Free and Forced Vibration Characteristics of a Composite Stiffened Plate Based on Energy Method

Abstract

The composite stiffened plate has garnered significant attention in the lightweight design of ship structures due to its superior mechanical properties. Although it reduces the structural weight, it also increases vibration sensitivity. Thus, investigating the vibration characteristics of the composite stiffened plate is crucial. This paper proposed a theoretical model based on the energy principle and the improved Fourier series method (IFSM) to analyze the composite stiffened plate vibration characteristics. The model demonstrates high reliability and accuracy, as confirmed through convergence analysis and comparison with experimental results from the published literature. Results indicate that geometry and material parameters significantly impact the natural frequency and can mitigate vibration responses by increasing thickness and stiffness. Additionally, ply design parameters markedly influence the vibration of composite stiffened plates, enhancing low-frequency vibration performance through optimal ply design. The structural parameters of the stiffeners, particularly the height and number of stiffeners, play a crucial role, enhancing stiffness and reducing the vibration responses of the composite stiffened plates.

1. Introduction

With rapid social progress and deepening globalization, governments are focusing on energy consumption and environmental protection. Specific requirements for energy conservation and emission reduction have been proposed with the aim of addressing climate change and promoting the realization of sustainable development. Shipping is one of the main sources of energy consumption and emissions in the transportation industry, thus presenting new requirements and challenges for the ship’s structural design. Lightweight structural design can effectively control the weight and gravity of the ship structure, which can not only improve navigational performance, but also reduce energy consumption and emissions. The Swedish government launched the LASS project and the LASS-C project [1,2], which focus on the development and application of composite materials in ship structural design. Composite materials are used in the structural design of ship superstructures such as a high-speed craft, a Ro-Pax ferry, and a cruise ship. The project investigates the application of composite materials in modern ship engineering from the technical, economic feasibility, and safety perspectives. The application of composite materials in ship engineering reduces the weight of hull structures, effectively lowers energy consumption, and enhances operational efficiency, providing new directions for future ship design and construction.

The composite stiffened plate combines the advantages of stiffened plate and composite material, offering significant advantages in lightweight ship design by meeting structural loading requirements while reducing the structural weight. However, the structural weight reduction typically heightens the sensitivity to structural vibration, and the use of composite materials introduces complexity to the vibration characteristics. Extensive research has been conducted on stiffened plate vibration, including the finite difference method [3], the spline composite strip method [4], the finite element method (FEM) [5], and the Rayleigh–Ritz method [6]. Peng et al. [7] developed a meshless theoretical model based on the first-order shear deformation theory (FSDT) to analyze the stiffened plate bulking and free vibration characteristics. Cho et al. [8,9,10,11,12] developed the VAPS software based on the assumed modal method to simulate the free and forced vibrations of the stiffened plate and explore the impact of different topologies, attachments, and arbitrary boundary conditions. Qin et al. [13] proposed an analytical model to investigate the static and dynamic characteristics of curvilinearly stiffened plates. Zhang and Lin [14] investigated Mindlin stiffened plate vibration characteristics based on the modal expansion method. Sahoo [15,16] applied the FEM to study the stiffened plate free vibration and the dynamic response under moving loads. Liu [17] proposed a theoretical model to examine the free vibration in stiffened plates with arbitrary cross-section beams. Qin et al. [18] established a meshless theoretical analytical model to analyze the circular stiffened plate structure bending and free vibration. Zhang et al. [19] presented an analytical solution for the rectangular orthogonal stiffened plate vibration response. Liu et al. [20] introduced a dynamic analysis method for stiffened plates under moving loads utilizing the FSDT along with Chebyshev polynomials for spatial discretization and penalty functions for various boundary conditions. Ko and Boo [21] proposed a condensed finite element matrix using the iterated improved reduced system (IIRS) method to address modal, frequency, and transient responses of stiffened plates. Guo et al. [22] investigated the concentric stiffened plate free vibration using a two-dimensional spectral Chebyshev technique. Shen et al. [23] proposed a meshless approach for analyzing the stiffened plate free and forced vibration and conducted structural vibration experiments. Wang [24] explored the free vibration of rectangular stiffened plates with cutouts. Gao et al. [25] developed a semi-analytical approach based on the domain decomposition method (DDM) to study the dynamic behavior of the stiffened plate under various conditions, complemented by experimental validations. Chen et al. [26] conducted an experimental study on the stiffened plate vibro-acoustic characteristics under various welding conditions.

Composite stiffened plate vibration characteristics have attracted extensive research due to their special structure and complex mechanical properties. Rikards et al. [27] explored the buckling and vibration characteristics of composite stiffened plates using the FEM. Ahmadian et al. [28] applied the super elements method to analyze the free vibration characteristics of the laminated stiffened plate with eccentric stiffeners. Qing et al. [29] developed a semi-analytical solution model for the laminate stiffened plate free vibration. Nayak and Bandyopadhyay [30] utilized the FEM to study the free vibration of doubly curved stiffened laminated shells. Prusty and Ray [31,32] analyzed the free vibration and buckling characteristics of composite stiffened plates and shells with various cross-section stiffeners using the FEM. Thinh and Quoc [33] used the FEM and experimental methods to study the composite stiffened plate buckling and vibration with various cross-section stiffeners. Mejdi and Atalla [34] proposed a semi-analytical model based on the modal expansion technique for the vibro-acoustic analysis of the composite stiffened plate. Damnjanović et al. [35,36] introduced a dynamic stiffness approach utilizing the FSDT and the HSDT (higher-order shear deformation theory) to analyze the stiffened and cracked laminated plate free vibration. Zhao and Kapania [37] developed an effective FEM to analyze the pre-stress vibration of curvilinearly stiffened composite plates under in-plane loads. Sinha et al. [38,39,40] conducted numerical and experimental studies on the free vibration of glass–fiber laminated composite stiffened plates. Chandra et al. [41,42] investigated the damping and structural dynamic responses of composite stiffened plates in a thermal environment. Wang et al. [43] formulated an analytical model for the structural dynamic characteristics of stiffened composite sandwich plates embedded with multilayer viscoelastic damping membranes and validated this model through a comparison with FEM results. Peng et al. [44] introduced a meshless method based on the FSDT to investigate the static and free vibration characteristics of the composite stiffened plate with varying parameters.

