Dynamic analysis of stepped cylindrical shells with internal bulkheads utilizing characteristic orthogonal polynomials

Abstract

This study investigates the dynamic properties of stepped cylindrical shells with internal bulkheads. A numerical analysis framework is formulated utilizing the Jacobi polynomials-Ritz method. The construction of the theoretical model incorporates a domain decomposition technique, virtual spring method, and the shear deformation theory, with the introduction of the characteristic orthogonal polynomials to delineate displacement functions. Both the free and forced vibrational behaviors of the structure are obtained by using the Ritz method. Moreover, the time-domain vibration response of the structure is ascertained utilizing the Newmark-β integral approach. This study offers a detailed examination of the influence exerted by factors including spring stiffness, truncation number, and the Jacobi parameter on the convergence of the presented method. The accuracy of the current approach is verified by juxtaposing its results with those derived from the finite element method. Additionally, the investigation explores the dynamic features of the stepped structure subject to different construction parameters, boundary conditions, and configurations of internal bulkheads, as demonstrated through a comprehensive set of numerical examples.

Keywords

Cylindrical shell bulkhead domain decomposition technique Jacobi polynomials dynamic characteristics

1. Introduction

As an exemplary lightweight structure, the cylindrical shells are extensively utilized in the maritime and aerospace industries, among others. The integration of bulkheads addresses the engineering requirements for spatial design and stability (Kim et al., 2023a, 2023b; Xie et al., 2017). Given that these structures often endure complex loads leading to potential damage, it is crucial to perform dynamic response analyses of cylindrical shells with internal bulkheads to inform structural stability design and practical engineering applications.

Significant research efforts have been directed toward understanding the vibration characteristics of solitary shells, largely due to the mathematical challenges associated with coordinating the requirements of coupled substructures. Leissa (1973) offered an exhaustive overview of the fundamental equations pertinent to thin shells and conducted an in-depth analysis of the vibration characteristics associated with cylindrical, conical, and spherical shells. Drawing on classical thin shell theory, Shu (1996) and Ng et al. (2003) analyzed the dynamic behaviors of conical shells under various edge restraints using the generalized differential quadrature method. Zhang et al. (2017) and Han et al. (2018) formulated equations for both free and forced vibrations based on Flügge shell theory, employing the wave propagation method to elaborate on the displacement functions of the cylindrical shell through Fourier series. Lee and Kwak (2015) developed a dynamic model of cylindrical shells via the Rayleigh–Ritz method and evaluated the distinctions among Sanders theory, Love theory, Reissner theory, and Flügge theory in estimating the natural frequency of structures. Xie et al. (2014, 2015) introduced the Haar wavelet approach to assess the dynamic characteristics of conical, spherical, and parabolic shells, focusing on the impacts of spring stiffness, material parameters, and structural parameters on the structure’s dynamics. Wang et al. (2018) and Shi et al. (2016) investigated the characteristics of free vibration in rotating shells under various boundary conditions employing the spectro-geometric-Ritz method. Several studies (Heydarpour et al., 2022; Malekzadeh and Heydarpour, 2012) were conducted to investigate the dynamic characteristics of composite cylindrical shells and spherical panels by using the differential quadrature method, and the effects of different material parameters, geometric parameters, and load parameters on the transient response of the structures were discussed. Qu et al. (2013b, 2013c) conducted an analysis on the dynamic attributes of stepped cylindrical and conical shells utilizing the domain decomposition technique, which allowed for the segmentation of the structure, thereby aiding in the detailed examination of high-order vibrational modes and responses. By adopting the linear plane wave assumption, Li and Hua (2009) scrutinized the dynamic response of composite cylindrical shells under step plane loads. In a study grounded in Donnell shell theory, Forouzesh and Jafari (2015) explored the dynamic behaviors of a cylindrical shell employing the Newmark integral approach to determine the responses of the structure in both time and frequency domains.

In the realm of vibration analysis for coupled structures, Yuan and Dickinson (1994) utilized the Ritz method to develop the dynamic model for cylindrical shells and plate systems, selecting orthogonal polynomials to streamline the choice of displacement functions. Subsequently, Missaoui et al. (1996) introduced a comprehensive formula to explore the dynamic response of cylindrical shells equipped with longitudinal partitions, employing virtual spring method to integrate the subsystems, and examining the impact of spring stiffness and longitudinal partitions on the structure’s overall vibrational response. Based on the variation principle, Cheng and Nicolas (1992) formulated an analytical model for the dynamic behavior of a circular cylindrical shell coupling with a circular plate, the coupling structure ensured by a continuous spring distribution. Lee et al. (2002) investigated the dynamic behaviors of interconnected cylindrical-spherical shell structure under various boundary conditions using the Rayleigh–Ritz method. Kang (2014) studied the dynamic characteristics of combined hemispherical-cylindrical shell structure with a top opening, employing the 3D dynamic equations and contrasting the findings with traditional 2D shell theory. Merz et al. (2007, 2009) employed a hybrid finite element/boundary element (FE/BE) method to examine the vibrational properties of stiffened cylindrical shells capped with finite rigid ends, incorporating bulkheads for segmenting the model. Ma et al. (2014, 2017) utilized the enhanced Fourier series approach for the dynamic analysis of various coupled structures, developing the dynamic model through the application of the variational principle. They further utilized the Ritz method for the evaluation of both free and forced vibrations of the structure. Caresta and Kessissoglou (2009, 2010) developed a simplified model of submarine with bulkhead to analyze its acoustic responses subject to harmonic force. Qu et al. (2013a, 2013d) and Wu et al. (2013) proposed an enhanced variational approach for the dynamic study of combined structures, utilizing Fourier series and Chebyshev polynomials as displacement functions, and presenting the influence of different construction parameters, boundary conditions, and load types on the vibrational behaviors.

