac{\partial w}{\partial x} \frac{\partial A_{11}}{\partial x} + \frac{\partial w}{\partial x} \frac{\partial A_{55}}{\partial x} + \frac{\partial w}{\partial x} \frac{\partial B_{11}}{\partial x} \frac{\partial ϕ_{x x}}{\partial x} + \frac{\frac{\partial w}{\partial x} \frac{\partial A_{12}}{\partial x} u}{x} + \frac{\cot α \frac{\partial w}{\partial x} \frac{\partial A_{12}}{\partial x} w}{x} + \frac{{(\frac{\partial w}{\partial x})}^{3} A_{11}}{2 x} + \frac{\frac{\partial u}{\partial x} \frac{\partial w}{\partial x} A_{11}}{x} + \frac{\partial^{2} u}{\partial x^{2}} \frac{\partial w}{\partial x} A_{11} + \frac{3}{2} {(\frac{\partial w}{\partial x})}^{2} \frac{\partial^{2} w}{\partial x^{2}} A_{11} + \frac{\partial u}{\partial x} \frac{\partial^{2} w}{\partial x^{2}} A_{11} + \frac{\frac{\partial w}{\partial x} \frac{\partial ϕ_{x x}}{\partial x} B_{11}}{x} + \frac{\partial^{2} w}{\partial x^{2}} \frac{\partial ϕ_{x x}}{\partial x} B_{11} + \frac{\partial w}{\partial x} \frac{\partial^{2} ϕ_{x x}}{\partial x^{2}} B_{11} + \frac{\cot α {(\frac{\partial w}{\partial x})}^{2} A_{12}}{2 x} - \frac{\cot α \frac{\partial u}{\partial x} A_{12}}{x} + \frac{\frac{\partial u}{\partial x} \frac{\partial w}{\partial x} A_{12}}{x} - \frac{\cot α \frac{\partial ϕ_{x x}}{\partial x} B_{12}}{x} + \frac{\frac{\partial w}{\partial x} \frac{\partial ϕ_{x x}}{\partial x} B_{12}}{x} + \frac{\frac{\partial^{2} w}{\partial x^{2}} u A_{12}}{x} + \frac{\cot α \frac{\partial^{2} w}{\partial x^{2}} w A_{12}}{x} - \frac{\cot α u A_{22}}{x^{2}} - \frac{\cot^{2} α w A_{22}}{x^{2}} + \frac{\frac{\partial w}{\partial x} A_{55}}{x} + \frac{\partial^{2} w}{\partial x^{2}} A_{55} + \frac{\partial ϕ_{x x}}{\partial x} A_{55} + \frac{\frac{\partial w}{\partial x} \frac{\partial B_{12}}{\partial x} ϕ_{x x}}{x} + \frac{\partial A_{55}}{\partial x} ϕ_{x x} + \frac{\frac{\partial^{2} w}{\partial x^{2}} B_{12} ϕ_{x x}}{x} - \frac{\cot α B_{22} ϕ_{x x}}{x^{2}} + \frac{A_{55} ϕ_{x x}}{x} = I_{0} \frac{\partial^{2} w}{\partial t^{2}}

(33)