The improved Fourier series method (IFSM) is widely used in the vibration analysis of beams, plates, and shells [45,46,47,48,49] due to its advantages in handling boundary conditions and achieving efficient computational convergence. Ye et al. [50,51] developed a unified model using IFSM for the free vibration of laminated composite shallow shells and moderately thick laminated plates under various boundary conditions. Zhang et al. [52] derived a series solution for orthogonal rectangular plate in-plane vibration with elastic constraints using IFSM. Wang et al. [53,54,55,56] applied IFSM to model the vibrations of laminate plates and shells, conducting comprehensive research to explore the effects of material parameters, shell types, and boundary conditions on structural dynamic characteristics. Shi et al. [57] investigated the free and forced vibration of moderately thick laminated plates under multi-point support boundary conditions through IFSM. Zhang et al. [58] studied the moderately thick laminated plate free vibration under non-uniform boundary conditions using IFSM. Cao [59] used IFSM to establish a theoretical model of rectangular stiffened plates with various numbers and lengths of stiffeners, analyzing the dynamic characteristics. Du et al. [60] formulated theoretical models to analyze the stiffened plate free and forced vibration based on IFSM and conducted experiments to validate the accuracy.

Through the above research, it can be found that the researchers focus on the isotropic stiffened plate and composite laminated shell bulking and free vibration characteristics. The study of the composite stiffened plate forced vibration response is also very important. This paper develops a theoretical model based on the FSDT, Timoshenko beam theory, and IFSM to analyze the free and forced vibration of the composite stiffened plate. Numerical examples demonstrate that the theoretical method presented in this paper has good reliability and accuracy. Utilizing this method, the parametric analysis of the composite stiffened plate is conducted, including geometry parameters, material parameters, ply parameters, stiffener parameters, etc. It aims to understand the vibration mechanisms of the composite stiffened plate and provide theoretical support for low-vibration design.

2. Theoretical Formulations

In this paper, the composite stiffened plate is taken as the research object, establishing the theoretical model based on the FSDT and Timoshenko beam theory. The composite stiffened plate is composed of a × b × h_p laminate panel and h_s × b_s laminate stiffener. A global coordinate system o-xyz is established based on the mid-plane xoy, and a local coordinate system o′-x′y′z′ is established based on the neutral axis o′x′, as shown in Figure 1 and Figure 2. u_p, v_p and w_p represent the displacements of the composite stiffened plate along the x-, y-, and z-axes, respectively, while ${\phi }_x$ and ${\phi }_y$ are the rotation around the y- and x-axes. θ is the angle between the kth ply direction and the x-axis. Z_k+1 and Z_k are the coordinates of the upper and lower surface of the kth ply along the z-axis. The artificial virtual springs uniformly distributed along each side of the composite stiffened plate simulate the boundary conditions. The stiffness values of the linear support springs (k_u, k_v, and k_w) and the twist springs (K_x and K_y) are adjusted to simulate various boundary conditions.

2.1. Kinematics and Stress–Strain Relations

2.1.1. Kinematics and Stress–Strain Relations of Laminate Panel

According to the FSDT, the displacement of the laminate panel is

$$\begin{array}{l}U\left(x,y,z,t\right)=u_p\left(x,y,t\right)+z{\phi }_x\left(x,y,t\right)\\ V\left(x,y,z,t\right)=v_p\left(x,y,t\right)+z{\phi }_y\left(x,y,t\right)\\ W\left(x,y,z,t\right)=w_p\left(x,y,t\right)\end{array}$$

(1)

where u_p, v_p, and w_p are the laminate panel displacements along the x-, y-, and z-axes, respectively. ${\phi }_x$ and ${\phi }_y$ are the rotations around the y- and x-axes. t is the time variable.

The liner strain–displacement relationship for the kth ply of the laminate panel is

$$\left[\begin{array}{c}{\epsilon }_{xx}\\ {\epsilon }_{yy}\\ {\gamma }_{xy}\end{array}\right]=\left[\begin{array}{c}{\epsilon }_{xx}^0\\ {\epsilon }_{yy}^0\\ {\gamma }_{xy}^0\end{array}\right]+z\left[\begin{array}{c}{\chi }_{xx}\\ {\chi }_{yy}\\ {\chi }_{xy}\end{array}\right]$$

(2)

$$\left[\begin{array}{c}{\gamma }_{xz}\\ {\gamma }_{yz}\end{array}\right]=\left[\begin{array}{c}{\gamma }_{xz}^0\\ {\gamma }_{yz}^0\end{array}\right]$$

(3)

where ${\epsilon }_{xx}$, ${\epsilon }_{yy}$, and ${\gamma }_{xy}$ are the in-plane normal strain components of the laminate panel, while ${\gamma }_{xz}$ and ${\gamma }_{yz}$ are the shear strain components along the panel thickness direction. ${\epsilon }_{xx}^0$, ${\epsilon }_{yy}^0$, and ${\gamma }_{xy}^0$ represent the membrane strains in the mid-plane. ${\chi }_{xx}$, ${\chi }_{yy}$, and ${\chi }_{xy}$ represent the changes in the curvature in the mid-plane. ${\gamma }_{xz}^0$ and ${\gamma }_{yz}^0$ are the shear strains in the thickness direction. The relationship expressions are

$${\left[\begin{array}{ccc}{\epsilon }_{xx}^0& {\epsilon }_{yy}^0& {\gamma }_{xy}^0\end{array}\right]}^{\top }={\left[\begin{array}{ccc}\frac{\partial u_p}{\partial x}& \frac{\partial v_p}{\partial y}& \frac{\partial u_p}{\partial y}+\frac{\partial v_p}{\partial x}\end{array}\right]}^{\top }$$

(4)

$${\left[\begin{array}{ccc}{\chi }_{xx}& {\chi }_{yy}& {\chi }_{xy}\end{array}\right]}^{\top }={\left[\begin{array}{ccc}\frac{\partial {\phi }_x}{\partial x}& \frac{\partial {\phi }_y}{\partial y}& \frac{\partial {\phi }_x}{\partial y}+\frac{\partial {\phi }_y}{\partial x}\end{array}\right]}^{\top }$$

(5)

$${\left[\begin{array}{cc}{\gamma }_{xz}^0& {\gamma }_{yz}^0\end{array}\right]}^{\top }={\left[\begin{array}{cc}\frac{\partial w_p}{\partial x}+{\phi }_x& \frac{\partial w_p}{\partial y}+{\phi }_y\end{array}\right]}^{\top }$$

(6)

According to Hooke’s Law, the stress–strain relationship for the kth ply of the laminate panel is

$$\left[\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\tau }_{xy}\\ {\tau }_{xz}\\ {\tau }_{yz}\end{array}\right]=\left[\begin{array}{ccccc}{\overline{Q}}_{11}^k& {\overline{Q}}_{12}^k& {\overline{Q}}_{16}^k& 0& 0\\ {\overline{Q}}_{12}^k& {\overline{Q}}_{22}^k& {\overline{Q}}_{26}^k& 0& 0\\ {\overline{Q}}_{16}^k& {\overline{Q}}_{26}^k& {\overline{Q}}_{66}^k& 0& 0\\ 0& 0& 0& {\overline{Q}}_{55}^k& {\overline{Q}}_{45}^k\\ 0& 0& 0& {\overline{Q}}_{45}^k& {\overline{Q}}_{44}^k\end{array}\right]\left[\begin{array}{c}{\epsilon }_{xx}\\ {\epsilon }_{yy}\\ {\gamma }_{xy}\\ {\gamma }_{xz}\\ {\gamma }_{yz}\end{array}\right]$$

(7)

where ${\sigma }_{xx}$ and ${\sigma }_{yy}$ represent the normal stresses in the x and y directions. ${\tau }_{yz}$, ${\tau }_{xz}$, and ${\tau }_{xy}$ represent the shear stresses in x, y, and z directions.