The review of existing literature reveals a substantial focus on the free vibration behaviors of individual structures; however, there is a notable paucity of research concerning the dynamic properties of the cylindrical shell with internal bulkheads, particularly with respect to the positioning of bulkheads and variations in shell thickness. This study endeavors to bridge this research gap by delving into the dynamic characteristics of stepped cylindrical shells equipped with internal bulkheads. An analytical framework is developed employing the Jacobi polynomials-Ritz method. The robustness and precision of the approach are enhanced by integrating a domain decomposition technique and incorporating a virtual spring method. Furthermore, the Jacobi polynomials are harnessed to formulate displacement functions. This research conducts both free and forced vibration analyses employing the Ritz method and adopts the Newmark-β integral approach to evaluate the structural time variation dynamic response. The methodological accuracy is validated through comparative analysis with FEM findings, leading to an exhaustive exploration of the dynamic analysis of stepped cylindrical shells containing internal bulkheads.

2. Theoretical formulations of the dynamic model

2.1. The description of model about stepped cylindrical shell

The theoretical model is composed of stepped cylindrical shell and annular plate, as shown in Figure 1. For convenience, the cylindrical coordinate system is selected to describe the model. The mean shell radius and length of the structure are, respectively, set as R and L. According to the domain decomposition technique, the cylindrical shell is segmented to N_r sections following axil direction, and h_i represents the thickness of each cylindrical shell segment.

Figure 1.

Calculation model of the stepped cylindrical shells with internal bulkheads.

Furthermore, the boundary conditions of the structure can be simulated by applying three groups of linear springs (k_u, k_v, and k_w along the axial, circumferential, and normal directions) and two groups of rotational springs (k_x and k_θ rotate about the axial and circumferential directions) on the two ends. For adjacent segments, the strong coupling relationship can be guaranteed by connective springs.

The constitutive equations of the structure can be acquired by integrating the stresses as follows:

[\begin{array}{l} N_{x}^{i} \\ N_{θ}^{i} \\ N_{x θ}^{i} \\ M_{x}^{i} \\ M_{θ}^{i} \\ M_{x θ}^{i} \end{array}] = [\begin{array}{l} A_{11} & A_{12} & 0 & 0 & 0 & 0 \\ A_{12} & A_{22} & 0 & 0 & 0 & 0 \\ 0 & 0 & A_{66} & 0 & 0 & 0 \\ 0 & 0 & 0 & D_{11} & D_{12} & 0 \\ 0 & 0 & 0 & D_{12} & D_{22} & 0 \\ 0 & 0 & 0 & 0 & 0 & D_{66} \end{array}] [\begin{array}{l} ε_{x}^{0, i} \\ ε_{θ}^{0, i} \\ γ_{x θ}^{0, i} \\ χ_{x}^{i} \\ χ_{θ}^{i} \\ χ_{x θ}^{i} \end{array}]

(1.a)

{\begin{cases} Q_{x z}^{i} \\ Q_{θ z}^{i} \end{cases}} = \bar{κ} [\begin{array}{l} A_{66} & 0 \\ 0 & A_{66} \end{array}] [\begin{array}{l} γ_{x z}^{0, i} \\ γ_{θ z}^{0, i} \end{array}]

(1.b)

where

\bar{κ}

is the shear correction factor, and assumption that the value is 5/6 according to the previous literatures. A_ij and D_ij are the extensional and bending stiffness which can be obtained by equation (2):

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

(2)

where Q_ij are elastic constants providing as:

Q_{11} = Q_{22} = \frac{E}{1 - μ^{2}}, Q_{12} = \frac{μ E}{1 - μ^{2}}, Q_{66} = \frac{E}{2 [1 + μ]}

(3)

Based on the FSDT (Bahranifard et al., 2022, 2023; Zhong et al., 2021), the strains and displacement of the structure are expressed as follows:

ε_{x}^{i} = ε_{x}^{0, i} + z χ_{x}^{i}, ε_{θ}^{i} = ε_{θ}^{0, i} + z χ_{θ}^{i}

(4.a)

γ_{x θ}^{i} = γ_{x θ}^{0, i} + z χ_{x θ}^{0, i}, γ_{θ z}^{i} = γ_{θ z}^{0, i}, γ_{x z}^{i} = γ_{x z}^{0, i}

