# Free Vibration Analysis of Moderately Thick Rectangular Plates with Variable Thickness and Arbitrary Boundary Conditions

Authors: Dongyan Shi; Qingshan Wang; Xianjie Shi; Fuzhen Pang
Journal: Shock and Vibration
 (2014)

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## Abstract

A generalized Fourier series solution based on the first-order shear deformation theory is presented for the free vibrations of moderately thick rectangular plates with variable thickness and arbitrary boundary conditions, a class of problem which is of practical interest and fundamental importance but rarely attempted in the literatures. Unlike in most existing studies where solutions are often developed for a particular type of boundary conditions, the current method can be generally applied to a wide range of boundary conditions with no need of modifying solution algorithms and procedures. Under the current framework, the one displacement and two rotation functions are generally sought, regardless of boundary conditions, as an improved trigonometric series in which several supplementary functions are introduced to remove the potential discontinuities with the displacement components and its derivatives at the edges and to accelerate the convergence of series representations. All the series expansion coefficients are treated as the generalized coordinates and solved using the Rayleigh-Ritz technique. The effectiveness and reliability of the presented solution are demonstrated by comparing the present results with those results published in the literatures and finite element method (FEM) data, and numerous new results for moderately thick rectangular plates with nonuniform thickness and elastic restraints are presented, which may serve as benchmark solution for future researches.


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## Body

## 1. Introduction

In comparison with vibration analysis of plates with uniform thickness, far less attention has been paid to the vibration problems of plates with variable thickness, which are commonly widely used in engineering applications as a stand-alone structure or a constituent structural component. By carefully designing the thickness distribution, a substantial increase in stiffness, buckling, and vibration capacities of the plate may be obtained over its uniform thickness counterpart. In addition, the moderately thick plates with variable thickness in these practical applications often work in complex environments and suffer from arbitrary edge restraints. Therefore a thorough dynamic study of moderately thick rectangular plates with variable thickness and arbitrary edge restraints is essential to assess and use the full potentials of plates.Over past decades, extensive investigations have been carried out to determine the vibration characteristics (natural frequencies, mode shapes, and so on) of moderately thick rectangular plates. A comprehensive review on the relevant studies done before 1995 was presented by Liew et al. [1]. The Rayleigh-Ritz method based on polynomials with properties corresponding to those of Timoshenko beam functions was used by Chung et al. [2] to study the free vibrations of orthotropic rectangular Mindlin plates with edges elastically restrained against rotation. Later, Cheung and Zhou [3] extended this solution for the free vibration problems of moderately rectangular plates. Wang [4] presented an exact formula for the vibration frequencies of simply supported Mindlin plates with the corresponding simply supported thin plate frequencies. Saha et al. [5] utilized the variational method to investigate the vibration characteristics of isotropic Mindlin plates with edges elastically restrained against rotation and translation. Gorman [6] presented accurate eigenvalues for shear-deformable plates resting on uniform elastic foundations with modified superposition-Galerkin method. Subsequently, Gorman [7, 8] utilized the superposition method to investigate the vibration problems of elastic restrained and point supported Mindlin plates. In his superposition method, the solution satisfied the differential equations exactly but approximated the boundary conditions. Xiang [9] and Xiang and Wei [10] employed the Levy solution approach in conjunction with the domain decomposition technique to analytically solve the free vibrations of rectangular Mindlin plates with two opposite edges simply supported. In [10], the influence of the step thickness ratios on the vibration behavior of rectangular Mindlin plates is studied. Yeh et al. [11] proposed a hybrid method which combines the finite difference method and the differential transformation method to analyse the free vibration of clamped and simply supported rectangular thin plates. Xiang et al. [12] extended the DSC-Ritz element method to solve the free vibration analysis of moderately thick rectangular plates with mixed supporting edges. Thai et al. [13–15] and Nguyen-Thoi et al. [16–19] have recently proposed the isogeometric analysis (IGA) and smoothed finite element method (S-FEM) to analyse the cracked Mindlin plate, laminated composite, and sandwich plates. In their research, the structural boundary was taken as uniform classic boundary conditions. In addition, Nguyen-Thanh et al. [20] utilized the alpha finite element method (α-FEM) in static, free vibration, and buckling analyses of Mindlin-Reissner plates using triangular elements only.A Rayleigh-Ritz formulation based onpb-2 functions was utilized by Liew et al. [21] to study the vibration behaviors of Mindlin plates with elastic restrained edges. Zhou et al. [22, 23] investigated the similar vibration problems in terms of a set of static Timoshenko beam functions. In their study, the free vibrations of rectangular Mindlin plates with internal line supports were also solved. Shen et al. [24] presented a new set of admissible functions, which satisfied both geometrical and natural boundary conditions, for the free and forced vibration problems of moderately thick plates with four free edges. Xing and Liu [25] presented the closed-form solutions for vibration problems of Mindlin plates with any combinations of simply supported and clamped edge conditions. Apart from aforementioned solutions, some other methods such as finite strip element method [26], spline strip method [27], finite element method [28], meshless method [29], Ritz method [30], differential quadrature method (DQM) [31–33], and Green function method [34, 35] had been developed to determine the vibration behavior of moderately thick plates.It is noteworthy that most of the previous researches on moderately thick rectangular plates are confined to the uniform thickness and classical boundary conditions, such as free, simply supported, clamped edges and their combinations. Only few studies have been found in the published literature about the vibrations of moderately thick rectangular plates with variable thickness and elastic restraints. A linear finite strip element method based on Mindlin’s plate theory was proposed by Gagnon et al. [26] to solve the vibration problems of rectangular thick plates, in which the thickness can vary in any direction. Huang et al. [36] developed a discrete method for solving the vibration problems of orthotropic rectangular plates with variable thickness and general classical boundary supports. Extended Kantorovich method was utilized to investigate the free vibrations of rectangular thick plates with variable thickness and different classical boundary conditions by Shufrin and Eisenberger [37]. Eftekhari and Jafari [38] proposed an efficient and accurate variational formulation for the vibration problems of variable thin and thick plates with elastic edge restraints.Although a large number of studies have been carried out based on Mindlin’s plate theory and methods, it appears that the information available about the vibration characteristics of nonuniform thickness moderately thick rectangular plates is very limited. Most of the contributions to moderately thick rectangular plates with classic boundary supports and elastic edges are confined to uniform thickness or bilinearly varying thickness. However, the engineering practices contain a variety of possible boundary conditions such as elastic restraints and nonlinearly varying thickness. The existing results are simply too scarce for engineering applications and comparative studies. Moreover, most of the available solution procedures in the open literature are often only customized for a specific set of restraint conditions, which may not be appropriate for practical application because there are hundreds of different combinations of boundary conditions for a plate. It is desirable to develop a unified and efficient method which is capable of dealing with more complicated problems involving arbitrary elastic edge restraints and nonuniform thickness.In view of those technical limitations and practical needs, this investigation sets out to present a modified Fourier solution technique for the free vibration analysis of moderately thick rectangular plates with variable thickness and arbitrary boundary conditions and to provide a unified and reasonable accurate alternative to other analytical and numerical techniques. This paper can be considered as an extension of the modified Fourier series method previously developed for modeling plates [39–41] and shells [42, 43]. The one displacement and two rotation functions are invariably expressed as the superposition of a 2D Fourier cosine series and four supplementary functions in the form of the product of a polynomial function and a single cosine series expansion, with all these unknown expansion coefficients treated as the generalized coordinates and determined using the Rayleigh-Ritz procedure. The change of the boundary conditions can be easily achieved by only varying the stiffness of the three sets of boundary springs along all edges of the rectangular plates without involving any change to the solution procedure. The current results are checked against with FEM results or existing results published in the literature for both uniform and nonuniform (linear and nonlinear variation) thickness plate cases, with good agreement achieved.


## 2. Theoretical Formulations

To perform the free vibration analysis of moderately thick rectangular plates with thickness varying in two directions subjected to the general elastic boundary conditions, the combination of the artificial spring technique together with Rayleigh-Ritz method is feasible. Consider a moderately thick rectangular plate, with the dimension ofa×b×h(x,y), and the coordinate of the moderately thick rectangular plate with elastically retrained edges is depicted in Figure 1. Three groups of boundary restraining springs (translation, rotational, and torsional springs) are arranged at all sides of the plate to separately simulate the boundary force. By assigning the stiffness of the boundary springs various values, it is equivalent to impose different boundary force on the midsurface of the plate. For example, the clamped boundary condition can be readily obtained by setting the spring coefficients into infinity (a very large number in practical calculation) for the translation, rotations, and torsional restraining springs along each edge.Figure 1
The general elastic boundary conditions of moderately thick rectangular plates with varying thickness in two directions.Based on the Mindlin plate theory, the displacements vectors with three directions are(1)u(x,y,z,t)=zψx(x,y,t),v(x,y,z,t)=zψy(x,y,t),w(x,y,z,t)=w(x,y,t),
where u, v, and w represent the x,  y, and z direction displacement functions and the ψx and ψy are the slop due to bending along in the respective planes. The relationship w with the slops ψx and ψy is ψx=-dw/dx and ψy=-dw/dy.For the moderately thick rectangular plates, making use of the strain-stress relationship defined in elasticity theory, the normal, shear strains and transverse shear strains can be expressed as follows:(2){εxxεyyγxyγxzγyz}={z∂ψx∂xz∂ψy∂yz(∂ψx∂y+∂ψy∂x)ψx+∂w∂xψy+∂w∂y},[σxxσyyτxyτxzτyz]=E2(1-μ2)[22μ0002μ2000001-μ00000κ(1-μ)00000κ(1-μ)]×[εxxεyyγxyγxzγyz],
where εxx, εyy, and γxy are the normal and shear strains in the x, y, and z coordinate system. The transverse shear strains γxz and γyz are constant through the thickness. The σxx and σyy are the normal stresses in the x, y directions; τxz, τyz, and τxy are shear stresses in the x, y, and z coordinate system. The E is Young’s modulus, μ is Poisson’s ration, and κ is the shear correction factor to account for the fact.In terms of transverse displacements and slope, the bending and twisting moments and the transverse shearing forces in plates can be expressed as(3)Mxx=∫-h/2h/2σxxzdz=D(∂ψx∂x+μ∂ψy∂y),Myy=∫-h/2h/2σyyzdz=D(∂ψy∂y+μ∂ψx∂x),Mxy=∫-h/2h/2τxyzdz=D(1-μ)2(∂ψx∂y+∂ψy∂x),Qx=κ∫-h/2h/2τxzdz=κGh(x,y)(ψx+∂w∂x),Qy=κ∫-h/2h/2τyzdz=κGh(x,y)(ψy+∂w∂y),
where h(x,y) is the thickness function, D=Eh(x,y)3/(12(1-μ2)) is the flexural rigidity, and G=E/(2(1+μ)) is the shear modulus.The boundary conditions for a general supported moderately thick rectangular plate can be expressed as the following forms based on the force equilibrium relationship on the four sides:(4)on⁡x=0,kx0w=-Qx,Kx0ψx=-Mxx,Kyx0ψy=-Mxy,on⁡x=a,kxaw=Qx,Kxaψx=Mxx,Kyxaψy=Mxy,on⁡y=0,ky0w=-Qy,Ky0ψy=-Myy,Kxy0ψx=-Mxy,on⁡y=b,kybw=Qy,Kybψy=Myy,Kxybψx=Mxy,
where kx0 and kxa (ky0 and kyb) are linear spring constants, Kx0 and Kxa (Ky0 and Kyb) are the rotational spring constants, and Kyx0 and Kyxa (Kxy0 and Kxyb) are the torsional spring constants at x=0 and a (y=0 and b), respectively. Therefore, arbitrary boundary conditions of the plate can be generated by assigning the linear springs, rotational springs, and torsional springs at proper stiffness. For instance, a clamped boundary (C) is achieved by simply setting the stiffness of the entire springs equal to infinity (which is represented by a very large number, 1014). Inversely, a free boundary (F) is gained by setting the stiffness of the entire springs equal to zero.Thus the total potential energy of the spring restrained plate which is composed of two parts, namely, the strain energy of the moderately thick rectangular plates and the potential energy stored in the boundary springs, can be expressed as(5)U=12∫0a∫0b{D[(∂ψx∂x+∂ψy∂y)2-2(1-μ)×(∂ψx∂x∂ψy∂y-14(∂ψx∂y+∂ψy∂x)2)]+κGh(x,y)×[(ψx+∂w∂x)2+(ψy+∂w∂y)2][(∂ψx∂x+∂ψy∂y)2]}dxdy+12∫0a[(ky0w2+Ky0ψy2+Kxy0ψx2)|y=0+(kybw2+Kybψy2+Kxybψx2)|y=b]dx+12∫0b[(kx0w2+Kx0ψx2+Kyx0ψy2)|x=0+(kxaw2+Kxaψx2+Kyxaψy2)|x=a]dy.As the springs are considered with no mass while retaining certain stiffness, the total kinetic energy of the moderately thick rectangular plates is(6)T=ρω22∫0b∫0a[h(x,y)w2+h3(x,y)(ψx2+ψy2)12]dxdy,
where ρ is the mass density and ω denotes the natural frequency of the plate.In view of satisfying arbitrarily supported boundary conditions of the moderately thick rectangular plate, the admissible functions expressed in the form of the improved Fourier series are introduced to remove the potential discontinuities with the functions and their derivatives. Thus, the moderately thick rectangular plate displacements and rotation are expressed as(7)ψx(x,y)=∑m=0∞∑n=0∞Amncos⁡⁡λamxcos⁡⁡λbny+∑l=12ζbl(y)∑m=0∞amlcos⁡⁡λamx+∑l=12ζal(x)∑n=0∞bnlcos⁡⁡λbny,(8)ψy(x,y)=∑m=0∞∑n=0∞Bmncos⁡⁡λamxcos⁡⁡λbny+∑l=12ζbl(y)∑m=0∞cmlcos⁡⁡λamx+∑l=12ζal(x)∑n=0∞dnlcos⁡⁡λbny,(9)w(x,y)=∑m=0∞∑n=0∞Cmncos⁡⁡λamxcos⁡⁡λbny+∑l=12ζbl(y)∑m=0∞emlcos⁡⁡λamx+∑l=12ζal(x)∑n=0∞fnlcos⁡⁡λbny,
where λam=mπ/a, λbn=nπ/b, and Amn, Bmn, and Cmn are the Fourier coefficients of two-dimensional Fourier series expansions for the displacements and rotation functions, respectively. aml, bml, cml, dml, eml, and fml are the supplemented coefficients of the auxiliary functions, where l=1,2. The specific expressions of the auxiliary functions ζal and ζbl are defined as
(10)ζa1(x)=a2πsin(πx2a)+a2πsin(3πx2a),ζa2(x)=-a2πcos⁡(πx2a)+a2πcos⁡(3πx2a),ζb1(y)=b2πsin(πy2b)+b2πsin(3πy2b),ζb2(y)=-b2πcos⁡(πy2b)+b2πcos⁡(3πy2b).As shown in (7)–(9), the supplementary functions ζa1(x), ζa2(x), ζb1(y), and ζb2(y) are used for the displacement and rotation expressions. The theoretical meaning of introducing these terms into the Fourier series is to remove the potential discontinuities and their derivatives throughout the whole plate structure including the boundaries and then to effectively enhance the convergence of the results. To ensure this continuity of selection expressions and corresponding derivatives at any point on the plate, the first-order derivatives of the x and y directions should exist, as indicated by (4). This requirement is guaranteed by the selected supplementary functions because it is easy to verify that
(11)ζa1(0)=ζa1(a)=ζa1′(a)=0,ζa1′(0)=1,ζa2(0)=ζa2(a)=ζa2′(0)=0,ζa2′(a)=1.Similar conditions exist for they-related polynomials, ζb1(y) and ζb2(y). It has to be mentioned that although the solution is theoretically exact for the superposition of infinite numbers of Fourier terms, in actual calculation we truncate the infinite series to M and N to obtain the results with acceptable accuracy.Since the energy expressions and admissible function of the plate have been established, the remaining task is to determine the Fourier expanded coefficients and supplemented coefficients in (7)–(9). The Lagrangian energy functional (L) of the plate is written as
(12)L=T-U.Then, the Lagrangian expression is minimized by taking its derivatives with respect to these coefficients:(13)∂L∂ϑ=0,ϑ={AmnamlbnlBmncmldnlCmnemlfnl}.Since the displacements and rotation components of the plate are chosen asM and N to obtain the results with acceptable accuracy, a total of 3*(M+1)*(N+1)+6*(M+N+2) equations are obtained.They can be summed up in a matrix form(14)(K-ω2M)E=0.The unknown coefficients in the displacement expressions can be expressed in the vector form asE, where(15)E={A00,A01,…,Am′0,Am′1,…,Am′n′,…,AMN,a01,…,aM1,a02,…,aM2,b01,…,bN1,b02,…,bN2B00,B01,…,Bm′0,Bm′1,…,Bm′n′,…,BMN,c01,…,cM1,c02,…,cM2,d01,…,dN1,d02,…,dN2C00,C01,…,Cm′0,Cm′1,…,Cm′n′,…,CMN,e01,…,eM1,e02,…,eM2,f01,…,fN1,f02,…,fN2}T.In (14), K is the stiffness matrix for the plate, and M is the mass matrix. They can be expressed separately as
(16)K=[K1-1K1-2K1-3⋯K1-9K1-2TK2-1K2-3⋯K2-9K1-3TK2-3TK3-3⋯K3-9⋮⋮⋮⋱⋮K1-9TK2-9TK3-9T⋯K9-9],M=[M1-1M1-2M1-3⋯M1-9M1-2TM2-1M2-3⋯M2-9M1-3TM2-3TM3-3⋯M3-9⋮⋮⋮⋱⋮M1-9TM2-9TM3-9T⋯M9-9].The specific expressions for the elements in (16) are given in Appendix A. Moreover, all the necessary expressions which will be used in the calculations of the eigenvalues and eigenvectors are given in Appendix B.Obviously, the natural frequencies and eigenvectors can now be readily obtained by solving a standard matrix eigenproblem. Since the components of each eigenvector are actually the expansion coefficients of the modified Fourier series, the corresponding mode shape can be directly determined from (14). In other words, once the coefficient eigenvector E is determined for a given frequency, the displacement functions of the plate can be determined by substituting the coefficients into (9). When the forced vibration is involved, by adding the work done by external force in the Lagrangian energy function and summing the loading vector F on the right side of (14), the characteristic equation for the forced vibration of the moderately thick rectangular plates is readily obtained.


