Traveling-wave vibration analysis of rotating composite laminated cylindrical shells with general elastic end restraints using a full layerwise theory

Abstract

Rotating laminated shells are widely used in engineering structures, where their vibration behavior is jointly affected by material anisotropy, centrifugal stiffening, gyroscopic coupling, and boundary flexibility. For moderately thick and thick shells, these effects become more complicated and may not be captured accurately by equivalent single-layer models. Accordingly, this paper presents a semi-analytical formulation for the traveling-wave vibration analysis of rotating laminated cylindrical shells with general elastic end restraints based on a full layerwise shell theory and the boundary spring technique. The governing equations are derived from Hamilton’s principle, in which the Fourier series and one-dimensional finite elements are employed to discretize the circumferential displacement field and axial domain, respectively, while a mass-weighted modal assurance criterion is introduced to track the continuous modal branches under rotation. The validity of the proposed method is verified through convergence studies, comparison with experimental results, and benchmark solutions available in the literature. The results show that the present formulation can accurately capture the forward and backward traveling-wave branches, together with the associated frequency splitting and mode veering phenomena induced by rotation. The effects of elastic end restraints and stacking sequence on the traveling-wave vibration characteristics are further clarified. The proposed method provides an efficient and accurate tool for the vibration analysis of rotating laminated cylindrical shells under general elastic support conditions.

Keywords

traveling-wave vibrations thick cylindrical shells layerwise theory rotation general boundary conditions

Introduction

Composite laminated cylindrical shells are key load-carrying and motion-transmitting components in many aerospace, turbomachinery, marine, and energy systems because of their high specific stiffness, tailorability, and favorable dynamic performance. In practical service, such shells often operate under rotation and are rarely subjected to ideal classical end conditions. Instead, the supports are more realistically represented by finite elastic restraints, and the resulting vibration response is strongly influenced by the combined effects of laminate anisotropy, gyroscopic coupling, centrifugal stiffening, and boundary compliance. These features make accurate prediction of the free-vibration characteristics of rotating laminated cylindrical shells both scientifically challenging and practically important.

The theoretical development and numerical analysis of plate and shell structures have been extensively documented in the fundamental works of Reddy,¹ Qatu,² Soedel,³ Amabili,⁴ Carrera,⁵ and Le.⁶ In recent years, a substantial amount of literature has been devoted to the vibration of composite laminates with increasingly complex geometries^7–15 and advanced material systems.^16–20 For the present problem, the most relevant developments mainly concern boundary modeling, rotation-induced vibration behavior, and refined laminate theories. Regarding boundary modeling, practical shell structures are rarely subjected to ideal classical constraints. Therefore, spring-based techniques have been widely used to represent elastic restraints, boundary flexibility, and coupling effects in a unified manner.²¹ In this context, a variety of semi-analytical methods have been proposed for shell structures with arbitrary or elastic boundaries. Jin et al.²² proposed a unified Fourier solution within the first-order shear deformation theory for the vibration analysis of composite laminated shells of revolution with general elastic restraints, in which artificial springs were introduced to represent both classical and non-classical boundary conditions in a unified framework. Zhang et al.²³ developed an improved Fourier-series method for the free and forced vibration of circular cylindrical double-shell structures under arbitrary boundary conditions, where boundary and coupling springs were employed to model elastic and rigid restraints conveniently. Choe et al.²⁴ presented a unified Jacobi-Ritz method for coupled composite laminated axisymmetric doubly curved shells with general boundary conditions, combining Jacobi polynomials in the meridional direction and Fourier series in the circumferential direction to achieve efficient semi-analytical modeling. Li et al.²⁵ proposed a semi-analytical Rayleigh-Ritz formulation for functionally graded porous cylindrical shells with arbitrary boundary restraints, in which unified Jacobi polynomials and Fourier series were adopted together with penalty-based spring modeling. Zhang et al.²⁶ developed a dynamic stiffness method for coupled conical-ribbed cylindrical-conical shell structures with general boundary conditions, where the global dynamic stiffness matrix was assembled from exact component solutions. Overall, these studies demonstrate that spring-based semi-analytical modeling provides an effective and flexible framework for shell vibration analysis under general boundary conditions, although most of these formulations are still based on equivalent single-layer (ESL)-type shell kinematics.

In many engineering applications, composite laminated structures operate under rotating service conditions. Vibration behavior under rotation becomes considerably more complicated than that of stationary ones, because centrifugal stiffening and gyroscopic coupling must be taken into account simultaneously. As a result, the modal frequencies may split into forward- and backward-traveling branches, accompanied by mode shifting, frequency veering, and mode-shape transfer.²⁷ Accordingly, increasing attention has been paid to rotating structures. Early studies mainly focused on isotropic^28–33 and laminated^34–36 rotating cylindrical structures, while later work was extended to functionally graded,^37,38 graphene/CNT-reinforced,^39,40 stepped,^41,42 sandwich,^43,44 and multi-component coupled shells.^45–47 In parallel, the methodological development evolved from conventional finite-element and shell-theory toward Rayleigh-Ritz, orthogonal-polynomial semi-analytical methods, and more recently nonlinear,^48–50 traveling-wave-based^51–53 and multi-mode coupling analyses.^54,55 Recent nonlinear studies on composite cylindrical shells have further shown that porosity, elastic foundations, magneto-electro-thermo-mechanical loads, thermal environments, and multi-harmonic excitations may induce complex amplitude-dependent responses, internal resonances, and nonlinear forced-vibration characteristics.^56–59 These studies highlight the significance of geometrically nonlinear models for composite shells under severe operating conditions, although the present work focuses on the linear small-amplitude free-vibration characteristics of rotating laminated cylindrical shells.

As indicated by the above literature review, although extensive studies have been conducted in this field, most existing formulations are based on ESL theories, including classical laminated shell theory (CLST), first-order shear deformation theory (FSDT), and higher-order shear deformation theory (HSDT). These ESL-based descriptions are generally adequate for thin shells, but may become insufficient for moderately thick and thick laminated shells, where transverse shear effects, thickness-wise deformation, and material heterogeneity become more significant.⁶⁰ To overcome these limitations, several refined laminate theories have been developed, such as Carrera’s Unified Formulation (CUF),⁶¹ zig-zag theories,⁶² and layerwise theories.⁶³ A recent layerwise study¹⁴ also showed that layerwise kinematics can improve free-vibration predictions for medium-thickness heterogeneous shells by more accurately describing thickness-wise deformation. CUF provides a general hierarchical framework for refined thickness expansions, while zig-zag theories efficiently represent slope discontinuities and transverse shear effects across layer interfaces with relatively few unknowns. In the present study, Reddy’s full layerwise shell theory is adopted not because it is universally superior to other advanced theories, but because it is suited to the rotating laminated cylindrical shell problem considered here. Its independent through-thickness interpolation of displacement components enables a direct description of thickness-wise deformation and interlaminar continuity in moderately thick and thick laminates. Moreover, the layerwise nodal variables allow the stiffness, centrifugal-stress-induced geometric stiffness, mass, and gyroscopic matrices to be constructed in a consistent and physically transparent manner, which is important for capturing forward- and backward-traveling-wave splitting and mode veering. In addition, LWST can be naturally combined with the present circumferential Fourier expansion, axial finite-element discretization, and end-spring modeling strategy. However, the combined treatment of full layerwise kinematics, rotation-induced gyroscopic/centrifugal effects, and spring-based general end restraints within an efficient Fourier-FE framework remains limited.

Accordingly, this study develops a layerwise semi-analytical formulation for the free vibration analysis of rotating composite laminated cylindrical shells under general elastic end restraints. The proposed formulation combines circumferential Fourier expansion with axial finite-element discretization, thereby retaining the high modal resolution and computational efficiency of semi-analytical methods while offering the geometric flexibility and systematic convergence properties of the finite-element method. This Fourier-FE framework is particularly advantageous for shell vibration problems because it naturally exploits the periodicity in the circumferential direction, reduces the computational cost of full two-dimensional discretization, and facilitates the treatment of traveling-wave characteristics in rotating shells. In addition, by introducing spring-based elastic restraints at the shell ends, the present model can represent a broad range of practical boundary conditions within a unified formulation, which further enhances its applicability to engineering shell structures.

The remainder of this paper is organized as follows. The second section presents the proposed Fourier-FE semi-analytical formulation based on full layerwise mechanics and Hamilton’s principle, followed by the solution procedure and the MAC-based mode-tracking method. The third section provides the convergence study and further verifies the accuracy and effectiveness of the proposed approach through comparisons with experimental results and benchmark solutions from the literature. The fourth section presents a parametric investigation of the effects of rotation speed, elastic end restraints, and stacking sequence on the free-vibration characteristics of rotating composite laminated cylindrical shells. Finally, the main conclusions are summarized in the last section.

Theoretical formulations

Displacements and strains of layerwise kinematics

The geometrical configuration, coordinates, through-thickness layerwise approximation, and end restraints of a rotating multilayered cylindrical shell are schematically depicted in Figure 1. Specifically, Figure 1(a) defines dimensions, and the adopted coordinate systems for the laminated shells under a constant rotational speed $Ω$ . The proposed layerwise discretization strategy is illustrated in Figure 1(b), where interpolation nodes are placed on the inner and outer surface as well as at all interlaminar interfaces. This arrangement enables an accurate through-thickness description of the displacement field and its gradients, thereby improving the fidelity of the predicted natural frequencies and mode shapes. Figure 1(c) presents spring-stiffness-based technique to model general elastic end restraints at both shell ends. Following Reddy’s full layerwise shell theory (LWST),¹ the displacement field of the rotating cylindrical shell, constructed using the above nodal scheme can be expressed as:

u (x, y, z, t) = \sum_{I = 1}^{N} U_{I} (x, y, t) ϕ_{I} (z)

(1a)

v (x, y, z, t) = \sum_{I = 1}^{N} V_{I} (x, y, t) ϕ_{I} (z)

(1b)

w (x, y, z, t) = \sum_{I = 1}^{N} W_{I} (x, y, t) ϕ_{I} (z)

(1c)

where

(u, v, w)

denote the total displacement components along the

(x, y, z)

x

is the axial coordinate;

y = R θ

is the circumferential coordinate (

R

being the radius of shell mid-surface and

θ

the angular position).

z

denotes the thickness coordinate.

