Active vibration control of rotating carbon nanotube reinforced composite cylindrical shells via piezoelectric patches

Abstract

This paper addresses the active vibration control of rotating carbon nanotube reinforced composite (CNTRC) cylindrical shells via piezoelectric actuator and sensor pairs. Considering circumferential initial stresses and Coriolis forces induced by rotation, an electromechanical coupling model of a simply supported CNTRC cylindrical shell, covered with surface-bonded piezoelectric sensors/actuators is established using the Lagrange equations and the model validation is carried out through a comparative analysis with existing literature. To suppress vibrations of rotating CNTRC cylindrical shells over a range of speeds, an LQR (Linear Quadratic Regulator) closed-loop controller is designed and its effectiveness is analyzed and evaluated through dynamic response analysis. Furthermore, the optimization of piezoelectric patch layout is performed by analyzing the performance of the controller for rotating CNTRC shells with typical piezoelectric sensors/actuators distributions. This paper presents and validates a strategy for vibration control of rotating CNTRC cylindrical shells using piezoelectric patches. The findings derived can offer guidance for vibration suppression of rotating thin-walled structures in practical engineering applications.

Keywords

Vibration control carbon nanotube reinforced composite rotating cylindrical shells piezoelectric patches

1. Introduction

In recent years, carbon nanotube-reinforced materials have garnered widespread attention owing to their high strength, exceptional toughness, and lightweight nature. As an advanced and high-quality material, carbon nanotube reinforced composites (CNTRC) find diverse applications, especially in aerospace. (Casati and Vedani, 2014; Liew et al., 2015). For example, in aero-engine rotor systems, numerous thin-walled shells are highly suitable for the use of carbon nanotube-reinforced composite materials. Rotating thin-walled shells exhibit intricate dynamic behaviors (Li et al., 2005; Sun et al., 2018). On the one hand, the centrifugal force generated by rotation induces circumferential stress in the structure, leading to an increase in structural stiffness. On the other hand, the Coriolis force arising from the non-collinearity of the structural deformation velocity vector and the rotational velocity vector results in the bifurcation of vibration frequencies. Consequently, investigating the dynamics and control of rotating CNTRC shells holds paramount importance.

The dynamics and control of CNTRC shells have garnered significant scholarly attention in recent years. Thomas and Roy (2016) studied the transient and steady state responses of CNTRC shell structures using finite element method and discussed the influences of volume fraction, distribution, and geometry of the shell on the dynamic characteristics of CNTRC shells. Free vibrations of conical and cylindrical shells and annular plates made of CNTRC materials were investigated by Civalek (2017) using discrete singular convolution method. Zghal et al. (2018) developed a high-order model of CNTRC shell structures and studied the free vibration behavior. Mellouli et al. (2020) also carried out the free vibration analysis of CNTRC shell structures, while a radial point interpolation method combined with the modified first-order shear deformation theory was proposed and utilized. Mallek et al. (2020) investigated the dynamic characteristics of functionally graded CNTRC plates and shell structures with surface-bonded piezoelectric layers and calculated the dynamic response of functionally graded CNTRC shells covered by two active layers. Forced vibration behaviors of CNTRC cylindrical shells were studied by Song et al. (2016b) employed Reddy’s high-order shear deformation theory for structural modeling, with considerations for thermal effects. The active vibration control of cylindrical shells has also been carried out in their following work reported (Song et al., 2016a). Given that shell-type structures are subjected to dynamic loads in applications of practical engineering, potentially resulting in large-amplitude vibrations, numerous studies (Li et al., 2021a, 2021b; Nguyen et al., 2019; Shen et al., 2020; Shen and Xiang, 2012; Wu et al., 2020) have been conducted on the nonlinear vibrations of CNTRC shells. The author of present study (Sun et al., 2022) also investigated nonlinear vibrations of CNTRC cylindrical shells resting on elastic foundations, considering nonlinear traveling wave modes.

The aforementioned research primarily focuses on stationary shells. With the increasing emphasis on the reliability of key components in rotating machinery, the study of the dynamics and control of high-speed rotating shells has become a recent hotspot. Several scholars (Chai et al., 2022, 2023; Sun et al., 2013; Sun and Liu, 2021a, 2021b; Zhao et al., 2023) conducted a series of investigations. Sun et al. (2013) proposed a general approach for the vibration studies of rotating cylindrical shells having arbitrary edges and extended the method to the vibration analysis of rotating combined thin-walled shells with multiple conical segments (Zhao et al., 2023). Nonlinear vibrations of rotating shells were also studied (Sun and Liu, 2021b), illustrating two kinds of multiple internal resonances (Sun and Liu, 2021a). In the study of Chai et al. (2022, 2023), the nonlinear free vibration of spinning cylindrical shells was investigated under arbitrary boundary conditions and a semi-analytical method for solving the frequency response of spinning cylindrical shells was developed. For rotating composite and sandwich shells, Qin et al. (2019) examined the free vibration characteristics using Chebyshev polynomials as admissible functions in the Rayleigh–Ritz method. Chai and Wang (2022) explored traveling wave vibrations of spinning graphene platelets reinforced metal foam joined conical-cylindrical shells. Afshari and Amirabadi (2022) conducted a comprehensive study on the free vibration analysis of rotating truncated conical shells reinforced with functionally graded agglomerated carbon nanotubes. Ghasemi and Meskini (2019) presented investigations on the free vibration of porous laminated rotating circular cylindrical shells with simply supported boundary conditions based on Love’s shell theory. An experimental study on the vibration of a rotating thin-walled cylinder was conducted by Fakkaew et al. (2019) and the dynamic behavior variations with rotational speed was investigated. Recently, several articles on vibration control of rotating shell have been reported. Kumar and Ray (2014) explored the active vibration control of thin rotating laminated composite truncated conical shells using an active constrained layer damping treatment. Rostami et al. (2021) investigated vibration control of the rotating sandwich cylindrical shell, considering functionally graded core and magneto-electro-elastic layers. Moghaddam and Ahmadi (2020) conducted nonlinear vibration analysis and investigated active vibration control of truncated conical shells under harmonic excitation using piezoelectric actuator. Brand and Cole (2021) employed surface-mounted actuators/sensors to suppress vibrations of rotating cylinders, proposing a mini-max optimization approach for effective vibration control over a specified speed range. Niasar et al. (2022, 2023) studied active control of free and forced vibration of a rotating functionally graded shell using piezoelectric patches and the optimization of piezoelectric patch positions was conducted to enhance vibration suppression performance.

The literature review highlights that most research pertaining to the dynamics and control of CNTRC shells primarily concentrates on stationary structures, leaving a notable gap in the literature regarding the dynamics and control of high-speed rotating shells made of CNTRC. To the authors’ knowledge, it may be the first time to investigate the active vibration control of rotating CNTRC cylindrical shells over a range of speeds. New contributions of this paper are outlined as follows: (1) An electromechanical coupling model for a CNTRC cylindrical shell, covered with surface-bonded piezoelectric sensors/actuators, has been established, considering circumferential initial stresses and Coriolis forces induced by rotation. (2) An LQR (Linear Quadratic Regulator) controller has been designed, and its parameters have been optimized for rotating CNTRC cylindrical shells over a range of speeds. (3) The optimization of piezoelectric patch layout has been carried out by analyzing the performance of the controller for rotating CNTRC shells with typical piezoelectric sensors/actuators distributions.

2. Model description

The core layer material considered in the cylindrical shell is the carbon nanotube-reinforced ceramic matrix composite, which is assumed to consist of single-walled carbon nanotubes and the isotropic ceramic matrix. As shown in Figure 1, h, R, and L represent the thickness, radius, and length of the host cylindrical shell, respectively. The shell rotates around the symmetric and horizontal axis at a constant angular speed Ω. On the inner and outer sides of the cylindrical shell core layer, pairs of piezoelectric sensors, and actuators are attached, respectively. The fundamental layout comprises eight sets of piezoelectric patches. Four sets of piezoelectric pairs, covering the arc of π/4, are evenly distributed in the circumferential direction of the cylindrical shell. Additionally, two sets are arranged axially, avoiding nodal positions, and covering the entire length L of the cylindrical shell.

For CNTRC materials, four distribution forms of reinforcing materials along the thickness direction are introduced, namely UD, FG-V, FG-O, and FG-X. The relationship between the volume fractions of materials in various distribution forms and the total volume fraction of CNTs $V_{C N T}^{*}$ is presented as follows (Qin et al., 2019)

\begin{array}{l} V_{C N T} (z) = V_{C N T}^{*}, (UD) \\ V_{C N T} (z) = (1 + 2 \frac{z}{h}) V_{C N T}^{*}, (FG - V) \\ V_{C N T} (z) = 2 (1 - 2 \frac{| z |}{h}) V_{C N T}^{*}, (FG - O) \\ V_{C N T} (z) = 2 (\frac{2 | z |}{h}) V_{C N T}^{*} . (FG - X) \end{array}

(1)

According to the mixing principle of CNT-reinforced materials and matrix materials, the effective Young’s modulus, shear modulus, Poisson’s ratio, and mass density along the thickness direction for CNT-reinforced composite materials can be expressed as

\begin{array}{l} E_{11} (z) = η_{1} V_{C N T} (z) E_{11}^{C N T} + V_{m} E^{m}, \\ \frac{η_{2}}{E_{22} (z)} = \frac{V_{C N T} (z)}{E_{22}^{C N T}} + \frac{V_{m} (z)}{E^{m}}, \\ \frac{η_{3}}{G_{12} (z)} = \frac{V_{C N T} (z)}{G_{12}^{C N T}} + \frac{V_{m} (z)}{G^{m}}, \\ μ_{12} (z) = V_{C N T} (z) μ_{12}^{C N T} + V_{m} (z) μ^{m}, \\ μ_{21} (z) = \frac{μ_{12} (z)}{E_{11} (z)} E_{22} (z), \\ ρ (z) = V_{C N T} (z) ρ^{C N T} + V_{m} (z) ρ^{m} . \end{array}

(2)

Here

E_{11}^{C N T}

and

E_{22}^{C N T}

are Young’s moduli for carbon nanotube, while E^m represents Young’s modulus for matrix.

G_{12}^{C N T}

μ_{12}^{C N T}

, and ρ^CNT are shear modulus, Poisson’s ratio, and mass density of CNTs, respectively. G^m, μ^m, and ρ^m are the corresponding properties of the matrix. η₁, η₂, and η₃ are the CNT efficiency parameters. The volume fraction of CNT material V_CNT(z) and the volume fraction of isotropic material V_m(z) satisfy the relationship V_CNT(z)+V_m(z)=1.

3. Governing equations of the rotating CNTRC cylindrical shells via piezoelectric actuator and sensor pairs

3.1. Energy of the core (CNTRC cylindrical shells)

According to the first-order shear deformation shell theory, the core’s displacement fields can be written as

{\begin{cases} u^{c} (x, θ, z_{c}, t) = u_{0}^{c} (x, θ, t) + z_{c} ϕ_{x} (x, θ, t) \\ v^{c} (x, θ, z_{c}, t) = v_{0}^{c} (x, θ, t) + z_{c} ϕ_{θ} (x, θ, t) \\ w^{c} (x, θ, z_{c}, t) = w_{0}^{c} (x, θ, t) \end{cases},

(3)

where

u_{0}^{c}

v_{0}^{c}

, and

w_{0}^{c}

are the displacements of the points on the middle surface in x, θ, and z directions, respectively. ϕ_x and ϕ_θ are rotations of the transverse normal about θ and x axes. t is the time.

The kinetic energy of the core is then expressed as

T_{c} = \frac{1}{2} \int_{V_{c}} ρ (z_{c}) {{({\dot{u}}^{c})}^{2} + {({\dot{v}}^{c})}^{2} + {({\dot{w}}^{c})}^{2} + 2 Ω (w^{c} {\dot{v}}^{c} - v^{c} {\dot{w}}^{c}) + Ω^{2} [{(v^{c})}^{2} + {(w^{c})}^{2}]} d V .