## 3. Numerical Examples and Discussion

In this section, a systematic comparison between the current solutions and theoretical results published by other researchers or finite element method (FEM) results is carried out to validate the excellent accuracy, reliability, and feasibility of the modified Fourier method. A comprehensive study on the effects of elastic restraint parameters and varying thickness in two directions is also reported. New results are obtained for plates subjected to general elastic boundary restraints with nonlinear variable thickness in both directions. The discussion is arranged as follows. Firstly, the convergence of the modified Fourier solution is checked. In addition, the influence of the stiffness of boundary spring components is studied. Secondly, the uniform thickness plates with various combinations of classical boundary conditions, elastic boundary conditions, and different structure parameters are examined. Thirdly, the nonuniform thickness plate with linear variation in both directions, various combinations of classical boundary conditions, conditions, and different structure parameters are examined. Then, numerical solutions for moderately thick rectangular plates with elastic boundary conditions are presented. The effects of the elastic restraint parameters are also investigated. Finally, the nonuniform thickness plate with nonlinear variation in both directions and arbitrary boundary conditions is also studied.


### 3.1. Convergence and Stiffness Value Study

Since the Fourier series is numerically truncated and only finite terms are counted in actual calculations, the proposed solution should be understood as a solution with arbitrary precision. In this subsection, a uniform thickness square moderately thick rectangular plate with completely clamped boundary condition (C-C-C-C) and four edges equally elastically restrained against linear spring constants and rotation spring constants supports (E-E-E-E,KT=ki/(b3/D), KR=Ki/(b/D), KT=10, and KR=5) has been selected to demonstrate the convergence and accuracy of the modified Fourier method. In Table 1, the first eight frequency parameters Ω=ωb2(ρh/D)1/2/π2 for the considered uniform thickness square moderately thick rectangular plate with C-C-C-C and E-E-E-E boundary conditions are examined. The table shows that the proposed method has fast convergence behavior. The maximum discrepancy in the worst case between the 6×6 truncated configuration and the 8×8 one is less than 0.064%. In order to fully illustrate the convergence of the present method, Figures 3, 4, and 5 present the 1st and 8th frequency parameters Ω=ωb2(ρh/D)1/2/π2 with various truncated numbers M=N subjected to different boundary condition and aspect ratios. A highly desired convergence characteristic is observed such that (a) sufficiently accurate results can be obtained with only a small number of terms in the series expansions and (b) the solution is consistently refined as more terms are included in the expansions. However, this should not constitute a problem in practice because one can always verify the accuracy of the solution by increasing the truncation number until a desired numerical precision is achieved. As a matter of fact, this “quality control” scheme can be easily implemented automatically. In modal analysis, the natural frequencies for higher-order modes tend to converge slower (see Table 1). Thus, an adequate truncation number should be dictated by the desired accuracy of the largest natural frequencies of interest. In view of the excellent numerical behavior of the current solution, the truncation numbers will be simply set as M=N=12 in the following calculations.Table 1
Frequency parametersΩ=ωb2(ρh/D)1/2/π2 for square moderately thick plates with uniform thickness and different boundary conditions.
Boundary conditions
M, N
Model sequence
1
2
3
4
5
6
7
8
C-C-C-C
M
=
N
=
2
3.375
7.191
7.191
9.856
16.05
16.28
17.47
17.47
M
=
N
=
4
3.299
6.308
6.308
8.853
10.46
10.56
12.63
12.63
M
=
N
=
6
3.294
6.282
6.282
8.801
10.37
10.47
12.54
12.54
M
=
N
=
8
3.292
6.278
6.278
8.797
10.36
10.46
12.53
12.53
M
=
N
=
10
3.292
6.276
6.276
8.794
10.36
10.46
12.53
12.53
M
=
N
=
12
3.292
6.276
6.276
8.793
10.36
10.46
12.52
12.52
M
=
N
=
14
3.292
6.276
6.276
8.793
10.36
10.46
12.52
12.52
M
=
N
=
16
3.292
6.275
6.275
8.792
10.36
10.45
12.52
12.52
Reference [4]
3.292
6.276
6.276
8.792
10.36
10.46
12.52
12.52
E-E-E-E
M
=
N
=
2
0.6153
1.0861
1.0861
1.7898
2.7709
2.8417
3.5010
3.5010
M
=
N
=
4
0.6150
1.0799
1.0799
1.7785
2.7695
2.8405
3.4914
3.4914
M
=
N
=
6
0.6149
1.0796
1.0796
1.7773
2.7694
2.8403
3.4898
3.4898
M
=
N
=
8
0.6149
1.0796
1.0796
1.7771
2.7693
2.8403
3.4893
3.4893
M
=
N
=
10
0.6149
1.0796
1.0796
1.7770
2.7693
2.8403
3.4892
3.4892
M
=
N
=
1
2
0.6149
1.0796
1.0796
1.7770
2.7693
2.8403
3.4892
3.4892
M
=
N
=
1
4
0.6149
1.0796
1.0796
1.7770
2.7693
2.8403
3.4892
3.4892
M
=
N
=
1
6
0.6149
1.0796
1.0796
1.7770
2.7693
2.8403
3.4891
3.4891
Reference [33]
0.6153
1.0861
1.0861
1.7898
2.7709
2.8417
3.5010
3.5010
FEM
0.6150
1.7097
1.7097
1.7774
2.7694
2.8406
3.4898
3.4898As far as the accuracy of the present method is concerned, the converged solutions of the present method are in excellent agreement with both the results reported by reference data and the finite element results. For C-C-C-C boundary conditions, the max discrepancy between the present results and the reference data does not exceed 0.011% for the worst case and in most cases is 0%. Comparing the results with exact solutions [4], it is observed that eight terms are sufficient to obtain accurate results. Moreover, with regard to the E-E-E-E boundary condition, the max discrepancy between the present results and the reference data does not exceed 0.31% for the worst case and in most cases is 0.1%. Regarding the results with DQM solutions [33], it can be seen that the six terms are sufficient to obtain enough accurate results. In addition, it is clear that the results of the present approach with just 663 DOFs (M×N=12×12) can predict the vibration characteristics accurately. Most of them are identical to those obtained from finite element method (FEM) with 102,01 DOFs (S4R: 0.01 m × 0.01 m). That is to say, it needs only 6.62% DOFs compared with FEM to obtain the same precision solutions for the considered case. On the same hardware (Intel i7-3.9 GHz), the computing time of the present formulation for the solution (M×N=12×12) implemented in optimized MATLAB scripts is about 2.125 s, whereas the finite element solution consumes 34.578 s, that is, at least 16 times more CPU time than the present method for the same problem.As mentioned earlier, in the current modeling framework, all the classical boundary conditions and their combinations can be conveniently viewed as special cases when the stiffness for the normal and tangential boundary springs becomes zero and/or infinitely large. Thus, the effects of the stiffness of the translation (ki), rotation (Ki), and torsional springs (Ki,j) on the modal characteristics should be investigated. As shown in the Figures 2(a)–2(c), the first and the third frequency parameters are separately obtained by varying the stiffness of one group of the boundary springs from extremely large (1014) to extremely small (100) while assigning the other group of the springs infinite stiffness (1014). It can be found in Figure 2(a) that the frequency parameter almost keeps at a level when the stiffness of the translation springs is larger than 1012 or smaller than 107. In Figure 2(b), the influences of the rotation springs on frequency parameters are given. It is shown that the frequency curves change greatly within the stiffness range (106 to 1010) while out of this range the frequency curves separately keep level. In Figure 2(c), the influences of the torsional springs on frequency parameters are given. It is shown that the frequency curves almost change when the stiffness changes in the whole range.The effect of boundary spring stiffness on the natural frequenciesΩ: (a) translation spring, (b) rotation spring, and (c) torsional spring.
(a)
(b)
(c)The effect of numerically truncated numbersM=N on the natural frequencies Ω for C-C-C-C boundary condition.
(a)
The 1st order
(b)
The 8th orderThe effect of numerically truncated numbersM=N on the natural frequencies Ω for F-F-F-F boundary condition.
(a)
The 1st order
(b)
The 8th orderThe effect of numerically truncated numbersM=N on the natural frequencies Ω with four edges elastically restrained against translation and rotation (KT=ki/(b3/D), KR=Ki/(b/D), KT=100, and KR=100).
(a)
The 1st order
(b)
The 8th orderBased on the analysis, it can be found that the torsional springs almost have no effect on the structure. Also, the relationship between the rotation springs and twisting moment can be seen from the boundary condition expression. Then, the twisting moments have small effect on the vibration characteristics of the structure. However, in this paper, in order to get a more accurate prediction of the vibration characteristics of the structure, the twisting moment on boundary edges is taken into account. In the latter study, in addition to the fact that free boundary is not unexpected considering the torsion spring, the other boundary conditions are introduced into a torsion spring, and the spring stiffness takes infinity. Form Figure2 analysis, it also concluded that the translation spring has a wider influence range than the rotation spring on the frequency parameters; that is, for the translation springs, the stable frequency parameters appear when the stiffness is more than 1012 or less 107 while for the rotation springs, when the stiffness value is assigned more than 1010 the frequency parameters become smooth. Thus, it is suitable to use 1014 to simulate the infinite stiffness value in the model validation parts and in the following examples.


