Effects of non-uniformity in thickness and volume fraction of agglomerated CNTs on the dynamics of a spinning CNT-reinforced nanocomposite truncated conical shell

Abstract

In the presented work, the free vibration analysis of a rotating truncated conical shell whose thickness varies in the meridional direction is examined. The material that the shell is made of is a polymeric matrix (epoxy) enriched with carbon nanotubes (CNTs) whose volume fraction varies in the meridional direction. CNTs agglomeration is considered and the density and elastic constants of such a two-phase nanocomposite are calculated sequentially based on the Eshelby–Mori–Tanaka approach alongside the rule of mixture. The conical shell is modeled using the first-order shear deformation theory (FSDT) including the relative acceleration, Coriolis acceleration, and centrifugal acceleration along with the initial hoop tension, and the governing equations are obtained utilizing Hamilton’s principle. The solution of governing equations is performed via the combination of an analytical solution in the circumferential direction and a numerical one in the meridional direction via the differential quadrature method (DQM). The impacts of several parameters on the forward and backward natural frequencies of such a rotating shell are studied such as chirality, mass fraction and dispersion pattern of the CNTs, thickness variation parameters, and boundary conditions. It is observed that higher natural frequencies and critical rotational speeds can be achieved for CNT-reinforced conical shells when the volume fraction of the CNTs and the thickness of the shell increase from its small radius to the large one.

Keywords

Conical shell rotating shell variable thickness carbon nanotubes free vibration

Introduction

There is a wide and extended range of usages and applications for rotating and non-rotating cylindrical and conical shells in mechanical, civil, aerospace, and nuclear engineering, such as pressure vessels, aircraft jet engines, gas turbines, and centrifugal separators.^1–3 Due to the variable radius of conical shells, there is a non-uniform stress distribution created by thermo-mechanical loads such as internal pressure, axial load, rotation, and temperature elevation. Thus, it is not an optimal design to utilize a constant thickness for a conical shell. Compared to the dynamic analysis of conical shells with uniform thickness, the dynamic analysis of those with variable thickness is poorly investigated. The natural frequencies of a conical shell with variable thickness were estimated and reported by Irie et al.⁴ They presented benchmark results for a shell whose thickness changes parabolically, linearly, or exponentially. Sivadas and Ganesa⁵ hired a higher-order shell theory and focused on the free vibration analysis of a clamped-clamped thick composite conical shell with variable thickness. They compared the natural frequencies with those predicted based on the classical thin shell theory^6,7 and discussed the effects of the employed shell theory on the estimated natural frequencies. Liu et al.⁸ inspected the free vibration analysis of a submerged conical shell with variable thickness stiffened with a ring. They discussed the impacts of several parameters on the natural frequencies including the geometric parameters of the ring and shell. Hu et al.⁹ examined the free vibration of laminated cylindrical and conical shells and sector plates with variable fiber curvature and variable thickness. They studied the effects of thickness variation parameters on the natural frequencies of these structures. Zarei and Rahimi¹⁰ focused on the influences of variable shell thickness and boundary conditions on the natural frequencies of a grid-stiffened composite conical shell. They carried out an experimental analysis to check the precision of their model. The free vibration and aeroelastic stability analyses of a truncated conical shell with variable thickness were inspected by Amirabadi et al.¹¹ They studied the dependency of the flutter boundaries (critical aerodynamic pressure) on the thickness variation parameters.

Compared to the works regarding the dynamic analysis of non-rotating conical shells, there are fewer works in the field of the dynamic analysis of rotating conical shells. In most of these works, the conical shells with uniform thickness are investigated. Qinkai and Fulei¹² focused on the free vibration analysis of a rotating isotropic homogenous truncated conical shell. They studied the impacts of the rotational speed and boundary conditions on the natural frequencies. In a similar work, they inspected the dynamic stability analysis of a rotating truncated conical shell exposed to the action of a harmonic axial load.¹³ They investigated the impacts of semi-vertex angle and rotational speed on stable and unstable regions. The free vibration analysis of a rotating non-homogenous truncated conical shell made of functionally graded (FG) material was discussed by Malekzadeh and Heydarpour.¹⁴ For several boundary conditions, they discussed the impacts of geometric parameters, graded index, and rotational speed on the natural frequencies. The free vibration characteristics of a rotating FG CNT-reinforced truncated conical shell were investigated by Heydarpour et al.¹⁵ They studied the dependency of the natural frequencies on the dispersion pattern and percentage of the CNTs. Dai et al.¹⁶ presented an approximate solution for the free vibrational characteristics of a rotating isotropic homogenous truncated conical shell. They employed the FSDT and examined the impacts of geometric parameters, boundary conditions, and rotational speed on the natural frequencies of rotating moderately thick shells. Shakouri¹⁷ inspected the influences of thermal loading on the natural frequencies of a rotating truncated conical shell made of FG material. He discussed the impacts of thermal conditions, geometric parameters, and rotational speed on the natural frequencies. Singha et al.¹⁸ examined the free vibrational behavior of a rotating pretwisted sandwich truncated conical shell with two graphene-reinforced layers and an isotropic homogenous core subjected to thermal loading. They focused on the impacts of rotational speed, pretwist angle, and geometric parameters on the natural frequencies. Afshari¹⁹ provided a parametric investigation on the free vibration behavior of a rotating truncated conical shell strengthened with graphene nanoplatelets (GNPs). He examined the dependency of the natural frequencies on several parameters including semi-vertex angle, rotational speed, mass fraction of the GNPs, and dispersion pattern of the GNPs. In a similar work, Afshari and Amiabadi²⁰ explored the free vibration of a rotating CNT-reinforced truncated conical shell. They investigated the impacts of dispersion pattern, mass fraction, chirality indexes, and agglomeration parameters of the CNTs on the natural frequencies. Utilizing the finite element method (FEM), Banerjee et al.²¹ studied the free vibration analysis of a rotating hybrid multiscale CNT-fiber-reinforced nanocomposite conical shell. They focused on the impacts of the mass fraction of CNTs and rotational speed on the fundamental frequency of the shell. Utilizing a higher-order shear deformation theory, Aris and Ahmadi²² inspected the free vibration analysis of a rotating stiffened truncated conical shell made of FG material exposed to thermal loading. The impacts of rotational speed, external and internal stiffeners, semi-vertex angle, and temperature elevation on the natural frequencies of rotating thick shells were inspected by them. He et al.²³ conducted a parametric investigation on the free vibrational analysis of a rotating laminated truncated conical shell with non-uniform thickness. They assumed that each lamina is made of three phases including a polymer enriched with GNPs and reinforced with glass fibers. They inspected the effects of the thickness variation parameters on the natural frequencies and the critical rotational speeds of the shell.