\frac{\partial D_{11}}{\partial x} \frac{\partial ϕ_{x x}}{\partial x} + \frac{\csc α \frac{\partial D_{12}}{\partial x} \frac{\partial ϕ_{θ θ}}{\partial θ}}{x} + \frac{\frac{\partial ϕ_{x x}}{\partial x} D_{11}}{x} + \frac{\partial^{2} ϕ_{x x}}{\partial x^{2}} D_{11} + \frac{\csc α \frac{\partial^{2} ϕ_{θ θ}}{\partial x \partial θ} D_{12}}{x} - \frac{\csc α \frac{\partial ϕ_{θ θ}}{\partial θ} D_{22}}{x^{2}} + \frac{\csc^{2} α \frac{\partial^{2} ϕ_{x x}}{\partial θ^{2}} D_{66}}{x^{2}} - \frac{\csc α \frac{\partial ϕ_{θ θ}}{\partial θ} D_{66}}{x^{2}} + \frac{\csc α \frac{\partial^{2} ϕ_{θ θ}}{\partial x \partial θ} D_{66}}{x} + \frac{\frac{\partial D_{12}}{\partial x} ϕ_{x x}}{x} - \frac{D_{22} ϕ_{x x}}{x^{2}} + \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2} \frac{\partial B_{11}}{\partial x} + \frac{\partial u}{\partial x} \frac{\partial B_{11}}{\partial x} + \frac{\csc^{2} α {(\frac{\partial w}{\partial θ})}^{2} \frac{\partial B_{12}}{\partial x}}{2 x^{2}} + \frac{\csc α \frac{\partial v}{\partial θ} \frac{\partial B_{12}}{\partial x}}{x} + \frac{\frac{\partial B_{12}}{\partial x} u}{x} + \frac{\cot α \frac{\partial B_{12}}{\partial x} w}{x} + \frac{{(\frac{\partial w}{\partial x})}^{2} B_{11}}{2 x} + \frac{\frac{\partial u}{\partial x} B_{11}}{x} + \frac{\partial^{2} u}{\partial x^{2}} B_{11} + \frac{\partial w}{\partial x} \frac{\partial^{2} w}{\partial x^{2}} B_{11} - \frac{{(\frac{\partial w}{\partial x})}^{2} B_{12}}{2 x} - \frac{\csc^{2} α {(\frac{\partial w}{\partial θ})}^{2} B_{12}}{2 x^{3}} + \frac{\cot α \frac{\partial w}{\partial x} B_{12}}{x} + \frac{\csc α \frac{\partial^{2} v}{\partial x \partial θ} B_{12}}{x} + \frac{\csc^{2} α \frac{\partial w}{\partial θ} \frac{\partial^{2} w}{\partial x \partial θ} B_{12}}{x^{2}} - \frac{\csc^{2} α {(\frac{\partial w}{\partial θ})}^{2} B_{22}}{2 x^{3}} - \frac{\csc α \frac{\partial v}{\partial θ} B_{22}}{x^{2}} - \frac{u B_{22}}{x^{2}} - \frac{\cot α w B_{22}}{x^{2}} + \frac{\csc^{2} α \frac{\partial^{2} u}{\partial θ^{2}} B_{66}}{x^{2}} - \frac{\csc α \frac{\partial v}{\partial θ} B_{66}}{x^{2}} + \frac{\csc^{2} α \frac{\partial w}{\partial x} \frac{\partial^{2} w}{\partial θ^{2}} B_{66}}{x^{2}} + \frac{\csc α \frac{\partial^{2} v}{\partial x \partial θ} B_{66}}{x} + \frac{\csc^{2} α \frac{\partial w}{\partial θ} \frac{\partial^{2} w}{\partial x \partial θ} B_{66}}{x^{2}} - \frac{\partial w}{\partial x} A_{55} - A_{55} ϕ_{x x} = I_{1} \ddot{u} + I_{2} {\ddot{ϕ}}_{x x}

(34)

\frac{\csc α B_{12} \frac{\partial^{2} u}{\partial x \partial θ}}{x} + \frac{\csc α B_{66} \frac{\partial^{2} u}{\partial x \partial θ}}{x} + \frac{\csc α B_{12} \frac{\partial w}{\partial x} \frac{\partial^{2} w}{\partial x \partial θ}}{x} + \frac{\csc α B_{66} \frac{\partial w}{\partial x} \frac{\partial^{2} w}{\partial x \partial θ}}{x} + \frac{\csc α D_{12} \frac{\partial^{2} ϕ_{x x}}{\partial x \partial θ}}{x} + \frac{\csc α D_{66} \frac{\partial^{2} ϕ_{x x}}{\partial x \partial θ}}{x} + \frac{\csc α B_{22} \frac{\partial u}{\partial θ}}{x^{2}} + \frac{\csc α B_{66} \frac{\partial u}{\partial θ}}{x^{2}} + \frac{\csc α \frac{\partial B_{66}}{\partial x} \frac{\partial u}{\partial θ}}{x} + \frac{\csc^{2} α B_{22} \frac{\partial^{2} v}{\partial θ^{2}}}{x^{2}} - \frac{B_{66} v}{x^{2}} + B_{66} \frac{\partial^{2} v}{\partial x^{2}} + \frac{\cot α A_{44} v}{x} + \frac{\partial B_{66}}{\partial x} \frac{\partial v}{\partial x} - \frac{\frac{\partial B_{66}}{\partial x} v}{x} + \frac{B_{66} \frac{\partial v}{\partial x}}{x} + \frac{\csc^{3} α B_{22} \frac{\partial w}{\partial θ} \frac{\partial^{2} w}{\partial θ^{2}}}{x^{3}} + \frac{\csc α B_{66} \frac{\partial w}{\partial θ} \frac{\partial^{2} w}{\partial x^{2}}}{x} + \frac{\csc α B_{66} \frac{\partial w}{\partial x} \frac{\partial w}{\partial θ}}{x^{2}} + \frac{\cot α \csc α B_{22} \frac{\partial w}{\partial θ}}{x^{2}} + \frac{\csc α \frac{\partial B_{66}}{\partial x} \frac{\partial w}{\partial x} \frac{\partial w}{\partial θ}}{x} - \frac{\csc α A_{44} \frac{\partial w}{\partial θ}}{x} + \frac{\csc α D_{22} \frac{\partial ϕ_{x x}}{\partial θ}}{x^{2}} + \frac{\csc α D_{66} \frac{\partial ϕ_{x x}}{\partial θ}}{x^{2}} + \frac{\csc^{2} α D_{22} \frac{\partial^{2} ϕ_{θ θ}}{\partial θ^{2}}}{x^{2}} - \frac{D_{66} ϕ_{θ θ}}{x^{2}} + D_{66} \frac{\partial^{2} ϕ_{θ θ}}{\partial x^{2}} + \frac{\csc α \frac{\partial D_{66}}{\partial x} \frac{\partial ϕ_{x x}}{\partial θ}}{x} + \frac{\partial D_{66}}{\partial x} \frac{\partial ϕ_{θ θ}}{\partial x} + \frac{D_{66} \frac{\partial ϕ_{θ θ}}{\partial x}}{x} - \frac{\frac{\partial D_{66}}{\partial x} ϕ_{θ θ}}{x} - A_{44} ϕ_{θ θ} = I_{1} \ddot{v} + I_{2} {\ddot{ϕ}}_{θ θ}