(U_{I}, V_{I}, W_{I})

are the nodal values of

(u, v, w)

I_{t h}

interpolation node.

ϕ_{I} (z)

denotes the Lagrangian interpolation function associated with node

I

, used to approximate for both the in-plane and transverse displacement components.

N

is the number of through-thickness interpolation nodes. With the proposed nodal scheme,

N

satisfies

N = N_{n l} + 1

, where

N_{n l}

denotes the number of numerical sublayers.

Figure 1.

Schematic of the rotating laminated cylindrical shell model: (a) geometry and coordinate systems; (b) layerwise through-thickness discretization; (c) spring-stiffness model for general elastic end restraints.

The strain components of the cylindrical shells associated with the full layerwise theory⁶³ of equations (1a)–(1c) are derived as follows:

ε_{x x} = U_{I, x} ϕ_{I}

(2a)

ε_{y y} = (V_{I, y} + \frac{W_{I}}{R}) ϕ_{I}

(2b)

ε_{z z} = W_{I} ϕ_{I, z}

(2c)

γ_{y z} = V_{I} ϕ_{I, z} + (W_{I, y} - \frac{V_{I}}{R}) ϕ_{I}

(2d)

γ_{x z} = U_{I} ϕ_{I, z} + W_{I, x} ϕ_{I}

(2e)

γ_{x y} = (U_{I, y} + V_{I, x}) ϕ_{I}

(2f)

where the summation over the subscript

I

is implied, and a comma followed by subscripts denotes differentiation with respect to the corresponding coordinate, e.g.,

U_{I, x} = \frac{\partial U_{I}}{\partial x}

, and similarly for the other terms.

Hamilton’s principle and equations of motion

The governing equations for free vibration of rotating laminated cylindrical shells with general elastic end restraints based on the full layerwise shell theory can be derived by means of Hamilton’s principle,

\int_{0}^{T} (δ U - δ M) d t = 0

(3)

where the total virtual potential energy

δ U

consists of the elastic strain energy

δ U_{ε}

, the elastic end restraint energy

δ U_{s p r}

, and the strain energy induced by initial centrifugal stress

δ U_{g e o}

The virtual elastic strain energy $δ U_{ε}$ for the present full layerwise shell theory is given by:

\begin{array}{l} \begin{array}{l} δ U_{ε} \end{array} & \begin{array}{l} = \end{array} & \begin{array}{l} \int_{Ω_{0}} \int_{- \frac{H}{2}}^{\frac{H}{2}} (σ_{x x} δ ε_{x x} + σ_{y y} δ ε_{y y} + σ_{z z} δ ε_{z z} + \\ σ_{y z} δ γ_{y z} + σ_{x z} δ γ_{x z} + σ_{x y} δ γ_{x y}) d z d y d x \end{array} \\ \begin{array}{l} = \end{array} & \begin{array}{l} \int_{Ω_{0}} [N_{x x}^{I} δ U_{I, x} + N_{y y}^{I} δ V_{I, y} + \frac{N_{y y}^{I}}{R} {δ W}_{I} + {\bar{Q}}_{z z}^{I} δ W_{I} + ({\bar{Q}}_{y z}^{I} - \frac{Q_{y z}^{I}}{R}) δ V_{I} + \\ + Q_{y z}^{I} {δ W}_{I, y} + {\bar{Q}}_{x z}^{I} δ U_{I} + Q_{x z}^{I} {δ W}_{I, x} + N_{x y}^{I} δ U_{I, y} + N_{x y}^{I} δ V_{I, x}] d y d x \end{array} \end{array}

(4)

where the stress resultants are expressed as:

{\begin{cases} N_{x x}^{I} \\ N_{y y}^{I} \\ N_{x y}^{I} \end{cases}} = \int_{- \frac{H}{2}}^{\frac{H}{2}} {\begin{cases} σ_{x x} \\ σ_{y y} \\ σ_{x y} \end{cases}} ϕ_{I} d z

(5a)

{\begin{cases} {\bar{Q}}_{y z}^{I} \\ {\bar{Q}}_{x z}^{I} \\ {\bar{Q}}_{z z}^{I} \end{cases}} = \int_{- \frac{H}{2}}^{\frac{H}{2}} {\begin{cases} σ_{y z} \\ σ_{x z} \\ σ_{z z} \end{cases}} \frac{d ϕ_{I}}{d z} d z, {\begin{cases} Q_{y z}^{I} \\ Q_{x z}^{I} \end{cases}} = \int_{- \frac{H}{2}}^{\frac{H}{2}} {\begin{cases} σ_{y z} \\ σ_{x z} \end{cases}} ϕ_{I} d z

(5b)

The virtual elastic end restraint energy $δ U_{s p r}$ for the present full layerwise shell theory is given by:

\begin{array}{l} \begin{array}{l} δ U_{s p r} = \int_{0}^{2 π R} \int_{- H / 2}^{H / 2} (k_{u 0} u_{0} δ u_{0} + k_{v 0} v_{0} δ v_{0} + k_{w 0} w_{0} δ w_{0}) d z d y + \\ \int_{0}^{2 π R} \int_{- H / 2}^{H / 2} (k_{u L} u_{L} δ u_{L} + k_{v L} v_{L} δ v_{L} + k_{w L} w_{L} δ w_{L}) d z d y \end{array} \\ \begin{array}{l} = \int_{0}^{2 π R} [K_{u 0}^{I J} U_{J} (0, y, t) δ U_{I} (0, y, t) + K_{u L}^{I J} U_{J} (L, y, t) δ U_{I} (L, y, t) + \\ K_{v 0}^{I J} V_{J} (0, y, t) δ V_{I} (0, y, t) + K_{v L}^{I J} V_{J} (L, y, t) δ V_{I} (L, y, t) + \\ K_{w 0}^{I J} W_{J} (0, y, t) δ W_{I} (0, y, t) + K_{w L}^{I J} W_{J} (L, y, t) δ W_{I} (L, y, t)] d y \end{array} \end{array}

(6)

where the subscripts

(0, L)

indicate the two ends of the cylindrical shell, and the elastic restraint stiffnesses at two ends are expressed as:

K_{u 0}^{I J} = k_{u 0} \int_{- H / 2}^{H / 2} ϕ_{I} ϕ_{J} d z, K_{u L}^{I J} = k_{u 0} \int_{- H / 2}^{H / 2} ϕ_{I} ϕ_{J} d z

(7)

the virtual strain energy induced by initial centrifugal stress

δ U_{g e o}

is given by

\begin{array}{l} δ U_{g e o} = \int_{Ω_{0}} \int_{- H / 2}^{- H / 2} ρ Ω^{2} w_{, y} δ w_{, y} d z d y d x \\ = \int_{Ω_{0}} G_{(Ω)}^{I J} w_{J, y} δ w_{I, y} d y d x \end{array}

(8)

where the geometric stiffness induced by centrifugal prestress

G_{(Ω)}^{I J}

is expressed as:

G_{(Ω)}^{I J} = \int_{- H / 2}^{H / 2} ρ Ω^{2} ϕ_{I} ϕ_{J} d z

(9)

The virtual kinetic energy $δ M$ for the rotating cylindrical shell can be described as follows:

\begin{array}{l} \begin{array}{l} δ M = \int_{Ω_{0}} \int_{- \frac{H}{2}}^{\frac{H}{2}} [ρ (\dot{u} δ \dot{u} + \dot{v} δ \dot{v} + \dot{w} δ \dot{w}) + 2 Ω (\dot{w} δ v - \dot{v} δ w) - Ω^{2} (v δ v + w δ w)] \\ d z d y d x \end{array} \\ \begin{array}{l} = I^{I J} \int_{Ω_{0}} [({\dot{U}}_{J} {δ \dot{U}}_{I} + {\dot{V}}_{J} {δ \dot{V}}_{I} + {\dot{W}}_{J} {δ \dot{W}}_{I}) + 2 Ω ({\dot{W}}_{J} δ V_{I} - {\dot{V}}_{J} δ W_{I}) - Ω^{2} (V_{J} δ V_{I} + W_{J} δ W_{I})] \\ d y d x \end{array} \end{array}

(10)

where the mass moment matrix

I^{I J}

is expressed as:

I^{I J} = \int_{- H / 2}^{H / 2} ρ ϕ_{I} ϕ_{J} d z

(11)

Substituting for the $δ U$ and $δ M$ from equations (4)–(11) into the Hamilton’s principle in equation (3), and integrating by parts to relieve the virtual generalized displacements, $δ U_{I}$ , $δ V_{I}$ , and $δ W_{I}$ , the governing equations of free vibration for the rotating cylindrical shell with general elastic end restraint can be derived as:

\begin{array}{l} δ U_{I} : & N_{x x, x}^{I} + N_{x y, y}^{I} - {\bar{Q}}_{x z}^{I} = I^{I J} {\ddot{U}}_{J} \end{array}

(12a)

\begin{array}{l} δ V_{I} : & N_{y y, y}^{I} + N_{x y, x}^{I} - {\bar{Q}}_{y z}^{I} + \frac{Q_{y z}^{I}}{R} = I^{I J} {\ddot{V}}_{J} + 2 Ω I^{I J} {\dot{W}}_{J} - Ω^{2} I^{I J} V_{J} \end{array}

(12b)

\begin{array}{l} δ W_{I} : & Q_{x z, x}^{I} + Q_{y z, y}^{I} - {\bar{Q}}_{z z}^{I} - \frac{N_{y y}^{I}}{R} + G_{(Ω)}^{I J} w_{J, y y} = I^{I J} {\ddot{W}}_{J} - 2 Ω I^{I J} {\dot{V}}_{J} - Ω^{2} I^{I J} W_{J} \end{array}

(12c)

The natural boundary conditions of the shell are:

\begin{array}{l} n_{x} (x_{b}) N_{x x}^{I} (x_{b}, y, t) + \sum_{J = 1}^{N} K_{u}^{I J} (x_{b}) U_{J} (x_{b}, y, t) = 0, \\ n_{x} (x_{b}) N_{x y}^{I} (x_{b}, y, t) + \sum_{J = 1}^{N} K_{v}^{I J} (x_{b}) V_{J} (x_{b}, y, t) = 0, & x_{b} \in {0, L} \\ n_{x} (x_{b}) Q_{x z}^{I} (x_{b}, y, t) + \sum_{J = 1}^{N} K_{w}^{I J} (x_{b}) W_{J} (x_{b}, y, t) = 0, \end{array}

(13)

where

n_{x}

is the x-component of the outward unit normal vector n on the end surfaces of the cylindrical shell.