As observed, in most of the works concerning the dynamics of rotating shells, those with uniform thickness are investigated. The works associated with the dynamics of the shells with non-uniform thickness are limited to non-rotating shells. Also, in the works regarding the dynamics of CNT-reinforced shells, it is assumed that the volume fraction of the CNTs changes in the thickness direction rather than the length direction. Based on the authors’ best knowledge, the free vibration analysis of a rotating conical shell with variable thickness reinforced with non-uniformly distributed nanofillers has not been studied yet. Thus, for the first time, the natural frequencies and critical rotational speeds of a rotating CNT-reinforced truncated conical shell with variable thickness are studied in the presented research. The shell is modeled based on the FSDT including the relative acceleration, Coriolis acceleration, and centrifugal acceleration along with the initial hoop tension. The impacts of the chirality, distribution pattern and mass fraction of the CNTs, thickness variation parameters, and boundary conditions on the forward and backward natural frequencies and critical rotational speeds are discussed.

Governing equations

Description and mathematical modeling of the problem

As shown in Figure 1(a), truncated conical shell rotating at a constant rotational speed Ω is considered. The shell is of semi-vertex angle λ, length L, thickness h = h(x), small radius a, and large radius b = a+Lsinλ and is made of a polymeric matrix enriched with agglomerated CNTs.

Figure 1.

A rotating CNT-reinforced conical shell with variable thickness.

In classical shell theories such as Love, Donnell, Flugge, etc., the shear stress in the thickness direction and rotational inertia are ignored. Therefore, the predicted results based on these theories are applicable only to thin shells. In the FSDT, the shear stress in the thickness direction and rotational inertia are included and the results predicted based on this theory are applicable to thin shells to moderately thick ones. It is noteworthy that there are some higher-order shear deformation theories such as the third-order shear deformation theory (TSDT) which expands the applicability of the results to thick shells. As shown by Yousefi et al.,²⁴ the differences between the results predicted based on the FSDT and the TSDT are sensible only for shells with high values of the thickness-to-radius ratio. Since utilizing the TSDT results in a larger amount of equation and higher computational effort, in the presented work the FSDT is utilized. Based on the FSDT, the deformation inside the shell is described as the relations below^25,26:

\begin{array}{l} u_{x} (t, z, x, θ) = u (t, x, θ) + z ϕ_{x} (t, x, θ), \\ u_{θ} (t, z, x, θ) = v (t, x, θ) + z ϕ_{θ} (t, x, θ), \\ u_{z} (t, z, x, θ) = w (t, x, θ), \end{array}

(1)

in which u_x, u_θ, and u_z respectively represent displacement along x, θ, and z directions, the corresponding displacements at the neutral surface are demonstrated by u, v, and w, and ϕ_θ and ϕ_x sequentially show the rotation about x- and θ-axes.

The variation of the thickness obeys the relation below²⁷:

h (x) = \frac{1 + β}{1 + β + α} h_{a v} [1 + α {(\frac{x}{L})}^{β}] .

(2)

Equation (2) is regulated to result in the same average (average) thickness h_av of the shell for all values of β = 0,1,2,… and α > −1. It can be confirmed utilizing the relation below:

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

(3)

For α = ±0.25, and β = 0,1,2,3,4, the variations of the thickness of the shell in the meridional direction are illustrated in Figure 2.

Figure 2.

The variation of the thickness of the shell in the meridional direction.

Strain and stress tensors

Utilizing the assumption of shallow shells, the relations below are presented for normal (ε_ij) and shear (γ_ij = 2ε_ij) components of the strain tensor^28,29:

\begin{array}{l} \begin{array}{l} ε_{x x} = \frac{\partial u}{\partial x} + z \frac{\partial ϕ_{x}}{\partial x}, & ε_{θ θ} = \frac{1}{R} [(\frac{\partial v}{\partial θ} + w \cos λ + u \sin λ) + z (\frac{\partial ϕ_{θ}}{\partial θ} + ϕ_{x} \sin λ)], \end{array} \\ γ_{x θ} = 2 ε_{x θ} = \frac{1}{R} (\frac{\partial u}{\partial θ} - v \sin λ) + \frac{\partial v}{\partial x} + z (\frac{1}{R} \frac{\partial ϕ_{x}}{\partial θ} - \frac{ϕ_{θ}}{R} \sin λ + \frac{\partial ϕ_{θ}}{\partial x}), \\ \begin{array}{l} γ_{x z} = 2 ε_{x z} = ϕ_{x} + \frac{\partial w}{\partial x}, & γ_{θ z} = 2 ε_{θ z} = - \frac{v}{R} \cos λ + ϕ_{θ} + \frac{1}{R} \frac{\partial w}{\partial θ}, \end{array} \end{array}

(4)

where R(x) = a+xsinλ is the radius of the shell and is shown in Figure 1.

An isotropic polymeric matrix enriched with randomly oriented orthotropic CNTs is an isotropic two-phase structure. Accordingly, normal (σ_ij) and shear (τ_ij) components of the stress tensor are presented as

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

(5)

where k_s stands for the shear correction factor taken as k_s = 5/6, and Q₁₁-Q₆₆ are defined as

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

(6)

where E and G sequentially represent the elastic and shear moduli, and ν is the Poisson’s ratio.

Effective mechanical properties

Three patterns are considered for the dispersion pattern of the CNTs in the meridional direction. The volume fraction of CNTs in these patterns varies in the meridional direction as follows:

\begin{array}{c} Type 1 : & V_{r} (x) = V_{r}^{*}, \\ Type 2 : & V_{r} (x) = (1 + p) {(\frac{x}{L})}^{p} V_{r}^{*}, \\ Type 3 : & V_{r} (x) = (1 + \frac{1}{p}) [1 - {(\frac{x}{L})}^{p}] V_{r}^{*}, \end{array}

(7)

where p = 0,1,2,3,…, and the volume fraction of the matrix is achievable as V_m = 1-V_r. In Equation (7),

V_{r}^{*}

is the average volume fraction of the CNTs which is described in the relation below in terms of the density of the CNTs (ρ_r), the density of the polymeric matrix (ρ_m), and the mass fraction of CNTs (w_r)³⁰:

V_{r}^{*} = \frac{1}{1 + \frac{ρ_{r}}{ρ_{m}} (\frac{1}{w_{r}} - 1)} .

(8)

To bring about a fair comparison between the dispersion patterns, Equation (7) is regulated to result in the same average volume fraction and consequently the same mass fraction of CNTs for all dispersion patterns and various values of p.

The variations of volume fraction of the CNTs in the meridional direction are illustrated in Figure 3 for types 2 and 3 and p = 1,2,3,4 alongside type 1. It is obvious that type 1 (uniform distribution of the CNTs) is a specific case of type 2 with p = 0.

