A semi-analytical method for vibration analysis of polygonal plates based on Jacobi-Ritz method

Abstract

Aiming at the vibration analysis of irregular shape plates, a semi-analytical method for dynamic vibration analysis of arbitrary polygonal plate structures under elastic boundary conditions is proposed based on the Jacobi-Ritz method. Based on the Kirchhoff thin plate theory, the vibration theoretical model of polygonal plate is established. Aiming at the difficult problem caused by the complex integral region involved in the energy functional of polygonal plate system, the simplified form of the energy functional equation of polygonal plate system is derived by introducing the divergence theorem. The displacement tolerance function of the polygonal plate is constructed by using the Jacobi orthogonal polynomial, and the complex boundary conditions of the polygonal plate are simulated by the penalty parameter method. A vibration analysis method of the polygonal plate based on the Jacobi-Ritz method is proposed. Finally, the effectiveness of the proposed method is verified by comparing with the simulation results and the reference data.

Keywords

Jacobi-Ritz method free vibration polygonal plate general boundary condition

Introduction

As the most basic structural unit in engineering, the vibration problem of plate structure is a classic subject in the study of plate and shell structure dynamics. Domestic and foreign scholars have carried out a lot of fruitful research on the vibration of plate structure, and put forward analytical method, semi-analytical method, and numerical method. The analytical method is the most accurate method for solving the vibration of plate structures. Its basic connotation is to directly solve the control equations through strict mathematical derivation, and to obtain accurate analytical solutions. Representative methods include variable separation method (Senjanović et al., 2013; Xing et al., 2022; Xing and Liu, 2009a, 2009b; Xing and Xu, 2013), superposition method (Gorman, 1978; 2004a; 2004b, 2005, 2006), and Fourier series method (Leng et al., 2022; Li et al., 2009, 2024a; Du et al., 2007; Du et al., 2011; Zhang and Li, 2009). For example, Navier used the double Fourier series method to expand the bending deflection of the plate into a double sine series, and solved the exact solution of the bending vibration of the rectangular plate with four edges simply supported. Levy used the single Fourier series method to solve the bending vibration of two rectangular plates with two opposite edges simply supported, which made a great contribution to the study of the bending vibration of thin plates. The advantage of the analytical method is that the results are accurate and can directly reflect the relationship between vibration characteristics and parameters. However, its universality is not strong, and it is only suitable for plate structures with regular geometric shapes and simple boundary constraints, which is difficult to meet the actual engineering requirements.

The semi-analytical method (Gao et al., 2024; Li et al., 2025; Tang et al., 2024) combines the characteristics of analytical method and numerical method. Its basic connotation is to transform the partial differential equation of the system into ordinary differential equation by introducing trial function and energy principle, which takes into account the accuracy of analytical method and the efficiency of numerical method. Representative methods include Ritz method (Dozio, 2011; Eftekhari and Jafari, 2013; Jia et al., 2022; Li et al., 2019, 2024b; Nagino et al., 2008; Pang et al., 2018, 2024; Shi et al., 2016; Song et al., 2022; Xu et al., 2010), spectral method (Bediz, 2018; Filiz et al., 2012), and so on. Among them, the Ritz method is based on the energy principle, and the bending vibration problem is transformed into an eigenvalue problem by selecting an appropriate trial function. It has the advantages of high computational efficiency and wide application range. It is suitable for plate structures with complex geometric shapes and boundary conditions, so it has been widely used. Shi et al. (2016) studied the free vibration characteristics of circular, annular and fan-shaped thin plates with arbitrary boundary conditions by using the improved Fourier-Ritz method. The displacement function of the plate is composed of standard Fourier series and auxiliary functions, and the artificial boundary spring technique is used to simulate the boundary conditions of the plate structure. Jia et al. (2022) used the Jacobi-Ritz method to analyze the vibration of rectangular plates under general boundary conditions. Based on the first-order shear deformation theory and the domain decomposition method, the structural energy functional was established. The Jacobi polynomial was used to construct the displacement tolerance function of the rectangular plate, and the free vibration of the rectangular plate was solved by the Ritz method.