(4.b)

where the strains are defined as:

ε_{x}^{0, i} = \frac{\partial u_{0}^{i}}{\partial x}, ε_{θ}^{0, i} = \frac{1}{R} \frac{\partial v_{0}^{i}}{\partial θ} + \frac{w_{0}^{i}}{R}

(5.a)

γ_{x θ}^{0, i} = \frac{1}{R} \frac{\partial u_{0}^{i}}{\partial θ} + \frac{\partial v_{0}^{i}}{\partial x}, γ_{θ z}^{0, i} = \frac{1}{R} \frac{\partial w_{0}^{i}}{\partial θ} - \frac{v_{0}^{i}}{R} + ϕ_{θ}^{i}, γ_{x z}^{0, i} = \frac{\partial w_{0}^{i}}{\partial x} + ϕ_{x}^{i}

(5.b)

χ_{x}^{i} = \frac{\partial ϕ_{x}^{i}}{\partial x}, χ_{θ}^{i} = \frac{1}{R} \frac{\partial ϕ_{θ}^{i}}{\partial θ}, χ_{x θ}^{i} = \frac{1}{R} \frac{\partial ϕ_{x}^{i}}{\partial θ} + \frac{\partial ϕ_{θ}^{i}}{\partial x}

(5.c)

where the displacements of middle surface in the x, θ, and z directions are represented by u, v, and w, respectively. The ith structure strain energy is expressed as:

U_{V}^{i} = \frac{1}{2} ∭_{V} (N_{x}^{i} ε_{x}^{0, i} + N_{θ}^{i} ε_{θ}^{0, i} + N_{x θ}^{i} γ_{x θ}^{0, i} + M_{x}^{i} χ_{x}^{i} + M_{θ}^{i} χ_{θ}^{i} + M_{x θ}^{i} χ_{x θ}^{i} + Q_{x z}^{i} γ_{x z}^{0, i} + Q_{θ z}^{i} γ_{θ z}^{0, i}) R d x d θ d z

(6)

The strain energy can be obtained by substituting equations (1a) and (5a) into equation (6):

U_{V}^{i} = \frac{1}{2} \iint_{S} {\begin{cases} A_{11} {(\frac{\partial u_{0}^{i}}{\partial x})}^{2} + A_{22} {(\frac{1}{R} \frac{\partial v_{0}^{i}}{\partial θ} + \frac{w_{0}^{i}}{R})}^{2} + A_{66} {(\frac{1}{R} \frac{\partial u_{0}^{i}}{\partial θ} + \frac{\partial v_{0}^{i}}{\partial x})}^{2} \\ + \bar{κ} A_{66} {(\frac{\partial w_{0}^{i}}{\partial x} + ϕ_{x}^{i})}^{2} + 2 A_{12} (\frac{\partial u_{0}^{i}}{\partial x}) (\frac{1}{R} \frac{\partial v_{0}^{i}}{\partial θ} + \frac{w_{0}^{i}}{R}) \\ + \bar{κ} A_{66} {(\frac{1}{R} \frac{\partial w_{0}^{i}}{\partial θ} - \frac{v_{0}^{i}}{R} + ϕ_{θ}^{i})}^{2} + D_{11} {(\frac{\partial ϕ_{x}^{i}}{\partial x})}^{2} + 2 D_{12} (\frac{\partial ϕ_{x}^{i}}{\partial x}) (\frac{1}{R} \frac{\partial ϕ_{θ}^{i}}{\partial θ}) \\ + D_{22} {(\frac{1}{R} \frac{\partial ϕ_{θ}^{i}}{\partial θ})}^{2} + D_{66} {(\frac{1}{R} \frac{\partial ϕ_{x}^{i}}{\partial θ} + \frac{\partial ϕ_{θ}^{i}}{\partial x})}^{2} \end{cases}} R d x d θ

(7)

The potential energy U_b stored in the terminal boundary springs is shown as:

U_{b} = \frac{1}{2} \int_{0}^{2 π} {\begin{cases} {[k_{u, x 0} u_{0}^{2} + k_{v, x 0} v_{0}^{2} + k_{w, x 0} w_{0}^{2} + k_{x, x 0} ϕ_{x}^{2} + k_{θ, x 0} ϕ_{θ}^{2}]}_{x = 0} \\ + {[k_{u, x L} u_{0}^{2} + k_{v, x L} v_{0}^{2} + k_{w, x L} w_{0}^{2} + k_{x, x L} ϕ_{x}^{2} + k_{θ, x L} ϕ_{θ}^{2}]}_{x = L} \end{cases}} R d θ

(8)

For adjacent segments, the potential energy reserved in the connective spring can be written as:

U_{s}^{i} = \frac{1}{2} \int_{0}^{2 π} {\begin{cases} k_{u} {(u_{0}^{i} - u_{0}^{i + 1})}^{2} + k_{v} {(v_{0}^{i} - v_{0}^{i + 1})}^{2} + k_{w} {(w_{0}^{i} - w_{0}^{i + 1})}^{2} \\ + k_{x} {(ϕ_{x}^{i} - ϕ_{x}^{i + 1})}^{2} + k_{θ} {(ϕ_{θ}^{i} - ϕ_{θ}^{i + 1})}^{2} \end{cases}}_{i, i + 1} R d θ