Figure 3.

The variation of volume fraction of the CNTs in the meridional direction.

Based on the rule of mixture, the relation below provides the density of a polymer enriched with CNTs³⁰:

ρ = ρ_{m} V_{m} + ρ_{r} V_{r},

(9)

The rule of mixture has a low accuracy in estimating the elastic constants of a CNT-reinforced polymer and cannot incorporate the CNTs agglomeration. Thus, in what follows the elastic constants are calculated utilizing the Eshelby–Mori–Tanaka approach,³¹ and CNTs agglomeration is modeled via a two-parameter model proposed by Shi et al.³²

Because of the high values of the length-to-diameter of the CNTs, they suffer from a low bending stiffness. Due to this weakness and the interfacial bonding between the polymeric matrix and the CNTs, the CNTs have a propensity to aggregate as illustrated in Figure 4. As observed, a certain amount of the CNTs is dispersed within the polymeric matrix, and the other ones are aggregated in the clusters. This inevitable phenomenon is called CNTs agglomeration.³³

Figure 4.

CNTs agglomeration.

To consider the CNTs agglomeration, Shi et al.³² proposed a two-parameter model. In this model, the intensity of the CNTs agglomeration is described via two parameters as

\begin{array}{l} 0 < η = \frac{V_{A g g}}{V_{CNT}} \leq 1, & 0 < μ = \frac{V_{C}}{V} \leq η, \end{array}

(10)

in which V, V_C, V_Agg, and V_CNT respectively are the volume of the shell, the volume of the clusters, the volume of the agglomerated CNTs, and the volume of the whole CNTs (whether agglomerated or not). CNTs agglomeration reduces the elastic constants of a CNT-reinforced polymer and its effects increase by a growth in η or a reduction in μ.³⁴ According to Equation (10) a growth in the agglomeration parameter η means a higher number of the CNTs are agglomerated and a reduction in the agglomeration parameter μ means there is a higher intensity of the agglomeration inside the clusters.

The relations below describe the elastic modulus and the Poisson’s ratio of an isotropic material are described in terms of the bulk (K) and shear moduli:

\begin{array}{l} E = \frac{9 K G}{3 K + G}, & ν = \frac{3 K - 2 G}{6 K + 2 G}, \end{array}

(11)

in which the shear and bulk moduli are affected by CNTs agglomeration as stated in the relations below³⁵:

\begin{array}{l} G = G_{o u t} [1 + \frac{μ (\frac{G_{i n}}{G_{o u t}} - 1)}{1 + (1 - μ) \frac{8 - 10 ν_{o u t}}{15 (1 - ν_{o u t})} (\frac{G_{i n}}{G_{o u t}} - 1)}], & K = K_{o u t} [1 + \frac{μ (\frac{K_{i n}}{K_{o u t}} - 1)}{1 + (1 - μ) \frac{1 + ν_{o u t}}{3 (1 - ν_{o u t})} (\frac{K_{i n}}{K_{o u t}} - 1)}], \end{array}

(12)

where subscripts “out” and “in” consecutively represent the outside and inside of the clusters and the elastic constants outside and inside of the clusters are presented as³⁵

\begin{array}{l} \begin{array}{l} G_{o u t} = G_{m} + \frac{(1 - η) (η_{r} - 2 G_{m} β_{r}) V_{r}}{2 [1 - μ + V_{r} (β_{r} - 1) (1 - η)]}, K_{o u t} = K_{m} + \frac{(δ_{r} - 3 K_{m} α_{r}) (1 - η) V_{r}}{3 [1 - μ + V_{r} (α_{r} - 1) (1 - η)]}, \end{array} \\ \begin{array}{l} G_{i n} = G_{m} + \frac{η (η_{r} - 2 G_{m} β_{r}) V_{r}}{2 [μ + V_{r} (β_{r} - 1) η]}, K_{i n} = K_{m} + \frac{η (δ_{r} - 3 K_{m} α_{r}) V_{r}}{3 [μ + V_{r} (α_{r} - 1) η]}, ν_{o u t} = \frac{3 K_{o u t} - 2 G_{o u t}}{6 K_{o u t} + 2 G_{o u t}}, \end{array} \end{array}

(13)

where G_m and K_m sequentially represent the shear and bulk moduli of the polymeric matrix and are presented in the relations below:

\begin{array}{l} G_{m} = \frac{0.5 E_{m}}{1 + ν_{m}}, & K_{m} = \frac{E_{m}}{3 (1 - 2 ν_{m})}, \end{array}

(14)

where E_m and ν_m consecutively indicate the elastic modulus and the Poisson’s ratio of the polymeric matrix and the definitions of other parameters are presented as follows³⁵:

\begin{array}{l} α_{r} = \frac{3 (G_{m} + K_{m}) + k_{r} + l_{r}}{3 (k_{r} + G_{m})}, \\ β_{r} = \frac{1}{5} [\frac{1}{3} \frac{4 G_{m} + l_{r} + 2 k_{r}}{k_{r} + G_{m}} + \frac{4 G_{m}}{G_{m} + p_{r}} + \frac{4 G_{m} (4 G_{m} + 3 K_{m})}{(3 K_{m} + G_{m}) G_{m} + (3 K_{m} + 7 G_{m}) m_{r}}], \\ δ_{r} = \frac{1}{3} [2 l_{r} + n_{r} + \frac{(3 K_{m} + G_{m} - l_{r}) (2 k_{r} + l_{r})}{k_{r} + G_{m}}], \\ η_{r} = \frac{1}{5} [\frac{2}{3} (n_{r} - l_{r}) + \frac{2}{3} \frac{(l_{r} + 2 G_{m}) (k_{r} - l_{r})}{k_{r} + G_{m}} + \frac{8 p_{r} G_{m}}{p_{r} + G_{m}} + \frac{8 m_{r} G_{m} (3 K_{m} + 4 G_{m})}{3 K_{m} (m_{r} + G_{m}) + (G_{m} + 7 m_{r}) G_{m}}], \end{array}

(15)

in which p_r, n_r, m_r, l_r, k_r are five independent elastic constants known as Hill’s elastic moduli.³⁶ These constants are reported in Table 1 in GPa alongside the density for some selected single-walled carbon nanotubes (SWCNT) of several chiral indexes.

Table 1.