(4)

The strain-displacement relationships are given by

ε^{c} = {\begin{cases} ε_{x x}^{c} & ε_{θ θ}^{c} & γ_{x θ}^{c} & γ_{x z}^{c} & γ_{θ z}^{c} \end{cases}}^{T} = ε_{0}^{c} + z_{c} κ^{c},

(5)

where

\begin{array}{l} ε_{0}^{c} = {\begin{cases} \frac{\partial u_{0}^{c}}{\partial x} & \frac{\partial v_{0}^{c}}{R \partial θ} + \frac{w_{0}^{c}}{R} & \frac{\partial u_{0}^{c}}{R \partial θ} + \frac{\partial v_{0}^{c}}{\partial x} & ϕ_{x} + \frac{\partial w_{0}^{c}}{\partial x} & ϕ_{θ} + \frac{\partial w_{0}^{c}}{R \partial θ} \end{cases}}^{T}, \\ κ^{c} = {\begin{cases} \frac{\partial ϕ_{x}}{\partial x} & \frac{\partial ϕ_{θ}}{R \partial θ} & \frac{\partial ϕ_{x}}{R \partial θ} + \frac{\partial ϕ_{θ}}{\partial x} & 0 & 0 \end{cases}}^{T} . \end{array}

(6)

The stress–strain relationships are given by

σ^{c} = {\begin{cases} σ_{x x}^{c} & σ_{θ θ}^{c} & τ_{x θ}^{c} & τ_{x z}^{c} & τ_{θ z}^{c} \end{cases}}^{T} = Q^{c} ε^{c},

(7)

where

Q^{c} = [\begin{array}{l} Q_{11} & Q_{12} & 0 & 0 & 0 \\ Q_{12} & Q_{22} & 0 & 0 & 0 \\ 0 & 0 & Q_{66} & 0 & 0 \\ 0 & 0 & 0 & Q_{55} & 0 \\ 0 & 0 & 0 & 0 & Q_{44} \end{array}],

(8)

in which

Q_{11} = \frac{E_{11} (z)}{1 - μ_{12} (z) μ_{21} (z)}, Q_{22} = \frac{E_{22} (z)}{1 - μ_{12} (z) μ_{21} (z)}, Q_{12} = \frac{μ_{21} (z) E_{11} (z)}{1 - μ_{12} (z) μ_{21} (z)}, Q_{44} = G_{23} (z), Q_{55} = G_{13} (z), Q_{66} = G_{12} (z) .

(9)

Here, the shear moduli are assumed to be G₁₃(z)=G₂₃(z)=1.2G₁₂(z).

The strain energy of stretching and bending of the core can be expressed as

U_{s} = \frac{1}{2} \int_{V_{c}} [σ_{x x}^{c} ε_{x x}^{c} + σ_{θ θ}^{c} ε_{θ θ}^{c} + τ_{x θ}^{c} γ_{x θ}^{c} + K_{x} τ_{x z}^{c} γ_{x z}^{c} + K_{θ} τ_{θ z}^{c} γ_{θ z}^{c}] d V,

(10)

where K_x and K_θ are the shear correction factors. A typical value K_x=K_θ= 5/6 is used in the following calculation.

The additional strain energy of the core due to rotation can be expressed as

U_{r} = \frac{Ω^{2}}{2} \int_{V_{c}} ρ (z_{c}) [{(\frac{\partial u^{c}}{\partial θ})}^{2} + {(\frac{\partial v^{c}}{\partial θ} + w^{c})}^{2} + {(v^{c} - \frac{\partial w^{c}}{\partial θ})}^{2}] d V .

(11)

The total strain energy of the core is then given by

U_{c} = U_{s} + U_{r} .

(12)

3.2. Energy of piezoelectric sensor/actuator pairs

Based on Loves’ shell theory, the piezoelectric pairs’ displacement fields can be written as

{\begin{cases} u^{p} (x, θ, z_{p}, t) = u_{0}^{p} (x, θ, t) - z_{p} \frac{\partial w_{0}^{p}}{\partial x} \\ v^{p} (x, θ, z_{p}, t) = v_{0}^{p} (x, θ, t) + \frac{z_{p}}{R_{p}} (v_{0}^{p} - \frac{\partial w_{0}^{p}}{\partial θ}) \\ w^{p} (x, θ, z_{p}, t) = w_{0}^{p} (x, θ, t) \end{cases}, (p = a, s)

(13)

where a and s represent actuators and sensors, respectively.

Figure 2 depicts the deformation relationship between piezoelectric layers and the core. As illustrated in the figure, perfect bonding is assumed at the interface of the piezoelectric layers and core, which yields the compatibility equations as follows

{\begin{cases} {u^{c} (x, θ, z_{c}, t) |}_{z_{c} = h_{c} / 2} = {u^{a} (x, θ, z_{a}, t) |}_{z_{a} = - h_{a} / 2} \\ {v^{c} (x, θ, z_{c}, t) |}_{z_{c} = h_{c} / 2} = {v^{a} (x, θ, z_{a}, t) |}_{z_{a} = - h_{a} / 2} \\ {w^{c} (x, θ, z_{c}, t) |}_{z_{c} = h_{c} / 2} = {w^{a} (x, θ, z_{a}, t) |}_{z_{a} = - h_{a} / 2} \end{cases}, {\begin{cases} {u^{c} (x, θ, z_{c}, t) |}_{z_{c} = - h_{c} / 2} = {u^{s} (x, θ, z_{s}, t) |}_{z_{s} = h_{s} / 2} \\ {v^{c} (x, θ, z_{c}, t) |}_{z_{c} = - h_{c} / 2} = {v^{s} (x, θ, z_{s}, t) |}_{z_{s} = h_{s} / 2} \\ {w^{c} (x, θ, z_{c}, t) |}_{z_{c} = - h_{c} / 2} = {w^{s} (x, θ, z_{s}, t) |}_{z_{s} = h_{s} / 2} \end{cases} .

(14)

Substituting (3) and (13) into (14), yields

{\begin{cases} u_{0}^{a} (x, θ, t) = u_{0}^{c} (x, θ, t) + \frac{h_{c}}{2} ϕ_{x} (x, θ, t) - \frac{h_{a}}{2} \frac{\partial w_{0}^{c}}{\partial x} \\ v_{0}^{a} (x, θ, t) = v_{0}^{c} (x, θ, t) + \frac{h_{c}}{2} ϕ_{θ} (x, θ, t) - \frac{h_{a}}{2 R_{a}} \frac{\partial w_{0}^{c}}{\partial θ} \\ w_{0}^{a} (x, θ, t) = w_{0}^{c} (x, θ, t) \end{cases}, {\begin{cases} u_{0}^{s} (x, θ, t) = u_{0}^{c} (x, θ, t) - \frac{h_{c}}{2} ϕ_{x} (x, θ, t) + \frac{h_{s}}{2} \frac{\partial w_{0}^{c}}{\partial x} \\ v_{0}^{s} (x, θ, t) = v_{0}^{c} (x, θ, t) - \frac{h_{c}}{2} ϕ_{θ} (x, θ, t) + \frac{h_{s}}{2 R_{s}} \frac{\partial w_{0}^{c}}{\partial θ} \\ w_{0}^{s} (x, θ, t) = w_{0}^{c} (x, θ, t) \end{cases} .

(15)

The kinetic energy of the piezoelectric sensor/actuator pairs is then expressed as

T_{p} = \frac{1}{2} \sum_{i = 1}^{n p} \int_{V_{a i} + V_{s i}} ρ_{p} {{({\dot{u}}^{p})}^{2} + {({\dot{v}}^{p})}^{2} + {({\dot{w}}^{p})}^{2} + 2 Ω (w^{p} {\dot{v}}^{p} - v^{p} {\dot{w}}^{p}) + Ω^{2} [{(v^{p})}^{2} + {(w^{p})}^{2}]} d V .

(16)

The strain at an arbitrary point strain of the piezoelectric pairs can be expressed as

ε^{p} = ε_{0}^{p} + z_{p} κ^{p},

(17)

where

ε_{0}^{p} = {\begin{cases} ε_{x x}^{0} \\ ε_{θ θ}^{0} \\ γ_{x θ}^{0} \end{cases}}^{p} = {\begin{cases} \frac{\partial u_{0}^{p}}{\partial x} \\ \frac{\partial v_{0}^{p}}{R_{p} \partial θ} + \frac{w_{0}^{p}}{R_{p}} \\ \frac{\partial u_{0}^{p}}{R_{p} \partial θ} + \frac{\partial v_{0}^{p}}{\partial x} \end{cases}}, κ^{p} = {\begin{cases} κ_{x x} \\ κ_{θ θ} \\ κ_{x θ} \end{cases}}^{p} = {\begin{cases} - \frac{\partial^{2} w_{0}^{p}}{\partial x^{2}} \\ \frac{1}{{R_{p}}^{2}} (\frac{\partial v_{0}^{p}}{\partial θ} - \frac{\partial^{2} w_{0}^{p}}{\partial θ^{2}}) \\ \frac{2}{R_{p}} (\frac{\partial v_{0}^{p}}{\partial x} - \frac{\partial^{2} w_{0}^{p}}{\partial x \partial θ}) \end{cases}} .

(18)

The constitutive equations of the piezoelectric materials are given by (Song et al., 2016a)

\begin{array}{l} σ^{p} = {\begin{cases} σ_{x x} \\ σ_{θ θ} \\ τ_{x θ} \end{cases}}^{p} = Q^{p} ε^{p} - e^{T} E^{p} = [\begin{array}{l} C_{11} & C_{12} & 0 \\ C_{12} & C_{22} & 0 \\ 0 & 0 & C_{66} \end{array}] {\begin{cases} ε_{x x} \\ ε_{θ θ} \\ γ_{x θ} \end{cases}}^{p} - {[\begin{array}{l} 0 & 0 & 0 \\ 0 & 0 & 0 \\ e_{31} & e_{32} & 0 \end{array}]}^{T} {\begin{cases} 0 \\ 0 \\ V_{0} (t) / h_{p} \end{cases}}, \\ D^{p} = {\begin{cases} 0 \\ 0 \\ D_{z} \end{cases}} = e ε^{p} + Ξ E^{p} = [\begin{array}{l} 0 & 0 & 0 \\ 0 & 0 & 0 \\ e_{31} & e_{32} & 0 \end{array}] {\begin{cases} ε_{x x} \\ ε_{θ θ} \\ γ_{x θ} \end{cases}}^{p} + [\begin{array}{l} Ξ_{11} \\ Ξ_{22} \\ Ξ_{33} \end{array}] {\begin{cases} 0 \\ 0 \\ V_{0} (t) / h_{p} \end{cases}} . \end{array}

(19)

where σ^p is the stress vector, D^p is the electric displacement vector. e and Ξ, respectively, denote the piezoelectric and dielectric constant matrices of the piezoelectric material. Q^p is the stiffness coefficient matrix where C₁₁=C₂₂=E/(1−μ²),C₁₂=μE/(1−μ²),C₆₆=E/[2(1+μ)]. E and μ are the Young’s modulus and Poisson’s ratio of piezoelectric materials, respectively. E^p is the electric field intensity vector where V₀(t) is the external voltage.

The strain energy of the piezoelectric sensor/actuator pairs is then given by

U_{p} = \frac{1}{2} \sum_{i = 1}^{n p} \int_{V_{a i} + V_{s i}} {(ε^{p})}^{T} σ^{p} d V - \frac{1}{2} \sum_{i = 1}^{n p} \int_{V_{a i}} {(D^{p})}^{T} E^{p} d V .

(20)

3.3. Boundary conditions and displacement functions

The geometric boundary conditions of simply supported functionally graded CNTRC cylindrical shells with piezoelectric sensor/actuator pairs are

{v (x, θ, t) |}_{x = 0, L} = {w (x, θ, t) |}_{x = 0, L} = {ϕ_{θ} (x, θ, t) |}_{x = 0, L} = 0 .

(21)

The displacement functions that satisfy the geometric boundary conditions are then expressed as

{\begin{cases} u (x, θ, t) = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} [u_{c, m n} (t) \cos n θ + u_{s, m n} (t) \sin n θ] \cos (λ_{m} x) = U (x, θ) q^{u} (t) \\ v (x, θ, t) = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} [v_{c, m n} (t) \cos n θ + v_{s, m n} (t) \sin n θ] \sin (λ_{m} x) = V (x, θ) q^{v} (t) \\ w (x, θ, t) = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} [w_{c, m n} (t) \cos n θ + w_{s, m n} (t) \sin n θ] \sin (λ_{m} x) = W (x, θ) q^{w} (t) \\ ϕ_{x} (x, θ, t) = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} [φ_{c, m n}^{x} (t) \cos n θ + φ_{s, m n}^{x} (t) \sin n θ] \cos (λ_{m} x) = Φ_{x} (x, θ) q^{x} (t) \\ ϕ_{θ} (x, θ, t) = \sum_{m = 1}^{\infty} \sum_{n = 0}^{\infty} [φ_{c, m n}^{θ} (t) \cos n θ + φ_{s, m n}^{θ} (t) \sin n θ] \sin (λ_{m} x) = Φ_{θ} (x, θ) q^{θ} (t) \end{cases},

(22)

where λ_m=mπ/L, and m and n are half axial wave numbers and circumferential wave numbers in x and θ directions, respectively. In equations (21) and (22), the superscript “c” and subscript “0” of u, v, and w are omitted for convenience.