The relationship between stiffness coefficients ${\overline{Q}}_{ij}^k\left(i,j=1,2,4,5,6\right)$ and material coefficients $Q_{ij}^k\left(i,j=1,2,4,5,6\right)$ for kth ply is

$$\left[\begin{array}{l}\overline{Q_{11}^k}\\ \overline{Q_{12}^k}\\ \overline{Q_{16}^k}\\ \overline{Q_{22}^k}\\ \overline{Q_{26}^k}\\ \overline{Q_{66}^k}\end{array}\right]=\left[\begin{array}{cccc}{m}^4& 2m^2{n}^2& n^4& 4m^2{n}^2\\ m^2{n}^2& \left(m^4+n^4\right)& m^2{n}^2& -4m^2{n}^2\\ m^{3}n& mn\left(n^2-m^2\right)& -m{n}^3& 2mn\left(n^2-m^2\right)\\ n^4& 2m^2{n}^2& m^4& 4m^2{n}^2\\ m{n}^3& mn\left(m^2-n^2\right)& -m^{3}n& 2mn\left(m^2-n^2\right)\\ m^2{n}^2& -2m^2{n}^2& m^2{n}^2& {\left(m^2-n^2\right)}^2\end{array}\right]\left[\begin{array}{c}{Q}_{11}^k\\ Q_{12}^k\\ Q_{22}^k\\ Q_{66}^k\end{array}\right]$$

(8)

$$\left[\begin{array}{c}\overline{Q_{44}^k}\\ \overline{Q_{45}^k}\\ \overline{Q_{55}^k}\end{array}\right]=\left[\begin{array}{cc}{m}^2& n^2\\ -mn& mn\\ n^2& m^2\end{array}\right]\left[\begin{array}{c}{Q}_{44}^k\\ Q_{55}^k\end{array}\right]$$

(9)

where $m=\cos \theta$ and $n=\sin \theta$. $\theta$ is the angle between the kth ply direction and the x-axis.

The kth ply material coefficients $Q_{ij}^k\left(i,j=1,2,4,5,6\right)$ are

$$\begin{array}{l}{Q}_{11}^k=\frac{E_1}{1-{\mu }_{12}{\mu }_{21}},Q_{12}^k={\mu }_{21}{Q}_{11}^k,Q_{22}^k=\frac{E_2}{1-{\mu }_{12}{\mu }_{21}},\\ Q_{44}^k=G_{23},Q_{55}^k=G_{13},Q_{66}^k=G_{12}\end{array}$$

(10)

where E₁ and E₂ are the longitudinal and transverse moduli. μ₁₂ is the major Poisson’s ratio, μ₂₁ = μ₁₂E₁/E₂. G₁₂, G₁₃, and G₂₃ are the shear moduli. In the case of isotropic materials, the elastic modulus is E = E₁ = E₂, and the shear modulus is G = G₁₂ = G₁₃ = G₂₃ = E/2(1 + μ).

Integrating along the thickness direction, the internal forces and bending moments in the laminate panel are

$$\left[\begin{array}{c}{N}_x\\ N_y\\ N_{xy}\end{array}\right]={\displaystyle \sum _{k=1}^N{\displaystyle {\int }_{z_k}^{z_{k+1}}\left[\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\tau }_{xy}\end{array}\right]}d}z$$

(11)

$$\left[\begin{array}{c}{M}_x\\ M_y\\ M_{xy}\end{array}\right]={\displaystyle \sum _{k=1}^N{\displaystyle {\int }_{z_k}^{z_{k+1}}\left[\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\tau }_{xy}\end{array}\right]}zdz}$$

(12)

$$\left[\begin{array}{c}{Q}_{xz}\\ Q_{yz}\end{array}\right]=\kappa {\displaystyle \sum _{k=1}^N{\displaystyle {\int }_{z_k}^{z_{k+1}}\left[\begin{array}{c}{\gamma }_{xz}\\ {\gamma }_{yz}\end{array}\right]}}dz$$

(13)

where N_x, N_y, and N_xy represent the in-plane force vectors. M_x, M_y, and M_xy represent the bending moment vectors. Q_x and Q_y represent the shear force vectors. κ = 5/6. Substituting Equation (7) into Equations (11)–(13), the relationship between the internal forces and strain components of the laminate panel is

$$\left[\begin{array}{c}{N}_x\\ N_y\\ N_{xy}\\ M_x\\ \begin{array}{l}{M}_y\\ M_{xy}\end{array}\end{array}\right]=\left[\begin{array}{cccccc}{A}_{11}& A_{12}& A_{16}& B_{11}& B_{12}& B_{16}\\ A_{12}& A_{22}& A_{26}& B_{12}& B_{22}& B_{26}\\ A_{16}& A_{26}& A_{66}& B_{16}& B_{26}& B_{66}\\ B_{11}& B_{12}& B_{16}& D_{11}& D_{12}& D_{16}\\ B_{12}& B_{22}& B_{26}& D_{12}& D_{22}& D_{26}\\ B_{16}& B_{26}& B_{66}& D_{16}& D_{26}& D_{66}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{xx}^0\\ {\epsilon }_{yy}^0\\ {\gamma }_{xy}^0\\ {\chi }_{xx}\\ \begin{array}{l}{\chi }_{yy}\\ {\chi }_{xy}\end{array}\end{array}\right]$$

(14)

$$\left[\begin{array}{c}{Q}_{xz}\\ Q_{yz}\end{array}\right]=\kappa \left[\begin{array}{cc}{A}_{55}& A_{45}\\ A_{45}& A_{44}\end{array}\right]\left[\begin{array}{c}{\gamma }_{xz}\\ {\gamma }_{yz}\end{array}\right]$$

(15)

where A_ij, B_ij, and D_ij are the tensile stiffness, tensile-bending stiffness, and bending stiffness, respectively,

$$\begin{array}{l}\begin{array}{cc}{A}_{ij}={\displaystyle \sum _{k=1}^N{\overline{Q}}_{ij}^k}\left(Z_{k+1}-Z_k\right)& \left(i,j=1,2,4,5,6\right)\end{array}\\ \begin{array}{cc}{B}_{ij}=\frac{1}{2}{\displaystyle \sum _{k=1}^N{\overline{Q}}_{ij}^k}\left(Z_{k+1}^2-Z_k^2\right)& \left(i,j=1,2,6\right)\end{array}\\ \begin{array}{cc}{D}_{ij}=\frac{1}{3}{\displaystyle \sum _{k=1}^N{\overline{Q}}_{ij}^k}\left(Z_{k+1}^3-Z_k^3\right)& \left(i,j=1,2,6\right)\end{array}\end{array}$$