(35)

where A_ij, B_ij, and D_ij are

(A_{i j} (x), B_{i j} (x), D_{i j} (x)) = \int_{- h / 2}^{h / 2} Q_{i j} (x . z) (1, z, z^{2}) d z

(36)

The boundary conditions of the shell which simply supported on both edges are considered as follows:

a t x = 0, L v = w = N_{x x} = M_{x x} = M_{x θ} = 0

(37)

The geometric boundary conditions are satisfied by the following approximate function:

u (x, θ, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} U (t) \cos (\frac{m π (x - L_{1})}{L}) \cos (n θ)

(38)

v (x, θ, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} V (t) \sin (\frac{m π (x - L_{1})}{L}) \sin (n θ)

(39)

w (x, θ, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} W (t) \sin (\frac{m π (x - L_{1})}{L}) \cos (n θ)

(40)

ϕ_{x x} (x, θ, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} X (t) \cos (\frac{m π (x - L_{1})}{L}) \cos (n θ)

(41)

ϕ_{θ θ} (x, θ, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} Y (t) \sin (\frac{m π (x - L_{1})}{L}) \sin (n θ)

(42)

where n and m refer to circumferential and longitudinal wave numbers, respectively. For the sake of generality of the formulations, a set of non-dimensional parameters are introduced as follows:

(\bar{U}, \bar{V}, \bar{W}) = \frac{(U, V, W)}{h}, \bar{x} = \frac{x}{L}, \bar{L_{1}} = \frac{L_{1}}{L}, \bar{L_{2}} = \frac{L_{2}}{L}, H = \frac{h}{L} (a_{i j}, b_{i j}, d_{i j}) = (\frac{A_{i j} h^{2}}{D_{c}}, \frac{B_{i j} h}{D_{c}}, \frac{D_{i j}}{D_{c}}), (I_{0}, I_{1}, I_{2}) = (\frac{I_{0}}{I_{c}}, \frac{I_{1}}{I_{c} h}, \frac{I_{2}}{I_{c} h^{2}}), τ = \frac{t}{h} \sqrt{\frac{D_{c}}{I_{c}}}

(43)

By substituting equations (38)–(42) into equations (31)–(35) and employing Galerkin approach, the nonlinear partial differential equations of motion are transformed into the time-dependent nonlinear ordinary differential equations in the following form:

L_{11} \bar{U} + L_{12} \bar{V} + L_{13} \bar{W} + L_{14} X + L_{15} Y + N_{11} {\bar{W}}^{2} = 0

(44)

L_{21} \bar{U} + L_{22} \bar{V} + L_{23} \bar{W} + L_{24} X + L_{25} Y + N_{21} {\bar{W}}^{2} = 0

(45)

L_{31} \bar{U} + L_{32} \bar{V} + L_{33} \bar{W} + L_{34} X + L_{35} Y + N_{31} \bar{U} \bar{W} + N_{32} \bar{V} \bar{W} + N_{33} {\bar{W}}^{2} + N_{34} {\bar{W}}^{3} + N_{35} X \bar{W} + N_{36} Y \bar{W} + L_{36} \frac{\partial^{2} \bar{W}}{\partial τ^{2}} = 0

(46)

