Research Article  Open Access
A Series Solution for the Vibration of Mindlin Rectangular Plates with Elastic Point Supports around the Edges
Abstract
A series solution for the transverse vibration of Mindlin rectangular plates with elastic point supports around the edges is studied. The series solution for the problem is obtained using improved Fourier series method, in which the vibration displacements and the crosssectional rotations of the midplane are represented by a double Fourier cosine series and four supplementary functions. The supplementary functions are expressed as the combination of trigonometric functions and a single cosine series expansion and are introduced to remove the potential discontinuities associated with the original admissible functions along the edges when they are viewed as periodic functions defined over the entire  plane. This series solution is approximately accurate in the sense that it explicitly satisfies, to any specified accuracy, both the governing equations and the boundary conditions. The convergence, accuracy, stability, and efficiency of the proposed method have been examined through a series of numerical examples. Some numerical examples about the nondimensional frequency and mode shapes of Mindlin rectangular plates with different pointsupported edge conditions are given.
1. Introduction
Rectangular plate is of great importance in various engineering branches, such as aerospace, electronics, and mechanical, nuclear, and marine engineering. A better understanding of its dynamic characteristics is meaningful when dealing with the design of a platetype structure. In the early stage of research about rectangular plate, the majority of research was about thin plate based on classical Kirchhoff hypothesis. However, for the shortage of consideration about transverse shear deformation and rotary inertia, vibration frequencies of thick plates are overestimated. More and more investigations about Mindlin plates began to attract attention after the Mindlin firstorder plate theory was proposed by Mindlin [1]. The vibration characteristics of Mindlin plates have been well investigated by researches with classical and elastic edge support [2–5]. Few literatures focused on the vibration behavior of Mindlin plates with elastic point supports around the edges. However, the boundary conditions of the plates are not always classical and elastic around the edges in practical engineering applications. And the boundary conditions of elastic point supports around the edges do exist. So it is much of great significance to study the vibration behavior of Mindlin plates with elastic point supports around the edges.
In the field of vibration analysis of structures subject to elastic edge/point supports, Takashi and Jin [6] used the collocation method to investigate Mindlin plates with constant thickness and two opposite edges simply supported. Jiarang and Jianqiao [7] established the threedimensional state equation for the laminated thick orthotropic plate with simply supported edges and obtained the numerical results. Ohya et al. [3] investigated the free vibration characteristics of rectangular Mindlin plates which have simultaneous elastic edge and internal supports via the superposition method. HosseiniHashemi et al. [8] proposed an exact closedform procedure to solve free vibration of moderately thick rectangular plates with two opposite edges simply supported. Dozio et al. [9] developed a Ritz method using a set of trigonometric functions to obtain accurate modal properties of rectangular plates with arbitrary thickness. Maretic [10] analyzed the transverse vibration of a circular plate loaded by uniform pressure along its edge. Based on the shear deformable plate theory, Bahmyari [11] used the ElementFree Galerkin Method to analyze the free vibration of inhomogeneous moderately thick plates with point supports resting on a twoparameter elastic foundation. The results show that the vibration results obtained are in a very good agreement with the available literatures in spite of using low numbers of nodes. Bashmal et al. [12] investigated the inplane free vibrations of an elastic and isotropic annular disk with elastic constraints at the inner and outer boundaries. And the inner and outer boundaries are applied either along the entire periphery of the disk or at a point. Esendemir [13] studied polymermatrix composite beam of arbitrary orientation supported from two ends acted upon with a force at the midpoint by an analytical elastoplastic stress analysis. Foyouzat et al. [14] presented an analyticalnumerical approach to determine the dynamic response of thin plates resting on multiple elastic point supports with timevarying stiffness. Gan et al. [15] studied the effect of intermediate elastic support on the vibration of functionally graded EulerBernoulli beams excited by a moving point load. Kucuk et al. [16] carried out the analytical elasticplastic stress analysis on an aluminum metalmatrix composite beam reinforced by unidirectional steel fibers supported at the ends acted upon with a force at the midpoint. HosseiniHashemi et al. [17–19] used the generalized differential quadrature method to study the buckling analysis and dynamic transverse vibration characteristics. Setoodeh and Karami [20] employed a threedimensional elasticity based layerwise finite element method (FEM) to study the static, free vibration, and buckling responses of general laminated thick composite plates. In this paper, the elastic line and point supports are successfully incorporated for thick plates. On the basis of threedimensional elasticity theory, Xu and Zhou [21] studied the bending of simplesupported rectangular plate on point supports, line supports, and elastic foundation. Wu [22] analyzed the free vibration of arbitrary quadrilateral thick plates with internal columns and elastic edge supports by using the powerful pb2 Ritz method and Reddy’s thirdorder shear deformation plate theory. Wu and Lu [23] studied the free vibration of rectangular plates with internal columns and elastic edge supports using the powerful pb2 Ritz method. Other literatures related to this field are shown in [24–28].