(16)

2.1.2. Kinematics and Stress–Strain Relations of Laminate Stiffener

According to the Timoshenko beam theory, the laminate stiffener displacement is

$$\begin{array}{l}{U}_s\left(x^{\prime },z^{\prime },t\right)=u_s\left(x^{\prime },t\right)+z^{\prime }{\varphi }_{x^{\prime }}\left(x^{\prime },t\right)\\ W_s\left(x^{\prime },z^{\prime },t\right)=w_s\left(x^{\prime },t\right)-y^{\prime }{\varphi }_{y^{\prime }}\left(x^{\prime },t\right)\end{array}$$

(17)

where u_s and w_s represent the laminate stiffener displacements along the x’- and z’-axes. ${\varphi }_{x^{\prime }}$ and ${\varphi }_{y^{\prime }}$ are the rotations around the y’- and x’-axes.

The strain–displacement relationship for the laminate stiffener is

$$\left[\begin{array}{c}{\epsilon }_{x^{\prime }}\\ {\gamma }_{x^{\prime }{z}^{\prime }}\end{array}\right]=\left[\begin{array}{c}\frac{\partial u_s}{\partial x^{\prime }}+z^{\prime }\frac{\partial {\varphi }_{x^{\prime }}}{\partial x^{\prime }}\\ \frac{\partial w_s}{\partial x^{\prime }}-y^{\prime }\frac{\partial {\varphi }_{y^{\prime }}}{\partial x^{\prime }}+{\varphi }_{x^{\prime }}\end{array}\right]$$

(18)

The stress–strain relationship for the laminate stiffener is

$$\left[\begin{array}{c}{\sigma }_{x^{\prime }}\\ {\tau }_{x^{\prime }{z}^{\prime }}\end{array}\right]=\left[\begin{array}{c}{\tilde{Q}}_{11}^k{\epsilon }_{x^{\prime }}\\ {\tilde{Q}}_{55}^k{\gamma }_{x^{\prime }{z}^{\prime }}\end{array}\right]$$

(19)

where

$${\tilde{Q}}_{11}^k={\overline{Q}}_{11}^k-{\left[\begin{array}{c}{\overline{Q}}_{12}^k\\ {\overline{Q}}_{16}^k\end{array}\right]}^{\top }{\left[\begin{array}{cc}{\overline{Q}}_{22}^k& {\overline{Q}}_{26}^k\\ {\overline{Q}}_{26}^k& {\overline{Q}}_{66}^k\end{array}\right]}^{-1}\left[\begin{array}{c}{\overline{Q}}_{12}^k\\ {\overline{Q}}_{16}^k\end{array}\right]$$

(20)

$${\tilde{Q}}_{55}^k={\overline{Q}}_{55}^k-\frac{{\left({\overline{Q}}_{45}^k\right)}^2}{{\overline{Q}}_{44}^k}$$

(21)

when the stiffener is made of isotropic material, ${\tilde{Q}}_{11}^k=E$, and ${\tilde{Q}}_{55}^k=G=E/2(1+\mu )$.

Considering the eccentricity of the stiffener, the displacement continuity condition is satisfied between the laminate panel and the laminate stiffener. The laminate stiffener displacement can be expressed in terms of the laminate panel.

$$\left[\begin{array}{c}{u}_s\\ w_s\\ {\varphi }_{x^{\prime }}\\ {\varphi }_{y^{\prime }}\end{array}\right]=\left[\begin{array}{ccccc}\cos \alpha & \sin \alpha & 0& -e\cos \alpha & -e\sin \alpha \\ 0& 0& 1& 0& 0\\ 0& 0& 0& \cos \alpha & \sin \alpha \\ 0& 0& 0& \sin \alpha & \cos \alpha \end{array}\right]\left[\begin{array}{c}{u}_p\\ v_p\\ w_p\\ {\phi }_x\\ {\phi }_y\end{array}\right]$$

(22)

where α is the angle between the stiffener direction and the x-axis. e is the distance between the laminate panel mid-plane and the stiffener neutral axis, defined as e = (h_p + h_s)/2.

2.2. Admissible Displacement Functions

When the derivative of the displacement function represented by the traditional Fourier series is an odd function, discontinuities or jumps will occur at the end-point of the interval [−L, L], as shown in Figure 3a. The improved Fourier series method introduces auxiliary functions to the traditional Fourier series to avoid the phenomenon of discontinuities or jumps and to improve the poor convergence of the series [60].

The displacement functions of the composite stiffened plate can be expressed as

$$u_0\left(x,y,t\right)={\displaystyle \sum _{m=-2}^{\infty }{\displaystyle \sum _{n=-2}^{\infty }{A}_{mn}{\xi }_{am}\left(x\right){\xi }_{bn}\left(y\right)}}$$

(23)

$$v_0\left(x,y,t\right)={\displaystyle \sum _{m=-2}^{\infty }{\displaystyle \sum _{n=-2}^{\infty }{B}_{mn}{\xi }_{am}\left(x\right){\xi }_{bn}\left(y\right)}}$$

(24)

$$w_0\left(x,y,t\right)={\displaystyle \sum _{m=-2}^{\infty }{\displaystyle \sum _{n=-2}^{\infty }{C}_{mn}{\xi }_{am}(x){\xi }_{bn}(y)}}$$

(25)

$${\phi }_x\left(x,y,t\right)={\displaystyle \sum _{m=-2}^{\infty }{\displaystyle \sum _{n=-2}^{\infty }{D}_{mn}{\xi }_{am}\left(x\right){\xi }_{bn}\left(y\right)}}$$

(26)

$${\phi }_y\left(x,y,t\right)={\displaystyle \sum _{m=-2}^{\infty }{\displaystyle \sum _{n=-2}^{\infty }{E}_{mn}{\xi }_{am}\left(x\right){\xi }_{bn}\left(y\right)}}$$

(27)

where ${\xi }_{am}(x)$ and ${\xi }_{bn}(y)$ are

$${\xi }_{am}\left(x\right)=\left\{\begin{array}{l}\cos {\lambda }_{am}x\begin{array}{cc},& m\ge 0\end{array}\\ \sin {\lambda }_{am}x\begin{array}{cc},& m<0\end{array}\end{array}\right.$$