L_{41} \bar{U} + L_{42} \bar{V} + L_{43} \bar{W} + L_{44} X + L_{45} Y + N_{41} {\bar{W}}^{2} = 0

(47)

L_{51} \bar{U} + L_{52} \bar{V} + L_{53} \bar{W} + L_{54} X + L_{55} Y + N_{51} {\bar{W}}^{2} = 0

(48)

where $L_{i j}$ and N_ij are the non-dimensional coefficients of linear and nonlinear terms, respectively, which are related to the geometry and mechanical properties of the shell. In above equations, in-plane inertia effects and rotary inertia effects can be ignored (Hashemi and Jafari, 2020b). By solving equations (44), (45), (47), and (48), the terms $\bar{U}$ , $\bar{V}$ , $X$ , and $Y$ are calculated in terms of $\bar{W}$ . Then, by inserting the results in equation (46), finally, the nonlinear time-dependent equation of the transverse motion of the shell is obtained as follows:

\ddot{\bar{W}} + μ \bar{W} + β {\bar{W}}^{2} + γ {\bar{W}}^{3} = 0

(49)

where $μ$ , $β$ , and $γ$ are in terms of $L_{i j}$ and N_ij coefficients.

4. Improved Lindstedt–Poincaré method

In this section, in order to analytically solve the nonlinear equation (49), the improved Lindstedt–Poincaré method is used (Hashemi and Jafari, 2021). For this purpose, equation (49) should be rewritten in the following form:

\ddot{\bar{W}} + μ \bar{W} + ϵ β {\bar{W}}^{2} + ϵ γ {\bar{W}}^{3} = 0

I n i t i a l C o n d i t i o n : \bar{W} (0) = \frac{W_{\max}}{h} = A, \frac{d \bar{W} (0)}{d τ} = 0

(50)

where A is the non-dimensional maximum vibration amplitude, and

ϵ

is bookkeeping parameter (Hashemi and Jafari, 2021). The linear natural frequency is calculated from:

ω_{L} = \sqrt{μ}

(51)

For the necessity of proposed method, the coefficients

μ

and

\bar{W} (τ)

can be written as a power series of it (

ϵ

\bar{W} = {\bar{W}}_{0} + ϵ {\bar{W}}_{1} + ϵ^{2} {\bar{W}}_{2} + \dots

(52)

μ = ω^{2} + ϵ c_{1} + ϵ^{2} c_{2} + \dots

(53)

The coefficients $c_{i} a n d ω$ in equation (53) are unknown which are calculated later. By substituting equations (52) and (53) into equation (49), and then equating coefficients $ϵ^{0}, ϵ^{1}, ϵ^{2}$ to zero, yield the following equations:

ϵ^{0} : \ddot{\bar{W}} + ω^{2} {\bar{W}}_{0} = 0, {\bar{W}}_{0} (0) = A, \frac{d {\bar{W}}_{0}}{d τ} (0) = 0

(54)

ϵ^{1} : {\ddot{\bar{W}}}_{1} + ω^{2} {\bar{W}}_{1} = - c_{1} {\bar{W}}_{0} - β {\bar{W}}_{0}^{2} - γ {\bar{W}}_{0}^{3}, {\bar{W}}_{1} (0) = 0, \frac{d {\bar{W}}_{1}}{d τ} (0) = 0

(55)

{ϵ^{2} : \ddot{\bar{W}}}_{2} + ω^{2} {\bar{W}}_{2} = - c_{2} {\bar{W}}_{0} - c_{1} {\bar{W}}_{1} - 2 β {\bar{W}}_{0} {\bar{W}}_{1} - 3 γ {\bar{W}}_{1} {\bar{W}}_{0}^{2} {\bar{W}}_{2} (0) = 0, \frac{d {\bar{W}}_{2}}{d τ} (0) = 0

(56)

The solution of equation (54) is

{\bar{W}}_{0} = A \cos (ω τ)

(57)

By substituting equation (57) into equation (55):

{\ddot{\bar{W}}}_{1} + ω^{2} {\bar{W}}_{1} = (- c_{1} A - \frac{3}{4} γ A^{3}) \cos (ω τ) - \frac{1}{2} β A^{2} \cos (2 ω τ) - \frac{β A^{2}}{2}

(58)

By setting the secular term equal to zero in equation (58), the coefficient

c_{1}

is obtained:

c_{1} = - \frac{3}{4} γ A^{2}

(59)

The solution of equation (58) is as follows:

{\bar{W}}_{1} = (\frac{β A^{2}}{3 ω^{2}} - \frac{γ A^{3}}{32 ω^{2}}) \cos (ω τ) + \frac{β A^{2}}{6 ω^{2}} \cos (2 ω τ) + \frac{γ A^{3}}{32 ω^{2}} \cos (3 ω τ) - \frac{β A^{2}}{2 ω^{2}}

