Improved non-dimensional dynamic influence function method for vibration analysis of arbitrarily shaped plates with clamped edges

Abstract

A new formulation for the non-dimensional dynamic influence function method, which was developed by the authors, is proposed to efficiently extract eigenvalues and mode shapes of clamped plates with arbitrary shapes. Compared with the finite element and boundary element methods, the non-dimensional dynamic influence function method yields highly accurate solutions in eigenvalue analysis problems of plates and membranes including acoustic cavities. However, the non-dimensional dynamic influence function method requires the uneconomic procedure of calculating the singularity of a system matrix in the frequency range of interest for extracting eigenvalues because it produces a non-algebraic eigenvalue problem. This article describes a new approach that reduces the problem of free vibrations of clamped plates to an algebraic eigenvalue problem, the solution of which is straightforward. The validity and efficiency of the proposed method are illustrated through several numerical examples.

Introduction

For vibration problems of plates with arbitrary shapes, numerical approximate methods such as the finite element method (FEM) and boundary element method (BEM) are usually applied.^1–5 However, these methods cannot be expected to give very accurate results because a large number of numerical calculations are required as the number of nodes increases. To obtain highly accurate results, the non-dimensional dynamic influence function (NDIF) method for free vibration analysis of arbitrarily shaped membranes has been introduced^6,7 by the authors and extended to arbitrarily shaped acoustic cavities ⁸ and clamped plates. ⁹ It has then been applied to arbitrarily shaped plates with a mixed boundary composed of simply supported and clamped edges ¹⁰ and to arbitrarily shaped plates with free edges. ¹¹

Recently, many researchers have studied various meshless methods, which discretize the entire region of a plate unlike the NDIF method. For instance, Bert and Malik ¹² developed a methodology for the extension of the differential quadrature method to irregularly shaped plates using the mapping concept of a square region to a general curvilinear region. Misra ¹³ proposed the radial basis function (RBF) method using multiple linear regression analysis to obtain eigenvalues of simply supported and clamped plates, which were compared with the eigenvalues presented by the authors’ previous research. ⁹ Krowiak ¹⁴ applied the RBF-pseudospectral method to the free vibration analysis of two-dimensional structures, which combines the meshless feature of the RBF method and the simplicity of the pseudospectral method. He also used eigenvalues ⁶ presented by the authors to show the accuracy of his method. Bui and Nguyen ¹⁵ developed a moving Kriging-interpolation-based meshfree method having Kronecker’s delta property. Finally, Rodrigues et al. ¹⁶ presented a meshless technique based on the RBF-QR algorithm for the static and vibration analyses of plates. The above methods^12–16 still have a limitation in accuracy owing to the large number of numerical calculations compared with the NDIF method. ⁹

Although the NDIF method yields highly accurate results compared with FEM, BEM, and other numerical methods, ⁹ its final system matrix equation does not have a form of an algebraic eigenvalue problem. As a result, the NDIF method needs the inefficient procedure of searching frequency values that make the system matrix singular in the frequency range of interest. To overcome this limitation, we have recently developed an improved NDIF method for membranes^17,18 and acoustic cavities ¹⁹ with arbitrary shapes. In this article, the basic concept of the improved NDIF method^17–19 is extended to arbitrarily shaped plates with clamped edges. The proposed method is validated by comparing its results with those of other methods such as the NDIF method, FEM (ANSYS), and the exact method. It is revealed that the proposed method gives as accurate results as the NDIF method and needs much shorter calculation time than the NDIF method. Note that the computational speed of the improved NDIF method was reported in detail in the authors’ previous paper. ¹⁷

NDIF method review

The equation of motion for free flexural vibration of a uniform thin plate is written as ²⁰

$$D{\nabla }^{4}w+{\rho }_s\frac{{\partial }^{2}w}{\partial t^2}=0$$

(1)

where $w=w(\mathbf{r},t)$ is the transverse deflection, ${\rho }_s$ is the surface density, and D is the flexural rigidity expressed as $D=E{h}^3/12(1-{\nu }^2)$ in terms of Young’s modulus E, Poisson’s ratio $\nu$ , and the plate thickness h. Assuming a harmonic motion $w(\mathbf{r},t)=W(\mathbf{r})e^{j\omega t}$ where $\omega$ is the circular frequency, equation (1) becomes

$${\nabla }^{4}W-{\Lambda }^{4}W=0$$

(2)

where the frequency parameter $\Lambda$ is

$$\Lambda ={\left(\frac{{\rho }_s{\omega }^2}{D}\right)}^{1/4}$$

(3)