For the vibration problem of irregular plates with more complex shapes, it is difficult to solve it by analytical method, coordinate transformation method, and domain decomposition method. Therefore, numerical method has become the mainstream in current research. The numerical method solves complex problems by discretizing the solution domain and approximate calculation. Representative methods include finite element method (Mukherjee and Mukhopadhyay, 1986), differential quadrature method (Bert and Malik, 1996), and differential volume method (Civan, 1994; Liu and Liew, 1998). Although the numerical method shows strong versatility in the vibration analysis of complex shape plates and can deal with complex geometric shapes, non-uniform materials, and complex boundary conditions, it has certain limitations in analysis efficiency and pre-processing workload. Compared with numerical methods, theoretical analysis methods such as analytical methods and semi-analytical methods have unique advantages in the study of structural vibration. They can reveal the inherent laws of structural vibration from the essence of mathematics and physics, and provide a theoretical basis for further understanding the dynamic characteristics of structures. In addition, the theoretical analysis method is convenient for parametric analysis and optimization design. It can efficiently study the influence of geometric parameters, material parameters, boundary conditions, and other factors on the vibration characteristics of the structure, and provide reliable theoretical support for structural design and optimization in engineering practice.

Therefore, in order to solve the problem of vibration analysis of polygonal plates, based on the Jacobi-Ritz method, the simplified form of the energy functional equation of the polygonal plate system is derived based on the Kirchhoff thin plate theory and the divergence theorem. The Jacobi orthogonal polynomial is used to construct the displacement tolerance function of the polygonal plate, and the penalty parameter method is used to simulate the complex boundary conditions of the polygonal plate. A vibration analysis method of polygonal plates based on Jacobi-Ritz method is proposed. Finally, the effectiveness of the proposed method is verified by comparing with the simulation results and the reference data. It aims to provide a new method for vibration analysis of irregular structures.

Mathematical formulation

General polygonal plate description

In this paper, the free vibration and dynamic characteristics of general polygonal isotropic thin plates with any edge are studied. The general polygonal plate model as shown in Figure 1, set the coordinate system is located in the plate, and let the direction of the vertical polygonal plate be z direction. H represents the number of edges of the polygonal plate. L_i denotes the i-th edge of the polygonal plate, the normal vector of L_i is n_i. x_i and y_i, respectively, represent the coordinates of the i-th vertex. S means the integral domain of the plate.

In order to simulate the complex boundary conditions, based on the penalty function method, a set of artificial springs are added to the edge of the polygonal plate, including the linear springs (k) and rotational springs (K). K_i and k_i represent the boundary spring stiffness of the i-th edge, respectively. By setting different boundary spring stiffness values to simulate different boundary conditions, the simulation of various classical and elastic boundary conditions is realized.

Displacement admissible function

Jacobi orthogonal polynomial is used as the displacement admissible function of general polygonal plate. The Jacobi orthogonal polynomial is expanded to represent the displacement field in the z direction of the model. The Jacobi polynomials satisfy the following recurrence relations in the interval [-1,1]:

P_{0}^{(α, β)} (ϕ) = 1

(1.a)

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

(1.b)

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

(1.c)

where

ϕ \in [- 1, 1]

;

α, β > - 1

;

i = 2, 3 . . .

According to Jacobi orthogonal polynomials, the displacement $w$ of the polygonal plate in the z direction can be expressed as:

w (x, y, t) = A (f_{M} (x) \otimes g_{M} (y)) e^{i ω t}

(2.a)

A = [A_{0}, A_{1}, A_{2}, \dots, A_{M}]

(2.b)

f_{M} (x) = {[P_{0}^{(α, β)} (x), P_{1}^{(α, β)} (x), P_{2}^{(α, β)} (x), \dots, P_{M}^{(α, β)} (x)]}^{T}

(2.c)

g_{M} (y) = {[P_{0}^{(α, β)} (y), P_{1}^{(α, β)} (y), P_{2}^{(α, β)} (y), \dots, P_{M}^{(α, β)} (y)]}^{T}

(2.d)

where

A

is the unknown Jacobi expansion coefficient;

M

denotes the polynomial truncation number;

f_{M} (x)

and

g_{M} (y)

represent the orthogonal polynomials in x and y directions;

ω

is the angular frequency.