(60)

Similarly, substituting equations (57) and (60) into equation (56) and eliminating the secular term, the coefficient

c_{2}

is obtained as follows:

c_{2} = \frac{β γ A^{3}}{4 ω^{2}} - \frac{3 γ^{2} A^{4}}{128 ω^{2}} + \frac{5 β^{2} A^{2}}{6 ω^{2}} - \frac{3 β γ A^{3}}{4 ω^{2}} + \frac{3 γ^{2} A^{4}}{64 ω^{2}}

(61)

Finally, from equation (53), the nonlinear natural frequency of the BDFG truncated conical shell is obtained:

ω_{N L} = \sqrt{\frac{(μ + \frac{3}{4} γ ϵ A^{2}) + Λ}{2}} Λ = \sqrt{{(μ + \frac{3}{4} ϵ γ A^{2})}^{2} + (2 β γ A^{3} - \frac{10 β^{2} A^{2}}{3} - \frac{3 γ^{2} A^{4}}{32}) ϵ^{2}}

(62)

5. Results and discussion

5.1. Validation

The present study has investigated the nonlinear vibrations analysis of BDFG truncated conical shells based on FSDT and using an analytical approach. Before presenting the parametric study, it is necessary to validate the results. For this purpose, the non-dimensional frequency (

λ = ω_{L} R_{2} \sqrt{ρ (1 - ν^{2}) / E}

) of an isotropic truncated conical shell with m = 1, h/R₂ = 0.01, L sin(α)/R₂ = 0.25 for various circumferential wave number (n) and semi-vertex angles (α) are compared with previously published papers and reported in Tables 2 and 3. These tables demonstrate that the results have good agreement with those outlined in literatures.

Table 2.

Frequency parameter results for isotropic conical shells for different values of circumferential wave number (m = 1, h/R₂ = 0.01, L sin(α)/R₂ = 0.25, ν = 0.3, α = 45^o,60°).

	$λ$
	α = 45°					α = 60°
n	Present	1^*	2^*	3^*	4^*	Present	1^*	2^*	3^*	4^*
2	0.7661	0.7656	0.6879	0.7642	0.7655	0.6367	0.6364	0.5722	0.6342	0.6348
3	0.7261	0.7253	0.6973	0.7211	0.7212	0.6300	0.6296	0.6001	0.6236	0.6238
4	0.6849	0.6838	0.6664	0.6747	0.6739	0.6267	0.6259	0.6054	0.6146	0.6145
5	0.6513	0.6500	0.6304	0.6336	0.6323	0.6305	0.6295	0.6077	0.6113	0.6111
6	0.6320	0.6306	0.6032	0.6049	0.6035	0.6447	0.6435	0.6159	0.6172	0.6171

1^*: (Mohammadrezazadeh and Jafari, 2021a); 2^*: (Irie, 1984); 3^*: (Li et al., 2009); 4^*: (Lam and Hua, 1999).

Table 3.

Comparison of the frequency parameter results of this study with literature for isotropic conical shells (m = 1, h/R₂ = 0.01, L sin(α)/R₂ = 0.25, ν = 0.3, α = 30°).

	$λ$
n	Present	1^*	2^*	3^*
2	0.8425	0.7910	0.8416	0.8420
3	0.7429	0.7284	0.7415	0.7376
4	0.6483	0.6352	0.6468	0.6362

1^*: (Lam and Hua, 1999); 2^*: (Mohammadrezazadeh and Jafari, 2021b); 3^*: (Pradhan and Reddy, 2004).

5.2. Parametric study

The present section concentrates on the nonlinear vibrations analysis of BDFG truncated conical shell with simply supported boundary conditions. The studied BDFG conical shell is made of two materials: Aluminum (Al) as the metal phase and alumina (Al₂O₃) as the ceramic phase of FGM, whose material properties have been given in Table 1.

Figure 2 illustrates the effect of semi-vertex angle (α) on the nonlinear vibration response (backbone curve) of a BDFG truncated conical shell for constant coefficients of m = n = 1, h/R₂ = 0.01, L₂ = 2, N_x = N_z = 1. According to this figure, it is observed that by increasing the semi-vertex angle, the linear natural frequency of the BDFG conical shell decreases. However, the hardening behavior of the shell impressively increases by increasing semi-vertex angle.