Laminate constitutive equations

The 3D stress-strain relations of the $k_{t h}$ orthotropic lamina, in the shell coordinate system take the form:

{\begin{cases} \begin{array}{l} σ_{x x} \\ σ_{y y} \\ σ_{z z} \end{array} \\ \begin{array}{l} σ_{y z} \\ σ_{x z} \\ σ_{x y} \end{array} \end{cases}}^{k} = {[\begin{array}{l} {\bar{C}}_{11} & {\bar{C}}_{12} & {\bar{C}}_{13} & 0 & 0 & {\bar{C}}_{16} \\ {\bar{C}}_{12} & {\bar{C}}_{22} & {\bar{C}}_{23} & 0 & 0 & {\bar{C}}_{26} \\ {\bar{C}}_{13} & {\bar{C}}_{23} & {\bar{C}}_{33} & 0 & 0 & {\bar{C}}_{36} \\ 0 & 0 & 0 & {\bar{C}}_{45} & {\bar{C}}_{45} & 0 \\ 0 & 0 & 0 & {\bar{C}}_{45} & {\bar{C}}_{55} & 0 \\ {\bar{C}}_{16} & {\bar{C}}_{26} & {\bar{C}}_{36} & 0 & 0 & {\bar{C}}_{66} \end{array}]}^{k} {\begin{cases} \begin{array}{l} ε_{x x} \\ ε_{y y} \\ ε_{z z} \end{array} \\ \begin{array}{l} γ_{y z} \\ γ_{x z} \\ γ_{x y} \end{array} \end{cases}}^{k}

(14)

where

[{\bar{C}}_{i j}]

are the transformed elastic coefficients with respect to the shell coordinate system.

By integrating the stresses through the shell thickness and using the constitutive relation in equation (14), the stress resultants appearing in equations (5a)–(5b) are defined as:

{\begin{cases} N_{x x}^{I} \\ N_{y y}^{I} \\ N_{x y}^{I} \end{cases}} = [\begin{array}{l} A_{11}^{I J} & A_{12}^{I J} & {\bar{A}}_{13}^{I J} & A_{16}^{I J} \\ A_{12}^{I J} & A_{22}^{I J} & {\bar{A}}_{23}^{I J} & A_{26}^{I J} \\ A_{16}^{I J} & A_{26}^{I J} & {\bar{A}}_{36}^{I J} & A_{66}^{I J} \end{array}] {\begin{cases} U_{J, x} \\ V_{J, y} + \frac{W_{J}}{R} \\ U_{J, y} + V_{J, x} \end{cases}}

(15)

{\begin{cases} {\bar{Q}}_{y z}^{I} \\ {\bar{Q}}_{x z}^{I} \end{cases}} = [\begin{array}{l} {\tilde{A}}_{45}^{I J} & {\tilde{A}}_{44}^{I J} \\ {\tilde{A}}_{55}^{I J} & {\tilde{A}}_{45}^{I J} \end{array}] {\begin{cases} U_{J} \\ V_{J} \end{cases}} + [\begin{array}{l} {\bar{A}}_{45}^{I J} & {\bar{A}}_{44}^{I J} \\ {\bar{A}}_{55}^{I J} & {\bar{A}}_{45}^{I J} \end{array}] {\begin{cases} W_{J, x} \\ W_{J, y} - \frac{V_{J}}{R} \end{cases}}

(16)

{\begin{cases} Q_{y z}^{I} \\ Q_{x z}^{I} \end{cases}} = [\begin{array}{l} {\bar{A}}_{45}^{I J} & {\bar{A}}_{44}^{I J} \\ {\bar{A}}_{55}^{I J} & {\bar{A}}_{45}^{I J} \end{array}] {\begin{cases} U_{J} \\ V_{J} \end{cases}} + [\begin{array}{l} A_{45}^{I J} & A_{44}^{I J} \\ A_{55}^{I J} & A_{45}^{I J} \end{array}] {\begin{cases} W_{J, x} \\ W_{J, y} - \frac{V_{J}}{R} \end{cases}}

(17)

{\bar{Q}}_{z z}^{I} = {\bar{A}}_{13}^{I J} U_{J, x} + {\bar{A}}_{23}^{I J} (V_{J, y} + \frac{W_{J}}{R}) + {\tilde{A}}_{33}^{I J} W_{J} + {\bar{A}}_{36}^{I J} (U_{J, y} + V_{J, x})

(18)

The laminate stiffnesses are defined as follows:

A_{i j}^{I J} = \sum_{k = 1}^{N_{p l}} \int_{h_{k}} {\bar{C}}_{i j}^{k} ϕ_{I} ϕ_{J} d z

(19a)

{\bar{A}}_{i j}^{I J} = \sum_{k = 1}^{N_{p l}} \int_{h_{k}} {\bar{C}}_{i j}^{k} ϕ_{I} ϕ_{J, z} d z

(19b)

{\tilde{A}}_{i j}^{I J} = \sum_{k = 1}^{N_{p l}} \int_{h_{k}} {\bar{C}}_{i j}^{k} ϕ_{I, z} ϕ_{J, z} d z

(19c)

where

N_{p l}

is the number of physical layers in the laminated shell, which satisfies

N_{p l} = N - 1

under the proposed nodal arrangement.

h_{k}

denotes the thickness of the

k_{t h}

physical layer.

Fourier-FE formulation

To efficiently solve the Euler-Lagrange equations in equations (12a, 12b, 12c), a semi-analytical Fourier-1D FE strategy is adopted. The circumferential direction of a closed cylindrical shell is periodic; therefore, Fourier expansion naturally represents the circumferential wave field, achieves wavenumber-wise decoupling, and facilitates the identification of forward and backward traveling waves. The axial direction is finite and affected by the elastic end restraints, so one-dimensional finite elements are used to impose general end springs conveniently and allow mesh refinement. The proposed discretization functions are defined as follows:

U_{I} = \sum_{K = 1}^{N_{n e}} ψ_{K} U_{I K} e^{i β y} e^{i w t}

(20a)

V_{I} = \sum_{K = 1}^{N_{n e}} ψ_{K} V_{I K} e^{i β y} e^{i w t}

(20b)

W_{I} = \sum_{K = 1}^{N_{n e}} ψ_{K} W_{I K} e^{i β y} e^{i w t}

(20c)

where

ψ_{K}

is the 1D interpolation shape function used to discretize the axial direction.

{U_{I K}, V_{I K}, W_{I K}}

are the amplitudes of nodal values of

{U_{I}, V_{I}, W_{I}}

K_{t h}

node 1-D element.

N_{n e}

denotes the number of nodes in each 1-D element.

β = \frac{n}{R}

, and

n

denotes the circumferential wavenumber. In the present implementation, two-node linear Lagrange finite elements are adopted in the axial direction, with interpolation functions

ψ_{1} (ξ) = (1 - ξ) / 2

and

ψ_{2} (ξ) = (1 + ξ) / 2

in the parent coordinate

ξ \in [- 1, 1]

Substituting the proposed Fourier-FE discretization equations equations (20a)–(20c) into the Hamilton’s principle in equation (3), the semi-analytical FE governing equations yield the following time-domain equation of motion for a given circumferential mode n and rotation speed $Ω$ :

M \ddot{q} + C_{gyro} \dot{q} + K_{total} q = 0

(21)

where the vector

q (t)

denotes all global nodal displacements.

The mass matrix $M$ in equation (21) takes the form:

M_{e} = [\begin{array}{l} M_{e}^{I J K L} \\ M_{e}^{I J K L} \\ M_{e}^{I J K L} \end{array}]

(22)

where

M_{e}^{I J K L} = 2 π R I^{I J} \int_{x_{e}} ψ_{K} ψ_{L} d x

(23)

The gyroscopic coupling matrix $C_{gyro}$ is a skew-symmetric matrix due to the Coriolis effect, which is given by,

C_{gyro, e} = [\begin{array}{l} 0 & 0 & 0 \\ 0 & 0 & {2 Ω M}_{e}^{I J K L} \\ 0 & - {2 Ω M}_{e}^{I J K L} & 0 \end{array}]

(24)

K_{total}

is the total stiffness matrix composed of centrifugal stiffening stiffness matrix

K_{c e n} (Ω^{2})

, initial hoop stress geometric stiffness

K_{g e o} (n, Ω)

, boundary end restraint stiffness matrix

K_{s p r}

, and strain stiffness matrix

K_{ε} (n)

The centrifugal stiffening stiffness matrix $K_{c e n} (Ω^{2})$ is defined as:

K_{cen, e} = [\begin{array}{l} 0 & 0 & 0 \\ 0 & {Ω^{2} M}_{e}^{I J K L} & 0 \\ 0 & 0 & {Ω^{2} M}_{e}^{I J K L} \end{array}]

(25)

The initial hoop stress geometric stiffness $K_{g e o} (n, Ω)$ is defined as:

K_{g e o, e} (n, Ω) = [\begin{array}{l} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & {{(\frac{n}{R})}^{2} Η}_{e}^{I J K L} \end{array}]

(26)

where

Η_{e}^{I J K L} = 2 π R \int_{x_{e}} ψ_{K} Ω^{2} I^{I J} ψ_{L} d x

(27)

The boundary end restraint stiffness matrix $K_{s p r}$ is defined as:

K_{s p r} = [\begin{array}{l} K_{s p r, U U}^{I J K L} & 0 & 0 \\ 0 & Κ_{s p r, V V}^{I J K L} & 0 \\ 0 & 0 & Κ_{s p r, W W}^{I J K L} \end{array}]

(28)

where

K_{s p r, U U}^{I J K L} = 2 π R \sum_{x_{b} \in {0, L}} ψ_{K} (x_{b}) K_{u}^{I J} (x_{b}) ψ_{L} (x_{b})

(29a)

K_{s p r, V V}^{I J K L} = 2 π R \sum_{x_{b} \in {0, L}} ψ_{K} (x_{b}) K_{v}^{I J} (x_{b}) ψ_{L} (x_{b})

(29b)

K_{s p r, W W}^{I J K L} = 2 π R \sum_{x_{b} \in {0, L}} ψ_{K} (x_{b}) K_{w}^{I J} (x_{b}) ψ_{L} (x_{b})

(29c)

The strain stiffness matrix $K_{ε} (n)$ takes the form:

K_{ε, e} = [\begin{array}{l} K_{U U}^{I J K L} & K_{U V}^{I J K L} & K_{U W}^{I J K L} \\ K_{U V}^{I J K L} & K_{V V}^{I J K L} & K_{V W}^{I J K L} \\ K_{U W}^{I J K L} & K_{V W}^{I J K L} & K_{W W}^{I J K L} \end{array}]

(30)

where the detailed expressions of each stiffness sub-matrix are provided in Appendix A.