From the literatures review, we can know that most researches focused on the classical plates. Only few literatures studied the vibration characteristics of Mindlin plates with point supports around the edges. And these literatures mainly used the collocation method, analyticalnumerical method, and so on. To the best of the authors’ knowledge, there are no published literatures focused on vibration characteristics of Mindlin plates with point supports around the edges by the method of modified Fourier. Thus, a unified, efficient, and accurate formulation to deal with the free vibration of Mindlin plates subjected to arbitrary pointsupported boundary condition is necessary and much of great significance.
In this paper, a modified Fourier solution for the free vibration of Mindlin rectangular plates with elastic point supports around the edges is proposed. The vibration displacements and the crosssectional rotations of the midplane are represented by a double Fourier cosine series and four supplementary functions. The supplementary functions are in the form of the combination between trigonometric functions and a single cosine series expansion. By importing supplementary functions, the discontinuity of the boundary condition is overcome. The change of the boundary conditions can be easily achieved by only varying the stiffness of the boundary springs around all edges of the plates without involving any change to the solution procedure. The natural frequencies of Mindlin rectangular plates are obtained by using the Rayleigh–Ritz method. Effectiveness and stability of this method will be verified by comparing with the results of FE modeling. Some numerical examples of free vibration for Mindlin rectangular plates with different aspect ratio and thickness are conducted under different pointsupported conditions.
2. Theoretical Formulations
2.1. PointSupported Edge Conditions
The rectangular Mindlin plate with arbitrary elastic point edge supports is shown in Figure 1. The boundary conditions are presented by three kinds of restraining springs [29–34], namely, translational, rotational, and torsional springs. Springs are evenly arranged on each edge of Mindlin plate. Through changing the stiffness of springs, different boundary conditions can be achieved [35, 36]. The governing differential equations of a Mindlin plate about its free vibration are as follows:
In the formulas above, transverse displacement is presented by , is the slope along direction, and is the slope along direction. means the mass density, is Poisson’s ratio, is the shear correction factor, and the thickness of the plate is . is the shear modulus, is Poisson’s ratio, and is the flexural rigidity.
Besides, the bending moment is expressed by (4) and (5), twisting moment is calculated by (6), and the transverse shearing forces in plates are expressed by (7) and (8).
There are three kinds of forces along every edge, namely, the bending moment, the twisting moment, and the shearing forces. Rotational, torsional, and translational springs along every edge are the counterparts to these three forces in this paper. The boundary conditions for an elastically restrained rectangular plate are as follows.
At ,
, , and represent the stiffness of the translation spring, rotational spring, and torsional spring, respectively, on the boundary edge ; the corresponding expression results in where is the elastic point supports numbers on edge , is the Dirac delta function, and , , and are, respectively, the stiffness of the translational, rotational, and torsional springs at the support located at . Similarly, the remaining boundary conditions on the other three edges can be expressed as follows.
At ,At ,At ,
Equations (9)–(32) represent a set of general boundary conditions by setting the spring stiffness to appropriate values. Based on (4)–(8), the boundary condition can be finally written as follows.