The effect of thickness to large radius ratio (h/R₂) on the backbone curve of a BDFG truncated conical shell is shown in Figure 3. This figure is plotted for constant parameters of m = n = 1, α = 60°, L₂ = 2, N_x = N_z = 1. According to this figure, it is revealed that by increasing thickness to large radius ratio, both linear natural frequency and hardening behavior of the BDFG conical shell significantly increases. The reason is that by increasing the thickness to large radius ratio, while the other parameters are assumed to be constant, the shell becomes thicker and consequently stiffer, and as a result, its natural frequency and hardening behavior increases.

Figure 4 depicts the efficacy of the length ratio (L₂/L) on the nonlinear vibration response of a BDFG truncated conical shell for constant parameters of m = n = 1, h/R₂ = 0.01, α = 60°, N_x = N_z = 1. This figure represents that the linear natural frequency, nonlinear frequency ratio, and hardening behavior of the shell increases as the length ratio of the shell increases.

Figure 5 reveals the nonlinear vibration response diagram of a BDFG truncated conical shell for different circumferential wave number (n) when other parameters are m = 1, L₂/L = 2, h/R₂ = 0.01, α = 60°, N_x = N_z = 1. The results of this figure indicate that increasing the circumferential wave number from 2 to 6 leads to an increase in the hardening behavior of the shell. Furthermore, by increasing the circumferential wave number from 2 to 4, the shell’s linear natural frequency decreases while by increasing the circumferential wave number from 4 to 6, the shell’s linear natural frequency increases.

The effect of two-directional functionally graded materials on the nonlinear behavior of the cylindrical shells are revealed in Figures 6 to 11. The constant parameters of m = n = 1, h/R₂ = 0.01, L₂ = 2, and α = 60 are considered in these figures.

The effect of the transverse gradient index (N_z) on the amplitude-frequency curves of a BDFG truncated conical shell for different longitudinal gradient index (N_x) are examined in Figures 6 to 8 As indicated in Figures 6 to 8, it is figured out that by increasing the transverse gradient index from 0 to 4, the dimensionless linear frequency of the BDFG truncated conical shell remarkably decrease by about 32%, 26%, and 20% for N_x = 0,1, and 2, respectively. It is also to be noted that maximum frequency corresponds to N_z = N_x = 0 which is the case of the homogeneous isotropic truncated conical shell with pure ceramic material constituents. In addition, these figures indicate that by increasing the transverse gradient index from 0 to 4, the dimensionless nonlinear frequency of the BDFG truncated conical shell significantly decreases by about 55%, 50%, and 44% for N_x = 0,1, and 2, respectively. The results illustrate that the effect of transverse gradient index on the nonlinear frequency is significantly more than linear natural frequency of the shell. In fact, by increasing the transverse gradient index, the volume fraction of ceramic phase of the shell decreases while those of metal phase of the shell increases that it causes to decrease in total stiffness, and consequently the linear and nonlinear natural frequency of the structure. Furthermore, these figures illustrate that by increasing vibration amplitude, the nonlinear frequency of the BDFG truncated conical shell increases, which means hardening behavior of the shell, whereas linear frequency remains constant.

The influence of the longitudinal gradient index (N_x) on the amplitude-frequency diagrams of a BDFG truncated conical shell for different transverse gradient index (N_z) are investigated in Figures 9 to 11. According to these figures, it is found that by increasing the longitudinal gradient index from 0 to 4, the dimensionless linear frequency of the BDFG truncated conical shell remarkably decrease by about 39%, 32%, and 26% for N_z = 0, 1, and 2, respectively. Another fact that is observed in these figures is that by increasing the longitudinal gradient index from 0 to 4, the dimensionless nonlinear frequency of the BDFG truncated conical shell significantly decreases by about 38%, 24%, and 17% for N_z = 0,1, and 2, respectively. In addition, by comparing Figures 6 to 11, it is revealed that the effect of transverse gradient index on the nonlinear frequency of the system is more than longitudinal gradient index.

6. Conclusion

In this research, the nonlinear free vibrations analysis of bidirectional functionally graded (BDFG) truncated conical shells based on first-order shear deformation theory (FSDT) was conducted. The materials were assumed to be isotropic and inhomogeneous through the length and thickness of truncated conical shell. The system of partial differential equations was converted into the set of ordinary differential equations by Galerkin approach and then solved through improved Lindstedt–Poincaré method. Previously published data were employed to verify results for isotropic truncated conical shell, and results show outstanding agreement with them. Eventually, the effects of some system parameters on the linear and nonlinear frequency parameters of the BDFG truncated conical shell were carried out. The main important outcomes of the present study could be listed as follows:

• As the semi-vertex angle increases, the linear natural frequency of the BDFG conical shell decreases while the hardening behavior of the shell significantly increases.