(28)

$${\xi }_{bn}\left(y\right)=\left\{\begin{array}{l}\cos {\lambda }_{bn}y\begin{array}{cc},& n\ge 0\end{array}\\ \sin {\lambda }_{bn}y\begin{array}{cc},& n<0\end{array}\end{array}\right.$$

(29)

where ${\lambda }_{am}=m\pi /a$ and ${\lambda }_{bn}=n\pi /b$. ${\xi }_{am}(x)$ and ${\xi }_{bn}(y)$ (m, n = −2, −1) are the auxiliary functions, and the derivatives satisfy the following initial conditions at the boundaries.

$${\xi }_{am}\left(0\right)={\xi }_{am}\left(a\right)={\xi }_{am}^{\prime }\left(a\right)=0,{\xi }_{am}^{\prime }\left(0\right)=1$$

(30)

$${\xi }_{bn}\left(0\right)={\xi }_{bn}\left(b\right)={\xi }_{bn}^{\prime }\left(b\right)=0,{\xi }_{bn}^{\prime }\left(0\right)=1$$

(31)

2.3. Governing Equation and Solution

Utilizing the energy principle, the Lagrange energy functional L for the composite stiffened plate is

$$L=T_p+T_s-U_p-U_s-V_{sp}-W$$

(32)

where U_p and U_s are the potential energy of the laminate panel and the laminate stiffener. T_p and T_s are the kinetic energy of the laminate panel and the laminate stiffener. V_sp is the elastic potential energy of the boundary spring. W is the work done by the external force.

The laminate panel potential energy U_p and kinetic energy T_p are

$$U_p=\frac{1}{2}{\displaystyle {\int }_0^a{\displaystyle {\int }_0^b\left(\begin{array}{l}{N}_x{\epsilon }_{xx}^0+N_y{\epsilon }_{yy}^0+N_{xy}{\epsilon }_{xy}^0+M_x{\chi }_{xx}+\\ M_y{\chi }_{yy}+M_{xy}{\chi }_{xy}+Q_{xz}{\gamma }_{xz}+Q_{yz}{\gamma }_{yz}\end{array}\right)dxdy}}$$

(33)

$$T_p=\frac{1}{2}{\displaystyle \iiint \left({\dot{U}}^2+{\dot{V}}^2+{\dot{W}}^2\right)dV}$$

(34)

The laminate stiffener potential energy U_s and kinetic energy T_s are

$$U_s=\frac{1}{2}{\displaystyle {\int }_0^L{\displaystyle {\int }_S\left({\sigma }_{x^{\prime }}{\epsilon }_{x^{\prime }}+{\tau }_{x^{\prime }{z}^{\prime }}{\gamma }_{x^{\prime }{z}^{\prime }}\right)dSd{x}^{\prime }}}$$

(35)

$$T_s=\frac{1}{2}\rho {\displaystyle {\int }_0^L{\displaystyle {\int }_S\left({\dot{U}}_s^2+{\dot{W}}_s^2\right)dSd{x}^{\prime }}}$$

(36)

The boundary spring elastic potential energy V_sp is

$$\begin{gathered} V_{sp}=\frac{1}{2}{\displaystyle {\int }_0^b{\displaystyle {\int }_{-h/2}^{h/2}\left\{\begin{array}{l}{\left[k_{x_0}^u{u}_0^2+k_{x_0}^v{v}_0^2+k_{x_0}^w{w}_0^2+K_{x_0}^x{\phi }_x^2+K_{x_0}^y{\phi }_y^2\right]}_{x=0}+\\ {\left[k_{x_a}^u{u}_0^2+k_{x_a}^v{v}_0^2+k_{x_a}^w{w}_0^2+K_{x_a}^x{\phi }_x^2+K_{x_a}^y{\phi }_y^2\right]}_{x=a}\end{array}\right\}dydz}} \\ +\frac{1}{2}{\displaystyle {\int }_0^a{\displaystyle {\int }_{-h/2}^{h/2}\left\{\begin{array}{l}{\left[k_{y_0}^u{u}_0^2+k_{y_0}^v{v}_0^2+k_{y_0}^w{w}_0^2+K_{y_0}^x{\phi }_x^2+K_{y_0}^y{\phi }_y^2\right]}_{y=0}+\\ {\left[k_{y_b}^u{u}_0^2+k_{y_b}^v{v}_0^2+k_{y_b}^w{w}_0^2+K_{y_b}^x{\phi }_x^2+K_{y_b}^y{\phi }_y^2\right]}_{y=b}\end{array}\right\}dxdz}} \end{gathered}$$

(37)

W is the work done by the external force.

$$W={\displaystyle {\iint }_S\left(f_u{u}_0+f_v{v}_0+f_w{w}_0+f_{{\phi }_x}{\phi }_x+f_{{\phi }_y}{\phi }_y\right)dxdy}$$

(38)

As per Rayleigh–Ritz method, the composite stiffened plate energy functional L is partialized by unknown Fourier coefficients G.

$$\begin{array}{cc}\frac{\partial L}{\partial G}=0,& G={\left[A_{mn},B_{mn},C_{mn},D_{mn},E_{mn}\right]}^{\top }\end{array}$$

(39)

The governing equation for the composite stiffened plate is

$$\left(K-{\omega }^{2}M\right)G=F$$

(40)

where F = 0; setting m = M and n = N, Equation (41) transforms into a set of linear Equation (40) with 5 × (M + 1) × (N + 1) + 10 × (M + N + 2) unknown coefficients. The natural frequencies and mode shapes of the composite stiffened plate can be obtained by solving the eigenvalues and eigenvectors of Equation (41).

$$\left(K-{\omega }^{2}M\right)G=0$$

(41)

3. Numerical Results and Discussions

In this section, a numerical procedure is developed using MATLAB 2020 based on the theoretical model in the previous section. Several numerical examples are conducted to demonstrate that the method in this paper exhibits good convergence, accuracy, and reliability. Initially, the convergence of the method is investigated, and a suitable spring stiffness value is selected. Subsequently, the accuracy of the method is verified through comparisons with the FEM results and experimental data from the published literature. Finally, the impact of various design parameters on the vibration characteristics of the composite stiffened plate is discussed.

3.1. Convergency Analysis

This section investigates the convergence of the numerical procedure developed in this paper. The laminate panel dimensions are a/b = 1 and h_p/b = 0.01, and the laminate stiffener are h_s/h_p = 1 and b_s/b = 0.01. The longitudinal laminate stiffener, aligned parallel to the y-axis, is positioned at x = a/2. The material parameters are E₂ = 10 GPa, E₁ = 20 E₂, G₁₂ = 0.5 E₂, G₁₃ = 0.5 E₂, G₂₃ = 0.33 E₂, μ₁₂ = 0.25, and ρ = 1500 kg/m³. The ply schemes of the laminate panel and laminate stiffener are [0/90] and [0]₁₀, respectively.

