New Analytical Free Vibration Solutions of Thin Plates Using the Fourier Series Method

Abstract

This article aims at analytically solving the free vibration problem of rectangular thin plates with one corner free and its opposite two adjacent edges rotationally-restrained, which is difficult to handle by conventional semi-inverse approaches such as the Levy solution and Naiver solution, etc. Based on the classical Fourier series theory, this work presents a first endeavor to treat the two-dimensional half-sinusoidal Fourier series, which is quite similar to the Navier’s form solution, as the solution form of plate deflection. By utilizing the orthogonality of the present trial function and the Stoke’s transformation technique, the present solution procedure converts the complicated plate problem into solving sets of linear algebra equations, which heavily decreases the difficulties. Therefore, the present approach enables one to solve the title problem in a unified, simple and straightforward way, which is very easily implemented by researchers. Another advantage of the present method over other analytical approaches is that it has general applicability to various boundary conditions through utilizing different types of Fourier series and it can be extended for further dynamic/static analysis of plates under different shear deformation theories. Moreover, without any extra derivation processes, new, precise analytical free vibration solutions for plates under three non-Levy-type boundary conditions are also obtained by choosing different rotating fixed coefficients. Consequently, we present more than 400 comprehensive free vibration results for plates with classical/non-classical boundaries, all the present results are confirmed by FEM/analytical solutions and can be used as benchmark data for further research.

1. Introduction

A rectangular thin plate is considered to be the basic structural element in practical applications because of its relevance in various engineering fields such as civil and structural engineering, mechanical engineering, naval and aerospace, etc. The extensive application of such structures requires the investigation of the dynamic characteristics of thin plates to establish an accurate and reliable design. It is confirmed that free vibration of plates causes a reduction in structure stiffness which reduces its load carrying capacity, as a result may cause premature structure failure. Simultaneously, a clear understanding of natural frequencies and associated mode shapes is the basis for reliable design toward avoiding a plate’s resonance problem. However, it is difficult to deal with the free vibration problem of plates under various boundary conditions, especially for the ones with complex non-classical boundaries. Numerous approaches have been developed to solve such types of intractable problems, all of them face the challenge of pursuing exact solutions that satisfy the governing partial differential equations (PDEs) under specific boundaries. Analytical methods are indispensable since they can provide theoretical, reliable and benchmark results, but are far from completed due to the complicated mathematical procedure involved. The literature review reveals that exact analytical solutions of plates under non-classical boundary restraints are not as easily available as those of Levy-type plates (two simply supported parallel edges). The boundary condition investigated in the present article is one of the most representative challenging cases, in which both the free corner and its opposite two adjacent rotationally restrained edges increase the solving difficulty of the vibration problem of plate. Accordingly, very few analytical benchmark results are available for the title problem. Consequently, seeking effective and simple analytical approaches becomes more and more crucial.

The aforementioned situation motivates the development of numerical/approximate approaches. Since there are excellent capabilities of the advanced numerical approaches such as simplicity, versatility, high precision and good convergence, they have been continuously applied to solve intractable mechanical problems of plates. Here we introduce some typical studies to illustrate new developments in this field. Wang [1] predicts the nonlinear dynamic behaviors of elastically restrained composite panels by using the classical Rayleigh–Ritz approach, the proposed solving procedure allowed one to transform the complicated nonlinear PDEs into handling several nonlinear ordinary differential equations. On basis of first-order shear deformation theory, Qin [2] developed a novel Jacobi–Ritz approach to predict free vibration responses of laminated plates with different edge conditions, excellent convergent and precise approximate results were presented for further investigation. Huang [3] provided new, precise buckling and vibration analysis for internally cracked square plates through combining the conventional Ritz method with the moving least square approach, in which novel enriched basis functions are selected as a solution for in-plane and out-of-plane displacements. It is promising to extend such an effective approach to investigate more intractable stability/dynamic problems of plates since it exhibits a powerful mathematical ability. Bidzard [4] carried out the FEM analysis on the dynamic response of multilayer FG-GPLRC toroidal panels under non-classical boundaries. The present FEM results revealed that frequency parameters decrease with the decreasing of rotational spring factor and the GPLs volume fractions. Amoushahi [5] investigated vibration and buckling behavior of composite plates by applying the traditional finite strip approach, in which the effects of moisture and temperature are well illustrated. Nguyen [6] employed a novel isogeometric Bézier finite element approach to solve the vibration problem of functionally gradient piezoelectric plates, all the numerical results obtained were precise enough since the proposed approach inherited all the advantages of the accurate geometry of isogeometric analysis and traditional FEM. Within the framework of the isogeometric approach, Alesadi [7] adopted the Carrera’s unified formulation as the approximate field solution, which yielded accurate numerical results for free vibration and buckling problems of cross-ply laminated plates. Jassas [8] developed the coupled motion equations for concrete slabs reinforced by agglomerated SiO2 nanoparticles through using the third order shear deformation theory. The authors provided new, precise approximate results for forced vibration of concrete slabs by combining the Newmark method and the harmonic differential quadrature approach. Results indicated that linear frequencies of such type structures increased when the volume percent of SiO2 nanoparticles grew up to 0.37. Civalek explored new, precise numerical solutions of plate [9,10,11] and shell [12] problems by adopting the advanced discrete singular convolution approach which was treated as a reliable and applicable tool for its excellent accuracy and efficiency.

Compared with the numerical/approximate methods, the progress made by analytical solutions is relatively few owing to the mathematical difficulty in solving the governing PDEs under the given boundaries. Semi-inverse approaches on the basis of Fourier series theory [13,14,15] are treated as the most popular analytical ways to solve plate problems, in which the preselected solutions are mainly in the form of single or double Sine series and the majority of boundaries investigated are mainly restricted to simple classical boundaries, such as the fully simply supported boundaries and Levy-type boundaries. For example, Li [16] employed a modified Fourier series method to acquire new exact analytical free vibration solutions for rectangular plates with general elastic boundary supports. Based on nonlocal elasticity theory and the generalized heat conduction model with phase delays, Ahmed [17] provided Sinusoidal form solutions for the thermomechanical problem of rotating size-dependent nanobeams with clamped–clamped boundary conditions. Except for the classical single/double Sine series solutions for plate problems, various types of trial functions such as Quasi-Green’s function [18], beam function [19], power series [20], displacement potential functions [21], etc. allow researchers to pursue accurate results for problems of plates with some certain boundaries. Based on the generalized thermoelasticity theory, Abbas [22] obtained the analytical solution for the temperature, displacement components, and stresses of thermoelastic material plates by utilizing an eigenvalue approach. By using the same method, the author [23] also provided new accurate analytical solutions for the free vibration problem of thermoelastic hollow sphere, in which the dispersion relations for various types of possible vibration modes of the hollow sphere were well derived. Ungbhakorn [24] developed the extended Kantorovich approach to handle the buckling problem of laminated composite plates. Liu [25] proposed a new Spectral stiffness approach for buckling analysis of Winkler foundation plates with general boundaries. By using an extended separation-of-variable approach, Xing [26,27] obtained close-form solutions of thin plates with several non-Levy boundaries, the given solution procedure enables one to solve eigenvalue equations simultaneously. Gorman [28] developed a modified Superposition-Galerkin approach to investigate vibration behaviors of thin plates, in which the difficulties in determining vast families of roots of traditional superposition method can be avoided. Hashemi [29] derived an exact in-plane vibration solution for Mindlin plates through utilizing the Helmholtz decomposition. The classical finite integral transform approach [30,31,32,33,34,35,36,37] and the new developed symplectic superposition approach [38,39,40,41] allowing one to solve plate and shell problems without the requirement of trial functions. However, many of the mentioned analytical approaches are applied in handling problems of plate under classical boundaries.

Considerable literatures confirm that the Navier’s form solutions are still accurate analytical solutions of the ideal form because of their simplicity and excellent orthogonality. It is well acknowledgement that Navier’s solution is most commonly used in analyzing static and dynamic problems of plates since its simple solution procedure. However, Navier’s solution is only suitable for plates with all edges simply supported. Consequently, it is of great significance to extent such effective classical approach to solve the problem of plates under more complex boundaries. Motivated by the above situation, a two-dimensional improved Fourier series approach is developed to provide exact vibration analysis of plates under the complicated non-classical boundary restraints. For the first time, we treat the two-dimensional half-sinusoidal Fourier series as the trial function for the deflection of plates. Interestingly, the adopted trial function is quite similar to the Navier’s form solution and automatically satisfies the zero deflection at rotationally restrained edges, zero effective shear force at free edges and zero twisting moment at the free corner. We also make a first endeavor to introduce the Stoke’s transformation technique on deriving detailed expressions for the first four order derivatives of the deflection, in which some associated unknown constants with obvious physical meaning can also be acquired. The obtained constants can be solved by dealing with sets of linear simultaneous equations after putting the obtained derivatives of the deflection into the remain boundaries. The present solution procedure and new Fourier series expansions, which can be directly adopted for solving mechanical problems of thick plates, are the main contribution of the present study. Another advantage of the present study is that new, precise free vibration results for rectangular thin plates under classical boundaries can be obtained by selecting different coefficients of the rotational springs introduced. Therefore, four different types of boundaries including RRFF, CCFF, CSFF and SSFF are considered for the investigated plate, where the free, rotationally restrained, simply supported and clamped edge are, respectively, designated as F, R, S and C. The present work chooses a clockwise representation for the boundaries studied, starting from the edges of x = 0. Finally, we list new precise results for the dimensionless natural frequency parameters and plot the corresponding mode shapes, all of which are testified to be qualified as new exact benchmarks after comparing with the ones solved by the conventional finite element method and other analytical methods.