3.4. Electro-mechanical equations of motion

Provided that an external load is applied on the surface of the shell in radial direction, the virtual work done by the external load can be expressed as

W_{Q} = \int_{A} Q_{z} (t) w (x, θ, t) d A .

(23)

where Q_z(t)=f₀cos(ω_dt) denotes the external load. f₀ and ω_d are the amplitude and frequency of external load, respectively.

Substituting displacement functions (22) into equation (3), and considering the kinetic and strain energy of the core given by equations (4) and (12), the piezoelectric sensor/actuator pairs given by equations (16) and (20), and the virtual work done by the external load (23), the electro-mechanical equations of motion for the rotating CNTRC cylindrical shells with piezoelectric sensor/actuator pairs can be derived using the Lagrange equations

(Μ^{c} + Μ^{a} + Μ^{s}) \ddot{q} + (G + C) \dot{q} + (K_{0}^{c} + K_{Ω}^{c} + K^{a} + K^{s}) q + K^{e} V^{a} (t) = F_{d},

(24)

where

q = {q^{u} {(t)}^{T}, q^{v} {(t)}^{T}, q^{w} {(t)}^{T}, q^{x} {(t)}^{T}, q^{θ} {(t)}^{T}}^{T}

is the generalized coordinate vector, the external voltage vector can be expressed as

V^{a} (t) = {V_{01} (t), V_{02} (t), \dots, V_{0 n p} (t)}^{T}

, and

F_{d} = {\begin{cases} 0 & 0 & \int_{A} Q_{z} (t) W^{T} d A & 0 & 0 \end{cases}}^{T}

is the force vector. Μ^c, Μ^a, and Μ^s are the mass matrices associated with core, piezoelectric actuator and sensor. K^c (including

K_{0}^{c}

the stiffness matrix of stationary cylindrical core and

K_{Ω}^{c}

the additional stiffness matrix induced by rotation), K^a and K^s are the corresponding stiffness matrices. G is the gyroscopic matrix, C is the assumed damping matrix, and K^e is the electromechanical coupling matrix. The expressions of Μ^c, Μ^a, Μ^s,

K_{0}^{c}

K_{Ω}^{c}

, K^a, K^s, G, C, and K^e are given in Appendix A.

4. Design of the controller

The piezoelectric pair employed for active control consists of a set of sensor and actuator, where the sensor is utilized for vibration detection and charge generation. The charge generated by the ith sensor can be calculated by (Song et al., 2016a)

Q_{i}^{s} (t) = \int_{A_{i}^{s}} D_{z} d A = {\hat{Γ}}_{i}^{s} q,

(25)

where

A_{i}^{s}

is the surface area of the ith sensor and

{\hat{Γ}}_{i}^{s}

is the coefficient row vector. The expression of

{\hat{Γ}}_{i}^{s}

is given in Appendix A. The induced voltage can be expressed in terms of charges as follows

V_{i}^{s} (t) = \frac{h_{p}}{Ξ_{33} A_{i}^{s}} Q_{i}^{s} (t),

(26)

Substitute Eq. (25) to Eq. (26)

V_{i}^{s} (t) = \frac{h_{p}}{Ξ_{33} A_{i}^{s}} {\hat{Γ}}_{i}^{s} q,

(27)

where

Γ_{i}^{s} = \frac{h_{p}}{Ξ_{33} A_{i}^{s}} {\hat{Γ}}_{i}^{s}

, then the sensor voltage can be simplified as

V_{i}^{s} (t) = Γ_{i}^{s} q,

(28)

For the whole control system, the sensor voltage can be computed by

V^{s} (t) = {[\begin{array}{l} Γ_{1}^{s} & Γ_{2}^{s} & \dots & Γ_{i}^{s} & \dots & Γ_{n p}^{s} \end{array}]}^{T} q = Γ^{s} q .

(29)

To suppress the vibration of rotating CNTRC cylindrical shells, the LQR controller is designed. Firstly, the electro-mechanical model of motion is rewritten in the form of the state-space equation

\begin{array}{l} \dot{Z} = A Z + B^{a} U + B^{f} F_{d}, \\ Y = C^{s} Z, \end{array}

(30)

where

\begin{array}{l} Z = {\begin{cases} q^{T} & {\dot{q}}^{T} \end{cases}}^{T}, \\ A = [\begin{array}{l} 0 & I \\ - {(M^{c} + M^{a} + M^{s})}^{- 1} (K_{0}^{c} + K_{Ω}^{c} + K^{a} + K^{s}) & - {(M^{c} + M^{a} + M^{s})}^{- 1} (G + C) \end{array}], \\ B^{a} = [\begin{array}{l} 0 \\ - {(M^{c} + M^{a} + M^{s})}^{- 1} K^{e} \end{array}], \\ B^{f} = [\begin{array}{l} 0 \\ {(M^{c} + M^{a} + M^{s})}^{- 1} \end{array}], \\ C^{s} = [\begin{array}{l} Γ^{s} & 0 \end{array}] . \end{array}

(31)

Here U is the controller input and its voltage can be calculated by the following equation

U = - G_{l} Z,

(32)

where G_l is the control gain matrix. The performance indicator can be defined as

J = \int_{0}^{\infty} [Z^{T} Q Z + U^{T} R U] d t .

(33)

where Q is the state weight matrix and R is the control weight matrix. The optimal input U should minimize the performance indicator. Then the expression of G_l can be calculated from

G_{l} = R^{- 1} {(B^{a})}^{T} P,

(34)

in which the matrix P satisfies the Riccati equation

P A + A^{T} P - P Β^{a} R^{- 1} {(B^{a})}^{T} P + Q = 0 .

(35)

5. Results and discussions

5.1. Model validation

In this section, model validation is performed through comparisons with available literature data. As shown in Table 1, the present calculation of the traveling wave frequencies is compared with the findings of Qin et al.(2019). The results obtained in this article closely align with the literature data, confirming the accuracy of the theoretical formulation and numerical calculation presented in the paper. Any slight difference observed can be attributed to the utilization of different shell theories between this paper and the reference. Moreover, the data presented in Table 1 were manually picked points from figures in Qin’s article, which may introduce some deviations.

Table 1.

Comparison of non-dimensional frequency $ω_{m n} (R^{2} / h) \sqrt{ρ^{m} / E^{m}}$ for flexural vibration mode (m = 1, n = 2) of rotating CNTRC shells (h = 1, R/h = 10, L = 2R, $V_{C N T}^{*}$ = 0.17).

Rotating speed Ω(rps)	Distribution pattern	Backward wave frequency ω_b		Forward wave frequency ω_f
Rotating speed Ω(rps)	Distribution pattern	Qin et al. (2019)	Present	Qin et al. (2019)	Present
5	UD	8.66	8.68	5.40	5.44
	FGV	8.40	8.48	5.18	5.26
	FGA	8.17	8.27	4.90	5.04
	FGX	9.12	9.16	5.83	5.90
10	UD	11.63	11.64	4.92	4.97
	FGV	11.41	11.47	4.75	4.83
	FGA	11.23	11.30	4.53	4.64
	FGX	11.98	12.04	5.24	5.34
15	UD	15.11	15.12	4.69	4.74
	FGV	14.95	14.99	4.55	4.64
	FGA	14.79	14.85	4.37	4.49
	FGX	15.39	15.45	4.95	5.04

5.2. Dynamic response

The geometrical and material parameters of the CNTRC cylindrical shell and piezoelectric patches are listed in Table 2. The default distribution of piezoelectric patches depicted in Figure 1 is employed in the following discussions, except as otherwise noted.

Table 2.

Geometrical and material parameters of the CNTRC cylindrical shells and piezoelectric patches.

Parameter	Cylindrical shell			Piezoelectric patches
Parameter	Symbol	Value	Unit	Symbol	Value	Unit
Thickness	h	10	mm	h _p	0.2	mm
Radius	R	200	mm	-	-	-
Length	L	1200	mm	-	-	-
Mass density of the matrix	ρ ^m	3970	kg/m³	ρ ^p	7600	kg/m³
Young’s modulus of the matrix	E ^m	2.5×10⁹	N/m²	E	63×10⁹	kg/m³
Poisson’s ratio of the matrix	μ ^m	0.26		μ	0.3
Young’s modulus of CNTs	$E_{11}^{C N T}$	5.6466	TPa	-	-	-
Young’s modulus of CNTs	$E_{22}^{C N T}$	7.0800	TPa	-	-	-
Young’s modulus of CNTs	$E_{12}^{C N T}$	1.9445	TPa	-	-	-
Dielectric constant	-	-	-	Ξ₃₃	1.5×10^-8	F/m
Strain constants	-	-	-	d₃₁, d₃₂	245×10^-12	m/V

Figure 3 gives the Campbell diagram and mode shapes of rotating CNTRC cylindrical shells. As shown in the figure, the frequencies of vibration mode with specific (m, n) combination bifurcate to two frequencies due to the Coriolis acceleration induced by rotation. The lower one is associated with the forward wave mode, exhibiting a decreasing trend with increasing speeds. Conversely, the higher one represents the backward wave frequency, which increases with rotation speeds. This study mainly focuses on the speed range of 0 to 200 rps. Within this research scope, six sets of vibration modes can be observed and the corresponding mode shapes are displayed in the figure. By employing the mode superposition method, the traveling wave vibration responses at each point on the rotating CNTRC cylindrical shell can be derived.

As shown in Figure 4, convergence analysis of dynamic responses considering different number of vibration modes is carried out. Without loss of generality, the response calculation takes the position (x = L/5, θ = 0) on the rotating CNTRC cylindrical shell as an example, the rotational speed is 200 rps. In the convergence analysis and subsequent response calculations, FG-X with volume fractions $V_{C N T}^{*} = 0.14$ is the type of CNTs distributions, and the CNT efficiency parameters are η₁=0.150, η₂=η₃=0.941. The amplitude f₀ of harmonic excitation is taken as 8000N in the forced vibration calculation, unless otherwise stated. As shown in the figure, the resonance excitation frequency corresponding to the peak amplitude in Figure 4, aligns with the frequency value at 200rps of the Campbell diagram. Notably, it can be observed that when n = 1∼5, the two curves associated with m = 1∼3 and m = 1∼4 are agree with each other. Therefore, vibration modes with n = 1∼5 and m = 1∼3 are considered in the following calculation of dynamic responses.

Figures 5 and 6 display the active control efficiency of forced and free vibration, respectively. In Figure 5, the influence of weighting ratio Q/R on the forced vibration responses and maximum voltage of actuators are presented. It is worth noting that there are significant differences in the control efficiency and maximum control voltage for the listed values of Q/R. Vibration suppression notably improves as the weighting ratio of Q/R increases. However, the actuators require a higher control voltage to achieve an enhancement in control efficiency. Consequently, considering the maximum voltage and control effect comprehensively, the value of Q/R is suggested to be 10⁵. In Figure 6, the controlled and uncontrolled free vibration responses are illustrated using the same value of Q/R, demonstrating effective control of free vibration. Additionally, Figure 6 also displays the maximum voltage of actuators required is adequate. These findings validate the appropriateness of the selected value of Q/R, which can be used for subsequent calculations and discussions.

Figures 7 and 8 display the control efficiency of the dynamic responses for rotating CNTRC cylindrical shells with different patterns of CNT distributions and rotational speeds, respectively. As shown in Figure 7, three patterns of CNT distributions, namely, FG-O, UD, and FG-V are considered. The comparison results indicate that the vibration of cylindrical shell with all these distributions of CNTRC materials can be effectively control for both high and low-frequency vibrations. As shown in Figure 8, the control efficiency of the forced vibration responses over the speed range of 0 to 200 rps is depicted. The comparisons show that the vibration over the speed range of 0 to 200 rps can be effectively suppressed and there is no significant difference in the control efficiency at each speed under the selected value of Q/R mentioned earlier.

