# A Modified Fourier–Ritz Formulation for Vibration Analysis of Arbitrarily Restrained Rectangular Plate with Cutouts

Authors: Yiming Liu; Zhuang Lin; Hu Ding; Guoyong Jin; Sensen Yan
Journal: Shock and Vibration
 (2018)

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## Abstract

A modified Fourier–Ritz method is developed for the flexural and in-plane vibration analysis of plates with two rectangular cutouts with arbitrary boundary conditions, aiming to provide a unified solving process for cases that the plate has various locations or sizes of cutout, and different kinds of boundary conditions. Under the current framework, modifying the position of the cutout or the boundary conditions of the plate is just as changing the geometric parameters of the plate, and there is no need to change the solution procedures. The arbitrary boundary conditions can be obtained by setting the stiffness constant of the boundary springs which are fixed uniformly along the edges of the plate at proper values. The strain and kinetic energy functions of a plate with rectangular cutout are derived in detail. The convergence and accuracy of the present method are demonstrated by comparing the present results with those obtained from the FEM software. In this paper, free in-plane and flexural vibration characteristics of the plate with rectangular cutout under general boundary conditions are studied. From the results, it can be found that the geometric parameters and positions of the cutout and the boundary conditions of the plate will obviously influence the natural vibration characteristics of the structures.


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## Body

## 1. Introduction

As one of the most commonly used structures, the rectangular plate has been in study for a long time. It is well known that, in a vibrating thin plate, there exist three types of modes: bending mode, longitudinal mode, and shear mode. The bending mode is referred to the out-of-plane vibration, while the longitudinal mode and shear mode are referred to the in-plane vibration [1]. The flexural vibration of thin plates with various boundary conditions has been attracting researchers for a long time. And a large number of papers on the flexural vibration have been published, such as the highly referenced [2–12]. In these papers, various methods including numerical, analytical, and semianalytical methods have been developed to analyze the vibration characteristics of plates under various boundary conditions. Maury et al. [3] studied the response of excited panels by the wave propagation method. Gorman [4, 5] and Mochida [8] used the superposition-Galerkin method to study the vibration characteristics of rectangular plates under free boundary condition. Li [9] proposed an exact series solution for the transverse vibration of beams under general elastic boundary restraints. Besides, flexural vibration of thin plates is also studied by other researchers with FE method, DQ method DSC method [10–12], and so on.Probably since the first nonzero natural frequency of in-plane vibration of a thin plate is much higher than the flexural vibration of the plate, compared with the transverse vibration problem, the in-plane vibration problem of plates is far less investigated in the past. As the in-plane vibration plays an important role in structural vibration, the in-plane vibration analysis has attracted attention of researchers in recent years. Many researchers have investigated the in-plane vibrations of plates with various methods [13–20]. Gorman [13, 14] analyzed the free in-plane vibration characteristics of plates with different kinds of boundary conditions by the superposition method. Farag and Pan [17, 18] studied the modal characteristics of the plate which has two clamped parallel edges and two completely free parallel edges. It should be mentioned that Xing and Liu [19] solved all possible exact solutions about in-plane vibrations of rectangular plates.Due to practical engineering requirements, it is often necessary to open a hole on a plate, for example, opening a hole on a steel plate to reduce the weight of the structure. So, it is necessary to study the in-plane and flexural vibration of rectangular plates with cutouts. As the transverse vibration of the plate has been studied for a long time, the transverse vibration of the rectangular plate with all kinds of cutouts also has been extensively studied. Several ways have been applied to handle the vibration problem of plates with cutouts. As illustrated in [21], the cutout on the plate is considered as an extremely thin plate. Kwaka and Han [22] used the coordinate coupling method to study the vibration of the plate with cutout under free boundary condition. Laura et al. [23] studied the out-of-plane vibration of the plate with rectangular cutout with simple supported boundary condition. And the out-of-plane vibration of plates with two rectangular cutouts is studied in Reference [24]. In this paper, plates with different kinds of rectangular cutouts are studied.Just as illustrated previously, neither the in-plane vibrations of the plate nor the plate with different kinds of cutouts is seldom studied. Among the available papers, Shufrin and Esienberger [25] used the multiterm extended Kantorovich method to study the in-plane vibration of the plate with rectangular cutouts. In their paper, the in-plane vibration of the rectangular plate with inner or corner cutout is studied. Chen et al. [26] used the Chebyshev–Lagrangian method to study the out-of-plane and in-plane vibration of the rectangular plate with rectangular cutout.In this paper, a modified Fourier–Ritz method is employed to analyze the vibration of the rectangular plate with cutout under arbitrary boundary conditions, aiming to provide a unified procedure for many cases, such as plate with various locations (inner corner or edge) or sizes of rectangular cutout, or different kinds of boundary conditions. The arbitrary boundary conditions can be obtained by setting the stiffness constant of the boundary springs at proper values. Under the current framework, the in-plane and flexural displacements of the rectangular plate with cutout are expanded in the form of standard Fourier series and eight auxiliary polynomials. The use of the auxiliary polynomial functions can eliminate potential discontinuities and accelerate convergence of the expanded Fourier series. The natural frequencies and mode results of plates with rectangular cutouts are obtained by using the Rayleigh–Ritz procedure.


## 2. Theoretical Formulations

### 2.1. The Model and Stress-Strain Relations of the Restrained Plate

Consider an elastically restrained plate with lengthLx, width Ly, and thickness h, as depicted in Figure 1. A Descartes coordinate is fixed at the middle surface of the rectangular plate. x, y, and z represent the directions of length, width, and thickness, respectively. u, v, and w are employed to denote the displacements at the midplane of the rectangular plate in directions of x, y, and z, separately. Under the current framework, the arbitrary boundary conditions of a rectangular plate can be simulated by setting three groups of linear springs (ku,kv,kw) and a group of rotational spring (kw) which are fixed uniformly along the boundaries of the rectangular plate at proper values. And arbitrary boundary conditions can be achieved by setting the stiffness constants of the four sets of the springs at proper values. According to Kirchhoff assumptions, the displacements of a thin plate in x, y, and z directions can be written as follows:(1)Ux,y,z=ux,y−z∂wx,y∂x,Vx,y,z=vx,y−z∂wx,y∂y,Wx,y,z=wx,y,where U, V, and W represent the displacements of the plate in directions of x, y, and z, respectively. Based on the thin plate theory, the relationship between the strains and displacements of thin plates can be written as follows:(2)εx=εx0+zχx,εy=εy0+zχy,γxy=γxy0+zχxy,where εx0 and εy0 are the normal strains in the midplane of the rectangular plate and γxy0 is the shear strain in the midplane of the rectangular plate. χx and χy represent curvature changes and χxy represents twist change, separately. They can be expressed as(3)εx0=∂u∂x,(4)εβ0=∂v∂y,(5)γxy0=∂v∂x+∂u∂y,(6)χx=−∂2w∂x2,(7)χy=−∂2w∂y2,(8)χxy=−2∂2w∂x∂y.Figure 1
Rectangular plate with elastic boundary restraints.Equations (3)–(8) represent the relationship between the strain and displacement of a thin plate in Descartes coordinates. Based on Hook’s law, the stress-strain equations are described as(9)σxσyσzτyzτxzτxy=Q11Q12Q13Q14Q15Q16Q21Q22Q23Q24Q21Q26Q31Q32Q33Q34Q31Q36Q41Q42Q43Q44Q41Q46Q51Q52Q53Q54Q51Q56Q61Q62Q63Q64Q61Q66εxεyεzγyzγxzγxy,where(10)Q11Q12Q13Q14Q15Q16Q21Q22Q23Q24Q21Q26Q31Q32Q33Q34Q31Q36Q41Q42Q43Q44Q41Q46Q51Q52Q53Q54Q51Q56Q61Q62Q63Q64Q61Q66=E1−μ2μE1−μ2⋯0μE1−μ2E1−μ2⋯0⋮⋮⋱⋮00⋯E21+μ,where σx,σy, and σz represent the normal stress of a rectangular plate in the direction of x, y, and z, respectively. E and μ are Young’s modulus and Poisson’s ratio of the plate, respectively.Performing the integration of stresses (σx,σy,τxy) over the thickness of the plate, we get(11)NxNyNxy=∫−h/2h/2σxσyτxydz,(12)MxMyMxy=∫−h/2h/2σxσyτxyzdz.Performing Equations (11) and (12) yields, we get(13)NxNyNxyMxMyMxy=A11A12A16B11B12B16A21A22A26B21B22B26A61A62A66B61B62B66B11B12B16D11D12D16B21B22B26D21D22D26B61B62B66D61D62D66εx0εy0γxy0χxχyχxy,where(14)A11A12A16B11B12B16A21A22A26B21B22B26A61A62A66B61B62B66B11B12B16D11D12D16B21B22B26D21D22D26B61B62B66D61D62D66=Eh1−μ2μEh1−μ20000μEh1−μ2Eh1−μ2000000μEh21+μ000000Eh3121−μ2μEh3121−μ20000μEh3121−μ2Eh3121−μ2000000Eh361+μ,where Nx and Ny are defined as the force resultants and Nxy is defined as the shear force resultant. MxandMy are defined as bending resultants and Mxy is defined as the twist moment resultant, respectively.By now, all the stress-strain relations and other needed equations are obtained, and they will be used to derive the energy functions of the rectangular plate with small deformation.