2. Basic Equations

Consider a rectangular thin plate with length a, width b and uniform thickness h, and the coordinate of plates under title boundary conditions is depicted in Figure 1. Through the derivation with the consideration of the classic Kirchhoff assumptions, the free vibration governing equation can be expressed as:

$$\frac{{\partial }^{4}w\left(x,y,t\right)}{\partial x^4}+2\frac{{\partial }^{4}w\left(x,y,t\right)}{\partial x^2\partial y^2}+\frac{{\partial }^{4}w\left(x,y,t\right)}{\partial y^4}+\frac{\rho h}{D}\frac{{\partial }^{2}w\left(x,y,t\right)}{\partial t^2}=0$$

(1)

where $\mu$ is Poisson’s ratio, $D=E{h}^3/\left[12\left(1-{\mu }^2\right)\right]$ is flexural rigidity, E is elasticity modulus and $\rho$ is the mass density of the plate. Resultant internal forces including bending moments, effective shear forces and twisting moments can be described by the deflection w (x, y) as follow:

$$\begin{array}{l}{M}_x=-D\left(\frac{{\partial }^{2}w}{\partial x^2}+\mu \frac{{\partial }^{2}w}{\partial y^2}\right)\hspace{1em},\hspace{1em}{V}_x=-D\left[\frac{{\partial }^{3}w}{\partial x^3}+\left(2-\mu \right)\frac{{\partial }^{3}w}{\partial x\partial y^2}\right]\\ M_y=-D\left(\frac{{\partial }^{2}w}{\partial y^2}+\mu \frac{{\partial }^{2}w}{\partial x^2}\right)\hspace{1em},\hspace{1em}{V}_y=-D\left[\frac{{\partial }^{3}w}{\partial y^3}+\left(2-\mu \right)\frac{{\partial }^{3}w}{\partial x^2\partial y}\right]\\ M_{xy}=-D\left(1-\mu \right)\frac{{\partial }^{2}w}{\partial x\partial y}\end{array}$$

(2)

On basis of the classical vibration theory, the deflection for a free vibration plate could be preselected as $w\left(x,y,t\right)=\sin \left(\omega t\right)w\left(x,y\right)$. Then one will obtain

$$\frac{{\partial }^{4}w\left(x,y\right)}{\partial x^4}+2\frac{{\partial }^{4}w\left(x,y\right)}{\partial x^2\partial y^2}+\frac{{\partial }^{4}w\left(x,y\right)}{\partial y^4}-\frac{\rho h{\omega }^2}{D}w\left(x,y\right)=0$$

(3)

Expressions for the RRFF plate, in which rotationally-restrained at edges $x=0$ and $y=0$, and its opposite corner free, are described as:

$${\left.w\right|}_{x=0}={\left.w\right|}_{y=0}=0,\hspace{1em}{\left.V_x\right|}_{x=a}={\left.V_y\right|}_{y=b}=0,\hspace{1em}{\left.M_{xy}\right|}_{x=a,y=b}={\left.-D\left(1-\mu \right)\frac{{\partial }^{2}w}{\partial x\partial y}\right|}_{x=a,y=b}=0$$

(4)

$$\begin{array}{l}{\left.M_x\right|}_{x=0}=-D{\left.\frac{{\partial }^{2}w}{\partial x^2}\right|}_{x=0}=-R_{x0}{\left.\frac{\partial w}{\partial x}\right|}_{x=0}\hspace{1em},\hspace{1em}{\left.M_x\right|}_{x=a}=-D{\left.\left(\frac{{\partial }^{2}w}{\partial x^2}+\mu \frac{{\partial }^{2}w}{\partial y^2}\right)\right|}_{x=a}=0\hspace{1em}\\ {\left.M_y\right|}_{y=0}=-D{\left.\frac{{\partial }^{2}w}{\partial y^2}\right|}_{y=0}=-R_{y0}{\left.\frac{\partial w}{\partial y}\right|}_{y=0}\hspace{1em},\hspace{1em}{\left.M_y\right|}_{y=b}=-D{\left.\left(\frac{{\partial }^{2}w}{\partial y^2}+\mu \frac{{\partial }^{2}w}{\partial x^2}\right)\right|}_{y=b}=0\end{array}$$

(5)

where $R_{x0},R_{y0}$ are the rotational springs that are reflected through setting the rotating fixed factor $r$ from reference [36] as follows:

$$\frac{a\cdot R_{x0}}{D_x}=\frac{3r_{x0}}{1-r_{x0}}\hspace{1em},\hspace{1em}\frac{b\cdot R_{y0}}{D_y}=\frac{3r_{y0}}{1-r_{y0}}$$

(6)

As to problems of rectangular thin plates, Fourier series is usually adopted as the trial function for the deflection $w\left(x,y\right)$ due to its simple form and orthogonality properties. To deal with title problem, $w\left(x,y\right)$ is assumed to have the following form:

$$w\left(x,y\right)={\displaystyle \sum _{m=1,3}^{\infty }{\displaystyle \sum _{n=1,3,}^{\infty }{w}_{mn}\sin \frac{{\delta }_{m}x}{2}\sin \frac{{\lambda }_{n}y}{2}}}\hspace{1em}\hspace{1em}m,n=1,3,5,\cdots$$

(7)

where ${\delta }_m=\frac{m\pi }{a}$, ${\lambda }_n=\frac{n\pi }{b}$; $w_{mn}=\frac{4}{ab}{\displaystyle {\mathrm{\int }}_0^a{\displaystyle {\mathrm{\int }}_0^{b}w\left(x,y\right)\sin \frac{{\delta }_{m}x}{2}\sin \frac{{\lambda }_{n}y}{2}\mathrm{d}x\mathrm{d}y}}$ is the unknown Fourier coefficient for the deflection.

The term-by-term differentiation of the trial function is achieved by utilizing the Stoke’s transformation [42], which is the main contribution of the current study. Now utilizing the Stoke’s transformation, the detail expressions for the first four order derivatives of the deflection are given as below:

$$\begin{array}{l}\frac{\partial w}{\partial x}={\displaystyle \sum _{m=1,3}^{\infty }{\displaystyle \sum _{n=1,3,}^{\infty }\left(-\frac{2}{a}{c}_n+\frac{{\delta }_m}{2}{w}_{mn}\right)\cos \frac{{\delta }_{m}x}{2}\sin \frac{{\lambda }_{n}y}{2}}}\\ \frac{{\partial }^{2}w}{\partial x^2}={\displaystyle \sum _{m=1,3}^{\infty }{\displaystyle \sum _{n=1,3,}^{\infty }\left\{\frac{2}{a}\left[{\left(-1\right)}^{\frac{m-1}{2}}{d}_n+\frac{{\delta }_m}{2}{c}_n\right]-{\left(\frac{{\delta }_m}{2}\right)}^2{w}_{mn}\right\}\sin \frac{{\delta }_{m}x}{2}\sin \frac{{\lambda }_{n}y}{2}}}\\ \frac{{\partial }^{3}w}{\partial x^3}={\displaystyle \sum _{m=1,3}^{\infty }{\displaystyle \sum _{n=1,3,}^{\infty }\left\{\frac{2}{a}\left[-e_n+{\left(-1\right)}^{\frac{m-1}{2}}\frac{{\delta }_m}{2}{d}_n+{\left(\frac{{\delta }_m}{2}\right)}^2{c}_n\right]-{\left(\frac{{\delta }_m}{2}\right)}^3{w}_{mn}\right\}\cos \frac{{\delta }_{m}x}{2}\sin \frac{{\lambda }_{n}y}{2}}}\\ \frac{{\partial }^{4}w}{\partial x^4}={\displaystyle \sum _{m=1,3}^{\infty }{\displaystyle \sum _{n=1,3,}^{\infty }\left\{\frac{2}{a}\left[\begin{array}{l}{\left(-1\right)}^{\frac{m-1}{2}}{f}_n+\frac{{\delta }_m}{2}{e}_n\\ -{\left(-1\right)}^{\frac{m-1}{2}}{\left(\frac{{\delta }_m}{2}\right)}^2{d}_n-{\left(\frac{{\delta }_m}{2}\right)}^3{c}_n\end{array}\right]+{\left(\frac{{\delta }_m}{2}\right)}^4{w}_{mn}\right\}\sin \frac{{\delta }_{m}x}{2}\sin \frac{{\lambda }_{n}y}{2}}}\end{array}$$