• By increasing thickness to large radius ratio, linear natural frequency and hardening behavior of the BDFG conical shell significantly increases.

• As the length ratio of the shell increases, the linear natural frequency, nonlinear frequency ratio, and hardening behavior of the shell increases.

• By increasing the circumferential wave number from 2 to 4, the shell’s linear natural frequency decreases, while by increasing the circumferential wave number from 4 to 6, the shell’s linear natural frequency increases. However, increasing the circumferential wave number from 2 to 6 leads to an increase in the hardening behavior of the shell.

• Independently of all the system parameters, increasing the vibration amplitude leads to an increase in the nonlinear to linear frequency ratio of the plate.

• By enlarging the longitudinal and transverse gradient indexes, the linear and nonlinear frequency of the BDFG truncated conical shells decrease.

• The influence of transverse gradient index on the nonlinear frequency is significantly more than linear natural frequency of the shell.

• The effect of transverse gradient index on the nonlinear frequency of the system is more than longitudinal gradient index.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Mahmoud Shadmani

References

Allahkarami

Ghassabzadeh Saryazdi

Tohidi

(2020a) Dynamic buckling analysis of bi-directional functionally graded porous truncated conical shell with different boundary conditions. Composite Structures 252: 112680.

Allahkarami

Tohidi

Dimitri

, et al. (2020b) Dynamic stability of bi-directional functionally graded porous cylindrical shells embedded in an elastic foundation. Applied Sciences 10(4): 1345.

Amirabadi

Farhatnia

Civalek

(2021) Frequency response of rotating two-directional functionally graded GPL-reinforced conical shells on elastic foundation. Journal of the Brazilian Society of Mechanical Sciences and Engineering 43(7): 349–424.

Bahadori

Najafizadeh

(2015) Free vibration analysis of two-dimensional functionally graded axisymmetric cylindrical shell on Winkler–Pasternak elastic foundation by First-order Shear Deformation Theory and using Navier-differential quadrature solution methods. Applied Mathematical Modelling 39(16): 4877–4894.

Civalek

Avcar

(2022) Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method. Engineering with Computers 38(1): 489–521.

Civalek

Dastjerdi

Akgöz

(2022) Buckling and free vibrations of CNT-reinforced cross-ply laminated composite plates. Mechanics Based Design of Structures and Machines 50(6): 1914–1931.

Dastjerdi

Akgöz

Civalek

, et al. (2020) On the non-linear dynamics of torus-shaped and cylindrical shell structures. International Journal of Engineering Science 156: 103371.

Dong

Wang

(2022) A comprehensive study on the coupled multi-mode vibrations of cylindrical shells. Mechanical Systems and Signal Processing 169: 108730.

Hashemi

Jafari

(2020a) An analytical solution for nonlinear vibrations analysis of functionally graded plate using modified Lindstedt–Poincare method. International Journal of Applied Mechanics 12(01): 2050003.

10.

Hashemi

Jafari

(2020b) Nonlinear free and forced vibrations of in-plane bi-directional functionally graded rectangular plate with temperature-dependent properties. International Journal of Structural Stability and Dynamics 20(08): 2050097.

11.

Hashemi

Jafari

(2021) An analytical solution for nonlinear vibration analysis of functionally graded rectangular plate in contact with fluid. Advances in Applied Mathematics and Mechanics 13(4): 914–941.

12.

Hashemi

Shahri

Beigzadeh

, et al. (2022) Nonlinear free vibration analysis of In-plane Bi-directional functionally graded plate with porosities resting on elastic foundations. International Journal of Applied Mechanics 14(01): 2150131.

13.

Hashemi

Zamani

Eftekhari

, et al. (2023) On the vibration of functionally graded annular plate with elastic edge supports and resting on Winkler foundation. Australian Journal of Mechanical Engineering 21(3): 926–941.

14.

Chen

, et al. (2018) A solar responsive photocatalytic fuel cell with the membrane electrode assembly design for simultaneous wastewater treatment and electricity generation. Journal of Hazardous Materials 358: 346–354.

15.

Irie

Yamada

Tanaka

(1984) Natural frequencies of truncated conical shells. Journal of Sound and Vibration 92(3): 447–453.

16.

Jafari Mehrabadi

Sobhani Aragh

(2013) On the thermal analysis of 2-D temperature-dependent functionally graded open cylindrical shells. Composite Structures 96: 773–785.

17.

Kazemzadeh-Parsi

Ammar

Chinesta

(2023) Parametric analysis of thick FGM plates based on 3D thermo-elasticity theory: a proper generalized decomposition approach. Materials 16(4): 1753.

18.