Assuming harmonic motion $q (t) = \hat{q} e^{i w t}$ , the time-domain equation of motion in equation (21) can be transformed into the frequency-domain governing equation for free vibration eigenproblem:

(K_{struct} (n) + K_{cen} (Ω) + K_{geo} (n, Ω) + K_{spr} + i ω C_{gyro} - ω^{2} M) \hat{q} = 0

(31)

where

ω

is the natural circular frequency and

\hat{q}

the associated mode shape. Non-trivial solutions exist only when the above characteristic equation is satisfied, which provides the natural frequencies and modes for each

(n, Ω)

Equation (31) yields the natural angular frequency $ω$ in rad/s. The corresponding cyclic frequency is shown in results as $f = ω / 2 π$ in Hz.

Solution algorithm and MAC-based mode-tracking technique

This section describes the computational procedure adopted in this work, including the Fourier-FE solution algorithm and the mode tracking technique used for rotating shells. The governing equation in equation (31) is reduced by a circumferential Fourier expansion and discretized in the axial direction using one-dimensional finite elements, leading to a quadratic eigenvalue problem characterized by the circumferential index n and rotational speed $Ω$ . The overall implementation is summarized in Algorithm 1.

Algorithm 1 provides, at each prescribed rotation speed $Ω_{k}$ , a set of eigenpairs ${ω_{r} (Ω_{k}), {\hat{q}}_{r} (Ω_{k})}$ associated with equation (31). To construct continuous mode branches over the rotation-speed grid and to avoid branch switching in the presence of frequency crossings or mode veering, a robust mode tracking technique is required. In rotating shells, gyroscopic coupling and rotation-induced initial stresses may cause strong modal interactions, under which a simple “nearest-frequency” criterion can lead to erroneous mode assignment. Therefore, the present study employs a Mode Assurance Criterion (MAC) based tracking strategy⁶⁴ that matches modes primarily by eigenvector similarity. Compared with the standard MAC based on direct eigenvector dot products, the mass-weighted MAC uses the mass matrix to account for the kinetic-energy distribution of different displacement components and through-thickness nodal variables, thereby providing a more physically meaningful criterion for branch tracking in the present layerwise rotating-shell formulation. The MAC-based mode-tracking procedure is summarized as follows.

First, to remove the arbitrary scaling of eigenvectors and enable a consistent similarity evaluation between adjacent speed points, all retained eigenvectors are mass-normalized using the consistent mass matrix M,

{\hat{q}}_{r}^{T} (n, Ω_{k}) M {\hat{q}}_{r} (n, Ω_{k}) = 1

(32)

The similarity between mode $r$ at $Ω_{k}$ and mode $s$ at $Ω_{k + 1}$ is quantified by the mass-weighted MAC as follows,

{MAC}_{r s} (Ω_{k}, Ω_{k + 1}) = \frac{{| {\hat{q}}_{r}^{T} (n, Ω_{k}) M {\hat{q}}_{s} (n, Ω_{k + 1}) |}^{2}}{{\hat{q}}_{r}^{T} (n, Ω_{k}) M {\hat{q}}_{r} (n, Ω_{k}) {\hat{q}}_{s}^{T} (n, Ω_{k + 1}) M {\hat{q}}_{s} (n, Ω_{k + 1})} \in (0, 1)

(33)

The ${MAC}_{r s}$ value directly quantifies the similarity between two mass-normalized mode shapes. The complete tracking procedure is summarized in Algorithm 2, while Figure 2 schematically illustrates the one-to-one MAC-based matching between two adjacent rotation speeds. Briefly, at the initial speed $Ω_{1}$ , the modes are ordered by ascending frequency and the resulting order is used to initialize branch indices. For each subsequent step $Ω_{k} \to Ω_{k + 1}$ , all modes at $Ω_{k + 1}$ are initially marked as unused. For a given branch $r$ , MAC values are computed between ${\hat{q}}_{r} (n, Ω_{k})$ and ${\hat{q}}_{s} (n, Ω_{k + 1})$ . The unused mode yielding the maximum MAC is selected as the match, which will be assigned to branch $r$ at $Ω_{k + 1}$ .The matched mode is then marked as used to enforce a one-to-one correspondence between branches and candidate modes.

Figure 2.

Schematic of MAC-based mode tracking between two adjacent rotation speeds.

In the present calculations, ${MAC}_{\min} = 0.90$ is adopted as the mode-matching threshold. When the maximum MAC is below this value or an abrupt frequency jump occurs, the corresponding speed interval is regarded as a possible mode-veering region shown in Figure 2. The local speed increment $Δ Ω$ is then reduced by half. The branch assignment is performed automatically without manual reassignment, and manual inspection is used only to verify the final continuity of the tracked branches.

Convergence study and model verification

Convergence study

To assess the numerical convergence of the proposed layerwise semi-analytical Fourier-finite element formulation, a computer program implementing the numerical algorithm is developed to analyze the free vibration characteristics of moderately-thick composite laminated cylindrical shell with general elastic end restraints. The considered shell is a symmetric laminate with stacking sequence [45°/-45°/45°/-45°]_s, and all plies are assumed to have equal thickness. Its geometric parameters are L = 0.48 m, R = 0.12 m and total thickness H = 0.5 mm. The mechanical properties of the selected orthotropic lamina are $E_{1} = 160 GPa$ , $E_{2} = E_{3} = 12 GPa$ , $G_{12} = 6 GPa$ , $G_{13} = G_{23} = 4 GP a$ , $ν_{12} = ν_{23} = ν_{13} = 0.3$ , $ρ = 1350 kg / m^{3}$ . Convergence is examined with respect to: (i) the axial 1D finite element discretization, (ii) the number of through-thickness layerwise interpolation nodes, and (iii) the stiffness parameters of the end spring. The results are used to determine the appropriate numerical settings for the subsequent computations and to demonstrate the robustness and advantage of the proposed method.

Figure 3(a) and (b) show the non-dimensional natural frequency versus the number of axial elements $N_{e l}$ under fixed boundary conditions at both ends. The fixed boundary conditions are implemented by setting the all-end spring stiffness to an extremely large value. It can be seen that the predicted frequencies decrease monotonically approaching asymptotic values as $N_{e l}$ increases, indicating a typical finite element convergence behavior with respect to the axial discretization. For all considered circumferential wave numbers, the variation becomes negligible beyond $N_{e l} = 48$ , and the curves at $N_{e l} = 48$ and $N_{e l} = 96$ are almost indistinguishable.

Figure 3.

Convergence of the proposed layerwise Fourier-finite element formulation with respect to the number of axial elements $N_{e l}$ for (a) m = 1 and (b) m = 2 with n = 1, 2, 3, 4.

Moreover, higher circumferential wave numbers exhibit a stronger sensitivity to coarse axial meshes, particularly for both axial wave numbers, whereas the low-n modes converge more quickly. Based on these observations, $N_{e l} = 48$ is adopted in the subsequent analyses as a good compromise between accuracy and computational efficiency.

The convergence with respect to the through-thickness layerwise resolution is examined by increasing the number of through-thickness interpolation nodes $N$ for three representative thickness ratios, $\frac{R}{H} = 60$ (thin), $\frac{R}{H} = 30$ (moderately thick), and $\frac{R}{H} = 10$ (thick). Figure 4 depicts the normalized frequency $\tilde{f} = \frac{f}{f_{N = 4}}$ for mode (1, 3), where $f_{N = 4}$ is the reference value obtained with $N = 4$ . The thin shell shows negligible sensitivity to $N$ , indicating that a coarse through-thickness description is sufficient when transverse shear and thickness-wise warping effects are weak. In contrast, the moderately thick and thick shells exhibit a clear dependence on $N$ that the frequencies are noticeably over-predicted for small $N$ and rapidly converge as $N$ increases. For all three thickness ratios, the results become essentially converged for $N \geq 3$ . Therefore, that $N = 3$ is adopted in the subsequent investigations.

Figure 4.

Convergence of the proposed layerwise Fourier-finite element formulation with respect to the number of through-thickness interpolation modes $N$ for three representative thickness ratios.

To further quantify the advantage of the present layerwise formulation over ESL-type models, a representative comparison with an FSDT-based model is conducted for the laminated cylindrical shell used in the above study. The converged present solution is taken as the reference, and the relative difference of the FSDT prediction is defined as: $\frac{| f_{F S D T} - f_{L W S T} |}{f_{L W S T}} \times 100 %$ .

As shown in Table 1, the FSDT results remain close to the LWST predictions for relatively thin shells, but the difference becomes more evident for moderately thick and thick configurations. This is because FSDT describes the laminate through an equivalent single-layer kinematic assumption and cannot fully represent the thickness-wise deformation and interlaminar displacement variation in heterogeneous laminates. In contrast, the present LWST directly interpolates the displacement components through the thickness and therefore provides a more refined description of transverse shear and thickness-wise deformation. The observed behavior highlights the necessity of employing a layerwise through-thickness kinematics for moderately thick and thick laminates.

Table 1.

Natural frequency comparison between present method and FSDT for laminated cylindrical shells with different thickness ratios.

$R / h$	Mode	Present (Hz)	FSDT (Hz)	Difference (%)
60	(1,4)	573.9	576.2	0.4
30	(1,4)	733.1	744.2	1.5
10	(1,4)	1787.3	1903.5	6.5

The convergence with respect to the elastic end restraints is next investigated by varying the end spring stiffness over several orders of magnitude. For each stiffness level, the relative error to the asymptotic (fully restrained) limit is evaluated as $ϵ_{f} (k) = \frac{| f (k) - f (\infty) |}{f (\infty)} \times 100 %$ , where $f (\infty)$ is approximated by the solution at a sufficiently large stiffness. In parallel, the convergence of mode shapes is quantified by the mass-weighted MAC described in equation (33) with respect to the same asymptotic mode. Figure 5(a) and (b) correspond to the $(m, n) = (1, 1)$ mode and $(m, n) = (1, 5)$ mode. In both cases, $ϵ_{f}$ decreases and the MAC increases toward unity as $K_{spr}$ increases, showing that both the natural frequencies and mode shapes converge to stable limiting values. It is observed that all stiffness components become effectively converged at approximately $K_{spr} = 1 \times 10^{12} N / m^{3}$ , beyond which further increases produce negligible changes in either $ϵ_{f}$ or MAC. Therefore, $K_{spr} = 1 \times 10^{12} N / m^{3}$ is adopted to represent the fixed-end condition in the subsequent studies.