(8)

$$\begin{array}{l}\frac{\partial w}{\partial y}={\displaystyle \sum _{m=1,3}^{\infty }{\displaystyle \sum _{n=1,3,}^{\infty }\left(-\frac{2}{b}{g}_m+\frac{{\lambda }_n}{2}{w}_{mn}\right)\sin \frac{{\delta }_{m}x}{2}\cos \frac{{\lambda }_{n}y}{2}}}\\ \frac{{\partial }^{2}w}{\partial y^2}={\displaystyle \sum _{m=1,3}^{\infty }{\displaystyle \sum _{n=1,3,}^{\infty }\left\{\frac{2}{b}\left[{\left(-1\right)}^{\frac{n-1}{2}}{h}_m+\frac{{\lambda }_n}{2}{g}_m\right]-{\left(\frac{{\lambda }_n}{2}\right)}^2{w}_{mn}\right\}\sin \frac{{\delta }_{m}x}{2}\sin \frac{{\lambda }_{n}y}{2}}}\\ \frac{{\partial }^{3}w}{\partial y^3}={\displaystyle \sum _{m=1,3}^{\infty }{\displaystyle \sum _{n=1,3,}^{\infty }\left\{\frac{2}{b}\left[-i_m+{\left(-1\right)}^{\frac{n-1}{2}}\frac{{\lambda }_n}{2}{h}_m+{\left(\frac{{\lambda }_n}{2}\right)}^2{g}_m\right]-{\left(\frac{{\lambda }_n}{2}\right)}^3{w}_{mn}\right\}\sin \frac{{\delta }_{m}x}{2}\cos \frac{{\lambda }_{n}y}{2}}}\\ \frac{{\partial }^{4}w}{\partial y^4}={\displaystyle \sum _{m=1,3}^{\infty }{\displaystyle \sum _{n=1,3,}^{\infty }\left\{\frac{2}{b}\left[\begin{array}{l}{\left(-1\right)}^{\frac{n-1}{2}}{j}_m+\frac{{\lambda }_n}{2}{i}_m\\ -{\left(-1\right)}^{\frac{n-1}{2}}{\left(\frac{{\lambda }_n}{2}\right)}^2{h}_m-{\left(\frac{{\lambda }_n}{2}\right)}^3{g}_m\end{array}\right]+{\left(\frac{{\lambda }_n}{2}\right)}^4{w}_{mn}\right\}\sin \frac{{\delta }_{m}x}{2}\sin \frac{{\lambda }_{n}y}{2}}}\end{array}$$

(9)

$$\begin{array}{l}\frac{{\partial }^{2}w}{\partial x\partial y}={\displaystyle \sum _{m=1,3}^{\infty }{\displaystyle \sum _{n=1,3,}^{\infty }\left(-\frac{2}{b}\frac{{\delta }_m}{2}{g}_m+\frac{{\delta }_m}{2}\frac{{\lambda }_n}{2}{w}_{mn}\right)\cos \frac{{\delta }_{m}x}{2}\cos \frac{{\lambda }_{n}y}{2}}}\\ \frac{{\partial }^{3}w}{\partial x\partial y^2}={\displaystyle \sum _{m=1,3}^{\infty }{\displaystyle \sum _{n=1,3,}^{\infty }\left\{-\frac{2}{a}{l}_n+\frac{2}{b}\frac{{\delta }_m}{2}\left[{\left(-1\right)}^{\frac{n-1}{2}}{h}_m+\frac{{\lambda }_n}{2}{g}_m\right]-\frac{{\delta }_m}{2}{\left(\frac{{\lambda }_n}{2}\right)}^2{w}_{mn}\right\}\cos \frac{{\delta }_{m}x}{2}\sin \frac{{\lambda }_{n}y}{2}}}\\ \frac{{\partial }^{3}w}{\partial x^2\partial y}={\displaystyle \sum _{m=1,3}^{\infty }{\displaystyle \sum _{n=1,3,}^{\infty }\left\{-\frac{2}{b}{k}_m+\frac{2}{a}\frac{{\lambda }_n}{2}\left[{\left(-1\right)}^{\frac{m-1}{2}}{d}_n+\frac{{\delta }_m}{2}{c}_n\right]-{\left(\frac{{\delta }_m}{2}\right)}^2\frac{{\lambda }_n}{2}{w}_{mn}\right\}\sin \frac{{\delta }_{m}x}{2}\cos \frac{{\lambda }_{n}y}{2}}}\\ \frac{{\partial }^{4}w}{\partial x^2\partial y^2}={\displaystyle \sum _{m=1,3}^{\infty }}{\displaystyle \sum _{n=1,3,}^{\infty }\left\{\begin{array}{l}\frac{4}{ab}{\left(-1\right)}^{\frac{m-1}{2}}\left[{\left.{\left(-1\right)}^{\frac{n-1}{2}}\frac{{\partial }^{2}w}{\partial x\partial y}\right|}_{x=a,y=b}+\frac{{\lambda }_n}{2}{\left.\frac{\partial w}{\partial x}\right|}_{x=a,y=0}\right]\\ -\frac{2}{a}\left[{\left(-1\right)}^{\frac{m-1}{2}}{\left(\frac{{\lambda }_n}{2}\right)}^2{d}_n-\frac{2}{b}\frac{{\delta }_m}{2}{l}_n\right]\\ -\frac{2}{b}{\left(\frac{{\delta }_m}{2}\right)}^2\left[{\left(-1\right)}^{\frac{n-1}{2}}{h}_m+\frac{{\lambda }_n}{2}{g}_m\right]+{\left(\frac{{\delta }_m}{2}\right)}^2{\left(\frac{{\lambda }_n}{2}\right)}^2{w}_{mn}\end{array}\right\}\sin \frac{{\delta }_{m}x}{2}\sin \frac{{\lambda }_{n}y}{2}}\end{array}$$

(10)

where

$$\begin{array}{l}{c}_n=\hspace{1em}\frac{2}{b}{\displaystyle {\mathrm{\int }}_0^b{\left.w\right|}_{x=0}}\sin \frac{{\lambda }_{n}y}{2}dy\hspace{1em},\hspace{1em}{d}_n=\frac{2}{b}{\displaystyle {\mathrm{\int }}_0^b{\left.\frac{\partial w}{\partial x}\right|}_{x=a}}\sin \frac{{\lambda }_{n}y}{2}dy\\ e_n=\hspace{1em}\frac{2}{b}{\displaystyle {\mathrm{\int }}_0^b{\left.\frac{{\partial }^{2}w}{\partial x^2}\right|}_{x=0}}\sin \frac{{\lambda }_{n}y}{2}dy\hspace{1em},\hspace{1em}{f}_n=\frac{2}{b}{\displaystyle {\mathrm{\int }}_0^b{\left.\frac{{\partial }^{3}w}{\partial x^3}\right|}_{x=a}}\sin \frac{{\lambda }_{n}y}{2}dy\hspace{1em}\\ \\ g_m=\frac{2}{a}{\displaystyle {\mathrm{\int }}_0^a{\left.w\right|}_{y=0}}\sin \frac{{\delta }_{m}x}{2}dx\hspace{1em},\hspace{1em}{h}_m=\frac{2}{a}{\displaystyle {\mathrm{\int }}_0^a{\left.\frac{\partial w}{\partial y}\right|}_{y=b}}\sin \frac{{\delta }_{m}x}{2}dx\\ i_m=\frac{2}{a}{\displaystyle {\mathrm{\int }}_0^a{\left.\frac{{\partial }^{2}w}{\partial y^2}\right|}_{y=0}}\sin \frac{{\delta }_{m}x}{2}dx\hspace{1em},\hspace{1em}{j}_m=\frac{2}{a}{\displaystyle {\mathrm{\int }}_0^a{\left.\frac{{\partial }^{3}w}{\partial y^3}\right|}_{y=b}}\sin \frac{{\delta }_{m}x}{2}dx\\ k_m=\frac{2}{a}{\displaystyle {\mathrm{\int }}_0^a{\left.\frac{{\partial }^{2}w}{\partial x^2}\right|}_{y=0}}\sin \frac{{\delta }_{m}x}{2}dx\hspace{1em},\hspace{1em}{l}_n=\frac{2}{b}{\displaystyle {\mathrm{\int }}_0^b{\left.\frac{{\partial }^{2}w}{\partial y^2}\right|}_{x=0}}\sin \frac{{\lambda }_{n}y}{2}dy\end{array}$$

(11)

