The effects of various axially graded nanofillers on the dynamic characteristics of a rotating cylindrical shell with non-uniform thickness

Abstract

This paper studies the dynamics of a rotating circular cylindrical shell with non-uniform thickness. The base material of the shell is a polymer which is strengthened with either carbon nanotubes (CNTs), graphene nanoplatelets (GNPs), or graphene oxide powders (GOPs) which are distributed functionally graded (FG) with volume fractions that change in the axial direction. The first-order shear deformation theory (FSDT) is utilized to conduct the mathematical modeling of the shell and the Coriolis, centrifugal, and relative accelerations, and the initial circumferential tension are incorporated. The governing equations are attained via Hamilton’s principle and are solved through a semi-analytical solution. The dependency of the natural frequencies and critical angular velocity on several factors are discussed. It is discovered that by considering the same mass fractions for CNTs, GNPs, and GOPs, the GNPs bring about the best reinforcing effect and the CNTs have the weakest reinforcing effect.

Keywords

Free vibration rotating shell carbon nanotubes graphene nanoplatelets graphene oxide powders

Introduction

There is an extended range of industrial usages and applications for circular cylindrical shells such as aerospace, mechanical, and civil engineering. A cylindrical shell is an accurate model to investigate the mechanical behavior of aircraft jet engines, pressure vessels, and centrifugal separators. Although there is a fair amount of research concerning the vibrational analysis of cylindrical shells of uniform thickness, the vibrational characteristics of those with non-uniform thickness have not been significantly examined. Sivadas and Ganesan¹ conducted a numerical investigation on the free vibrational analysis of clamped and simply supported isotropic homogeneous cylindrical shells whose thicknesses vary in the axial direction according to linear and quadratic profiles. Through a similar work, they inspected the free vibration analysis of an orthotropic cylindrical shell whose thickness changes in the axial direction.² They prepared benchmark results for the shells of clamped and simply supported edges. In another similar work, they investigated the asymmetric free vibrational behavior of a cantilever (clamped-free) isotropic homogeneous cylindrical shell with non-uniform thickness.³ They provided results for the shells whose thicknesses vary in axial direction based on various types of thickness variation profiles and regulated these profiles to result in the same mass of the shell. In Refs. 1-3, The authors used Love’s shell theory (LST) which limits the reliability of their results to thin shells. To provide more accurate natural frequencies, especially for thick shells, Sivadas and Ganesan⁴ employed a refined shell theory including shear deformations and thickness stretching, and analyzed the axisymmetric free vibration of a thick cylindrical shell whose thickness varies in the axial direction. They compared the natural frequencies with those reported based on low-accurate theories. Duan and Koh⁵ proposed an exact solution for the axisymmetric free vibrational analysis of clamped and simply supported cylindrical shells whose thicknesses change in the axial direction according to an arbitrary power form. They provided exact benchmark results for other researchers to check the precision of their models. In two similar works, Viswanathan et al.^6,7 investigated the free vibrational analysis of symmetric angle-ply cylindrical thin and thick shells with non-uniform thickness. They discussed the dependency of the natural frequencies on the thickness variation factors and geometric factors of the shell. El-Kaabazi and Kennedy⁸ provided the natural frequencies and vibrational modes of a cylindrical shell with non-uniform thickness. They presented a parametric examination to discover the influences of thickness variation factors on the natural frequencies and vibrational modes of the shell. An exact analytical solution was performed by Taati et al.⁹ for the free vibrational analysis of rotating a thin cylindrical shell with non-uniform thickness. They examined the dependency of the natural frequencies of the rotating shell on the thickness variation factors. Zheng et al.¹⁰ inspected the free vibration behavior of a cylindrical shell with arbitrary non-uniform thickness and elastically restrained edges. They presented some benchmark results for the natural frequencies of cylindrical shells with non-uniform thickness under various boundary conditions. Quoc et al.¹¹ conducted work on the free vibrational analysis of a rotating metal-ceramic FGM cylindrical shell with non-uniform thickness subjected to a thermal load (temperature elevation). They inspected the influences of boundary conditions, material gradation, temperature elevation, and angular velocity on the natural frequencies of the rotating shell. Phu et al.¹² conducted analytical nonlinear analyses on the dynamic buckling and forced vibrational analyses of a metal-ceramic FGM cylindrical shell with non-uniform thickness exposed to mechanical loading. They inspected the influences of geometric factors and material gradation on the dynamic deflection and dynamic stability regions of the shell. A semi-analytical solution was prepared by Thang et al.¹³ to analyze the free vibration of a bi-directional metal-ceramic FGM cylindrical shell with non-uniform thickness. They inspected the effects of geometric parameters, material gradation, and thickness variation factors on the natural frequencies.

As observed, the vibration analysis of a cylindrical shell with non-uniform thickness has been investigated poorly and most of the present works are limited to non-rotating shells. The authors’ investigations show that the free vibration characteristics of rotating axially FG CNT-reinforced, GNP-reinforced, or GOP-reinforced cylindrical shells with non-uniform thickness have not been investigated yet. As a result, the present work is devoted to this subject for the first time. The motivation for providing this work is to present a more general model and consequently more general benchmark results for free vibrational characteristics of rotating nanocomposite cylindrical shells by considering a non-uniform thickness, non-uniform distribution of the nanofillers and examining three different types of common nanofillers. The modeling of the cylindrical shell is conducted using the FSDT including the Coriolis, centrifugal, relative accelerations, and the initial circumferential tension. The dependencies of the natural frequencies of the rotating shell on several factors are discussed including type, mass fraction, and distribution pattern of the nanofillers. The main objective of this work is to find the best profiles for the variation of the thickness and the volume fraction of the nanofillers to increase the critical angular velocity of the shell. Also, a comparison is made between CNT-reinforced, GNP-reinforced, and GOP-reinforced rotating cylindrical shells to find the best type of nanofillers which provide the highest increase in the critical angular velocity of the shell. As the novelty of the present work, it should be noted that the present work is the first theoretical research regarding the vibration analysis of nanocomposite cylindrical shells which considers variations in the thickness and volume fraction of the nanofillers and investigates three different types of nanofillers. As a result, comprehensive benchmark results are provided by the authors which can be used to validate future works.

Mathematical modeling

The mathematical modeling of the problem is conducted in this section.

Description of the problem

As exhibited in Figure 1, a circular cylindrical shell whose thickness varies in the axial direction (h = h(x)) is considered. The shell is of length L, radius R, and rotates at angular velocity Ω. The base material of the shell is polymer strengthened with either CNTs, GNPs, or GOPs in which the volume fraction of the nanofillers varies in the axial direction. It should be noticed that among all various types of nanofillers, CNTs, GNPs, and GOPs have better mechanical properties which make them better options to be used as a reinforcing phase in two-phase nanocomposite.¹⁴

Figure 1.

A rotating axially FG nanocomposite cylindrical shell with non-uniform thickness.

According to the FSDT, the relations below describe the displacement field in the shell¹⁵:

{\begin{cases} u_{x} (t, x, θ, z) \\ u_{θ} (t, x, θ, z) \\ u_{z} (t, x, θ, z) \end{cases}} = {\begin{cases} u (t, x, θ) \\ v (t, x, θ) \\ w (t, x, θ) \end{cases}} + z {\begin{array}{c} α_{x} (t, x, θ) \\ α_{θ} (t, x, θ) \\ 0 \end{array}},

(1)

where u_x, u_θ, and u_z sequentially stand for displacement along x, θ, and z directions, the corresponding displacements at the neutral surface are shown by u, v, and w. Also, α_θ and α_x respectively show rotation about x- and θ-axes.

As Figure 1 shows, the thickness of the shell varies in the axial direction. This variation is considered in the general form presented in equation (2)^16,17:

h (x) = \frac{1 + q}{1 + q + ζ} h_{a v} [1 + ζ {(\frac{x}{L})}^{q}],

(2)

in which this equation is adjusted to generate the same mean (average) thickness h_av for any desired values of q = 0,1,2,… and ζ ≥ 0. It can be asserted through the equation below:

\frac{1}{L} \int_{0}^{L} h (x) d x = h_{a v} .

(3)

For several values of the thickness variation factors q and ζ, the variations of the thickness in the axial direction are exhibited in Figure 2.

Figure 2.

The thickness variation profiles.