At ,At ,At ,At ,
According to the Mindlin plate theory, the transverse displacement of the plate middle surface and the rotations of the cross section, respectively, along the direction and the direction are utilized. Based on the traditional solution method, the admissible functions usually express a Fourier series expansion, because the Fourier functions constitute a complete set and exhibit an excellent numerical stability in the previous study [37–43]. We found the conventional Fourier series expression to have some defects which contain the convergence problem along the boundary edges except for a few simple boundary conditions, and the derivatives of a Fourier series cannot be obtained simply through termbyterm differentiation.
In this study, in order to overcome the shortcoming with the conventional Fourier series expression, the admissible functions will be expressed as a more wholesome form of Fourier series expansion:where , , , , , , , , and represent the unknown coefficients, , . The specific expressions of the auxiliary functions and are defined as
As shown in (45)–(47), the supplementary functions , , , and are used for the and direction displacement expressions. It is easy to find that
Similar conditions are present in the supplementary function in direction, namely, and . Whereas these conditions are not necessary, the existence of these conditions is purely for the sake of simplifying followup mathematical expressions and derivation process.
Examining the admissible functions in (45)–(47), one will notice that, besides the standard 2D Fourier series defined over the domain , four additional 1D Fourier expansions are also included in the admissible functions expressions. In light of (52), it is not difficult to see that the fourth term (or the third single Fourier series) on the right side of (45) is equal to the normal derivative of the displacement function at edge . Thus, this term will actually inherit any potential discontinuity associated with the normal derivative at the boundary . Similarly, the other three terms are used to take care of the possible discontinuities at the remaining edges. Thus, the 2D Fourier expansion will now represent a periodic (residual displacement) function defined over the entire plane. The convergence characteristic for the Fourier series representation of such a function has been well established in mathematics. Namely, the Fourier series will converge with a speed of at least.
By substituting (45)–(47) into boundary condition (33) and expanding all the related terms, except for those containing , into Fourier cosine series, one will have
The expressions for Fourier cosine expansion coefficient of related functions can be found in Appendix A. To establish the relationship between the Fourier coefficients in (54), the terms on the left side shall also be expressed in the form of cosine series, resulting inwhere and are Kronecker delta. Using similar procedures, the other seven boundary condition equations can be obtained. Thus, a total of twelve constraint equations can be derived as follows.
At ,At ,At ,At ,
In this paper, all the series expansions have been uniformly truncated to and . Equations (58)–(69) can be rewritten in matrix form aswhere and
All the elements in matrices and can be directly derived from (58)–(69). For example, Appendix B shows the explicit expressions that correspond to the th equation in (58). Equations (58)–(69) represent linear algebraic equations against a total of unknown Fourier coefficients. To solve for all these unknowns, additional equations will have to be derived from the governing equations as described below.
2.2. Solving Governing Differential Equations
By substituting (45)–(47) into governing differential equations (1)–(3) and then expanding all the resulting equations and sine functions into cosine series and comparing the like terms, the following equations in matrix form will be deduced:where and
The elements in the matrices , , , and can be directly obtained from the results of substitution process. The coefficients in cannot be treated as independent variables and need to be eliminated from (74) by making use of (70). By doing so, the final characteristic equation can be written as where and , and the modal frequencies and the corresponding eigenvectors can now be easily determined by solving a standard matrix eigenproblem. The elements in each eigenvector are actually the Fourier coefficients for the corresponding mode whose mode shape in the physical space can be simply calculated using (45), (46), (47), and (70).
3. Numerical Examples and Discussion
In this section, some results and discussions about the free vibration of Mindlin rectangular plates are presented to verify the accuracy and flexibility of the proposed method. Based on that, some new results are obtained for Mindlin rectangular plates with various aspect ratios and thicknesses subjected to different pointsupported conditions. The discussion is arranged as follows. Firstly, the convergence of the present method is checked when the plate is square rectangular and pointsupported. Then, the accuracy of this method is compared with those obtained by FEM commercial program ABAQUS (S4R model). And the following results obtained by FEM are all based on the commercial program ABAQUS (S4R model) unless otherwise stated. In addition, pointsupported plates with different geometry parameters and various thicknesses are calculated by the present method. Then, the method is extended to multipointsupported plate. After the convergence of the multipointsupported plate is verified, more multipointsupported plates with different geometry parameters and number of clamped points are calculated through this method.