(9)

The total potential energy of boundary springs and connective springs is expressed in equation (10):

U_{B S} = U_{b} + \sum_{i = 1}^{N_{r} - 1} U_{s}^{i}

(10)

The ith structure kinetic energy is described below:

\begin{array}{l} T^{i} = \frac{1}{2} ∭_{V} ρ_{T} [{(\frac{\partial u^{i}}{\partial t})}^{2} + {(\frac{\partial v^{i}}{\partial t})}^{2} + {(\frac{\partial w^{i}}{\partial t})}^{2}] (1 + \frac{z}{R}) R d x d θ d z \\ = \frac{1}{2} \int_{0}^{l} \int_{0}^{2 π} {\begin{cases} I_{1} [{(\frac{\partial u_{0}^{i}}{\partial t})}^{2} + {(\frac{\partial v_{0}^{i}}{\partial t})}^{2} + {(\frac{\partial w_{0}^{i}}{\partial t})}^{2}] + I_{3} [{(\frac{\partial ϕ_{x}^{i}}{\partial t})}^{2} + {(\frac{\partial ϕ_{θ}^{i}}{\partial t})}^{2}] \\ + 2 I_{2} [(\frac{\partial u_{0}^{i}}{\partial t}) (\frac{\partial ϕ_{x}^{i}}{\partial t}) + (\frac{\partial v_{0}^{i}}{\partial t}) (\frac{\partial ϕ_{θ}^{i}}{\partial t})] \end{cases}} R d x d θ \end{array}

(11)

The mass inertia terms of the structure are expressed as:

(I_{1}, I_{2}, I_{3}) = \int_{- h / 2}^{h / 2} ρ (1 + \frac{z}{R}) (1, z, z^{2}) d z

(12)

The virtual work of the ith structure done by the external concentrated force:

W^{i} = w_{i} f (t) \int_{- \infty}^{\infty} δ (x - x_{0}) d x

(13)

where w_i represents the displacement along normal direction,

f (t)

is the external mechanical force in the normal direction, and x₀ is the excitation position. The triangular pulse load is considered for the transient vibration analysis in this study, and the time-dependent load being characterized as follows:

f (t) = {\begin{cases} \frac{q_{0}}{t_{0}} * t & 0 \leq t < t_{0} \\ q_{0} (2 - \frac{1}{t_{0}} t) & t_{0} \leq t \leq 2 t_{0} \\ 0 \end{cases}

(14)

where q₀ and t₀ are the peak value and duration time, respectively.

2.2. Coupling relationship between bulkhead and cylindrical shell

The bulkhead can be represented by an annular plate in this study, which is displayed in Figure 2.

Figure 2.

Bulkhead and its coupling relationship with cylindrical shell.

The annular plate can be described by applying cylindrical coordinate system (r, θ, z); the radial, circumferential, and thickness directions of the structure are represented by r, θ, and z, respectively. In order to satisfy the continuity between annular plate segments, five sets of springs (k_u, k_v, k_w, k_r, and k_θ) are introduced at the corresponding position. The extensional strain energy and bending strain energy of the bulkhead are expressed as:

U_{S}^{i} = \frac{1}{2} \int_{0}^{R} \int_{0}^{2 π} {\begin{cases} A_{11} {(\frac{\partial u_{0}^{i}}{\partial r})}^{2} + A_{22} {(\frac{u_{0}^{i}}{r} + \frac{1}{r} \frac{\partial v_{0}^{i}}{\partial θ})}^{2} + A_{66} {(\frac{\partial v_{0}^{i}}{\partial r} + \frac{1}{r} \frac{\partial u_{0}^{i}}{\partial θ} - \frac{v_{0}^{i}}{r})}^{2} \\ + 2 A_{12} (\frac{\partial u_{0}^{i}}{\partial r}) (\frac{u_{0}^{i}}{r} + \frac{1}{r} \frac{\partial v_{0}^{i}}{\partial θ}) + \bar{κ} A_{44} {(\frac{1}{r} \frac{\partial w_{0}^{i}}{\partial θ} + ϕ_{θ}^{i})}^{2} + \bar{κ} A_{55} {(\frac{\partial w_{0}^{i}}{\partial r} + ϕ_{r}^{i})}^{2} \end{cases}} r d r d θ

(15)

U_{B}^{i} = \frac{1}{2} \int_{0}^{R} \int_{0}^{2 π} {\begin{cases} D_{66} {(\frac{\partial ϕ_{θ}^{i}}{\partial r} + \frac{1}{r} \frac{\partial ϕ_{r}^{i}}{\partial θ} - \frac{ϕ_{θ}^{i}}{r})}^{2} + 2 D_{12} (\frac{\partial ϕ_{r}^{i}}{\partial r}) (\frac{1}{r} \frac{\partial ϕ_{θ}^{i}}{\partial θ} + \frac{ϕ_{r}^{i}}{r}) \\ + D_{22} {(\frac{1}{r} \frac{\partial ϕ_{θ}^{i}}{\partial θ} + \frac{ϕ_{r}}{r})}^{2} + D_{11} {(\frac{\partial ϕ_{r}^{i}}{\partial r})}^{2} \end{cases}} r d r d θ