Hill’s elastic moduli (GPa) and density for some selected SWCNTs.³⁵

Chiral indexes (m,n)	k _r	l _r	m _r	n _r	p _r	ρ_r (kg/m³)
(5,5)	536	184	132	2143	791	1400
(10,10)	271	88	17	1089	442
(15,15)	181	58	5	726	301
(20,20)	136	43	2	545	227
(50,50)	55	17	0.1	218	92

It is noteworthy that details on the manufacturing aspects of CNT-reinforced nanocomposites can be found in previously published papers.^37–39

Hamilton’s principle

The boundary conditions and governing equations can be derived through the relation below known as Hamilton’s principle^19,20:

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

(16)

in which t stands for time, t₁ and t₂ represent two arbitrary moments, δ is called the variational operator, T indicates kinetic energy, U_s is the strain energy, U_h stands for potential energy due to the initial hoop tension, and W_ext represents the work done by non-conservative external loads.

The relation below provides the kinetic energy:

T = \frac{1}{2} \underset{V}{∭} ρ \vec{v} . \vec{v} d V,

(17)

where the volume of the shell can be presented as

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

(18)

where S represents the surface of the shell and is defined in the relation below:

d S = R (x) d x d θ,

(19)

In equation (17),

\vec{v}

indicates the absolute velocity vector which is calculated as^19,20

\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},

(20)

in which

\begin{array}{l} \vec{Ω} = Ω (- \vec{i} \cos λ + \vec{k} \sin λ) . & \vec{R} = u_{x} \vec{i} + u_{θ} \vec{j} + u_{z} \vec{k} . \end{array}

(21)

Substituting equation (21) into equation (20), results in

\vec{v} = (\frac{\partial u_{x}}{\partial t} - Ω u_{θ} \sin λ) \vec{i} + [\frac{\partial u_{θ}}{\partial t} + Ω (u_{x} \sin λ + u_{z} \cos λ)] \vec{j} + (\frac{\partial u_{z}}{\partial t} - Ω u_{θ} \cos λ) \vec{k},

(22)

and inserting equations (1) and (22) into equation (17) and considering equation (18) result in the relation below:

\begin{array}{l} T = \frac{1}{2} \iint_{S} {I_{0} [{(\frac{\partial u}{\partial t})}^{2} + {(\frac{\partial v}{\partial t})}^{2} + {(\frac{\partial w}{\partial t})}^{2}] + I_{2} [{(\frac{\partial ϕ_{x}}{\partial t})}^{2} + {(\frac{\partial ϕ_{θ}}{\partial t})}^{2}] + 2 Ω [I_{0} (\frac{\partial v}{\partial t} u \sin λ \\ + \frac{\partial v}{\partial t} w \cos λ - \frac{\partial u}{\partial t} v \sin λ - \frac{\partial w}{\partial t} v \cos λ) + I_{2} (\frac{\partial ϕ_{θ}}{\partial t} ϕ_{x} \sin λ - \frac{\partial ϕ_{x}}{\partial t} ϕ_{θ} \sin λ)] \\ + Ω^{2} [I_{0} (u^{2} \sin^{2} λ + v^{2} + w^{2} \cos^{2} λ + u w \sin 2 λ) + I_{2} (ϕ_{x}^{2} \sin^{2} λ + ϕ_{θ}^{2})]} d S . \end{array}

(23)

where the inertia terms are defined in the relation below:

{\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}},

(24)

and the following relation is utilized

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

(25)

The equation below provides the strain energy of the shell:

U_{s} = \frac{1}{2} \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,

(26)

which can be represented by utilizing equations (4) and (18) as

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

(27)

where N_ij, Q_ij, and M_ij represent the stress resultants and are considered as follows:

\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}

(28)

Inserting equations (4) and (5) into equation (28) brings about the equations below:

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

(29)

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

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

(30)

and the following relation is utilized

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

(31)

Regardless of vibration, rotation provides a stationary circumferential stress in the shell known as the initial hoop tension. The corresponding potential energy is presented in the relation below^19,20:

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

(32)

where, as defined in equation (33),

σ_{θ θ}^{0}

and

ε_{θ θ}^{N L}

respectively indicate the initial hoop stress and the nonlinear part of circumferential normal strain^19,20:

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

(33)

Utilizing equations (1), (18), (24), (25), (32), and (33), the relation below can be achieved:

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

(34)

The work done by non-conservative external loads is considered equal to zero in the free vibration analysis (W_ext = 0). Accordingly, by inserting equations (19), (23), (27), and (34) into equation (16), the governing equations are derived as the relations below:

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

(35)

and the boundary conditions at x = 0 and x = L are stated in the relations below:

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

(36)

It is noteworthy that in the set of governing equations (35), the terms containing second-time derivatives (∂²/∂t²) are associated with the relative acceleration, those containing the square of rotational speed (Ω²) are associated with either the initial hoop tension or the centrifugal acceleration, and the terms containing rotational speed (Ω) are associated with the Coriolis parts of acceleration.¹⁹

Solution procedure

Analytical solution in the circumferential direction

Inserting equation (29) into equation (35) and utilizing the solution below^19,20:

\begin{array}{l} \begin{array}{l} u (t, x, θ) = \sum_{n = 0}^{\infty} U (x) \cos (ω t + n θ), & v (t, x, θ) = \sum_{n = 0}^{\infty} V (x) \sin (ω t + n θ), \end{array} \\ w (t, x, θ) = \sum_{n = 0}^{\infty} W (x) \cos (ω t + n θ), \\ \begin{array}{l} ϕ_{x} (t, x, θ) = \sum_{n = 0}^{\infty} X (x) \cos (ω t + n θ), & ϕ_{θ} (t, x, θ) = \sum_{n = 0}^{\infty} Θ (x) \sin (ω t + n θ), \end{array} \end{array}

(37)

result in the following relations:

\begin{array}{l} I_{0} ω^{2} U + 2 I_{0} Ω ω V \sin λ + A_{11} U^{″} + (\frac{A_{11} \sin λ}{R} + A_{11}^{'}) U^{'} - (\frac{A_{22} \sin^{2} λ + A_{66} n^{2}}{R^{2}} - \frac{A_{12}^{'} \sin λ}{R} \\ + I_{0} n^{2} Ω^{2}) U + \frac{(A_{12} + A_{66}) n}{R} V^{'} - n [\frac{(A_{66} + A_{22}) \sin λ}{R^{2}} - \frac{A_{12}^{'}}{R} + 2 I_{0} Ω^{2} \sin λ] V + \frac{A_{12} \cos λ}{R} W^{'} \\ - \cos λ (\frac{A_{22} \sin λ}{R^{2}} - \frac{A_{12}^{'}}{R}) W = 0, \end{array}