Khorsand

Tang

(2019) Multi-directional functionally graded materials for enhancing the durability of shell structures. International Journal of Pressure Vessels and Piping 175: 103926.

19.

Kim

Seo

Park

JHY

, et al. (2023) Ca²⁺-Permeable TRPV1 receptor mediates neuroprotective effects in a mouse model of Alzheimer's disease via BDNF/CREB signaling pathway. Molecules and Cells 46(1): 319–328.

20.

Lam

Hua

(1999) On free vibration of a rotating truncated circular orthotropic conical shell. Composites Part B: Engineering 30(2): 135–144.

21.

Kishimoto

Huang

(2009) The calculations of natural frequencies and forced vibration responses of conical shell using the Rayleigh–Ritz method. Mechanics Research Communications 36(5): 595–602.

22.

Lian

Sun

, et al. (2022) A semi-analytical method for free vibration of semi-closed shells of revolution with variable curvature. Journal of Vibration and Control 29: 2198–2213.

23.

Lindholm

(1966) Non-symmetric transverse vibrations of truncated conical shells. International Journal of Mechanical Sciences 8(9): 561–579.

24.

Mohammadrezazadeh

Jafari

(2021a) Nonlinear vibration suppression of laminated composite conical shells on elastic foundations with magnetostrictive layers. Composite Structures 258: 113323.

25.

Mohammadrezazadeh

Jafari

(2021b) Nonlinear vibration analysis of laminated composite angle-ply cylindrical and conical shells. Composite Structures 255: 112867.

26.

Mohammadzadeh

Najafizadeh

Nejati

(2013) Buckling of 2D-FG cylindrical shells under combined external pressure and axial compression. Advances in Applied Mathematics and Mechanics 5(3): 391–406.

27.

Nasr Esfahani

Hashemian

Aghadavoudi

(2022) The vibration study of a sandwich conical shell with a saturated FGP core. Scientific Reports 12(1): 4950.

28.

Nguyen

Nguyen-Thoi

Tran

, et al. (2021) Nonlinear vibration of full-filled fluid corrugated sandwich functionally graded cylindrical shells. Journal of Vibration and Control 27(9–10): 1020–1035.

29.

Pilafkan

Folkow

Darvizeh

, et al. (2013) Three dimensional frequency analysis of bidirectional functionally graded thick cylindrical shells using a radial point interpolation method (RPIM). European Journal of Mechanics - A: Solids 39: 26–34.

30.

Pradhan

Reddy

(2004) Vibration control of composite shells using embedded actuating layers. Smart Materials and Structures 13(5): 1245–1257.

31.

Qin

Chu

(2017) Free vibrations of cylindrical shells with arbitrary boundary conditions: a comparison study. International Journal of Mechanical Sciences 133: 91–99.

32.

Shariyat

Asgari

(2013) Nonlinear thermal buckling and postbuckling analyses of imperfect variable thickness temperature-dependent bidirectional functionally graded cylindrical shells. International Journal of Pressure Vessels and Piping 111–112: 310–320.

33.

Sobhani

Arbabian

Civalek

, et al. (2021) The free vibration analysis of hybrid porous nanocomposite joined hemispherical–cylindrical–conical shells. Engineering with Computers 38: 3125–3152.

34.

Sobhani

Masoodi

Civalek

, et al. (2022) Agglomerated impact of CNT vs. GNP nanofillers on hybridization of polymer matrix for vibration of coupled hemispherical-conical-conical shells. Aerospace Science and Technology 120: 107257.

35.

Sofıyev

Kuruoglu

(2015) Large-amplitude vibration of the geometrically imperfect FGM truncated conical shell. Journal of Vibration and Control 21(1): 142–156.

36.

Tahouneh

Naei

(2016) The effect of multi-directional nanocomposite materials on the vibrational response of thick shell panels with finite length and rested on two-parameter elastic foundations. International Journal of Advanced Structural Engineering (IJASE) 8(1): 11–28.

37.

Weingarten

(1966) The effect of internal or external pressure on the free vibrations of conical shells. International Journal of Mechanical Sciences 8(2): 115–124.

38.

Huang

(2020) A semianalytical finite element method for stress and deformation analyses of bi-directional functionally graded truncated conical shells. Mechanics Based Design of Structures and Machines 48(4): 433–458.

39.

(2021) A semi-analytical FE method for the 3D bending analysis of nonhomogeneous orthotropic toroidal shells. Steel and Composite Structures 39(3): 291–306.

40.

Xue

Jin

Zhang

, et al. (2023) Free vibration analysis of functionally graded porous cylindrical panels and shells with porosity distributions along the thickness and length directions. Thin-Walled Structures 184: 110448.