Substituting the assumed $w\left(x,y\right)$ and its fourth high-order derivatives, that appeared in Equations (8)–(10), into Equation (3) yields the following expression for the free vibration governing equation:

$${\displaystyle \sum _{m=1,3}^{\infty }{\displaystyle \sum _{n=1,3}^{\infty }\left\{\begin{array}{l}\left[{\left(\frac{{\delta }_m^2}{4}+\frac{{\lambda }_n^2}{4}\right)}^2-\frac{\rho h{\omega }^2}{D}\right]w_{mn}\\ +\frac{2}{a}\left[{\left(-1\right)}^{\frac{m-1}{2}}{f}_n+\frac{{\delta }_m}{2}{e}_n-{\left(-1\right)}^{\frac{m-1}{2}}{\left(\frac{{\delta }_m}{2}\right)}^2{d}_n-{\left(\frac{{\delta }_m}{2}\right)}^3{c}_n\right]\\ +\frac{2}{b}\left[{\left(-1\right)}^{\frac{n-1}{2}}{j}_m+\frac{{\lambda }_n}{2}{i}_m-{\left(-1\right)}^{\frac{n-1}{2}}{\left(\frac{{\lambda }_n}{2}\right)}^2{h}_m-{\left(\frac{{\lambda }_n}{2}\right)}^3{g}_m\right]\\ +\frac{8}{ab}{\left(-1\right)}^{\frac{m-1}{2}}\left[{\left.{\left(-1\right)}^{\frac{n-1}{2}}\frac{{\partial }^{2}w}{\partial x\partial y}\right|}_{x=a,y=b}+\frac{{\lambda }_n}{2}{\left.\frac{\partial w}{\partial x}\right|}_{x=a,y=0}\right]\\ -\frac{4}{a}\left[{\left(-1\right)}^{\frac{m-1}{2}}{\left(\frac{{\lambda }_n}{2}\right)}^2{d}_n-\frac{2}{b}\frac{{\delta }_m}{2}{l}_n\right]-\frac{4}{b}{\left(\frac{{\delta }_m}{2}\right)}^2\left[{\left(-1\right)}^{\frac{n-1}{2}}{h}_m+\frac{{\lambda }_n}{2}{g}_m\right]\end{array}\right\}\sin \frac{{\alpha }_{m}x}{2}\sin \frac{{\beta }_{n}y}{2}\hspace{1em}}}=0$$

(12)

It is obvious that ${\left.\frac{{\partial }^{2}w}{\partial x\partial y}\right|}_{x=a,y=b}$ is zero since the zero-twisting moment at free corner. Similarly, due to the zero deflection at rotationally-restrained edges ($x=0$, $y=0$) and zero effective shear force at free edges ($x=a$, $y=b$), it is easy to derive that

$${\left.\frac{{\partial }^{2}w}{\partial y^2}\right|}_{x=0}=0\hspace{1em},\hspace{1em}{\left.\frac{\partial w}{\partial x}\right|}_{y=0}={\left.\frac{{\partial }^{2}w}{\partial x^2}\right|}_{y=0}=0$$

(13)

$$c_n=0\hspace{1em},\hspace{1em}{l}_n=0\hspace{1em},\hspace{1em}{g}_m=0\hspace{1em},\hspace{1em}{k}_m=0\hspace{1em},\hspace{1em}$$

(14)

$$f_n=\left(2-\mu \right){\left(\frac{{\lambda }_n}{2}\right)}^2{d}_n\hspace{1em},\hspace{1em}{j}_m=\left(2-\mu \right){\left(\frac{{\delta }_m}{2}\right)}^2{h}_m$$

(15)

It is clearly that $d_n \mathrm{and} h_m$ are the Fourier coefficients of the slope along the free edges. Obviously $-D\cdot i_m$ and $-D\cdot e_n$ are the Fourier coefficients of bending moments along the rotationally restrained edges, the corresponding bending moments can be directly acquired by the following formula:

$${\left.M_x\right|}_{x=0}={\left.-D\frac{{\partial }^{2}w}{\partial x^2}\right|}_{x=0}=-D{\displaystyle \sum _{n=1,3}^{\infty }{e}_n\sin \frac{{\lambda }_{n}y}{2}}\hspace{1em},\hspace{1em}{\left.\hspace{1em}{M}_y\right|}_{y=0}={\left.-D\frac{{\partial }^{2}w}{\partial y^2}\right|}_{y=0}=-D{\displaystyle \sum _{n=1,3}^{\infty }{i}_m\sin \frac{{\delta }_{m}x}{2}}$$

(16)

One can obtain the following simplified expression for the free vibration governing equations by substituting Equations (13)–(15) into the Equation (12), which is shown below:

$${\displaystyle \sum _{m=1,3}^{\infty }{\displaystyle \sum _{n=1,3}^{\infty }\left\{\begin{array}{l}\left[{\left(\frac{{\delta }_m^2}{4}+\frac{{\lambda }_n^2}{4}\right)}^2-\frac{\rho h{\omega }^2}{D}\right]w_{mn}\\ -\frac{2}{b}{\left(-1\right)}^{\frac{n-1}{2}}\left[\mu {\left(\frac{{\delta }_m}{2}\right)}^2+{\left(\frac{{\lambda }_n}{2}\right)}^2\right]h_m+\frac{2}{b}\frac{{\lambda }_n}{2}{i}_m\\ -\frac{2}{a}{\left(-1\right)}^{\frac{m-1}{2}}\left[{\left(\frac{{\delta }_m}{2}\right)}^2+\mu {\left(\frac{{\lambda }_n}{2}\right)}^2\right]d_n+\frac{2}{a}\frac{{\delta }_m}{2}{e}_n\end{array}\right\}\sin \frac{{\alpha }_{m}x}{2}\sin \frac{{\beta }_{n}y}{2}\hspace{1em}}}=0$$

(17)

After the simple derivation, one can obtain the expression of $w_{mn}$ with $d_n$, $e_n$, $h_m$ and $i_m$ to be determined.

$$w_{mn}={\xi }_{mn}\left[\frac{2}{b}{\chi }_{2mn}{h}_m-\frac{2}{b}\frac{{\lambda }_n}{2}{i}_m+\frac{2}{a}{\chi }_{1mn}{d}_n-\frac{2}{a}\frac{{\delta }_m}{2}{e}_n\right]$$

(18)

where

$${\xi }_{mn}=\frac{1}{{\left(\frac{{\delta }_m^2}{4}+\frac{{\lambda }_n^2}{4}\right)}^2-\frac{\rho h{\omega }^2}{D}}\hspace{1em},\hspace{1em}{\chi }_{1mn}={\left(-1\right)}^{\frac{m-1}{2}}\left[\frac{{\delta }_m^2}{4}+\mu \frac{{\lambda }_n^2}{4}\right]\hspace{1em},\hspace{1em}{\chi }_{2mn}={\left(-1\right)}^{\frac{n-1}{2}}\left[\mu \frac{{\delta }_m^2}{4}+\frac{{\lambda }_n^2}{4}\right]$$

(19)

Through the above solving procedure, it is clear that the preselected trial function satisfies the boundaries in Equation (4) automatically. The assumed deflection is still required to satisfy the boundaries in Equation (5), which can determine the unknown constants in $w_{mn}$. The boundary condition at the rotationally restrained edge will be satisfied by substituting the Equation (16) and the slopes in Equations (8) and (9) into Equation (5), which produces Equations (20) and (22). Similarly, utilizing the second-order derivatives of $w\left(x,y\right)$ in Equations (8) and (9) to acquire the bending moments at free edges and letting them to be equal to zero, which can satisfy the remain bending moments requirements and produce Equations (21) and (23).

$$-D{\displaystyle \sum _{n=1,3}^{\infty }{e}_n\sin \frac{{\lambda }_{n}y}{2}}=-R_{x0}{\displaystyle \sum _{n=1,3}^{\infty }{\displaystyle \sum _{m=1,3}^{\infty }\frac{{\delta }_m}{2}{w}_{mn}}}\sin \frac{{\lambda }_{n}y}{2}$$

(20)

$$-D{\displaystyle \sum _{n=1,3}^{\infty }{\displaystyle \sum _{m=1,3}^{\infty }{\left(-1\right)}^{\frac{m-1}{2}}\left\{\begin{array}{l}\frac{2}{a}{\left(-1\right)}^{\frac{m-1}{2}}{d}_n+\frac{2}{b}\mu {\left(-1\right)}^{\frac{n-1}{2}}{h}_m\\ -\left(\frac{{\delta }_m^2}{4}+\mu \frac{{\lambda }_n^2}{4}\right)w_{mn}\end{array}\right\}}}\sin \frac{{\lambda }_{n}y}{2}=0$$

(21)

$$-D{\displaystyle \sum _{m=1,3}^{\infty }{i}_m\sin \frac{{\delta }_{m}x}{2}}=-R_{y0}{\displaystyle \sum _{n=1,3}^{\infty }{\displaystyle \sum _{m=1,3}^{\infty }\frac{{\lambda }_n}{2}{w}_{mn}\sin \frac{{\delta }_{m}x}{2}}}$$