Stress and strain tensors

The following relations are presented for normal components of the strain tensor (ε_ij) and shear ones (γ_ij = 2ε_ij)¹¹:

\begin{array}{l} \begin{array}{l} ε_{x x} = \frac{\partial u}{\partial x} + z \frac{\partial α_{x}}{\partial x}, & ε_{θ θ} = \frac{1}{R} (\frac{\partial v}{\partial θ} + w + z \frac{\partial α_{θ}}{\partial θ}), \end{array} \\ \begin{array}{l} γ_{x θ} = \frac{\partial v}{\partial x} + \frac{1}{R} \frac{\partial u}{\partial θ} + z (\frac{\partial α_{θ}}{\partial x} + \frac{1}{R} \frac{\partial α_{x}}{\partial θ}), \\ γ_{θ z} = - \frac{v}{R} + \frac{1}{R} \frac{\partial w}{\partial θ} + α_{θ}, γ_{x z} = α_{x} + \frac{\partial w}{\partial x}, \end{array} \end{array}

(4)

Since the nanofillers are orientated in random directions, the fabricated two-phase nanocomposite is an isotropic structure. Thus, the normal (σ_ij) and shear (τ_ij) components of the stress tensor are presented in equation (5):

\begin{array}{l} σ_{x x} = C_{11} ε_{x x} + C_{12} ε_{θ θ}, & σ_{θ θ} = C_{21} ε_{x x} + C_{22} ε_{θ θ}, & τ_{θ z} = k_{s} C_{44} γ_{θ z}, & τ_{x z} = k_{s} C_{55} γ_{x z}, & τ_{x θ} = C_{66} γ_{x θ}, \end{array}

(5)

where the shear correction parameter is chosen as k_s = 5/6, and the elastic coefficients C₁₁-C₆₆ are presented in the equations below:

\begin{array}{l} C_{11} = C_{22} = \frac{E}{1 - ν^{2}}, & C_{21} = C_{12} = ν C_{11}, & C_{44} = C_{55} = C_{66} = G = \frac{0.5 E}{1 + ν}, \end{array}

(6)

where G and E respectively represent shear and elastic moduli, and ν represents Poisson’s ratio.

Mechanical properties

As stated, three types of nanofillers are considered in this paper including CNTs, GNPs, and GOPs. It should be noted that CNTs are cylindrical-shaped nanofillers, GNPs are rectangular-shaped nanofillers, and GOPs are circular-shaped nanofillers. It is assumed the volume fraction of the nanofillers varies in the axial direction. In this paper, three general forms are chosen for the dispersion pattern of the nanofillers. The variations of the volume fraction of nanofillers in these patterns are presented in the relations below¹⁸:

\begin{array}{l} Type 1 : & F_{n f} (x) = F_{n f}^{*}, \\ Type 2 : & F_{n f} (x) = (1 + ξ) {(\frac{x}{L})}^{ξ} F_{n f}^{*}, \\ Type 3 : & F_{n f} (x) = (1 + \frac{1}{ξ}) [1 - {(\frac{x}{L})}^{ξ}] F_{n f}^{*}, \end{array}

(7)

in which ξ = 0,1,2,3,…, and the volume fraction of the matrix is achievable as F_m = 1-F_nf.

It is obvious that considering ξ = 0 in type 2 brings about type 1 (uniform distribution). In equation (7), $F_{n f}^{*}$ stands for the mean volume fraction of the nanofillers and is described in terms of the density of the nanofillers (ρ_nf), the density of the polymeric matrix (ρ_m), and the mass fraction of nanofillers (w_nf) as¹⁹

F_{n f}^{*} = \frac{1}{1 + \frac{ρ_{n f}}{ρ_{m}} (\frac{1}{w_{n f}} - 1)} .

(8)

Equation (7) is adjusted to result in the same mean volume fraction of the nanofillers and consequently the same mass fraction of the nanofillers for all types of dispersion patterns and various values of ξ. For types 2 and 3 and ξ = 1,2,3,4, the variations of volume fraction of the nanofillers in the axial direction are exhibited in Figure 3 along with type 1.

Figure 3.

The variations of the volume fraction of the nanofillers in the axial direction.

The density and the Poisson’s ratio of the shell are stated in the relation below utilizing the rule of mixture¹⁹:

\begin{array}{l} ρ = ρ_{n f} F_{n f} + ρ_{m} F_{m}, & ν = ν_{n f} F_{n f} + ν_{m} F_{m}, \end{array}

(9)

As described in the Halpin-Tsai model, the following relation presents the elastic modulus of a polymer reinforced with nanofillers^20,21:

\begin{array}{l} GNP - reinforced & CNT - reinforced : & E = (0.375 \frac{1 + ξ_{1} η_{1} F_{n f}}{1 - η_{1} F_{n f}} + 0.625 \frac{1 + ξ_{2} η_{2} F_{n f}}{1 - η_{2} F_{n f}}) E_{m}, \\ GOP - reinforced : & E = (0.49 \frac{1 + ξ_{1} η_{1} F_{n f}}{1 - η_{1} F_{n f}} + 0.51 \frac{1 + ξ_{2} η_{2} F_{n f}}{1 - η_{2} F_{n f}}) E_{m}, \end{array}

(10)

where

\begin{array}{l} CNT - reinforced : & η_{1} = \frac{η - β}{η + ξ_{1} β}, & η_{2} = \frac{η - β}{η + ξ_{2} β}, & ξ_{1} = \frac{2 l_{C N T}}{d_{C N T}}, & ξ_{2} = 2, & β = \frac{d_{C N T}}{4 t_{C N T}} . \\ GNP - reinforced : & η_{1} = \frac{η - 1}{η + ξ_{1}}, & η_{2} = \frac{η - 1}{η + ξ_{2}}, & ξ_{1} = \frac{2 l_{G N P}}{t_{G N P}}, & ξ_{2} = \frac{2 w_{G N P}}{t_{G N P}} . \\ GOP - reinforced : & η_{1} = \frac{η - 1}{η + ξ_{1}}, & η_{2} = \frac{η - 1}{η + ξ_{2}}, & ξ_{1} = ξ_{2} = \frac{2 d_{G O P}}{t_{G O P}} . \end{array}

(11)

in which, l_CNT, d_CNT, and t_CNT sequentially show the length, diameter, and thickness of an individual CNT. l_GNP, w_GNP, and t_GNP consecutively indicate the length, width, and thickness of an individual GNP. d_GOP and t_GOP sequentially represent the diameter and thickness of an individual GOP. Also, in all cases, η is defined as follows:

η = \frac{E_{n f}}{E_{m}} .

(12)

where, E_nf and E_m are the elastic modulus of the nanofillers and the polymeric matrix, respectively. It should be noted that for the CNTs, E_nf is the elastic modulus when the CNT is under longitudinal tensile loading. Meanwhile, for the GNPs and the GOPs, E_nf is the elastic modulus when the nanofiller is under in-plane tensile loading. The geometric parameters and material properties of the selected nanofillers are presented in Table 1.

Table 1.

The material properties and geometric factors of the selected nanofillers.^22–25

CNTs	GNPs	GOPs
E_nf = 640 GPa, ν_nf = 0.33	E_nf = 1010 GPa, ν_nf = 0.186	E_nf = 448.8 GPa, ν_nf = 0.165
ρ_nf = 1350 kg/m³	ρ_nf = 1062.5 kg/m³	ρ_nf = 1090 kg/m³
l_CNT = 25 μm, d_CNT = 1.4 nm, t_CNT = 0.34 nm	l_GNP = 2.5 μm, w_GNP = 1.5 μm, t_GNP = 1.5 nm	d_GOP = 0.5 μm, t_GOP = 0.95 nm

Hamilton’s principle

By introducing δ as the variational operator, t as time, and t₁ and t₂ as two desired moments, the governing equations concerning the dynamics of a rotating shell and associated boundary conditions are achievable utilizing Hamilton’s principle through the equation below²⁶:

\int_{t_{1}}^{t_{2}} (δ T - δ U_{s} - δ U_{h} + δ W) d t = 0,

(13)

where T is the kinetic energy, U_s denotes the strain energy of the shell, U_h shows the strain energy created by the initial circumferential tension, and W is the work done by external non-conservative loads.

The following relation provides the kinetic energy:

T = 0.5 \underset{V}{∭} ρ \vec{v} . \vec{v} d V,

(14)

where

\underset{V}{∭} () d V = \iint_{S} \int_{- \frac{h}{2}}^{\frac{h}{2}} () d z d S,

(15)

where dS = Rdxdθ shows the surface of the shell.