(16)

The kinetic energy and spring potential energy of the bulkhead can refer to the cylindrical shell. The coupling relationship of the bulkhead and cylindrical shell is shown in Figure 2. Assuming that the bulkhead is fixed at $x_{a} = L_{a}$ , the displacement relationships of the bulkhead and cylindrical shell are defined as:

u_{0, l}^{i} |_{x_{a} = L_{a}} = - w_{0, a}^{i}, v_{0, l}^{i} |_{x_{a} = L_{a}} = v_{0, a}^{i}, w_{0, l}^{i} |_{x_{a} = L_{a}} = u_{0, a}^{i}, ϕ_{x, l}^{i} |_{x_{a} = L_{a}} = ϕ_{r, a}^{i}, ϕ_{θ, l}^{i} |_{x_{a} = L_{a}} = ϕ_{θ, a}^{i}

(17)

where the subscripts a and l, respectively, indicate the bulkhead and cylindrical shell. The strong coupling condition between bulkhead and cylindrical shell can be also guaranteed by springs (k_{al_u}, k_{al_v}, k_{al_w}, k_{al_r}, and k_{al_θ}), and the coupling boundary meets the following relationship:

k_{a l_u} (u_{0, l}^{i} |_{x_{a} = L_{a}} + w_{0, a}^{i}) = N_{x}^{l} |_{x_{a} = L_{a}}

(18.a)

k_{a l_v} (v_{0, l}^{i} |_{x_{a} = L_{a}} - v_{0, a}^{i}) = N_{x θ}^{l} |_{x_{a} = L_{a}}

(18.b)

k_{a l_w} (w_{0, l}^{i} |_{x_{a} = L_{a}} - u_{0, a}^{i}) = Q_{θ}^{l} |_{x_{a} = L_{a}}

(18.c)

k_{a l_r} (ϕ_{x, l}^{i} |_{x_{a} = L_{a}} - ϕ_{r, a}^{i}) = M_{x}^{l} |_{x_{a} = L_{a}}

(18.d)

k_{a l_θ} (ϕ_{θ, l}^{i} |_{x_{a} = L_{a}} - ϕ_{θ, a}^{i}) = M_{x θ}^{l} |_{x_{a} = L_{a}}

(18.e)

The corresponding potential energy reserved in the coupling springs is:

V_{l a} = \frac{1}{2} \int_{0}^{2 π} {\begin{cases} k_{a l_u} {(u_{0, l}^{i} |_{x_{a} = L_{a}} + w_{0, a}^{i})}^{2} + k_{a l_v} {(v_{0, l}^{i} |_{x_{a} = L_{a}} - v_{0, a}^{i})}^{2} + \\ k_{a l_w} {(w_{0, l}^{i} |_{x_{a} = L_{a}} - w_{0, a}^{i})}^{2} + k_{a l_r} {(ϕ_{x, l}^{i} |_{x_{a} = L_{a}} - ϕ_{r, a}^{i})}^{2} \\ + k_{a l_θ} {(ϕ_{θ, l}^{i} |_{x_{a} = L_{a}} - ϕ_{θ, a}^{i})}^{2} \end{cases}} R d θ

(19)

The coupling energy of each bulkhead can be added to the total energy equation if there are multiple bulkheads.

2.3. Characteristic orthogonal polynomials

The Fourier series combined with characteristic orthogonal polynomials are expanded to represent the displacements of the structure. As known that, different characteristic orthogonal polynomials can be obtained by changing Jacobi polynomials parameter.

Figure 3 displays different characteristic orthogonal polynomials, indicating that displacement functions can be generalized through the application of Jacobi polynomials. The Jacobi orthogonal polynomials are formulated as follows:

P_{0}^{(α, β)} (ψ) = 1

(20.a)

P_{1}^{(α, β)} (ψ) = \frac{α + β + 2}{2} ψ - \frac{α - β}{2}

(20.b)

\begin{array}{l} P_{i}^{(α, β)} (ψ) = \frac{(α + β + 2 i - 1) {α^{2} - β^{2} + ψ (α + β + 2 i) (α + β + 2 i - 2)}}{2 i (α + β + i) (α + β + 2 i - 2)} P_{i - 1}^{(α, β)} (ψ) \\ - \frac{(α + i - 1) (β + i - 1) (α + β + 2 i)}{i (α + β + i) (α + β + 2 i - 2)} P_{i - 2}^{(α, β)} (ψ) i = 2, 3 . . . . . . \end{array}

(20.c)

where

ψ \in [- 1, 1]

α, β > - 1

Figure 3.