The effects of layout of piezoelectric patches on control efficiency are discussed. Figure 9 presents five typical layouts of piezoelectric patches bonding on the rotating CNTRC cylindrical shell. For the first three layouts, the axial direction adopts the basic distribution as shown in Figure 1 and four, eight, and sixteen piezoelectric patches are used in circumferential direction, respectively. For the last two layouts, the circumferential distribution utilizes the basic layout with four uniformly distributed piezoelectric patches. The number of axial piezoelectric pairs in Case 4 and Case 5 is taken as four and eight, respectively. It is worth noting that the optimization of piezoelectric layout studied in this article is conducted for the typical layouts proposed. Piezoelectric sensors and actuators for all these five cases have the same location and total area of piezoelectric patches which is a limitation of the optimization. To mitigate the limitation regarding size and position, further research will be conducted in subsequent studies.

Figures 10 and 11 discuss the effects of circumferential distribution of piezoelectric patches on the control efficiency. The forced vibration responses of rotating CNTRC cylindrical shells for the Case 1∼3 are shown in Figure 10. The time history of dynamic responses under excitation with fundamental frequency (ω_d= 555 Hz) is displayed in Figure 11. It can be observed from these figures that the vibration suppression in Case 1 is notably superior to other cases. Moreover, in cases where a greater number of piezoelectric pairs are arranged in the circumferential direction, the vibration suppression effect tends to weaken. It can be concluded that arranging fewer piezoelectric patches circumferentially leads to a significant control effect when the total area of the piezoelectric patches is constant.

Figures 12 and 13 discuss the effects of axial distribution of piezoelectric patches on the control efficiency. The forced vibration responses of rotating CNTRC cylindrical shells for the Case 1, Case 4, and Case 5 are presented in Figure 12. The time history of dynamic responses under excitation with fundamental frequency (ω_d= 555 Hz) is displayed in Figure 13. By comparing the cases with a different number of axially distributed piezoelectric patches, it is evident that the control is most effective when the number of piezoelectric patches distributed in the axial direction is lowest. In other words, as the number of piezoelectric patches arranged along the axis increases, the control effect decreases and the vibration suppression is not markedly significant.

6. Conclusions

This paper conducts active control of rotating CNTRC cylindrical shell using piezoelectric actuator/sensor pairs. An electromechanical coupling model for a CNTRC cylindrical shell covered with piezoelectric patches is established utilizing the Lagrange equations. To suppress vibrations of rotating CNTRC cylindrical shells over a range of speeds, an LQR (Linear Quadratic Regulator) closed-loop controller is designed and the effectiveness of the controller is analyzed and evaluated by the dynamic response. Furthermore, the optimization of piezoelectric patch layout is carried out by analyzing the performance of the controller for rotating CNTRC shells with typical piezoelectric sensors/actuators distributions.

The main findings are summarized as follows: (1) The electromechanical coupling model established for a CNTRC cylindrical shell, covered with surface-bonded piezoelectric sensors/actuators, proves to be effective and suitable for vibration analysis and active control investigation. (2) The designed LQR controller demonstrates high effectiveness in suppressing the free vibration and forced vibration response of rotating CNTRC cylindrical shells over a speed range. And the efficiency of this control hinges on the choice of the weighting ratio Q/R employed in the controller. As long as the Q/R ratio is well-optimized, there is no significant change in control efficiency for different patterns of CNT distributions and rotational speeds. (3) The arrangement of piezoelectric patches plays a crucial role in influencing the efficiency of active control. Specifically, when piezoelectric patches are distributed in the circumferential direction and the total area remains constant, a higher control efficiency is achieved with fewer patches. Similarly, in the axial arrangement, control effectiveness is more pronounced when a smaller number of piezoelectric patches are used.

Footnotes

Acknowledgements

The authors are grateful to the National Natural Science Foundation of China (Grant Nos. 12172307 and 11802129), Shandong Provincial Natural Science Foundation of China (Grant No. ZR2020QA039), the Opening Project of Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province (Grant No. SZDKF-202103) and Fundamental Research Funds for the Central Universities in Southwest Jiaotong University of China (No. 2682023ZTPY028) for financial support in this study.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors are grateful to the National Natural Science Foundation of China (Grant Nos. 12172307 and 11802129), Shandong Provincial Natural Science Foundation of China (Grant No. ZR2020QA039), the Opening Project of Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province (Grant No. SZDKF-202103) and Fundamental Research Funds for the Central Universities in Southwest Jiaotong University of China (No. 2682023ZTPY028) for financial support in this study.

ORCID iDs

Shupeng Sun

Dengqing Cao

Data Availability Statement

The data are available from the corresponding author on reasonable request*.

Appendix

Appendix A

The matrix M^c in Eq. (24) is

M^{c} = [\begin{array}{c} I_{0} R_{c} \int_{0}^{2 π} \int_{0}^{L} U^{T} U d x d θ & 0 & 0 & I_{1} R_{c} \int_{0}^{2 π} \int_{0}^{L} U^{T} Φ_{x} d x d θ & 0 \\ 0 & I_{0} R_{c} \int_{0}^{2 π} \int_{0}^{L} V^{T} V d x d θ & 0 & 0 & I_{1} R_{c} \int_{0}^{2 π} \int_{0}^{L} V^{T} Φ_{θ} d x d θ \\ 0 & 0 & I_{0} R_{c} \int_{0}^{2 π} \int_{0}^{L} W^{T} W d x d θ & 0 & 0 \\ I_{1} R_{c} \int_{0}^{2 π} \int_{0}^{L} {Φ_{x}}^{T} U d x d θ & 0 & 0 & I_{2} R_{c} \int_{0}^{2 π} \int_{0}^{L} {Φ_{x}}^{T} Φ_{x} d x d θ & 0 \\ 0 & I_{1} R_{c} \int_{0}^{2 π} \int_{0}^{L} Φ_{θ}^{T} V d x d θ & 0 & 0 & I_{2} R_{c} \int_{0}^{2 π} \int_{0}^{L} Φ_{θ}^{T} Φ_{θ} d x d θ \end{array}],

where

\int_{- h_{c} / 2}^{h_{c} / 2} ρ (z_{c}) d z_{c} = I_{0}, \int_{- h_{c} / 2}^{h_{c} / 2} ρ (z_{c}) z_{c} d z_{c} = I_{1}, \int_{- h_{c} / 2}^{h_{c} / 2} ρ (z_{c}) {z_{c}}^{2} d z_{c} = I_{2} .

The matrix M^a in Eq. (24) is

M^{a} = [\begin{array}{l} M_{u u}^{a} & 0 & M_{u w}^{a} & M_{u x}^{a} & 0 \\ 0 & M_{v v}^{a} & M_{v w}^{a} & 0 & M_{v θ}^{a} \\ {(M_{u w}^{a})}^{T} & {(M_{v w}^{a})}^{T} & M_{w w}^{a} & M_{w x}^{a} & M_{w θ}^{a} \\ {(M_{u x}^{a})}^{T} & 0 & {(M_{w x}^{a})}^{T} & M_{x x}^{a} & 0 \\ 0 & {(M_{v θ}^{a})}^{T} & {(M_{w θ}^{a})}^{T} & 0 & M_{θ θ}^{a} \end{array}],

where

M_{j k}^{a} = \frac{1}{2} \sum_{i = 1}^{n p} \int_{V_{a i}} ρ_{a} m_{j k}^{a} d V = \frac{1}{2} \sum_{i = 1}^{n p} \int_{{\bar{θ}}_{i}}^{{\tilde{θ}}_{i}} \int_{{\bar{x}}_{i}}^{{\tilde{x}}_{i}} \int_{- h_{a} / 2}^{h_{a} / 2} ρ_{a} m_{j k}^{a} R_{a} d z_{a} d x d θ (j, k = u, v, w, x, θ),

in which

\begin{array}{l} m_{u u}^{a} = 2 U^{T} U, m_{u w}^{a} = - (h_{a} + 2 z_{a}) U^{T} \frac{\partial W}{\partial x}, m_{u x}^{a} = h_{c} U^{T} Φ_{x}, m_{v v}^{a} = \frac{2 {(R_{a} + z_{a})}^{2}}{{R_{a}}^{2}} V^{T} V, m_{v w}^{a} = - \frac{[(h_{a} + 2 z_{a}) R_{a} + h_{a} z_{a}] (R_{a} + z_{a})}{{R_{a}}^{3}} V^{T} \frac{\partial W}{\partial θ}, \\ m_{v θ}^{a} = \frac{h_{c} {(R_{a} + z_{a})}^{2}}{{R_{a}}^{2}} V^{T} Φ_{θ}, m_{w w}^{a} = \frac{{[(h_{a} + 2 z_{a}) R_{a} + h_{a} z_{a}]}^{2}}{2 {R_{a}}^{4}} \frac{\partial W^{T}}{\partial θ} \frac{\partial W}{\partial θ} + \frac{{(h_{a} + 2 z_{a})}^{2}}{2} \frac{\partial W^{T}}{\partial x} \frac{\partial W}{\partial x} + 2 W^{T} W, m_{w x}^{a} = - \frac{(h_{a} + 2 z_{a}) h_{c}}{2} \frac{\partial W^{T}}{\partial x} Φ_{x}, \\ m_{w θ}^{a} = - \frac{[(h_{a} + 2 z_{a}) R_{a} + h_{a} z_{a}] (R_{a} + z_{a}) h_{c}}{2 {R_{a}}^{3}} \frac{\partial W^{T}}{\partial θ} Φ_{θ}, m_{x x}^{a} = \frac{{h_{c}}^{2}}{2} Φ_{x}^{T} Φ_{x}, m_{θ θ}^{a} = \frac{{(R_{a} + z_{a})}^{2} {h_{c}}^{2}}{2 {R_{a}}^{2}} Φ_{θ}^{T} Φ_{θ} . \end{array}

The matrix M^s in Eq. (24) is

M^{s} = [\begin{array}{l} M_{u u}^{s} & 0 & M_{u w}^{s} & M_{u x}^{s} & 0 \\ 0 & M_{v v}^{s} & M_{v w}^{s} & 0 & M_{v θ}^{s} \\ {(M_{u w}^{s})}^{T} & {(M_{v w}^{s})}^{T} & M_{w w}^{s} & M_{w x}^{s} & M_{w θ}^{s} \\ {(M_{u x}^{s})}^{T} & 0 & {(M_{w x}^{s})}^{T} & M_{x x}^{s} & 0 \\ 0 & {(M_{v θ}^{s})}^{T} & {(M_{w θ}^{s})}^{T} & 0 & M_{θ θ}^{s} \end{array}],

where

M_{j k}^{s} = \frac{1}{2} \sum_{i = 1}^{n p} \int_{V_{s i}} ρ_{s} m_{j k}^{s} d V = \frac{1}{2} \sum_{i = 1}^{n p} \int_{{\bar{θ}}_{i}}^{{\tilde{θ}}_{i}} \int_{{\bar{x}}_{i}}^{{\tilde{x}}_{i}} \int_{- h_{s} / 2}^{h_{s} / 2} ρ_{s} m_{j k}^{s} R_{s} d z_{s} d x d θ (j, k = u, v, w, x, θ),

in which

\begin{array}{l} m_{u u}^{s} = 2 U^{T} U, m_{u w}^{s} = (h_{s} - 2 z_{s}) U^{T} \frac{\partial W}{\partial x}, m_{u x}^{s} = - h_{c} U^{T} Φ_{x}, m_{v v}^{s} = \frac{2 {(R_{s} + z_{s})}^{2}}{{R_{s}}^{2}} V^{T} V, m_{v w}^{s} = - \frac{[(2 z_{s} - h_{s}) R_{s} - h_{s} z_{s}] (R_{s} + z_{s})}{{R_{s}}^{3}} V^{T} \frac{\partial W}{\partial θ}, \\ m_{v θ}^{s} = \frac{- h_{c} {(R_{s} + z_{s})}^{2}}{{R_{s}}^{2}} V^{T} Φ_{θ}, m_{w w}^{s} = \frac{{[(2 z_{s} - h_{s}) R_{s} - h_{s} z_{s}]}^{2}}{2 {R_{s}}^{4}} \frac{\partial W^{T}}{\partial θ} \frac{\partial W}{\partial θ} + \frac{{(- h_{s} + 2 z_{s})}^{2}}{2} \frac{\partial W^{T}}{\partial x} \frac{\partial W}{\partial x} + 2 W^{T} W, m_{w x}^{s} = \frac{(2 z_{s} - h_{s}) h_{s}}{2} \frac{\partial W^{T}}{\partial x} Φ_{x}, \\ m_{w θ}^{s} = \frac{[(2 z_{s} - h_{s}) R_{s} - h_{s} z_{s}] (R_{s} + z_{s}) h_{s}}{2 {R_{s}}^{3}} \frac{\partial W^{T}}{\partial θ} Φ_{θ}, m_{x x}^{s} = \frac{{h_{c}}^{2}}{2} Φ_{x}^{T} Φ_{x}, m_{θ θ}^{s} = \frac{{(R_{s} + z_{s})}^{2} {h_{c}}^{2}}{2 {R_{s}}^{2}} Φ_{θ}^{T} Φ_{θ} . \end{array}