In equation (14), $\vec{v}$ is the absolute velocity vector which is presented in equation (16)²⁶:

\vec{v} = \vec{Ω} \times \vec{R} + \frac{\partial u_{x}}{\partial t} \vec{i} + \frac{\partial u_{θ}}{\partial t} \vec{j} + \frac{\partial u_{z}}{\partial t} \vec{k},

(16)

where

\begin{array}{l} \vec{R} = u_{x} \vec{i} + u_{θ} \vec{j} + u_{z} \vec{k}, & \vec{Ω} = - \vec{i} Ω . \end{array}

(17)

Inserting equation (17) into equation (16) leads to equation (18):

\vec{v} = \frac{\partial u_{x}}{\partial t} \vec{i} + (\frac{\partial u_{θ}}{\partial t} + Ω u_{z}) \vec{j} + (\frac{\partial u_{z}}{\partial t} - Ω u_{θ}) \vec{k} .

(18)

Inserting equations (1) and (18) into equation (14) and applying equation (15) provides the relation below:

\begin{array}{l} T = 0.5 \iint_{S} {I_{0} [{(\frac{\partial w}{\partial t})}^{2} + {(\frac{\partial v}{\partial t})}^{2} + {(\frac{\partial u}{\partial t})}^{2}] + I_{2} [{(\frac{\partial α_{x}}{\partial t})}^{2} + {(\frac{\partial α_{θ}}{\partial t})}^{2}] \\ + 2 Ω [I_{0} (\frac{\partial v}{\partial t} w - \frac{\partial w}{\partial t} v)] + Ω^{2} [I_{0} (v^{2} + w^{2}) + I_{2} α_{θ}^{2}]} d S . \end{array}

(19)

in which the translational inertia (mass) per unit area and rotational inertia per unit area are defined in equation (20):

{\begin{cases} I_{0} (x) \\ I_{2} (x) \end{cases}} = \int_{- \frac{h}{2}}^{\frac{h}{2}} ρ (x) {\begin{cases} 1 \\ z^{2} \end{cases}} d z = ρ (x) {\begin{cases} h \\ \frac{h^{3}}{12} \end{cases}},

(20)

and the relation below is utilized:

\int_{- \frac{h}{2}}^{\frac{h}{2}} ρ (x) z d z = 0,

(21)

The equation below provides the strain energy:

U_{s} = 0.5 \underset{V}{∭} (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}) d V,

(22)

which leads to the following relation utilizing equation (4) and (15):

\begin{array}{l} δ U_{s} = \iint_{S} {N_{x x} \frac{\partial δ u}{\partial x} + M_{x x} \frac{\partial δ α_{x}}{\partial x} + \frac{N_{θ θ}}{R} (\frac{\partial δ v}{\partial θ} + δ w) + \frac{M_{θ θ}}{R} \frac{\partial δ α_{θ}}{\partial θ} + N_{x θ} (\frac{\partial δ v}{\partial x} + \frac{1}{R} \frac{\partial δ u}{\partial θ}) \\ + M_{x θ} (\frac{1}{R} \frac{\partial δ α_{x}}{\partial θ} + \frac{\partial δ α_{θ}}{\partial x}) + Q_{x z} (δ α_{x} + \frac{\partial δ w}{\partial x}) + Q_{θ z} (- \frac{δ v}{R} + \frac{1}{R} \frac{\partial δ w}{\partial θ} + δ α_{θ})} d S, \end{array}

(23)

in which N_ij, M_ij, and Q_ij are the stress resultants and are described in equation (24):

\begin{array}{l} {\begin{cases} N_{x x} \\ N_{θ θ} \\ N_{x θ} \end{cases}} = \int_{- \frac{h}{2}}^{\frac{h}{2}} {\begin{cases} σ_{x x} \\ σ_{θ θ} \\ τ_{x θ} \end{cases}} d z, & {\begin{cases} Q_{x z} \\ Q_{θ z} \end{cases}} = \int_{- \frac{h}{2}}^{\frac{h}{2}} {\begin{cases} τ_{x z} \\ τ_{θ z} \end{cases}} d z, & {\begin{cases} M_{x x} \\ M_{θ θ} \\ M_{x θ} \end{cases}} = \int_{- \frac{h}{2}}^{\frac{h}{2}} {\begin{cases} σ_{x x} \\ σ_{θ θ} \\ τ_{x θ} \end{cases}} z d z . \end{array}

(24)

Inserting equations (4) and (5) into equation (24) generates the relations below for stress resultants:

\begin{array}{l} \begin{array}{l} N_{x x} = A_{11} \frac{\partial u}{\partial x} + \frac{A_{12}}{R} (\frac{\partial v}{\partial θ} + w), & M_{x x} = D_{11} \frac{\partial α_{x}}{\partial x} + \frac{D_{12}}{R} \frac{\partial α_{θ}}{\partial θ}, \end{array} \\ \begin{array}{l} N_{θ θ} = A_{12} \frac{\partial u}{\partial x} + \frac{A_{22}}{R} (\frac{\partial v}{\partial θ} + w), & M_{θ θ} = D_{12} \frac{\partial α_{x}}{\partial x} + \frac{D_{22}}{R} \frac{\partial α_{θ}}{\partial θ}, \end{array} \\ \begin{array}{l} N_{x θ} = A_{66} (\frac{1}{R} \frac{\partial u}{\partial θ} + \frac{\partial v}{\partial x}), & M_{x θ} = D_{66} (\frac{1}{R} \frac{\partial α_{x}}{\partial θ} + \frac{\partial α_{θ}}{\partial x}), \end{array} \\ \begin{array}{l} Q_{θ z} = A_{44} (- \frac{v}{R} + \frac{1}{R} \frac{\partial w}{\partial θ} + α_{θ}), & Q_{x z} = A_{55} (\frac{\partial w}{\partial x} + α_{x}), \end{array} \end{array}

(25)

where the stiffness coefficients of the shell (shear, extensional, and flexural) are described in relations below:

\begin{array}{l} {\begin{cases} A_{i j} (x) \\ D_{i j} (x) \end{cases}} = \int_{- \frac{h}{2}}^{\frac{h}{2}} C_{i j} (x) {\begin{cases} 1 \\ z^{2} \end{cases}} d z = C_{i j} (x) {\begin{cases} h \\ \frac{h^{3}}{12} \end{cases}}, & i & j = 1, 2, 6, \\ A_{i j} (x) = k_{s} \int_{- \frac{h}{2}}^{\frac{h}{2}} C_{i j} (x) d z = k_{s} h C_{i j} (x), & i & j = 4, 5 . \end{array}

(26)

and the relation below is utilized:

\int_{- \frac{h}{2}}^{\frac{h}{2}} C_{i j} (x) z d z = 0 .

(27)

Regardless of vibration, rotation provides an initial stationary stress in the circumferential direction known as the initial circumferential tension. The corresponding strain energy is presented as follows^27,28:

U_{h} = \underset{V}{∭} ε_{θ θ}^{N L} σ_{θ θ}^{0} d V,

(28)

in which, with the relations below, $σ_{θ θ}^{0}$ and $ε_{θ θ}^{N L}$ sequentially stand for the initial circumferential stress and the nonlinear part of the normal strain in the circumferential direction ( $ε_{θ θ}$ )^27,28:

\begin{array}{l} σ_{θ θ}^{0} = ρ R^{2} Ω^{2}, & ε_{θ θ}^{N L} = \frac{1}{2 R^{2}} [{(\frac{\partial u_{x}}{\partial θ})}^{2} + {(\frac{\partial u_{θ}}{\partial θ} + u_{z})}^{2} + {(\frac{\partial u_{z}}{\partial θ} - u_{θ})}^{2}] . \end{array}

(29)

Using equation (1), (15), (20), (21), (28) and (29), the equation below can be obtained:

\begin{array}{l} U_{h} = 0.5 Ω^{2} \iint_{S} {I_{0} [{(\frac{\partial u}{\partial θ})}^{2} + {(\frac{\partial v}{\partial θ})}^{2} + {(\frac{\partial w}{\partial θ})}^{2} + v^{2} + w^{2} + 2 (w \frac{\partial v}{\partial θ} - v \frac{\partial w}{\partial θ})] \\ + I_{2} [{(\frac{\partial α_{x}}{\partial θ})}^{2} + α_{θ}^{2} + {(\frac{\partial α_{θ}}{\partial θ})}^{2}]} d S, \end{array}

(30)