### 3.2. Uniform Thickness Moderately Thick Plates with Classical and Elastic Boundary Conditions

In this subsection, the modified Fourier solution is applied to deal with vibration problems of uniform thickness moderately thick rectangular plates subject to the classical boundary conditions and arbitrary elastic boundary conditions. In present work, three groups of continuously distributed boundary springs are introduced to simulate the given or typical boundary conditions. As mentioned earlier, the stiffness of these boundary springs can take any value from zero to infinity to better model many real-world restraint conditions. Taking edgex=0, for example, the corresponding spring stiffness for the three types of classical boundaries and elastic boundaries iscompletely free:Qx=0, Mxx=0, and Mxy=0(17a)kx0=0,Kx0=0,Kyx0=0,simply supported:w=0, Mxx=0, and ψy=0(17b)kx0=1014,Kx0=0,Kyx0=1014,completely clamped:w=0, ψx=0, and ψy=0(17c)kx0=1014,Kx0=1014,Kyx0=1014,arbitrarily elastic boundary:w≠0, ψx≠0, ψy≠0, Qx≠0, Mxx≠0, and Mxy≠0(17d)kx0=Γwx0,Kx0=Γxx0,Kyx0=1014.The appropriateness of the three classical boundaries defined in (17a)–(17c) will be proved by several examples given in following, the arbitrary elastic boundaries are also defined in (17d), and the Γwi (Γwi=ki/k0, k0=1×109 N/m2, i=x0, xa, y0, yb) and Γxi (Γxi=Ki/K0, K0=1×109 Nm/rad, i=x0, xa, y0, yb) elastic restraint parameters represent corresponding spring stiffness. For the sake of simplicity, a four-letter string is employed to represent the restraint condition of a plate, such that F-C-S-E identifies the plate with edges x=0, y=0, x=a, and y=b having free, clamped, shear-supported restrained, and elastic boundary conditions, respectively.As for the first case, a uniform thickness moderately thick plate with different classical boundaries and structure parameters is investigated here. In Table2, the comparison of the first eight frequency parameters Ω=ωb2(ρh/D)1/2/π2 of the considered plate is presented. The S-S-S-S, C-F-F-F, S-S-F-F, F-F-F-F, and S-C-S-C boundary conditions are performed in the comparison. Excellent agreements are observed between the solutions obtained by the modified Fourier method, the referential data, and finite element method (FEM) results for the uniform thickness moderately thick rectangular plates. It is also verified that the definition of the three types of classical boundaries in (17a)–(17c) is appropriate. In addition, the elastic boundary conditions (17d) are also verified. In the next two examples, we will account for the vibration of moderately thick plate with elastic edge supports. The first model considered is an S-S-S-S square moderately thick plate with all edges elastically rotationally restrained. That is, Kx0=Kxa=Ky0=Kyb=Γx, and all the other restraining springs are set to have an infinite stiffness (namely, represented by 1014 in numerical calculation). The six frequency parameters Ω=ωb2(ρh/D)1/2/π2 are given in Table 3 for several different restraining coefficient values; the finite element method (FEM) results calculated using ABAQUS are also listed in this table as a reference. The second model concerns a complete square moderately thick plate with all edges elastically restrained. That is, kx0=kxa=ky0=kyb=Γw and Kx0=Kxa=Ky0=Kyb=Γx. The six frequency parameters Ω=ωb2(ρh/D)1/2/π2 are given in Table 4 for several different restraining coefficient values; the finite element method (FEM) results are also listed in Table 4 as a reference. It can be clearly seen that the comparison is extremely good, which implies that the current method is able to make correct predictions for the modal characteristics of moderately thick plate with not only classical boundary conditions but also elastic edge restraints.Table 2
Frequency parametersΩ=ωb2(ρh/D)1/2/π2 for uniform thickness moderately thick plates with different classical boundaries and structure parameters.
Boundary condition
a/b
Model sequence
1
2
3
4
5
6
7
8
S-S-S-S
1
1.931
4.605
4.605
7.064
8.605
8.605
10.792
10.792
(1.933a)
(4.611)
(4.611)
(7.067)
(8.607)
(8.607)
(10.799)
(10.780)
(1.931b)
(4.605)
(4.605)
(7.064)
(8.605)
(8.605)
(10.792)
(10.792)
3/2
1.408
2.648
4.127
4.605
5.261
7.064
7.153
8.184
(1.410a)
(2.651)
(4.128)
(4.608)
(5.267)
(7.074)
(7.155)
(8.188)
(1.408b)
(2.648)
(4.127)
(4.605)
(5.261)
(7.064)
(7.151)
(8.184)
C-F-F-F
1
0.348
0.817
2.035
2.583
2.862
4.816
5.478
5.774
(0.348a)
(0.817)
(2.038)
(2.586)
(2.865)
(4.821)
(5.504)
(5.793)
(0.348b)
(0.816)
(2.034)
(2.582)
(2.860)
(4.811)
(5.477)
(5.772)
3/2
0.155
0.502
0.941
1.665
2.292
2.613
3.303
3.498
(0.154a)
(0.502)
(0.941)
(1.663)
(2.295)
(2.618)
(3.305)
(3.500)
(0.154b)
(0.501)
(0.940)
(1.662)
(2.292)
(2.612)
(3.298)
(3.494)
S-S-F-F
1
0.333
1.678
1.875
3.559
4.720
4.946
6.474
6.632
(0.335a)
(1.680)
(1.878)
(3.600)
(4.721)
(4.949)
(6.478)
(6.642)
(0.333b)
(1.677)
(1.874)
(3.557)
(4.718)
(4.945)
(6.472)
(6.631)
3/2
0.222
0.938
1.631
2.363
2.573
4.082
4.482
4.753
(0.223a)
(0.940)
(1.632)
(2.364)
(2.575)
(4.092)
(4.483)
(4.759)
(0.222b)
(0.938)
(1.631)
(2.361)
(2.571)
(4.077)
(4.479)
(4.752)
F-F-F-F
1
1.290
1.919
2.363
3.235
3.235
5.605
5.605
5.646
(1.291a)
(1.920)
(2.365)
(3.239)
(3.239)
(5.607)
(5.607)
(6.648)
3/2
0.865
0.948
1.961
2.166
2.461
2.843
3.551
4.026
(0.865a)
(0.949)
(1.963)
(2.168)
(2.465)
(2.844)
(3.556)
(4.034)
S-C-S-C
1
2.700
4.971
5.990
7.973
8.787
10.250
11.334
12.024
(2.701a)
(4.971)
(5.993)
(7.975)
(8.788)
(10.253)
(11.335)
(12.028)
3/2
2.348
3.253
4.972
5.646
6.489
7.381
7.974
9.930
(2.348a)
(3.255)
(4.974)
(5.647)
(6.590)
(7.384)
(7.801)
(9.932)
aResults in parentheses are taken from FEM.
bResults in parentheses are taken from [11].Table 3
Frequency parametersΩ=ωb2(ρh/D)1/2/π2 for S-S-S-S moderately thick plates (a/b=1) with uniform thickness and elastic rotation support (Kx0=Kxa=Ky0=Kyb=Γx).
Γ
x
a
h
0
/
a
Model sequence
1
2
3
4
5
6
0
0.05
1.945
4.834
4.834
7.591
9.515
9.519
(1.942b)
(4.836)
(4.836)
(7.584)
(9.547)
(9.552)
0.1
1.856
4.500
4.500
6.829
8.499
8.513
(1.858b)
(4.510)
(4.510)
(6.843)
(8.539)
(8.554)
0.15
1.758
4.114
4.114
6.040
7.421
7.445
(1.761b)
(4.126)
(4.126)
(6.060)
(7.462)
(7.486)
0.2
1.656
3.733
3.733
5.335
6.468
6.499
(1.660b)
(3.747)
(3.747)
(5.357)
(6.508)
(6.538)
0.5
0.05
3.541
7.071
7.071
10.227
12.318
12.399
(3.546b)
(7.091)
(7.091)
(10.253)
(12.386)
(12.468)
0.1
3.274
6.242
6.242
8.719
10.302
10.421
(3.282b)
(6.267)
(6.267)
(8.754)
(10.370)
(10.488)
0.15
2.947
5.361
5.361
7.290
8.465
8.606
(2.956b)
(5.387)
(5.387)
(7.327)
(8.524)
(8.666)
0.2
2.624
4.599
4.599
6.148
7.057
7.199
(2.634b)
(4.624)
(4.624)
(6.182)
(7.107)
(7.250)
1
0.05
3.541
7.072
7.072
10.228
12.320
12.401
(3.547b)
(7.093)
(7.093)
(10.254)
(12.388)
(12.470)
0.1
3.278
6.248
6.248
8.727
10.311
10.429
(3.286b)
(6.274)
(6.274)
(8.762)
(10.378)
(10.497)
0.15
2.957
5.374
5.374
7.305
8.478
8.620
(2.966b)
(5.400)
(5.400)
(7.341)
(8.537)
(8.680)
0.2
2.641
4.617
4.617
6.166
7.070
7.215
(2.651b)
(4.643)
(4.643)
(6.200)
(7.121)
(7.268)
1.5
0.05
3.542
7.073
7.073
10.229
12.320
12.401
(3.547b)
(7.093)
(7.093)
(10.254)
(12.388)
(12.471)
0.1
3.280
6.251
6.251
8.729
10.313
10.432
(3.287b)
(6.276)
(6.276)
(8.765)
(10.381)
(10.500)
0.15
2.960
5.379
5.379
7.309
8.482
8.625
(2.969b)
(5.405)
(5.405)
(7.346)
(8.541)
(8.685)
0.2
2.647
4.623
4.623
6.172
7.074
7.221
(2.657b)
(4.649)
(4.649)
(6.206)
(7.125)
(7.274)
2
0.05
3.542
7.073
7.073
10.229
12.321
12.402
(3.547b)
(7.093)
(7.093)
(10.255)
(12.389)
(12.471)
0.1
3.280
6.252
6.252
8.730
10.315
10.434
(3.288b)
(6.277)
(6.277)
(8.766)
(10.382)
(10.501)
0.15
2.962
5.381
5.381
7.312
8.484
8.628
(2.971b)
(5.407)
(5.407)
(7.348)
(8.543)
(8.688)
0.2
2.650
4.627
4.627
6.175
7.076
7.224
(2.660b)
(4.652)
(4.652)
(6.210)
(7.128
(7.277)
a
Γ
x
=
K
i
/
K
0
,
(
K
0
=
1
×
1
0
9
Nm
/
rad, i=x0,xa,y0,yb).
bResults in parentheses are taken from FEM.Table 4
Frequency parametersΩ=ωb2(ρh/D)1/2/π2 for moderately thick plates (a/b=1) with uniform thickness and all edge elastic restraints (Kx0=Kxa=Ky0=Kyb=Γx and kx0=kxa=ky0=kyb=Γw).
Γ
x
a
Γ
w
a
h
0
/
a
Model sequence
1
2
3
4
5
6
0.5
0
0.05
0.986
1.967
3.893
3.893
4.850
4.850
(0.988b)
(1.971)
(3.897)
(3.897)
(4.852)
(4.852)
0.1
0.916
1.835
3.515
3.522
4.375
4.375
(0.917b)
(1.837)
(3.519)
(3.528)
(4.341)
(4.341)
0.15
0.789
1.635
2.983
3.027
3.771
3.771
(0.792b)
(1.638)
(2.988)
(3.029)
(3.775)
(3.775)
0.2
0.646
1.444
2.497
2.601
3.249
3.249
(0.652b)
(1.448)
(2.508)
(2.621)
(3.254)
(3.254)
1
0.5
0.05
3.417
3.417
4.431
5.294
5.461
6.448
(3.419b)
(3.419)
(4.435)
(5.303)
(5.478)
(6.456)
0.1
1.593
1.593
2.401
3.849
3.861
4.729
(1.611b)
(1.611)
(2.408)
(3.852)
(3.866)
(4.767)
0.15
1.122
1.122
1.913
3.272
3.290
4.038
(1.127b)
(1.227)
(1.924)
(3.284)
(3.314)
(4.069)
0.2
0.890
0.890
1.640
2.781
2.829
3.476
(0.899b)
(0.899)
(1.654)
(2.787)
(2.834)
(3.497)
1.5
1
0.05
4.325
4.325
5.619
6.273
6.571
7.619
(4.334b)
(4.334)
(5.681)
(6.299)
(6.642)
(7.642)
0.1
2.013
2.013
2.812
4.088
4.116
4.993
(2.024b)
(2.024)
(2.827)
(4.122)
(4.187)
(4.994)
0.15
1.338
1.338
2.108
3.430
3.443
4.196
(1.342b)
(1.342)
(2.112)
(3.467)
(3.482)
(4.200)
0.2
1.038
1.038
1.772
2.936
2.965
3.613
(1.042b)
(1.042)
(1.786)
(2.937)
(2.977)
(3.624)
2
2
0.05
5.231
5.231
6.919
7.630
8.002
9.220
(5.238b)
(5.238)
(6.931)
(7.637)
(8.011)
(9.241)
0.1
2.594
2.594
3.439
4.470
4.544
5.428
(2.597b)
(2.597)
(3.442)
(4.495)
(4.557)
(5.438)
0.15
1.653
1.653
2.403
3.613
3.632
4.392
(1.662b)
(1.662)
(2.414)
(3.627)
(3.646)
(4.404)
0.2
1.238
1.238
1.949
3.077
3.099
3.751
(1.244b)
(1.244)
(1.964)
(3.094)
(3.116)
(3.788)
a
Γ
x
=
K
i
/
K
0
,
(
K
0
=
1
×
1
0
9
Nm/rad,  i=x0,xa,y0,yb),  Γw=ki/k0,(k0=1×109N/m,  i=x0,xa,y0,yb).
bResults in parentheses are taken from FEM.The excellent agreements between the solution obtained by the modified Fourier method and the referential data for the moderately thick plate subjected to the combinations of classical boundary conditions and elastic boundary conditions given in Tables2–4 indicate that the proposed method is sufficiently accurate to deal with uniform thickness moderately thick plate with arbitrary boundary conditions.


### 3.3. Linearly Variation Thickness Moderately Thick Plates with Classical and Elastic Boundary Conditions

In the theoretical formulations, this paper concerns the varying thickness moderately thick plates with classical and elastic boundary conditions. The varying thickness functionh(x,y) can be expressed as h0(1+αxs)(1+βyt), in which the h0, α, and β represent the initial thickness, gradient in x direction, and gradient in y direction. When the indexes s and t take the value s=t=1, the analytical model imitates the linearly variation thickness moderately thick plates structure. In order to unify the description and facilitate the analytical calculations of the involved integrals, all the thickness variation functions can be expanded into either 1D or 2D Fourier cosine series, resulting in
(18)h(x,y)=∑m=0∞∑n=0∞h0qmncos⁡⁡mπxacos⁡⁡mπyb,
where
(19)qmn=4ab∫0b∫0a(1+αxs)(1+βyt)cos⁡⁡mπxacos⁡⁡mπybdxdy.In order to prove the validity of the present method for the vibration of linearly variation thickness moderately thick plates with arbitrary boundary conditions, the typical classical boundary conditions as the first case will be considered. In Table5, the comparison of the first six frequency parameters Ω=ωa2(ρh0/D0)1/2 of the moderately thick plates with linearly varying thickness is presented. The S-S-S-S, C-F-F-F, S-S-F-F, C-C-C-C, and S-C-S-C boundary conditions are performed in the comparison. Excellent agreements are observed between the solutions obtained by the modified Fourier method and finite element method (FEM) results for the moderately thick plates with linear variation thickness. To investigate the influence of the aspect ratio on the uniform thickness and nonuniform thickness moderately thick plates, the effect on the frequency parameters for plates with S-S-S-S boundary conditions is presented in Figure 6. The thickness functions are h0 and h0(1+0.5×x)(1+0.5×y), respectively. It is seen from Figure 6 that the influence of aspect ratios on the frequency parameters for nonuniform thickness moderately thick plates is more complicated.Table 5