### 2.2. Energy Functions

In this paper, the Rayleigh–Ritz procedure is employed to solve the vibration problems of a rectangular plate. Thus, the energy functions of the plate should be obtained first. For vibration analysis, the Lagrange energy function (L) of the rectangular plate is defined as(15)L=T−Us−Usp,where T,Us, and Usp are kinetic energy of the rectangular plate, potential energy of the rectangular plate, and the potential energy stored in the boundary springs, respectively.Based on the classical thin plate theory (CPT), the potential energy (Us) of a rectangular plate in Equation (15) can be rewritten as(16)Us=12∫0a∫0bNxεx0+Nyεy0+Nxyγxy0+Mxχx+Myχy+Mxyχxydxdy.Inserting Equation (13) into Equation (16), the potential energy equation of the rectangular plate can be described as(17)Us=12∫0a∫0bKεx02+2μεx0εy0+εy02+1−μ2γxy02dxdy+12∫0a∫0bDχx2+2μχxχy+χy2+1−μ2χxy2dxdy.And the kinetic energy (T) of the rectangular plates is written as(18)T=ρh2∫0a∫0b∂u∂t2+∂v∂t2+∂w∂t2dxdy,where ρ and h are the density and the thickness of the rectangular plate, respectively.As Laura et al. illustrated in [23], a rectangular plate with a hole can be regarded as a structure combined by two kinds of different materials. And the material properties of the hole on the plate can be considered as ρ=0kg/m3, E=0GPa, and μ=0. Thus, the Lagrange energy equation of a rectangular plate with rectangular cutout can be obtained in this method. In order to describe this problem clearly, one specific example is presented here. Consider a rectangular plate with one inner cutout, as shown in Figure 2. The length and width of the plate are Lx and Ly. The location of the cutout can be described by the coordinates of the corners of the cutout, and the coordinates of the corners are ALx0,Ly0 and BLx1,Ly1. Thus, the potential energy equations of the structure depicted in Figure 2 can be written as(19)Us=12K∫0Ly0∫0Lx+∫Ly0Ly1∫0Lx0+∫Ly0Ly1∫Lx1Lx+∫Ly1Ly∫0LxM1dxdy+12D∫0Ly0∫0Lx+∫Ly0Ly1∫0Lx0+∫Ly0Ly1∫Lx1Lx+∫Ly1Ly∫0LxM2dxdy,where(20)M1=εx02+2μεx0εy0+εy02+1−μ2γxy02,(21)M2=χx2+2μχxχy+χy2+1−μ2χxy2.Figure 2
Models of plate with rectangular cutout: (a) plate with inner cutout; (b) plate with corner cutout; and (c) plate with edge cutout.
(a)
(b)
(c)As illustrated previously, the general boundary conditions can be achieved by setting the stiffness constant of boundary springs at proper values. In this paper,kψu,kψv, kψw, and kψr are adopted to represent the stiffness constants of the boundary springs which are fixed along the four edges of the rectangular plate x = 0, y = 0, x = a, and y = b, respectively. Thus, the potential energy stored in the boundary springs can be written as(22)Usp=12∫0bkx0uu2+kx0vv2+kx0ww2+kx0r∂w∂x2x=0+kx1uu2+kx1vv2+kx1ww2+kx1r∂w∂x2x=ady+12∫0aky0uu2+ky0vv2+ky0ww2+ky0r∂w∂y2y=0+ky1uu2+ky1vv2+ky1ww2+ky1r∂w∂y2y=bdx.According to the Rayleigh–Ritz procedure, choosing suitable displacement functions is of great importance to make the approximate solutions converge to the exact solutions.The displacement of a rectangular plate can be theoretically expressed in any group of orthogonal polynomials. However, since computers have the round-off errors, the high-order polynomials generally become numerically instable. For the plates under arbitrary boundary conditions, the displacement functions expressed in the form of standard Fourier cosine or sine series can avoid this problem, but discontinuities of the displacement functions and their derivatives will potentially exist along the boundaries of the rectangular plates. This kind of problem is demonstrated in detail [27–33]. In order to overcome this difficulty, the modified Fourier series is employed, and the displacement of the plate is expressed as follows:(23)ux,y=∑m=0∞∑n=0∞Amncosλamxcosλbny+Px,y,(24)vx,y=∑m=0∞∑n=0∞Bmncosλamxcosλbny+Qx,y,(25)wx,y=∑m=0∞∑n=0∞Cmncosλamxcosλbny+Ex,y,(26)Px,y=∑m=0∞∑l=12Amlcosλamxξly+∑l=12∑n=0∞Alnξlxcosλbny,(27)Qx,y=∑m=0∞∑l=12Bmlcosλamxξly+∑l=12∑n=0∞Blnξlxcosλbny,(28)Ex,y=∑m=0∞∑l=14Emlcosλamxζly+∑l=14∑n=0∞Elnζlxcosλbny,where λam=mπ/Lx,λbn=nπ/Ly, and Amn,Bmn,Cmn are the coefficients of expansions in Equations (23)–(25); Aml,Aln,Bml,Bln,Cml,andCln are the coefficients of eight auxiliary polynomials in Equations (26)–(28); and ξl and ζl are eight auxiliary polynomial functions. Displacements in the midplane expressed in the form of standard cosine Fourier series plus eight auxiliary polynomials can eliminate the potential discontinuity along the boundary of the plate. The eight auxiliary polynomial functions are(29)ξ1α=ααLα−12,ξ2α=α2LααLα−1,ζ1α=9Lα4πsinπα2Lα−Lα12πsin3πα2Lα,ζ2α=−9Lα4πcosπα2Lα−Lα12πcos3πα2Lα,ζ3α=Lα3π3sinπα2Lα−Lα33π3sin3πα2Lα,ζ4α=−Lα3π3cosπα2Lα−Lα33π3cos3πα2Lα.Specifically, symbolsα and Lα (α=x,y, Lα=Lx,Ly) represent independent variables and length, respectively. And it is obvious that(30)ζ1′0=ζ3′″0=ζ2′L=ζ4′″L=1,ξ10=ξ1L=ξ1′L=0,ξ1′0=1,ξ20=ξ2L=ξ2′0=0,ξ2′L=1.As the Lagrange energy function and the displacement functions of the rectangular plate are established, the next task is to obtain the value of the coefficients in the displacement functions. Substituting Equations (16)–(20) into the Lagrange energy function and minimizing the Lagrange energy functions by making the derivatives of the unknown variables of expansions equal to zero as(31)∂L∂E=0,E=Amn,Aml,Aln,Bmn,Bml,Bln,Cmn,Cml,Cln.Finally, totaled3∗M+1∗N+1+12∗M+1+12∗N+1 equations are obtained and can be written in the form of matrix as(32)K−ω2MD=0,where K,M, and D are stiffness matrix, mass matrix, and coefficient matrix, respectively. They are described as(33)K=K1,1K1,2K1,3⋯K1,9K2,1K2,2K2,3⋯K2,9K3,1K3,2K3,3⋯K3,9⋮⋮⋮⋱⋮K9,1K9,2K9,3⋯K9,9,(34)M=M1,1M1,2M1,3⋯M1,9M2,1M2,2M2,3⋯M2,9M3,1M3,2M3,3⋯M3,9⋮⋮⋮⋱⋮M9,1M9,2M9,3⋯M9,9,(35)D=Du,Dv,Dw.The element in stiffness and mass matrix is given in Appendix. The vectorD can be written as follows:(36)Du=A00,…,Amn,A01,…,Aml,…,A10,…,Aln,Dv=B00,…,Bmn,B01,…,Bml,…,B10,…,Bln,Dw=C00,…,Cmn,C01,…,Cml,…,C10,…,Cln.Based on the classical thin plate theory (CPT), the in-plane and out-of-plane vibrations are decoupled, and thus the out-of-plane and in-plane vibration of a plate structure can be studied separately. When studying the flexural vibrations of the plate structure, the stiffness mass and coefficients matrix of the structure can be defined as(37)Kf=K7,7K7,8K7,9K8,7K8,8K8,9K9,7K9,8K9,9,Mf=M7,7M7,8M7,9M8,7M8,8M8,9M9,7M9,8M9,9,Df=Dw.When studying the in-plane vibrations of a rectangular plate, the stiffness mass and coefficients matrix can be written as(38)Kin=K1,1K1,2K1,3⋯K1,6K2,1K2,2K2,3⋯K2,6K3,1K3,2K3,3⋯K3,6⋮⋮⋮⋱⋮K6,1K6,2K6,3⋯K6,6,Min=M1,1M1,2M1,3⋯M1,6M2,1M2,2M2,3⋯M2,6M3,1M3,2M3,3⋯M3,6⋮⋮⋮⋱⋮M6,1M6,2M6,3⋯M6,6,Din=Du,Dv.