Figure 5.

Convergence of the proposed layerwise Fourier-finite element formulation with respect to the end spring stiffness $K_{spr}$ for (a): $(m, n) = (1, 1)$ mode and (b): $(m, n) = (1, 5)$ mode.

Importantly, the stiffness sensitivity is mode dependent. For the (1, 1) mode in Figure 5(a), the frequency is most sensitive to the radial restraint $K_{w}$ , whereas the axial restraint $K_{u}$ exhibits a moderate influence and the circumferential restraint $K_{v}$ has a negligible effect over the considered range. In contrast, for the (1, 5) mode in Figure 5(b), the axial stiffness $K_{u}$ becomes the dominant factor, while $K_{w}$ plays a secondary role and $K_{v}$ again remains weakly influential. The inset mode shapes further illustrate the gradual transition from a compliant to an effectively restrained boundary as $K_{spr}$ increases, as well as the associated changes in modal characteristics. These results demonstrate that the proposed formulation not only captures the asymptotic convergence of elastic restraints, but also reveals how different modal families respond to individual stiffness components, which is essential for reliable modelling of general elastic end constraints.

Model verification

The previous subsection was performed to identify the suitable numerical settings through convergence study. With these parameters fixed, this subsection focuses on numerical verification of the proposed semi-analytical method by comparing the predicted natural frequencies and representative mode-shape characteristics with the results reported in the open literature.

The proposed semi-analytical method for natural frequency analysis of cylindrical shells is first validated against experimental modal tests reported by Choe et al.²⁴ The selected structure is an isotropic cylindrical shell with free-free boundary conditions. The geometric and material parameters are L = 0.25 m, R = 0.05 m, h = 0.004,

E = 216 GPa

ν = 0.3

, and

ρ = 7800 kg / m^{3}

. Table 2 compares the measured natural frequencies with the present predictions, together with the results obtained using FSDT-based finite element method (FSDT-FEM). As shown in Table 2, the present method agrees well with the experimental results. Furthermore, representative mode shape comparison is provided in Table 3, where the present mode patterns are consistent with experimentally identified shapes and also match the FEM-FSDT results. Overall, both the frequency and mode shape comparisons confirm the accuracy of the proposed layerwise solution. It is also noted that the FEM-FSDT predictions exhibit slight deviations from the experimental data for certain modes, showing that the present approach provides improved agreement for the selected numerical example.

Table 2.

Comparison of natural frequencies of isotropic cylindrical shells.

Mode	Experiments (Hz)²⁴	Present (Hz)	Error (%)	FEM-FSDT (Hz)	Error (%)
(1,1)	474.73	480	1.1	481.74	1.46
(2,2)	566.55	547.11	3.55	549.2	3.16
(1,3)	1425.4	1449.5	1.66	1467.1	2.84
(2,3)	2501.1	2585.6	3.27	2678.4	6.62
(3,3)	2663.37	2691.7	1.05	2770.9	3.88

Table 3.

Comparison of mode shapes of isotropic cylindrical shells.

Mode (m, n)	Experimental²⁴	Present	FEM-FSDT
(1,2)
(2,2)

The subsequent verification example is conducted for the free vibration of an isotropic cylindrical shell in the rotating state with clamp-free (C-F) boundary conditions. Following Ref. 46, the shell geometry is L = 143 mm, R = 171.1 mm, and h = 2.2 mm. The material properties are

E = 206 GPa

ν = 0.3

, and

ρ = 7850 kg / m^{3}

. Tables 4 and 5 compare the present rotational natural frequencies and corresponding mode shapes with the results from Ref. 46, where the rotating shell eigenproblem is formulated based on Sanders’ shell theory with orthogonal-polynomial assumed mode, and solved by the state-space approach. The rotational speed is set to

Ω = 1000 rad / s

, and the axial mode is fixed at m = 1.

Table 4.

Comparison of natural frequencies of a rotational cylindrical shell.

n	Forward traveling			Backward traveling
n	Present	Ref. 46	Error (%)	Present	Ref. 46	Error (%)
3	1388.0	1371.5	1.20	1560.8	1545.3	1.00
4	1119.6	1109.6	0.90	1267.7	1251.4	1.30
5	1099.5	1083.3	1.50	1211.4	1201.8	0.80
6	1223.8	1215.3	0.70	1304.7	1316.5	0.90
7	1425.6	1441.5	1.10	1508.2	1529.6	1.40

Table 5.

Comparison of mode shapes of a rotational cylindrical shell.

n	2	3	4	5	6
Ref. 46
Present

Specifically, Table 4 compares the natural frequencies of the forward traveling and backward traveling branches for several circumferential wave numbers n. The present predictions agree closely with Ref. 46, with deviations generally within about 1.5% for both FW and BW branches, showing that the proposed model accurately captures the rotation-induced frequency splitting into FW and BW branches. In addition, Table 5 compares the corresponding mode-shape patterns. It is obvious that the present predictions exhibit the same modal characteristics and circumferential wave distributions as those in Ref. 46, providing further validation of the proposed approach for rotating cylindrical shells.

Parametric investigation

This section presents a parametric investigation of the free vibration characteristics of rotating laminated cylindrical shells using the proposed layerwise semi-analytical formulation. The cylindrical shell considered in this section adopts the same geometric dimensions and material properties as those used in the $N_{e l}$ convergence study. A symmetric unidirectional laminate, [0°/0°/0°/0°]_s is taken as the baseline stacking sequence. First, the influence of the rotation speed $Ω$ is examined to quantify the evolution of natural frequencies and the rotation-induced splitting into forward and backward traveling wave branches. Next, the influence of elastic end restraints is investigated by varying the boundary spring stiffness components, emphasizing how boundary compliance modifies both the stationary and rotating responses and providing practical guidance for selecting stiffness levels that represent limiting end conditions. Finally, the effects of laminate stacking sequence are investigated to demonstrate how tailoring the ply orientations can be exploited to tune modal characteristics and improve vibration performance within the proposed theoretical framework.

Influence of rotation speed

Figure 6(a) and (b) show the variation of natural frequencies with the rotation speed $Ω$ for m = 1 and m = 2, respectively. For each $(m, n)$ pair, the forward-traveling wave (FW) and backward-traveling wave (BW) are clearly distinguished. As $Ω$ increases, the frequencies exhibit an evident rotation-induced splitting between the FW and BW branches. This phenomenon is mainly caused by the gyroscopic coupling in the rotating shell, which introduces a direction-dependent contribution to the eigenvalue problem. As a result, the BW branch generally increases more rapidly, whereas the FW branch shows a weaker dependence on $Ω$ , leading to a progressively enlarged FW/BW separation at higher rotation speeds.

Figure 6.

The influence of rotational speed on the natural frequency of the shell with C-C boundary conditions: (a) m = 1; (b) m = 2.

It can also be observed that higher circumferential wavenumbers are more sensitive to the rotation speed. For both axial modes, the frequency shift and the FW/BW splitting are more pronounced for higher n than for lower n. From a mechanical viewpoint, higher circumferential wavenumbers correspond to shorter circumferential wavelengths and larger circumferential curvature. Therefore, the modal deformation becomes more strongly affected by hoop stress, circumferential stiffness, and Coriolis coupling. This explains why high-n traveling waves are more sensitive to rotation effects and require particular attention in the modal prediction of high-speed rotating shells.

To better illustrate the behavior at lower rotation speeds, Figure 7 provides a zoomed-in view of the low- $Ω$ range. It is obvious that even for relatively small $Ω$ , the FW/BW separation is already observable, showing that the Coriolis effect influences the modal characteristic from the onset of rotation. More importantly, the FW and BW branches behave clearly differently initial trends. For the FW branch, the frequency first decreases with increasing $Ω$ , and then turns to increase as $Ω$ becomes higher. By contrast, the BW branch increases monotonically over the entire speed interval shown.

Figure 7.

The zoomed-in diagram of natural frequency variation under low rotational speed: (a) m = 1, n = 4; (b) m = 1, n = 5; (c) m = 1, n = 6; (d) m = 2, n = 4; (e) m = 2, n = 5; (f) m = 2, n = 6.

This asymmetric behavior results from the competition between gyroscopic coupling and centrifugal stiffening. The gyroscopic matrix in equation (24) produces opposite frequency shifts for waves traveling in opposite directions: it tends to lower the FW frequency at small $Ω$ , while increasing the BW frequency. As $Ω$ further increases, the centrifugal-stress-induced stiffness in equation (25) becomes more influential and provides a positive stiffening effect for both branches. Consequently, the FW branch eventually reverses its initial decreasing trend and increases with $Ω$ , while the BW branch continues to increase. This low-to-moderate speed regime is therefore important because even modest rotation may lead to non-negligible and branch-dependent frequency shifts.

Figure 8 further presents the variation of the center frequency $f_{c}$ for m = 1 and m = 2, respectively, where $f_{c}$ represents the average frequency of the corresponding FW and BW branches. Since the center frequency removes the antisymmetric part of the gyroscopic splitting, it provides a clearer measure of the net rotation-induced stiffening effect. As observed in Figure 8(a) and (b), $f_{c}$ increases monotonically with $Ω$ for all considered modes, and growth rate tends to be higher for larger n, reflecting stronger rotation sensitivity for higher circumferential wave numbers. This confirms that the centrifugal stiffening effect becomes more significant for modes with shorter circumferential wavelengths.

Figure 8.

The center frequency variation curves: (a) m = 1; (b) m = 2.