The matrix $K_{0}^{c}$ in Eq. (24) is

K_{0}^{c} = [\begin{array}{l} K_{0 u u}^{c} & K_{0 u v}^{c} & K_{0 u w}^{c} & K_{0 u x}^{c} & K_{0 u θ}^{c} \\ {(K_{0 u v}^{c})}^{T} & K_{0 v v}^{c} & K_{0 v w}^{c} & K_{0 v x}^{c} & K_{0 v θ}^{c} \\ {(K_{0 u w}^{c})}^{T} & {(K_{0 v w}^{c})}^{T} & K_{0 w w}^{c} & K_{0 w x}^{c} & K_{0 w θ}^{c} \\ {(K_{0 u x}^{c})}^{T} & {(K_{0 v x}^{c})}^{T} & {(K_{0 w x}^{c})}^{T} & K_{0 x x}^{c} & K_{0 x θ}^{c} \\ {(K_{0 u θ}^{c})}^{T} & {(K_{0 v θ}^{c})}^{T} & {(K_{0 w θ}^{c})}^{T} & {(K_{0 x θ}^{c})}^{T} & K_{0 θ θ}^{c} \end{array}],

where

\begin{array}{l} K_{0 u u}^{c} = \frac{R}{2} \int_{0}^{2 π} \int_{0}^{L} (2 a_{1} \frac{\partial U^{T}}{\partial x} \frac{\partial U}{\partial x} + \frac{2 d_{1}}{R^{2}} \frac{\partial U^{T}}{\partial θ} \frac{\partial U}{\partial θ}) d x d θ, K_{0 u v}^{c} = \frac{R}{2} \int_{0}^{2 π} \int_{0}^{L} (\frac{2 b_{1}}{R} \frac{\partial U^{T}}{\partial x} \frac{\partial V}{\partial θ} + \frac{2 d_{1}}{R} \frac{\partial U^{T}}{\partial θ} \frac{\partial V}{\partial x}) d x d θ, \\ K_{0 u w}^{c} = \frac{R}{2} \int_{0}^{2 π} \int_{0}^{L} (\frac{2 b_{1}}{R} \frac{\partial U^{T}}{\partial x} W) d x d θ, K_{0 u x}^{c} = \frac{R}{2} \int_{0}^{2 π} \int_{0}^{L} (a_{2} \frac{\partial U^{T}}{\partial x} \frac{\partial Φ_{x}}{\partial x} + \frac{d_{2}}{R^{2}} \frac{\partial U^{T}}{\partial θ} \frac{\partial Φ_{x}}{\partial θ}) d x d θ, \\ K_{0 u θ}^{c} = \frac{R}{2} \int_{0}^{2 π} \int_{0}^{L} (\frac{b_{2}}{R} \frac{\partial U^{T}}{\partial x} \frac{\partial Φ_{θ}}{\partial θ} + \frac{d_{2}}{R} \frac{\partial U^{T}}{\partial θ} \frac{\partial Φ_{θ}}{\partial x}) d x d θ, K_{0 v v}^{c} = \frac{R}{2} \int_{0}^{2 π} \int_{0}^{L} (\frac{2 c_{1}}{R^{2}} \frac{\partial V^{T}}{\partial θ} \frac{\partial V}{\partial θ} + 2 d_{1} \frac{\partial V^{T}}{\partial x} \frac{\partial V}{\partial x}) d x d θ, \\ K_{0 v w}^{c} = \frac{R}{2} \int_{0}^{2 π} \int_{0}^{L} (\frac{2 c_{1}}{R^{2}} \frac{\partial V^{T}}{\partial θ} W) d x d θ, K_{0 v x}^{c} = \frac{R}{2} \int_{0}^{2 π} \int_{0}^{L} (\frac{b_{2}}{R} \frac{\partial V^{T}}{\partial θ} \frac{\partial Φ_{x}}{\partial x} + \frac{d_{2}}{R} \frac{\partial V^{T}}{\partial x} \frac{\partial Φ_{x}}{\partial θ}) d x d θ, \\ K_{0 v θ}^{c} = \frac{R}{2} \int_{0}^{2 π} \int_{0}^{L} (\frac{c_{2}}{R^{2}} \frac{\partial V^{T}}{\partial θ} \frac{\partial Φ_{θ}}{\partial θ} + d_{2} \frac{\partial V^{T}}{\partial x} \frac{\partial Φ_{θ}}{\partial x}) d x d θ, K_{0 w x}^{c} = \frac{R}{2} \int_{0}^{2 π} \int_{0}^{L} (\frac{b_{2}}{R} W^{T} \frac{\partial Φ_{x}}{\partial x} + 2 e_{1} \frac{\partial W^{T}}{\partial x} Φ_{x}) d x d θ, \\ K_{0 w θ}^{c} = \frac{R}{2} \int_{0}^{2 π} \int_{0}^{L} (\frac{c_{2}}{R^{2}} W^{T} \frac{\partial Φ_{θ}}{\partial θ} + \frac{2 f_{1}}{R} \frac{\partial W^{T}}{\partial θ} Φ_{θ}) d x d θ, K_{0 x x}^{c} = \frac{R}{2} \int_{0}^{2 π} \int_{0}^{L} (2 a_{3} \frac{\partial Φ_{x}^{T}}{\partial x} \frac{\partial Φ_{x}}{\partial x} + \frac{2 d_{3}}{R^{2}} \frac{\partial Φ_{x}^{T}}{\partial θ} \frac{\partial Φ_{x}}{\partial θ} + 2 e_{1} Φ_{x}^{T} Φ_{x}) d x d θ, \\ K_{0 x θ}^{c} = \frac{R}{2} \int_{0}^{2 π} \int_{0}^{L} (\frac{2 b_{3}}{R} \frac{\partial Φ_{x}^{T}}{\partial x} \frac{\partial Φ_{θ}}{\partial θ} + \frac{2 d_{3}}{R} \frac{\partial Φ_{x}^{T}}{\partial θ} \frac{\partial Φ_{θ}}{\partial x}) d x d θ, K_{0 θ θ}^{c} = \frac{R}{2} \int_{0}^{2 π} \int_{0}^{L} (2 d_{3} \frac{\partial Φ_{θ}^{T}}{\partial x} \frac{\partial Φ_{θ}}{\partial x} + 2 f_{1} Φ_{θ}^{T} Φ_{θ} + \frac{2 c_{3}}{R^{2}} \frac{\partial Φ_{θ}^{T}}{\partial θ} \frac{\partial Φ_{θ}}{\partial θ}) d x d θ, \\ K_{0 w w}^{c} = \frac{R}{2} \int_{0}^{2 π} \int_{0}^{L} (2 c_{1} \frac{W^{T} W}{R^{2}} + 2 e_{1} \frac{\partial W^{T}}{\partial x} \frac{\partial W}{\partial x} + \frac{2 f_{1}}{R^{2}} \frac{\partial W^{T}}{\partial θ} \frac{\partial W}{\partial θ}) d x d θ, \end{array}

in which

\begin{array}{l} a_{1} = \int_{- h_{c} / 2}^{h_{c} / 2} Q_{11} d z_{c}, a_{2} = 2 \int_{- h_{c} / 2}^{h_{c} / 2} Q_{11} z_{c} d z_{c}, a_{3} = \int_{- h_{c} / 2}^{h_{c} / 2} Q_{11} {z_{c}}^{2} d z_{c}, b_{1} = \int_{- h_{c} / 2}^{h_{c} / 2} Q_{12} d z_{c}, b_{2} = 2 \int_{- h_{c} / 2}^{h_{c} / 2} Q_{12} z_{c} d z_{c}, b_{3} = \int_{- h_{c} / 2}^{h_{c} / 2} Q_{12} {z_{c}}^{2} d z_{c}, \\ c_{1} = \int_{- h_{c} / 2}^{h_{c} / 2} Q_{22} d z_{c}, c_{2} = 2 \int_{- h_{c} / 2}^{h_{c} / 2} Q_{22} z_{c} d z_{c}, c_{3} = \int_{- h_{c} / 2}^{h_{c} / 2} Q_{22} {z_{c}}^{2} d z_{c}, d_{1} = \int_{- h_{c} / 2}^{h_{c} / 2} G_{12} d z_{c}, d_{2} = 2 \int_{- h_{c} / 2}^{h_{c} / 2} G_{12} z_{c} d z_{c}, d_{3} = \int_{- h_{c} / 2}^{h_{c} / 2} G_{12} {z_{c}}^{2} d z_{c}, \\ e_{1} = K_{x} \int_{- h_{c} / 2}^{h_{c} / 2} G_{13} d z_{c}, e_{2} = 2 K_{x} \int_{- h_{c} / 2}^{h_{c} / 2} G_{13} z_{c} d z_{c}, e_{3} = K_{x} \int_{- h_{c} / 2}^{h_{c} / 2} G_{13} {z_{c}}^{2} d z_{c}, f_{1} = K_{θ} \int_{- h_{c} / 2}^{h_{c} / 2} G_{23} d z_{c}, f_{2} = 2 K_{θ} \int_{- h_{c} / 2}^{h_{c} / 2} G_{23} z_{c} d z_{c}, \\ f_{3} = K_{θ} \int_{- h_{c} / 2}^{h_{c} / 2} G_{23} {z_{c}}^{2} d z_{c} . \end{array}

The matrix $K_{Ω}^{c}$ in Eq. (24) is

K_{Ω}^{c} = [\begin{array}{l} K_{Ω u u}^{c} & K_{Ω u v}^{c} & K_{Ω u w}^{c} & K_{Ω u x}^{c} & K_{Ω u θ}^{c} \\ {(K_{Ω u v}^{c})}^{T} & K_{Ω v v}^{c} & K_{Ω v w}^{c} & K_{Ω v x}^{c} & K_{Ω v θ}^{c} \\ {(K_{Ω u w}^{c})}^{T} & {(K_{Ω v w}^{c})}^{T} & K_{Ω w w}^{c} & K_{Ω w x}^{c} & K_{Ω w θ}^{c} \\ {(K_{Ω u x}^{c})}^{T} & {(K_{Ω v x}^{c})}^{T} & {(K_{Ω w x}^{c})}^{T} & K_{Ω x x}^{c} & K_{Ω x θ}^{c} \\ {(K_{Ω u θ}^{c})}^{T} & {(K_{Ω v θ}^{c})}^{T} & {(K_{Ω w θ}^{c})}^{T} & {(K_{Ω x θ}^{c})}^{T} & K_{Ω θ θ}^{c} \end{array}],

where

\begin{array}{l} K_{Ω u u}^{c} = Ω^{2} R_{c} I_{0} \int_{0}^{2 π} \int_{0}^{L} (\frac{\partial U^{T}}{\partial θ} \frac{\partial U}{\partial θ}) d x d θ, K_{Ω u x}^{c} = Ω^{2} R_{c} I_{1} \int_{0}^{2 π} \int_{0}^{L} (\frac{\partial U^{T}}{\partial θ} \frac{\partial Φ_{x}}{\partial θ}) d x d θ, K_{Ω v v}^{c} = Ω^{2} R_{c} I_{0} \int_{0}^{2 π} \int_{0}^{L} (\frac{\partial V^{T}}{\partial θ} \frac{\partial V}{\partial θ}) d x d θ, \\ K_{Ω v w}^{c} = Ω^{2} R_{c} I_{0} \int_{0}^{2 π} \int_{0}^{L} (\frac{\partial V^{T}}{\partial θ} W - V^{T} \frac{\partial W}{\partial θ}) d x d θ, K_{Ω v θ}^{c} = Ω^{2} R_{c} I_{1} \int_{0}^{2 π} \int_{0}^{L} (\frac{\partial V^{T}}{\partial θ} \frac{\partial Φ_{θ}}{\partial θ}) d x d θ, K_{Ω w w}^{c} = Ω^{2} R_{c} I_{0} \int_{0}^{2 π} \int_{0}^{L} (\frac{\partial W^{T}}{\partial θ} \frac{\partial W}{\partial θ}) d x d θ, \\ K_{Ω w θ}^{c} = Ω^{2} R_{c} I_{1} \int_{0}^{2 π} \int_{0}^{L} (W^{T} \frac{\partial Φ_{θ}}{\partial θ} - \frac{\partial W^{T}}{\partial θ} Φ_{θ}) d x d θ, K_{Ω x x}^{c} = Ω^{2} R_{c} I_{2} \int_{0}^{2 π} \int_{0}^{L} (\frac{\partial Φ_{x}^{T}}{\partial θ} \frac{\partial Φ_{x}}{\partial θ}) d x d θ, K_{Ω θ θ}^{c} = Ω^{2} R_{c} I_{2} \int_{0}^{2 π} \int_{0}^{L} (\frac{\partial Φ_{θ}^{T}}{\partial θ} \frac{\partial Φ_{θ}}{\partial θ}) d x d θ, \\ K_{Ω u v}^{c} = K_{Ω u w}^{c} = K_{Ω u θ}^{c} = K_{Ω v x}^{c} = K_{Ω w x}^{c} = K_{Ω x θ}^{c} = 0 . \end{array}