## 2.1. The Model and Stress-Strain Relations of the Restrained Plate

Consider an elastically restrained plate with lengthLx, width Ly, and thickness h, as depicted in Figure 1. A Descartes coordinate is fixed at the middle surface of the rectangular plate. x, y, and z represent the directions of length, width, and thickness, respectively. u, v, and w are employed to denote the displacements at the midplane of the rectangular plate in directions of x, y, and z, separately. Under the current framework, the arbitrary boundary conditions of a rectangular plate can be simulated by setting three groups of linear springs (ku,kv,kw) and a group of rotational spring (kw) which are fixed uniformly along the boundaries of the rectangular plate at proper values. And arbitrary boundary conditions can be achieved by setting the stiffness constants of the four sets of the springs at proper values. According to Kirchhoff assumptions, the displacements of a thin plate in x, y, and z directions can be written as follows:(1)Ux,y,z=ux,y−z∂wx,y∂x,Vx,y,z=vx,y−z∂wx,y∂y,Wx,y,z=wx,y,where U, V, and W represent the displacements of the plate in directions of x, y, and z, respectively. Based on the thin plate theory, the relationship between the strains and displacements of thin plates can be written as follows:(2)εx=εx0+zχx,εy=εy0+zχy,γxy=γxy0+zχxy,where εx0 and εy0 are the normal strains in the midplane of the rectangular plate and γxy0 is the shear strain in the midplane of the rectangular plate. χx and χy represent curvature changes and χxy represents twist change, separately. They can be expressed as(3)εx0=∂u∂x,(4)εβ0=∂v∂y,(5)γxy0=∂v∂x+∂u∂y,(6)χx=−∂2w∂x2,(7)χy=−∂2w∂y2,(8)χxy=−2∂2w∂x∂y.Figure 1
Rectangular plate with elastic boundary restraints.Equations (3)–(8) represent the relationship between the strain and displacement of a thin plate in Descartes coordinates. Based on Hook’s law, the stress-strain equations are described as(9)σxσyσzτyzτxzτxy=Q11Q12Q13Q14Q15Q16Q21Q22Q23Q24Q21Q26Q31Q32Q33Q34Q31Q36Q41Q42Q43Q44Q41Q46Q51Q52Q53Q54Q51Q56Q61Q62Q63Q64Q61Q66εxεyεzγyzγxzγxy,where(10)Q11Q12Q13Q14Q15Q16Q21Q22Q23Q24Q21Q26Q31Q32Q33Q34Q31Q36Q41Q42Q43Q44Q41Q46Q51Q52Q53Q54Q51Q56Q61Q62Q63Q64Q61Q66=E1−μ2μE1−μ2⋯0μE1−μ2E1−μ2⋯0⋮⋮⋱⋮00⋯E21+μ,where σx,σy, and σz represent the normal stress of a rectangular plate in the direction of x, y, and z, respectively. E and μ are Young’s modulus and Poisson’s ratio of the plate, respectively.Performing the integration of stresses (σx,σy,τxy) over the thickness of the plate, we get(11)NxNyNxy=∫−h/2h/2σxσyτxydz,(12)MxMyMxy=∫−h/2h/2σxσyτxyzdz.Performing Equations (11) and (12) yields, we get(13)NxNyNxyMxMyMxy=A11A12A16B11B12B16A21A22A26B21B22B26A61A62A66B61B62B66B11B12B16D11D12D16B21B22B26D21D22D26B61B62B66D61D62D66εx0εy0γxy0χxχyχxy,where(14)A11A12A16B11B12B16A21A22A26B21B22B26A61A62A66B61B62B66B11B12B16D11D12D16B21B22B26D21D22D26B61B62B66D61D62D66=Eh1−μ2μEh1−μ20000μEh1−μ2Eh1−μ2000000μEh21+μ000000Eh3121−μ2μEh3121−μ20000μEh3121−μ2Eh3121−μ2000000Eh361+μ,where Nx and Ny are defined as the force resultants and Nxy is defined as the shear force resultant. MxandMy are defined as bending resultants and Mxy is defined as the twist moment resultant, respectively.By now, all the stress-strain relations and other needed equations are obtained, and they will be used to derive the energy functions of the rectangular plate with small deformation.