Energy functional variational analysis

Based on the geometric relationship and basic assumptions in elasticity, the displacement relationship of the plate structure is as follows:

{\begin{cases} ε_{x} = \frac{\partial u}{\partial x} = - \frac{\partial^{2} w}{\partial x^{2}} z \\ ε_{y} = \frac{\partial v}{\partial y} = - \frac{\partial^{2} w}{\partial y^{2}} z \\ γ_{x y} = \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} = - 2 \frac{\partial^{2} w}{\partial x \partial y} z \end{cases}

(3)

The physical equation of small deflection bending problem of thin plate is:

{\begin{cases} σ_{x} = \frac{E}{1 - μ^{2}} (ε_{x} + μ ε_{y}) \\ σ_{y} = \frac{E}{1 - μ^{2}} (ε_{y} + μ ε_{x}) \\ τ_{x y} = \frac{E}{2 (1 + μ)} γ_{x y} \end{cases}

(4)

Substituting equation (3) into equation (4), the expression of the stress component on the deflection of the plate is as follows:

{\begin{cases} σ_{x} = - \frac{E z}{1 - μ^{2}} (\frac{\partial^{2} w}{\partial x^{2}} + μ \frac{\partial^{2} w}{\partial y^{2}}) \\ σ_{y} = - \frac{E z}{1 - μ^{2}} (\frac{\partial^{2} w}{\partial y^{2}} + μ \frac{\partial^{2} w}{\partial x^{2}}) \\ τ_{x y} = - \frac{E z}{1 + μ} \frac{\partial^{2} w}{\partial x \partial y} \end{cases}

(5)

According to the theory of elasticity, the strain energy of a linear elastomer is:

V_{p} = \frac{1}{2} ∭ (σ_{x} ε_{x} + σ_{y} ε_{y} + σ_{z} ε_{z} + τ_{x y} γ_{x y} + τ_{y z} γ_{y z} + τ_{z x} γ_{z x}) d V

(6)

The deformation component can be ignored in the bending problem of thin plate. Therefore, equation (6) can be simplified as:

V_{p} = \frac{1}{2} ∭ (σ_{x} ε_{x} + σ_{y} ε_{y} + τ_{x y} γ_{x y}) d V

(7)

Substituting equation (3) and equation (4) into equation (7) can obtain:

V_{p} = \frac{E}{2 (1 - μ^{2})} ∭ {{(\nabla^{2} w)}^{2} - 2 (1 - μ) [\frac{\partial^{2} w}{\partial x^{2}} \frac{\partial^{2} w}{\partial y^{2}} - {(\frac{\partial^{2} w}{\partial x \partial y})}^{2}]} z^{2} d V

(8)

For the polygonal thin plate with equal thickness studied in this paper, z is integrated along the plate thickness, and the bending vibration strain energy of polygonal plate is expressed as:

V_{p} = \frac{D}{2} \iint_{S} [{(\frac{\partial^{2} w (x, y)}{\partial x^{2}})}^{2} + {(\frac{\partial^{2} w (x, y)}{\partial y^{2}})}^{2} + 2 μ (\frac{\partial^{2} w (x, y)}{\partial x^{2}}) (\frac{\partial^{2} w (x, y)}{\partial y^{2}}) + 2 (1 - μ) {(\frac{\partial^{2} w (x, y)}{\partial x \partial y})}^{2}] d x d y

(9)

where

D = \frac{E h^{3}}{12 (1 - μ^{2})}

is the bending stiffness of the polygonal plate;

E

μ

, and

h

are the Young ‘s modulus, Poisson ‘s ratio and thickness of the polygonal plate, respectively.