The matrix K^a in Eq. (24) is

K^{a} = [\begin{array}{l} K_{u u}^{a} & K_{u v}^{a} & K_{u w}^{a} & K_{u x}^{a} & K_{u θ}^{a} \\ {(K_{u v}^{a})}^{T} & K_{v v}^{a} & K_{v w}^{a} & K_{v x}^{a} & K_{v θ}^{a} \\ {(K_{u w}^{a})}^{T} & {(K_{v w}^{a})}^{T} & K_{w w}^{a} & K_{w x}^{a} & K_{w θ}^{a} \\ {(K_{u x}^{a})}^{T} & {(K_{v x}^{a})}^{T} & {(K_{w x}^{a})}^{T} & K_{x x}^{a} & K_{x θ}^{a} \\ {(K_{u θ}^{a})}^{T} & {(K_{v θ}^{a})}^{T} & {(K_{w θ}^{a})}^{T} & {(K_{x θ}^{a})}^{T} & K_{θ θ}^{a} \end{array}],

where

K_{j k}^{a} = \frac{1}{2} \sum_{i = 1}^{n p} \int_{V_{a i}} k_{j k}^{a} d V = \frac{1}{2} \sum_{i = 1}^{n p} \int_{{\bar{θ}}_{i}}^{{\tilde{θ}}_{i}} \int_{{\bar{x}}_{i}}^{{\tilde{x}}_{i}} \int_{- h_{a} / 2}^{h_{a} / 2} k_{j k}^{a} R_{a} d z_{a} d x d θ (j, k = u, v, w, x, θ),

in which

\begin{array}{l} k_{u u}^{a} = 2 C_{11} \frac{\partial U^{T}}{\partial x} \frac{\partial U}{\partial x} + \frac{2 C_{66}}{{R_{a}}^{2}} \frac{\partial U^{T}}{\partial θ} \frac{\partial U}{\partial θ}, \\ k_{u v}^{a} = \frac{2}{{R_{a}}^{2}} [C_{66} (R_{a} + 2 z_{a}) \frac{\partial U^{T}}{\partial θ} \frac{\partial V}{\partial x} + C_{12} (R_{a} + z_{a}) \frac{\partial U^{T}}{\partial x} \frac{\partial V}{\partial θ}], \\ k_{u w}^{a} = - C_{11} (2 z_{a} + h_{a}) \frac{\partial U^{T}}{\partial x} \frac{\partial^{2} W}{\partial x^{2}} - \frac{C_{12} (R_{a} (2 z_{a} + h_{a}) + z_{a} h_{a})}{{R_{a}}^{3}} \frac{\partial U^{T}}{\partial x} \frac{\partial^{2} W}{\partial θ^{2}} + \frac{2 C_{12}}{R_{a}} \frac{\partial U^{T}}{\partial x} W - \frac{2 C_{66} (R_{a} (2 z_{a} + h_{a}) + z_{a} h_{a})}{{R_{a}}^{3}} \frac{\partial U^{T}}{\partial θ} \frac{\partial^{2} W}{\partial x \partial θ}, \\ k_{u x}^{a} = C_{11} h_{c} \frac{\partial U^{T}}{\partial x} \frac{\partial Φ_{x}}{\partial x} + \frac{C_{66} h_{c}}{{R_{a}}^{2}} \frac{\partial U^{T}}{\partial θ} \frac{\partial Φ_{x}}{\partial θ}, \\ k_{u θ}^{a} = \frac{C_{12} h_{c} (z_{a} + R_{a})}{{R_{a}}^{2}} \frac{\partial U^{T}}{\partial x} \frac{\partial Φ_{θ}}{\partial θ} + \frac{C_{66} h_{c} (2 z_{a} + R_{a})}{{R_{a}}^{2}} \frac{\partial U^{T}}{\partial θ} \frac{\partial Φ_{θ}}{\partial x}, \\ k_{v v}^{a} = \frac{2 C_{22} {(z_{a} + R_{a})}^{2}}{{R_{a}}^{4}} \frac{\partial V^{T}}{\partial θ} \frac{\partial V}{\partial θ} + \frac{2 C_{66} {(2 z_{a} + R_{a})}^{2}}{{R_{a}}^{2}} \frac{\partial V^{T}}{\partial x} \frac{\partial V}{\partial x}, \\ k_{v w}^{a} = - \frac{C_{22} (z_{a} + R_{a}) [R_{a} (2 z_{a} + h_{a}) + z_{a} h_{a}]}{{R_{a}}^{5}} \frac{\partial V^{T}}{\partial θ} \frac{\partial^{2} W}{\partial θ^{2}} - \frac{2 C_{66} [R_{a} (2 z_{a} + h_{a}) + z_{a} h_{a}] (2 z_{a} + R_{a})}{{R_{a}}^{3}} \frac{\partial V^{T}}{\partial x} \frac{\partial^{2} W}{\partial x \partial θ} - \frac{C_{12} (z_{a} + R_{a}) (2 z_{a} + h_{a})}{{R_{a}}^{2}} \frac{\partial V^{T}}{\partial θ} \frac{\partial^{2} W}{\partial x^{2}} \\ + \frac{2 C_{22} (z_{a} + R_{a})}{{R_{a}}^{3}} \frac{\partial V^{T}}{\partial θ} W, \\ k_{v x}^{a} = \frac{C_{66} h_{c} (2 z_{a} + R_{a})}{{R_{a}}^{2}} \frac{\partial V^{T}}{\partial x} \frac{\partial Φ_{x}}{\partial θ} + \frac{C_{12} h_{c} (z_{a} + R_{a})}{{R_{a}}^{2}} \frac{\partial V^{T}}{\partial θ} \frac{\partial Φ_{x}}{\partial x}, \end{array}

\begin{array}{l} k_{v θ}^{a} = \frac{C_{22} h_{c} {(z_{a} + R_{a})}^{2}}{{R_{a}}^{4}} \frac{\partial V^{T}}{\partial θ} \frac{\partial Φ_{θ}}{\partial θ} + \frac{C_{66} h_{c} {(2 z_{a} + R_{a})}^{2}}{{R_{a}}^{2}} \frac{\partial V^{T}}{\partial x} \frac{\partial Φ_{θ}}{\partial x}, \\ k_{w w}^{a} = \frac{C_{11} {(2 z_{a} + h_{a})}^{2}}{2} \frac{\partial^{2} W^{T}}{\partial x^{2}} \frac{\partial^{2} W}{\partial x^{2}} + \frac{{(R_{a} (2 z_{a} + h_{a}) + z_{a} h_{a})}^{2} C_{22}}{2 {R_{a}}^{6}} \frac{\partial^{2} W^{T}}{\partial θ^{2}} \frac{\partial^{2} W}{\partial θ^{2}} + \frac{2 {(R_{a} (2 z_{a} + h_{a}) + z_{a} h_{a})}^{2} C_{66}}{{R_{a}}^{4}} \frac{\partial^{2} W^{T}}{\partial x \partial θ} \frac{\partial^{2} W}{\partial x \partial θ} + \frac{2 C_{22}}{{R_{a}}^{2}} W^{T} W \\ + \frac{C_{12} [R_{a} (2 z_{a} + h_{a}) + z_{a} h_{a}] (2 z_{a} + h_{a})}{2 {R_{a}}^{3}} (\frac{\partial^{2} W^{T}}{\partial x^{2}} \frac{\partial^{2} W}{\partial θ^{2}} + \frac{\partial^{2} W^{T}}{\partial θ^{2}} \frac{\partial^{2} W}{\partial x^{2}}) - \frac{C_{12} (2 z_{a} + h_{a})}{R_{a}} (\frac{\partial^{2} W^{T}}{\partial x^{2}} W + W^{T} \frac{\partial^{2} W}{\partial x^{2}}) \\ - \frac{C_{22} (R_{a} (2 z_{a} + h_{a}) + z_{a} h_{a})}{{R_{a}}^{4}} (\frac{\partial^{2} W^{T}}{\partial θ^{2}} W + W^{T} \frac{\partial^{2} W}{\partial θ^{2}}), \\ k_{w x}^{a} = - \frac{C_{12} h_{c} (R_{a} (2 z_{a} + h_{a}) + z_{a} h_{a})}{2 {R_{a}}^{3}} \frac{\partial^{2} W^{T}}{\partial θ^{2}} \frac{\partial Φ_{x}}{\partial x} - \frac{C_{66} h_{c} (R_{a} (2 z_{a} + h_{a}) + z_{a} h_{a})}{{R_{a}}^{3}} \frac{\partial^{2} W^{T}}{\partial x \partial θ} \frac{\partial Φ_{x}}{\partial θ} - \frac{C_{11} h_{c} R_{a} (2 z_{a} + h_{a})}{2 R_{a}} \frac{\partial^{2} W^{T}}{\partial x^{2}} \frac{\partial Φ_{x}}{\partial x} + \frac{C_{12} h_{c}}{R_{a}} W^{T} \frac{\partial Φ_{x}}{\partial x}, \\ k_{w θ}^{a} = - \frac{C_{22} h_{c} (z_{a} + R_{a}) (R_{a} (2 z_{a} + h_{a}) + z_{a} h_{a})}{2 {R_{a}}^{5}} \frac{\partial^{2} W^{T}}{\partial θ^{2}} \frac{\partial Φ_{θ}}{\partial θ} - \frac{C_{66} h_{c} (R_{a} (2 z_{a} + h_{a}) + z_{a} h_{a}) (2 z_{a} + R_{a})}{{R_{a}}^{3}} \frac{\partial^{2} W^{T}}{\partial x \partial θ} \frac{\partial Φ_{θ}}{\partial x} \\ - \frac{C_{12} h_{c} (2 z_{a} + h_{a}) (z_{a} + R_{a})}{2 {R_{a}}^{2}} \frac{\partial^{2} W^{T}}{\partial x^{2}} \frac{\partial Φ_{θ}}{\partial θ} + \frac{C_{22} h_{c} (z_{a} + R_{a})}{{R_{a}}^{3}} W^{T} \frac{\partial Φ_{θ}}{\partial θ}, \\ k_{x x}^{a} = \frac{C_{11} {h_{c}}^{2}}{2} \frac{\partial Φ_{x}^{T}}{\partial x} \frac{\partial Φ_{x}}{\partial x} + \frac{C_{66} {h_{c}}^{2}}{2 {R_{a}}^{2}} \frac{\partial Φ_{x}^{T}}{\partial θ} \frac{\partial Φ_{x}}{\partial θ}, \\ k_{x θ}^{a} = \frac{C_{12} {h_{c}}^{2} (z_{a} + R_{a})}{2 {R_{a}}^{2}} \frac{\partial Φ_{x}^{T}}{\partial x} \frac{\partial Φ_{θ}}{\partial θ} + \frac{C_{66} {h_{c}}^{2} (2 z_{a} + R_{a})}{2 {R_{a}}^{2}} \frac{\partial Φ_{x}^{T}}{\partial θ} \frac{\partial Φ_{θ}}{\partial x}, \\ k_{θ θ}^{a} = \frac{C_{22} {h_{c}}^{2} {(z_{a} + R_{a})}^{2}}{2 {R_{a}}^{4}} \frac{\partial Φ_{θ}^{T}}{\partial θ} \frac{\partial Φ_{θ}}{\partial θ} + \frac{C_{66} {h_{c}}^{2} {(2 z_{a} + R_{a})}^{2}}{2 {R_{a}}^{2}} \frac{\partial Φ_{θ}^{T}}{\partial x} \frac{\partial Φ_{θ}}{\partial x} . \end{array}