## 2.2. Energy Functions

In this paper, the Rayleigh–Ritz procedure is employed to solve the vibration problems of a rectangular plate. Thus, the energy functions of the plate should be obtained first. For vibration analysis, the Lagrange energy function (L) of the rectangular plate is defined as(15)L=T−Us−Usp,where T,Us, and Usp are kinetic energy of the rectangular plate, potential energy of the rectangular plate, and the potential energy stored in the boundary springs, respectively.Based on the classical thin plate theory (CPT), the potential energy (Us) of a rectangular plate in Equation (15) can be rewritten as(16)Us=12∫0a∫0bNxεx0+Nyεy0+Nxyγxy0+Mxχx+Myχy+Mxyχxydxdy.Inserting Equation (13) into Equation (16), the potential energy equation of the rectangular plate can be described as(17)Us=12∫0a∫0bKεx02+2μεx0εy0+εy02+1−μ2γxy02dxdy+12∫0a∫0bDχx2+2μχxχy+χy2+1−μ2χxy2dxdy.And the kinetic energy (T) of the rectangular plates is written as(18)T=ρh2∫0a∫0b∂u∂t2+∂v∂t2+∂w∂t2dxdy,where ρ and h are the density and the thickness of the rectangular plate, respectively.As Laura et al. illustrated in [23], a rectangular plate with a hole can be regarded as a structure combined by two kinds of different materials. And the material properties of the hole on the plate can be considered as ρ=0kg/m3, E=0GPa, and μ=0. Thus, the Lagrange energy equation of a rectangular plate with rectangular cutout can be obtained in this method. In order to describe this problem clearly, one specific example is presented here. Consider a rectangular plate with one inner cutout, as shown in Figure 2. The length and width of the plate are Lx and Ly. The location of the cutout can be described by the coordinates of the corners of the cutout, and the coordinates of the corners are ALx0,Ly0 and BLx1,Ly1. Thus, the potential energy equations of the structure depicted in Figure 2 can be written as(19)Us=12K∫0Ly0∫0Lx+∫Ly0Ly1∫0Lx0+∫Ly0Ly1∫Lx1Lx+∫Ly1Ly∫0LxM1dxdy+12D∫0Ly0∫0Lx+∫Ly0Ly1∫0Lx0+∫Ly0Ly1∫Lx1Lx+∫Ly1Ly∫0LxM2dxdy,where(20)M1=εx02+2μεx0εy0+εy02+1−μ2γxy02,(21)M2=χx2+2μχxχy+χy2+1−μ2χxy2.Figure 2
Models of plate with rectangular cutout: (a) plate with inner cutout; (b) plate with corner cutout; and (c) plate with edge cutout.
(a)
(b)
(c)As illustrated previously, the general boundary conditions can be achieved by setting the stiffness constant of boundary springs at proper values. In this paper,kψu,kψv, kψw, and kψr are adopted to represent the stiffness constants of the boundary springs which are fixed along the four edges of the rectangular plate x = 0, y = 0, x = a, and y = b, respectively. Thus, the potential energy stored in the boundary springs can be written as(22)Usp=12∫0bkx0uu2+kx0vv2+kx0ww2+kx0r∂w∂x2x=0+kx1uu2+kx1vv2+kx1ww2+kx1r∂w∂x2x=ady+12∫0aky0uu2+ky0vv2+ky0ww2+ky0r∂w∂y2y=0+ky1uu2+ky1vv2+ky1ww2+ky1r∂w∂y2y=bdx.According to the Rayleigh–Ritz procedure, choosing suitable displacement functions is of great importance to make the approximate solutions converge to the exact solutions.The displacement of a rectangular plate can be theoretically expressed in any group of orthogonal polynomials. However, since computers have the round-off errors, the high-order polynomials generally become numerically instable. For the plates under arbitrary boundary conditions, the displacement functions expressed in the form of standard Fourier cosine or sine series can avoid this problem, but discontinuities of the displacement functions and their derivatives will potentially exist along the boundaries of the rectangular plates. This kind of problem is demonstrated in detail [27–33]. In order to overcome this difficulty, the modified Fourier series is employed, and the displacement of the plate is expressed as follows:(23)ux,y=∑m=0∞∑n=0∞Amncosλamxcosλbny+Px,y,(24)vx,y=∑m=0∞∑n=0∞Bmncosλamxcosλbny+Qx,y,(25)wx,y=∑m=0∞∑n=0∞Cmncosλamxcosλbny+Ex,y,(26)Px,y=∑m=0∞∑l=12Amlcosλamxξly+∑l=12∑n=0∞Alnξlxcosλbny,(27)Qx,y=∑m=0∞∑l=12Bmlcosλamxξly+∑l=12∑n=0∞Blnξlxcosλbny,(28)Ex,y=∑m=0∞∑l=14Emlcosλamxζly+∑l=14∑n=0∞Elnζlxcosλbny,where λam=mπ/Lx,λbn=nπ/Ly, and Amn,Bmn,Cmn are the coefficients of expansions in Equations (23)–(25); Aml,Aln,Bml,Bln,Cml,andCln are the coefficients of eight auxiliary polynomials in Equations (26)–(28); and ξl and ζl are eight auxiliary polynomial functions. Displacements in the midplane expressed in the form of standard cosine Fourier series plus eight auxiliary polynomials can eliminate the potential discontinuity along the boundary of the plate. The eight auxiliary polynomial functions are(29)ξ1α=ααLα−12,ξ2α=α2LααLα−1,ζ1α=9Lα4πsinπα2Lα−Lα12πsin3πα2Lα,ζ2α=−9Lα4πcosπα2Lα−Lα12πcos3πα2Lα,ζ3α=Lα3π3sinπα2Lα−Lα33π3sin3πα2Lα,ζ4α=−Lα3π3cosπα2Lα−Lα33π3cos3πα2Lα.Specifically, symbolsα and Lα (α=x,y, Lα=Lx,Ly) represent independent variables and length, respectively. And it is obvious that(30)ζ1′0=ζ3′″0=ζ2′L=ζ4′″L=1,ξ10=ξ1L=ξ1′L=0,ξ1′0=1,ξ20=ξ2L=ξ2′0=0,ξ2′L=1.As the Lagrange energy function and the displacement functions of the rectangular plate are established, the next task is to obtain the value of the coefficients in the displacement functions. Substituting Equations (16)–(20) into the Lagrange energy function and minimizing the Lagrange energy functions by making the derivatives of the unknown variables of expansions equal to zero as(31)∂L∂E=0,E=Amn,Aml,Aln,Bmn,Bml,Bln,Cmn,Cml,Cln.Finally, totaled3∗M+1∗N+1+12∗M+1+12∗N+1 equations are obtained and can be written in the form of matrix as(32)K−ω2MD=0,where K,M, and D are stiffness matrix, mass matrix, and coefficient matrix, respectively. They are described as(33)K=K1,1K1,2K1,3⋯K1,9K2,1K2,2K2,3⋯K2,9K3,1K3,2K3,3⋯K3,9⋮⋮⋮⋱⋮K9,1K9,2K9,3⋯K9,9,(34)M=M1,1M1,2M1,3⋯M1,9M2,1M2,2M2,3⋯M2,9M3,1M3,2M3,3⋯M3,9⋮⋮⋮⋱⋮M9,1M9,2M9,3⋯M9,9,(35)D=Du,Dv,Dw.The element in stiffness and mass matrix is given in Appendix. The vectorD can be written as follows:(36)Du=A00,…,Amn,A01,…,Aml,…,A10,…,Aln,Dv=B00,…,Bmn,B01,…,Bml,…,B10,…,Bln,Dw=C00,…,Cmn,C01,…,Cml,…,C10,…,Cln.Based on the classical thin plate theory (CPT), the in-plane and out-of-plane vibrations are decoupled, and thus the out-of-plane and in-plane vibration of a plate structure can be studied separately. When studying the flexural vibrations of the plate structure, the stiffness mass and coefficients matrix of the structure can be defined as(37)Kf=K7,7K7,8K7,9K8,7K8,8K8,9K9,7K9,8K9,9,Mf=M7,7M7,8M7,9M8,7M8,8M8,9M9,7M9,8M9,9,Df=Dw.When studying the in-plane vibrations of a rectangular plate, the stiffness mass and coefficients matrix can be written as(38)Kin=K1,1K1,2K1,3⋯K1,6K2,1K2,2K2,3⋯K2,6K3,1K3,2K3,3⋯K3,6⋮⋮⋮⋱⋮K6,1K6,2K6,3⋯K6,6,Min=M1,1M1,2M1,3⋯M1,6M2,1M2,2M2,3⋯M2,6M3,1M3,2M3,3⋯M3,6⋮⋮⋮⋱⋮M6,1M6,2M6,3⋯M6,6,Din=Du,Dv.


## 3. Validation and Discussion

With the formulations derived in Section2, several numerical results together with those available in the literature will be presented in this section to validate the accuracy and convergence of the current method. Besides, the vibration characteristics of the structure will also be studied.