The insets in Figure 8 also reveal mode shifting and frequency veering between neighboring circumferential branches. As the rotation speed increases, adjacent modal branches approach each other but do not cross directly; instead, they undergo an avoided crossing and exchange their relative frequency ordering. This behavior indicates strong modal interaction in the corresponding speed range. From the viewpoint of mode shapes, such veering is usually accompanied by an exchange of modal characteristics between the interacting branches. In other words, the branch that follows the lower-frequency path before the veering region may gradually acquire the deformation pattern of the neighboring branch after passing through the veering region, and vice versa. This modal-character exchange explains why simple frequency-based sorting may lead to incorrect branch assignment near veering regions. Therefore, the MAC-based mode-tracking strategy is necessary to preserve the continuity of modal branches. From an engineering perspective, such localized modal interactions should be considered carefully, because operating near a veering region may cause abrupt changes in modal characteristics and vibration sensitivity.

Influence of boundary spring stiffness

In this subsection, the influence of the elastic end restraints is investigated by varying the radial stiffness component $K_{w}$ while simultaneously sweeping the rotation speed $Ω$ . This choice is motivated by the stiffness-sensitivity results in Figure 5. For the low-order radial-dominant mode (1,1), $K_{w}$ has the most pronounced influence, while $K_{u}$ is moderate and $K_{v}$ is weak. For the higher circumferential mode, $K_{u}$ becomes dominant, $K_{w}$ remains influential, and is still weak. Therefore, $K_{w}$ is selected as a representative boundary parameter for the following parametric maps.

Figures 9 and 10 illustrate the combined effects of $K_{w}$ and $Ω$ on the BW and FW natural frequencies for m = 1, with circumferential wave number n = 2, 3, and 4 shown in subplots (a)-(c). For all cases, increasing $K_{w}$ produces a pronounced stiffening effect and raises the corresponding natural frequencies, especially in the low-to intermediate stiffness range where the boundary condition transitions from compliant to effectively restrained. Once $K_{w}$ becomes sufficiently large, the frequency surfaces gradually level off, indicating that the response approaches the clamped-end limit and further increases in stiffness have a diminishing impact. In addition, at a fixed $K_{w}$ , increasing $Ω$ leads to the expected rotation-induced evolution of BW and FW branches, and the combined $K_{w}$ - $Ω$ dependence becomes more prominent for higher circumferential wave numbers.

Figure 9.

The influence of elastic end stiffness and rotational speed on the BW natural frequencies for m = 1: (a) n = 2; (b) n = 3; (c) n = 4.

Figure 10.

The influence of elastic end stiffness and rotational speed on the FW natural frequency for m = 1: (a) n = 2; (b) n = 3; (c) n = 4.

The corresponding result for m = 2 is presented in Figures 11 and 12 for the BW and FW, respectively. Similar qualitative trends are observed: the frequencies increase monotonically with $K_{w}$ and exhibit a transition from a boundary-compliance-dominated regime at small $K_{w}$ to an asymptotic, stiffness-insensitive regime at large $K_{w}$ . Compared with m = 1, the m = 2 family generally shows stronger overall frequency levels and a clear coupled influence of $K_{w}$ and $Ω$ , indicating the increased axial deformation content and the enhanced role of end restraints for higher axial modal order.

Figure 11.

The influence of elastic end stiffness and rotational speed on the BW natural frequency for m = 2: (a) n = 2; (b) n = 3; (c) n = 4.

Figure 12.

The influence of elastic end stiffness and rotational speed on the FW natural frequency for m = 2: (a) n = 2; (b) n = 3; (c) n = 4.

In addition to the frequency maps, the influence of the end-support stiffness on the modal patterns is also listed in Table 6, where representative mode shapes are shown for three stiffness levels (

K_{w} = 10^{2}

10^{6}

, and

10^{11}

). As

K_{w}

increases from a highly compliant support to an effectively restrained end condition, the deformation field near the shell ends is progressively suppressed and the mode shapes evolve toward the fixed-end limit. For small

K_{w}

, noticeable end compliance leads to stronger boundary-layer deformation and more pronounced global bending features, where for intermediate stiffness the modal patterns exhibit a gradual transition accompanied by a redistribution of nodal lines. When

K_{w}

becomes sufficiently large, the mode shapes stabilize and the modal-line topology remains essentially unchanged, consistent with the convergence of the frequency maps and the MAC values approaching unity. This qualitative agreement between the stiffness-dependent frequency trends and the corresponding mode-shape evolution further confirms that the proposed formulation captures not only the asymptotic frequency limits but also the associated modal transitions induced by elastic end restraints.

Table 6.

Influence of the end-support stiffness on the selected mode shapes.

Mode order	$K_{w}$	100	1e6	1e11
m = 1	n = 2
	n = 3
	n = 4
m = 2	n = 2
	n = 3
	n = 4

From a mechanical viewpoint, increasing the end-spring stiffness progressively suppresses the displacement components at the shell ends, thereby increasing the effective constraint level and raising the natural frequencies. The influence is more pronounced in the low-to-intermediate stiffness range because the boundary condition changes rapidly from a compliant support to a nearly restrained state. Once the stiffness is sufficiently large, the boundary behaves close to the clamped limit, and further stiffness increase produces only negligible frequency changes. The different sensitivities to the radial, axial, and circumferential springs are mode-dependent, because each modal family contains different proportions of radial bending, axial deformation, and circumferential deformation.

Influence of stacking sequence

This subsection investigates how laminate tailoring affects the traveling-wave frequencies of the rotating cylindrical shell. Four baseline symmetric stacking sequences are considered, namely [0°/0°/0°/0°] _s, [45°/-45°/0°/90°] _s, [45°/-45°/45°/-45°] _s, and [90°/90°/90°/90°]_s. Figures 13–15 summarize the dependence of the natural frequencies on the circumferential wave number n for three representative speeds, $Ω =$ 100, 600, and 1200 rad/s, respectively. Across all rotation speeds, the stacking sequence has a pronounced influence on both the absolute frequency level and its variation with n. Both BW and FW branches show a non-monotonic trend with a shallow minimum around n≈3-4, followed by a rapid increase toward higher n. Importantly, BW and FW branches retain nearly identical n-dependent shapes for a given laminate, indicating that the stacking sequence does not alter the modal family itself, but rather affects its effective stiffness and hence the frequency level.

Figure 13.

Variation of natural frequencies with circumferential wavenumber n for different stacking sequences at $Ω =$ 100 rad/s: (a) m = 1, BW; (b) m = 1, FW; (c) m = 2, BW; (d) m = 2, FW.

Figure 14.

Variation of natural frequencies with circumferential wavenumber n for different stacking sequences at $Ω =$ 600 rad/s: (a) m = 1, BW; (b) m = 1, FW; (c) m = 2, BW; (d) m = 2, FW.

Figure 15.

Variation of natural frequencies with circumferential wavenumber n for different stacking sequences at $Ω =$ 1200 rad/s: (a) m = 1, BW; (b) m = 1, FW; (c) m = 2, BW; (d) m = 2, FW.

The ranking among laminates is strongly n-dependent. The hoop-dominant laminate [90°/90°/90°/90°] _s becomes increasingly stiff with growing n and ultimately yields the highest frequencies at large circumferential wave numbers. This behavior is consistent with high-n modes being governed by circumferential deformation, for which hoop-reinforced laminates provide the largest effective circumferential stiffness. In contrast, the axial-dominant laminate [0°/0°/0°/0°] _s generally forms a lower bound for the frequencies, especially at higher n, reflecting its comparatively reduced resistance to circumferential deformation. The quasi-isotropic [45°/-45°/0°/90°] _s and the shear-dominant [45°/-45°/45°/-45°] _s cases lie between these two extremes and may interchange their ordering depending on n and m. Notably, for lower n = 1, 2 and higher axial order m = 2, the quasi-isotropic laminate can exceed the hoop-dominant case, suggesting that the governing stiffness contributions in this regime involve coupled axial-circumferential effects rather than purely hoop-controlled response.

As $Ω$ increases from 100 to 600 rad/s shown in Figures 13 and 14, the laminate ranking is unchanged, but differences grow for the high-n branches, indicating that rotation enhances the role of directional stiffness at short circumferential wavelengths, while laminate effects remain weaker around the shallow trough of the f-n curves. At $Ω$ = 1200 rad/s shown in Figure 15, this behavior is amplified: the high-n branches show the largest laminate-induced spread, with [90°/90°/90°/90°]_s clearly dominating the high-n response and [0°/0°/0°/0°]_s remaining the lowest-frequency configuration, demonstrating that stacking sequence affects the rotating shell’s natural frequencies primarily through directional stiffness tailoring, most prominently at high n and high $Ω$ .

The stacking-sequence effect can be interpreted through the directional stiffness contribution of different ply orientations. For higher circumferential wavenumbers, the modal deformation varies more rapidly along the circumferential direction, and the response becomes more sensitive to the effective hoop stiffness. Therefore, hoop-dominant laminates tend to produce higher frequencies at large n. In contrast, axial-dominant laminates provide less resistance to circumferential deformation and usually lead to lower frequencies in the high-n range. For low circumferential wavenumbers and higher axial modes, axial-circumferential coupling becomes more relevant, which explains why the frequency ranking among different laminates may change with m, n, and $Ω$ .

Conclusions

A layerwise semi-analytical Fourier-finite element formulation has been presented for the free vibration analysis of rotating composite laminated cylindrical shells with general elastic end restraints. The method combines full layerwise shell kinematics with circumferential Fourier expansion and axial finite-element discretization, and a mass-weighted MAC-based mode tracking strategy is introduced to identify continuous modal branches under rotation. The main conclusions are as follows.

(1) The proposed formulation exhibits stable convergence with respect to the axial discretization, the through-thickness layerwise resolution, and the end-spring stiffness. The results also indicate that, although ESL-type descriptions may remain adequate for thin shells, a layerwise treatment is required for moderately thick and thick laminated shells to achieve reliable vibration predictions.

(2) The present method is validated through comparison with experimental results for stationary laminated shells and benchmark solutions for rotating cylindrical shells. Good agreement is obtained for both frequencies and mode shapes, confirming the accuracy and effectiveness of the proposed formulation.

(3) Rotation leads to distinct splitting of the natural frequencies into forward- and backward-traveling branches. With increasing rotational speed, the branch separation becomes more pronounced, and mode veering and branch interaction may occur over certain speed ranges. These features highlight the necessity of reliable mode tracking in the vibration analysis of rotating laminated cylindrical shells.