In the free vibrational analysis of a system, it is assumed that there is no external load applied to the structure (δW = 0). Thus, inserting equation (19), (23) and (30) into equation (13) results in the following equations as the set of governing equations:

\begin{array}{l} \frac{\partial N_{x x}}{\partial x} + \frac{1}{R} \frac{\partial N_{x θ}}{\partial θ} + I_{0} Ω^{2} \frac{\partial^{2} u}{\partial θ^{2}} - I_{0} \frac{\partial^{2} u}{\partial t^{2}} = 0, \\ \frac{1}{R} \frac{\partial N_{θ θ}}{\partial θ} + \frac{\partial N_{x θ}}{\partial x} + \frac{Q_{θ z}}{R} + I_{0} Ω^{2} (\frac{\partial^{2} v}{\partial θ^{2}} + 2 \frac{\partial w}{\partial θ}) - 2 I_{0} Ω \frac{\partial w}{\partial t} - I_{0} \frac{\partial^{2} v}{\partial t^{2}} = 0, \\ \frac{\partial Q_{x z}}{\partial x} - \frac{N_{θ θ}}{R} + \frac{1}{R} \frac{\partial Q_{θ z}}{\partial θ} + I_{0} Ω^{2} (\frac{\partial^{2} w}{\partial θ^{2}} - 2 \frac{\partial v}{\partial θ}) + 2 I_{0} Ω \frac{\partial v}{\partial t} - I_{0} \frac{\partial^{2} w}{\partial t^{2}} = 0, \\ \frac{\partial M_{x x}}{\partial x} + \frac{1}{R} \frac{\partial M_{x θ}}{\partial θ} - Q_{x z} + I_{2} Ω^{2} \frac{\partial^{2} α_{x}}{\partial θ^{2}} - I_{2} \frac{\partial^{2} α_{x}}{\partial t^{2}} = 0, \\ \frac{1}{R} \frac{\partial M_{θ θ}}{\partial θ} + \frac{\partial M_{x θ}}{\partial x} - Q_{θ z} + I_{2} Ω^{2} \frac{\partial^{2} α_{θ}}{\partial θ^{2}} - I_{2} \frac{\partial^{2} α_{θ}}{\partial t^{2}} = 0; \end{array}

(31)

and the boundary conditions at the ends of the shell are described in the following relations:

\begin{array}{l} \begin{array}{l} Clamped (C) : & u = w = v = 0, & α_{θ} = α_{x} = 0 . \end{array} \\ \begin{array}{l} Free (F) : & N_{x x} = N_{x θ} = Q_{x z} = 0, & M_{x x} = M_{x θ} = 0 . \end{array} \\ \begin{array}{l} Simply supported (S) : & v = w = 0, & α_{θ} = 0, & N_{x x} = 0, & M_{x x} = 0 . \end{array} \end{array}

(32)

Semi-analytical solution

Analytical solution

By inserting equation (25) into equation (31) and considering the solution presented in equation (33)^27,28:

{\begin{cases} u (t, x, θ) \\ v (t, x, θ) \\ w (t, x, θ) \\ α_{x} (t, x, θ) \\ α_{θ} (t, x, θ) \end{cases}} = \sum_{n = 0}^{\infty} {\begin{cases} U_{n} (x) \cos (ω t + n θ) \\ V_{n} (x) \sin (ω t + n θ) \\ W_{n} (x) \cos (ω t + n θ) \\ X_{n} (x) \cos (ω t + n θ) \\ Θ_{n} (x) \sin (ω t + n θ) \end{cases}},

(33)

the governing equations are represented in the form of the relations below:

\begin{array}{l} \begin{array}{l} I_{0} ω^{2} U_{n} + A_{11} U_{n}^{″} + A_{11}^{'} U_{n}^{'} - (\frac{A_{66} n^{2}}{R^{2}} + I_{0} n^{2} Ω^{2}) U_{n} + \frac{(A_{12} + A_{66}) n}{R} V_{n}^{'} + \frac{A_{12}^{'} n}{R} V_{n} + \frac{A_{12}}{R} W_{n}^{'} + \frac{A_{12}^{'}}{R} W_{n} = 0, \\ I_{0} ω^{2} V_{n} + 2 I_{0} Ω ω W_{n} - \frac{(A_{12} + A_{66}) n}{R} U_{n}^{'} - \frac{A_{66}^{'} n}{R} U_{n} + A_{66} V_{n}^{″} + A_{66}^{'} V_{n}^{'} \\ - (\frac{A_{22} n^{2} + A_{44}}{R^{2}} + I_{0} n^{2} Ω^{2}) V_{n} - n (\frac{A_{22} + A_{44}}{R^{2}} + 2 I_{0} Ω^{2}) W_{n} + \frac{A_{44}}{R} Θ = 0, \\ I_{0} ω^{2} W_{n} + 2 I_{0} Ω ω V_{n} - \frac{A_{12}}{R} U_{n}^{'} - n (\frac{A_{22} + A_{44}}{R^{2}} + 2 I_{0} Ω^{2}) V_{n} + A_{55} W_{n}^{″} + A_{55}^{'} W_{n}^{'} \\ - (\frac{A_{22} + A_{44} n^{2}}{R^{2}} + I_{0} n^{2} Ω^{2}) W_{n} + A_{55} X_{n}^{'} + A_{55}^{'} X_{n} + \frac{A_{44} n}{R} Θ_{n} = 0, \end{array} \\ \begin{array}{l} I_{2} ω^{2} X_{n} - A_{55} W_{n}^{'} + D_{11} X_{n}^{″} + D_{11}^{'} X_{n}^{'} - (\frac{D_{66} n^{2}}{R^{2}} + A_{55} + I_{2} n^{2} Ω^{2}) X_{n} + \frac{(D_{12} + D_{66}) n}{R} Θ_{n}^{'} + \frac{D_{12}^{'} n}{R} Θ_{n} = 0, \\ I_{2} ω^{2} Θ_{n} + \frac{A_{44}}{R} V + \frac{A_{44} n}{R} W_{n} - \frac{(D_{12} + D_{66}) n}{R} X_{n}^{'} - \frac{D_{66}^{'} n}{R} X_{n} \\ + D_{66} Θ_{n}^{″} + D_{66}^{'} Θ_{n}^{'} - (\frac{D_{22} n^{2}}{R^{2}} + A_{44} + I_{2} n^{2} Ω^{2}) Θ_{n} = 0, \end{array} \end{array}

(34)

where the prime is utilized to imply derivative with respect to x.

In equation (33) ω and n = 0,1,2,3,… respectively imply the natural frequency and the circumferential wave number.

Inserting equations (25) and (33) into the boundary conditions described in equation (32) results in the following relations:

\begin{array}{l} \begin{array}{l} Clamped (C) : & \begin{array}{l} U_{n} = 0, & W_{n} = 0, & V_{n} = 0, & Θ_{n} = 0, & X_{n} = 0, \end{array} \end{array} \\ \begin{array}{l} Simply Supported (S) : & \begin{array}{l} V_{n} = 0, & W_{n} = 0, & U_{n}^{'} = 0, & Θ_{n} = 0, & X_{n}^{'} = 0, \end{array} \end{array} \\ \begin{array}{l} Free (F) : & \begin{array}{l} A_{11} U_{n}^{'} + \frac{A_{12}}{R} (n V_{n} + W_{n}) = 0, & \frac{n}{R} U_{n} - V_{n}^{'} = 0, \end{array} \end{array} \\ \begin{array}{l} W_{n}^{'} + X_{n} = 0, & D_{11} X_{n}^{'} + \frac{D_{12} n}{R} Θ_{n} = 0, & \frac{n}{R} X_{n} - Θ_{n}^{'} = 0 . \end{array} \end{array}

(35)

Numerical solution

It is unlikely that an exact solution can be found for the governing equation (34) for any combination of various boundary conditions (35) at both edges of the shell. Thus, an approximate solution is performed in the current section utilizing the differential quadrature method (DQM). The basic idea behind the DQM is to discretize the domain of the problem to N pre-selected points and estimate all necessary derivatives of a function like F(x) at each point at these points as described in equation (36):

{\frac{d^{p} F}{d x^{p}}} = [Λ^{(p)}] {F},

(36)

where [Λ^(p)] is a matrix of order N known as the weighting coefficient matrix concerning the pth order derivative. The weighting coefficient matrices are calculated via the equations below²⁹:

\begin{array}{l} Λ_{i j}^{(1)} = {\begin{cases} \begin{array}{l} \sum_{\begin{array}{l} m = 1 \\ m \neq i \end{array}}^{N} \frac{1}{x_{i} - x_{m}} & i = j \\ \frac{\prod_{\begin{array}{l} m = 1 \\ m \neq i, j \end{array}}^{N} (x_{i} - x_{m})}{\prod_{\begin{array}{l} m = 1 \\ m \neq j \end{array}}^{N} (x_{j} - x_{m})} & i \neq j \end{array} & i & j = 1, 2, . . ., N \end{cases}, \\ \begin{array}{l} [Λ^{(p)}] = [Λ^{(1)}] [Λ^{(p - 1)}], & p = 2, 3, \dots, N - 1 . \end{array} \end{array}

(37)

The convergence of the presented approximate solution via the DQM is significantly affected by the dispersion pattern of the grid points. A well-acknowledged distribution pattern of the grid points is a non-uniform one called the Chebyshev-Gauss–Lobatto pattern. This pattern is presented in the equation below for 0 ≤ x≤L²⁹:

x_{i} = 0.5 L [1 - \cos (\frac{1 - i}{1 - N} π)], i = 1, 2,, . . ., N .