\begin{array}{l} I_{0} ω^{2} V + 2 I_{0} Ω ω (U \sin λ + W \cos λ) - \frac{n (A_{12} + A_{66})}{R} U^{'} - n [(A_{22} + A_{66}) \frac{\sin λ}{R^{2}} + \frac{A_{66}^{'}}{R} + 2 I_{0} Ω^{2} \sin λ] U \\ + A_{66} V^{″} + (\frac{A_{66} \sin λ}{R} + A_{66}^{'}) V^{'} - (\frac{A_{22} n^{2} + A_{44} \cos^{2} λ + A_{66} \sin^{2} λ}{R^{2}} + \frac{A_{66}^{'} \sin λ}{R} + I_{0} n^{2} Ω^{2}) V \\ - n \cos λ (\frac{A_{22} + A_{44}}{R^{2}} + 2 I_{0} Ω^{2}) W + \frac{A_{44} \cos λ}{R} Θ = 0, \end{array}

\begin{array}{l} I_{0} ω^{2} W + 2 I_{0} Ω ω V \cos λ - \frac{A_{12} \cos λ}{R} U^{'} - \frac{A_{22} \sin λ \cos λ}{R^{2}} U \\ - n \cos λ (\frac{A_{22} + A_{44}}{R^{2}} + 2 I_{0} Ω^{2}) V + A_{55} W^{″} + (\frac{A_{55} \sin λ}{R} + A_{55}^{'}) W^{'} \\ - (\frac{A_{22} \cos^{2} λ + A_{44} n^{2}}{R^{2}} + I_{0} n^{2} Ω^{2}) W + A_{55} X^{'} + (\frac{A_{55} \sin λ}{R} + A_{55}^{'}) X + \frac{A_{44} n}{R} Θ = 0, \end{array}

\begin{array}{l} I_{2} ω^{2} X + 2 I_{2} Ω ω Θ \sin λ - A_{55} W^{'} + D_{11} X^{″} + (\frac{D_{11} \sin λ}{R} + D_{11}^{'}) X^{'} \\ - (\frac{D_{22} \sin^{2} λ + D_{66} n^{2}}{R^{2}} - \frac{D_{12}^{'} \sin λ}{R} + A_{55} + I_{2} n^{2} Ω^{2}) X + \frac{(D_{12} + D_{66}) n}{R} Θ^{'} \\ - n [\frac{(D_{22} + D_{66}) \sin λ}{R^{2}} - \frac{D_{12}^{'}}{R} + 2 I_{2} Ω^{2} \sin λ] Θ = 0, \end{array}

\begin{array}{l} I_{2} ω^{2} Θ + 2 I_{2} Ω ω X \sin λ + \frac{A_{44} \cos λ}{R} V + \frac{A_{44} n}{R} W - \frac{(D_{12} + D_{66}) n}{R} X^{'} \\ - n [\frac{(D_{22} + D_{66}) \sin λ}{R^{2}} + \frac{D_{66}^{'}}{R} + 2 I_{2} Ω^{2} \sin λ] X + D_{66} Θ^{″} + (\frac{D_{66} \sin λ}{R} + D_{66}^{'}) Θ^{'} \\ - (\frac{D_{22} n^{2} + D_{66} \sin^{2} λ}{R^{2}} + \frac{D_{66}^{'} \sin λ}{R} + A_{44} + I_{2} n^{2} Ω^{2}) Θ = 0; \end{array}

(38)

where the prime is hired to indicate derivative with respect to x. It is noteworthy that in equation (37), ω is the natural frequency and n is the circumferential wave number.

By inserting equations (29) and (37) into equation (36), the relations below can be stated for the boundary conditions at x = 0 & L:

\begin{array}{l} Clamped (C) : & U = V = W = 0, & X = Θ = 0, \end{array}

\begin{array}{l} \begin{array}{l} Simply Supported (S) : & V = W = 0, & Θ = 0, \end{array} \\ \begin{array}{l} A_{11} U^{'} + \frac{A_{12} \sin λ}{R} U = 0, & D_{11} X^{'} + \frac{D_{12} \sin λ}{R} X = 0, \end{array} \end{array}

\begin{array}{l} \begin{array}{l} Free (F) : & \begin{array}{l} A_{11} U^{'} + \frac{A_{12}}{R} (U \sin λ + n V + W \cos λ) = 0, & \frac{n}{R} U + \frac{V}{R} \sin λ - V^{'} = 0, \end{array} \end{array} \\ \begin{array}{l} X + W^{'} = 0, & D_{11} X^{'} + \frac{D_{12}}{R} (X \sin λ + n Θ) = 0, & \frac{n}{R} X - Θ^{'} + \frac{\sin λ}{R} Θ = 0 . \end{array} \end{array}

(39)

Approximate solution in the meridional direction

Mathematical complexities in the governing equations (38) and boundary conditions (39) reduce the possibility of finding an analytical solution in the meridional direction. Therefore, an approximate solution is presented in the current section via the DQM. The main idea of this numerical method is to discretize the domain of the problem to N pre-selected points and estimate derivatives of a function like g(x) at each point using the values of the function at these points as

{\frac{d^{s} g}{d x^{s}}}_{N \times 1} = {[D^{(s)}]}_{N \times N} {g}_{N \times 1},

(40)

in which [D^(s)] represents the weighting coefficient matrix for the sth order derivative and is calculated via the relations below⁴⁰:

\begin{array}{l} D_{i j}^{(1)} = {\begin{cases} \begin{array}{l} \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 \\ \sum_{\begin{array}{l} m = 1 \\ m \neq i \end{array}}^{N} {(x_{i} - x_{m})}^{- 1} & i = j \end{array} & i, j = 1, 2, 3, . . ., N \end{cases}, \\ \begin{array}{l} [D^{(s)}] = [D^{(1)}] [D^{(s - 1)}], & s = 2, 3, \dots, N - 1 . \end{array} \end{array}

(41)

The dispersion pattern of discrete points has a significant impact on the convergence and precision of the solution. A well-accepted one is the Gauss–Lobatto–Chebyshev pattern which is presented in the relation below for 0 ≤ x≤L⁴⁰:

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

(42)

Utilizing equations (42) and (40), the governing equations are described in the algebraic form below:

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

(43)

where, as defined in Appendix A,

{q}

is the total displacement vector, and [M], [G], and [K] consecutively represent the mass matrix, gyroscopic matrix, and stiffness matrix.