If not mentioned, the material constants are chosen as follows in this study: Gpa, kg/m^{3}, and . To avoid roundoff results, nondimensional frequency is used and stated as follows: . The shear correction factor is selected as . In addition, the nondimensional translational, rotational, and torsional springs parameters are defined as and , , where N/m and N·m/rad.
3.1. Convergence Studies
A Mindlin plate with arbitrary elastic point edge supports is shown in Figure 1. The first example is about pointsupported square rectangular plate, namely, case 1 in Figure 2. In Table 1, the first eight nondimensional frequency parameters for the square plate of case 1 are given with different restraining stiffness (, ) for each point support, and the results are compared with the FEA results. FEA results of Tables 1 and 2 are both obtained by FEM commercial program ABAQUS (S4R model, 18720 elements).


The geometric dimensions of case 1 are as follows: m, m. Besides, series expansion truncated number . It is noted that the current results compare well with the FEA results. As shown in Table 1, when nondimensional springs parameters of translational and rotational springs are in the range of ~, nondimensional frequency parameter is of convergence. In other words, nondimensional springs parameters at this range can guarantee rigid boundary conditions. When nondimensional springs parameters are less than 10, the boundary conditions can be seen free. To understand the modes of square rectangular plate in free boundary conditions better, the first six mode shapes of square rectangular plates with completely free boundary conditions calculated by the present method and FEA are presented in Figures 3 and 4. And when the nondimensional springs parameters are in the range of 10^{2} and 10^{6}, the boundary conditions are elastic. As shown in Table 2, the first eight nondimensional frequency parameters become quickly converged at for the given 5digit precision. For simplicity, the displacement expansion will be truncated to in all the subsequent calculations.
(a) The 1st mode
(b) The 2nd mode
(c) The 3rd mode
(d) The 4th mode
(e) The 5th mode
(f) The 6th mode
(a) The 1st mode
(b) The 2nd mode
(c) The 3rd mode
(d) The 4th mode
(e) The 5th mode
(f) The 6th mode
To eliminate the particularity of square rectangular plate, the second example is about pointsupported rectangular plate 1b in Figure 2. The geometric dimensions of case 2 are as follows: m, m, and m. Besides, series expansion truncated number . In Table 3, the first eight nondimensional frequency parameters about 1b calculated by the new method and FEA are given. Meanwhile, the comparison in Table 3 shows the accuracy of the new method to ordinary rectangular Mindlin plate. FEA results of plate 1b in Table 3 and Figure 8 are obtained by FEM commercial program ABAQUS (S4R model, 37500 elements).

3.2. PointSupported Plate
The accuracy and nice convergence characteristic of this method have been proved when the plate is pointsupported. In Table 4, the first eight nondimensional frequency parameters are given with different stiffness values, aspect ratios, and thickness ratios. The first six mode shapes of square rectangular plates with completely elastic boundary conditions (, ) calculated by the present method and FEA are presented in Figures 5 and 6. FEA results of plate 1a in Figure 6 are obtained by FEM commercial program ABAQUS (S4R model, 18720 elements). The first six mode shapes matched well between current method and FEA. It can be seen from Table 4 that nondimensional frequency parameters under elastic boundary conditions tend to decrease with the aspect ratio, whereas the first three modes in each case basically represent the rigidbody motions. The change of these three modes is smaller than higher modes. With the increase of nondimensional spring parameters, the boundary conditions change from elastic to clamped. When the spring stiffness is set as , , the first six mode shapes of 1a and 1b which are displayed in Figures 7 and 8 can be seen as point clamped.