### 3.1. Flexural Vibration Analysis of Rectangular Plate with Rectangular Cutouts

First, the flexural vibration of rectangular plate with one inner, edge, or corner cutout will be studied. These three kinds of models are depicted in Figure2. The length and width of the considered rectangular plate are Lx=Ly=2m, and the thickness of the plate is h=0.003m. And the material parameters of the plate structure are defined as E=216GPa, μ=0.3, and ρ=7900kg/m3. The coordinate of the points shown in Figure 2 are (a) A10.5m,0.5m B11.5m,1.5m, (b) A21m,1m, and (c) A30.5m,1m B31.5m,1m. For simplicity, a string consisting of several letters is adopted to represent the boundary conditions of the structure depicted in Figure 2. For example, “CCCC-FFFF” denotes the boundary conditions of the edges corresponding to number one to eight, and “C” and “F” represent clamped and free boundary conditions, respectively.The first six nonzero natural frequencies of flexural vibration of plate structures with one cutout are listed in Table1. The results obtained by software ANSYS are also presented here to verify the correctness of the present method. Comparing the results listed in Table 1, it can be observed that these two sets of results are very close. Besides, it can be also found that the position of the cutouts has an effect on the vibration characteristics of the structure. For example, frequency parameters in Table 1 show that the plate with one corner cutout has the largest nonzero first frequency while the plate with one edge cutout has the smallest nonzero first frequency; structures considered here are under completely free boundary conditions, and all the cutouts are also kept the same size.Table 1
Frequency parameters for flexural vibration of elastic-restrained plate with one rectangular cutout.
Mode order
1
2
3
4
5
6
Plate with inner cutout
FFFF-FFFF
Present
2.013
2.875
4.211
5.966
5.966
9.438
ANSYS
2.008
2.870
4.204
5.949
5.949
9.298
CCCC-FFFF
Present
12.510
14.680
14.750
20.080
20.600
27.870
ANSYS
12.347
14.442
14.442
19.530
20.172
27.466
Plate with corner cutout
FFF-FFF
Present
2.976
3.254
5.145
6.395
11.399
11.468
ANSYS
2.976
3.230
5.150
6.376
11.398
11.493
CCF-FCC
Present
6.327
12.616
15.225
22.902
23.115
26.817
ANSYS
6.325
12.529
15.210
22.811
22.997
26.769
Plate with edge cutout
FFFF-FFFF
Present
1.461
3.541
3.653
5.349
7.552
7.937
ANSYS
1.445
3.537
3.626
5.345
7.436
7.921
CCF-FCCC
Present
7.036
13.522
17.631
20.311
22.268
26.834
ANSYS
6.959
13.479
17.551
20.116
21.801
26.732By solving Equation (32), the natural frequencies and eigenvectors (D) can be obtained simultaneously. By inserting the eigenvector into the displacement expressions (Equations (23)–(25)), the mode shapes of the plate with one inner cutout can be acquired. The mode shapes of CCCC-FFFF plate with one inner cutout are compared with those obtained by ANSYS in Figure 3 (the right one). It can be seen that the mode shapes obtained by the present method match with those acquired by software ANSYS very well. Besides, it can also be found that with the increase of the frequency, the mode shapes of the structure become more and more complex.Figure 3
Mode shapes for flexural vibration of the CCCC-FFFF plate structure with one rectangular cutout locating inside of the plate: the (a) first mode shape; (b) second mode shape; (c) fourth mode shape; (d) eighth mode shape; (e) sixteenth mode shape; (f) thirty-second mode shape; (g) hundredth mode shape; and (h) two-hundredth mode shape.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)The nonzero frequency parameters of the rectangular plate with one rectangular cutout in Table1 are calculated by selecting the truncate number as M=N=14. In order to check the convergence of the current method, the natural frequencies of the plate with two inner cutouts are calculated with different truncation numbers (i.e., Mf=Nf=7−13). The models of the rectangular plate with two cutouts are depicted in Figure 4. In the following discussion, the geometric parameters of the plate with two cutouts are considered as 2 m in length, 2 m in width, and 0.003 m in thickness, and the material properties of the plate are defined the same as previous. The coordinates of the points shown in Figure 4 are (a) A0.4m,0.75m B0.9m,1.25m C1.1m,0.75m D1.6m,1.25m, (b) A0.75m,1.5m B1.25m,1.5m C0.75m,0.5m D1.25m,0.5m, and (c) A0.5m,1.5m B1.5m,1.5m. The frequency results of the plate with two inner cutouts are listed in Table 2. It shows that the natural frequency converges at M=N=13, and the convergence and numerical stability of the current method are validated.Figure 4
Models of rectangular plate with two cutouts: (a) rectangular plate with two inner cutouts; (b) rectangular plate two with edge cutouts; and (c) rectangular plate with two corner cutouts.
(a)
(b)
(c)Table 2
Natural frequencies of FFFF-FFFF-FFFF rectangular plate with two inner cutouts in different truncation schemes.
M = N
1
2
3
4
5
6
7
8
7
2.30
3.26
4.07
6.22
6.45
11.04
11.13
12.12
8
2.30
3.25
4.01
6.21
6.43
11.01
11.13
11.96
9
2.29
3.25
4.01
6.20
6.43
10.84
10.98
11.96
10
2.29
3.24
4.00
6.20
6.43
10.84
10.97
11.90
11
2.29
3.24
4.00
6.19
6.42
10.60
10.95
11.90
12
2.29
3.24
3.98
6.18
6.42
10.59
10.95
11.89
13
2.29
3.24
3.98
6.18
6.41
10.45
10.93
11.89
14
2.29
3.23
3.97
6.18
6.41
10.45
10.93
11.84
15
2.29
3.23
3.97
6.17
6.41
10.38
10.92
11.84
16
2.29
3.22
3.96
6.17
6.41
10.38
10.92
11.83
17
2.29
3.22
3.96
6.17
6.41
10.38
10.92
11.83
18
2.29
3.22
3.96
6.17
6.41
10.38
10.92
11.83As mentioned previously, the current method can be used to study the vibration characteristics of the rectangular plate with rectangular cutout under general elastic-restrained boundary. And the general elastic restrained boundary can be obtained by setting the stiffness constants of the boundary springs at proper values. For example, the clamped boundary conditions can be achieved by setting the stiffness constants of the linear and rotational springs as infinite. In practical calculations, a sufficiently large number is always employed to represent the “infinite.” In the first example, the clamped boundary conditions are obtained by setting the stiffness constant at 1012. So, effects of the stiffness constant of boundary springs on the solution results ought to be studied. The frequency parameters for EEEE-FFFF-FFFF plate with two inner rectangular cutouts are listed in Table 3, and the alphabet “E” represents elastic boundary condition. It can be found that as the stiffness constants of the springs increase, the natural frequencies of the structure increase gradually, and when the stiffness constant is beyond 1012, the natural frequency remain unchanged. Thus, in the following discussion, 1012 is selected to represent the value of “infinite.”Table 3
Frequency parameters of an EEEE-FFFF-FFFF plate with two inner rectangular cutouts (Ku=Kv=Kw=Kr=k).
Mode order
k
1
2
3
4
5
6
7
8
105
6.66
10.33
11.92
14.35
16.40
17.67
19.55
20.50
106
7.35
12.08
14.84
18.17
21.53
25.67
27.32
30.82
107
7.44
12.30
15.23
18.76
22.19
26.92
28.67
32.81
108
7.44
12.32
15.27
18.82
22.26
27.04
28.81
33.02
109
7.44
12.33
15.28
18.83
22.26
27.06
28.83
33.04
1010
7.44
12.33
15.28
18.83
22.26
27.06
28.83
33.04
1011
7.45
12.33
15.28
18.83
22.26
27.06
28.83
33.04
1012
7.45
12.33
15.28
18.83
22.26
27.06
28.83
33.04
1012
7.45
12.33
15.28
18.83
22.26
27.06
28.83
33.04
1014
7.45
12.33
15.28
18.83
22.26
27.06
28.83
33.04The mode shapes of the structure that rectangular cutouts locate inside of the plate or on the boundary of the plate or on the corner of the plate are shown in Figures5–7, and the geometric and material parameters of the plate are kept the same as mentioned previously. Figures 5–7 show that structures shown in Figure 4 has similar mode shapes in low frequency. Besides, the mode shapes of these structures have symmetric characteristics.Figure 5
Mode shapes for flexural vibration of the CCCC-FFFF-FFFF plate structure with two rectangular cutouts locating inside of the plate: the (a) first mode shape; (b) second mode shape; (c) third mode shape; (d) fourth mode shape; (e) fifth mode shape; (f) sixth mode shape; (g) fiftieth mode shape; (h) hundredth mode shape; and (i) two-hundredth mode shape.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figure 6
Mode shapes for flexural vibration of the CCCC-FFFF plate structure with two rectangular cutouts locating on corner of the plate: the (a) first mode shape; (b) second mode shape; (c) third mode shape; (d) fourth mode shape; (e) fifth mode shape; (f) sixth mode shape; (g) fiftieth mode shape; (h) hundredth mode shape; and (i) two-hundredth mode shape.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figure 7
Mode shapes for flexural vibration of the CCCCCC-FFFFFF plate structure with two rectangular cutouts locating on boundary of the plate: the (a) first mode shape; (b) second mode shape; (c) third mode shape; (d) fourth mode shape; (e) fifth mode shape; (f) sixth mode shape; (g) fiftieth mode shape; (h) hundredth mode shape; and (i) two-hundredth mode shape.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)As illustrated previously, the current method is able to handle the cases with arbitrary boundary conditions. The first six nonzero frequency parameters of the structure with one rectangular cutout under four kinds of boundary conditions are listed in Table4. The rectangular plate considered is in the length of 2 m, width of 1 m, and thickness of 0.003 m, and the corner points of the cutout are located at A (0.8 m, 0.4 m) and B (1.2 m, 0.6 m). The material properties of the structure are E=216GPa, μ=0.3, and ρ=7900kg/m3. It should be mentioned that the alphabet “S” and “C” are used to represent the simply supported and clamped boundary conditions, respectively. From Table 4, it can be found that the present results are very close to the results obtained by software ANSYS.Table 4
Frequency parameters of the structure with one cutout located inside of the plate under different kinds of boundary conditions.
Mode order
1
2
3
4
5
6
7
8
FFFF-FFFF
Present
3.904
4.815
10.977
11.171
15.429
18.809
19.560
22.139
ANSYS
3.880
4.808
10.927
11.110
15.330
18.783
19.529
22.140
CFFF-FFFF
Present
0.634
2.710
4.004
8.997
11.299
16.489
17.105
21.917
ANSYS
0.645
2.706
3.984
8.962
11.230
16.383
17.068
21.887
CCFF-FFFF
Present
3.183
6.898
13.680
16.725
21.370
24.889
29.914
39.087
ANSYS
3.166
6.856
13.620
16.640
21.290
24.748
29.760
38.754
CSFF-FFFF
Present
1.572
5.839
11.533
13.173
17.061
24.273
26.066
36.552
ANSYS
1.580
5.816
11.470
13.100
16.930
24.188
25.939
34.347