(22)

$$-D{\displaystyle \sum _{n=1,3}^{\infty }{\displaystyle \sum _{m=1,3}^{\infty }{\left(-1\right)}^{\frac{n-1}{2}}\left\{\begin{array}{l}\frac{2}{a}\mu {\left(-1\right)}^{\frac{m-1}{2}}{d}_n+\frac{2}{b}{\left(-1\right)}^{\frac{n-1}{2}}{h}_m\\ -\left(\mu \frac{{\delta }_m^2}{4}+\frac{{\lambda }_n^2}{4}\right)w_{mn}\end{array}\right\}}}\sin \frac{{\delta }_{m}x}{2}=0$$

(23)

Multiplying each side of Equations (20) and (21) by $\sin \frac{{\lambda }_{n}y}{2}\mathrm{d}y$ and integrating from 0 to b gives Equations (24) and (25) while multiplying each side of Equations (22) and (23) by $\sin \frac{{\delta }_{m}x}{2}\mathrm{d}x$ and integrating from 0 to a gives Equations (26) and (27), which are shown as follows:

$$\begin{array}{l}{\displaystyle \sum _{m=1,3}^{\infty }\frac{2}{b}{S}_{1mn}{\chi }_{2mn}{h}_m-{\displaystyle \sum _{m=1,3}^{\infty }\frac{2}{b}{S}_{1mn}\frac{{\chi }_n}{2}{i}_m}+{\displaystyle \sum _{m=1,3}^{\infty }\frac{2}{a}{S}_{1mn}{\chi }_{1mn}{d}_n}}\\ -\left[2a\left(1-r_{x0}\right)+{\displaystyle \sum _{m=1,3}^{\infty }\frac{2}{a}{S}_{1mn}\frac{{\delta }_m}{2}}\right]e_n=0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}n=1,3,5\cdots \end{array}$$

(24)

$$\begin{array}{l}{\displaystyle \sum _{m=1,3}^{\infty }\frac{2}{b}{\left(-1\right)}^{\frac{m-1}{2}}\left[S_{2mn}{\chi }_{2mn}-{\left(-1\right)}^{\frac{n-1}{2}}\mu \right]}{h}_m-{\displaystyle \sum _{m=1,3}^{\infty }\frac{2}{b}{\left(-1\right)}^{\frac{m-1}{2}}{S}_{2mn}\frac{{\lambda }_n}{2}}{i}_m\\ +{\displaystyle \sum _{m=1,3}^{\infty }\frac{2}{a}\left[{\left(-1\right)}^{\frac{m-1}{2}}{S}_{2mn}{\chi }_{1mn}-1\right]}{d}_n\\ -{\displaystyle \sum _{m=1,3}^{\infty }\frac{2}{a}{\left(-1\right)}^{\frac{m-1}{2}}{S}_{2mn}\frac{{\delta }_m}{2}}{e}_n=\hspace{1em}0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}n=1,3,5\cdots \hspace{1em}\end{array}$$

(25)

$$\begin{array}{l}{\displaystyle \sum _{n=1,3}^{\infty }\frac{2}{b}{S}_{3mn}{\chi }_{2mn}{h}_m-\left[2b\left(1-r_{y0}\right)+{\displaystyle \sum _{n=1,3}^{\infty }\frac{2}{b}{S}_{3mn}\frac{{\chi }_n}{2}}\right]i_m+{\displaystyle \sum _{n=1,3}^{\infty }\frac{2}{a}{S}_{3mn}{\chi }_{1mn}{d}_n}}\\ -{\displaystyle \sum _{n=1,3}^{\infty }\frac{2}{a}{S}_{3mn}\frac{{\delta }_m}{2}}{e}_n=0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}m=1,3,5\cdots \hspace{1em}\end{array}$$

(26)

$$\begin{array}{l}{\displaystyle \sum _{n=1,3}^{\infty }\frac{2}{b}\left[{\left(-1\right)}^{\frac{n-1}{2}}{S}_{4mn}{\chi }_{2mn}-1\right]}{h}_m-{\displaystyle \sum _{n=1,3}^{\infty }\frac{2}{b}{\left(-1\right)}^{\frac{n-1}{2}}{S}_{4mn}\frac{{\lambda }_n}{2}}{i}_m\\ +{\displaystyle \sum _{n=1,3}^{\infty }\frac{2}{a}{\left(-1\right)}^{\frac{n-1}{2}}\left[S_{4mn}{\chi }_{1mn}-{\left(-1\right)}^{\frac{m-1}{2}}\mu \right]}{d}_n\\ -{\displaystyle \sum _{n=1,3}^{\infty }\frac{2}{a}{\left(-1\right)}^{\frac{n-1}{2}}{S}_{4mn}\frac{{\delta }_m}{2}}{e}_n=\hspace{1em}0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}m=1,3,5\cdots \hspace{1em}\end{array}$$

(27)

where

$$\begin{array}{l}{S}_{1mn}=\frac{3r_{x0}{\delta }_m}{{\left(\frac{{\delta }_m^2}{4}+\frac{{\lambda }_n^2}{4}\right)}^2-\frac{\rho h{\omega }^2}{D}}\hspace{1em},\hspace{1em}{S}_{2mn}=\frac{\frac{{\delta }_m^2}{4}+\mu \frac{{\lambda }_n^2}{4}}{{\left(\frac{{\delta }_m^2}{4}+\frac{{\lambda }_n^2}{4}\right)}^2-\frac{\rho h{\omega }^2}{D}}\\ S_{3mn}=\frac{3r_{y0}{\lambda }_n}{{\left(\frac{{\delta }_m^2}{4}+\frac{{\lambda }_n^2}{4}\right)}^2-\frac{\rho h{\omega }^2}{D}}\hspace{1em},\hspace{1em}{S}_{4mn}=\frac{\mu \frac{{\delta }_m^2}{4}+\frac{{\lambda }_n^2}{4}}{{\left(\frac{{\delta }_m^2}{4}+\frac{{\lambda }_n^2}{4}\right)}^2-\frac{\rho h{\omega }^2}{D}}\end{array}$$

(28)

The complicated title problem can be turned into dealing with sets of infinite linear algebra equations composed of Equations (24)–(27) via the above derivations. A zero determinant of above matrix is needed since the requirement of non-zero solutions, which gives the natural frequencies. Subsequently, the constants $d_n$, $e_n$, $h_m$ and $i_m$ ($m=1,3,\cdots ,M; n=1,3,\cdots ,N$) can be acquired by substituting the obtained natural frequencies into the corresponding matrix. Finally, the associate free vibration mode shapes will be determined through using the $w_{mn}$ determined by the acquired constants. It is theoretically possible for one to obtain exact results when m and n are close to infinity. Actually, convergent results with desired accuracy can be obtained by solving sets of finite equations containing the identical count of unknowns. In the present study, the identical terms s is selected for m and n, with their upper limit setting at $M=\frac{2s-1}{2}$ and $N=\frac{2s-1}{2}$.

3. Comprehensive Frequency Parameter and Mode Shape Results

Aimed at testifying the capability of the proposed approach in predicting free vibration response for rectangular thin plates with classic/non-classic type edge conditions, new precise comprehensive results including frequency parameters and the associated mode shapes are in comparison with FEM results and the existing analytical solutions [16,43]. It is worth noting that the reliable FEM results for comparison are acquired by using ABAQUS 6.13 (2013), where 4-node thin shell element S4R is employed. To guarantee the accuracy of the FEM results, we testify the convergence for the FEM results of CCFF plate with aspect ratio b/a = 1, as is shown in

Table 1. Convergence of the dimensionless frequency parameter $\varphi$ obtained by FEM for the CCFF square plates with different mesh sizes.

	Mode
	First	Second	Third	Fourth	Fifth	Sixth	Seventh	Eighth	Ninth	Tenth
1/100	6.9190	23.905	26.590	47.654	62.711	65.572	85.726	88.377	121.50	124.21
1/200	6.9187	23.901	26.585	47.646	62.711	65.539	85.694	88.341	121.38	124.08
1/300	6.9185	23.900	26.584	47.642	62.705	65.532	85.686	88.332	121.35	124.06
1/400	6.9184	23.900	26.583	47.640	62.702	65.529	85.682	88.328	121.34	124.05
1/500	6.9183	23.900	26.583	47.639	62.702	65.529	85.682	88.327	121.34	124.05

. It is found that most of the obtained FEM results for $\varphi$ converge when the mesh size is 1/400 of the plate length. Thus, the uniform mesh size with short edge length of 1/400 is adopted throughout the present study. It is notable that the investigation on classic boundaries is attained through changing the coefficient r, in which $r=0.999999$ means clamped and $r=0.000001$ means simply-supported. Meanwhile, effects on free vibration of plates aroused by aspect ratio b/a and different degrees of boundary constraint are also conducted, in the present study Poisson’s ratio $\mu$ equals 0.3, the aspect ratio $\chi =b/a$ of the CCFF, CSFF and SSFF rectangular plate ranges from 0.5 to 5 and the coefficient $r$ of the RRFF square plate ranges from 0.1 to 0.9.