(38)

Utilizing equation (36), one can write the governing equations (34) in the form below:

ω^{2} [M] {r} + ω [G] {r} + [K] {r} = {0},

(39)

where, as presented in Appendix A,

{r}

is displacement vector, and [K], [M], and [G] sequentially represents stiffness, mass, and gyroscopic matrices.

Using equations (35) and and (36), and the definition of ${r}$ described in equation (46), one can write the boundary conditions in the form below:

[ψ] {r} = {0},

(40)

where [ψ] is presented in Appendix B.

The natural frequencies of the rotating shell are achievable by the solution of equations (39) and (40). However, it results in an inconsistency in the numbers of algebraic equations and undetermined variables.³⁰ It can be eliminated by dividing the grid points into two sets: the boundary points which include x₁ and x_N and the domain points which include x₂-x_N-1. Neglecting to satisfy equation (39) can be neglected at the boundary points and this relation can be rewritten as equation (41):

ω^{2} [M_{t r}] {r} + ω [G_{t r}] {r} + [K_{t r}] {r} = {0},

(41)

where the subscript tr implies the corresponding truncated non-square matrix.

By separating the columns of the matrices associated with the domain and boundary points, equations (40) and (41) can be partitioned and rearranged as the following equations:

ω^{2} ({[M_{t r}]}_{b} {r}_{b} + {[M_{t r}]}_{d} {r}_{d}) + ω ({[G_{t r}]}_{b} {r}_{b} + {[G_{t r}]}_{d} {r}_{d}) + {[K_{t r}]}_{b} {r}_{b} + {[K_{t r}]}_{d} {r}_{d} = {0},

(42a)

{[ψ]}_{d} {r}_{d} + {[ψ]}_{b} {r}_{b} = {0},

(42b)

in which the subscripts “b” and “d” sequentially imply the boundary and domain points.

Inserting equations (42-b) into (42-a) provides the following eigenvalue equation:

ω^{2} [M_{t}] {r}_{d} + ω [G_{t}] {r}_{d} + [K_{t}] {r}_{d} = {0},

(43)

where

\begin{array}{l} [M_{t}] = {[M_{t r}]}_{d} + {[M_{t r}]}_{b} [y], & [G_{t}] = {[G_{t r}]}_{d} + {[G_{t r}]}_{b} [y], \\ [K_{t}] = {[K_{t r}]}_{d} + {[K_{t r}]}_{b} [y], & [y] = - {[ψ]}_{b}^{- 1} {[ψ]}_{d} . \end{array}

(44)

The natural frequencies of the rotating shell are available through the solution of the eigenvalue equation (43). Among these eigenvalues, positive ones are called the forward (F) frequencies and negative ones are called the backward (B) frequencies.^27,28 It should be noted that in some research the reverse definition is considered.

Numerical results

Numerical examples and associated physical explanations are provided in the current section to examine the impacts of several factors on the critical angular velocities and natural frequencies of the shell. The boundary conditions are denoted by two capital letters (equation (32)) that sequentially indicate the conditions at x = 0 & L and forward and backward frequencies are sequentially denoted by F and B. The natural frequencies are represented by ω_nm in which, as stated in equation (33) n is the circumferential wave number, and m = 1,2,3,… is hired to indicate the vibrational modes in the axial direction. The following dimensionless definitions are utilized for the angular velocity and natural frequencies:

\begin{array}{l} γ = Ω R \sqrt{\frac{ρ_{m}}{E_{m}} (1 - ν_{m}^{2})}, & λ_{n m} = ω_{n m} R \sqrt{\frac{ρ_{m}}{E_{m}} (1 - ν_{m}^{2})} . \end{array}

(45)

Except in cases that are stated directly, the numerical examples are presented for a CC rotating cylindrical shell of R = 0.25 m, h_av/R = 0.02, and L/R = 3 whose thickness varies along longitudinal direction with thickness variation factors as ζ = 0.1 and q = 1. Epoxy (ν_m = 0.34, E_m = 2.1 GPa, ρ_m = 1150 kg/m³)³¹ is used for the polymeric matrix which is enriched with either CNTs, GNPs, or GOPs based on type 2 with ξ = 2 and w_g = 0.05 = 5%.

Convergence and verification

For a GNP-reinforced shell rotating at γ = 0.25, the variations of the forward and backward frequencies associated with n = 4 and m = 1,2,3,4 versus the variations of the number of grid points are exhibited in Figure 4. As shown, the presented approximate solution in the axial direction via the DQM convergences rapidly. Hereafter, the results are provided for N = 15.

Figure 4.

Convergence analysis of the presented approximate solution via the DQM

Two examples are presented in this section to inspect the precision and accuracy of the present work. As the first one, consider an SS rotating cylindrical thin shell made of a homogenous isotropic material (ν = 0.3) with a uniform thickness as h/R = 0.002 and a length-to-radius ratio as L/R = 5. For four selected values of angular velocity, forward and backward frequencies can be found in Table 2 for n = 3 and m = 1 along with the corresponding results reported by other authors.^32–34 The results tabulated in this table confirm the accuracy of the presented research.

Table 2.

Dimensionless natural frequencies for a simply supported (SS) homogenous isotropic rotating cylindrical shell with uniform thickness (ν = 0.3, h/R = 0.002, L/R = 5, n = 3, m = 1).

	Backward modes				Forward modes
γ	Present	Ref. 32	Ref. 33	Ref. 34	Present	Ref. 32	Ref. 33	Ref. 34
0	0.0382	0.0382	0.0386	0.0384	0.0382	0.0382	0.0386	0.0384
0.01	0.0391	0.0399	0.0402	0.0400	0.0511	0.0519	0.0523	0.0521
0.03	0.0633	0.0673	0.0674	0.0673	0.0995	0.1035	0.1036	0.1035
0.05	0.0955	0.1026	0.1027	0.1026	0.1559	0.1630	0.1631	0.1630

As the second example, consider an SS rotating cylindrical thin shell made of a homogenous isotropic material (ν = 0.33) of R = 0.5 m, and L/R = 5 whose thickness varies as h = h₀[1+sin(πx/L)]. For h₀/R = 0.01 and some selected values of angular velocity, forward and backward frequencies are reported in Hz in Table 3 for n = 3 and m = 1 along with those predicted by Taati et al.⁹ As observed, the results are in high agreement which confirms the accuracy of the presented research.

Table 3.

Dimensionless natural frequencies (Hz) for an SS homogenous isotropic rotating cylindrical shell with non-uniform thickness (ν = 0.33, R = 0.5 m, L/R = 5, h = h₀[1 + sin(πx/L)], h₀/R = 0.01, n = 3, m = 1).

Ω (rev/s)	Backward modes		Forward modes
Ω (rev/s)	Present	Ref. 9	Present	Ref. 9
0	95.84	96.31	95.84	96.31
5	93.57	94.01	99.61	100.07
10	92.75	93.20	104.83	105.28
15	93.30	93.73	111.42	111.85
20	95.07	95.48	119.23	119.65
25	97.92	98.31	128.12	128.51
30	101.67	102.04	137.92	138.29
35	106.20	106.54	148.49	148.83
40	111.36	111.67	159.70	160.01

Tables 2 and 3 show that in all cases, forward and backward frequencies obtained in the present work are lower than those reported by Refs. 9,32-34. The main reason for this discrepancy is the difference between the employed shell theories. LST is utilized in Refs. 9,32-34, and a more accurate one (the FSDT) is employed in the present work. It should be noted that shear deformations and rotational inertia are not incorporated in LST which leads to higher stiffness and lower inertia which provides higher (low accurate) natural frequencies.

