Research Article | Open Access

Ilwook Park, Taehyun Kim, Usik Lee, "Frequency Domain Spectral Element Model for the Vibration Analysis of a Thin Plate with Arbitrary Boundary Conditions", *Mathematical Problems in Engineering*, vol. 2016, Article ID 9475397, 20 pages, 2016. https://doi.org/10.1155/2016/9475397

# Frequency Domain Spectral Element Model for the Vibration Analysis of a Thin Plate with Arbitrary Boundary Conditions

**Academic Editor:**Giovanni Garcea

#### Abstract

We propose a new spectral element model for finite rectangular plate elements with arbitrary boundary conditions. The new spectral element model is developed by modifying the boundary splitting method used in our previous study so that the four corner nodes of a finite rectangular plate element become active. Thus, the new spectral element model can be applied to any finite rectangular plate element with arbitrary boundary conditions, while the spectral element model introduced in the our previous study is valid only for finite rectangular plate elements with four fixed corner nodes. The new spectral element model can be used as a generic finite element model because it can be assembled in any plate direction. The accuracy and computational efficiency of the new spectral element model are validated by a comparison with exact solutions, solutions obtained by the standard finite element method, and solutions from the commercial finite element analysis package ANSYS.

#### 1. Introduction

The plate is a representative structural element that is widely used in many engineering fields such as mechanical, civil, aerospace, shipbuilding, and structural engineering. Severe or unwanted vibration of a plate is a very important engineering problem. Thus, it is required to accurately predict the vibration characteristics of a plate during the design phase. Exact solutions are available only for Levy-type plates [1, 2]. Thus, numerous computational methods have been developed for the vibrations of plates during the last two centuries.

The finite element method (FEM) is one of the most widely used computational methods that can be applied to various complex structures including the plates. The FEM in general provides reliable solutions in the low frequency range, but poor solutions in the high frequency range. Thus, to improve the solution accuracy in the high frequency range, a finite element must be divided into many smaller finite elements so that their sizes are smaller than the wavelengths of the highest vibration mode of interest. However, this will result in a significant increase in computation cost. Thus, as an alternative to FEM, we can consider the spectral element method (SEM) for the vibration analysis of plates.

The SEM considered in this study is the fast Fourier transform- (FFT-) based frequency domain analysis method [3, 4]. The spectral element matrix (or exact dynamic stiffness matrix) used in the SEM is formulated from free wave solutions that satisfy governing differential equations of motion in the frequency domain. Thus, compared with FEM, the SEM can provide exact solutions by representing a uniform structure as a single finite element, regardless of the size of the uniform structure. Accordingly the SEM is known as an exact solution method that has the flexibility of FEM and the exactness of continuum elements [3].

Despite the outstanding features of the SEM, it is mostly used in one-dimensional (1D) structures [3, 4]. The SEM application to two-dimensional (2D) structures such as plates has been limited to very specific geometries and boundary conditions, for example, Levy-type plates [6–10] and infinite or semi-infinite plates [11–15]. Some researchers [16, 17] have introduced the spectral super element method (SSEM) for rectangular plates with prespecified boundary conditions on two parallel edges in one direction (say, the -direction). However, as their spectral element models can be assembled only in another direction (the -direction), their applications must be limited to very specific boundary conditions. Recently, Park et al. [5, 18, 19] developed spectral element models for rectangular membrane, isotropic plate, and orthotropic laminated composite plate elements by using two methods in combination: the boundary splitting method [20] and the spectral super element method (SSEM) [16]. They derived frequency-dependent shape functions by applying a Kantorovich method-based finite strip element method in one direction and the SEM in another direction in combination, and* vice versa*. Accordingly, their spectral element models can be assembled in both the - and -directions. However, the spectral element models by Park et al. [5, 18, 19] still have some limitations because they are valid only for finite rectangular membranes or plate elements whose four corner nodes are fixed. To the authors’ best knowledge, there have been no reports on a generic type of spectral element model that can be assembled in any direction of a plate subjected to arbitrary boundary conditions.