Characteristic orthogonal polynomials. (a) Legendre polynomials (α=β=0) and (b) Chebyshev polynomials of the first type (α=β=-1/2).

The displacement function components of the structure are expressed in equation (21a):

u_{0} (x, θ, t) = \sum_{m = 0}^{M} \sum_{n = 0}^{N} P_{m}^{(α, β)} (x) [A_{m} (t) \cos (n θ) + B_{m} (t) \sin (n θ)]

(21.a)

v_{0} (x, θ, t) = \sum_{m = 0}^{M} \sum_{n = 0}^{N} P_{m}^{(α, β)} (x) [C_{m} (t) \sin (n θ) + D_{m} (t) \cos (n θ)]

(21.b)

w_{0} (x, θ, t) = \sum_{m = 0}^{M} \sum_{n = 0}^{N} P_{m}^{(α, β)} (x) [E_{m} (t) \cos (n θ) + F_{m} (t) \sin (n θ)]

(21.c)

ϕ_{x} (x, θ, t) = \sum_{m = 0}^{M} \sum_{n = 0}^{N} P_{m}^{(α, β)} (x) [G_{m} (t) \cos (n θ) + H_{m} (t) \sin (n θ)]

(21.d)

ϕ_{θ} (x, θ, t) = \sum_{m = 0}^{M} \sum_{n = 0}^{N} P_{m}^{(α, β)} (x) [I_{m} (t) \sin (n θ) + J_{m} (t) \cos (n θ)]

(21.e)

where

A_{m}

B_{m}

C_{m}

D_{m}

E_{m}

F_{m}

G_{m}

H_{m}

I_{m}

, and

J_{m}

indicate the generalized coordinate variables; n and m represent the wave number in circumferential and axis direction. N and M denote the highest circumferential wavenumber and truncated number, respectively.

2.4. Solution procedure

The total Lagrange energy function of the structure can be expressed in equation (22):

L = \sum_{1}^{N_{r}} (W^{i} + T^{i} - U_{V}^{i}) - U_{B S}

(22)

The variational form regarding unknown coefficient is expressed as:

\begin{array}{l} \frac{\partial L}{\partial ϑ} = 0 & ϑ denotes unknown coefficients : A_{m}, B_{m}, C_{m}, D_{m}, E_{m}, F_{m}, G_{m}, H_{m}, I_{m}, J_{m} \end{array}

(23)

According to the above structural energy equation combined with the characteristic orthogonal polynomials, the corresponding dynamic equations can be written as matrix form:

(K - ω^{2} M) Q = F

(24)

where K denotes the stiffness matrices, M represents the mass matrices of the structure, Q denotes the global coordinate vector, and F represents the external excitation matrices. The time-domain response of the structure can be obtained by utilizing Newmark time integration method (Bahranifard et al., 2024), and the Rayleigh damping matrix is defined:

C = a K + b M

(25)

where a and b are denoted as follows:

a = \frac{2 w_{i} w_{j}}{w_{i} + w_{j}} ξ, b = \frac{2 ξ}{w_{i} + w_{j}}

(26)

where w_i and w_j denote the ith and jth frequency of the structure, and

ξ

denotes the damping ratio and it is defined as 0.05 in this study.

3. Comparison and verification

The boundary conditions of free, simply-supported, clamped, elastic support, and shear-diaphragm can be represented by the symbols F, S, C, E, and SD. The material properties and convergence parameters are E = 210 GPa, ρ = 7850 kg/m³, and μ = 0.3. In terms of parameters of the structure, h₁ = 0.01 m, h₁/h₂/h₃/h₄/ = 1/1/2/2, R = 0.5 m, and L = 3 m; for both ends bulkheads: h_p1 = h_p2 = 0.02 m.

3.1. Convergence study

To ascertain the validity and effectiveness of the proposed methodology, this section provides a numerical exploration of the structure. The domain decomposition technique and virtual spring method, integral to this approach, facilitate the examination of method convergence, which is influenced by factors such as spring stiffness, section number, and orthogonal polynomials parameters.

Figure 4 depicts the variation in free vibration results of the structure, considering an array of spring stiffness values. Notably, a free boundary condition is achieved when the spring stiffness is set to zero, whereas a clamped boundary condition emerges when the stiffness value lies between 10^14 and 10^15. A marked increase in natural frequencies is observed as spring stiffness escalates from 10^6 to 10^10, indicating the establishment of an elastic boundary condition. The value of the connective spring in this study is 10^15 to ensure the strong coupling relationship between bulkhead and cylindrical shell. The specific boundary restraints are cataloged in Table 1, underscoring the method’s capacity to accurately simulate various boundary scenarios through the adjustment of spring stiffness.

Figure 4.

Natural frequency with different boundary parameters (N_r = 4, α = β = 0, M = N = 8). (a) Boundary spring and (b) connective spring.

Table 1.

The general boundary restraints with the spring stiffness.