Utilizing equations (45), (40), and (A-1), the boundary conditions are represented in the algebraic form below:

[Λ] {q} = {0},

(44)

where [Λ] is described in Appendix B.Simultaneous solutions of algebraic equation (43) and (44) provide the natural frequencies, but it results in an inconsistency in the numbers of undetermined variables and equations.²⁸ This inconsistency can be overcome by dividing the grid points into two sets: the boundary points (x₁ and x_N) and domain ones (x₂-x_N-1). To overcome the inconsistency satisfying equation (43) can be neglected at the boundary points and this equation can be represented as the relation below:

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

(45)

in which, for each matrix, the subscript t means a truncated non-square matrix of order (5N-10)×5N created by removing 10 rows related to the boundary points. It includes 1th, Nth, (N+1)th, 2Nth, (2N+1)th,…, and 5Nth rows. By separating the domain and boundary points,equations (44) and (45) can be partitioned and rearranged as the relations below:

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

(46-a)

{[Λ]}_{b} {q}_{b} + {[Λ]}_{d} {q}_{d} = {0},

(46-b)

where the subscripts “b” and ''d'' respectively imply the boundary and domain points. Substituting equation (46) into equation (46-a) results in the eigenvalue equation below:

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

(47)

where

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

(48)

The natural frequencies of the rotating shell can be found through the solution of the eigenvalue equation (47). Positive eigenvalues are called forward natural frequencies and negative ones are called backward natural frequencies.^19,20 It is noteworthy that the reverse definition is utilized by some authors.

Numerical examples and discussions

Numerical examples are presented in this section to investigate the impacts of several parameters on the natural frequencies and critical rotational speeds. The boundary conditions are denoted by two letters that respectively depict the boundary conditions at the small and large radii of the shell. Also, the forward and backward natural frequencies are denoted by F and B, sequentially. The natural frequencies are represented by ω_nm where n represents the circumferential wave number (Eq. (37)) and m = 1,2,… determines the sequence of vibrational modes in the meridional direction. The natural frequencies and the rotational speed of the shell are reported in dimensionless forms defined in the relations below:

\begin{array}{l} ω_{n m}^{*} = ω_{n m} a \sqrt{\frac{ρ_{m}}{E_{m}}}, & Ω^{*} = Ω a \sqrt{\frac{ρ_{m}}{E_{m}}} . \end{array}

(49)

Except in cases which are stated directly, numerical results are reported for an SC rotating CNT-reinforced truncated conical shell of λ = 30°, L/a = 3, h_av/a = 0.02, a = 0.25 m, α = 0.5, and β = 1 rotating at Ω^* = 0.2. Epoxy is chosen for the polymeric matrix with ρ_m = 1150 kg/m³, ν_m = 0.34, and E_m = 2.1 GPa.³⁵ This matrix is strengthened with SWCNT (10,10) based on type 2 with p = 1, w_r = 0.01 (1%), η = 0.8 and μ = 0.2.

Convergence and verification

For n = 2, the variations of the estimated forward and backward natural frequencies versus the variations of the number of grid points (N is equation (40) are investigated in Figure 5. As observed, the presented numerical solution via the DQM benefits from a high convergence. Hereafter, the results are reported for N = 13.

Figure 5.

Convergence analysis of the provided numerical solution via the DQM

In what follows, three examples are presented to prove the precision of the presented work. As the first one, consider a rotating homogenous isotropic (ν = 0.3) conical shell with uniform thickness (α = β = 0) of geometric parameters L/a = 6, λ = 30°, and h/a = 0.01. For two different boundary conditions and two values of the rotational speed (Ω^** = Ωb [ρ(1-ν²)/E]^0.5), the forward and backward natural frequencies (ω^** = ωb [ρ(1-ν²)/E]^0.5) are tabulated in Table 2 for n = m = 1 along with the corresponding results reported by Dai et al.¹⁶ The results presented in this table verify the precision of the presented work.

Table 2.

Dimensionless forward and backward natural frequencies (ω^**) of an isotropic homogenous rotating conical shell with uniform thickness (λ = 30°, L/a = 6, h/a = 0.01, ν = 0.3).

		Ω^** = 0.2		Ω^** = 0.3
		Forward	Backward	Forward	Backward
CC	Present	0.8855	0.5990	0.9443	0.5198
CC	Dai et al¹⁶	0.8836	0.6018	0.9260	0.5265
SS	Present	0.7217	0.5390	0.7562	0.4480
SS	Dai et al¹⁶	0.7290	0.5331	0.7724	0.4350

As the second example, consider a rotating (Ω^* = 0.1) CS CNT-reinforced conical shell with uniform thickness (α = β = 0) of geometric parameters L/a = 4, λ = 20°, and h/a = 0.1. The polymeric (epoxy) matrix of the shell is reinforced with SWCNT (20,20) based on type 1 (uniform) with w_r = 0.05 (5%), η = 0.8 and μ = 0.4. For n = 1 and m = 1,2,3,4 the dimensionless forward and backward natural frequencies (ω^*_nm) are presented in Table 3 along with the corresponding ones reported by Afshari and Amirabadi.²⁰ The results presented in this table verify the precision of the presented work.

Table 3.

Dimensionless forward and backward natural frequencies of a rotating CS CNT-reinforced conical shell with uniform thickness (L/a = 4, λ = 20°, h/a = 0.1, SWCNT (20,20), type 1, w_r = 0.05, η = 0.8, μ = 0.4, Ω* = 0.1, n = 1).

		m = 1	m = 2	m = 3	m = 4
Forward	Present	0.5529	0.5887	0.7615	0.8646
Forward	Afshari and Amirabadi²⁰	0.5489	0.5835	0.7552	0.8577
Backward	Present	0.3653	0.5798	0.7136	0.8050
Backward	Afshari and Amirabadi²⁰	0.3612	0.5747	0.7069	0.7981

As the third example, consider a non-rotating (Ω^* = 0) SC isotropic homogenous (ν = 0.3) conical shell with non-uniform thickness (α = 0.25, β = 1) of geometric parameters L/a = 4, λ = 75°, and h/a = 0.01. For n = 3 and m = 1,2,3,4,5,6 the dimensionless natural frequencies (ω^*_nm) are presented in Table 4 along with those reported by Amirabadi et al.¹¹ The results presented in this table confirm the accuracy of the presented work.

Table 4.

Dimensionless natural frequencies of a non-rotating SC isotropic homogenous conical shell with non-uniform thickness (ν = 0.3, L/a = 4, λ = 75°, h/a = 0.01, α = 0.25, β = 1, n = 3).

	m = 1	m = 2	m = 3	m = 4	m = 5	m = 6
Present	0.0398	0.0612	0.0753	0.0909	0.1093	0.1288
Amirabadi et al¹¹	0.0398	0.0612	0.0753	0.0909	0.1093	0.1288

Parametric study

For n = 0,1,2,3 and m = 1,2,3,4 the variations of the natural frequencies versus the variations of the rotational speed are depicted in Figure 6 which is called the Campbell diagram. The Campbell diagram is the most important tool in the vibrational analysis of a rotating system. It not only presents the variations of both forward and backward natural frequencies versus the variation of the rotational speeds but also helps engineers to predict the critical rotational speeds of the rotating system which is discussed in the following.

Figure 6.