Substituting equation (2) into equation (9) can be obtained:

V_{p} = \frac{D}{2} \iint_{S} A [\begin{array}{l} (\frac{\partial^{2} f_{M} (x)}{\partial x^{2}} \frac{\partial^{2} f_{M}^{T} (x)}{\partial x^{2}}) \otimes (g_{M} (y) g_{M}^{T} (y)) \\ + (f_{M} (x) f_{M}^{T} (x)) \otimes (\frac{\partial^{2} g_{M} (y)}{\partial y^{2}} \frac{\partial^{2} g_{M}^{T} (y)}{\partial y^{2}}) \\ + 2 μ (\frac{\partial^{2} f_{M} (x)}{\partial x^{2}} f_{M}^{T} (x)) \otimes (g_{M} (y) \frac{\partial^{2} g_{M}^{T} (y)}{\partial y^{2}}) \\ + 2 (1 - μ) (\frac{\partial f_{M} (x)}{\partial x} \frac{\partial f_{M}^{T} (x)}{\partial x}) \otimes (\frac{\partial g_{M} (y)}{\partial y} \frac{\partial g_{M}^{T} (y)}{\partial y}) \end{array}] A^{T} d x d y

(10)

The elastic potential energy generated by the boundary spring of the polygonal plate is expressed as:

V_{s} = \sum_{i = 1}^{H} \frac{1}{2} \int_{L_{i}} [k_{i} w {(x, y)}^{2} + K_{i} {(\frac{\partial w (x, y)}{\partial n_{i}})}^{2}] d L_{i}

(11)

By substituting equation (2) into equation (11), this formula can be rewritten as:

V_{s} = \sum_{i = 1}^{H} \frac{1}{2} \int_{L_{i}} A [\begin{array}{l} k_{i} (f_{M} (x) f_{M}^{T} (x)) \otimes (g_{M} (y) \otimes g_{M}^{T} (y)) \\ + K_{i} (\begin{array}{l} (\frac{\partial f_{M} (x)}{\partial x} \frac{\partial f_{M}^{T} (x)}{\partial x}) \otimes (g_{M} (y) g_{M}^{T} (y)) \cos^{2} θ_{i} \\ + (f_{M} (x) f_{M}^{T} (x)) \otimes (\frac{\partial g_{M} (y)}{\partial y} \frac{\partial g_{M}^{T} (y)}{\partial y}) \sin^{2} θ_{i} \\ + (\frac{\partial f_{M} (x)}{\partial x} f_{M}^{T} (x)) \otimes (g_{M} (y) \frac{\partial g_{M}^{T} (y)}{\partial y}) \cos θ_{i} \sin θ_{i} \\ + (\frac{\partial f_{M}^{T} (x)}{\partial x} f_{M} (x)) \otimes (g_{M}^{T} (y) \frac{\partial g_{M} (y)}{\partial y}) \cos θ_{i} \sin θ_{i} \end{array}) \end{array}] A^{T} d L_{i}

(12)

where

θ_{i}

is the direction angle of

n_{i}

The kinetic energy of the polygonal plate is:

T = \frac{1}{2} ρ h \iint_{S} {(\frac{\partial w (x, y, t)}{\partial t})}^{2} d x d y

(13)

By substituting equation (2) into formula equation (13), we can obtain:

T = \frac{1}{2} ρ h ω^{2} \iint_{S} A [(f_{M} (x) f_{M}^{T} (x)) \otimes (g_{M} (y) g_{M}^{T} (y))] A^{T} d x d y

(14)

where

ρ

is the mass density of the general polygonal plate.

The energy functional of the system can be expressed as:

Π = V_{p} + V_{s} - T

(15)

Since the integral region of the polygonal plate is an irregular region, the solution of the energy equation will be complicated. Here, the divergence theorem is introduced to convert the two-dimensional continuous domain integral of the energy functional of the system into a one-dimensional integral.

Assuming that there exists a function $F (x, y) = P (x, y) i + Q (x, y) j$ , the divergence theorem can be described as:

\iint_{S} \nabla \cdot F (x, y) d x d y = \int_{L} F (x, y) \cdot n d L

(16)

where L is the boundary of the integral domain S and

n

is the normal vector of the boundary L.