### 3.2. In-Plane Vibration Analysis of Rectangular Plate with Cutouts

The flexural vibration of the rectangular plate with cutouts has been studied in the previous section. It proves that the current method is capable of investigating the flexural vibration characteristics of the plate with rectangular cutout. Next, the in-plane vibration of the structure will be investigated. The natural frequencies and mode shapes for in-plane vibration of rectangular plate with cutouts will be presented in this section.In order to verify the correctness of the present method, plate with one cutout is studied firstly. The first example is a classical case: plate with one cutout under the clamped boundary condition. The clamped boundary conditions can be obtained by setting the stiffness constants of the springs at 1012. For this example, the geometric parameters of the rectangular plate considered here are Lx=Ly=3m and h=0.0025m, the material properties of the plate are defined as E=70GPa, μ=0.33, and ρ=2700kg/m3, and the truncated numbers are selected as Min=Nin=24. The coordinates of the opposite corners of the cutouts shown in Figure 2 are (a) A10.25m,0.25m B10.75m,0.75m, (b) A22.5m,2.5m, (c) A31.25m,2.5m B31.75m,2.5m. The first six frequency parameters for in-plane vibration of structures with one cutout are presented in Table 5. And the mode shapes of the CCCC-FFFF plate with one inner cutout are also presented here. From Table 5 and Figure 8, it can be seen that the present results show an excellent agreement with results acquired by software ANSYS.Table 5
Frequency parameters of rectangular plate with one cutout.
Mode order
1
2
3
4
5
6
7
8
Plate with inner cutout
CCCC-FFFF
964.7a
1004.1a
1157.5a
1436.8a
1618.7a
1640.3a
1658.5a
1906.3a
959.27b
1003.8b
1155.9b
1432.0b
1615.5b
1637.1b
1658.1b
1902.0b
Plate with corner cutout
CCF-FCC
978.0a
1008.9a
1177.5a
1426.6a
1623.8a
1650.5a
1657.2a
1859.1a
977.8b
1008.9b
1177.3b
1426.3b
1623.2b
1650.4b
1657.3b
1858.8b
Plate with edge cutout
CCFF-FCCC
946.0a
993.4a
1143.6a
1468.1a
1592.9a
1614.2a
1624.3a
1892.1a
942.4b
991.2b
1141.0b
1467.7b
1589.6b
1613.5b
1622.9b
1891.3b
apresent results; bANSYS results.Figure 8
Mode shapes for in-plane vibration of the CCCC-FFFF plate structure with one rectangular cutout locating inside of the plate: the (a) first mode shape; (b) second mode shape; (c) third mode shape; (d) fourth mode shape; (e) fifth mode shape; (f) sixth mode shape; (g) twentieth mode shape; (h) twenty-first mode shape; and (i) twenty-second mode shape.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)The effects of the boundary conditions on the vibration of the plate with rectangular cutout are considered here. According to [17], there exist two types of simply supported boundary conditions for in-plane vibration. The first type can be considered as stiffness constants of the tangential springs and normal springs are infinite and zero, respectively, while the second one can be seen as stiffness constants of the tangential and normal springs are zero and infinite, separately. For simplicity, these two kinds of boundary conditions are denoted by “St”and “Sn,” respectively. The geometric and material parameters of the plate are same as the ones defined in Table 5. Table 6 shows the natural frequencies of the plate with rectangular cutout located inside of the plate under different kinds of boundary conditions. It can be clearly found that boundary conditions influence the vibration characteristics of the plate structure obviously. The first natural frequency of the same structure varies with different boundary conditions.Table 6
Frequency parameters of the plate with one inner cutout.
Mode order
1
2
3
4
5
6
CFCF-FFFF
Present
451.80
806.28
851.34
937.11
1031.56
1059.58
ANSYS
448.66
803.85
847.42
934.39
1029.41
1055.32
CCCF-FFFF
Present
613.80
830.60
914.19
1188.37
1308.05
1349.43
ANSYS
609.35
827.98
907.89
1182.01
1299.73
1347.12
CSnStF-FFFF
Present
401.86
513.41
705.11
930.53
1065.76
1153.31
ANSYS
400.68
509.44
704.71
921.95
1065.40
1153.22
S
n
S
n
S
n
S
n-FFFF
Present
697.20
836.04
884.95
1130.19
1155.51
1222.97
ANSYS
696.86
830.35
884.93
1125.6
1155.8
1219.3
S
t
S
t
S
t
S
t-FFFF
Present
449.99
523.70
733.39
1009.46
1046.24
1111.86
ANSYS
441.34
523.73
733.36
1004.20
1046.51
1107.60Next, the convergence of the current method for in-plane vibration will be studied. Structures considered here have the same geometric and material parameters as the plate with two cutouts defined in Section3.1. Table 7 shows the frequency parameters of the FFFFFF-FFF-FFF plate with two edge rectangular cutouts determined by different truncated numbers. It can be found that as the truncated number increases the natural frequencies for in-plane vibrations change little at first, and when the truncated number is beyond 28, the natural frequencies remain unchanged. The desired convergence and numerical stability of the current method for in-plane vibration are validated.Table 7
Frequency parameters of the FFFFFF-FFF-FFF plate with two edge rectangular cutouts.
M = N
1
2
3
4
5
6
7
8
39
409.2
674.6
773.4
877
1618.9
1195.5
1347.7
1420.8
40
409.1
674.2
772.8
876.8
1195.4
1347.6
1420.7
1471.4
41
408.9
674
772.4
876.4
1195.4
1347.4
1420.6
1471.3
42
408.8
673.9
772.1
876.1
1195.4
1347.3
1420.5
1471.2
43
408.7
673.5
771.6
875.9
1195.3
1347.1
1420.4
1471.1
44
408.6
673.4
771.4
875.6
1195.3
1347
1420.3
1471.1
45
408.5
673.3
771.1
875.4
1195.2
1346.9
1420.2
1470.9
46
408.5
673.3
771.1
875.4
1195.2
1346.9
1420.2
1470.9As mentioned previously, the mode shapes of the plate with cutouts can be obtained by inserting the eigenvector into the displacement expressions. The first six mode shapes for in-plane vibration of the plate with two cutouts are shown in Figures9–11, respectively. Compared with the flexural vibration of the plate with two cutouts, it can be found that not all mode shapes for in-plane vibration of the structure have symmetric characteristic.Figure 9
Mode shapes of in-plane vibration of the CCCC-FFFF-FFFF plate structure with two rectangular cutouts locating inside of the plate: the (a) first mode shape; (b) second mode shape; (c) third mode shape; (d) fourth mode shape; (e) fifth mode shape; and (f) sixth mode shape.
(a)
(b)
(c)
(d)
(e)
(f)Figure 10
Mode shapes of in-plane vibration of the CCCC-FFFF plate structure with two rectangular cutouts locating on the corner of the plate: the (a) first mode shape; (b) second mode shape; (c) third mode shape; (d) fourth mode shape; (e) fifth mode shape; and (f) sixth mode shape.
(a)
(b)
(c)
(d)
(e)
(f)Figure 11
Mode shapes of in-plane vibration of the CCCCCC-FFFFFF plate structure with two rectangular cutouts locating on the boundary of the plate: the (a) first mode shape; (b) second mode shape; (c) third mode shape; (d) fourth mode shape; (e) fifth mode shape; and (f) sixth mode shape.
(a)
(b)
(c)
(d)
(e)
(f)