Table 1. Convergence of the first 8 dimensionless frequency parameters $\mathrm{\Omega }$ for a fully free composite stiffened plate.

M × N	Mode
	1	2	3	4	5	6	7	8
8 × 8	4.797	12.910	14.782	16.028	16.139	28.417	35.808	38.122
10 × 10	4.776	12.909	14.769	16.015	16.127	28.369	35.793	38.100
12 × 12	4.767	12.908	14.762	16.009	16.121	28.334	35.786	38.090
14 × 14	4.763	12.908	14.755	16.005	16.117	28.318	35.782	38.084
16 × 16	4.760	12.908	14.751	16.002	16.115	28.299	35.781	38.079
18 × 18	4.758	12.908	14.746	16.000	16.112	28.291	35.780	38.076
20 × 20	4.756	12.908	14.743	15.998	16.111	28.277	35.779	38.074
FEM	4.753	12.901	14.571	15.982	16.102	28.028	35.767	38.052

presents the first eight dimensionless frequency parameters of the composite stiffened plate under fully free boundary conditions. The composite stiffened plate dimensionless frequency parameter is $\mathrm{\Omega }=\omega b^2/h_p\sqrt{\rho /E_2}$. Table 1 reveals that the frequency parameters gradually converge as the mode truncation number increases, indicating that the method exhibits good convergence and accuracy. Considering both computational efficiency and accuracy, the mode truncation number for subsequent numerical calculations is M = N = 16.

In this paper, artificial virtual spring stiffness is adjusted to simulate the composite stiffened plate boundary conditions. For the free boundary condition, the spring stiffness value is set to 0. For the clamped boundary condition, the spring stiffness value is ideally set to $\infty$, but it cannot be achieved in numerical calculations. Therefore, it is necessary to select suitable values for the spring stiffness K (k_u, k_v, k_w, K_x, and K_y) to simulate the clamped boundary condition.

Figure 4 illustrates the first three dimensionless frequency parameters of the composite stiffened plate for varying spring stiffness K. The geometrical and material parameters of the composite stiffened plate depicted in Figure 4 are consistent with Table 1. The boundary conditions of the composite stiffened plate are FCFE, i.e., x = 0 and x = a are the free boundaries (F), y = 0 is the clamped support boundary (C), and y = b is the elastic support boundary (E). The spring stiffness varies from K₀ (0) to K₁₅ (10¹⁵). According to Figure 4, when the spring stiffness value is below K₅, the frequency parameter is minimally influenced by the spring stiffness. As the spring stiffness value increases from K₅ to K₁₀, the frequency parameter rises rapidly, indicating strengthened boundary constraints. Within this range, the boundary condition is considered as elastic support. Above K₁₀, the frequency parameter remains essentially unchanged, which can be regarded as clamped support. In this study, the spring stiffness is set to K₁₃ to simulate the clamped support.

Table 2. Spring stiffness values K for classic boundary condition at x = 0.

Boundary Condition	ku	kv	kw	Kx	Ky
F	0	0	0	0	0
C	1013	1013	1013	1013	1013
S	1013	1013	1013	0	0

lists the spring stiffness values K for classic boundary conditions at x = 0.

3.2. Method Verification

In this section, the theoretical results are compared with the FEM results and the experimental results published in the literature to verify the accuracy of the proposed method.

3.2.1. Composite Stiffened Plate Vibration Characteristics Verification

Rikards [27] explored the free vibration of the composite stiffened plate with a central stiffener under fully clamped boundary conditions. The dimensions are a = 0.25 m, b = 0.5 m, h_p = 0.00104 m, h_s = 0.0105 m, and b_s = 0.00364 m. The central stiffener, positioned at x = 1/2 a, is parallel to the y-axis. The material parameters are E₁ = 128 GPa, E₂ = 11 GPa, G₁₂ = 4.48 GPa, G₁₃ = 4.48 GPa, G₂₃ = 1.53 GPa, μ₁₂ = 0.25, and ρ = 1500 kg/m³. The ply schemes for the laminate panel and laminate stiffener are [0/ ± 45/90]_s and [0₇/90₇]_s, respectively. Each ply thickness is 0.13 mm. The FEM model of the composite stiffened plate is established using Patran. For the finite element analysis (FEA) of the composite stiffened plate, the upper frequency limit is set at 1000 Hz. Based on the bending wavelength formulation, the maximum element size should not exceed 0.02 m. Both the laminate panel and the laminate stiffener are modeled using shell elements, with a maximum element size of 0.0125 m, and the number of total meshes is 840. Nastran is used to analyze the composite stiffened plate free and forced vibration characteristics.

Table 3. Composite stiffened plate natural frequency (Hz).

Mode	1	2	3	4	5
Present	209.24	221.38	258.63	311.01	341.27
FEM	212.83	221.25	269.40	309.71	352.99
Rikards [27]	215.00	235.50	274.50	315.40	361.40

presents the first five natural frequencies of the composite stiffened plate. The average difference between the theoretical results and the FEM results is 2.15% and the maximum error does not exceed 5%. Figure 5 shows the composite stiffened plate vibration displacement response subjected to single-point excitation. The excitation position is (a/2, b/2) and the excitation amplitude is 1 N. It can be found that the theoretical results and the FEM results of displacement response are in good agreement.

3.2.2. Steel Stiffened Plate Vibration Characteristics Verification

GAO [25] conducted an experimental study on the steel stiffened plate free and forced vibration. The dimensions of the steel stiffened plate are a = 0.5 m, b = 0.5 m, h_p = 0.008 m, h_s = 0.01 m, and b_s = 0.01 m. The stiffeners are arranged at x = 1/4 a and x = 3/4 a, parallel to the y-axis. The material parameters are E = 210 GPa, μ = 0.3, and ρ = 7800 kg/m³. The boundary condition is FCFC, i.e., x = 0 and x = a are the free boundaries, and y = 0 and y = b are the clamped support boundaries. Patran is used to create the FEM model for the steel stiffened plate. The calculation upper frequency is 1000 Hz. According to the bending wavelength formulation, the maximum element size must not exceed 0.046 m. The panel and stiffeners are modeled using shell elements and beam elements, respectively, with a maximum element size of 0.0125 m. The total number of shell elements is 1600, and there are 80 beam elements. Nastran is employed to analyze the steel stiffened plate vibration characteristics. GAO [25] performed a modal experiment using the multi-point excitation and single-point pickup techniques. Both sides of the steel stiffened plate are fixed to simulate the clamped boundary conditions. An acceleration sensor is arranged at the center of the steel stiffened plate. A force hammer is used to knock the excitation points one by one, and the natural frequencies and the mode shapes are obtained through frequency analysis. Furthermore, GAO also conducted a forced vibration experiment of the steel stiffened plate under single-point excitation. A vibration exciter, which induces a linear sweep excitation at an amplitude of 1 N, is positioned beneath the center of the stiffened plate and is connected to the stiffened plate through a force sensor. The force sensor is used to record the excitation information from the vibration exciter. Acceleration sensors, arranged at the excitation point (a/2, b/2) and the observation point (a/2, b/5), are used to measure the vibration acceleration response through the data acquisition system post-processing. Figure 6 shows the experiment arrangements of the vibration exciter and sensors on the steel stiffened plate.