The Campbell diagram of the rotating conical shell for n = 0,1,2,3.

As shown in Figure 6, for a non-rotating shell forward and backward natural frequencies are the same in each vibrational mode. As the rotational speed grows, the gyroscopic effect and the initial hoop tension increase. The gyroscopic effect leads to higher forward natural frequencies and lower backward natural frequencies and initial hoop tension results in a higher stiffness and increases both forward and backward natural frequencies. Accordingly, as demonstrated in Figure 6 for n ≥ 1, by growing the rotational speed, all forward natural frequencies grow due to compatible effects of the initial hoop tension and the gyroscopic effect, and backward ones experience different trends due to incompatible impacts of the initial hoop tension and the gyroscopic effect.

Although the shell is not exposed to an external load, any residual unbalance-which is usually inevitable-generates a harmonic excitation whose excitation frequency is equal to the rotational speed of the shell. As a result, the bisector Ω^* = ω^*_nm is depicted in Figure 6. Intersections of this bisector with the Campbell diagram determine the critical rotational speeds of the shell which result in resonance phenomenon and abrupt failure. At these critical rotational speeds, any residual unbalance increases the amplitude of vibration and leads to a failure. As Figure 6 shows, the critical rotational speeds of the shell are created by the intersection of the bisector Ω^* = ω^*_nm with the backward frequencies. It reveals the high importance of incorporating the backward modes along with the forward ones in the vibration analysis of a rotating system.

As demonstrated, critical rotational speeds can be seen only for n = 0,1,2 and vanish for n ≥ 3 which is an advantage of the initial hoop tension. Hereafter, the impacts of several parameters on the critical rotational speeds (Campbell diagram) and the natural frequencies are explored. To reduce the length of the paper, these investigations are limited to two selected vibrational modes ω₁₁ and ω₁₂.

Figures 7 and 8 are presented to explore the impacts of the thickness variation parameters on the Campbell diagram. These figures demonstrate that to achieve higher natural frequencies and critical rotational speeds, it is better to use higher values of the thickness variation parameter β and the higher positive values of the thickness variation parameter α. According to Figure 2, it means that higher natural frequencies and critical rotational speeds are attained by growing the thickness from the small radius to the large one. The reason for this superiority is that stiffness of the shell diminishes at the points with larger radius and increasing the thickness of the shell at these points leads to higher improvement in stiffness of the shell. It is noteworthy that as these figures show, thickness variation parameters weakly affect the Campbell diagram which can be explained by Equation (2). As stated, this equation is regulated to result in the same average thickness of the shell for all values of α and β.

Figure 7.

The impacts of the thickness variation parameter α on the Campbell diagram.

Figure 8.

The impacts of the thickness variation parameter β on the Campbell diagram.

The impacts of the mass fraction (percentage) of the CNTs on the Campbell diagram are studied in Figure 9. Compared to the epoxy (matrix), CNTs have higher elastic constants and approximately the same density. Thus, subjoining more CNTs to the polymeric matrix increases the stiffness of the shell and as observed in Figure 9, it dramatically results in higher natural frequencies and critical rotational speeds.

Figure 9.

The impacts of the mass fraction of the CNTs on the Campbell diagram.

Figure 10 shows the dependency of the Campbell diagram on the chirality of the CNTs. CNTs with lower chirality indexes have smaller diameters which result in a shorter distance between the carob atoms. Accordingly, CNTs with lower chirality indexes benefit from higher elastic constants (Table 1). As a result, utilizing CNTs with lower chirality indexes provides higher stiffness of the shell and as observed in Figure 10, brings about higher natural frequencies and critical rotational speeds.

Figure 10.

Impacts of the chirality of the CNTs on the Campbell diagram.

Figure 11 shows the impacts of the dispersion pattern of the CNTs on the Campbell diagram. As this figure demonstrates, the highest natural frequencies and critical rotational speeds belong to the shell reinforced based on type 2 and the lowest ones belong to the shell reinforced based on type 3. Based on Figure 3, it can be concluded that the CNTs distributed near the large radius of the shell (x = L) have a better enriching effect than those distributed near the small radius of the shell (x = 0). In other words, to achieve higher natural frequencies and critical rotational speeds, it is better to distribute most of the CNTs close to the large radius of the shell. The reason behind the superiority of such a distribution pattern is that the stiffness of the shell decreases at the points with a larger radius and subjoining most of the CNTs nearer to these points provides a better reinforcing effect and brings about higher stiffness of the shell.

Figure 11.

The impacts of the dispersion pattern of the CNTs on the Campbell diagram.

Figure 12 is devoted to inspecting the impacts of the dispersion pattern parameter p on the Campbell diagram for both types 2 and 3. As observed, to achieve higher natural frequencies and critical rotational speeds, it is better to utilize low values of p for type 2 and high values of p for type 3. In conjunction with Figure 3, it is revealed that more gentle variation in the volume fraction of the CNTs along the meridional direction improves the stiffness of the shell and results in higher natural frequencies and critical rotational speeds.

Figure 12.

The impacts of the dispersion pattern parameter p on the Campbell diagram.

The impacts of the boundary conditions on the Campbell diagram are inspected in Figure 13. As demonstrated in this figure, the highest critical rotational speeds belong to the CC shell. It means that utilizing more restricted supports provides higher stiffness and results in higher critical rotational speeds. As demonstrated in Figure 13, depending on the vibrational modes, the lowest critical rotational speeds belong to the FC or SS ones. It reveals that the critical rotational speeds of the shell are more affected by the condition at the large radius of the shell (x = L) rather than the condition at the small one (x = 0). The main reason is the higher perimeter and longer edge of the shell at x = L.

Figure 13.

The impacts of the boundary conditions on the Campbell diagram.

Along with the critical rotational speeds associated with the resonance phenomenon, there are other critical rotational speeds. At each of these critical rotational speeds, a backward natural frequency reaches zero which results in unstable oscillations. For example, as demonstrated in Figure 13, such a critical situation can be seen for an FC shell at Ω* = 0.4285 for n = m = 1. As Figure 6 shows, the initial hoop tension removes these types of critical rotational speeds for high values of the circumferential wave number (n ≥ 2). This is similar to what happens for the critical rotational speeds associated with the resonance phenomenon.

Conclusions

The free vibration analysis of a rotating conical shell with variable thickness enriched with CNTs was discussed in this research. The shell was modeled based on the FSDT including the relative acceleration, Coriolis acceleration, and centrifugal acceleration along with the initial hoop tension. The impacts of several parameters on the natural frequencies and critical rotational speeds were explored. The main findings and conclusions of this work are declared in the statements below:

• As the rotational speed grows, natural frequencies in forward modes increase, but experience different trends in backward ones.