We first carry out vibration analysis on evaluating the non-dimensional frequency parameter, $\varphi =\omega a^2\sqrt{\rho h/D}$, for CSFF, SSFF and CCFF plates under 10 different aspect ratios such that 100 results are present for each case, which are tableted in Table 1,

Table 2. Frequency parameter $\varphi$ of CCFF rectangular plates with defined rotational fixity factors $r_{x0}=r_{y0}=0.999999$ and aspect ratio $\chi$ changing from 0.5 to 5.

$$\mathit{\chi }$$		Mode
		First	Second	Third	Fourth	Fifth	Sixth	Seventh	Eighth	Ninth	Tenth
0.5	Present	17.118	36.372	73.402	90.902	115.20	131.62	158.98	208.74	219.42	248.45
	Ref. [43]	17.118	36.372	73.402	90.902	115.20	-	-	-	-	-
	FEM	17.116	36.363	73.391	90.894	115.17	131.60	158.93	208.73	219.36	248.46
1	Present	6.9191	23.903	26.587	47.651	62.706	65.531	85.697	88.342	121.35	124.05
	Ref. [43]	6.9191	23.903	26.587	47.651	62.706	-	-	-	-	-
	FEM	6.9184	23.900	26.583	47.640	62.702	65.529	85.682	88.328	121.34	124.05
1.5	Present	4.9677	13.235	23.278	30.112	34.138	52.236	56.033	62.667	72.788	78.511
	FEM	4.9673	13.233	23.275	30.110	34.132	52.227	56.029	62.666	72.781	78.498
2	Present	4.2795	9.0929	18.351	22.726	28.800	32.904	39.745	52.184	54.855	62.112
	Ref. [43]	4.2795	9.0929	18.351	22.726	28.800	-	-	-	-	-
	FEM	4.2793	9.0918	18.349	22.724	28.795	32.901	39.738	52.181	54.846	62.111
2.5	Present	3.9726	7.1296	13.038	21.728	22.798	26.402	33.479	34.675	43.607	50.053
	FEM	3.9725	7.1287	13.037	21.726	22.796	26.399	33.474	34.675	43.600	50.049
3	Present	3.8149	6.0344	10.143	16.364	22.227	24.570	25.416	30.169	35.642	37.386
	FEM	3.8148	6.0338	10.142	16.362	22.225	24.567	25.414	30.164	35.641	37.380
3.5	Present	3.7249	5.3602	8.3946	12.940	19.077	22.202	24.211	26.928	28.171	33.423
	FEM	3.7248	5.3598	8.3937	12.939	19.076	22.200	24.208	26.925	28.167	33.417
4	Present	3.6692	4.9175	7.2540	10.721	15.406	21.228	22.251	23.714	26.642	28.644
	FEM	3.6692	4.9171	7.2533	10.720	15.405	21.226	22.249	23.712	26.639	28.642
4.5	Present	3.6327	4.6127	6.4673	9.2024	12.885	17.543	22.045	23.047	23.585	25.704
	FEM	3.6328	4.6125	6.4668	9.2015	12.884	17.541	22.043	23.046	23.585	25.701
5	Present	3.6075	4.3952	5.9014	8.1166	11.086	14.844	19.383	22.081	23.066	24.665
	FEM	3.6075	4.3949	5.9009	8.1158	11.085	14.843	19.382	22.080	23.064	24.663

and

Table 3. Frequency parameter $\varphi$ of CSFF rectangular plates with defined rotational fixity factors $r_{x0}=0.999999, r_{y0}=0.000001$ and aspect ratio $\chi$ changing from 0.5 to 5.

$$\mathit{\chi }$$		Mode
		First	Second	Third	Fourth	Fifth	Sixth	Seventh	Eighth	Ninth	Tenth
0.5	Present	8.5061	30.954	64.140	70.962	92.922	128.79	139.47	199.83	202.51	208.60
	Ref. [43]	8.5062	30.955	64.142	70.962	92.927	-	-	-	-	-
	FEM	8.5030	30.943	64.137	70.950	92.899	128.79	139.45	199.81	202.50	208.59
1	Present	5.3508	19.073	24.673	43.087	52.706	63.754	77.488	83.660	106.29	120.34
	Ref. [43]	5.3509	19.075	24.671	43.087	52.707	-	-	-	-	-
	FEM	5.3499	19.071	24.675	43.076	52.706	63.754	77.476	83.652	106.28	120.33
1.5	Present	4.4122	10.870	22.852	25.572	32.347	48.278	49.368	62.439	71.359	72.932
	FEM	4.4117	10.869	22.850	25.570	32.340	48.270	49.364	62.437	71.351	72.920
2	Present	4.0292	7.8658	15.849	22.573	27.800	29.251	37.927	47.071	51.891	61.980
	Ref. [43]	4.0294	7.8661	15.850	22.576	27.801	-	-	-	-	-
	FEM	4.0290	7.8649	15.847	22.572	27.798	29.250	37.924	47.071	51.887	61.979
2.5	Present	3.8434	6.4013	11.522	19.657	22.430	25.953	31.191	32.724	41.880	46.016
	FEM	3.8433	6.4005	11.520	19.656	22.428	25.949	31.189	32.718	41.872	46.015
3	Present	3.7409	5.5654	9.1527	14.784	21.990	22.884	24.843	29.525	32.902	36.284
	FEM	3.7409	5.5650	9.1523	14.785	21.990	22.883	24.841	29.523	32.901	36.281
3.5	Present	3.6792	5.0420	7.7087	11.830	17.544	22.149	23.969	25.093	27.675	32.667
	FEM	3.6791	5.0416	7.7079	11.829	17.543	22.147	23.966	25.092	27.671	32.662
4	Present	3.6393	4.6931	6.7580	9.9093	14.261	19.828	22.154	23.584	26.245	26.958
	FEM	3.6392	4.6928	6.7574	9.9084	14.260	19.826	22.152	23.583	26.242	26.956
4.5	Present	3.6121	4.4497	6.0968	8.5895	12.010	16.405	21.614	22.259	23.298	25.486
	FEM	3.6121	4.4494	6.0962	8.5887	12.009	16.404	21.613	22.257	23.297	25.483
5	Present	3.5927	4.2735	5.6176	7.6419	10.402	13.945	18.278	22.041	22.938	23.570
	FEM	3.5928	4.2733	5.6171	7.6412	10.401	13.944	18.278	22.040	22.936	23.569

. The corresponding vibration mode shapes for the CSFF, CCFF and SSFF square plates are shown in Figure 2, Figure 3 and Figure 4. As to the RRFF square plates, two different cases are considered: (1) the coefficient $r_{x0}=r_{y0}=r$ ranges from 0.1 to 0.9, new exact results for $\varphi$ are given in

Table 4. Frequency parameter $\varphi$ of SSFF rectangular plates with defined rotational fixity factors $r_{x0}=r_{y0}=0.000001$ and aspect ratio $\chi$ changing from 0.5 to 5.

$$\mathit{\chi }$$		Mode
		First	Second	Third	Fourth	Fifth	Sixth	Seventh	Eighth	Ninth	Tenth
0.5	Present	6.6436	25.374	58.740	65.177	89.093	113.14	131.50	185.64	189.60	202.12
	Ref. [16]	6.644	25.370	58.739	65.183	89.084	113.17	-	-	-	-
	Ref. [43]	6.6437	25.375	58.741	65.179	89.096	-	-	-	-	-
	FEM	6.6401	25.366	58.729	65.175	89.065	113.13	131.47	185.63	189.55	202.13
1	Present	3.3668	17.314	19.292	38.208	51.034	53.485	72.964	74.626	104.71	107.22
	Ref. [16]	3.370	17.321	19.293	38.203	51.032	53.497	-	-	-	-
	Ref. [43]	3.3670	17.316	19.293	38.210	51.035	-	-	-	-	-
	FEM	3.3659	17.313	19.292	38.199	51.033	53.485	72.945	74.614	104.71	107.21
1.5	Present	2.2328	9.5393	16.679	24.543	26.993	44.058	48.175	51.149	61.061	68.835
	FEM	2.2321	9.5374	16.678	24.540	26.987	44.047	48.173	51.148	61.052	68.821
2	Present	1.6607	6.3435	14.684	16.293	22.273	28.291	32.875	46.416	47.401	50.531
	Ref. [43]	1.6609	6.3438	14.685	16.295	22.274	-	-	-	-	-
	FEM	1.6605	6.3427	14.682	16.292	22.271	28.291	32.870	46.415	47.395	50.530
2.5	Present	1.3195	4.7306	10.316	15.789	18.845	20.100	27.184	30.659	37.051	45.407
	FEM	1.3192	4.7296	10.314	15.789	18.843	20.097	27.179	30.658	37.043	45.406
3	Present	1.0937	3.7679	7.8075	13.722	15.734	18.629	21.949	23.967	31.060	32.321
	FEM	1.0935	3.7674	7.8068	13.720	15.734	18.627	21.947	23.966	31.058	32.320
3.5	Present	0.93401	3.1304	6.2428	10. 680	15.513	16.705	17.882	21.828	24.305	27.372
	FEM	0.93375	3.1297	6.2418	10. 679	15.512	16.704	17.880	21.825	24.303	27.367
4	Present	0.81491	2.6776	5.1872	8.6529	13.220	15.555	17.240	19.093	20.477	24.784
	FEM	0.81468	2.6770	5.1863	8.6518	13.219	15.555	17.239	19.092	20.475	24.780
4.5	Present	0.72280	2.3397	4.4321	7.2357	10.892	15.228	15.742	16.910	19.424	21.101
	FEM	0.72260	2.3391	4.4314	7.2346	10.891	15.227	15.742	16.909	19.422	21.100
5	Present	0.64944	2.0779	3.8673	6.2000	9.1985	12.915	15.471	16.562	17.498	18.740
	FEM	0.64926	2.0775	3.8667	6.1990	9.1975	12.914	15.471	16.562	17.497	18.738