The above equation can be described as:

\iint_{S} \frac{\partial Q (x, y)}{\partial x} d x d y = \int_{L} Q (x, y) j \cdot n d L

(17)

After modifying the integrand function, the above equation can be converted to:

\iint_{S} Q (x, y) d x d y = \int_{L} (\int Q (x, y) d y) j \cdot n d L

(18)

For polygonal plates, the above expression can be expressed as:

\iint_{S} Q (x, y) d x d y = \sum_{i = 1}^{H} \int_{L_{i}} (\int Q (x, y) d y) j \cdot n_{i} d L_{i}

(19)

where

n_{i} = d x / j d L_{i}

. Substitution (13) can be obtained:

\iint_{S} Q (x, y) d x d y = {\sum_{i = 1}^{H} - \int_{x_{i}}^{x_{i + 1}} (\int Q (x, y) d y) |}_{y = y_{i} (x)} d x

(20)

where

y = y_{i} (x)

represents the function expression of y with respect to x on the i-th edge.

The divergence theorem is applied to the energy equation of general polygonal plate, and the strain energy $V_{p}$ and kinetic energy $T$ of the system can be further expressed as:

V_{p} = \frac{D}{2} {\sum_{i = 1}^{H} - \int_{x_{i}}^{x_{i + 1}} (\int A [\begin{array}{l} (\frac{\partial^{2} f_{M} (x)}{\partial x^{2}} \frac{\partial^{2} f_{M}^{T} (x)}{\partial x^{2}}) \otimes (g_{M} (y) g_{M}^{T} (y)) \\ + (f_{M} (x) \otimes f_{M}^{T} (x)) \otimes (\frac{\partial^{2} g_{M} (y)}{\partial y^{2}} \frac{\partial^{2} g_{M}^{T} (y)}{\partial y^{2}}) \\ + μ (\begin{array}{l} (\frac{\partial^{2} f_{M} (x)}{\partial x^{2}} f_{M}^{T} (x)) \otimes (g_{M} (y) \frac{\partial^{2} g_{M}^{T} (y)}{\partial y^{2}}) + \\ (\frac{\partial^{2} f_{M}^{T} (x)}{\partial x^{2}} f_{M} (x)) \otimes (g_{M}^{T} (y) \frac{\partial^{2} g_{M} (y)}{\partial y^{2}}) \end{array}) \\ + 2 (1 - μ) (\frac{\partial f_{M} (x)}{\partial x} \frac{\partial f_{M}^{T} (x)}{\partial x}) \otimes (\frac{\partial g_{M} (y)}{\partial y} \frac{\partial g_{M}^{T} (y)}{\partial y}) \end{array}] A^{T} d y) |}_{y = y_{i} (x)} d x

(21)

T = \frac{1}{2} ρ h ω^{2} {\sum_{i = 1}^{H} - \int_{x_{i}}^{x_{i + 1}} (\int A [(f_{M} (x) f_{M}^{T} (x)) \otimes (g_{M} (y) g_{M}^{T} (y))] A^{T} d y) |}_{y = y_{i} (x)} d x

(22)

Based on the Ritz method, the extremum of the unknown coefficients in the energy functional expression of the system is obtained:

\frac{\partial Π}{\partial A} = 0

(23)

By sorting out the variational results, the dynamic response equation of the polygonal plate can be expressed as:

(K - ω^{2} M) A = F

(24)

where

K

and

M

represent the stiffness matrix and mass matrix of the polygonal plate, respectively.

F

denotes the excitation force matrix.

When the polygonal plate is not subjected to the excitation force, that is, the free vibration of the polygonal plate, then equation (24) can be rewritten as follows:

(K - ω^{2} M) A = 0

(25)

By solving equation (25), the free vibration results of the general polygonal plate can be calculated.