Boundary restraints	k_u (N/m)	k_v (N/m)	k_w (N/m)	k_x (Nm/rad)	k_θ (Nm/rad)
F	0	0	0	0	0
E	1^8	10^8	10^8	10^8	10^8
S	10^15	10^15	10^15	0	0
SD	0	10^15	10^15	0	10^15
C	10^15	10^15	10^15	10^15	10^15

Figure 5 displays the convergence for the truncation number of segments and orthogonal polynomials under Clamped-Clamped and Clamped-Free boundary conditions. The results exhibit rapid convergence as the truncation coefficients are increased, achieving a convergent outcome when the truncation coefficients N_r ≥ 4 and M ≥ 6; however, large convergence parameters will make the matrix order too high and reduce the computational efficiency. Therefore, in the subsequent calculation and analysis, the truncation number of segments and orthogonal polynomials are selected as: N_r = 4 and M = 8.

Figure 5.

Convergence for the free vibration of the structure (α = β = 0, N = 8).

As elucidated earlier, through the modification of the orthogonal polynomial parameters, the Jacobi orthogonal polynomial can metamorphose into either Legendre or Chebyshev polynomials. Figure 6 showcases the relative percentage errors of the structure subjected to diverse Jacobi parameters, employing α = β = 0 as the foundational benchmark for comparative analysis. Notably, the maximal relative percentage error maintains a minimal margin across various α and β values. This phenomenon highlights the versatility and widespread applicability of Jacobi orthogonal polynomials choosing as displacement functions, marking a pivotal discovery of this investigation.

Figure 6.

Absolute percentage error with different orthogonal polynomial parameters (N_r = 4, M = N = 8).

3.2. Validation study

Figures 7 and 8 present the free and forced vibration of the structure utilizing the presented method and finite element analysis software (ABAQUS). The mesh size of the FE model is set as 0.03 m, which contains 12772 nodes, 38 linear triangular elements of type S3, and 12751 linear quadrilateral elements of type S4R.

Figure 7.

Comparison of structural free vibration.

Figure 8.

FE mode and the comparison of structural forced vibration (N_r = 4, α = β = 0, M = 8). (a) FE model of the structure, (b) steady vibration, and (c) transient vibration.

From Figure 7, it can be found that whether it is vibration mode or natural frequency, the current approach is consistent with FEM when solving the modal of the structure. The thickness of the structure displays a step change in this study, resulting in a large difference in its stiffness. In the modal analysis, it can be found that the vibration mode is more likely to appear in low stiffness regions, such as n = 4, m = 1 and n = 5, m = 2.

The parameters for the triangular pulse load in Figure 8 are specified as follows: peak value q₀ = 1N, duration time t₀ = 0.005s, and Δt = 0.001s. The locations for excitation and observation within the structure are illustrated in Figure 1, with the excitation point positioned at (0.1 L, 0, R), and the observation point at (0.9 L, 0, R). Notably, the comparison reveals the overall trend of the current approach and FEM largely align. Furthermore, the findings indicate that different maximum circumferential modes have varying impacts on the structure’s dynamic response. Incorporating a broader range of circumferential modes enhances the accuracy of dynamic response predictions. However, selecting an excessively high number of modes increases computational demands. Consequently, this study opts for the highest circumferential wavenumber N = 8 to balance precision with computational efficiency.

4. Discussions

The impact of varying boundary conditions on the free vibration and dynamic response of stepped cylindrical shells with internal bulkheads is elucidated in Figure 9. It is evident that boundary conditions markedly influence the dynamic behaviors of the structure, with the sequence of increasing stiffness from lowest to highest being E-SD, C-F, S-S, and C-C. This progression suggests that strengthening the boundary conditions enhances the structural stiffness. Notably, the peak of the steady vibration response curve coincides with the structure’s natural frequencies. The resonance peak frequency diminishes as boundary conditions are lessened, whereas the count of resonance peaks incrementally rises. Furthermore, the period and amplitude of the transient vibration response diminish as boundary conditions are intensified.

Figure 9.

Dynamic characteristics of the structure with different boundary conditions (N_r = 4, α = β = 0, M = N = 8, h₁ = 0.01 m, h₁/h₂/h₃/h₄/ = 1/1/2/2, R = 0.5 m, L = 3 m, x_a1 = 0, x_a2 = L, and h_p1 = h_p2 = 0.02 m).

The analysis also reveals that the dynamic response under C-C and S-S boundary conditions exhibits a high degree of similarity. In contrast, the behavior of free vibrations under these conditions shows marked differences; specifically, under C-C conditions, vibration modes associated with the bulkhead are prominent in the third and fourth modes, whereas under S-S conditions, these modes are evident in the first and fourth modes. This observation suggests a notable alignment between the natural frequencies and vibration modes of the cylindrical shell across both sets of boundary conditions, as evidenced by the resonance peak in the steady vibration response. This indicates the significant influence of the cylindrical shell’s circumferential mode on the forced vibration response.

Figure 10 presents the dynamic characteristics of the structure under C-C boundary condition across different length-to-radius ratios, showing a decrease in overall stiffness with an increase in this ratio, leading to a general decline in natural frequencies, leftward shifts of resonance peaks, and increased period and amplitude of the transient vibration response.