The matrix K^s in Eq. (24) is

K^{s} = [\begin{array}{l} K_{u u}^{s} & K_{u v}^{s} & K_{u w}^{s} & K_{u x}^{s} & K_{u θ}^{s} \\ {(K_{u v}^{s})}^{T} & K_{v v}^{s} & K_{v w}^{s} & K_{v x}^{s} & K_{v θ}^{s} \\ {(K_{u w}^{s})}^{T} & {(K_{v w}^{s})}^{T} & K_{w w}^{s} & K_{w x}^{s} & K_{w θ}^{s} \\ {(K_{u x}^{s})}^{T} & {(K_{v x}^{s})}^{T} & {(K_{w x}^{s})}^{T} & K_{x x}^{s} & K_{x θ}^{s} \\ {(K_{u θ}^{s})}^{T} & {(K_{v θ}^{s})}^{T} & {(K_{w θ}^{s})}^{T} & {(K_{x θ}^{s})}^{T} & K_{θ θ}^{s} \end{array}],

where

K_{j k}^{s} = \frac{1}{2} \sum_{i = 1}^{n p} \int_{V_{s i}} k_{j k}^{s} d V = \frac{1}{2} \sum_{i = 1}^{n p} \int_{{\bar{θ}}_{i}}^{{\tilde{θ}}_{i}} \int_{{\bar{x}}_{i}}^{{\tilde{x}}_{i}} \int_{- h_{s} / 2}^{h_{s} / 2} k_{j k}^{s} R_{s} d z_{s} d x d θ (j, k = u, v, w, x, θ),

in which

\begin{array}{l} k_{u u}^{s} = 2 C_{11} \frac{\partial U^{T}}{\partial x} \frac{\partial U}{\partial x} + \frac{2 C_{66}}{{R_{s}}^{2}} \frac{\partial U^{T}}{\partial θ} \frac{\partial U}{\partial θ}, \\ k_{u v}^{s} = \frac{2}{{R_{s}}^{2}} [C_{66} (R_{s} + 2 z_{s}) \frac{\partial U^{T}}{\partial θ} \frac{\partial V}{\partial x} + C_{12} (R_{s} + z_{s}) \frac{\partial U^{T}}{\partial x} \frac{\partial V}{\partial θ}], \\ k_{u w}^{s} = - C_{11} (2 z_{s} - h_{s}) \frac{\partial U^{T}}{\partial x} \frac{\partial^{2} W}{\partial x^{2}} - \frac{C_{12} (R_{s} (2 z_{s} - h_{s}) - h_{s} z_{s})}{{R_{s}}^{3}} \frac{\partial U^{T}}{\partial x} \frac{\partial^{2} W}{\partial θ^{2}} + \frac{2 C_{12}}{R_{s}} \frac{\partial U^{T}}{\partial x} W - \frac{2 C_{66} (R_{s} (2 z_{s} - h_{s}) - h_{s} z_{s})}{{R_{s}}^{3}} \frac{\partial U^{T}}{\partial θ} \frac{\partial^{2} W}{\partial x \partial θ}, \\ k_{u x}^{s} = - C_{11} h_{c} \frac{\partial U^{T}}{\partial x} \frac{\partial Φ_{x}}{\partial x} - \frac{C_{66} h_{c}}{{R_{s}}^{2}} \frac{\partial U^{T}}{\partial θ} \frac{\partial Φ_{x}}{\partial θ}, \\ k_{u θ}^{s} = - \frac{C_{12} h_{c} (z_{s} + R_{s})}{{R_{s}}^{2}} \frac{\partial U^{T}}{\partial x} \frac{\partial Φ_{θ}}{\partial θ} - \frac{C_{66} h_{c} (2 z_{s} + R_{s})}{{R_{s}}^{2}} \frac{\partial U^{T}}{\partial θ} \frac{\partial Φ_{θ}}{\partial x} . \end{array}

\begin{array}{l} k_{v v}^{s} = \frac{2 C_{22} {(z_{s} + R_{s})}^{2}}{{R_{s}}^{4}} \frac{\partial V^{T}}{\partial θ} \frac{\partial V}{\partial θ} + \frac{2 C_{66} {(2 z_{s} + R_{s})}^{2}}{{R_{s}}^{2}} \frac{\partial V^{T}}{\partial x} \frac{\partial V}{\partial x}, \\ k_{v w}^{s} = - \frac{C_{22} (z_{s} + R_{s}) [R_{s} (2 z_{s} - h_{s}) - h_{s} z_{s}]}{{R_{s}}^{5}} \frac{\partial V^{T}}{\partial θ} \frac{\partial^{2} W}{\partial θ^{2}} - \frac{2 C_{66} [R_{s} (2 z_{s} - h_{s}) - h_{s} z_{s}] (2 z_{s} + R_{s})}{{R_{s}}^{3}} \frac{\partial V^{T}}{\partial x} \frac{\partial^{2} W}{\partial x \partial θ} - \frac{C_{12} (z_{s} + R_{s}) (2 z_{s} - h_{s})}{{R_{s}}^{2}} \frac{\partial V^{T}}{\partial θ} \frac{\partial^{2} W}{\partial x^{2}} \\ + \frac{2 C_{22} (z_{s} + R_{s})}{{R_{s}}^{3}} \frac{\partial V^{T}}{\partial θ} W, \\ k_{v x}^{s} = - \frac{C_{66} h_{c} (2 z_{s} + R_{s})}{{R_{s}}^{2}} \frac{\partial V^{T}}{\partial x} \frac{\partial Φ_{x}}{\partial θ} - \frac{C_{12} h_{c} (z_{s} + R_{s})}{{R_{s}}^{2}} \frac{\partial V^{T}}{\partial θ} \frac{\partial Φ_{x}}{\partial x}, \\ k_{v θ}^{s} = - \frac{C_{22} h_{c} {(z_{s} + R_{s})}^{2}}{{R_{s}}^{4}} \frac{\partial V^{T}}{\partial θ} \frac{\partial Φ_{θ}}{\partial θ} - \frac{C_{66} h_{c} {(2 z_{s} + R_{s})}^{2}}{{R_{s}}^{2}} \frac{\partial V^{T}}{\partial x} \frac{\partial Φ_{θ}}{\partial x}, \\ k_{w w}^{s} = \frac{C_{11} {(2 z_{s} - h_{s})}^{2}}{2} \frac{\partial^{2} W^{T}}{\partial x^{2}} \frac{\partial^{2} W}{\partial x^{2}} + \frac{C_{22} {[R_{s} (2 z_{s} - h_{s}) - h_{s} z_{s}]}^{2}}{2 {R_{s}}^{6}} \frac{\partial^{2} W^{T}}{\partial θ^{2}} \frac{\partial^{2} W}{\partial θ^{2}} + \frac{2 C_{66} {[R_{s} (2 z_{s} - h_{s}) - h_{s} z_{a}]}^{2}}{{R_{s}}^{4}} \frac{\partial^{2} W^{T}}{\partial x \partial θ} \frac{\partial^{2} W}{\partial x \partial θ} + \frac{2 C_{22}}{{R_{s}}^{2}} W^{T} W \\ + \frac{C_{12} [R_{s} (2 z_{s} - h_{s}) - h_{s} z_{s}] (2 z_{s} - h_{s})}{2 {R_{s}}^{3}} (\frac{\partial^{2} W^{T}}{\partial x^{2}} \frac{\partial^{2} W}{\partial θ^{2}} + \frac{\partial^{2} W^{T}}{\partial θ^{2}} \frac{\partial^{2} W}{\partial x^{2}}) - \frac{C_{12} (2 z_{s} - h_{s})}{R_{s}} (\frac{\partial^{2} W^{T}}{\partial x^{2}} W + W^{T} \frac{\partial^{2} W}{\partial x^{2}}) \\ - \frac{C_{22} [R_{s} (2 z_{s} - h_{s}) - h_{s} z_{s}]}{{R_{s}}^{4}} (\frac{\partial^{2} W^{T}}{\partial θ^{2}} W + W^{T} \frac{\partial^{2} W}{\partial θ^{2}}), \\ k_{w x}^{s} = \frac{C_{12} h_{c} [R_{s} (2 z_{s} - h_{s}) - h_{s} z_{s}]}{2 {R_{s}}^{3}} \frac{\partial^{2} W^{T}}{\partial θ^{2}} \frac{\partial Φ_{x}}{\partial x} + \frac{C_{66} h_{c} [(R_{s} (2 z_{s} - h_{s}) - h_{s} z_{s})]}{{R_{s}}^{3}} \frac{\partial^{2} W^{T}}{\partial x \partial θ} \frac{\partial Φ_{x}}{\partial θ} + \frac{C_{11} h_{c} R_{s} (2 z_{s} - h_{s})}{2 R_{s}} \frac{\partial^{2} W^{T}}{\partial x^{2}} \frac{\partial Φ_{x}}{\partial x} - \frac{C_{12} h_{c}}{R_{s}} W^{T} \frac{\partial Φ_{x}}{\partial x}, \\ k_{w θ}^{s} = \frac{C_{22} h_{c} (z_{s} + R_{s}) [R_{s} (2 z_{s} - h_{s}) - h_{s} z_{s}]}{2 {R_{s}}^{5}} \frac{\partial^{2} W^{T}}{\partial θ^{2}} \frac{\partial Φ_{θ}}{\partial θ} + \frac{C_{66} h_{c} [R_{s} (2 z_{s} - h_{s}) - h_{s} z_{s}] (2 z_{s} + R_{s})}{{R_{s}}^{3}} \frac{\partial^{2} W^{T}}{\partial x \partial θ} \frac{\partial Φ_{θ}}{\partial x} \\ + \frac{C_{12} h_{c} (2 z_{s} - h_{s}) (z_{s} + R_{s})}{2 {R_{s}}^{2}} \frac{\partial^{2} W^{T}}{\partial x^{2}} \frac{\partial Φ_{θ}}{\partial θ} - \frac{C_{22} h_{c} (z_{s} + R_{s})}{{R_{s}}^{3}} W^{T} \frac{\partial Φ_{θ}}{\partial θ}, \\ k_{x x}^{s} = \frac{C_{11} {h_{c}}^{2}}{2} \frac{\partial Φ_{x}^{T}}{\partial x} \frac{\partial Φ_{x}}{\partial x} + \frac{C_{66} {h_{c}}^{2}}{2 {R_{s}}^{2}} \frac{\partial Φ_{x}^{T}}{\partial θ} \frac{\partial Φ_{x}}{\partial θ}, \\ k_{x θ}^{s} = \frac{C_{12} {h_{c}}^{2} (z_{s} + R_{s})}{2 {R_{s}}^{2}} \frac{\partial Φ_{x}^{T}}{\partial x} \frac{\partial Φ_{θ}}{\partial θ} + \frac{C_{66} {h_{c}}^{2} (2 z_{s} + R_{s})}{2 {R_{s}}^{2}} \frac{\partial Φ_{x}^{T}}{\partial θ} \frac{\partial Φ_{θ}}{\partial x}, \\ k_{θ θ}^{s} = \frac{C_{22} {h_{c}}^{2} {(z_{s} + R_{s})}^{2}}{2 {R_{s}}^{4}} \frac{\partial Φ_{θ}^{T}}{\partial θ} \frac{\partial Φ_{θ}}{\partial θ} + \frac{C_{66} {h_{c}}^{2} {(2 z_{s} + R_{s})}^{2}}{2 {R_{s}}^{2}} \frac{\partial Φ_{θ}^{T}}{\partial x} \frac{\partial Φ_{θ}}{\partial x} . \end{array}

The matrix G in Eq. (24) is

G = [\begin{array}{c} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 Ω R_{c} I_{0} \int_{0}^{2 π} \int_{0}^{L} (V^{T} W) d x d θ & 0 & 0 \\ 0 & - 2 Ω R_{c} I_{0} \int_{0}^{2 π} \int_{0}^{L} (W^{T} V) d x d θ & 0 & 0 & - 2 Ω R_{c} I_{1} \int_{0}^{2 π} \int_{0}^{L} (W^{T} Φ_{θ}) d x d θ \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 Ω R_{c} I_{1} \int_{0}^{2 π} \int_{0}^{L} (Φ_{θ}^{T} W) d x d θ & 0 & 0 \end{array}] .