Numerical analysis

In this paper, the free vibration and dynamic analysis of some polygonal plates are carried out, and the model test of polygonal plates is carried out. By comparing the calculation results of the method described in this paper with the finite element results, literature results and experimental results, the effectiveness and accuracy of the method are verified. At the same time, the free vibration characteristics of polygonal plates such as triangular plates, four-deformed plates, pentagonal plates, and six-deformed plates under different boundary conditions, as well as the steady-state forced vibration and transient response of polygonal plates under excitation are obtained.

In this paper, F, C, S, and E are used to represent free, clamped, simply supported, and elastic boundary conditions, respectively. When there is no special point, the parameters of the polygonal plate material and the displacement allowable function are selected as: E = 210 GPa, ρ = 7850 kg/m³, μ = 0.3, h = 0.01 m, M = 10, α = β = 1. In addition, the dimensionless frequency parameter is defined as: $Ω = ω d^{2} \sqrt{ρ h / D}$ , where d is the length of the longest edge of the polygonal plate.

Convergence study

In this section, the convergence of the method proposed in this paper is analyzed by taking a simply supported square plate as an example. The square plate has a side length of 1 m, and it should be emphasized that the conclusions drawn hereinafter are also applicable to arbitrary polygonal plates under other boundary conditions. Since the method in this paper integrates air springs and orthogonal Jacobian polynomials, the convergence of the algorithm is determined by three key factors: the selection of spring stiffness, the truncation order of the displacement admissible function, and the choice of Jacobi parameters.

The dimensionless frequency parameters of the rectangular plate under different boundary spring stiffness values are shown in the following figure. Here, the dimensionless frequency parameter is selected $Ω = ω d^{2} \sqrt{ρ h / D}$ , where $ω$ is the circular frequency, d is the edge length of the plate, $ρ$ is the density, h is the thickness of the plate, and D is the bending stiffness. Figure 2(a) only considers the influence of the translational spring stiffness, and the rotational spring stiffness value is set to 0. Figure 2(b) considers the influence of both translational spring and rotational spring, and assumes that the stiffness values of translational spring and rotational spring are the same. For the penalty parameter method, when the translational and rotational spring stiffness values are zero, it is equivalent to no boundary constraint, that is, free boundary condition. At the same time, the rigid fixed boundary condition means that the structure does not move and rotate at the boundary, and the spring stiffness value can be regarded as infinite.

It can be seen from Figure 2 that for the boundary spring, when the spring stiffness value is in the range of 10^-1∼10³, the frequency parameter value is stable, the spring stiffness value is small, and the plate is not constrained by the boundary spring, which is equivalent to the free boundary. When the spring stiffness value is greater than 10¹⁰, the frequency parameter is stable, which can be regarded as the spring stiffness infinity, that is, equivalent to the rigid fixed boundary condition; when the spring stiffness value is in the range of 10³∼10¹⁰, the frequency parameters change significantly and can be regarded as elastic boundary conditions. Therefore, the spring stiffness values corresponding to different boundary conditions are set as shown in Table 1.

Table 1.

The spring stiffness values of the general edge conditions.

Boundary conditions	k	K
F	0	0
E	10⁷	10⁷
S	10¹⁵	0
C	10¹⁵	10¹⁵

Figure 3 shows the relative error percentage of the frequency parameters of the quadrilateral plate with respect to the Jacobi parameter. Taking the calculated results of the Jacobi parameter α = β = 1 as the reference value, the influence of different Jacobi parameters on the frequency parameters of the plate structure can be intuitively expressed. It can be seen from the following figure that all Jacobi parameters have almost the same effect on the frequency parameters. Therefore, the Jacobi parameters are used in the subsequent calculation as α = β = 1.

In order to take into account the calculation accuracy and calculation efficiency, it is necessary to cut off the displacement allowable function reasonably. Figure 4 explores the influence of the number of displacement allowable function segments on the frequency parameters. From the graph curve, it can be seen that when the number of displacement allowable function segments reaches 8, the frequency parameters converge stably, and the influence of increasing the number of segments on the calculation accuracy can be ignored.