Frequency parameterΩ=ωa2(ρh0/D0)1/2 for moderately thick plates with linearly varying thickness in different boundary conditions (h(x,y)=h0(1+0.5x)(1+0.5y)).
Boundarycondition
a
/
b
h
0
/
a
Model sequence
1
2
3
4
5
6
S-S-S-S
1
0.1
27.261
62.397
62.926
92.423
111.537
111.732
(27.263a)
(62.410)
(63.842)
(92.445)
(111.867)
(111.921)
0.2
23.103
47.010
47.191
65.018
75.831
75.865
(23.489a)
(47.209)
(47.349)
(65.057)
(76.605)
(76.473)
3/2
0.1
59.338
99.101
138.798
151.102
166.710
204.855
(59.874a)
(99.249)
(138.177)
(151.362)
(166.644)
(205.381)
0.2
41.175
63.765
63.765
73.598
83.956
89.084
(41.586a)
(63.313)
(63.669)
(73.661)
(83.918)
(89.547)
C-F-F-F
1
0.1
4.173
10.331
27.407
37.826
38.809
64.095
(4.856a)
(10.680)
(27.513)
(37.203)
(39.135)
(64.858)
0.2
3.983
8.702
21.403
29.443
30.286
46.705
(4.118a)
(9.306)
(21.990)
(29.670)
(30.602)
(47.156)
3/2
0.1
4.868
15.488
35.620
63.523
89.969
100.089
(4.887a)
(15.320)
(35.175)
(63.353)
(89.159)
(100.783)
0.2
4.469
9.763
22.338
42.188)
50.104
56.331
(4.944a)
(9.636)
(22.087)
(42.177)
(50.580)
(56.559)
S-S-F-F
1
0.1
4.117
24.795
26.216
48.567)
68.195
69.667
(4.789a)
(24.149)
(26.751)
(49.324)
(68.842)
(70.575)
0.2
3.820
20.516
21.924
37.523
48.980
50.327
(4.103a)
(20.041)
(22.125)
(37.221)
(48.894)
(50.622)
3/2
0.1
7.935
39.454
65.063
92.877
92.877
136.176
(7.856a)
(40.145)
(65.162)
(92.462)
(93.022)
(136.336)
0.2
6.389
26.149
35.627
49.539
56.844
65.170
(6.406a)
(26.536)
(35.626)
(49.831)
(57.144)
(65.981)
C-C-C-C
1
0.1
44.471
80.299
80.681
109.422
126.871
128.207
(45.102a)
(80.271)
(80.656)
(110.015)
(126.328)
(128.888)
0.2
32.162
52.781
52.933
69.410
78.474
79.482
(32.240a)
(52.629)
(53.166)
(70.063)
(79.059)
(80.266)
3/2
0.1
82.975
115.587
151.992
161.579
176.601
212.330
(83.139a)
(115.838)
(152.200)
(162.193)
(177.058)
(212.849)
0.2
48.577
66.022
83.468
90.830
97.474
98.157
(49.187a)
(66.259)
(84.065)
(91.749)
(97.827)
(98.777)
S-C-S-C
1
0.1
36.847
66.664
76.965
101.045
113.594
125.551
(37.901a)
(68.568)
(78.356)
(101.200)
(113.978)
(125.651)
0.2
27.673
48.469
51.220
67.027
76.358
78.241
(27.897a)
(48.685)
(51.491)
(67.447)
(77.530)
(79.573)
3/2
0.1
77.378
106.848
150.079
154.569
173.120
208.391
(78.003a)
(107.058)
(150.991)
(155.171)
(173.099)
(208.402)
0.2
45.895
64.732
73.393
83.282
90.325
97.089
(45.803a)
(64.131)
73.656
83.801
90.518
97.015
aResults in parentheses are taken from FEM.The effect of aspect ratioa/b on the natural frequencies Ω for S-S-S-S boundary condition: (a) uniform thickness and (b) nonuniform thickness.
(a)
(b)In the next two examples, we also account for the vibrations of moderately thick plate with linear variation thickness and elastic edge supports. The first model considered is an S-S-S-S square moderately thick plate with all edges elastically restrained. That is,Kx0=Kxa=Ky0=Kyb=Γx. The six frequency parameters Ω=ωa2(ρh0/D0)1/2 are given in Table 6 for several different restraining coefficient values; the finite element method (FEM) results calculated using ABAQUS are also presented as a reference. The second model is a complete square moderately thick plate with all edges elastically restrained. That is, kx0=kxa=ky0=kyb=Γw and Kx0=Kxa=Ky0=Kyb=Γx. The six frequency parameters Ω=ωa2(ρh0/D0)1/2 are given in Table 7 for several different restraining coefficient values; the finite element method (FEM) results are also listed as a reference. It can be clearly seen that the comparison is extremely good, which implies that the current method is able to make correct predictions for the modal characteristics of linearly varying thickness moderately thick plate with not only classical boundary conditions but also elastic edge restraints.Table 6
Frequency parameterΩ=ωa2(ρh0/D0)1/2 for S-S-S-S moderately thick plates (a/b=1) with linearly varying thickness and elastic rotation support (h(x,y)=h0(1+0.5x)(1+0.5y), Kx0=Kxa=Ky0=Kyb=Γx).
Γ
x
a
h
0
/
a
Model sequence
1
2
3
4
5
6
0
0.05
28.809
70.080
70.891
108.967
134.884
135.295
(28.897b)
(70.334)
(71.129)
(109.112)
(135.254)
(135.772)
0.1
27.261
62.398
62.926
92.423
111.540
111.735
(26.606b)
(61.784)
(62.303)
(91.193)
(111.323)
(111.648)
0.15
25.231
54.192
54.505
77.118
91.228
91.312
(25.204b)
(55.297)
(55.595)
(77.503)
(91.871)
(91.941)
0.2
23.103
47.010
47.191
65.018
75.832
75.866
(23.947b)
(47.071)
(47.208)
(65.404)
(76.531)
(76.589)
0.5
0.05
23.103
47.010
47.191
65.018
75.832
75.866
(23.210b)
(47.245)
(47.358)
(65.241)
(76.122)
(76.345)
0.1
48.377
94.650
95.597
135.484
162.635
163.501
(48.524b)
(94.891)
(95.992)
(135.845)
(167.012)
(164.024)
0.15
36.239
70.939
71.298
99.997
118.584
119.009
(36.647b)
(71.214)
(71.687)
(100.024)
(119.114)
(119.457)
0.2
28.749
56.937
57.037
79.214
93.009
93.162
(29.142b)
(57.034)
(57.544)
(79.654)
(93.668)
(93.754)
1
0.05
49.651
96.759
97.619
138.089
165.368
166.304
(49.755b)
(97.245)
(97.964)
(138.544)
(165.864)
(166.878)
0.1
38.951
73.735
74.251
102.748
121.079
121.709
(39.021b)
(74.145)
(74.664)
(102.874)
(121.613)
(121.817)
0.15
30.630
58.406
58.533
80.432
93.984
94.255
(30.638b)
(58.684)
(58.644)
(80.874)
(94.121)
(94.478)
0.2
25.548
48.636
48.674
66.165
76.708
76.824
(25.687b)
(48.788)
(48.992)
(66.544)
(76.824)
(76.944)
1.5
0.05
50.118
97.561
98.374
139.096
166.421
167.382
(50.247b)
(97.877)
(98.661)
(139.247)
(166.875)
(167.662)
0.1
40.310
75.216
75.783
104.229
122.402
123.164
(40.366b)
(75.542)
(75.921)
(104.557)
(122.874)
(123.385)
0.15
31.837
59.352
59.530
81.245
94.611
94.983
(31.902b)
(59.451)
(59.871)
(81.375)
(94.784)
(95.105)
0.2
26.302
49.119
49.156
66.528
76.954
77.130
(26.418b)
(49.354)
(49.334)
(66.921)
(77.010)
(77.470)
2
0.05
50.360
97.983
98.769
139.631
166.979
167.954
(50.472b)
(98.005)
(99.114)
(140.154)
(167.188)
(167.986)
0.1
41.130
76.146
76.724
105.165
123.229
124.082
(41.421b)
(76.642)
(78.114)
(105.661)
(123.278)
(124.974)
0.15
32.687
60.025
60.245
81.832
95.052
95.508
(32.778b)
(60.244)
(60.549)
(82.146)
(95.114)
(95.842)
0.2
26.893
49.492
49.539
66.815
77.138
77.370
(26.922b)
(49.874)
(49.924)
(66.987)
(77.367)
(77.588)
a
Γ
x
=
K
i
/
K
0
,
(
K
0
=
1
×
1
0
9
Nm/rad,  i=x0,xa,y0,yb).
bResults in parentheses are taken from FEM.Table 7
Frequency parameterΩ=ωa2(ρh0/D0)1/2 for moderately thick plates (a/b=1) with linearly varying thickness and elastic rotation and translation support (h(x,y)=h0(1+0.5x)(1+0.5y), Kx0=Kxa=Ky0=Kyb=Γx, and kx0=kxa=ky0=kyb=Γw).
Γ
x
a
Γ
w
a
h
0
/
a
Model sequence
1
2
3
4
5
6
0.5
0
0.05
14.751
14.917
28.232
55.251
55.397
69.512
(14.780b)
(14.822)
(29.252)
(55.476)
(55.505)
(70.904)
0.1
11.811
11.910
23.857
44.462
45.037
55.453
(11.818b)
(11.897)
(24.711)
(44.376)
(45.257)
(56.511)
0.15
8.487
8.514
20.011
34.783
36.335
44.970
(8.500b)
(8.544)
(20.738)
(34.404)
(36.663)
(45.625)
0.2
6.073
6.073
17.431
28.679
30.805
38.010
(6.096b)
(6.106)
(18.039)
(28.129)
(31.165)
(38.311)
1
0.5
0.05
21.225
32.173
32.173
44.381
64.061
65.222
(20.708b)
(32.097)
(32.285)
(45.110)
(64.182)
(65.355)
0.1
8.140
16.878
16.941
28.100
48.663
49.015
(7.943b)
(16.838)
(16.970)
(28.888)
(48.700)
(49.171)
0.15
4.493
11.859
11.878
22.429
38.352
39.247
(4.383b)
(11.867)
(11.891)
(23.125)
(38.177)
(39.488)
0.2
2.932
8.690
8.691
18.919
31.108
32.600
(3.540b)
(8.791)
(8.791)
(16.229)
(27.520)
(27.889)
1.5
1
0.05
27.819
40.919
41.430
54.595
70.792
72.896
(27.146b)
(41.008)
(41.434)
(55.256)
(70.895)
(73.051)
0.1
11.285
20.068
20.230
31.101
51.046
51.429
(11.012b)
(20.025)
(20.270)
(31.849)
(51.126)
(51.569)
0.15
6.292
13.912
13.922
24.132
40.500
41.120
(6.139b)
(13.912)
(13.934)
(24.808)
(40.426)
(41.316)
0.2
4.121
10.321
10.331
20.051
32.796
33.921
(4.021b)
(10.329)
(10.357)
(20.635)
(32.540)
(34.178)
2
2
0.05
34.615
51.972
53.088
68.255
81.344
84.510
(33.806b)
(52.100)
(53.157)
(68.891)
(81.433)
(84.761)
0.1
15.416
24.659
24.829
35.615
53.879
54.494
(15.045b)
(24.625)
(24.888)
(36.303)
(53.970)
(54.626)
0.15
8.740
16.420
16.481
26.353
42.417
42.946
(8.529b)
(16.414)
(16.499)
(26.996)
(42.398)
(43.114)
0.2
5.761
12.127
12.127
21.434
34.286
35.192
(5.622b)
(12.134)
(12.148)
(21.998)
(34.110)
(35.415)
a
Γ
x
=
K
i
/
K
0
,
(
K
0
=
1
×
1
0
9
Nm/rad
,
i
=
x
0
,
x
a
,
y
0
,
y
b
),  Γw=ki/k0,(k0=1×109N/m,  i=x0,xa,y0,yb).
bResults in parentheses are taken from FEM.As the last case of this subsection, the influence of the gradientα and β on the fundamental frequency parameters for a linearly varying thickness moderately thick plate is investigated. The model is a square moderately thick plate with all edges elastically restrained. That is, kx0=kxa=ky0=kyb=Γw=2 and Kx0=Kxa=Ky0=Kyb=Γx=2. The six frequency parameters Ω=ωa2(ρh0/D0)1/2 are given in Table 8 for several different α and β values; the finite element method (FEM) results are also listed as a reference. Again, good agreement can be observed. Through Table 8, it is also found that the frequency parameter increases with increasing gradient parameters.Table 8
Frequency parameterΩ=ωa2(ρh0/D0)1/2 for elastic support moderately thick plates (a/b=1) with different gradient (kx0=kxa=ky0=kyb=Γw=2,   Kx0=Kxa=Ky0=Kyb=Γx=2).
α, β
h
0
/
a
Model sequence
1
2
3
4
5
6
α
=
β
=
0.1
0.05
31.275
52.615
52.657
68.954
76.146
79.850
(31.303a)
(52.664)
(52.675)
(69.033)
(76.174)
(79.948)
0.1
17.270
25.244
25.250
33.949
46.050
46.625
(17.271a)
(25.255)
(25.258)
(34.089)
(46.122)
(46.708)
0.15
10.190
16.266
16.274
24.164
37.327
37.526
(10.189a)
(16.269)
(16.280)
(24.323)
(37.401)
(37.622)
0.2
6.780
12.258
12.260
19.685
31.452
31.754
(6.780a)
(12.265)
(12.270)
(19.851)
(31.514)
(31.864)
α
=
β
=
0.2
0.05
32.530
52.947
53.141
68.954
77.034
80.589
(32.577a)
(53.014)
(53.203)
(69.172)
(77.088)
(80.773)
0.1
16.882
24.979
24.979
34.154
48.057
48.575
(16.883a)
(24.983)
(25.001)
(34.433)
(48.134)
(48.667)
0.15
9.804
16.284
16.309
24.683
38.844
39.085
(9.803a)
(16.283)
(16.322)
(24.980)
(38.909)
(39.191)
0.2
6.499
12.290
12.295
20.133
32.354
32.770
(6.498a)
(12.296)
(12.307)
(20.420)
(32.387)
(32.895)
α
=
β
=
0.3
0.05
33.516
52.845
53.308
68.704
78.203
81.561
(33.563a)
(52.939)
(53.389)
(69.055)
(78.271)
(81.789)
0.1
16.419
24.799
24.830
34.527
50.061
50.583
(16.419a)
(24.790)
(24.869)
(34.947)
(50.143)
(50.688)
0.15
9.430
16.334
16.378
25.236
40.193
40.506
(9.429a)
(16.329)
(16.395)
(25.660)
(40.243)
(40.625)
0.2
6.235
12.288
12.293
20.572
33.102
33.663
(6.235a)
(12.296)
(12.306)
(20.965)
(33.090)
(33.811)
α
=
β
=
0.4
0.05
34.215
52.463
53.264
68.430
79.654
82.865
(34.252a)
(52.575)
(53.346)
(68.921)
(79.732)
(83.113)
0.1
15.921
24.695
24.793
35.027
52.011
52.571
(15.921a)
(24.674)
(24.842)
(35.583)
(52.098)
(52.688)
0.15
9.074
16.388
16.444
25.797
41.379
41.789
(9.074a)
(16.381)
(16.464)
(26.336)
(41.404)
(41.928)
0.2
5.990
12.236
12.238
21.002
33.735
34.461
(5.989a)
(12.249)
(12.250)
(21.487)
(33.653)
(34.641)
α
=
β
=
0.5
0.05
34.615
51.972
53.088
68.255
81.344
84.510
(33.806a)
(52.100)
(53.157)
(68.891)
(81.433)
(84.761)
0.1
15.416
24.659
24.829
35.615
53.879
54.494
(15.045a)
(24.625)
(24.888)
(36.303)
(53.970)
(54.626)
0.15
8.740
16.420
16.481
26.353
42.417
42.946
(8.529a)
(16.414)
(16.499)
(26.996)
(42.398)
(43.114)
0.2
5.761
12.127
12.127
21.434
34.286
35.192
(5.622a)
(12.134)
(12.148)
(21.998)
(34.110)
(35.415)
aResults in parentheses are taken from FEM.The above studies are given as linearly varying thickness moderately thick plates with several of boundary condition and different structure parameters. In the next section, vibration results for the plates subjected to nonlinear variation thickness will be presented.