Thus, the purpose of this study is to develop a new spectral element model for finite rectangular plate elements that can be applied to any plate subjected to arbitrary boundary conditions. The new spectral element model is developed by modifying the boundary splitting technique used in our previous study [5] so that the four corner nodes of a finite rectangular plate element become active. The performance of the new spectral element model is evaluated by comparison with exact solutions, FEM solutions, and solutions using the commercial finite element analysis package ANSYS [21].

#### 2. Spectral Element Model for a Finite Plate Element

##### 2.1. Governing Equations in the Frequency Domain

The time domain equation of motion and the boundary conditions of plate structures with transverse vibrations are described in [2]. The time domain equation of motion of a plate can be transformed into a frequency domain equation of motion of the plate by using the FFT as follows [5]:where is the transverse displacement in spectral form, is the external force in spectral form, is the mass per unit area of the plate, and is the flexural bending rigidity of the plate where is the modulus of elasticity,* ν* is Poisson’s ratio, and

*h*is the plate thickness. Similarly, the time domain boundary conditions can be transformed into frequency domain boundary conditions as follows:where and are the dimensions of a finite plate in the - and -directions, respectively; and are the resultant moments; and and are the resultant transverse shear forces defined byAnd and are the slopes defined by

We need to obtain frequency domain free wave solutions for a homogeneous equation of motion in order to formulate the spectral element model for a finite plate element. To realize this, the homogeneous equation of motion is considered by removing external force in (1) as follows:

The weak form of (5) can be obtained in the following form:

A free vibration solution satisfying the weak form given in (6) can be obtained approximately by using two combined methods: the boundary splitting method [20] and the spectral super element method (SSEM) [16]. The SSEM uses a combination of the Kantorovich method (based on the finite strip element method) and the frequency domain 1D spectral element method.

The concept of the boundary splitting method is illustrated in Figures 1 and 2. Figure 1 indicates the concept used in our previous study [5]. Figure 2 indicates the concept used in the present study. The original problems, shown in Figures 1(a) and 2(a), are represented by the sum of two partial problems, Problem and Problem . In Figures 1 and 2, the geometric boundary conditions of the original problems are presented in simple forms by using the following definitions:

In our previous study [5], Problem , shown in Figure 1(b), has fixed (null) boundary conditions on the parallel edges at and . Problem , shown in Figure 1(c), has fixed (null) boundary conditions on the parallel edges at and . As a result, the spectral element model developed in [5] is valid only for finite rectangular plate elements whose four corner nodes are fixed, as shown in Figure 1(d). Accordingly, an application of this approach should be limited to very specific problems as considered in [5].

We propose a new boundary splitting method by modifying the boundary splitting method used in [5] such that the four corner nodes of a finite plate element become active. Problem , shown in Figure 2(b), has arbitrary boundary conditions rather than fixed boundary conditions on the parallel edges at and , and its solution is represented by . Problem , shown in Figure 2(c), has fixed (null) boundary conditions on the parallel edges at and . However, the boundary conditions at and in Problem must be specified such that the sum of the boundary conditions at and in Problem and those in Problem is identical to the boundary conditions at and in the original problem. The solution of Problem is represented by . Then, the solution of the original problem can be obtained by summing the solutions to Problem and Problem as follows:Accordingly, compared to the spectral element model developed in our previous study [5] based on the boundary splitting shown in Figure 1, the present spectral element model that was developed based on the boundary splitting shown in Figure 2 has four active corner nodes. Thus, it can be used as a generic finite element model that can be assembled in both the - and -directions of a plate with arbitrary boundary conditions.

##### 2.2. Derivation of

To obtain the solution for Problem by using the SSEM, a rectangular finite plate element is divided into finite strip elements in the -direction, as shown in Figure 3(a). The th finite strip element, which has a width of in the -direction, is shown in Figure 3(b).