The matrix C in Eq. (24) is

C = {(Φ^{T})}^{- 1} [\begin{array}{l} 2 ζ ω_{1} & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ 0 & \dots & 2 ζ ω_{N} \end{array}] Φ^{- 1},

where ω_i(i=1,⋯,N) denotes the natural frequencies, ζ is the assumed damping coefficient and the modal matrix Φ can be obtained by solving the eigenvalue problem defined by the mass matrix Μ^c and the stiffness matrix K^c.

The matrix K^e in Eq. (24) is

K^{e} = [\begin{array}{l} K_{u 1}^{e} & K_{u 2}^{e} & \dots & K_{u n p}^{e} \\ K_{v 1}^{e} & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋮ \\ K_{θ 1}^{e} & \dots & \dots & K_{θ n p}^{e} \end{array}],

where

K_{j i}^{e} = \int_{V_{a i}} k_{j i}^{e} d V = \int_{{\bar{θ}}_{i}}^{{\tilde{θ}}_{i}} \int_{{\bar{x}}_{i}}^{{\tilde{x}}_{i}} \int_{- h_{a} / 2}^{h_{a} / 2} k_{j i}^{e} R_{a} d z_{a} d x d θ (j = u, v, w, x, θ; i = 1, 2, \dots, n p),

in which

\begin{array}{l} k_{u i}^{e} = - \frac{e_{31}}{h_{a}} \frac{\partial U^{T}}{\partial x}, k_{v i}^{e} = - \frac{e_{32}}{h_{a}} \frac{(z_{a} + R_{a})}{{R_{a}}^{2}} \frac{\partial V^{T}}{\partial θ}, k_{w i}^{e} = + \frac{e_{32} [R_{a} (2 z_{a} + h_{a}) + z_{a} h_{a}]}{2 h_{a} {R_{a}}^{3}} \frac{\partial^{2} W^{T}}{\partial θ^{2}} + \frac{e_{31} (2 z_{a} + h_{a})}{2 h_{a}} \frac{\partial^{2} W^{T}}{\partial x^{2}} - \frac{e_{32} W^{T}}{h_{a} R_{a}}, \\ k_{x i}^{e} = - \frac{e_{31} h_{c}}{2 h_{a}} \frac{\partial Φ_{x}^{T}}{\partial x}, k_{θ i}^{e} = - \frac{e_{32} h_{c} (z_{a} + R_{a})}{2 h_{a} {R_{a}}^{2}} \frac{\partial Φ_{θ}^{T}}{\partial θ} . \end{array}

The matrix ${\hat{Γ}}_{i}^{s}$ in Eq. (24) is

{\hat{Γ}}_{i}^{s} = [\begin{array}{l} {\hat{Γ}}_{1 i}^{s} & {\hat{Γ}}_{2 i}^{s} & {\hat{Γ}}_{3 i}^{s} & {\hat{Γ}}_{4 i}^{s} & {\hat{Γ}}_{5 i}^{s} \end{array}],

in which

\begin{array}{l} {\hat{Γ}}_{1 i}^{s} = \int_{A_{i}^{s}} (e_{31} \frac{\partial U}{\partial x}) d A \\ {\hat{Γ}}_{2 i}^{s} = \int_{A_{i}^{s}} (e_{32} \frac{\partial V}{R_{s} \partial θ} + e_{32} \frac{z_{s}}{{R_{s}}^{2}} \frac{\partial V}{\partial θ}) d A \\ {\hat{Γ}}_{3 i}^{s} = \int_{A_{i}^{s}} (e_{31} \frac{h_{s}}{2} \frac{\partial^{2} W}{\partial x^{2}} - e_{31} z_{s} \frac{\partial^{2} W}{\partial x^{2}} + e_{32} \frac{h_{s}}{2 {R_{s}}^{2}} \frac{\partial^{2} W}{\partial θ^{2}} + e_{32} \frac{W}{R_{s}} + e_{32} \frac{z_{s} h_{s}}{2 {R_{s}}^{3}} \frac{\partial^{2} W}{\partial θ^{2}} - e_{32} \frac{z_{s}}{{R_{s}}^{2}} \frac{\partial^{2} W}{\partial θ^{2}}) d A \\ {\hat{Γ}}_{4 i}^{s} = \int_{A_{i}^{s}} (- e_{31} \frac{h_{c}}{2} \frac{\partial Φ_{x}}{\partial x}) d A \\ {\hat{Γ}}_{5 i}^{s} = \int_{A_{i}^{s}} (- e_{32} \frac{h_{c}}{2 R_{s}} \frac{\partial Φ_{θ}}{\partial θ} - e_{32} \frac{h_{c}}{2} \frac{z_{s}}{{R_{s}}^{2}} \frac{\partial Φ_{θ}}{\partial θ}) d A \end{array}

References

Afshari

Amirabadi

(2022) Vibration characteristics of rotating truncated conical shells reinforced with agglomerated carbon nanotubes. Journal of Vibration and Control 28(15-16): 1894–1914.

Brand

Cole

MOT

(2021) Mini-max optimization of actuator/sensor placement for flexural vibration control of a rotating thin-walled cylinder over a range of speeds. Journal of Sound and Vibration 506: 116105.

Casati

Vedani

(2014) Metal matrix composites reinforced by nano-particles-a review. Metals 4(1): 65–83.

Chai

Wang

(2022) Traveling wave vibration of graphene platelet reinforced porous joined conical-cylindrical shells in a spinning motion. Engineering Structures 252: 113718.

Chai

Wang

Teng

(2022) Nonlinear free vibration of spinning cylindrical shells with arbitrary boundary conditions. Applied Mathematics and Mechanics 43(8): 1203–1218.

Chai

Wang

Yang

(2023) Frequency response of spinning cylindrical shells with discontinuous conditions: a semi method. Thin-Walled Structures 182: 110253.

Civalek

(2017) Free vibration of carbon nanotubes reinforced (CNTR) and functionally graded shells and plates based on FSDT via discrete singular convolution method. Composites Part B: Engineering 111: 45–59.

Fakkaew

Cole

MOT

Chamroon

(2019) On the vibrational dynamics of rotating thin-walled cylinders: a theoretical and experimental study utilizing active magnetic bearings. International Journal of Mechanical Sciences 163: 105101.

Ghasemi

Meskini

(2019) Free vibration analysis of porous laminated rotating circular cylindrical shells. Journal of Vibration and Control 25(18): 2494–2508.

10.

Kumar

Ray

(2014) Control of smart rotating laminated composite truncated conical shell using ACLD treatment. International Journal of Mechanical Sciences 89: 123–141.

11.

Lam

K-Y

T-Y

(2005) Rotating Shell Dynamics: Elsevier Science Ltd.

12.

, et al. (2021) Nonlinear vibration characteristics of fibre reinforced composite cylindrical shells in thermal environment. Mechanical Systems and Signal Processing 156: 107665.

13.

Sun

, et al. (2021) Nonlinear vibrations of fiber-reinforced composite cylindrical shells with bolt loosening boundary conditions. Journal of Sound and Vibration 496: 115935.

14.

Liew

Lei

Zhang

(2015) Mechanical analysis of functionally graded carbon nanotube reinforced composites: a review. Composite Structures 120: 90–97.

15.

Mallek

Jrad

Wali

, et al. (2020) Dynamic analysis of functionally graded carbon nanotube-reinforced shell structures with piezoelectric layers under dynamic loads. Journal of Vibration and Control 26(13-14): 1157–1172.

16.

Mellouli

Jrad

Wali

, et al. (2020) Free vibration analysis of FG-CNTRC shell structures using the meshfree radial point interpolation method. Computers & Mathematics with Applications 79(11): 3160–3178.

17.

Moghaddam

SMF

Ahmadi

(2020) Active vibration control of truncated conical shell under harmonic excitation using piezoelectric actuator. Thin-Walled Structures 151: 106642.

18.

Nguyen

Quang

Anh

VTT

, et al. (2019) Nonlinear vibration of carbon nanotube reinforced composite truncated conical shells in thermal environment. International Journal of Structural Stability and Dynamics 19(12): 1950158.

19.

Niasar

Rahaghi

Jafari

(2022) Optimal location of FG actuator/sensor patches on an FG rotating conical shell for active control of vibration. Acta Mechanica 233(12): 5335–5357.

20.

Niasar

Jafari

Rahaghi

, et al. (2024) Active control of free and forced vibration of a rotating FG cylindrical shell FG piezoelectric patches. Mechanics Based Design of Structures and Machines 52: 3900–3924. DOI: 10.1080/15397734.2023.2212297.

21.

Qin

Pang

Safaei

, et al. (2019) Free vibration analysis of rotating functionally graded CNT reinforced composite cylindrical shells with arbitrary boundary conditions. Composite Structures 220: 847–860.

22.

Rostami

Rahaghi

Mohammadimehr

(2021) Vibration control of the rotating sandwich cylindrical shell considering functionally graded core and functionally graded magneto-electro-elastic layers by using differential quadrature method. Journal of Sandwich Structures & Materials 23(1): 132–173.

23.

Shen

Xiang

(2012) Nonlinear vibration of nanotube-reinforced composite cylindrical shells in thermal environments. Computer Methods in Applied Mechanics and Engineering 213-216: 196–205.

24.

Shen

Reddy

(2020) Large amplitude vibration of FG-CNTRC laminated cylindrical shells with negative Poisson's ratio. Computer Methods in Applied Mechanics and Engineering 360: 112727.

25.

Song

Zhang

Liew

(2016) Active vibration control of CNT-reinforced composite cylindrical shells via piezoelectric patches. Composite Structures 158: 92–100.

26.

Song

Zhang

Liew

(2016) Vibration analysis of CNT-reinforced functionally graded composite cylindrical shells in thermal environments. International Journal of Mechanical Sciences 115-116: 339–347.

27.

Sun

Liu

(2021) Multiple internal resonances in nonlinear vibrations of rotating thin-walled cylindrical shells. Journal of Sound and Vibration 510: 116313.

28.

Sun

Liu

(2021) Parametric study and stability analysis on nonlinear traveling wave vibrations of rotating thin cylindrical shells. Archive of Applied Mechanics 91(6): 2833–2851.

29.

Sun

Cao

Han

(2013) Vibration studies of rotating cylindrical shells with arbitrary edges using characteristic orthogonal polynomials in the Rayleigh-Ritz method. International Journal of Mechanical Sciences 68: 180–189.

30.

Sun

Liu

Cao

(2018) Nonlinear travelling wave vibrations of a rotating thin cylindrical shell. Journal of Sound and Vibration 431: 122–136.

31.

Sun

Guo

Feng

, et al. (2022) Nonlinear vibration analysis of CNT-reinforced functionally graded composite cylindrical shells resting on elastic foundations. International Journal of Non-Linear Mechanics 143: 104037.

32.

Thomas

Roy

(2016) Vibration analysis of functionally graded carbon nanotube-reinforced composite shell structures. Acta Mechanica 227(2): 581–599.

33.

Zhang

Yao

(2020) Nonlinear forced vibration of functionally graded carbon nanotube reinforced composite circular cylindrical shells. Acta Mechanica 231(6): 2497–2519.

34.

Zghal

Frikha

Dammak

(2018) Free vibration analysis of carbon nanotube-reinforced functionally graded composite shell structures. Applied Mathematical Modelling 53: 132–155.

35.

Zhao

Sun

Yang

, et al. (2023) Vibration analysis of rotating combined thin-walled shells with multiple conical segments. Journal of Vibration and Acoustics 145(1): 4055229.