### 3.4. Nonlinearly Varying Thickness Moderately Thick Plates with Classical and Elastic Boundary Conditions

In Section3.3, the linearly varying thickness moderately thick plates were studied. However, in the practical engineering applications, the varying thickness of a moderately thick plate may not always be linear variation in nature. A variety of possible thickness varying cases may be encountered in practice. Therefore, the moderately thick plates with nonlinear variation thickness subjected to general elastic edge restraints are examined in this subsection. For the sake of brevity, the indexes s and t will be chosen as 2 to imitate the nonlinearly varying thickness moderately thick plates structure in this section; that is, s=t=2. Also the thickness varying functions are also be expressed as Fourier cosine series according to (18) and (19).In order to validate the accuracy and reliability of the proposed method for predicting the vibration behavior of nonlinearly varying thickness moderately thick plates with arbitrary boundary conditions, the typical classical boundary conditions viewed as the special cases of elastically restrained edges will be considered. The comparison of the first six frequency parametersΩ=ωa2(ρh0/D0)1/2 for the moderately thick plates with nonlinearly varying thickness is presented in Table 9. The S-S-S-S, C-F-F-F, S-S-F-F, C-C-C-C, and S-C-S-C boundary conditions are performed in the comparison. The results adequately demonstrated the great accuracy of the modified Fourier method.Table 9
Frequency parameterΩ=ωa2(ρh0/D0)1/2 for moderately thick plates with linearly varying thickness in different boundary conditions (h(x,y)=h0(1+0.5x2)(1+0.5y2)).
Boundary condition
a
/
b
h
0
/
a
Model sequence
1
2
3
4
5
6
S-S-S-S
1
0.1
23.462
54.999
55.573
82.950
100.900
101.187
(23.741a)
(55.114)
(55.983)
(83.146)
(101.004)
(101.347)
0.2
18.606
42.725
42.971
60.040
71.167
71.761
(18.936a)
(43.014)
(43.148)
(60.247)
(71.642)
(71.993)
3/2
0.1
53.055
65.251
85.404
106.700
114.880
143.531
(53.467a)
(65.472)
(85.984)
(107.211)
(115.687)
(144.007)
0.2
17.439
21.531
32.170
44.987
59.438
64.179
(17.874a)
(21.748)
(32.651)
(45.264)
(60.188)
(64.879)
C-F-F-F
1
0.1
3.658
9.317
24.135
32.816
35.433
59.179
(3.749a)
(9.547)
(24.034)
(33.627)
(36.308)
(60.142)
0.2
3.520
7.986
19.591
26.959
27.491
43.882
(3.611a)
(8.176)
(20.075)
(27.125)
(27.625)
(43.984)
3/2
0.1
3.901
9.963
28.984
69.187
73.686
89.360
(3.930a)
(10.095)
(29.381)
(69.578)
(73.994)
(90.242)
0.2
3.292
4.439
13.385
23.947
40.850
44.758
(3.386a)
(4.498)
(14.012)
(24.068)
(41.221)
(45.142)
S-S-F-F
1
0.1
3.557
21.268
22.762
43.524
60.253
62.429
(3.649a)
(21.441)
(22.796)
(43.916)
(60.622)
(62.630)
0.2
3.171
17.822
19.512
33.616
45.042
46.833
(3.489a)
(18.006)
(20.430)
(33.815)
(45.247)
(49.004)
3/2
0.1
6.738
31.658
52.071
75.625
86.604
94.903
(6.957a)
(31.882)
(52.518)
(75.823)
(86.939)
(95.610)
0.2
3.653
12.930
20.955
26.517
44.097
48.058
(6.807a)
(13.160)
(21.312)
(26.864)
(44.422)
(48.873)
C-C-C-C
1
0.1
41.020
75.196
75.614
104.142
119.683
120.949
(41.405a)
(75.452)
(75.882)
(104.711)
(119.637)
(121.137)
0.2
31.237
51.686
51.868
68.411
76.868
77.864
(32.010a)
(52.960)
(52.145)
(69.335)
(77.481)
(78.099)
3/2
0.1
95.823
128.727
163.657
173.751
190.890
223.595
(95.364a)
(128.932)
(163.588)
(172.350)
(190.350)
(222.948)
0.2
53.306
71.372
89.384
102.404
104.660
120.908
(53.587a)
(71.759)
(90.103)
(102.503)
(105.174)
(121.275)
S-C-S-C
1
0.1
33.715
60.298
71.446
94.187
103.802
117.928
(34.148a)
(60.786)
(71.828)
(94.515)
(104.263)
(119.326)
0.2
26.143
45.499
49.458
64.253
72.941
76.191
(26.789a)
(46.622)
(50.680)
(64.329)
(73.474)
(76.843)
3/2
0.1
73.228
115.845
137.445
148.166
168.342
182.542
(73.761a)
(116.121)
(137.788)
(148.530)
(168.698)
(182.921)
0.2
41.302
53.648
65.630
72.202
81.632
88.105
(41.304a)
(53.757)
(65.842)
(72.794)
(82.033)
(88.474)
aResults in parentheses are taken from FEM.We now turn to elastically restrained moderately thick plates. The first one involves an S-S-S-S moderately thick square plate with a uniform rotational restraint along each edge; that is,Kx0=Kxa=Ky0=Kyb=Γx. The calculated frequency parameters Ω=ωa2(ρh0/D0)1/2 are given in Table 10 together with the FEM results. Since the elastically restrained plates with nonlinear variation thickness are rarely investigated, the FEM results are used as the reference. A good agreement is observed between the current and FEM results. The second example concerns a moderately thick square plate elastically supported along all edges. The stiffness of the linear and rotational restraints is set equal to kx0=kxa=ky0=kyb=Γw and Kx0=Kxa=Ky0=Kyb=Γx, respectively. The six frequency parameters Ω=ωa2(ρh0/D0)1/2 are shown in Table 11 for several different restraining coefficient values. It can also be noticed that the Fourier series method is able to predict the modal characteristics of nonlinearly varying thickness moderately thick plate with not only classical boundary conditions but also elastic edge restraints correctly.Table 10
Frequency parameterΩ=ωa2(ρh0/D0)1/2 for S-S-S-S moderately thick plates (a/b=1) with linearly varying thickness and elastic rotation support (h(x,y)=h0(1+0.5x2)(1+0.5y2), Kx0=Kxa=Ky0=Kyb=Γx).
Γ
x
a
h
0
/
a
Model sequence
1
2
3
4
5
6
0
0.05
23.462
54.999
55.573
82.950
100.900
101.187
(23.741b)
(55.114)
(55.983)
(83.146)
(101.004)
(101.347)
0.1
18.606
42.725
42.971
60.040
71.167
71.761
(18.936b)
(43.014)
(43.148)
(60.247)
(71.642)
(71.993)
0.15
53.055
65.251
85.404
106.700
114.880
143.531
(53.467b)
(65.472)
(85.984)
(107.211)
(115.687)
(144.007)
0.2
17.439
21.531
32.170
44.987
59.438
64.179
(17.874b)
(21.748)
(32.651)
(45.264)
(60.188)
(64.879)
0.5
0.05
6.904
13.374
13.514
19.400
22.904
23.056
(9.909b)
(13.548)
(13.651)
(19.564)
(23.088)
(23.379)
0.1
5.316
10.237
10.314
14.499
17.155
17.264
(5.374b)
(10.448)
(10.651)
(14.764)
(17.188)
(17.879)
0.15
4.136
8.201
8.220
11.474
13.578
13.671
(4.174b)
(8.348)
(8.651)
(11.764)
(13.688)
(13.879)
0.2
3.449
6.915
6.924
9.549
11.225
11.327
(3.474b)
(6.948)
(6.951)
(9.764)
(11.388)
(11.579)
1
0.05
7.040
13.617
13.745
19.726
23.233
23.402
(7.074b)
(13.748)
(13.851)
(19.964)
(23.488)
(23.879)
0.1
5.712
10.696
10.793
14.991
17.594
17.744
(5.774b)
(10.748)
(10.851)
(15.054)
(17.788)
(17.879)
0.15
4.489
8.507
8.540
11.759
13.803
13.922
(4.514b)
(8.548)
(8.611)
(11.864)
(13.988)
(14.119)
0.2
3.677
7.084
7.090
9.692
11.330
11.445
(3.684b)
(7.148)
(7.191)
(9.764)
(11.588)
(11.879)
1.5
0.05
7.089
13.708
13.828
19.847
23.355
23.530
(7.124b)
(13.748)
(13.951)
(19.967)
(23.588)
(23.879)
0.1
5.895
10.923
11.024
15.239
17.813
17.988
(5.874b)
(10.748)
(11.651)
(15.264)
(17.818)
(17.979)
0.15
4.697
8.694
8.737
11.938
13.790
14.083
(4.774b)
(8.748)
(8.951)
(11.864)
(13.688)
(14.879)
0.2
3.836
7.201
7.210
9.792
11.403
11.530
(3.874b)
(7.748)
(7.651)
(9.864)
(11.588)
(11.879)
2
0.05
7.114
13.754
13.871
19.911
23.418
23.597
(7.174b)
(12.848)
(13.851)
(19.864)
(23.588)
(23.879)
0.1
6.001
11.060
11.159
15.391
17.944
18.136
(6.004b)
(11.148)
(11.151)
(15.464)
(17.988)
(18.379)
0.15
4.836
8.821
8.872
12.061
14.037
14.189
(4.874b)
(8.848)
(8.951)
(12.164)
(14.188)
(14.879)
0.2
3.954
7.289
7.301
9.871
11.457
11.594
(3.974b)
(7.348)
(7.651)
(9.964)
(11.488)
(11.679)
a
Γ
x
=
K
i
/
K
0
,
(
K
0
=
1
×
1
0
9
Nm/rad,  i=x0,xa,y0,yb).
bResults in parentheses are taken from FEM.Table 11
Frequency parameterΩ=ωa2(ρh0/D0)1/2 for moderately thick plates (a/b=1) with linearly varying thickness and elastic rotation and translation support (h(x,y)=h0(1+0.5x2)(1+0.5y2),  Kx0=Kxa=Ky0=Kyb=Γx,  and  kx0=kxa=ky0=kyb=Γw).
Γ
x
a
Γ
w
a
h
0
/
a
Model sequence
1
2
3
4
5
6
0.5
0
0.05
12.926
13.041
25.090
47.935
47.943
61.761
(12.931b)
(13.053)
25.099
(47.947)
(47.952)
(61.788)
0.1
10.900
10.988
21.545
39.980
40.147
50.322
(10.904b)
(10.992)
(21.557)
(39.984)
(40.166)
(50.334)
0.15
8.245
8.298
18.081
31.503
32.617
40.950
(8.247b)
(8.230)
(18.092)
(31.508)
(32.622)
(40.955)
0.2
6.106
6.124
15.639
25.631
27.601
34.458
(6.110b)
(6.125)
(15.642)
(25.637)
(27.617)
(34.462)
1
0.5
0.05
21.575
31.615
31.644
42.573
57.889
58.607
(21.580b)
(31.617)
(31.644)
(42.581)
(57.893)
(58.610)
0.1
8.451
16.024
16.210
25.969
43.839
43.867
(8.451b)
(16.025)
(16.217)
(25.972)
(43.844)
(43.870)
0.15
4.682
11.479
11.480
20.642
35.172
35.598
(4.684b)
(11.480)
(11.481)
(20.644)
(35.178)
(35.604)
0.2
3.059
8.621
8.648
17.320
28.487
29.638
(3.062b)
(8.619)
(8.644)
(17.327)
(28.492)
(29.641)
1.5
1
0.05
27.713
40.203
40.978
52.951
65.352
66.815
(27.714b)
(40.207)
(40.988)
(52.972)
(65.368)
(66.822)
0.1
11.667
19.322
19.615
29.122
46.115
46.183
(11.674b)
(19.327)
(19.619)
(29.137)
(46.120)
(46.188)
0.15
6.544
13.416
13.481
22.405
37.238
37.430
(6.547b)
(13.420)
(13.488)
(22.410)
(37.241)
(37.434)
0.2
4.295
10.175
10.190
18.543
30.330
31.056
(4.296b)
(10.177)
(10.194)
(18.544)
(30.336)
(31.060)
2
2
0.05
33.453
50.854
52.088
66.311
76.610
78.947
(33.455b)
(50.852)
(50.097)
(66.321)
(76.627)
(78.971)
0.1
15.832
24.130
24.372
33.924
49.118
49.362
(15.830b)
(24.127)
(24.374)
(33.931)
(49.147)
(49.382)
0.15
9.060
15.956
16.099
24.754
39.134
39.249
(9.064b)
(15.961)
(16.102)
(24.755)
(39.138)
(39.452)
0.2
5.992
11.955
11.971
20.034
31.908
32.392
(5.991b)
(11.957)
(11.978)
(20.042)
(31.910)
(32.398)
a
Γ
x
=
K
i
/
K
0
,
(
K
0
=
1
×
1
0
9
Nm/rad,  i=x0,xa,y0,yb),  Γw=ki/k0,(k0=1×109N/m,  i=x0,xa,y0,yb).
bResults in parentheses are taken from FEM.Finally, the influence of the gradient on the fundamental frequency parameters of a nonlinearly varying thickness moderately thick plate is investigated. The plate is elastically restrained, in which the stiffness of the boundary springs is taken askx0=kxa=ky0=kyb=Γw=2 and Kx0=Kxa=Ky0=Kyb=Γx=2, respectively. The six frequency parameters Ω=ωa2(ρh0/D0)1/2 are presented in Table 12 for several different slop values. It can be seen that the fundamental frequency parameters will decrease with the increase of the parameter h0/a.Table 12
Frequency parameterΩ=ωa2(ρh0/D0)1/2 for elastic support moderately thick plates (a/b=1) with different gradient (kx0=kxa=ky0=kyb=Γw=2, Kx0=Kxa=Ky0=Kyb=Γx=2).
α, β
h
0
/
a
Model sequence
1
2
3
4
5
6
α
=
β
=
0.1
0.05
30.187
51.102
51.141
67.066
73.954
77.412
(30.188a)
(51.105)
(51.144)
(67.068)
(73.961)
(77.477)
0.1
16.995
24.719
24.724
33.071
44.343
44.819
(16.996a)
(24.725)
(24.729)
(33.084)
(44.347)
(44.822)
0.15
10.077
15.872
15.883
23.443
35.937
36.065
(10.081a)
(15.877)
(15.892)
(23.447)
(35.941)
(36.084)
0.2
6.713
11.953
11.958
19.086
30.372
30.604
(6.713a)
(11.957)
(11.965)
(19.092)
(30.378)
(30.612)
α
=
β
=
0.2
0.05
31.206
51.436
51.614
67.058
74.398
77.595
(31.208a)
(51.438)
(51.619)
(67.064)
(74.402)
(77.596)
0.1
16.786
24.503
24.508
33.144
45.583
45.923
(16.790a)
(24.510)
(24.513)
(33.147)
(45.587)
(45.932)
0.15
9.821
15.865
15.906
23.766
36.897
36.990
(9.824a)
(15.870)
(15.907)
(23.768)
(36.894)
(36.997)
0.2
6.523
11.978
11.992
19.356
30.912
31.162
(6.524a)
(11.981)
(11.997)
(19.362)
(30.916)
(31.174)
α
=
β
=
0.3
0.05
32.097
51.462
51.900
66.832
74.991
77.863
(32.098a)
(51.466)
(51.907)
(66.849)
(74.996)
(77.872)
0.1
16.507
24.332
24.383
33.330
46.806
47.074
(16.511a)
(24.327)
(24.380)
(33.339)
(46.810)
(47.086)
0.15
9.562
15.885
15.964
24.103
37.748
37.828
(9.562a)
(15.892)
(15.977)
(21.105)
(37.751)
(37.832)
0.2
6.338
11.992
12.013
19.603
31.338
31.634
(6.340a)
(11.994)
(12.011)
(19.612)
(31.344)
(31.639)
α
=
β
=
0.4
0.05
32.851
51.239
52.046
66.547
75.735
78.303
(33.455a)
(50.852)
(50.097)
(66.321)
(76.627)
(78.971)
0.1
16.182
24.209
24.343
33.598
47.990
48.230
(33.455a)
(50.852)
(50.097)
(66.321)
(76.627)
(78.971)
0.15
9.307
15.919
16.034
24.436
38.491
38.579
(33.455a)
(50.852)
(50.097)
(66.321)
(76.627)
(78.971)
0.2
6.161
11.986
12.008
19.828
31.666
32.040
(33.455a)
(50.852)
(50.097)
(66.321)
(76.627)
(78.971)
α
=
β
=
0.5
0.05
33.453
50.854
52.088
66.311
76.610
78.947
(33.455a)
(50.852)
(50.097)
(66.321)
(76.627)
(78.971)
0.1
15.832
24.130
24.372
33.924
49.118
49.362
(15.830a)
(24.127)
(24.374)
(33.931)
(49.147)
(49.382)
0.15
9.060
15.956
16.099
24.754
39.134
39.249
(9.064a)
(15.961)
(16.102)
(24.755)
(39.138)
(39.452)
0.2
5.992
11.955
11.971
20.034
31.908
32.392
(5.991a)
(11.957)
(11.978)
(20.042)
(31.910)
(32.398)
aResults in parentheses are taken from FEM.In the above examples, it has been demonstrated that the presented method can be universally applied to nonlinearly varying thickness moderately thick plates with several boundary conditions and different structure parameters. New results are obtained for plates with nonlinear variation thickness in both directions subjected to general elastic boundary restraints, which may be used for benchmarking of researchers in the field. In addition, it is interesting to see that the nature frequency decreases with the increase of the index for thickness function.


## 3.1. Convergence and Stiffness Value Study

Since the Fourier series is numerically truncated and only finite terms are counted in actual calculations, the proposed solution should be understood as a solution with arbitrary precision. In this subsection, a uniform thickness square moderately thick rectangular plate with completely clamped boundary condition (C-C-C-C) and four edges equally elastically restrained against linear spring constants and rotation spring constants supports (E-E-E-E,KT=ki/(b3/D), KR=Ki/(b/D), KT=10, and KR=5) has been selected to demonstrate the convergence and accuracy of the modified Fourier method. In Table 1, the first eight frequency parameters Ω=ωb2(ρh/D)1/2/π2 for the considered uniform thickness square moderately thick rectangular plate with C-C-C-C and E-E-E-E boundary conditions are examined. The table shows that the proposed method has fast convergence behavior. The maximum discrepancy in the worst case between the 6×6 truncated configuration and the 8×8 one is less than 0.064%. In order to fully illustrate the convergence of the present method, Figures 3, 4, and 5 present the 1st and 8th frequency parameters Ω=ωb2(ρh/D)1/2/π2 with various truncated numbers M=N subjected to different boundary condition and aspect ratios. A highly desired convergence characteristic is observed such that (a) sufficiently accurate results can be obtained with only a small number of terms in the series expansions and (b) the solution is consistently refined as more terms are included in the expansions. However, this should not constitute a problem in practice because one can always verify the accuracy of the solution by increasing the truncation number until a desired numerical precision is achieved. As a matter of fact, this “quality control” scheme can be easily implemented automatically. In modal analysis, the natural frequencies for higher-order modes tend to converge slower (see Table 1). Thus, an adequate truncation number should be dictated by the desired accuracy of the largest natural frequencies of interest. In view of the excellent numerical behavior of the current solution, the truncation numbers will be simply set as M=N=12 in the following calculations.Table 1
Frequency parametersΩ=ωb2(ρh/D)1/2/π2 for square moderately thick plates with uniform thickness and different boundary conditions.
Boundary conditions
M, N
Model sequence
1
2
3
4
5
6
7
8
C-C-C-C
M
=
N
=
2
3.375
7.191
7.191
9.856
16.05
16.28
17.47
17.47
M
=
N
=
4
3.299
6.308
6.308
8.853
10.46
10.56
12.63
12.63
M
=
N
=
6
3.294
6.282
6.282
8.801
10.37
10.47
12.54
12.54
M
=
N
=
8
3.292
6.278
6.278
8.797
10.36
10.46
12.53
12.53
M
=
N
=
10
3.292
6.276
6.276
8.794
10.36
10.46
12.53
12.53
M
=
N
=
12
3.292
6.276
6.276
8.793
10.36
10.46
12.52
12.52
M
=
N
=
14
3.292
6.276
6.276
8.793
10.36
10.46
12.52
12.52
M
=
N
=
16
3.292
6.275
6.275
8.792
10.36
10.45
12.52
12.52
Reference [4]
3.292
6.276
6.276
8.792
10.36
10.46
12.52
12.52
E-E-E-E
M
=
N
=
2
0.6153
1.0861
1.0861
1.7898
2.7709
2.8417
3.5010
3.5010
M
=
N
=
4
0.6150
1.0799
1.0799
1.7785
2.7695
2.8405
3.4914
3.4914
M
=
N
=
6
0.6149
1.0796
1.0796
1.7773
2.7694
2.8403
3.4898
3.4898
M
=
N
=
8
0.6149
1.0796
1.0796
1.7771
2.7693
2.8403
3.4893
3.4893
M
=
N
=
10
0.6149
1.0796
1.0796
1.7770
2.7693
2.8403
3.4892
3.4892
M
=
N
=
1
2
0.6149
1.0796
1.0796
1.7770
2.7693
2.8403
3.4892
3.4892
M
=
N
=
1
4
0.6149
1.0796
1.0796
1.7770
2.7693
2.8403
3.4892
3.4892
M
=
N
=
1
6
0.6149
1.0796
1.0796
1.7770
2.7693
2.8403
3.4891
3.4891
Reference [33]
0.6153
1.0861
1.0861
1.7898
2.7709
2.8417
3.5010
3.5010
FEM
0.6150
1.7097
1.7097
1.7774
2.7694
2.8406
3.4898
3.4898As far as the accuracy of the present method is concerned, the converged solutions of the present method are in excellent agreement with both the results reported by reference data and the finite element results. For C-C-C-C boundary conditions, the max discrepancy between the present results and the reference data does not exceed 0.011% for the worst case and in most cases is 0%. Comparing the results with exact solutions [4], it is observed that eight terms are sufficient to obtain accurate results. Moreover, with regard to the E-E-E-E boundary condition, the max discrepancy between the present results and the reference data does not exceed 0.31% for the worst case and in most cases is 0.1%. Regarding the results with DQM solutions [33], it can be seen that the six terms are sufficient to obtain enough accurate results. In addition, it is clear that the results of the present approach with just 663 DOFs (M×N=12×12) can predict the vibration characteristics accurately. Most of them are identical to those obtained from finite element method (FEM) with 102,01 DOFs (S4R: 0.01 m × 0.01 m). That is to say, it needs only 6.62% DOFs compared with FEM to obtain the same precision solutions for the considered case. On the same hardware (Intel i7-3.9 GHz), the computing time of the present formulation for the solution (M×N=12×12) implemented in optimized MATLAB scripts is about 2.125 s, whereas the finite element solution consumes 34.578 s, that is, at least 16 times more CPU time than the present method for the same problem.As mentioned earlier, in the current modeling framework, all the classical boundary conditions and their combinations can be conveniently viewed as special cases when the stiffness for the normal and tangential boundary springs becomes zero and/or infinitely large. Thus, the effects of the stiffness of the translation (ki), rotation (Ki), and torsional springs (Ki,j) on the modal characteristics should be investigated. As shown in the Figures 2(a)–2(c), the first and the third frequency parameters are separately obtained by varying the stiffness of one group of the boundary springs from extremely large (1014) to extremely small (100) while assigning the other group of the springs infinite stiffness (1014). It can be found in Figure 2(a) that the frequency parameter almost keeps at a level when the stiffness of the translation springs is larger than 1012 or smaller than 107. In Figure 2(b), the influences of the rotation springs on frequency parameters are given. It is shown that the frequency curves change greatly within the stiffness range (106 to 1010) while out of this range the frequency curves separately keep level. In Figure 2(c), the influences of the torsional springs on frequency parameters are given. It is shown that the frequency curves almost change when the stiffness changes in the whole range.The effect of boundary spring stiffness on the natural frequenciesΩ: (a) translation spring, (b) rotation spring, and (c) torsional spring.
(a)
(b)
(c)The effect of numerically truncated numbersM=N on the natural frequencies Ω for C-C-C-C boundary condition.
(a)
The 1st order
(b)
The 8th orderThe effect of numerically truncated numbersM=N on the natural frequencies Ω for F-F-F-F boundary condition.
(a)
The 1st order
(b)
The 8th orderThe effect of numerically truncated numbersM=N on the natural frequencies Ω with four edges elastically restrained against translation and rotation (KT=ki/(b3/D), KR=Ki/(b/D), KT=100, and KR=100).
(a)
The 1st order
(b)
The 8th orderBased on the analysis, it can be found that the torsional springs almost have no effect on the structure. Also, the relationship between the rotation springs and twisting moment can be seen from the boundary condition expression. Then, the twisting moments have small effect on the vibration characteristics of the structure. However, in this paper, in order to get a more accurate prediction of the vibration characteristics of the structure, the twisting moment on boundary edges is taken into account. In the latter study, in addition to the fact that free boundary is not unexpected considering the torsion spring, the other boundary conditions are introduced into a torsion spring, and the spring stiffness takes infinity. Form Figure2 analysis, it also concluded that the translation spring has a wider influence range than the rotation spring on the frequency parameters; that is, for the translation springs, the stable frequency parameters appear when the stiffness is more than 1012 or less 107 while for the rotation springs, when the stiffness value is assigned more than 1010 the frequency parameters become smooth. Thus, it is suitable to use 1014 to simulate the infinite stiffness value in the model validation parts and in the following examples.