Table 4. Steel stiffened plate natural frequency.

Mode	1	2	3
Present	220.17 Hz	240.23 Hz	349.71 Hz
FEM	220.12 Hz	240.11 Hz	350.14 Hz
Exp [22]	230.40 Hz	247.70 Hz	353.30 Hz

displays the first three modes of the theoretical, FEM, and experimental results and shows that the mode shapes are consistent. The maximum discrepancy between the theoretical and experimental results is less than 5%. Figure 7 illustrates the steel stiffened plate vibration acceleration levels of the excitation and observation points. From Figure 7, it is obvious that the theoretical curves are in good agreement with the FEM curves. Both theoretical and experimental curves show similar trends, with relatively minor deviations at single peaks. It demonstrates that the theoretical method presented in this paper effectively analyzes the stiffened plate free and forced vibration.

3.3. Parameter Analysis

The previous section demonstrated that the theoretical method presented in this paper has good reliability and accuracy in studying the composite stiffened plate vibration characteristics. This section explores the impact of the geometric, material, and laminate ply parameters on the free and forced vibration characteristics of the composite stiffened plate. In this section, a composite stiffened plate with two stiffeners is taken as the research object. The stiffeners are located at x = 1/4 a and x = 3/4 a, parallel to the y-axis. Unless specially stated, the laminate material parameters are E₂ = 10 GPa, E₁ = 20 E₂, G₁₂ = 0.6 E₂, G₁₃ = 0.6 E₂, G₂₃ = 0.5 E₂, μ₁₂ = 0.25, and ρ = 1600 kg/m³, while the isotropic parameters are E = 210 GPa, μ = 0.3, and ρ = 7850 kg/m³. The boundary conditions are simply supported.

3.3.1. Free Vibration of Composite Stiffened Plate

Table 5. The first 8 dimensionless frequency parameters of the composite stiffened plate with various geometric parameters.

a/b	hp/b	Mode
		1	2	3	4	5	6	7	8
1	0.004	17.296	31.251	42.998	57.475	71.400	78.258	88.508	94.805
	0.006	14.577	28.325	38.064	56.861	60.123	69.609	75.925	94.564
	0.008	13.615	27.424	34.629	52.222	56.719	67.693	71.524	94.325
	0.01	13.175	27.040	32.466	48.139	56.638	65.901	69.545	89.673
1.5	0.004	13.223	20.751	29.578	30.920	44.150	54.949	58.010	58.462
	0.006	11.280	17.429	28.178	29.853	44.048	46.452	50.476	55.188
	0.008	10.503	16.289	27.788	28.777	40.523	43.978	45.990	55.001
	0.01	10.130	15.773	27.621	27.876	37.190	43.657	43.907	54.865
2	0.004	11.232	16.889	20.444	26.700	26.852	39.138	40.386	42.911
	0.006	10.079	13.843	18.637	26.527	26.632	38.294	39.700	39.830
	0.008	9.526	12.740	18.112	26.159	26.595	34.993	37.099	39.110
	0.01	9.244	12.228	17.891	25.795	26.562	32.619	35.246	38.991

presents the first eight dimensionless frequency parameters of the composite stiffened plate with various geometric parameters. The laminate panel aspect ratio a/b is 1, 1.5, and 2, and the laminate panel thickness–length ratio h_p/b is 0.004, 0.006, 0.008, and 0.01. The stiffener dimension is h_s = 0.01 b and b_s = 0.01 b. The ply schemes of the laminate panel and stiffeners are [0/90] and [0]₁₀, respectively. According to Table 5, it is evident that the composite stiffened plate frequency parameter decreases as the aspect ratio and panel thickness increase.

Table 6. The dimensionless frequency parameters of the composite stiffened plate with different anisotropic ratios.

a/b	E1/E2	Mode
		1	2	3	4	5	6	7	8
1	5	11.005	21.798	30.566	40.533	46.054	57.599	59.312	74.392
	10	13.444	26.549	36.823	47.556	57.003	70.504	71.136	89.122
	20	17.052	33.525	45.820	57.779	73.389	87.699	88.245	109.186
	30	19.890	38.964	52.879	65.977	85.950	100.363	101.743	125.247
1.5	5	8.270	13.830	22.373	24.690	34.583	35.041	42.004	45.955
	10	9.977	16.818	27.771	30.030	40.716	42.419	52.952	56.415
	20	12.520	21.295	35.374	37.495	50.173	53.201	68.495	72.111
	30	14.532	24.845	41.186	43.210	57.842	61.758	80.176	84.535
2	5	7.032	10.745	15.353	21.279	23.259	29.550	30.584	32.940
	10	8.454	12.948	19.239	26.428	28.708	34.794	37.643	40.713
	20	10.575	16.277	24.778	33.885	36.167	42.497	48.374	52.104
	30	12.258	18.936	29.035	39.443	42.127	48.664	56.875	60.918

presents the dimensionless frequency parameters of the composite stiffened plate with different anisotropic ratios. The dimensions are a/b = 1, 1.5, 2, h_p = 0.01 b, h_s = 0.01 b, and b_s = 0.01 b. The laminate panel material anisotropic ratios are 5, 10, 20, and 30. The laminate panel ply scheme is [45/−45]_s, while the stiffener material is made from isotropic material. As indicated in Table 6, both the stiffness and frequency parameters of the composite stiffened plate increase as the anisotropic ratio increases.

During designing the laminated composite structure, the ply scheme, ply angle, and ply number are the key design parameters. Reasonable ply design solutions can meet the special functional requirements of the actual project. Figure 8 illustrates the variation curves of the fundamental frequency parameters of the composite stiffened plates with different ply angles, which vary from 0° to 180° in 5° increments. The dimensions are a/b = 1, 1.5, 2, h_p/b = 0.01, h_s = 0.01 b, and b_s = 0.01 b. The ply schemes of the laminate panel are [θ]₄, [0/θ]₂, [0/θ]_s, and [θ/0]_s. The ply scheme of the laminate stiffener is [0]₁₀. According to Figure 8, the fundamental frequency parameter is symmetrical with respect to 90° across different geometries and ply schemes. The subsequent study on the ply angle can be discussed in the range of 0° to 90°. For the square composite stiffened plates, the fundamental frequency parameter initially increases and then decreases with the change in the ply angle. For the rectangular composite stiffened panels, the fundamental frequency parameter increases gradually. As the plate thickness increases, the trend of the fundamental frequency parameter with the changing ply angle remains consistent, although the value of the frequency parameter decreases. Figure 9 explores the impact of the ply number on the composite stiffened plate frequency parameter. The ply schemes are [0/90]_n, [30/−30]_n, [45/−45]_n, and [75/−75]_n. The thickness and material of each layer is the same. The results from Figure 9 indicate that the frequency parameter increases as the ply numbers increase. However, once the ply number reaches a certain level, the frequency parameter tends to stabilize and remain essentially unchanged.