(4) Both the elastic end restraints and the stacking sequence have significant effects on the vibration characteristics. Increasing the boundary stiffness drives the frequencies toward their clamped-limit values in a mode-dependent manner, whereas the stacking sequence mainly affects the frequency level through directional stiffness tailoring. This laminate effect becomes increasingly evident at high circumferential wave numbers and high rotational speeds.

Footnotes

ORCID iD

Yuchen Chen

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by the National Natural Science Foundation of China (Grant No.52501256) and the Aeronautical Science Foundation of China (No. 2019ZF067004).

Declaration of conflicting interests

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

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Appendix

K_{U U}^{I J K L} = 2 π R \int_{x e} [ψ_{K, x} (A_{11}^{I J} ψ_{L, x} + i β A_{16}^{I J} ψ_{L}) + ψ_{K} {\bar{A}}_{55}^{I J} ψ_{L} + ψ_{K} (i β A_{16}^{I J} ψ_{L, x} + β^{2} A_{66}^{I J} ψ_{L})] d x

K_{U V}^{I J K L} = 2 π R \int_{x e} [ψ_{K, x} (i β A_{12}^{I J} ψ_{L} + A_{16}^{I J} ψ_{L, x}) + ψ_{K} ({\bar{A}}_{45}^{I J} - {\tilde{A}}_{45}^{I J} \frac{1}{R}) ψ_{L} + ψ_{K} (β^{2} A_{26}^{I J} ψ_{L} + i β A_{66}^{I J} ψ_{L, x})] d x

K_{U W}^{I J K L} = 2 π R \int_{x e} [ψ_{K, x} ({\tilde{A}}_{13}^{I J} ψ_{L, x} + A_{12}^{I J} \frac{1}{R} ψ_{L}) + ψ_{K} ({\tilde{A}}_{55}^{I J} ψ_{L, x} + i β {\tilde{A}}_{45}^{I J} ψ_{L}) + i β ψ_{K} (A_{26}^{I J} \frac{1}{R} ψ_{L} + {\tilde{A}}_{36}^{I J} ψ_{L})] d x

K_{V U}^{I J K L} = 2 π R \int_{x e} [i β ψ_{K} (A_{12}^{I J} ψ_{L, x} + i β A_{26}^{I J} ψ_{L}) + ψ_{K} {\bar{A}}_{45}^{I J} ψ_{L} - \frac{1}{R} ψ_{K} ({\tilde{A}}_{45}^{I J} ψ_{L}) + ψ_{K, x} (A_{16}^{I J} ψ_{L, x} + i β A_{66}^{I J} ψ_{L})] d x

K_{V V}^{I J K L} = 2 π R \int_{x e} [i β ψ_{K} (i β A_{22}^{I J} ψ_{L} + A_{26}^{I J} ψ_{L, x}) + ψ_{K} ({\bar{A}}_{44}^{I J} ψ_{L} - \frac{1}{R} {\tilde{A}}_{44}^{I J} ψ_{L}) - \frac{1}{R} ψ_{K} ({\tilde{A}}_{44}^{I J} ψ_{L} - \frac{1}{R} A_{44}^{I J} ψ_{L}) + ψ_{K, x} (i β A_{26}^{I J} ψ_{L} + A_{66}^{I J} ψ_{L, x})] d x

K_{V W}^{I J K L} = 2 π R \int_{x e} [i β ψ_{K} (\frac{1}{R} A_{22}^{I J} ψ_{L} + {\tilde{A}}_{23}^{I J} ψ_{L} + i β A_{26}^{I J}) + ψ_{K} ({i β \tilde{A}}_{44}^{I J} ψ_{L} + {\tilde{A}}_{45}^{I J} ψ_{L, x}) - \frac{1}{R} ψ_{K} (i β A_{44}^{I J} ψ_{L} + A_{45}^{I J} ψ_{L, x}) + ψ_{K, x} (\frac{1}{R} A_{26}^{I J} ψ_{L} + {\tilde{A}}_{36}^{I J} ψ_{L})] d x

K_{W U}^{I J K L} = 2 π R \int_{x e} [\frac{1}{R} ψ_{K} (A_{12}^{I J} ψ_{L, x} + i β A_{26}^{I J} ψ_{L}) + i β ψ_{K} {\tilde{A}}_{45}^{I J} ψ_{L} + ψ_{K, x} {\tilde{A}}_{55}^{I J} ψ_{L} + ψ_{K} ({\tilde{A}}_{13}^{I J} ψ_{L, x} + i β {\tilde{A}}_{36}^{I J} ψ_{L})] d x

K_{W V}^{I J K L} = 2 π R \int_{x e} [\frac{1}{R} ψ_{K} (i β A_{22}^{I J} ψ_{L} + A_{26}^{I J} ψ_{L, x}) + i β ψ_{K} ({\tilde{A}}_{44}^{I J} ψ_{L} - \frac{1}{R} A_{44}^{I J} ψ_{L}) + ψ_{K, x} ({\tilde{A}}_{45}^{I J} ψ_{L} - \frac{1}{R} A_{45}^{I J} ψ_{L}) + ψ_{K} (i β {\tilde{A}}_{23}^{I J} ψ_{L} + {\tilde{A}}_{36}^{I J} ψ_{L, x})] d x

K_{W W}^{I J K L} = 2 π R \int_{x e} [\frac{1}{R} ψ_{K} (A_{22}^{I J} \frac{1}{R} ψ_{L} + {\tilde{A}}_{23}^{I J} ψ_{L}) + i β ψ_{K} (i β A_{44}^{I J} ψ_{L} + A_{45}^{I J} ψ_{L, x}) + ψ_{K, x} (i β A_{45}^{I J} ψ_{L} + A_{55}^{I J} ψ_{L, x}) + ψ_{K} (\frac{1}{R} {\tilde{A}}_{23}^{I J} ψ_{L} + {\bar{A}}_{33}^{I J} ψ_{L})] d x

References

Reddy

. Mechanics of laminated composite plates and shells: theory and analysis. CRC Press, 2003.

Qatu

. Vibration of laminated shells and plates. Elsevier, 2004.

Soedel

. Vibrations of shells and plates. CRC Press, 2004.

Amabili

. Nonlinear vibrations and stability of shells and plates. Cambridge University Press, 2008.

Carrera

Brischetto

Nali

. Plates and shells for smart structures: classical and advanced theories for modeling and analysis. John Wiley & Sons, 2011, vol 36.

. Vibrations of shells and rods. Springer Science & Business Media, 2012.

Chen

Long

, et al. A modified variational approach for vibration analysis of ring-stiffened conical-cylindrical shell combinations. Eur J Mech Solid 2013; 37: 200–215. https://doi.org/10.1016/j.euromechsol.2012.06.006

Zou

Liu

. An analytical formulation for the underwater acoustic radiation of a cylindrical shell with an internal flexural floor based on the reciprocity theorem. Appl Acoust 2019; 154: 18–27. https://doi.org/10.1016/j.apacoust.2019.04.017

Kolarević

Nefovska-Danilović

. Dynamic stiffness-based free vibration study of open circular cylindrical shells. J Sound Vib 2020; 486: 115600. https://doi.org/10.1016/j.jsv.2020.115600

10.

Zhu

. Free vibration of partially fluid-filled or fluid-surrounded composite shells using the dynamic stiffness method. Acta Mech 2020; 231(9): 3961–3978. https://doi.org/10.1007/s00707-020-02734-3

11.

Talezadehlari

. Free vibration analysis of perforated composite cylindrical shell and panel using multi-domain generalized differential quadrature (GDQ) method. Compos Struct 2022; 287: 115337. https://doi.org/10.1016/j.compstruct.2022.115337

12.

Jin

, et al. A unified vibration modeling of open cylindrical shell-rectangular plate coupling structures based on the dynamic stiffness method. J Sound Vib 2023; 563: 117870. https://doi.org/10.1016/j.jsv.2023.117870

13.

Rahnama

Hamzeloo

Morad Sheikhi

. Vibration analysis of anisogrid composite lattice sandwich truncated conical shells: theoretical and experimental approaches. J Compos Mater 2024; 58(22): 2429–2442. https://doi.org/10.1177/00219983241264364

14.

Zhao

Wang

Mei

, et al. Jacobi-Ritz method for free vibration analysis of medium-thickness sandwich cylindrical shells utilizing the layerwise theory and domain decomposition method. Ocean Eng 2025; 342: 123065. https://doi.org/10.1016/j.oceaneng.2025.123065

15.

Guenanou

Houmat

Hachemi

, et al. Geometrically nonlinear free vibration analysis of shear deformable laminated composite elliptic plates using the p-version FEM. J Compos Mater 2025; 59(26): 2991–3020. https://doi.org/10.1177/00219983251340016

16.

Wang

. Free vibration of functionally graded porous cylindrical shell using a sinusoidal shear deformation theory. Aerosp Sci Technol 2017; 66: 83–91. https://doi.org/10.1016/j.ast.2017.03.003

17.

Sun

Zhu

Wang

. Free vibration analysis of a hard-coating cantilever cylindrical shell with elastic constraints. Aerosp Sci Technol 2017; 63: 232–244. https://doi.org/10.1016/j.ast.2017.01.002

18.

Ninh

Bich

. Characteristics of nonlinear vibration of nanocomposite cylindrical shells with piezoelectric actuators under thermo-mechanical loads. Aerosp Sci Technol 2018; 77: 595–609. https://doi.org/10.1016/j.ast.2018.04.008

19.

Cao

Chen

. Three-dimensional free vibrations of piezoelectric spherical shells filled with Non-Newtonian fluids. J Sound Vib 2025; 618: 119294. https://doi.org/10.1016/j.jsv.2025.119294

20.

Çetin

Kurtaran

. Large amplitude free vibration of hemp fiber-reinforced laminated composite plates using generalized differential quadrature method. J Compos Mater 2025; 59(26): 2957–2980. https://doi.org/10.1177/00219983251341619

21.

Qin

Chu

. Free vibrations of cylindrical shells with arbitrary boundary conditions: a comparison study. Int J Mech Sci 2017; 133: 91–99. https://doi.org/10.1016/j.ijmecsci.2017.08.012

22.

Jin

Jia

, et al. A general Fourier solution for the vibration analysis of composite laminated structure elements of revolution with general elastic restraints. Compos Struct 2014; 109: 150–168. https://doi.org/10.1016/j.compstruct.2013.10.052

23.