; (2) the coefficient $r_{x0}=r$ ranging from 0.1 to 0.9 with $r_{y0}=0.5$, and new benchmark results for $\varphi$ are shown in

Table 5. Frequency parameter $\varphi$ of square RRFF plates with the defined rotational fixity factors $r_{x0}=r_{y0}=r$ changing from 0.1 to 0.9.

r		Mode
		First	Second	Third	Fourth	Fifth	Sixth	Seventh	Eighth	Ninth	Tenth
0.1	Present	3.6851	17.606	19.599	38.510	51.357	53.811	73.264	74.951	105.04	107.56
	FEM	3.6849	17.606	19.598	38.509	51.356	53.809	73.256	74.944	105.03	107.55
0.2	Present	4.0032	17.936	19.951	38.864	51.741	54.198	73.625	75.338	105.44	107.96
	FEM	4.0030	17.935	19.949	38.861	51.740	54.197	73.611	75.324	105.43	107.95
0.3	Present	4.3243	18.312	20.356	39.279	52.199	54.662	74.057	75.805	105.93	108.44
	FEM	4.3242	18.311	20.352	39.274	52.195	54.660	74.047	75.800	105.92	108.44
0.4	Present	4.6513	18.746	20.827	39.775	52.757	55.228	74.588	76.381	106.53	109.05
	FEM	4.6512	18.745	20.824	39.771	52.756	55.227	74.580	76.372	106.52	109.03
0.5	Present	4.9870	19.251	21.381	40.379	53.451	55.936	75.257	77.108	107.31	109.83
	FEM	4.9867	19.250	21.380	40.374	53.446	55.932	75.247	77.101	107.30	109.82
0.6	Present	5.3342	19.848	22.041	41.129	54.335	56.841	76.123	78.053	108.35	110.87
	FEM	5.3340	19.847	22.040	41.126	54.333	56.839	76.114	78.042	108.33	110.86
0.7	Present	5.6963	20.562	22.837	42.087	55.496	58.035	77.290	79.328	109.80	112.33
	FEM	5.6962	20.561	22.835	42.086	55.493	58.034	77.277	79.316	109.77	112.32
0.8	Present	6.0771	21.433	23.813	43.351	57.078	59.670	78.942	81.133	111.94	114.48
	FEM	6.0768	21.431	23.812	43.350	57.077	59.665	78.931	81.123	111.93	114.47
0.9	Present	6.4816	22.515	25.033	45.096	59.327	62.012	81.453	83.857	115.35	117.95
	FEM	6.4813	22.514	25.032	45.094	59.325	62.009	81.446	83.845	115.34	117.95

. The first five vibration modes for the second case of RRFF square plates are given in Figure 5, where the coefficient $r_{x0}=0.1, 0.3, 0.5 \mathrm{and} 0.8$, respectively.

As is shown in Table 2, Table 3 and Table 4, the non-dimensional frequency parameter decreases as the aspect ratio $\chi =b/a$ increases and vice versa. Therefore, the minimum values of $\varphi$ corresponds to the aspect ratio (5.0). The non-dimensional frequency parameter decreases rapidly when the aspect ratio is 1 or less than one. Additionally, it can be observed that the values of $\varphi$ changes very little when $\chi =b/a$ changes from 2.5 to 5. It can be concluded that the aspect ratio from 0.5 to 2.0 has more influence on the natural frequencies as compared to the aspect ratio from 2.5 to 5. Through Table 2, Table 3 and Table 4, it also found that boundary conditions have great influence on the non-dimensional frequency parameter. The acquired $\varphi$ for all boundary conditions is higher as the aspect ratio gets smaller than one. The magnitude of $\varphi$ is always greater for CCFF than those under CSFF and SSFF when aspect ratio $\chi =b/a$ is same. Moreover, the non-dimensional frequency parameter for CSFF is higher than the SSFF for all aspect ratio. It can be concluded that the plate with more clamped edges requires more energy to vibrate. Through the comparisons in Table 2, Table 3 and Table 4, it is clearly seen that all the results obtained are in good agreement with ones solved numerically or analytically, especially with those in Ref. [43]. Similarly, through Table 5 and

Table 6. Frequency parameter $\varphi$ of square RRFF plates with the defined rotational fixity factors $r_{y0}=0.5$ and $r_{x0}=r$ changing from 0.1 to 0.9.

r		Mode
		First	Second	Third	Fourth	Fifth	Sixth	Seventh	Eighth	Ninth	Tenth
0.1	Present	4.3770	18.259	20.678	39.456	52.080	55.202	74.159	76.145	105.79	109.09
	FEM	4.3769	18.257	20.675	39.453	52.077	55.200	74.149	76.137	105.78	109.08
0.2	Present	4.5172	18.481	20.789	39.628	52.371	55.294	74.370	76.301	106.10	109.17
	FEM	4.5168	18.480	20.781	39.622	52.366	55.290	74.351	76.287	106.08	109.16
0.3	Present	4.6651	18.723	20.933	39.832	52.699	55.425	74.618	76.499	106.46	109.29
	FEM	4.6648	18.720	20.931	39.825	52.693	55.421	74.609	76.485	106.45	109.27
0.4	Present	4.8214	18.981	21.123	40.078	53.064	55.622	74.910	76.758	106.87	109.49
	FEM	4.8211	18.979	21.122	40.073	53.060	55.620	74.899	76.736	106.86	109.48
0.5	Present	4.9870	19.251	21.381	40.379	53.451	55.936	75.257	77.108	107.31	109.83
	FEM	4.9867	19.250	21.380	40.374	53.446	55.932	75.247	77.101	107.30	109.82
0.6	Present	5.1627	19.525	21.737	40.755	53.829	56.453	75.670	77.602	107.74	110.44
	FEM	5.1623	19.521	21.735	40.750	53.827	56.450	75.655	77.589	107.72	110.43
0.7	Present	5.3496	19.797	22.228	41.240	54.161	57.300	76.171	78.332	108.09	111.54
	FEM	5.3493	19.795	22.222	41.236	54.157	57.295	76.156	78.322	108.07	111.53
0.8	Present	5.5487	20.065	22.897	41.885	54.438	58.638	76.802	79.454	108.36	113.40
	FEM	5.5482	20.063	22.896	41.883	54.435	58.632	76.788	79.437	108.33	113.39
0.9	Present	5.7614	20.340	23.797	42.778	54.686	60.699	77.674	81.256	108.58	116.55
	FEM	5.7611	20.337	23.792	42.775	54.680	60.691	77.660	81.244	108.57	116.54

, it is obvious that the obtained $\varphi$ increases with the increase of the rotating fixed coefficient r, which indicates that the RRFF plate with stricter boundary constraints needs more energy to vibrate. The aforesaid parametric analysis indicates that both the aspect ratios and the degree of boundary restraint effect free vibration behaviors of plates significantly.

As is shown in Figure 2, Figure 3 and Figure 4, it is easily found that all the obtained free vibration mode shapes for the CCFF, CSFF and SSFF plates match well with the ones provided by the Finite element method. It is also obviously found the present mode shapes strictly satisfy the boundary conditions, which confirm the accuracy of the present method. Through Figure 5, it is found that the mode shapes for the RRFF plate under different degree boundary restraints change gradually, which further confirm the affections of boundary conditions on the free vibration characteristics of plates.