## 3.2. Uniform Thickness Moderately Thick Plates with Classical and Elastic Boundary Conditions

In this subsection, the modified Fourier solution is applied to deal with vibration problems of uniform thickness moderately thick rectangular plates subject to the classical boundary conditions and arbitrary elastic boundary conditions. In present work, three groups of continuously distributed boundary springs are introduced to simulate the given or typical boundary conditions. As mentioned earlier, the stiffness of these boundary springs can take any value from zero to infinity to better model many real-world restraint conditions. Taking edgex=0, for example, the corresponding spring stiffness for the three types of classical boundaries and elastic boundaries iscompletely free:Qx=0, Mxx=0, and Mxy=0(17a)kx0=0,Kx0=0,Kyx0=0,simply supported:w=0, Mxx=0, and ψy=0(17b)kx0=1014,Kx0=0,Kyx0=1014,completely clamped:w=0, ψx=0, and ψy=0(17c)kx0=1014,Kx0=1014,Kyx0=1014,arbitrarily elastic boundary:w≠0, ψx≠0, ψy≠0, Qx≠0, Mxx≠0, and Mxy≠0(17d)kx0=Γwx0,Kx0=Γxx0,Kyx0=1014.The appropriateness of the three classical boundaries defined in (17a)–(17c) will be proved by several examples given in following, the arbitrary elastic boundaries are also defined in (17d), and the Γwi (Γwi=ki/k0, k0=1×109 N/m2, i=x0, xa, y0, yb) and Γxi (Γxi=Ki/K0, K0=1×109 Nm/rad, i=x0, xa, y0, yb) elastic restraint parameters represent corresponding spring stiffness. For the sake of simplicity, a four-letter string is employed to represent the restraint condition of a plate, such that F-C-S-E identifies the plate with edges x=0, y=0, x=a, and y=b having free, clamped, shear-supported restrained, and elastic boundary conditions, respectively.As for the first case, a uniform thickness moderately thick plate with different classical boundaries and structure parameters is investigated here. In Table2, the comparison of the first eight frequency parameters Ω=ωb2(ρh/D)1/2/π2 of the considered plate is presented. The S-S-S-S, C-F-F-F, S-S-F-F, F-F-F-F, and S-C-S-C boundary conditions are performed in the comparison. Excellent agreements are observed between the solutions obtained by the modified Fourier method, the referential data, and finite element method (FEM) results for the uniform thickness moderately thick rectangular plates. It is also verified that the definition of the three types of classical boundaries in (17a)–(17c) is appropriate. In addition, the elastic boundary conditions (17d) are also verified. In the next two examples, we will account for the vibration of moderately thick plate with elastic edge supports. The first model considered is an S-S-S-S square moderately thick plate with all edges elastically rotationally restrained. That is, Kx0=Kxa=Ky0=Kyb=Γx, and all the other restraining springs are set to have an infinite stiffness (namely, represented by 1014 in numerical calculation). The six frequency parameters Ω=ωb2(ρh/D)1/2/π2 are given in Table 3 for several different restraining coefficient values; the finite element method (FEM) results calculated using ABAQUS are also listed in this table as a reference. The second model concerns a complete square moderately thick plate with all edges elastically restrained. That is, kx0=kxa=ky0=kyb=Γw and Kx0=Kxa=Ky0=Kyb=Γx. The six frequency parameters Ω=ωb2(ρh/D)1/2/π2 are given in Table 4 for several different restraining coefficient values; the finite element method (FEM) results are also listed in Table 4 as a reference. It can be clearly seen that the comparison is extremely good, which implies that the current method is able to make correct predictions for the modal characteristics of moderately thick plate with not only classical boundary conditions but also elastic edge restraints.Table 2
Frequency parametersΩ=ωb2(ρh/D)1/2/π2 for uniform thickness moderately thick plates with different classical boundaries and structure parameters.
Boundary condition
a/b
Model sequence
1
2
3
4
5
6
7
8
S-S-S-S
1
1.931
4.605
4.605
7.064
8.605
8.605
10.792
10.792
(1.933a)
(4.611)
(4.611)
(7.067)
(8.607)
(8.607)
(10.799)
(10.780)
(1.931b)
(4.605)
(4.605)
(7.064)
(8.605)
(8.605)
(10.792)
(10.792)
3/2
1.408
2.648
4.127
4.605
5.261
7.064
7.153
8.184
(1.410a)
(2.651)
(4.128)
(4.608)
(5.267)
(7.074)
(7.155)
(8.188)
(1.408b)
(2.648)
(4.127)
(4.605)
(5.261)
(7.064)
(7.151)
(8.184)
C-F-F-F
1
0.348
0.817
2.035
2.583
2.862
4.816
5.478
5.774
(0.348a)
(0.817)
(2.038)
(2.586)
(2.865)
(4.821)
(5.504)
(5.793)
(0.348b)
(0.816)
(2.034)
(2.582)
(2.860)
(4.811)
(5.477)
(5.772)
3/2
0.155
0.502
0.941
1.665
2.292
2.613
3.303
3.498
(0.154a)
(0.502)
(0.941)
(1.663)
(2.295)
(2.618)
(3.305)
(3.500)
(0.154b)
(0.501)
(0.940)
(1.662)
(2.292)
(2.612)
(3.298)
(3.494)
S-S-F-F
1
0.333
1.678
1.875
3.559
4.720
4.946
6.474
6.632
(0.335a)
(1.680)
(1.878)
(3.600)
(4.721)
(4.949)
(6.478)
(6.642)
(0.333b)
(1.677)
(1.874)
(3.557)
(4.718)
(4.945)
(6.472)
(6.631)
3/2
0.222
0.938
1.631
2.363
2.573
4.082
4.482
4.753
(0.223a)
(0.940)
(1.632)
(2.364)
(2.575)
(4.092)
(4.483)
(4.759)
(0.222b)
(0.938)
(1.631)
(2.361)
(2.571)
(4.077)
(4.479)
(4.752)
F-F-F-F
1
1.290
1.919
2.363
3.235
3.235
5.605
5.605
5.646
(1.291a)
(1.920)
(2.365)
(3.239)
(3.239)
(5.607)
(5.607)
(6.648)
3/2
0.865
0.948
1.961
2.166
2.461
2.843
3.551
4.026
(0.865a)
(0.949)
(1.963)
(2.168)
(2.465)
(2.844)
(3.556)
(4.034)
S-C-S-C
1
2.700
4.971
5.990
7.973
8.787
10.250
11.334
12.024
(2.701a)
(4.971)
(5.993)
(7.975)
(8.788)
(10.253)
(11.335)
(12.028)
3/2
2.348
3.253
4.972
5.646
6.489
7.381
7.974
9.930
(2.348a)
(3.255)
(4.974)
(5.647)
(6.590)
(7.384)
(7.801)
(9.932)
aResults in parentheses are taken from FEM.
bResults in parentheses are taken from [11].Table 3
Frequency parametersΩ=ωb2(ρh/D)1/2/π2 for S-S-S-S moderately thick plates (a/b=1) with uniform thickness and elastic rotation support (Kx0=Kxa=Ky0=Kyb=Γx).
Γ
x
a
h
0
/
a
Model sequence
1
2
3
4
5
6
0
0.05
1.945
4.834
4.834
7.591
9.515
9.519
(1.942b)
(4.836)
(4.836)
(7.584)
(9.547)
(9.552)
0.1
1.856
4.500
4.500
6.829
8.499
8.513
(1.858b)
(4.510)
(4.510)
(6.843)
(8.539)
(8.554)
0.15
1.758
4.114
4.114
6.040
7.421
7.445
(1.761b)
(4.126)
(4.126)
(6.060)
(7.462)
(7.486)
0.2
1.656
3.733
3.733
5.335
6.468
6.499
(1.660b)
(3.747)
(3.747)
(5.357)
(6.508)
(6.538)
0.5
0.05
3.541
7.071
7.071
10.227
12.318
12.399
(3.546b)
(7.091)
(7.091)
(10.253)
(12.386)
(12.468)
0.1
3.274
6.242
6.242
8.719
10.302
10.421
(3.282b)
(6.267)
(6.267)
(8.754)
(10.370)
(10.488)
0.15
2.947
5.361
5.361
7.290
8.465
8.606
(2.956b)
(5.387)
(5.387)
(7.327)
(8.524)
(8.666)
0.2
2.624
4.599
4.599
6.148
7.057
7.199
(2.634b)
(4.624)
(4.624)
(6.182)
(7.107)
(7.250)
1
0.05
3.541
7.072
7.072
10.228
12.320
12.401
(3.547b)
(7.093)
(7.093)
(10.254)
(12.388)
(12.470)
0.1
3.278
6.248
6.248
8.727
10.311
10.429
(3.286b)
(6.274)
(6.274)
(8.762)
(10.378)
(10.497)
0.15
2.957
5.374
5.374
7.305
8.478
8.620
(2.966b)
(5.400)
(5.400)
(7.341)
(8.537)
(8.680)
0.2
2.641
4.617
4.617
6.166
7.070
7.215
(2.651b)
(4.643)
(4.643)
(6.200)
(7.121)
(7.268)
1.5
0.05
3.542
7.073
7.073
10.229
12.320
12.401
(3.547b)
(7.093)
(7.093)
(10.254)
(12.388)
(12.471)
0.1
3.280
6.251
6.251
8.729
10.313
10.432
(3.287b)
(6.276)
(6.276)
(8.765)
(10.381)
(10.500)
0.15
2.960
5.379
5.379
7.309
8.482
8.625
(2.969b)
(5.405)
(5.405)
(7.346)
(8.541)
(8.685)
0.2
2.647
4.623
4.623
6.172
7.074
7.221
(2.657b)
(4.649)
(4.649)
(6.206)
(7.125)
(7.274)
2
0.05
3.542
7.073
7.073
10.229
12.321
12.402
(3.547b)
(7.093)
(7.093)
(10.255)
(12.389)
(12.471)
0.1
3.280
6.252
6.252
8.730
10.315
10.434
(3.288b)
(6.277)
(6.277)
(8.766)
(10.382)
(10.501)
0.15
2.962
5.381
5.381
7.312
8.484
8.628
(2.971b)
(5.407)
(5.407)
(7.348)
(8.543)
(8.688)
0.2
2.650
4.627
4.627
6.175
7.076
7.224
(2.660b)
(4.652)
(4.652)
(6.210)
(7.128
(7.277)
a
Γ
x
=
K
i
/
K
0
,
(
K
0
=
1
×
1
0
9
Nm
/
rad, i=x0,xa,y0,yb).
bResults in parentheses are taken from FEM.Table 4
Frequency parametersΩ=ωb2(ρh/D)1/2/π2 for moderately thick plates (a/b=1) with uniform thickness and all edge elastic restraints (Kx0=Kxa=Ky0=Kyb=Γx and kx0=kxa=ky0=kyb=Γw).
Γ
x
a
Γ
w
a
h
0
/
a
Model sequence
1
2
3
4
5
6
0.5
0
0.05
0.986
1.967
3.893
3.893
4.850
4.850
(0.988b)
(1.971)
(3.897)
(3.897)
(4.852)
(4.852)
0.1
0.916
1.835
3.515
3.522
4.375
4.375
(0.917b)
(1.837)
(3.519)
(3.528)
(4.341)
(4.341)
0.15
0.789
1.635
2.983
3.027
3.771
3.771
(0.792b)
(1.638)
(2.988)
(3.029)
(3.775)
(3.775)
0.2
0.646
1.444
2.497
2.601
3.249
3.249
(0.652b)
(1.448)
(2.508)
(2.621)
(3.254)
(3.254)
1
0.5
0.05
3.417
3.417
4.431
5.294
5.461
6.448
(3.419b)
(3.419)
(4.435)
(5.303)
(5.478)
(6.456)
0.1
1.593
1.593
2.401
3.849
3.861
4.729
(1.611b)
(1.611)
(2.408)
(3.852)
(3.866)
(4.767)
0.15
1.122
1.122
1.913
3.272
3.290
4.038
(1.127b)
(1.227)
(1.924)
(3.284)
(3.314)
(4.069)
0.2
0.890
0.890
1.640
2.781
2.829
3.476
(0.899b)
(0.899)
(1.654)
(2.787)
(2.834)
(3.497)
1.5
1
0.05
4.325
4.325
5.619
6.273
6.571
7.619
(4.334b)
(4.334)
(5.681)
(6.299)
(6.642)
(7.642)
0.1
2.013
2.013
2.812
4.088
4.116
4.993
(2.024b)
(2.024)
(2.827)
(4.122)
(4.187)
(4.994)
0.15
1.338
1.338
2.108
3.430
3.443
4.196
(1.342b)
(1.342)
(2.112)
(3.467)
(3.482)
(4.200)
0.2
1.038
1.038
1.772
2.936
2.965
3.613
(1.042b)
(1.042)
(1.786)
(2.937)
(2.977)
(3.624)
2
2
0.05
5.231
5.231
6.919
7.630
8.002
9.220
(5.238b)
(5.238)
(6.931)
(7.637)
(8.011)
(9.241)
0.1
2.594
2.594
3.439
4.470
4.544
5.428
(2.597b)
(2.597)
(3.442)
(4.495)
(4.557)
(5.438)
0.15
1.653
1.653
2.403
3.613
3.632
4.392
(1.662b)
(1.662)
(2.414)
(3.627)
(3.646)
(4.404)
0.2
1.238
1.238
1.949
3.077
3.099
3.751
(1.244b)
(1.244)
(1.964)
(3.094)
(3.116)
(3.788)
a
Γ
x
=
K
i
/
K
0
,
(
K
0
=
1
×
1
0
9
Nm/rad,  i=x0,xa,y0,yb),  Γw=ki/k0,(k0=1×109N/m,  i=x0,xa,y0,yb).
bResults in parentheses are taken from FEM.The excellent agreements between the solution obtained by the modified Fourier method and the referential data for the moderately thick plate subjected to the combinations of classical boundary conditions and elastic boundary conditions given in Tables2–4 indicate that the proposed method is sufficiently accurate to deal with uniform thickness moderately thick plate with arbitrary boundary conditions.