Zhang

Jin

, et al. Vibration analysis of circular cylindrical double-shell structures under general coupling and end boundary conditions. Appl Acoust 2016; 110: 176–193. https://doi.org/10.1016/j.apacoust.2016.03.038

24.

Choe

Wang

Tang

, et al. Vibration analysis for coupled composite laminated axis-symmetric doubly-curved revolution shell structures by unified Jacobi-Ritz method. Compos Struct 2018; 194: 136–157. https://doi.org/10.1016/j.compstruct.2018.03.095

25.

Pang

Chen

, et al. Vibration analysis of functionally graded porous cylindrical shell with arbitrary boundary restraints by using a semi analytical method. Compos B Eng 2019; 164: 249–264. https://doi.org/10.1016/j.compositesb.2018.11.046

26.

Zhang

Jin

Wang

, et al. Dynamic stiffness formulation and vibration analysis of coupled conical-ribbed cylindrical-conical shell structure with general boundary condition. Ocean Eng 2021; 234: 109294. https://doi.org/10.1016/j.oceaneng.2021.109294

27.

Guo

Chu

Zheng

. The influence of rotation on vibration of a thick cylindrical shell. J Sound Vib 2001; 242(3): 487–505. https://doi.org/10.1006/jsvi.2000.3356

28.

Guo

Zheng

Chu

. Vibration analysis of spinning cylindrical shells by finite element method. Int J Solid Struct 2002; 39(3): 725–739. https://doi.org/10.1016/s0020-7683(01)00031-2

29.

Liew

Zhao

, et al. Harmonic reproducing kernel particle method for free vibration analysis of rotating cylindrical shells. Comput Methods Appl Mech Eng 2002; 191(37-38): 4141–4157. https://doi.org/10.1016/s0045-7825(02)00358-4

30.

Dong

Zhu

Wang

, et al. Nonlinear free vibration of graded graphene reinforced cylindrical shells: effects of spinning motion and axial load. J Sound Vib 2018; 437: 79–96. https://doi.org/10.1016/j.jsv.2018.08.036

31.

Liu

Chu

. Nonlinear vibrations of rotating thin circular cylindrical shell. Nonlinear Dyn 2012; 67(2): 1467–1479. https://doi.org/10.1007/s11071-011-0082-7

32.

Sun

Chu

Cao

. Vibration characteristics of thin rotating cylindrical shells with various boundary conditions. J Sound Vib 2012; 331(18): 4170–4186. https://doi.org/10.1016/j.jsv.2012.04.018

33.

Dong

Liu

, et al. Semi-analytical and experimental studies on travelling wave vibrations of a moderately thick cylindrical shell subject to a spinning motion. J Sound Vib 2022; 535: 117095. https://doi.org/10.1016/j.jsv.2022.117095

34.

Tzeng

. Viscoelastic analysis of composite cylinders subjected to rotation. J Compos Mater 2002; 36(2): 229–239. https://doi.org/10.1177/0021998302036002464

35.

Shariyat

Mohammadjani

. Three-dimensional stress field analysis of rotating thick bidirectional functionally graded axisymmetric annular plates with nonuniform loads and elastic foundations. J Compos Mater 2014; 48(23): 2879–2904. https://doi.org/10.1177/0021998313503389

36.

Song

Zhai

Chen

, et al. Traveling wave analysis of rotating cross-ply laminated cylindrical shells with arbitrary boundaries conditions via Rayleigh-Ritz method. Compos Struct 2015; 133: 1101–1115. https://doi.org/10.1016/j.compstruct.2015.08.015

37.

Malekzadeh

Heydarpour

. Free vibration analysis of rotating functionally graded cylindrical shells in thermal environment. Compos Struct 2012; 94(9): 2971–2981. https://doi.org/10.1016/j.compstruct.2012.04.011

38.

Sheng

Wang

. The non-linear vibrations of rotating functionally graded cylindrical shells. Nonlinear Dyn 2017; 87(2): 1095–1109. https://doi.org/10.1007/s11071-016-3100-y

39.

Dong

Chen

, et al. Vibration characteristics of functionally graded graphene reinforced porous nanocomposite cylindrical shells with spinning motion. Compos B Eng 2018; 145: 1–13. https://doi.org/10.1016/j.compositesb.2018.03.009

40.

Qin

Pang

Safaei

, et al. Free vibration analysis of rotating functionally graded CNT reinforced composite cylindrical shells with arbitrary boundary conditions. Compos Struct 2019; 220: 847–860. https://doi.org/10.1016/j.compstruct.2019.04.046

41.

Wang

, et al. Gegenbauer-Ritz method for free vibration analysis of rotating functionally graded graphene reinforced porous composite stepped cylindrical shells with arbitrary boundary conditions. Eng Struct 2024; 303: 117555. https://doi.org/10.1016/j.engstruct.2024.117555

42.

Wang

Zhai

, et al. Analytical analysis of traveling wave vibration for rotating bi-directional dual-functionally graded carbon nanotube reinforced composite stepped thin cylindrical shells. Thin-Walled Struct 2025; 210: 112956. https://doi.org/10.1016/j.tws.2025.112956

43.

Jian

Wang

Zhai

, et al. Traveling wave vibration characteristic analysis of FRC sandwich cylindrical shells with FGP-GPLRC core under discontinuous elastic boundary conditions. Compos Struct 2024; 348: 118487. https://doi.org/10.1016/j.compstruct.2024.118487

44.

Yang

, et al. Free vibration characteristics analysis of rotating composite fiber-reinforced honeycomb cylindrical shells with arbitrary boundary conditions. Compos Struct 2025; 380: 119986. https://doi.org/10.1016/j.compstruct.2025.119986

45.

Qin

Yang

, et al. Free vibration analysis of rotating cylindrical shells coupled with moderately thick annular plates. Int J Mech Sci 2018; 142–143: 127–139. https://doi.org/10.1016/j.ijmecsci.2018.04.044

46.

Sun

, et al. Vibration characteristics analysis for rotating bolted joined cylindrical shells considering the discontinuous variable-stiffness connection. Thin-Walled Struct 2022; 177: 109422. https://doi.org/10.1016/j.tws.2022.109422

47.

Zang

Wang

, et al. Vibration analysis of hole-containing laminated composite combined structures in electric aircraft using an area scan-fitting approach. J Sound Vib 2025; 619: 119399. https://doi.org/10.1016/j.jsv.2025.119399

48.

Yan

Liu

, et al. Veering of frequency loci and nonlinear coupling vibration analysis between traveling waves of rotational hard-coated cylindrical shells. J Sound Vib 2022; 536: 117152. https://doi.org/10.1016/j.jsv.2022.117152

49.

Zhao

, et al. Nonlinear dynamic analysis of a viscoelastic rotating cantilever beam under stochastic perturbation. J Sound Vib 2025; 619: 119428. https://doi.org/10.1016/j.jsv.2025.119428

50.

Song

Chai

Wang

, et al. Vibration performance of rotating thin-walled cylindrical shell with tip-rubbing excitation between drum and stator vane segment of aero-engine. J Sound Vib 2022; 525: 116759. https://doi.org/10.1016/j.jsv.2022.116759

51.

Sun

Hao

Zhang

, et al. Traveling wave vibration and critical rotating speed of rotating porous metal conical shell with elastic boundary conditions. Aerosp Sci Technol 2024; 148: 109091. https://doi.org/10.1016/j.ast.2024.109091

52.

Wan

Liu

Cheng

, et al. Traveling wave vibration analysis of rotating reinforced functionally graded conical shells. J Mech Sci Technol 2025; 39(5): 2417–2429. https://doi.org/10.1007/s12206-025-0402-3

53.

Wang

Zhai

, et al. Modeling and traveling wave vibration analysis of rotating multi-ring hard coating double thin-walled cylindrical shells coupled structure considering connection mistuning. J Sound Vib 2025; 594: 118668. https://doi.org/10.1016/j.jsv.2024.118668

54.

Dong

Wang

, et al. Nonlinear coupled multi-mode vibrations of simply-supported cylindrical shells: Comparison studies. Commun Nonlinear Sci Numer Simul 2024; 128: 107667. https://doi.org/10.1016/j.cnsns.2023.107667

55.

Zhao

Sun

Chu

, et al. Multi-mode vibration control and optimization of smart rotating cylindrical shells subjected to circumferential moving loads. Thin-Walled Struct 2025; 211: 113094. https://doi.org/10.1016/j.tws.2025.113094

56.

Liu

Qin

Chu

. Nonlinear forced vibrations of FGM sandwich cylindrical shells with porosities on an elastic substrate. Nonlinear Dyn 2021; 104(2): 1007–1021. https://doi.org/10.1007/s11071-021-06358-7

57.

Liu

Qin

Chu

. Nonlinear forced vibrations of rotating cylindrical shells under multi-harmonic excitations in thermal environment. Nonlinear Dyn 2022; 108(4): 2977–2991. https://doi.org/10.1007/s11071-022-07449-9

58.

Liu

Qin

Chu

. Investigation of magneto-electro-thermo-mechanical loads on nonlinear forced vibrations of composite cylindrical shells. Commun Nonlinear Sci Numer Simul 2022; 107: 106146. https://doi.org/10.1016/j.cnsns.2021.106146

59.

Liu

Vella

Qin

, et al. A review of nonlinear dynamics and vibration in lightweight composite structures: recent advances and challenges. Eng Struct 2026; 360: 122835. https://doi.org/10.1016/j.engstruct.2026.122835

60.

Lam

Loy

. Analysis of rotating laminated cylindrical shells by different thin shell theories. J Sound Vib 1995; 186(1): 23–35. https://doi.org/10.1006/jsvi.1995.0431

61.

Carrera

. Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Arch Comput Methods Eng 2003; 10(3): 215–296. https://doi.org/10.1007/bf02736224

62.

Carrera

. Historical review of zig-zag theories for multilayered plates and shells. Appl Mech Rev 2003; 56(3): 287–308. https://doi.org/10.1115/1.1557614

63.

Reddy

Starnes

Jr . General buckling of stiffened circular cylindrical shells according to a layerwise theory. Comput Struct 1993; 49(4): 605–616. https://doi.org/10.1016/0045-7949(93)90065-l

64.

Kim

. Mac-based mode-tracking in structural topology optimization. Comput Struct 2000; 74(3): 375–383. https://doi.org/10.1016/s0045-7949(99)00056-5