Parametric study

For n = 1,2,3,4 and m = 1,2,3,4, Figures 5–7 show the variations of natural frequencies in both forward and backward modes versus the variation of the angular velocity (Campbell diagram) for CNT-reinforced, GNP-reinforced, and GOP-reinforced shells. As observed, for all cases, in each vibrational mode forward and backward frequencies have the same values for a non-rotating shell and are different for a rotating shell. By increasing the angular velocity, both initial circumferential tension and gyroscopic effect increase. The initial circumferential tension enhances stiffness and tends to enhance both forward and backward frequencies and the gyroscopic tends to increase forward frequencies and decrease backward frequencies. Consequently, Figures 5–7 show that as the angular velocity increases, all forward frequencies increase due to the compatible influences of the gyroscopic effect and the initial circumferential tension, and backward ones experience different trends due to the incompatible influences of the gyroscopic effect and the initial circumferential tension.

Figure 5.

Campbell diagram of a CNT-reinforced rotating cylindrical shell for n = 1,2,3,4.

Figure 6.

Campbell diagram of a GNP-reinforced rotating cylindrical shell for n = 1,2,3,4.

Figure 7.

Campbell diagram of a GOP-reinforced rotating cylindrical shell for n = 1,2,3,4.

In Figures 5–7, the bisector λ_nm = γ is exhibited whose intersections with the curves of the Campbell diagram determine the critical angular velocities of the shell. The two lowest critical angular velocities are γ_cr = 0.3443 and γ_cr = 3838 for a CNT-reinforced shell, γ_cr = 0.5358 and γ_cr = 5994 for a GNP-reinforced shell, and γ_cr = 0.4157 and γ_cr = 4647 for a GOP-reinforced shell. Rotation of the shell at angular velocities close to any of these critical angular velocities leads to a destructive phenomenon. This phenomenon is known as resonance and leads to an abrupt and catastrophic failure of the structure. A comparison between the attained critical angular velocities shows that the highest critical angular velocities and natural frequencies belong to the GNP-reinforced shell and the lowest ones belong to the CNT-reinforced shell. To explain the advantages of the GNP-reinforced and GOP-reinforced rather than the CNT-reinforced, it should be noted that rectangular-shaped GNPs and circular-shaped GOPs have bigger specific surface areas in comparison with cylindrical-shaped CNTs. These privileges result in a stronger bonding between the polymeric matrix and the nanofillers which brings about a better reinforcing effect.

Figures 5–7 reveal that critical angular velocities exist only for small circumferential wave numbers (here n = 1,2) and vanishes for higher values of this parameter (here n ≥ 3). It can be remarked a positive effect of the initial circumferential tension which increases the stiffness of the shell and increases by enhancing the angular velocity. As a negative influence of the initial circumferential tension, it should be noted that rotation of the shell at a high angular velocity provides a high value of the initial circumferential tension which generates high circumferential stress and provides plastic regions in the shell.

Hereafter, the effects of several factors on the critical angular velocities and the natural frequencies (Campbell diagram) are examined. To diminish the length of this section, this parametric investigation is limited to two selected vibrational modes λ₁₁ and λ₂₁ (n = 1,2 and m = 1) which are associated with the lowest critical angular velocities.

The dependency of the Campbell diagram on the mass fraction of the nanofillers is examined in Figures 8–10. Compared to epoxy (matrix), all types of nanofillers (CNTs, GNPs, and GOPs) benefit from remarkably higher shear and elastic moduli and have the approximately same density as the epoxy. Accordingly, as shown in these figures, subjoining more nanofillers to the polymeric matrix leads to higher critical angular velocities and natural frequencies.

Figure 8.

Effects of the percentage of the CNTs on the campbell diagram.

Figure 9.

Effects of the percentage of the GNPs on the campbell diagram.

Figure 10.

Effects of the percentage of the GOPs on the campbell diagram.

The differences between the reinforcing effects of nanofillers (the CNTs, GNPs, and GOPs), the high dependency of the dynamic characteristics of the shell to the mass fraction of the nanofillers, as well as the differences between their prices form an interesting optimization problem to achieve the highest critical angular velocities of the shell and the lowest prices, simultaneously. It can be considered as a topic for future studies.

The influences of the dispersion pattern of the nanofillers on the Campbell diagram are inspected in Figures 11–13. As revealed, independent of the type of the nanofillers, the highest critical angular velocities and natural frequencies belong to the shell reinforced based on type 1 and the lowest critical angular velocities belong to the shell reinforced based on type 2. In conjunction with Figure 3, it is discovered that to improve the reinforcing effect of the nanofillers, it is more beneficial to distribute them uniformly. In conjunction with Figures 2 and 3, it is concluded that to reinforce a cylindrical shell with nano-uniform thickness, the worst decision is to distribute most of the nanofillers at the thicker parts of the shell.

Figure 11.

Effects of the dispersion pattern of the CNTs on the campbell diagram.

Figure 12.

Effects of the dispersion pattern of the GNPs on the campbell diagram.

Figure 13.

Effects of the dispersion pattern of the GOPs on the campbell diagram.

Figures 14–16 are presented to discover the effects of the dispersion pattern parameter ξ on the Campbell diagram. As observed, to attain higher critical angular velocities and natural frequencies, it is better to use lower values of ξ. In conjunction with Figure 3 (type 2), an intense variation in the variation of the volume fraction of the nanofillers results in lower critical angular velocities and natural frequencies. In other words, it can be concluded that utilizing a gentle variation in the variation of the volume fraction or using a uniform dispersion pattern of the nanofillers brings about higher critical angular velocities and higher natural frequencies. It is in complete agreement with what was concluded from Figures 11–13.

Figure 14.

Effects of the dispersion pattern parameter ξ on the campbell diagram of a CNT-reinforced shell.

Figure 15.

Effects of the dispersion pattern parameter ξ on the campbell diagram of a GNP-reinforced shell.

Figure 16.

Effects of the dispersion pattern parameter ξ on the campbell diagram of a GOP-reinforced shell.

Conclusions

The free vibration analysis of a rotating circular cylindrical shell with non-uniform thickness enriched with axially FG nanofillers was investigated in this paper. Three types of nanofillers were investigated including CNTs, GNPs, and GOPs. The shell was modeled via the FSDT including the Coriolis, centrifugal, and relative accelerations, and the initial circumferential tension. The influences of several factors on the critical angular velocities and natural frequencies were discovered. The main findings of the present work are outlined in the statements below:

• By increasing the angular velocity, all forward frequencies increase, and backward ones experience various trends.

• Critical angular velocities exist for small values of the circumferential wave number and vanish for high values of this parameter.

• By considering the same mass fractions for CNTs, GNPs, and GOPs, the GNPs have the best reinforcing effect, and the CNTs have the weakest reinforcing effect.

• An enhancement in the percentage (mass fraction) of the nanofillers brings about a significant increase in the critical angular velocities and natural frequencies.

• Variations in the volume fraction of nanofillers in the axial direction do not necessarily result in a better reinforcing effect. In other words, the uniform distribution of them provides a better reinforcing effect.

• To enrich a cylindrical shell with non-uniform thickness, the worst distribution pattern is to distribute more nanofillers at the thicker parts of the shell.

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

Hassan Afshari

Data availability statement

The data that support the findings of this study will be available upon reasonable request.*

Appendix A

(46)

\begin{array}{l} \begin{array}{l} {r}_{5 N \times 1} = {\begin{cases} {U_{n}} \\ {V_{n}} \\ {W_{n}} \\ {X_{n}} \\ {Θ_{n}} \end{cases}}, & [K] = [\begin{array}{l} k_{11} & k_{12} & k_{13} & [0] & [0] \\ k_{21} & k_{22} & k_{23} & [0] & k_{25} \\ k_{31} & k_{32} & k_{33} & k_{34} & k_{35} \\ [0] & [0] & k_{43} & k_{44} & k_{45} \\ [0] & k_{52} & k_{53} & k_{54} & k_{55} \end{array}], \end{array} \\ \begin{array}{l} [G] = [\begin{array}{l} [0] & [0] & [0] & [0] & [0] \\ [0] & [0] & g_{23} & [0] & [0] \\ [0] & g_{32} & [0] & [0] & [0] \\ [0] & [0] & [0] & [0] & [0] \\ [0] & [0] & [0] & [0] & [0] \end{array}], & [M] = [\begin{array}{l} m_{11} & [0] & [0] & [0] & [0] \\ [0] & m_{22} & [0] & [0] & [0] \\ [0] & [0] & m_{33} & [0] & [0] \\ [0] & [0] & [0] & m_{44} & [0] \\ [0] & [0] & [0] & [0] & m_{55} \end{array}], \end{array} \end{array}

where [0]_N×N indicates the zero matrix and k_ij, m_ij, and g_ij are described in the following relations: (47)