Stiffener structural parameters are crucial for the design of stiffened plate vibrations. Figure 10 shows the natural frequency variation in the composite stiffened plate with different stiffener heights. The dimensions are b = 0.5 m, a = 2 b, h_p = 0.01 b, b_s = 0.01 b, and h_s/h_p = 2, 4, 6, 8, 10. The ply schemes of the laminate panel and laminate stiffener are [0/90]₁₀ and [0]₁₀. According to Figure 10, the natural frequency of the composite stiffened plate increases as the stiffener height increases. Figure 11 investigates the influences of stiffener space on the natural frequency of the composite stiffened plate. The dimensions are b = 0.5 m, a = 2b, h_p = 0.01 b, h_s = 2 h_p, and b_s = 0.01 b. The stiffener space c is 0.3 m, 0.4 m, 0.5 m, and 0.6 m. As depicted in Figure 11, the fundamental frequency of the composite stiffened plate decreases as the stiffener space increases. Figure 12 explores the impact of the stiffener numbers on the composite stiffened plate natural frequency. The number of stiffeners is 2, 4, 6, and 8. The results, as shown in Figure 12, indicate that the frequency parameter increases with the stiffener numbers.

3.3.2. Forced Vibration of Composite Stiffened Plate

This section studies the vibration displacement response of the composite stiffened plate under single-point excitation, utilizing the theoretical method presented in this paper. The excitation position is (a/2, b/2) and the excitation amplitude is 1 N.

Figure 13 displays the displacement response curves of the composite stiffened plate with varying thicknesses. The dimensions are b = 0.5 m, a = 2 b, h_p/b = 0.004, 0.006, 0.008, 0.01, h_s = 0.01 b, and b_s = 0.01 b. The laminate ply schemes are consistent with Table 5. According to Figure 13, as the laminate panel thickness increases, the displacement response curve shifts to the higher frequency region, and the resonant peak value decreases. As the laminate panel thickness increases, the natural frequency and stiffness of the composite stiffened plate also increase.

Figure 14 illustrates the displacement response of the composite stiffened plate with different anisotropic ratios. The dimensions are b = 0.5 m, a = 2 b, h_p = 0.01 b, h_s = 0.01 b, and b_s = 0.01 b. The material and ply scheme parameters are consistent with Table 6. According to Figure 14, as the anisotropic ratio increases, the resonance peak of the displacement response shifts to a higher frequency, and its amplitude decreases. It is attributed to the improved structural stiffness of the composite stiffened plate, which leads to an increase in the natural frequency.

Figure 15 investigates the influence of ply angle on the displacement response of the composite stiffened plate. The dimensions are b = 0.5 m, a = 2 b, h_p = 0.01 b, h_s = 0.01 b, and b_s = 0.01 b. The ply scheme for the laminate panel is [θ]₄, with the ply angles varying from 0° to 90° in 15° increments. According to Figure 15, as the ply angle increases, the first-order resonance frequency shifts to a higher frequency region and the corresponding amplitude decreases. The increase in the ply angle significantly affects the higher-order modes, and there is uncertainty in the displacement response curve of the composite stiffened plate. Figure 16 presents the effect of the ply number on the vibration displacement response of the composite stiffened plate. The ply schemes are [0/90], [0/90]₂, [0/90]₃, [0/90]₅, [0/90]₁₀, and [0/90]₂₀. As illustrated in Figure 16, the displacement response curve shifts to a higher frequency region as the ply number increases. However, as the ply number increases to a certain level, the displacement response curves gradually converge and coincide.

Figure 17 illustrates the influence of the stiffener height on the displacement response of the composite stiffened plate. As the stiffener height increases, the displacement response curve shifts toward the higher frequency region. However, increasing the stiffener height does not uniformly reduce the structural vibration displacement response across all frequencies. This is because the higher stiffener increases the overall stiffness of the structure while changing the local mass distribution. Figure 18 shows the vibration displacement response of the composite stiffened plate with different stiffener spaces. It shows that as the stiffener space increases, the first resonance peak moves to a lower frequency region, while the first resonance peak value increases. The stiffener space leads to a change in the local stiffness and mass distribution, which has a complex influence on the composite stiffened plate vibration, with different performances at different frequency regions. Figure 19 analyzes the impact of varying stiffener numbers on the displacement response of the composite stiffened plate. As the number of stiffeners increases, the displacement response of the composite stiffened plate decreases, and the resonance peak shifts to a higher frequency region.

4. Conclusions

In this paper, a unified theoretical model for the composite stiffened plate is established utilizing the FSDT and Timoshenko beam theory. According to the energy principle and Rayleigh–Ritz method, the governing equation of the composite stiffened plate is derived and solved. The displacement functions are constructed using the improved Fourier series, which offers advantages in addressing boundary condition issues. The theoretical model is applied to explore the impact of various parameters including geometric dimensions, material properties, laminate ply design, and stiffener characteristics on the vibration behavior of the composite stiffened plate. The conclusions are as follows:

• The theoretical model developed in this paper is suitable for studying the free and forced vibration characteristics of isotropic and composite stiffened plates with good convergence efficiency and calculation accuracy.
• Increasing both the thickness and material stiffness of the laminate panel enhances the overall stiffness and natural frequency of the composite stiffened plate, subsequently reducing the vibration response.
• In designing the laminate ply scheme, increasing the ply angle can reduce the first-order resonance peak and enhance the low-order vibration response of a rectangular composite stiffened plate. Additionally, the vibration response of the composite stiffened plate can be minimized by increasing the ply number. However, as the ply number increases to a certain level, the vibration behavior of the composite stiffened plate remains unchanged.
• The natural frequency of the composite stiffened plate increases, and the vibration response peak shifts to a higher frequency region as the stiffener height increases. Conversely, with an increase in the stiffener space, the fundamental frequency of the composite stiffened plate decreases, while the amplitude of the first-order resonance peak increases. Additionally, as the number of stiffeners increases, the composite stiffened plate natural frequency increases, and the vibration response decreases.

In summary, the theoretical method presented in this paper demonstrates broad applicability and effectively predicts the free and forced vibration of composite stiffened plates. The parametric analysis confirms that optimal structural design can significantly mitigate vibration issues of composite stiffened plates. This study provides theoretical guidance for the low-vibration design and application of composite stiffened plates in ships.