## 3.1. Flexural Vibration Analysis of Rectangular Plate with Rectangular Cutouts

First, the flexural vibration of rectangular plate with one inner, edge, or corner cutout will be studied. These three kinds of models are depicted in Figure2. The length and width of the considered rectangular plate are Lx=Ly=2m, and the thickness of the plate is h=0.003m. And the material parameters of the plate structure are defined as E=216GPa, μ=0.3, and ρ=7900kg/m3. The coordinate of the points shown in Figure 2 are (a) A10.5m,0.5m B11.5m,1.5m, (b) A21m,1m, and (c) A30.5m,1m B31.5m,1m. For simplicity, a string consisting of several letters is adopted to represent the boundary conditions of the structure depicted in Figure 2. For example, “CCCC-FFFF” denotes the boundary conditions of the edges corresponding to number one to eight, and “C” and “F” represent clamped and free boundary conditions, respectively.The first six nonzero natural frequencies of flexural vibration of plate structures with one cutout are listed in Table1. The results obtained by software ANSYS are also presented here to verify the correctness of the present method. Comparing the results listed in Table 1, it can be observed that these two sets of results are very close. Besides, it can be also found that the position of the cutouts has an effect on the vibration characteristics of the structure. For example, frequency parameters in Table 1 show that the plate with one corner cutout has the largest nonzero first frequency while the plate with one edge cutout has the smallest nonzero first frequency; structures considered here are under completely free boundary conditions, and all the cutouts are also kept the same size.Table 1
Frequency parameters for flexural vibration of elastic-restrained plate with one rectangular cutout.
Mode order
1
2
3
4
5
6
Plate with inner cutout
FFFF-FFFF
Present
2.013
2.875
4.211
5.966
5.966
9.438
ANSYS
2.008
2.870
4.204
5.949
5.949
9.298
CCCC-FFFF
Present
12.510
14.680
14.750
20.080
20.600
27.870
ANSYS
12.347
14.442
14.442
19.530
20.172
27.466
Plate with corner cutout
FFF-FFF
Present
2.976
3.254
5.145
6.395
11.399
11.468
ANSYS
2.976
3.230
5.150
6.376
11.398
11.493
CCF-FCC
Present
6.327
12.616
15.225
22.902
23.115
26.817
ANSYS
6.325
12.529
15.210
22.811
22.997
26.769
Plate with edge cutout
FFFF-FFFF
Present
1.461
3.541
3.653
5.349
7.552
7.937
ANSYS
1.445
3.537
3.626
5.345
7.436
7.921
CCF-FCCC
Present
7.036
13.522
17.631
20.311
22.268
26.834
ANSYS
6.959
13.479
17.551
20.116
21.801
26.732By solving Equation (32), the natural frequencies and eigenvectors (D) can be obtained simultaneously. By inserting the eigenvector into the displacement expressions (Equations (23)–(25)), the mode shapes of the plate with one inner cutout can be acquired. The mode shapes of CCCC-FFFF plate with one inner cutout are compared with those obtained by ANSYS in Figure 3 (the right one). It can be seen that the mode shapes obtained by the present method match with those acquired by software ANSYS very well. Besides, it can also be found that with the increase of the frequency, the mode shapes of the structure become more and more complex.Figure 3
Mode shapes for flexural vibration of the CCCC-FFFF plate structure with one rectangular cutout locating inside of the plate: the (a) first mode shape; (b) second mode shape; (c) fourth mode shape; (d) eighth mode shape; (e) sixteenth mode shape; (f) thirty-second mode shape; (g) hundredth mode shape; and (h) two-hundredth mode shape.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)The nonzero frequency parameters of the rectangular plate with one rectangular cutout in Table1 are calculated by selecting the truncate number as M=N=14. In order to check the convergence of the current method, the natural frequencies of the plate with two inner cutouts are calculated with different truncation numbers (i.e., Mf=Nf=7−13). The models of the rectangular plate with two cutouts are depicted in Figure 4. In the following discussion, the geometric parameters of the plate with two cutouts are considered as 2 m in length, 2 m in width, and 0.003 m in thickness, and the material properties of the plate are defined the same as previous. The coordinates of the points shown in Figure 4 are (a) A0.4m,0.75m B0.9m,1.25m C1.1m,0.75m D1.6m,1.25m, (b) A0.75m,1.5m B1.25m,1.5m C0.75m,0.5m D1.25m,0.5m, and (c) A0.5m,1.5m B1.5m,1.5m. The frequency results of the plate with two inner cutouts are listed in Table 2. It shows that the natural frequency converges at M=N=13, and the convergence and numerical stability of the current method are validated.Figure 4
Models of rectangular plate with two cutouts: (a) rectangular plate with two inner cutouts; (b) rectangular plate two with edge cutouts; and (c) rectangular plate with two corner cutouts.
(a)
(b)
(c)Table 2
Natural frequencies of FFFF-FFFF-FFFF rectangular plate with two inner cutouts in different truncation schemes.
M = N
1
2
3
4
5
6
7
8
7
2.30
3.26
4.07
6.22
6.45
11.04
11.13
12.12
8
2.30
3.25
4.01
6.21
6.43
11.01
11.13
11.96
9
2.29
3.25
4.01
6.20
6.43
10.84
10.98
11.96
10
2.29
3.24
4.00
6.20
6.43
10.84
10.97
11.90
11
2.29
3.24
4.00
6.19
6.42
10.60
10.95
11.90
12
2.29
3.24
3.98
6.18
6.42
10.59
10.95
11.89
13
2.29
3.24
3.98
6.18
6.41
10.45
10.93
11.89
14
2.29
3.23
3.97
6.18
6.41
10.45
10.93
11.84
15
2.29
3.23
3.97
6.17
6.41
10.38
10.92
11.84
16
2.29
3.22
3.96
6.17
6.41
10.38
10.92
11.83
17
2.29
3.22
3.96
6.17
6.41
10.38
10.92
11.83
18
2.29
3.22
3.96
6.17
6.41
10.38
10.92
11.83As mentioned previously, the current method can be used to study the vibration characteristics of the rectangular plate with rectangular cutout under general elastic-restrained boundary. And the general elastic restrained boundary can be obtained by setting the stiffness constants of the boundary springs at proper values. For example, the clamped boundary conditions can be achieved by setting the stiffness constants of the linear and rotational springs as infinite. In practical calculations, a sufficiently large number is always employed to represent the “infinite.” In the first example, the clamped boundary conditions are obtained by setting the stiffness constant at 1012. So, effects of the stiffness constant of boundary springs on the solution results ought to be studied. The frequency parameters for EEEE-FFFF-FFFF plate with two inner rectangular cutouts are listed in Table 3, and the alphabet “E” represents elastic boundary condition. It can be found that as the stiffness constants of the springs increase, the natural frequencies of the structure increase gradually, and when the stiffness constant is beyond 1012, the natural frequency remain unchanged. Thus, in the following discussion, 1012 is selected to represent the value of “infinite.”Table 3
Frequency parameters of an EEEE-FFFF-FFFF plate with two inner rectangular cutouts (Ku=Kv=Kw=Kr=k).
Mode order
k
1
2
3
4
5
6
7
8
105
6.66
10.33
11.92
14.35
16.40
17.67
19.55
20.50
106
7.35
12.08
14.84
18.17
21.53
25.67
27.32
30.82
107
7.44
12.30
15.23
18.76
22.19
26.92
28.67
32.81
108
7.44
12.32
15.27
18.82
22.26
27.04
28.81
33.02
109
7.44
12.33
15.28
18.83
22.26
27.06
28.83
33.04
1010
7.44
12.33
15.28
18.83
22.26
27.06
28.83
33.04
1011
7.45
12.33
15.28
18.83
22.26
27.06
28.83
33.04
1012
7.45
12.33
15.28
18.83
22.26
27.06
28.83
33.04
1012
7.45
12.33
15.28
18.83
22.26
27.06
28.83
33.04
1014
7.45
12.33
15.28
18.83
22.26
27.06
28.83
33.04The mode shapes of the structure that rectangular cutouts locate inside of the plate or on the boundary of the plate or on the corner of the plate are shown in Figures5–7, and the geometric and material parameters of the plate are kept the same as mentioned previously. Figures 5–7 show that structures shown in Figure 4 has similar mode shapes in low frequency. Besides, the mode shapes of these structures have symmetric characteristics.Figure 5
Mode shapes for flexural vibration of the CCCC-FFFF-FFFF plate structure with two rectangular cutouts locating inside of the plate: the (a) first mode shape; (b) second mode shape; (c) third mode shape; (d) fourth mode shape; (e) fifth mode shape; (f) sixth mode shape; (g) fiftieth mode shape; (h) hundredth mode shape; and (i) two-hundredth mode shape.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figure 6
Mode shapes for flexural vibration of the CCCC-FFFF plate structure with two rectangular cutouts locating on corner of the plate: the (a) first mode shape; (b) second mode shape; (c) third mode shape; (d) fourth mode shape; (e) fifth mode shape; (f) sixth mode shape; (g) fiftieth mode shape; (h) hundredth mode shape; and (i) two-hundredth mode shape.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figure 7
Mode shapes for flexural vibration of the CCCCCC-FFFFFF plate structure with two rectangular cutouts locating on boundary of the plate: the (a) first mode shape; (b) second mode shape; (c) third mode shape; (d) fourth mode shape; (e) fifth mode shape; (f) sixth mode shape; (g) fiftieth mode shape; (h) hundredth mode shape; and (i) two-hundredth mode shape.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)As illustrated previously, the current method is able to handle the cases with arbitrary boundary conditions. The first six nonzero frequency parameters of the structure with one rectangular cutout under four kinds of boundary conditions are listed in Table4. The rectangular plate considered is in the length of 2 m, width of 1 m, and thickness of 0.003 m, and the corner points of the cutout are located at A (0.8 m, 0.4 m) and B (1.2 m, 0.6 m). The material properties of the structure are E=216GPa, μ=0.3, and ρ=7900kg/m3. It should be mentioned that the alphabet “S” and “C” are used to represent the simply supported and clamped boundary conditions, respectively. From Table 4, it can be found that the present results are very close to the results obtained by software ANSYS.Table 4
Frequency parameters of the structure with one cutout located inside of the plate under different kinds of boundary conditions.
Mode order
1
2
3
4
5
6
7
8
FFFF-FFFF
Present
3.904
4.815
10.977
11.171
15.429
18.809
19.560
22.139
ANSYS
3.880
4.808
10.927
11.110
15.330
18.783
19.529
22.140
CFFF-FFFF
Present
0.634
2.710
4.004
8.997
11.299
16.489
17.105
21.917
ANSYS
0.645
2.706
3.984
8.962
11.230
16.383
17.068
21.887
CCFF-FFFF
Present
3.183
6.898
13.680
16.725
21.370
24.889
29.914
39.087
ANSYS
3.166
6.856
13.620
16.640
21.290
24.748
29.760
38.754
CSFF-FFFF
Present
1.572
5.839
11.533
13.173
17.061
24.273
26.066
36.552
ANSYS
1.580
5.816
11.470
13.100
16.930
24.188
25.939
34.347