\begin{array}{l} k_{11} = [a_{11}] [Λ^{(2)}] + [d a_{11}] [Λ^{(1)}] - {\frac{n^{2}}{R^{2}} [a_{66}] + n^{2} Ω^{2} [J_{0}]}, \\ \begin{array}{l} k_{12} = \frac{n}{R} ([a_{12}] + [a_{66}]) [Λ^{(1)}] + \frac{n}{R} [d a_{12}], & k_{13} = \frac{1}{R} [a_{12}] [Λ^{(1)}] + \frac{1}{R} [d a_{12}], & m_{11} = [J_{0}], \end{array} \\ k_{21} = - \frac{n}{R} ([a_{12}] + [a_{66}]) [Λ^{(1)}] - \frac{n}{R} [d a_{66}], \\ k_{22} = [a_{66}] [Λ^{(2)}] + [d a_{66}] [Λ^{(1)}] - {\frac{1}{R^{2}} (n^{2} [a_{22}] + [a_{44}]) + n^{2} Ω^{2} [J_{0}]}, \\ \begin{array}{l} k_{23} = - n {\frac{1}{R^{2}} ([a_{22}] + [a_{44}]) + 2 Ω^{2} [J_{0}]}, & k_{25} = \frac{1}{R} [a_{44}], \end{array} \\ \begin{array}{l} g_{23} = 2 Ω [J_{0}], & m_{22} = [J_{0}], \end{array} \\ \begin{array}{l} k_{31} = - \frac{1}{R} [a_{12}] [Λ^{(1)}], & k_{32} = - n {\frac{1}{R^{2}} ([a_{22}] + [a_{44}]) + 2 Ω^{2} [J_{0}]}, \end{array} \\ k_{33} = [a_{55}] [Λ^{(2)}] + [d a_{55}] [Λ^{(1)}] - {\frac{1}{R^{2}} ([a_{22}] + n^{2} [a_{44}]) + n^{2} Ω^{2} [J_{0}]}, \\ \begin{array}{l} k_{34} = [a_{55}] [Λ^{(1)}] + [d a_{55}], & k_{35} = \frac{n}{R} [a_{44}], & g_{32} = 2 Ω [J_{0}], & m_{33} = [J_{0}], \end{array} \\ \begin{array}{l} k_{43} = - [a_{55}] [Λ^{(1)}], & k_{44} = [d_{11}] [Λ^{(2)}] + [d d_{11}] [Λ^{(1)}] - {\frac{n^{2}}{R^{2}} [d_{66}] + [a_{55}] + n^{2} Ω^{2} [J_{2}]}, \end{array} \\ \begin{array}{l} k_{45} = \frac{n}{R} ([d_{12}] + [d_{66}]) [Λ^{(1)}] + \frac{n}{R} [d d_{12}], & m_{44} = [J_{2}], \end{array} \\ \begin{array}{l} k_{52} = \frac{1}{R} [a_{44}], & k_{53} = \frac{n}{R} [a_{44}], & k_{54} = - \frac{n}{R} ([d_{12}] + [d_{66}]) [Λ^{(1)}] - \frac{n}{R} [d d_{66}], \end{array} \\ \begin{array}{l} k_{55} = [d_{66}] [Λ^{(2)}] + [d d_{66}] [Λ^{(1)}] - {\frac{n^{2}}{R^{2}} [d_{22}] + [a_{44}] + n^{2} Ω^{2} [J_{2}]}, & m_{55} = [J_{2}], \end{array} \end{array}

where I_N×N represents the identity matrix and as presented in equation (48), some diagonal matrices are defined: (48)

\begin{array}{l} {[a_{p q}]}_{i i} = A_{p q} (x_{i}), & {[d a_{p q}]}_{i i} = A_{p q}^{'} (x_{i}), \\ {[J_{0}]}_{i i} = I_{0} (x_{i}), & i = 1, 2, 3, \dots, N . \\ {[d_{p q}]}_{i i} = D_{p q} (x_{i}), & {[d d_{p q}]}_{i i} = D_{p q}^{'} (x_{i}), \\ {[J_{2}]}_{i i} = I_{2} (x_{i}), & p & q = 1, 2, 4, 5, 6 . \end{array}

Appendix B

(49)

{[ψ]}_{10 \times 5 N} = [\begin{array}{l} ψ_{11} & ψ_{12} & ψ_{13} & ψ_{14} & ψ_{15} \\ ψ_{21} & ψ_{22} & ψ_{23} & ψ_{24} & ψ_{25} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ ψ_{101} & ψ_{102} & ψ_{103} & ψ_{104} & ψ_{105} \end{array}],

where ψ₁₁- ψ₅₅ are related to the condition at x = 0 and are presented in the relations below for various classical boundary conditions: (50)

\begin{array}{l} Clamped (C) : \\ ψ_{11} = ψ_{22} = ψ_{33} = ψ_{44} = ψ_{55} = I_{1}, \\ ψ_{12} = ψ_{13} = ψ_{14} = ψ_{15} = ψ_{21} = ψ_{23} = ψ_{24} = ψ_{25} = ψ_{31} = ψ_{32} = ψ_{34} = \\ ψ_{35} = ψ_{41} = ψ_{42} = ψ_{43} = ψ_{45} = ψ_{51} = ψ_{52} = ψ_{53} = ψ_{54} = {0}_{1 \times N}, \\ \begin{array}{l} Simply supported (S) : \\ \begin{array}{l} ψ_{11} = Λ_{1}^{(1)}, & ψ_{44} = Λ_{1}^{(1)}, & ψ_{22} = ψ_{33} = ψ_{55} = I_{1}, \end{array} \\ ψ_{12} = ψ_{13} = ψ_{14} = ψ_{15} = ψ_{21} = ψ_{23} = ψ_{24} = ψ_{25} = ψ_{31} = ψ_{32} = ψ_{34} = \\ ψ_{35} = ψ_{41} = ψ_{42} = ψ_{43} = ψ_{45} = ψ_{51} = ψ_{52} = ψ_{53} = ψ_{54} = {0}_{1 \times N}, \\ \begin{array}{l} Free (F) : \\ \begin{array}{l} ψ_{11} = A_{11}^{(0)} Λ_{1}^{(1)}, & ψ_{12} = \frac{n A_{12}^{(0)}}{R} I_{1}, & ψ_{13} = \frac{A_{12}^{(0)}}{R} I_{1}, & ψ_{21} = \frac{n}{R} I_{1}, \end{array} \\ \begin{array}{l} ψ_{22} = - Λ_{1}^{(1)}, & ψ_{33} = Λ_{1}^{(1)}, & ψ_{34} = I_{1}, & ψ_{44} = D_{11}^{(0)} Λ_{1}^{(1)}, \end{array} \\ \begin{array}{l} ψ_{45} = \frac{n D_{12}^{(0)}}{R} I_{1}, & ψ_{54} = \frac{n}{R} I_{1}, & ψ_{55} = - Λ_{1}^{(1)}, \end{array} \\ ψ_{14} = ψ_{15} = ψ_{23} = ψ_{24} = ψ_{25} = ψ_{31} = ψ_{32} = ψ_{35} = ψ_{41} = ψ_{42} = ψ_{43} = ψ_{51} = ψ_{52} = ψ_{53} = {0}_{1 \times N}, \end{array} \end{array} \end{array}

where subscript 1 is employed to indicate the first row of each matrix and (51)

\begin{array}{l} A_{p q}^{(0)} = {A_{p q} |}_{x = 0}, & D_{p q}^{(0)} = {D_{p q} |}_{x = 0}, & p & q = 1, 2 . \end{array}