Free vibration analysis of general polygonal plate

In order to verify the effectiveness of the method in solving the free vibration of general polygon plates, several examples of right-angled triangle plates and regular quadrilateral plates with different boundary conditions are given. The right-angled side length of the triangle plate and the side length of the rectangular plate are both 1 m, and the finite element convergence grid scale is 0.001 m. In Tables 2 and 3, the dimensionless natural frequencies of right-angle triangular plates and rectangular plates under different boundary conditions calculated by this method are given, respectively, and the calculation results are compared with the published literature and finite element results. The comparative study shows that the calculation results of this method are smaller than those of other methods, which shows that the method in this paper has higher accuracy for the free vibration analysis of polygonal plates.

Table 2.

Comparison table of dimensionless frequency of right-angle triangular plate.

Boundary conditions	SSS			CCC			FFF
Mode no.	Present	Liew (1993)	FEM	Present	Liew (1993)	FEM	Present	Liew (1993)	FEM
1	49.348	49.348	49.116	93.789	93.766	93.506	19.068	19.068	19.023
2	98.696	98.696	98.187	157.787	157.71	157.066	29.123	29.123	29.027
3	128.305	128.30	127.504	194.767	194.67	193.713	45.397	45.97	45.178
4	167.796	167.79	166.757	242.849	242.62	241.247	49.475	49.475	49.276
5	197.390	197.39	195.981	277.791	277.46	275.670	72.349	72.350	72.102
6	246.773	246.74	244.703	335.919	335.47	332.951	84.052	84.052	83.567

Table 3.

Comparison table of dimensionless frequency of rectangular plate.

Boundary conditions	SSSS			CCCC			CSSF
Mode no.	Present	Li (2004)	FEM	Present	Li (2004)	FEM	Present	Li (2004)	FEM
1	19.739	19.74	19.655	35.985	35.986	35.950	16.797	16.785	16.672
2	49.348	49.35	49.196	73.394	73.395	73.282	31.117	31.115	30.985
3	49.348	49.35	49.196	73.394	73.395	73.282	51.412	51.392	51.324
4	78.957	78.96	78.553	108.218	108.22	107.950	64.027	64.016	63.812
5	98.696	98.70	98.464	131.582	131.58	131.306	67.552	67.549	67.285
6	98.696	98.70	98.468	132.207	132.21	131.940	101.129	101.21	100.592

At the same time, the first six vibration modes of a right-angled triangular plate under simply supported boundary conditions and a rectangular plate under rigidly fixed boundary conditions are presented in Figures 5 and 6, respectively. A comparative analysis with the modal contour plots derived from finite element simulations reveals that, even when the truncation order M of the displacement admissible function is set to a relatively small value of 8, the proposed method can still accurately capture the modal characteristics of polygonal plates. This demonstrates the effectiveness and precision of the method in addressing the free vibration problems of polygonal plates, including triangular and rectangular configurations.

Next, the free vibration characteristics of polygonal plates with various special shapes under different boundary conditions are considered, and the calculation results are compared with the finite element calculation results to verify that this method can be applied to the free vibration solution of polygonal plates with arbitrary shapes. The geometry of these polygonal plates is shown in Figure 7.

Tables 4 and 5 show the natural frequencies of irregular pentagonal plates and regular hexagonal plates under general boundary constraints. Compared with the finite element calculation results, it is not difficult to find that the calculation results in this paper also have high accuracy for the free vibration analysis of general polygonal plates. At the same time, by giving different boundary constraints of regular hexagonal plates, it can be seen that boundary constraints have a significant impact on the vibration characteristics of polygonal plates. With the increase of boundary conditions, the overall stiffness of the structure is improved, which leads to the increase of the natural frequency of polygonal plates.

Table 4.

Natural frequency table of irregular pentagonal plate.