## 3.2. In-Plane Vibration Analysis of Rectangular Plate with Cutouts

The flexural vibration of the rectangular plate with cutouts has been studied in the previous section. It proves that the current method is capable of investigating the flexural vibration characteristics of the plate with rectangular cutout. Next, the in-plane vibration of the structure will be investigated. The natural frequencies and mode shapes for in-plane vibration of rectangular plate with cutouts will be presented in this section.In order to verify the correctness of the present method, plate with one cutout is studied firstly. The first example is a classical case: plate with one cutout under the clamped boundary condition. The clamped boundary conditions can be obtained by setting the stiffness constants of the springs at 1012. For this example, the geometric parameters of the rectangular plate considered here are Lx=Ly=3m and h=0.0025m, the material properties of the plate are defined as E=70GPa, μ=0.33, and ρ=2700kg/m3, and the truncated numbers are selected as Min=Nin=24. The coordinates of the opposite corners of the cutouts shown in Figure 2 are (a) A10.25m,0.25m B10.75m,0.75m, (b) A22.5m,2.5m, (c) A31.25m,2.5m B31.75m,2.5m. The first six frequency parameters for in-plane vibration of structures with one cutout are presented in Table 5. And the mode shapes of the CCCC-FFFF plate with one inner cutout are also presented here. From Table 5 and Figure 8, it can be seen that the present results show an excellent agreement with results acquired by software ANSYS.Table 5
Frequency parameters of rectangular plate with one cutout.
Mode order
1
2
3
4
5
6
7
8
Plate with inner cutout
CCCC-FFFF
964.7a
1004.1a
1157.5a
1436.8a
1618.7a
1640.3a
1658.5a
1906.3a
959.27b
1003.8b
1155.9b
1432.0b
1615.5b
1637.1b
1658.1b
1902.0b
Plate with corner cutout
CCF-FCC
978.0a
1008.9a
1177.5a
1426.6a
1623.8a
1650.5a
1657.2a
1859.1a
977.8b
1008.9b
1177.3b
1426.3b
1623.2b
1650.4b
1657.3b
1858.8b
Plate with edge cutout
CCFF-FCCC
946.0a
993.4a
1143.6a
1468.1a
1592.9a
1614.2a
1624.3a
1892.1a
942.4b
991.2b
1141.0b
1467.7b
1589.6b
1613.5b
1622.9b
1891.3b
apresent results; bANSYS results.Figure 8
Mode shapes for in-plane vibration of the CCCC-FFFF plate structure with one rectangular cutout locating inside of the plate: the (a) first mode shape; (b) second mode shape; (c) third mode shape; (d) fourth mode shape; (e) fifth mode shape; (f) sixth mode shape; (g) twentieth mode shape; (h) twenty-first mode shape; and (i) twenty-second mode shape.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)The effects of the boundary conditions on the vibration of the plate with rectangular cutout are considered here. According to [17], there exist two types of simply supported boundary conditions for in-plane vibration. The first type can be considered as stiffness constants of the tangential springs and normal springs are infinite and zero, respectively, while the second one can be seen as stiffness constants of the tangential and normal springs are zero and infinite, separately. For simplicity, these two kinds of boundary conditions are denoted by “St”and “Sn,” respectively. The geometric and material parameters of the plate are same as the ones defined in Table 5. Table 6 shows the natural frequencies of the plate with rectangular cutout located inside of the plate under different kinds of boundary conditions. It can be clearly found that boundary conditions influence the vibration characteristics of the plate structure obviously. The first natural frequency of the same structure varies with different boundary conditions.Table 6
Frequency parameters of the plate with one inner cutout.
Mode order
1
2
3
4
5
6
CFCF-FFFF
Present
451.80
806.28
851.34
937.11
1031.56
1059.58
ANSYS
448.66
803.85
847.42
934.39
1029.41
1055.32
CCCF-FFFF
Present
613.80
830.60
914.19
1188.37
1308.05
1349.43
ANSYS
609.35
827.98
907.89
1182.01
1299.73
1347.12
CSnStF-FFFF
Present
401.86
513.41
705.11
930.53
1065.76
1153.31
ANSYS
400.68
509.44
704.71
921.95
1065.40
1153.22
S
n
S
n
S
n
S
n-FFFF
Present
697.20
836.04
884.95
1130.19
1155.51
1222.97
ANSYS
696.86
830.35
884.93
1125.6
1155.8
1219.3
S
t
S
t
S
t
S
t-FFFF
Present
449.99
523.70
733.39
1009.46
1046.24
1111.86
ANSYS
441.34
523.73
733.36
1004.20
1046.51
1107.60Next, the convergence of the current method for in-plane vibration will be studied. Structures considered here have the same geometric and material parameters as the plate with two cutouts defined in Section3.1. Table 7 shows the frequency parameters of the FFFFFF-FFF-FFF plate with two edge rectangular cutouts determined by different truncated numbers. It can be found that as the truncated number increases the natural frequencies for in-plane vibrations change little at first, and when the truncated number is beyond 28, the natural frequencies remain unchanged. The desired convergence and numerical stability of the current method for in-plane vibration are validated.Table 7
Frequency parameters of the FFFFFF-FFF-FFF plate with two edge rectangular cutouts.
M = N
1
2
3
4
5
6
7
8
39
409.2
674.6
773.4
877
1618.9
1195.5
1347.7
1420.8
40
409.1
674.2
772.8
876.8
1195.4
1347.6
1420.7
1471.4
41
408.9
674
772.4
876.4
1195.4
1347.4
1420.6
1471.3
42
408.8
673.9
772.1
876.1
1195.4
1347.3
1420.5
1471.2
43
408.7
673.5
771.6
875.9
1195.3
1347.1
1420.4
1471.1
44
408.6
673.4
771.4
875.6
1195.3
1347
1420.3
1471.1
45
408.5
673.3
771.1
875.4
1195.2
1346.9
1420.2
1470.9
46
408.5
673.3
771.1
875.4
1195.2
1346.9
1420.2
1470.9As mentioned previously, the mode shapes of the plate with cutouts can be obtained by inserting the eigenvector into the displacement expressions. The first six mode shapes for in-plane vibration of the plate with two cutouts are shown in Figures9–11, respectively. Compared with the flexural vibration of the plate with two cutouts, it can be found that not all mode shapes for in-plane vibration of the structure have symmetric characteristic.Figure 9
Mode shapes of in-plane vibration of the CCCC-FFFF-FFFF plate structure with two rectangular cutouts locating inside of the plate: the (a) first mode shape; (b) second mode shape; (c) third mode shape; (d) fourth mode shape; (e) fifth mode shape; and (f) sixth mode shape.
(a)
(b)
(c)
(d)
(e)
(f)Figure 10
Mode shapes of in-plane vibration of the CCCC-FFFF plate structure with two rectangular cutouts locating on the corner of the plate: the (a) first mode shape; (b) second mode shape; (c) third mode shape; (d) fourth mode shape; (e) fifth mode shape; and (f) sixth mode shape.
(a)
(b)
(c)
(d)
(e)
(f)Figure 11
Mode shapes of in-plane vibration of the CCCCCC-FFFFFF plate structure with two rectangular cutouts locating on the boundary of the plate: the (a) first mode shape; (b) second mode shape; (c) third mode shape; (d) fourth mode shape; (e) fifth mode shape; and (f) sixth mode shape.
(a)
(b)
(c)
(d)
(e)
(f)


## 4. Conclusions

A modified Fourier–Ritz formulation is presented for the flexural and in-plane vibration of plates with different kinds of cutouts (inner, edge, or corner). Under the current framework, the in-plane and flexural displacements of rectangular plates are expanded in the form of standard Fourier cosine series plus eight auxiliary polynomial functions. The current method is capable of universally dealing with cases with various positions of cutouts or different kinds of boundary restraint. The modification of the position or size of the cutouts is just like modifying the geometric parameters of the plate, and there is no need to change the solution procedure. The arbitrary elastic boundary conditions can be achieved by setting the stiffness constant of the boundary springs at proper values, for example, clamped boundary condition can be achieved by setting the stiffness constant of the boundary springs to infinite. Although, only uniform elastic restraints along the edge of the rectangular plate are studied in this paper, the modified Fourier–Ritz method can also be used for nonuniform elastic supports.Several examples are presented in this paper to validate the desired accuracy and convergence of the current method. The effects of stiffness constants of the boundary springs on the solution results are also studied. It should be mentioned that with current method, the vibration of the plates with different kinds of cutouts or boundary conditions can be studied in a unified solution procedure. The vibration characteristics of the plate structure with rectangular cutouts are studied. The results show that both the position of cutouts and boundary conditions of the plate will influence the vibration characteristics of the structure obviously. The mode shapes for flexural and in-plane vibration of the plates with cutouts located inside of the plate or on the edge of the plate or on the corner of the plate are plotted in this paper.



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*Source: 8759074-2018-10-16.xml*