Figure 10.

Dynamic characteristics of the structure with different length-to-radius ratio (N_r = 4, α = β = 0, M = N = 8, h₁ = 0.01 m, h₁/h₂/h₃/h₄/ = 1/1/2/2, R = 0.5 m, x_a1 = 0, x_a2 = L, and h_p1 = h_p2 = 0.02 m).

Figure 11 depicts the first five vibration modes of the structure at varying length-to-radius ratios, holding the radius constant. It is demonstrated that bulkhead vibration modes predominantly emerge when the structure’s length-to-radius ratio is minimal. As this ratio increases, the stiffness of the cylindrical shell lessens, facilitating the appearance of cylindrical shell vibration modes more prominently.

Figure 11.

The first five modes of the structure with different length-to-radius ratio.

Keeping the total mass of the cylindrical shell equal, Figure 12 delineates the influence of varying thickness distributions on the free vibration and dynamic response of a structure under C-C boundary condition. The findings reveal marked differences in dynamic characteristics attributable to changes in thickness distribution. Specifically, alterations in distribution form result in a crisscross pattern of natural frequency variations, highlighting the significant impact of step thickness distribution on the structure’s free vibration behavior. Despite modifications being limited to the thickness of the cylindrical shell, a common bulkhead vibration mode persists across different thickness distributions.

Figure 12.

Dynamic characteristics of the structure with different thickness distribution (N_r = 4, α = β = 0, M = N = 8, h₁ = 0.01 m, R = 0.5 m, L = 3 m, x_a1 = 0, x_a2 = L, and h_p1 = h_p2 = 0.02 m).

For the forced vibration response, with the excitation and examination points located on the first and fourth steps, respectively. It is noted that the vibration response amplitude is notably lower when the excitation point or the examination point is on a thicker step. These observations suggest that strategic structural design can effectively alter the dynamic characteristics of a structure in practical engineering applications, facilitating modal avoidance and vibration control through impedance variation across different thickness distributions.

Figures 13–14 compare the dynamic characteristics of structures with varying numbers and positions of bulkheads under C-C boundary condition. When bulkheads are positioned at both ends of the structure, their impact on the dynamic characteristics of the structure is minimal, with both free and forced vibration responses remaining essentially unchanged. Conversely, relocating bulkheads from the ends toward the center significantly elevates the natural frequency, enhances structural stiffness, reduces the amplitude of vibration response, and is possible to diversify the vibration modes of the structure.

Figure 13.

Dynamic characteristics of the structure with different number and position of bulkheads (N_r = 4, α = β = 0, M = N = 8, h₁ = 0.01 m, h₁/h₂/h₃/h₄/ = 1/1/2/2, R = 0.5 m, L = 3 m, and h_p1 = h_p2 = 0.02 m).

Figure 14.

Modes of the structure with different number and position of bulkheads.

Moreover, vibration modes are predominantly observed within the cylindrical shell segment flanked by two bulkheads, indicating that the emergence of the vibration modes in regions of lower stiffness. An increase in overall structural stiffness is observed when bulkhead placement is judiciously considered, and the incorporation of bulkheads effectively elevates the structural natural frequency, underscoring their importance in dynamic characteristic modulation.

5. Conclusions

In this article, an analytical model through utilizing the Jacobi polynomials-Ritz method is applied to explore the dynamic characteristics of stepped cylindrical shells with internal bulkheads. The convergence and efficacy of the method are enhanced by the incorporation of the FSDT, domain decomposition technique, and virtual spring method, with the adoption of Jacobi orthogonal polynomials expanding the repertoire of displacement function possibilities. This study further evaluates dynamic behaviors of revolution structures via the Ritz method, while the time-domain vibration response of the structure is derived by employing the Newmark-β integral approach. The veracity and dependability of the numerical model are verified by juxtaposing its results with those obtained from the FEM, thus providing critical insights into vibration control in stepped configurations. The key conclusions are as follows:

i. The proposed method demonstrates strong convergence when the truncation coefficients are set as: number of segments H = 4 and the highest wave number along the circumferential and axial directions M = 8 and N = 8, respectively.

ii. Boundary conditions and geometric dimensions significantly influence the dynamic characteristics of the structure, with the overall stiffness increasing as the length-to-radius ratio decreases and boundary conditions are strengthened.

iii. Strategic structural design can markedly alter the dynamic characteristics for practical engineering applications, enabling modal avoidance and vibration control through the differential impedance of varying thickness distributions.

iv. Vibration modes are predominantly observed within the cylindrical shell segment sandwiched between two bulkheads, indicating the manifestation of structural vibration modes in areas of reduced stiffness. Optimally situating bulkheads enhances the overall stiffness of the structure and effectively raises its natural frequency.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was funded by Postdoctoral Fellowship Program of CPSF (GZC20233427), Heilongjiang Postdoctoral Fund (LBH-Z23111), National Natural Science Foundation of China (52101351 and U2006229) and Fundamental Research Funds for the Central Universities (3072024XX0102).

ORCID iDs

Cong Gao

Haichao Li

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