Mode no.		1	2	3	4	5	6
CCCCC	Present	28.349	56.981	59.518	92.628	96.867	105.523
CCCCC	FEM	28.252	56.245	59.257	92.05	96.029	104.63
FSCSF	Present	4.541	17.795	18.837	38.732	43.014	51.053
FSCSF	FEM	4.516	17.731	18.783	38.56	42.916	50.935
FFSCS	Present	5.405	19.863	22.055	42.421	47.223	53.470
FFSCS	FEM	5.388	19.819	22.010	42.332	47.148	53.349
FFCSS	Present	7.307	20.222	25.910	43.497	49.571	56.390
FFCSS	FEM	7.288	20.151	25.833	43.35	49.423	56.202
SFSCC	Present	16.465	32.698	44.401	60.025	71.004	81.629
SFSCC	FEM	16.431	32.525	44.303	59.782	70.837	81.28

Table 5.

Natural frequency table of regular hexagon plate.

Mode no.		1	2	3	4	5	6
SSSSSS	Present	17.885	45.374	45.399	80.729	81.264	93.499
SSSSSS	FEM	17.667	44.974	44.975	80.547	80.549	93.166
FFFFFF	Present	15.904	15.904	26.586	33.962	40.086	59.269
FFFFFF	FEM	15.888	15.888	26.583	33.863	40.070	59.224
CFFCCF	Present	14.550	28.396	45.079	48.216	65.389	73.646
CFFCCF	FEM	14.466	28.258	44.731	47.804	64.945	72.780
CCFSFS	Present	18.259	34.307	45.125	60.827	69.499	89.606
CCFSFS	FEM	18.198	34.085	44.758	60.335	68.992	89.135
FFFFCC	Present	4.171	11.669	22.555	29.349	42.746	58.164
FFFFCC	FEM	4.160	11.562	22.478	29.208	42.374	57.784
SCFFFC	Present	7.062	20.418	26.899	45.921	55.912	57.067
SCFFFC	FEM	7.044	20.307	26.778	45.603	55.781	56.761
FFCSCS	Present	11.983	27.057	39.250	55.532	62.066	82.524
FFCSCS	FEM	11.909	26.871	39.023	55.205	61.694	81.828

At the same time, the first six modes of two polygonal plates are shown in Figures 8 and 9. Compared with the modal diagram of FEM calculation results, it shows that this method can better express the vibration modal characteristics of general polygonal plates. The calculation results show that this method is applicable to the free vibration analysis of general polygonal plates in addition to the more regular square and triangular plates.

Conclusions

Aiming at the problem of vibration analysis of irregular shape plates, a semi-analytical method for dynamic vibration analysis of arbitrary polygonal plate structures under elastic boundary conditions is proposed based on the Jacobi-Ritz method. Based on the Kirchhoff thin plate theory, the vibration theoretical model of polygonal plate is established. Aiming at the difficult problem caused by the complex integral region involved in the energy functional of polygonal plate system, the two-dimensional continuous domain integral in the energy functional of polygonal plate system is simplified into one-dimensional integral by introducing divergence theorem, and the simplified form of energy functional of polygonal plate system is obtained. Then, based on the existing Jacobi-Ritz method, the Jacobi orthogonal polynomial is used as the displacement tolerance function of the polygonal plate, which further improves the applicability and computational efficiency of the method. The complex boundary conditions of polygonal plates are simulated by penalty parameter method, and the vibration model of polygonal plates under elastic boundary conditions is established.

Taking the triangular plate, quadrilateral plate, pentagonal plate and hexagonal plate as the research object, through a large number of numerical calculations, and comparing with the simulation results and reference data, the results show that the maximum error of the first six natural frequencies of the polygonal plate is less than 2 %, and the modal shape of the polygonal plate structure is consistent with the simulation results, which verifies the effectiveness and accuracy of the vibration analysis method of the polygonal plate based on the Jacobi-Ritz method.

Footnotes

ORCID iD

Fuzhen Pang

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by National Natural Science Foundation of China (52471324, 52471322) and the Joint Training Fund Project of Hanjiang National Laboratory (No.HJLJ20240207).

Declaration of conflicting interests

The authors declare that there is no conflict of interest regarding the publication of this paper.

Data Availability Statement

The data used to support the findings of this study are included within the article.*

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