We finally carry out convergence study for the dimensionless frequency parameter $\varphi$ of all the five cases, as illustrated by

Table 7. Convergence of the dimensionless frequency parameter $\varphi$ for the CCFF plates with aspect ratio $\chi$ equaling 0.5 and 4.5, respectively.

b/a	s	Mode
		First	Second	Third	Fourth	Fifth	Sixth	Seventh	Eighth	Ninth	Tenth
0.5	30	17.116	36.377	73.408	90.929	115.23	131.65	159.01	208.76	219.46	248.47
	50	17.117	36.374	73.405	90.915	115.21	131.63	158.99	208.75	219.44	248.46
	100	17.118	36.372	73.403	90.906	115.20	131.62	158.98	208.74	219.43	248.45
	150	17.118	36.372	73.402	90.902	115.20	131.62	158.98	208.74	219.42	248.45
	200	17.118	36.372	73.402	90.902	115.20	131.62	158.98	208.74	219.42	248.45
4.5	30	3.6323	4.6126	6.4676	9.2037	12.886	17.547	22.052	23.053	23.591	25.712
	50	3.6325	4.6127	6.4674	9.2029	12.885	17.545	22.048	23.050	23.588	25.708
	100	3.6327	4.6127	6.4673	9.2025	12.885	17.543	22.046	23.048	23.586	25.705
	150	3.6327	4.6127	6.4673	9.2024	12.885	17.543	22.045	23.047	23.585	25.704
	200	3.6327	4.6127	6.4673	9.2024	12.885	17.543	22.045	23.047	23.585	25.704

,

Table 8. Convergence of the dimensionless frequency parameter $\varphi$ for the CSFF plates with aspect ratio $\chi$ equaling 0.5 and 4.5, respectively.

b/a	s	Mode
		First	Second	Third	Fourth	Fifth	Sixth	Seventh	Eighth	Ninth	Tenth
0.5	30	8.5065	30.961	64.141	70.967	92.929	128.83	139.49	199.85	202.56	208.64
	50	8.5063	30.958	64.141	70.964	92.924	128.81	139.49	199.85	202.53	208.62
	100	8.5062	30.955	64.141	70.963	92.923	128.80	139.48	199.84	202.52	208.61
	150	8.5061	30.954	64.140	70.962	92.922	128.79	139.47	199.83	202.51	208.60
	200	8.5061	30.954	64.140	70.962	92.922	128.79	139.47	199.83	202.51	208.60
4.5	30	3.6116	4.4492	6.0963	8.5890	12.009	16.404	21.615	22.264	23.305	25.493
	50	3.6119	4.4494	6.0965	8.5892	12.010	16.405	21.615	22.262	23.302	25.489
	100	3.6120	4.4496	6.0967	8.5894	12.010	16.405	21.614	22.260	23.299	25.487
	150	3.6121	4.4497	6.0968	8.5895	12.010	16.405	21.614	22.259	23.298	25.486
	200	3.6121	4.4497	6.0968	8.5895	12.010	16.405	21.614	22.259	23.298	25.486

,

Table 9. Convergence of the dimensionless frequency parameter $\varphi$ for the SSFF plates with aspect ratio $\chi$ equaling 0.5 and 4.5, respectively.

b/a	s	Mode
		First	Second	Third	Fourth	Fifth	Sixth	Seventh	Eighth	Ninth	Tenth
0.5	30	6.6432	25.372	58.738	65.175	89.089	113.13	131.49	185.63	189.58	202.11
	50	6.6434	25.373	58.739	65.176	89.092	113.14	131.49	185.63	189.59	202.12
	100	6.6435	25.374	58.740	65.176	89.092	113.14	131.50	185.64	189.60	202.12
	150	6.6436	25.374	58.740	65.177	89.093	113.14	131.50	185.64	189.60	202.12
	200	6.6436	25.374	58.740	65.177	89.093	113.14	131.50	185.64	189.60	202.12
4.5	30	0.72276	2.3395	4.4317	7.2351	10.891	15.227	15.742	16.910	19.423	21.100
	50	0.72278	2.3396	4.4319	7.2354	10.891	15.227	15.742	16.910	19.423	21.101
	100	0.72280	2.3396	4.4321	7.2356	10.892	15.228	15.742	16.910	19.424	21.101
	150	0.72280	2.3397	4.4321	7.2357	10.892	15.228	15.742	16.910	19.424	21.101
	200	0.72280	2.3397	4.4321	7.2357	10.892	15.228	15.742	16.910	19.424	21.101

,

Table 10. Convergence of the dimensionless frequency parameter $\varphi$ for the RRFF square plates with $r=r_{x0}=r_{y0}$ equaling 0.1 and 0.8, respectively.

$$\mathit{R} = {\mathit{r}}_{\mathit{x}0} = {\mathit{r}}_{\mathit{y}0}$$	s	Mode
		First	Second	Third	Fourth	Fifth	Sixth	Seventh	Eighth	Ninth	Tenth
0.1	30	3.6848	17.605	19.598	38.507	51.354	53.809	73.260	74.947	105.03	107.55
	50	3.6849	17.606	19.599	38.508	51.355	53.810	73.262	74.949	105.03	107.56
	100	3.6851	17.606	19.599	38.510	51.356	53.811	73.263	74.950	105.04	107.56
	150	3.6851	17.606	19.599	38.510	51.357	53.811	73.264	74.951	105.04	107.56
	200	3.6851	17.606	19.599	38.510	51.357	53.811	73.264	74.951	105.04	107.56
0.8	30	6.0770	21.436	23.816	43.352	57.077	59.670	78.939	81.131	111.95	114.49
	50	6.0771	21.434	23.814	43.352	57.077	59.670	78.941	81.132	111.94	114.49
	100	6.0771	21.433	23.813	43.351	57.078	59.670	78.942	81.133	111.94	114.49
	150	6.0771	21.433	23.813	43.351	57.078	59.670	78.942	81.133	111.94	114.48
	200	6.0771	21.433	23.813	43.351	57.078	59.670	78.942	81.133	111.94	114.48

and

Table 11. Convergence of the dimensionless frequency parameter $\varphi$ for the RRFF square plates with $r_{y0}=0.5$ and $r_{x0}$ equaling 0.1 and 0.8, respectively.

$${\mathit{R}}_{\mathit{x}0}$$	s	Mode
		First	Second	Third	Fourth	Fifth	Sixth	Seventh	Eighth	Ninth	Tenth
0.1	30	4.3767	18.256	20.676	39.453	52.078	55.200	74.152	76.140	105.77	109.08
	50	4.3768	18.257	20.677	39.454	52.079	55.201	74.156	76.142	105.78	109.08
	100	4.3770	18.259	20.678	39.455	52.080	55.201	74.158	76.144	105.79	109.08
	150	4.3770	18.259	20.678	39.456	52.080	55.202	74.159	76.145	105.79	109.09
	200	4.3770	18.259	20.678	39.456	52.080	55.202	74.159	76.145	105.79	109.09
0.8	30	5.5485	20.066	22.900	41.884	54.436	58.636	76.797	79.450	108.34	113.41
	50	5.5486	20.066	22.899	41.884	54.438	58.637	76.800	79.452	108.35	113.41
	100	5.5487	20.065	22.898	41.885	54.438	58.637	76.801	79.453	108.36	113.40
	150	5.5487	20.065	22.897	41.885	54.438	58.638	76.802	79.454	108.36	113.40
	200	5.5487	20.065	22.897	41.885	54.438	58.638	76.802	79.454	108.36	113.40

, in which the convergent results are precise up to five significant figures and are marked in bold. It should be pointed that all the given solutions converge at a slower rate due to the Fourier series form solution. However, the acquired simultaneous equations are easily handled through using the commercial software such as Mathematica 10.3. For each case, it is obviously seen that most of $\varphi$ are convergent at $s=150$, which leads to 150 series of terms that are adopted throughout the present work. Excellent agreement is found between the present numerical/graphical results with FEM solutions and other analytical solutions, which confirms the qualification of the present method in solving complex plate problems. Obviously, all the present the non-dimensional natural frequency parameters and the associated vibration mode shapes can be adopted for engineers and scientists for academic and practical applications. The laws indicated by the present parametric analysis enable designers to effectively avoid resonance problems of rectangular thin plate structures.

4. Conclusions

A new two-dimensional improved Fourier series approach is developed to seek an exact analytical free vibration solution of thin plates under classical/non-classical edge conditions. The significant merits of the present method differing from other analytical methods are: (1) it provides a more simple and straightforward solution procedure for precise plate free vibration analysis since it avoids some complicated mathematical manipulations such as the transformation procedure in finite integral transform method or superposition procedure in symplectic superposition approach; (2) within the framework of Fourier series theory, it reduces the mathematical difficulty of the plate problem by converting the boundary value problems of higher-order partial differential equations into solving linear algebra equations; (3) it provides more precise analytical solutions for mechanical problems of moderately-thick/thick plates under more complicated boundary conditions by using different types of Fourier series. The present trial function exactly satisfies both the governing vibration formula and the non-classical boundaries after determining the unknown constants with obvious physics meaning. In addition, new analytical solutions for the CCFF, CSFF and SSFF plates are also acquired by choosing introduced rotating fixed coefficients. Consequently, we provided new exact results, including 50 mode shapes and more than 400 natural frequencies, for the intractable title problem. In the present parametric analysis, it is found that both the aspect ratio and boundary restraint significantly influence the free vibration behaviors of plates. All the analytical results are precise enough since they agree well with the results obtained by using the FEM and other analytical methods, which are promising to validate solutions of other methods.