## 3.3. Linearly Variation Thickness Moderately Thick Plates with Classical and Elastic Boundary Conditions

In the theoretical formulations, this paper concerns the varying thickness moderately thick plates with classical and elastic boundary conditions. The varying thickness functionh(x,y) can be expressed as h0(1+αxs)(1+βyt), in which the h0, α, and β represent the initial thickness, gradient in x direction, and gradient in y direction. When the indexes s and t take the value s=t=1, the analytical model imitates the linearly variation thickness moderately thick plates structure. In order to unify the description and facilitate the analytical calculations of the involved integrals, all the thickness variation functions can be expanded into either 1D or 2D Fourier cosine series, resulting in
(18)h(x,y)=∑m=0∞∑n=0∞h0qmncos⁡⁡mπxacos⁡⁡mπyb,
where
(19)qmn=4ab∫0b∫0a(1+αxs)(1+βyt)cos⁡⁡mπxacos⁡⁡mπybdxdy.In order to prove the validity of the present method for the vibration of linearly variation thickness moderately thick plates with arbitrary boundary conditions, the typical classical boundary conditions as the first case will be considered. In Table5, the comparison of the first six frequency parameters Ω=ωa2(ρh0/D0)1/2 of the moderately thick plates with linearly varying thickness is presented. The S-S-S-S, C-F-F-F, S-S-F-F, C-C-C-C, and S-C-S-C boundary conditions are performed in the comparison. Excellent agreements are observed between the solutions obtained by the modified Fourier method and finite element method (FEM) results for the moderately thick plates with linear variation thickness. To investigate the influence of the aspect ratio on the uniform thickness and nonuniform thickness moderately thick plates, the effect on the frequency parameters for plates with S-S-S-S boundary conditions is presented in Figure 6. The thickness functions are h0 and h0(1+0.5×x)(1+0.5×y), respectively. It is seen from Figure 6 that the influence of aspect ratios on the frequency parameters for nonuniform thickness moderately thick plates is more complicated.Table 5
Frequency parameterΩ=ωa2(ρh0/D0)1/2 for moderately thick plates with linearly varying thickness in different boundary conditions (h(x,y)=h0(1+0.5x)(1+0.5y)).
Boundarycondition
a
/
b
h
0
/
a
Model sequence
1
2
3
4
5
6
S-S-S-S
1
0.1
27.261
62.397
62.926
92.423
111.537
111.732
(27.263a)
(62.410)
(63.842)
(92.445)
(111.867)
(111.921)
0.2
23.103
47.010
47.191
65.018
75.831
75.865
(23.489a)
(47.209)
(47.349)
(65.057)
(76.605)
(76.473)
3/2
0.1
59.338
99.101
138.798
151.102
166.710
204.855
(59.874a)
(99.249)
(138.177)
(151.362)
(166.644)
(205.381)
0.2
41.175
63.765
63.765
73.598
83.956
89.084
(41.586a)
(63.313)
(63.669)
(73.661)
(83.918)
(89.547)
C-F-F-F
1
0.1
4.173
10.331
27.407
37.826
38.809
64.095
(4.856a)
(10.680)
(27.513)
(37.203)
(39.135)
(64.858)
0.2
3.983
8.702
21.403
29.443
30.286
46.705
(4.118a)
(9.306)
(21.990)
(29.670)
(30.602)
(47.156)
3/2
0.1
4.868
15.488
35.620
63.523
89.969
100.089
(4.887a)
(15.320)
(35.175)
(63.353)
(89.159)
(100.783)
0.2
4.469
9.763
22.338
42.188)
50.104
56.331
(4.944a)
(9.636)
(22.087)
(42.177)
(50.580)
(56.559)
S-S-F-F
1
0.1
4.117
24.795
26.216
48.567)
68.195
69.667
(4.789a)
(24.149)
(26.751)
(49.324)
(68.842)
(70.575)
0.2
3.820
20.516
21.924
37.523
48.980
50.327
(4.103a)
(20.041)
(22.125)
(37.221)
(48.894)
(50.622)
3/2
0.1
7.935
39.454
65.063
92.877
92.877
136.176
(7.856a)
(40.145)
(65.162)
(92.462)
(93.022)
(136.336)
0.2
6.389
26.149
35.627
49.539
56.844
65.170
(6.406a)
(26.536)
(35.626)
(49.831)
(57.144)
(65.981)
C-C-C-C
1
0.1
44.471
80.299
80.681
109.422
126.871
128.207
(45.102a)
(80.271)
(80.656)
(110.015)
(126.328)
(128.888)
0.2
32.162
52.781
52.933
69.410
78.474
79.482
(32.240a)
(52.629)
(53.166)
(70.063)
(79.059)
(80.266)
3/2
0.1
82.975
115.587
151.992
161.579
176.601
212.330
(83.139a)
(115.838)
(152.200)
(162.193)
(177.058)
(212.849)
0.2
48.577
66.022
83.468
90.830
97.474
98.157
(49.187a)
(66.259)
(84.065)
(91.749)
(97.827)
(98.777)
S-C-S-C
1
0.1
36.847
66.664
76.965
101.045
113.594
125.551
(37.901a)
(68.568)
(78.356)
(101.200)
(113.978)
(125.651)
0.2
27.673
48.469
51.220
67.027
76.358
78.241
(27.897a)
(48.685)
(51.491)
(67.447)
(77.530)
(79.573)
3/2
0.1
77.378
106.848
150.079
154.569
173.120
208.391
(78.003a)
(107.058)
(150.991)
(155.171)
(173.099)
(208.402)
0.2
45.895
64.732
73.393
83.282
90.325
97.089
(45.803a)
(64.131)
73.656
83.801
90.518
97.015
aResults in parentheses are taken from FEM.The effect of aspect ratioa/b on the natural frequencies Ω for S-S-S-S boundary condition: (a) uniform thickness and (b) nonuniform thickness.
(a)
(b)In the next two examples, we also account for the vibrations of moderately thick plate with linear variation thickness and elastic edge supports. The first model considered is an S-S-S-S square moderately thick plate with all edges elastically restrained. That is,Kx0=Kxa=Ky0=Kyb=Γx. The six frequency parameters Ω=ωa2(ρh0/D0)1/2 are given in Table 6 for several different restraining coefficient values; the finite element method (FEM) results calculated using ABAQUS are also presented as a reference. The second model is a complete square moderately thick plate with all edges elastically restrained. That is, kx0=kxa=ky0=kyb=Γw and Kx0=Kxa=Ky0=Kyb=Γx. The six frequency parameters Ω=ωa2(ρh0/D0)1/2 are given in Table 7 for several different restraining coefficient values; the finite element method (FEM) results are also listed as a reference. It can be clearly seen that the comparison is extremely good, which implies that the current method is able to make correct predictions for the modal characteristics of linearly varying thickness moderately thick plate with not only classical boundary conditions but also elastic edge restraints.Table 6
Frequency parameterΩ=ωa2(ρh0/D0)1/2 for S-S-S-S moderately thick plates (a/b=1) with linearly varying thickness and elastic rotation support (h(x,y)=h0(1+0.5x)(1+0.5y), Kx0=Kxa=Ky0=Kyb=Γx).
Γ
x
a
h
0
/
a
Model sequence
1
2
3
4
5
6
0
0.05
28.809
70.080
70.891
108.967
134.884
135.295
(28.897b)
(70.334)
(71.129)
(109.112)
(135.254)
(135.772)
0.1
27.261
62.398
62.926
92.423
111.540
111.735
(26.606b)
(61.784)
(62.303)
(91.193)
(111.323)
(111.648)
0.15
25.231
54.192
54.505
77.118
91.228
91.312
(25.204b)
(55.297)
(55.595)
(77.503)
(91.871)
(91.941)
0.2
23.103
47.010
47.191
65.018
75.832
75.866
(23.947b)
(47.071)
(47.208)
(65.404)
(76.531)
(76.589)
0.5
0.05
23.103
47.010
47.191
65.018
75.832
75.866
(23.210b)
(47.245)
(47.358)
(65.241)
(76.122)
(76.345)
0.1
48.377
94.650
95.597
135.484
162.635
163.501
(48.524b)
(94.891)
(95.992)
(135.845)
(167.012)
(164.024)
0.15
36.239
70.939
71.298
99.997
118.584
119.009
(36.647b)
(71.214)
(71.687)
(100.024)
(119.114)
(119.457)
0.2
28.749
56.937
57.037
79.214
93.009
93.162
(29.142b)
(57.034)
(57.544)
(79.654)
(93.668)
(93.754)
1
0.05
49.651
96.759
97.619
138.089
165.368
166.304
(49.755b)
(97.245)
(97.964)
(138.544)
(165.864)
(166.878)
0.1
38.951
73.735
74.251
102.748
121.079
121.709
(39.021b)
(74.145)
(74.664)
(102.874)
(121.613)
(121.817)
0.15
30.630
58.406
58.533
80.432
93.984
94.255
(30.638b)
(58.684)
(58.644)
(80.874)
(94.121)
(94.478)
0.2
25.548
48.636
48.674
66.165
76.708
76.824
(25.687b)
(48.788)
(48.992)
(66.544)
(76.824)
(76.944)
1.5
0.05
50.118
97.561
98.374
139.096
166.421
167.382
(50.247b)
(97.877)
(98.661)
(139.247)
(166.875)
(167.662)
0.1
40.310
75.216
75.783
104.229
122.402
123.164
(40.366b)
(75.542)
(75.921)
(104.557)
(122.874)
(123.385)
0.15
31.837
59.352
59.530
81.245
94.611
94.983
(31.902b)
(59.451)
(59.871)
(81.375)
(94.784)
(95.105)
0.2
26.302
49.119
49.156
66.528
76.954
77.130
(26.418b)
(49.354)
(49.334)
(66.921)
(77.010)
(77.470)
2
0.05
50.360
97.983
98.769
139.631
166.979
167.954
(50.472b)
(98.005)
(99.114)
(140.154)
(167.188)
(167.986)
0.1
41.130
76.146
76.724
105.165
123.229
124.082
(41.421b)
(76.642)
(78.114)
(105.661)
(123.278)
(124.974)
0.15
32.687
60.025
60.245
81.832
95.052
95.508
(32.778b)
(60.244)
(60.549)
(82.146)
(95.114)
(95.842)
0.2
26.893
49.492
49.539
66.815
77.138
77.370
(26.922b)
(49.874)
(49.924)
(66.987)
(77.367)
(77.588)
a
Γ
x
=
K
i
/
K
0
,
(
K
0
=
1
×
1
0
9
Nm/rad,  i=x0,xa,y0,yb).
bResults in parentheses are taken from FEM.Table 7
Frequency parameterΩ=ωa2(ρh0/D0)1/2 for moderately thick plates (a/b=1) with linearly varying thickness and elastic rotation and translation support (h(x,y)=h0(1+0.5x)(1+0.5y), Kx0=Kxa=Ky0=Kyb=Γx, and kx0=kxa=ky0=kyb=Γw).
Γ
x
a
Γ
w
a
h
0
/
a
Model sequence
1
2
3
4
5
6
0.5
0
0.05
14.751
14.917
28.232
55.251
55.397
69.512
(14.780b)
(14.822)
(29.252)
(55.476)
(55.505)
(70.904)
0.1
11.811
11.910
23.857
44.462
45.037
55.453
(11.818b)
(11.897)
(24.711)
(44.376)
(45.257)
(56.511)
0.15
8.487
8.514
20.011
34.783
36.335
44.970
(8.500b)
(8.544)
(20.738)
(34.404)
(36.663)
(45.625)
0.2
6.073
6.073
17.431
28.679
30.805
38.010
(6.096b)
(6.106)
(18.039)
(28.129)
(31.165)
(38.311)
1
0.5
0.05
21.225
32.173
32.173
44.381
64.061
65.222
(20.708b)
(32.097)
(32.285)
(45.110)
(64.182)
(65.355)
0.1
8.140
16.878
16.941
28.100
48.663
49.015
(7.943b)
(16.838)
(16.970)
(28.888)
(48.700)
(49.171)
0.15
4.493
11.859
11.878
22.429
38.352
39.247
(4.383b)
(11.867)
(11.891)
(23.125)
(38.177)
(39.488)
0.2
2.932
8.690
8.691
18.919
31.108
32.600
(3.540b)
(8.791)
(8.791)
(16.229)
(27.520)
(27.889)
1.5
1
0.05
27.819
40.919
41.430
54.595
70.792
72.896
(27.146b)
(41.008)
(41.434)
(55.256)
(70.895)
(73.051)
0.1
11.285
20.068
20.230
31.101
51.046
51.429
(11.012b)
(20.025)
(20.270)
(31.849)
(51.126)
(51.569)
0.15
6.292
13.912
13.922
24.132
40.500
41.120
(6.139b)
(13.912)
(13.934)
(24.808)
(40.426)
(41.316)
0.2
4.121
10.321
10.331
20.051
32.796
33.921
(4.021b)
(10.329)
(10.357)
(20.635)
(32.540)
(34.178)
2
2
0.05
34.615
51.972
53.088
68.255
81.344
84.510
(33.806b)
(52.100)
(53.157)
(68.891)
(81.433)
(84.761)
0.1
15.416
24.659
24.829
35.615
53.879
54.494
(15.045b)
(24.625)
(24.888)
(36.303)
(53.970)
(54.626)
0.15
8.740
16.420
16.481
26.353
42.417
42.946
(8.529b)
(16.414)
(16.499)
(26.996)
(42.398)
(43.114)
0.2
5.761
12.127
12.127
21.434
34.286
35.192
(5.622b)
(12.134)
(12.148)
(21.998)
(34.110)
(35.415)
a
Γ
x
=
K
i
/
K
0
,
(
K
0
=
1
×
1
0
9
Nm/rad
,
i
=
x
0
,
x
a
,
y
0
,
y
b
),  Γw=ki/k0,(k0=1×109N/m,  i=x0,xa,y0,yb).
bResults in parentheses are taken from FEM.As the last case of this subsection, the influence of the gradientα and β on the fundamental frequency parameters for a linearly varying thickness moderately thick plate is investigated. The model is a square moderately thick plate with all edges elastically restrained. That is, kx0=kxa=ky0=kyb=Γw=2 and Kx0=Kxa=Ky0=Kyb=Γx=2. The six frequency parameters Ω=ωa2(ρh0/D0)1/2 are given in Table 8 for several different α and β values; the finite element method (FEM) results are also listed as a reference. Again, good agreement can be observed. Through Table 8, it is also found that the frequency parameter increases with increasing gradient parameters.Table 8
Frequency parameterΩ=ωa2(ρh0/D0)1/2 for elastic support moderately thick plates (a/b=1) with different gradient (kx0=kxa=ky0=kyb=Γw=2,   Kx0=Kxa=Ky0=Kyb=Γx=2).
α, β
h
0
/
a
Model sequence
1
2
3
4
5
6
α
=
β
=
0.1
0.05
31.275
52.615
52.657
68.954
76.146
79.850
(31.303a)
(52.664)
(52.675)
(69.033)
(76.174)
(79.948)
0.1
17.270
25.244
25.250
33.949
46.050
46.625
(17.271a)
(25.255)
(25.258)
(34.089)
(46.122)
(46.708)
0.15
10.190
16.266
16.274
24.164
37.327
37.526
(10.189a)
(16.269)
(16.280)
(24.323)
(37.401)
(37.622)
0.2
6.780
12.258
12.260
19.685
31.452
31.754
(6.780a)
(12.265)
(12.270)
(19.851)
(31.514)
(31.864)
α
=
β
=
0.2
0.05
32.530
52.947
53.141
68.954
77.034
80.589
(32.577a)
(53.014)
(53.203)
(69.172)
(77.088)
(80.773)
0.1
16.882
24.979
24.979
34.154
48.057
48.575
(16.883a)
(24.983)
(25.001)
(34.433)
(48.134)
(48.667)
0.15
9.804
16.284
16.309
24.683
38.844
39.085
(9.803a)
(16.283)
(16.322)
(24.980)
(38.909)
(39.191)
0.2
6.499
12.290
12.295
20.133
32.354
32.770
(6.498a)
(12.296)
(12.307)
(20.420)
(32.387)
(32.895)
α
=
β
=
0.3
0.05
33.516
52.845
53.308
68.704
78.203
81.561
(33.563a)
(52.939)
(53.389)
(69.055)
(78.271)
(81.789)
0.1
16.419
24.799
24.830
34.527
50.061
50.583
(16.419a)
(24.790)
(24.869)
(34.947)
(50.143)
(50.688)
0.15
9.430
16.334
16.378
25.236
40.193
40.506
(9.429a)
(16.329)
(16.395)
(25.660)
(40.243)
(40.625)
0.2
6.235
12.288
12.293
20.572
33.102
33.663
(6.235a)
(12.296)
(12.306)
(20.965)
(33.090)
(33.811)
α
=
β
=
0.4
0.05
34.215
52.463
53.264
68.430
79.654
82.865
(34.252a)
(52.575)
(53.346)
(68.921)
(79.732)
(83.113)
0.1
15.921
24.695
24.793
35.027
52.011
52.571
(15.921a)
(24.674)
(24.842)
(35.583)
(52.098)
(52.688)
0.15
9.074
16.388
16.444
25.797
41.379
41.789
(9.074a)
(16.381)
(16.464)
(26.336)
(41.404)
(41.928)
0.2
5.990
12.236
12.238
21.002
33.735
34.461
(5.989a)
(12.249)
(12.250)
(21.487)
(33.653)
(34.641)
α
=
β
=
0.5
0.05
34.615
51.972
53.088
68.255
81.344
84.510
(33.806a)
(52.100)
(53.157)
(68.891)
(81.433)
(84.761)
0.1
15.416
24.659
24.829
35.615
53.879
54.494
(15.045a)
(24.625)
(24.888)
(36.303)
(53.970)
(54.626)
0.15
8.740
16.420
16.481
26.353
42.417
42.946
(8.529a)
(16.414)
(16.499)
(26.996)
(42.398)
(43.114)
0.2
5.761
12.127
12.127
21.434
34.286
35.192
(5.622a)
(12.134)
(12.148)
(21.998)
(34.110)
(35.415)
aResults in parentheses are taken from FEM.The above studies are given as linearly varying thickness moderately thick plates with several of boundary condition and different structure parameters. In the next section, vibration results for the plates subjected to nonlinear variation thickness will be presented.