ψ₆₁- ψ₁₀₅ are concerned with the condition at x = L and are presented in equations below for various classical boundary conditions: (52)

\begin{array}{l} Clamped (C) : \\ ψ_{61} = ψ_{72} = ψ_{83} = ψ_{94} = ψ_{105} = I_{N}, \\ ψ_{62} = ψ_{63} = ψ_{64} = ψ_{65} = ψ_{71} = ψ_{73} = ψ_{74} = ψ_{75} = ψ_{81} = ψ_{82} = ψ_{84} = \\ ψ_{85} = ψ_{91} = ψ_{92} = ψ_{93} = ψ_{95} = ψ_{101} = ψ_{102} = ψ_{103} = ψ_{104} = {0}_{1 \times N}, \\ \begin{array}{l} Simply supported (S) : \\ \begin{array}{l} ψ_{61} = Λ_{N}^{(1)}, & ψ_{94} = Λ_{N}^{(1)}, & ψ_{72} = ψ_{83} = ψ_{105} = I_{N}, \end{array} \\ ψ_{62} = ψ_{63} = ψ_{64} = ψ_{65} = ψ_{71} = ψ_{73} = ψ_{74} = ψ_{75} = ψ_{81} = ψ_{82} = ψ_{84} = ψ_{85} = \\ ψ_{91} = ψ_{92} = ψ_{93} = ψ_{95} = ψ_{101} = ψ_{102} = ψ_{103} = ψ_{104} = {0}_{1 \times N}, \\ \begin{array}{l} Free (F) : \\ \begin{array}{l} ψ_{61} = A_{11}^{(L)} Λ_{N}^{(1)}, & ψ_{62} = \frac{n A_{12}^{(L)}}{R} I_{N}, & ψ_{63} = \frac{A_{12}^{(L)}}{R} I_{N}, & ψ_{71} = \frac{n}{R} I_{N}, \end{array} \\ \begin{array}{l} ψ_{72} = - Λ_{N}^{(1)}, & ψ_{83} = Λ_{N}^{(1)}, & ψ_{84} = I_{N}, & ψ_{94} = D_{11}^{(L)} Λ_{N}^{(1)}, \end{array} \\ \begin{array}{l} ψ_{95} = \frac{n D_{12}^{(L)}}{R} I_{N}, & ψ_{104} = \frac{n}{R} I_{N}, & ψ_{105} = - Λ_{N}^{(1)}, \end{array} \\ ψ_{64} = ψ_{65} = ψ_{73} = ψ_{74} = ψ_{75} = ψ_{81} = ψ_{82} = ψ_{85} = ψ_{91} = ψ_{92} = ψ_{93} = ψ_{101} = ψ_{102} = ψ_{103} = {0}_{1 \times N}, \end{array} \end{array} \end{array}

where subscript N is employed to indicate the last row of each matrix and (53)

\begin{array}{l} A_{p q}^{(L)} = {A_{p q} |}_{x = L}, & D_{p q}^{(L)} = {D_{p q} |}_{x = L}, & p & q = 1, 2 . \end{array}

References

Sivadas

Ganesan

. Free vibration of circular cylindrical shells with axially varying thickness. J Sound Vib 1991; 147: 73–85.

Ganesan

Sivadas

. Vibration analysis of orthotropic shells with variable thickness. Comput Struct 1990; 35: 239–248.

Ganesan

Sivadas

. Free vibration of cantilever circular cylindrical shells with variable thickness. Comput Struct 1990; 34: 669–677.

Sivadas

Ganesan

. Axisymmetric vibration analysis of thick cylindrical shell with variable thickness. J Sound Vib 1993; 160: 387–400.

Duan

Koh

. Axisymmetric transverse vibrations of circular cylindrical shells with variable thickness. J Sound Vib 2008; 317: 1035–1041.

Viswanathan

Javed

Aziz

, et al. Free vibration of symmetric angle-ply laminated cylindrical shells of variable thickness including shear deformation theory. Int J Phys Sci 2011; 6: 6098–6109.

Viswanathan

Lee

Aziz

, et al. Free vibration of symmetric angle-ply laminated cylindrical shells of variable thickness. Acta Mech 2011; 221: 309–319.

El-Kaabazi

Kennedy

. Calculation of natural frequencies and vibration modes of variable thickness cylindrical shells using the Wittrick–Williams algorithm. Comput Struct 2012; 104-105: 4–12.

Taati

Fallah

Ahmadian

. Closed-form solution for free vibration of variable-thickness cylindrical shells rotating with a constant angular velocity. Thin-Walled Struct 2021; 166: 108062.

10.

Zheng

Liu

. Vibration characteristics analysis of an elastically restrained cylindrical shell with arbitrary thickness variation. Thin-Walled Struct 2021; 165: 107930.

11.

Quoc

Huan

Phuong

. Vibration characteristics of rotating functionally graded circular cylindrical shell with variable thickness under thermal environment. Int J Pres Ves Pip 2021; 193: 104452.

12.

Phu

Bich

Doan

. Nonlinear forced vibration and dynamic buckling analysis for functionally graded cylindrical shells with variable thickness subjected to mechanical load. Iran J Sci Technol Trans Mech Eng 2022; 46: 649–665.

13.

Thang

Kim

. Free vibration analysis of bi-directional functionally graded cylindrical shells with varying thickness. Aero Sci Technol 2023; 137: 108271.

14.

Sobhani

Avcar

. The influence of various nanofiller materials (CNTs, GNPs, and GOPs) on the natural frequencies of nanocomposite cylindrical shells: a comparative study. Mater Today Commun 2022; 33: 104547.

15.

Naghdi

Cooper

. Propagation of elastic waves in cylindrical shells, including the effects of transverse shear and rotatory inertia. J Acoust Soc Am 1956; 28: 56–63.

16.

Amirabadi

Afshari

Afjaei

, et al. Effect of variable thickness on the aeroelastic stability boundaries of truncated conical shells. Waves Random Complex Media. 2022; 1–24. DOI: 10.1080/17455030.2022.2157517.

17.

Amirabadi

Afshari

Darakhsh

, et al. The free vibration analysis of a cantilever trapezoidal plate with non-uniform thickness reinforced with non-uniformly distributed nanofillers. J Vib Control. 2023. DOI: 10.1177/10775463231214023.

18.

Amirabadi

Afshari

Sarafraz

, et al. Effects of non-uniformity in thickness and volume fraction of agglomerated CNTs on the dynamics of a spinning CNT-reinforced nanocomposite truncated conical shell. Noise Vib Worldw. 2024; 09574565241235330. DOI: 10.1177/09574565241235330.

19.

Afshari

Adab

. Size-dependent buckling and vibration analyses of GNP reinforced microplates based on the quasi-3D sinusoidal shear deformation theory. Mech Base Des Struct Mach 2022; 50: 184–205.

20.

Affdl

JCH

Kardos

. The Halpin-Tsai equations: a review. Polym Eng Sci 1976; 16: 344–352.

21.

Amirabadi

Darakhsh

Sarafraz

, et al. Vibration characteristics of a cylindrical sandwich shell with an FG auxetic honeycomb core and nanocomposite face layers subjected to supersonic fluid flow. Noise & Vibration Worldwide. 2024. DOI: 10.1177/09574565241252994.

22.

Kim

Park

Y-B

Okoli

, et al. Processing, characterization, and modeling of carbon nanotube-reinforced multiscale composites. Compos Sci Technol 2009; 69: 335–342.

23.

Rafiee

Wang

, et al. Enhanced mechanical properties of nanocomposites at low graphene content. ACS Nano 2009; 3: 3884–3890.

24.

Lin

Richard Liu

Cheng

. Single-layer graphene oxide reinforced metal matrix composites by laser sintering: microstructure and mechanical property enhancement. Acta Mater 2014; 80: 183–193.

25.

Suk

Piner

, et al. Mechanical properties of monolayer graphene oxide. ACS Nano 2010; 4: 6557–6564.

26.

Chen

Jiang

, et al. Dynamics of a spinning three-phase polymer/fiber/GNP laminated nanocomposite conical shell with non-uniform thickness. Int J Struct Stabil Dynam 2023; 24: 2450118.

27.

Afshari

. Effect of graphene nanoplatelet reinforcements on the dynamics of rotating truncated conical shells. J Braz Soc Mech Sci Eng 2020; 42: 519.

28.

Afshari

Amirabadi

. Vibration characteristics of rotating truncated conical shells reinforced with agglomerated carbon nanotubes. J Vib Control 2022; 28: 1894–1914.

29.

Bert

Malik

. Differential quadrature method in computational mechanics: a review. Appl Mech Rev 1996; 49: 1–28.

30.

Afshari

. Free vibration analysis of GNP-reinforced truncated conical shells with different boundary conditions. Aust J Mech Eng 2022; 20: 1363–1378.

31.

Tornabene

Bacciocchi

Fantuzzi

, et al. Multiscale approach for three-phase CNT/polymer/fiber laminated nanocomposite structures. Polym Compos 2019; 40: E102–E126.

32.

Sun

Chu

Cao

. Vibration characteristics of thin rotating cylindrical shells with various boundary conditions. J Sound Vib 2012; 331: 4170–4186.

33.

Qinkai

Fulei

. Effect of rotation on frequency characteristics of a truncated circular conical shell. Arch Appl Mech 2013; 83: 1789–1800.

34.

Dai

Cao

Chen

. Frequency analysis of rotating truncated conical shells using the Haar wavelet method. Appl Math Model 2018; 57: 603–613.