• Because of the positive impact of the initial hoop tension, the critical rotational speeds associated with both resonance phenomenon and unstable oscillations do not exist for high values of the circumferential wave number.

• The following items can be utilized to achieve higher natural frequencies and critical rotational speeds of a rotating FG CNT-reinforced truncated conical shell with variable thickness:

-Increasing the thickness of the conical shell from its small radius to its large radius

-Increasing the mass fraction (percentage) of the subjoined CNTs

-Increasing the volume fraction of the subjoined CNTs gently from the small radius to the large one

-Utilizing the CNTs with lower chirality indexes

-Utilizing more restricted conditions (preferably clamped) at the edges of the shell, especially at the large radius

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

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

ORCID iD

Hassan Afshari

Data Availability Statement

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

Appendix

References

Maleki

Tahani

. An investigation into the static response of fiber-reinforced open conical shell panels considering various types of orthotropy. Proc IME C J Mech Eng Sci 2014; 228: 3–21.

Das

. Study on CNT reinforced functionally graded sandwich conical shell subjected to low-velocity impact under thermal environment. Noise Vib Worldw 2022; 53: 189–205.

Nasution

Syah

Ramdan

, et al. Modeling and computational simulation for supersonic flutter prediction of polymer/GNP/fiber laminated composite joined conical-conical shells. Arab J Chem 2022; 15: 103460.

Irie

Yamada

Kaneko

. Free vibration of a conical shell with variable thickness. J Sound Vib 1982; 82: 83–94.

Sivadas

Ganesan

. Vibration analysis of thick composite clamped conical shells of varying thickness. J Sound Vib 1992; 152: 27–37.

Sivadas

Ganesan

. Free vibration of cantilever conical shells with variable thickness. Comput Struct 1990; 36: 559–566.

Sivadas

Ganesan

. Vibration analysis of laminated conical shells with variable thickness. J Sound Vib 1991; 148: 477–491.

Liu

Cheng

. Free vibration of a fluid loaded ring-stiffened conical shell with variable thickness. J Vib Acoust 2014; 136: 051003.

Zhong

Wang

, et al. A strong-form Chebyshev-RPIM meshless solution for free vibration of conical shell panels with variable thickness and fiber curvature. Compos Struct 2022; 296: 115884.

10.

Zarei

Rahimi

. Effect of boundary condition and variable shell thickness on the vibration behavior of grid-stiffened composite conical shells. Appl Acoust 2022; 188: 108546.

11.

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.

12.

Qinkai

Fulei

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

13.

Qinkai

Fulei

. Parametric instability of a rotating truncated conical shell subjected to periodic axial loads. Mech Res Commun 2013; 53: 63–74.

14.

Malekzadeh

Heydarpour

. Free vibration analysis of rotating functionally graded truncated conical shells. Compos Struct 2013; 97: 176–188.

15.

Heydarpour

Aghdam

Malekzadeh

. Free vibration analysis of rotating functionally graded carbon nanotube-reinforced composite truncated conical shells. Compos Struct 2014; 117: 187–200.

16.

Dai

Cao

Chen

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

17.

Shakouri

. Free vibration analysis of functionally graded rotating conical shells in thermal environment. Compos B Eng 2019; 163: 574–584.

18.

Singha

Rout

Bandyopadhyay

, et al. Free vibration of rotating pretwisted FG-GRC sandwich conical shells in thermal environment using HSDT. Compos Struct 2021; 257: 113144.

19.

Afshari

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

20.

Afshari

Amirabadi

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

21.

Banerjee

Rout

Karmakar

, et al. A solution for free vibration of rotating hybrid multiscale nanocomposite conical shell. Noise Vib Worldw 2022; 53: 214–224.

22.

Aris

Ahmadi

. Using the higher-order shear deformation theory to analyze the free vibration of stiffened rotating FGM conical shells in a thermal environment. Thin-Walled Struct 2023; 183: 110366.

23.

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; 2450118. DOI: 10.1142/S0219455424501189.

24.

Yousefi

Memarzadeh

Afshari

, et al. Agglomeration effects on free vibration characteristics of three-phase CNT/polymer/fiber laminated truncated conical shells. Thin-Walled Struct 2020; 157: 107077.

25.

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.

26.

Asanjarani

Satouri

Alizadeh

, et al. Free vibration analysis of 2D-FGM truncated conical shell resting on Winkler–Pasternak foundations based on FSDT. Proc IME C J Mech Eng Sci 2015; 229: 818–839.

27.

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; 10775463231214023. DOI: 10.1177/10775463231214023.

28.

Afshari

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

29.

Maji

Rout

Karmakar

. Free vibration response of carbon nanotube reinforced pretwisted conical shell under thermal environment. Proc IME C J Mech Eng Sci 2020; 234: 770–783.

30.

Afshari

Ariaseresht

Rahimian Koloor

, et al. Supersonic flutter behavior of a polymeric truncated conical shell reinforced with agglomerated CNTs. Waves Random Complex Media. 2022; 1–25. DOI: 10.1080/17455030.2022.2082581.

31.

Eshelby

. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the royal society of London Series A Mathematical and physical sciences 1957; 241: 376–396.

32.

Shi

D-L

Feng

X-Q

Huang

, et al. The effect of nanotube waviness and agglomeration on the elastic property of carbon nanotube-reinforced composites. J Eng Mater Technol 2004; 126: 250–257.

33.

Yousefi

Memarzadeh

Afshari

, et al. Dynamic characteristics of truncated conical panels made of FRPs reinforced with agglomerated CNTs. In: Structures 2021. Amsterdam, Netherlands: Elsevier; 2021, pp. 4701–4717.

34.

Yousefi

Memarzadeh

Afshari

, et al. Optimization of CNT/polymer/fiber laminated truncated conical panels for maximum fundamental frequency and minimum cost. Mech Base Des Struct Mach 2023; 51: 3922–3944.

35.

Tornabene

Bacciocchi

Fantuzzi

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

36.

Hill

. Theory of mechanical properties of fibre-strengthened materials: I. Elastic behaviour. J Mech Phys Solid 1964; 12: 199–212.

37.

Parnian

D’Amore

. Fabrication of high-performance CNT reinforced polymer composite for additive manufacturing by phase inversion technique. Polymers 2021; 13: 4007.

38.

Gupta

Kumar

. Carbon nanotube-reinforced polymers. Amsterdam, Netherlands: Elsevier, 2018, pp. 61–81. Fabrication of carbon nanotube/polymer nanocomposites

39.

Soni

Thomas

Kar

. A comprehensive review on CNTs and CNT-reinforced composites: syntheses, characteristics and applications. Mater Today Commun 2020; 25: 101546.

40.

Bert

Malik

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