**(a)**

**(b)**

The displacement field in the th finite strip element can be represented bywhere is a one-by-four interpolation function matrix and are the nodal line degree of freedom (DOF) functions defined bywhere

By using (9), the displacement field over the entire domain of the finite plate element can be represented aswherewith

In (16), are functions defined bywhere is the Heaviside unit step function.

Substituting (13) into (5) yieldswherewith the following definitions:The constant matrices , , , and are provided in Appendix A.

Next, we assume solutions of (18) to be in the following form:where is a constant, is the wavenumber in the -direction, and

Substituting (21) into (18) gives the following eigenvalue problem:or

The dispersion relation (i.e., the frequency-wavenumber relationship) can be obtained from (24) as follows:

From (25), the wavenumbers can be computed as

By using the wavenumbers given by (26), we can write the general solution of (18) in the following form:wherewithIn (29), is the th eigenvector, which can be readily computed from (23) using .

The nodal DOFs at the nodes defined on the edges at and can be written in vector form aswhereHere, the superscripts* L* and* R* denote the nodal values on the left edge (i.e., at ) and the right edge (i.e., at ) of the plate, respectively. Superscript* A* denotes the quantities related to or contributed by .

By substituting (27) into (31), the nodal DOF vector can be written in terms of the constant vector as follows:where

The constant vector can be removed from (27) by using (32) to obtain the following expression:where

From (12), by using the nodal DOFs defined in Figure 4, we obtain the following expressions:By applying (36) to (14), we obtain the following expressions:Applying (37) to (31) gives

We rearrange the order of nodal DOFs in (38) to define a new nodal DOF vector as follows:

The nodal DOF vector can be related to the new nodal DOF vector as follows:where is the transformation matrix defined bywhere

The matrices and in (42) are ()-by-() block diagonal matrices defined bywhere

Applying (40) to (34) gives whereFinally, substituting (45) into (13) giveswhere

##### 2.3. Derivation of

We can find the solution to Problem by using a procedure similar to that used to obtain the solution for in Problem . Problem can be obtained from Problem by rotating the coordinate system 90° clockwise. However, the differences for Problem are as follows: (1) the fixed (null) boundary conditions are placed at and ; (2) the finite plate element is divided into finite strip elements in the -direction, as shown in Figure 5; and (3) should be determined to satisfy the boundary conditions at and in combination with the boundary values contributed by .

By following the solution procedure used for Problem , the displacement field in the* i*th finite strip element, which has a width of , can be written in the following form:where is a one-by-four interpolation function matrix and are the nodal line DOF functions defined bywhere

By using (49), the displacement field in the whole domain of the finite plate element can be represented bywhereIn (54), the following definition is used:where and is the Heaviside unit step function. Note that the null boundary conditions at and have been applied to obtain (52).

Substituting (52) into (5) giveswherewith the following definitions:The constant matrices , , , and are provided in Appendix B.

Now we assume solutions to (57) in the following form:where is a constant, is the wavenumber in the -direction, and

Substituting (60) into (57) gives the following eigenvalue problem:or

The dispersion relation can be obtained from (63) as follows:From (64), the wavenumbers can be readily computed in the following forms:

By using the wavenumbers computed from (64), we can write the general solution of (57) in the following form:wherewith

In (61), is the th eigenvector that can be readily computed from (63) using .

By using (47), the nodal values contributed by at the th nodes on the bottom edge at and the upper edge at can be related to the nodal DOF vector as follows:orwhere the primes () denote the derivatives with respect to or , and

The superscripts* B* and* U* denote the quantities at the bottom edge (i.e.*,* at ) and the upper edge (i.e., at ) of the plate, respectively. The superscripts* A* and* B* denote the quantities related to or contributed by and , respectively.

By using (70), the nodal values contributed by at all nodes on the bottom and upper edges, except for four corner nodes, can be written in vector form aswhere

By using (70), (72) can be written in terms of as follows:wherewith

Similarly, the nodal values contributed by at all nodes on the bottom and upper edges can be computed from (66), and they can be written in the vector form aswhere