## 3.4. Nonlinearly Varying Thickness Moderately Thick Plates with Classical and Elastic Boundary Conditions

In Section3.3, the linearly varying thickness moderately thick plates were studied. However, in the practical engineering applications, the varying thickness of a moderately thick plate may not always be linear variation in nature. A variety of possible thickness varying cases may be encountered in practice. Therefore, the moderately thick plates with nonlinear variation thickness subjected to general elastic edge restraints are examined in this subsection. For the sake of brevity, the indexes s and t will be chosen as 2 to imitate the nonlinearly varying thickness moderately thick plates structure in this section; that is, s=t=2. Also the thickness varying functions are also be expressed as Fourier cosine series according to (18) and (19).In order to validate the accuracy and reliability of the proposed method for predicting the vibration behavior of nonlinearly varying thickness moderately thick plates with arbitrary boundary conditions, the typical classical boundary conditions viewed as the special cases of elastically restrained edges will be considered. The comparison of the first six frequency parametersΩ=ωa2(ρh0/D0)1/2 for the moderately thick plates with nonlinearly varying thickness is presented in Table 9. The S-S-S-S, C-F-F-F, S-S-F-F, C-C-C-C, and S-C-S-C boundary conditions are performed in the comparison. The results adequately demonstrated the great accuracy of the modified Fourier method.Table 9
Frequency parameterΩ=ωa2(ρh0/D0)1/2 for moderately thick plates with linearly varying thickness in different boundary conditions (h(x,y)=h0(1+0.5x2)(1+0.5y2)).
Boundary condition
a
/
b
h
0
/
a
Model sequence
1
2
3
4
5
6
S-S-S-S
1
0.1
23.462
54.999
55.573
82.950
100.900
101.187
(23.741a)
(55.114)
(55.983)
(83.146)
(101.004)
(101.347)
0.2
18.606
42.725
42.971
60.040
71.167
71.761
(18.936a)
(43.014)
(43.148)
(60.247)
(71.642)
(71.993)
3/2
0.1
53.055
65.251
85.404
106.700
114.880
143.531
(53.467a)
(65.472)
(85.984)
(107.211)
(115.687)
(144.007)
0.2
17.439
21.531
32.170
44.987
59.438
64.179
(17.874a)
(21.748)
(32.651)
(45.264)
(60.188)
(64.879)
C-F-F-F
1
0.1
3.658
9.317
24.135
32.816
35.433
59.179
(3.749a)
(9.547)
(24.034)
(33.627)
(36.308)
(60.142)
0.2
3.520
7.986
19.591
26.959
27.491
43.882
(3.611a)
(8.176)
(20.075)
(27.125)
(27.625)
(43.984)
3/2
0.1
3.901
9.963
28.984
69.187
73.686
89.360
(3.930a)
(10.095)
(29.381)
(69.578)
(73.994)
(90.242)
0.2
3.292
4.439
13.385
23.947
40.850
44.758
(3.386a)
(4.498)
(14.012)
(24.068)
(41.221)
(45.142)
S-S-F-F
1
0.1
3.557
21.268
22.762
43.524
60.253
62.429
(3.649a)
(21.441)
(22.796)
(43.916)
(60.622)
(62.630)
0.2
3.171
17.822
19.512
33.616
45.042
46.833
(3.489a)
(18.006)
(20.430)
(33.815)
(45.247)
(49.004)
3/2
0.1
6.738
31.658
52.071
75.625
86.604
94.903
(6.957a)
(31.882)
(52.518)
(75.823)
(86.939)
(95.610)
0.2
3.653
12.930
20.955
26.517
44.097
48.058
(6.807a)
(13.160)
(21.312)
(26.864)
(44.422)
(48.873)
C-C-C-C
1
0.1
41.020
75.196
75.614
104.142
119.683
120.949
(41.405a)
(75.452)
(75.882)
(104.711)
(119.637)
(121.137)
0.2
31.237
51.686
51.868
68.411
76.868
77.864
(32.010a)
(52.960)
(52.145)
(69.335)
(77.481)
(78.099)
3/2
0.1
95.823
128.727
163.657
173.751
190.890
223.595
(95.364a)
(128.932)
(163.588)
(172.350)
(190.350)
(222.948)
0.2
53.306
71.372
89.384
102.404
104.660
120.908
(53.587a)
(71.759)
(90.103)
(102.503)
(105.174)
(121.275)
S-C-S-C
1
0.1
33.715
60.298
71.446
94.187
103.802
117.928
(34.148a)
(60.786)
(71.828)
(94.515)
(104.263)
(119.326)
0.2
26.143
45.499
49.458
64.253
72.941
76.191
(26.789a)
(46.622)
(50.680)
(64.329)
(73.474)
(76.843)
3/2
0.1
73.228
115.845
137.445
148.166
168.342
182.542
(73.761a)
(116.121)
(137.788)
(148.530)
(168.698)
(182.921)
0.2
41.302
53.648
65.630
72.202
81.632
88.105
(41.304a)
(53.757)
(65.842)
(72.794)
(82.033)
(88.474)
aResults in parentheses are taken from FEM.We now turn to elastically restrained moderately thick plates. The first one involves an S-S-S-S moderately thick square plate with a uniform rotational restraint along each edge; that is,Kx0=Kxa=Ky0=Kyb=Γx. The calculated frequency parameters Ω=ωa2(ρh0/D0)1/2 are given in Table 10 together with the FEM results. Since the elastically restrained plates with nonlinear variation thickness are rarely investigated, the FEM results are used as the reference. A good agreement is observed between the current and FEM results. The second example concerns a moderately thick square plate elastically supported along all edges. The stiffness of the linear and rotational restraints is set equal to kx0=kxa=ky0=kyb=Γw and Kx0=Kxa=Ky0=Kyb=Γx, respectively. The six frequency parameters Ω=ωa2(ρh0/D0)1/2 are shown in Table 11 for several different restraining coefficient values. It can also be noticed that the Fourier series method is able to predict the modal characteristics of nonlinearly varying thickness moderately thick plate with not only classical boundary conditions but also elastic edge restraints correctly.Table 10
Frequency parameterΩ=ωa2(ρh0/D0)1/2 for S-S-S-S moderately thick plates (a/b=1) with linearly varying thickness and elastic rotation support (h(x,y)=h0(1+0.5x2)(1+0.5y2), Kx0=Kxa=Ky0=Kyb=Γx).
Γ
x
a
h
0
/
a
Model sequence
1
2
3
4
5
6
0
0.05
23.462
54.999
55.573
82.950
100.900
101.187
(23.741b)
(55.114)
(55.983)
(83.146)
(101.004)
(101.347)
0.1
18.606
42.725
42.971
60.040
71.167
71.761
(18.936b)
(43.014)
(43.148)
(60.247)
(71.642)
(71.993)
0.15
53.055
65.251
85.404
106.700
114.880
143.531
(53.467b)
(65.472)
(85.984)
(107.211)
(115.687)
(144.007)
0.2
17.439
21.531
32.170
44.987
59.438
64.179
(17.874b)
(21.748)
(32.651)
(45.264)
(60.188)
(64.879)
0.5
0.05
6.904
13.374
13.514
19.400
22.904
23.056
(9.909b)
(13.548)
(13.651)
(19.564)
(23.088)
(23.379)
0.1
5.316
10.237
10.314
14.499
17.155
17.264
(5.374b)
(10.448)
(10.651)
(14.764)
(17.188)
(17.879)
0.15
4.136
8.201
8.220
11.474
13.578
13.671
(4.174b)
(8.348)
(8.651)
(11.764)
(13.688)
(13.879)
0.2
3.449
6.915
6.924
9.549
11.225
11.327
(3.474b)
(6.948)
(6.951)
(9.764)
(11.388)
(11.579)
1
0.05
7.040
13.617
13.745
19.726
23.233
23.402
(7.074b)
(13.748)
(13.851)
(19.964)
(23.488)
(23.879)
0.1
5.712
10.696
10.793
14.991
17.594
17.744
(5.774b)
(10.748)
(10.851)
(15.054)
(17.788)
(17.879)
0.15
4.489
8.507
8.540
11.759
13.803
13.922
(4.514b)
(8.548)
(8.611)
(11.864)
(13.988)
(14.119)
0.2
3.677
7.084
7.090
9.692
11.330
11.445
(3.684b)
(7.148)
(7.191)
(9.764)
(11.588)
(11.879)
1.5
0.05
7.089
13.708
13.828
19.847
23.355
23.530
(7.124b)
(13.748)
(13.951)
(19.967)
(23.588)
(23.879)
0.1
5.895
10.923
11.024
15.239
17.813
17.988
(5.874b)
(10.748)
(11.651)
(15.264)
(17.818)
(17.979)
0.15
4.697
8.694
8.737
11.938
13.790
14.083
(4.774b)
(8.748)
(8.951)
(11.864)
(13.688)
(14.879)
0.2
3.836
7.201
7.210
9.792
11.403
11.530
(3.874b)
(7.748)
(7.651)
(9.864)
(11.588)
(11.879)
2
0.05
7.114
13.754
13.871
19.911
23.418
23.597
(7.174b)
(12.848)
(13.851)
(19.864)
(23.588)
(23.879)
0.1
6.001
11.060
11.159
15.391
17.944
18.136
(6.004b)
(11.148)
(11.151)
(15.464)
(17.988)
(18.379)
0.15
4.836
8.821
8.872
12.061
14.037
14.189
(4.874b)
(8.848)
(8.951)
(12.164)
(14.188)
(14.879)
0.2
3.954
7.289
7.301
9.871
11.457
11.594
(3.974b)
(7.348)
(7.651)
(9.964)
(11.488)
(11.679)
a
Γ
x
=
K
i
/
K
0
,
(
K
0
=
1
×
1
0
9
Nm/rad,  i=x0,xa,y0,yb).
bResults in parentheses are taken from FEM.Table 11
Frequency parameterΩ=ωa2(ρh0/D0)1/2 for moderately thick plates (a/b=1) with linearly varying thickness and elastic rotation and translation support (h(x,y)=h0(1+0.5x2)(1+0.5y2),  Kx0=Kxa=Ky0=Kyb=Γx,  and  kx0=kxa=ky0=kyb=Γw).
Γ
x
a
Γ
w
a
h
0
/
a
Model sequence
1
2
3
4
5
6
0.5
0
0.05
12.926
13.041
25.090
47.935
47.943
61.761
(12.931b)
(13.053)
25.099
(47.947)
(47.952)
(61.788)
0.1
10.900
10.988
21.545
39.980
40.147
50.322
(10.904b)
(10.992)
(21.557)
(39.984)
(40.166)
(50.334)
0.15
8.245
8.298
18.081
31.503
32.617
40.950
(8.247b)
(8.230)
(18.092)
(31.508)
(32.622)
(40.955)
0.2
6.106
6.124
15.639
25.631
27.601
34.458
(6.110b)
(6.125)
(15.642)
(25.637)
(27.617)
(34.462)
1
0.5
0.05
21.575
31.615
31.644
42.573
57.889
58.607
(21.580b)
(31.617)
(31.644)
(42.581)
(57.893)
(58.610)
0.1
8.451
16.024
16.210
25.969
43.839
43.867
(8.451b)
(16.025)
(16.217)
(25.972)
(43.844)
(43.870)
0.15
4.682
11.479
11.480
20.642
35.172
35.598
(4.684b)
(11.480)
(11.481)
(20.644)
(35.178)
(35.604)
0.2
3.059
8.621
8.648
17.320
28.487
29.638
(3.062b)
(8.619)
(8.644)
(17.327)
(28.492)
(29.641)
1.5
1
0.05
27.713
40.203
40.978
52.951
65.352
66.815
(27.714b)
(40.207)
(40.988)
(52.972)
(65.368)
(66.822)
0.1
11.667
19.322
19.615
29.122
46.115
46.183
(11.674b)
(19.327)
(19.619)
(29.137)
(46.120)
(46.188)
0.15
6.544
13.416
13.481
22.405
37.238
37.430
(6.547b)
(13.420)
(13.488)
(22.410)
(37.241)
(37.434)
0.2
4.295
10.175
10.190
18.543
30.330
31.056
(4.296b)
(10.177)
(10.194)
(18.544)
(30.336)
(31.060)
2
2
0.05
33.453
50.854
52.088
66.311
76.610
78.947
(33.455b)
(50.852)
(50.097)
(66.321)
(76.627)
(78.971)
0.1
15.832
24.130
24.372
33.924
49.118
49.362
(15.830b)
(24.127)
(24.374)
(33.931)
(49.147)
(49.382)
0.15
9.060
15.956
16.099
24.754
39.134
39.249
(9.064b)
(15.961)
(16.102)
(24.755)
(39.138)
(39.452)
0.2
5.992
11.955
11.971
20.034
31.908
32.392
(5.991b)
(11.957)
(11.978)
(20.042)
(31.910)
(32.398)
a
Γ
x
=
K
i
/
K
0
,
(
K
0
=
1
×
1
0
9
Nm/rad,  i=x0,xa,y0,yb),  Γw=ki/k0,(k0=1×109N/m,  i=x0,xa,y0,yb).
bResults in parentheses are taken from FEM.Finally, the influence of the gradient on the fundamental frequency parameters of a nonlinearly varying thickness moderately thick plate is investigated. The plate is elastically restrained, in which the stiffness of the boundary springs is taken askx0=kxa=ky0=kyb=Γw=2 and Kx0=Kxa=Ky0=Kyb=Γx=2, respectively. The six frequency parameters Ω=ωa2(ρh0/D0)1/2 are presented in Table 12 for several different slop values. It can be seen that the fundamental frequency parameters will decrease with the increase of the parameter h0/a.Table 12
Frequency parameterΩ=ωa2(ρh0/D0)1/2 for elastic support moderately thick plates (a/b=1) with different gradient (kx0=kxa=ky0=kyb=Γw=2, Kx0=Kxa=Ky0=Kyb=Γx=2).
α, β
h
0
/
a
Model sequence
1
2
3
4
5
6
α
=
β
=
0.1
0.05
30.187
51.102
51.141
67.066
73.954
77.412
(30.188a)
(51.105)
(51.144)
(67.068)
(73.961)
(77.477)
0.1
16.995
24.719
24.724
33.071
44.343
44.819
(16.996a)
(24.725)
(24.729)
(33.084)
(44.347)
(44.822)
0.15
10.077
15.872
15.883
23.443
35.937
36.065
(10.081a)
(15.877)
(15.892)
(23.447)
(35.941)
(36.084)
0.2
6.713
11.953
11.958
19.086
30.372
30.604
(6.713a)
(11.957)
(11.965)
(19.092)
(30.378)
(30.612)
α
=
β
=
0.2
0.05
31.206
51.436
51.614
67.058
74.398
77.595
(31.208a)
(51.438)
(51.619)
(67.064)
(74.402)
(77.596)
0.1
16.786
24.503
24.508
33.144
45.583
45.923
(16.790a)
(24.510)
(24.513)
(33.147)
(45.587)
(45.932)
0.15
9.821
15.865
15.906
23.766
36.897
36.990
(9.824a)
(15.870)
(15.907)
(23.768)
(36.894)
(36.997)
0.2
6.523
11.978
11.992
19.356
30.912
31.162
(6.524a)
(11.981)
(11.997)
(19.362)
(30.916)
(31.174)
α
=
β
=
0.3
0.05
32.097
51.462
51.900
66.832
74.991
77.863
(32.098a)
(51.466)
(51.907)
(66.849)
(74.996)
(77.872)
0.1
16.507
24.332
24.383
33.330
46.806
47.074
(16.511a)
(24.327)
(24.380)
(33.339)
(46.810)
(47.086)
0.15
9.562
15.885
15.964
24.103
37.748
37.828
(9.562a)
(15.892)
(15.977)
(21.105)
(37.751)
(37.832)
0.2
6.338
11.992
12.013
19.603
31.338
31.634
(6.340a)
(11.994)
(12.011)
(19.612)
(31.344)
(31.639)
α
=
β
=
0.4
0.05
32.851
51.239
52.046
66.547
75.735
78.303
(33.455a)
(50.852)
(50.097)
(66.321)
(76.627)
(78.971)
0.1
16.182
24.209
24.343
33.598
47.990
48.230
(33.455a)
(50.852)
(50.097)
(66.321)
(76.627)
(78.971)
0.15
9.307
15.919
16.034
24.436
38.491
38.579
(33.455a)
(50.852)
(50.097)
(66.321)
(76.627)
(78.971)
0.2
6.161
11.986
12.008
19.828
31.666
32.040
(33.455a)
(50.852)
(50.097)
(66.321)
(76.627)
(78.971)
α
=
β
=
0.5
0.05
33.453
50.854
52.088
66.311
76.610
78.947
(33.455a)
(50.852)
(50.097)
(66.321)
(76.627)
(78.971)
0.1
15.832
24.130
24.372
33.924
49.118
49.362
(15.830a)
(24.127)
(24.374)
(33.931)
(49.147)
(49.382)
0.15
9.060
15.956
16.099
24.754
39.134
39.249
(9.064a)
(15.961)
(16.102)
(24.755)
(39.138)
(39.452)
0.2
5.992
11.955
11.971
20.034
31.908
32.392
(5.991a)
(11.957)
(11.978)
(20.042)
(31.910)
(32.398)
aResults in parentheses are taken from FEM.In the above examples, it has been demonstrated that the presented method can be universally applied to nonlinearly varying thickness moderately thick plates with several boundary conditions and different structure parameters. New results are obtained for plates with nonlinear variation thickness in both directions subjected to general elastic boundary restraints, which may be used for benchmarking of researchers in the field. In addition, it is interesting to see that the nature frequency decreases with the increase of the index for thickness function.


## 4. Conclusions

In this paper, a modified Fourier method has been presented to study the free vibration behaviors of moderately thick rectangular plates with variable thickness and arbitrary boundary conditions. The first-order shear deformation plate theory is adopted to formulate the theoretical model. The displacements and rotation components of the plate, regardless of boundary conditions, are invariantly expressed as the superposition of a 2D Fourier cosine series and four supplementary functions in the form of the product of a polynomial function and a single cosine series expansion to ensure and accelerate the convergence of the solution. At each edge of the plate, the general restraint conditions are implemented by introducing one group of linear springs and two groups of rotational springs, which are continuously distributed and determined by the stiffness of these springs. Instead of seeking a solution in strong forms in the previous studies, all the Fourier coefficients will be treated equally and independently as the generalized coordinates and solved directly from the Rayleigh-Ritz technique. The change of the boundary conditions can be easily achieved by only varying the stiffness of the three sets of boundary springs along all edges of the rectangular plates without involving any change to the solution procedure. The convergence of the present solution is examined and the excellent accuracy is validated by comparison with existing results published in the literature and FEM data. Excellent agreements are obtained from these comparisons. The proposed method provides a unified means for extracting the modal parameters and predicting the vibration behaviors of moderately thick plates with variable thickness variation functions and arbitrary elastic edge restraints. A variety of free vibration results for moderately thick rectangular plates with different thickness variation functions and boundary conditions are presented. New results for free vibration of moderately thick rectangular plates with various thickness variation functions and edge conditions are presented, which may be used for benchmarking of researchers in the field.



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*Source: 572395-2014-07-14.xml*