Considering an infinite plate with a point source located at point $P_s$ as shown in Figure 1, the deflection form of the plate depends on the scalar distance $|\mathbf{r}-{\mathbf{r}}_s|$ from the source point $P_s$ to the field point P. If the deflection at the source point is assumed to be regular in value (no singularity exists in the entire domain), a general solution of equation (2) is given by

$$W(\mathbf{r})=A{J}_0\left(\Lambda |\mathbf{r}-{\mathbf{r}}_s|\right)+B{I}_0\left(\Lambda |\mathbf{r}-{\mathbf{r}}_s|\right)$$

(4)

where $J_0$ and $I_0$ denote the Bessel function of the first kind and the modified Bessel function of the first kind, respectively. ²¹ The general solution is termed the NDIF of the infinite plate. It is a wave-type function that propagates omni-directionally from one point to infinity.

Figure 1.

Infinite plate including an arbitrarily shaped plate (dashed line).

For free vibration analysis of an arbitrarily shaped plate whose boundary is illustrated by the dotted line in Figure. 1, N nodes are first distributed at points $P_1$ , $P_2$ ,…, $P_N$ along the boundary. Assuming that harmonic displacements of amplitudes $A_1$ , $A_2$ ,…, $A_N$ are generated at points $P_1$ , $P_2$ ,…, $P_N$ , respectively, the total displacement response at point P may be obtained by the sum of responses (linear combination of NDIFs given in equation (4)) that have resulted from each boundary point as follows

$$W(\mathbf{r})=\sum _{k=1}^N{A}_k{J}_0\left(\Lambda |\mathbf{r}-{\mathbf{r}}_k|\right)+B_k{I}_0\left(\Lambda |\mathbf{r}-{\mathbf{r}}_k|\right)$$

(5)

Equation (5) is employed as an approximate solution for the free vibration of the finite-sized plate depicted by the dotted line in Figure 1. Note that the approximate solution also satisfies the governing equation (equation (2)) because the NDIF satisfies the governing equation.

If a clamped plate is considered, the boundary condition defined continuously along the dotted line in Figure 1 is discretized so as to be satisfied only at the N nodes as follows ²⁰

$$W({\mathbf{r}}_i)=0$$

(6)

$$\frac{\partial W({\mathbf{r}}_i)}{\partial n_i}=0,i=1,2,\dots ,N$$

(7)

where $n_i$ denotes the normal direction from the boundary at point $P_i$ in Figure 1. Substituting equation (5) into equations (6) and (7) gives

$$W(r_i)={\displaystyle \sum _{k=1}^N{A}_k{J}_0\left(\Lambda \left|r_i-r_k\right|\right)+B_k{I}_0\left(\Lambda \left|r_i-r_k\right|\right)}=0,i=1,2,\dots ,N$$

(8)

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{\partial W({\mathbf{r}}_i)}{\partial n_i}}& =\sum _{k=1}^N{A}_k\frac{\partial }{\partial n_i}{J}_0\left(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|\right)\hfill \\ \multicolumn{1}{c}{}& +B_k\frac{\partial }{\partial n_i}{I}_0\left(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|\right)=0,i=1,2,\dots ,N\hfill \end{array}$$

(9)

Equation (9) may be rewritten as

$$\begin{array}{c}\sum _{k=1}^N{A}_k\Lambda J_1\left(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|\right)\frac{\partial }{\partial n_i}|{\mathbf{r}}_i-{\mathbf{r}}_k|\hfill \\ -B_k\Lambda I_1\left(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|\right)\frac{\partial }{\partial n_i}|{\mathbf{r}}_i-{\mathbf{r}}_k|=0,i=1,2,\dots ,N\hfill \end{array}$$

(10)

Equations (8) and (10) can be rewritten as matrix forms

$$\mathbf{S}{\mathbf{M}}^{(J)}\mathbf{A}+\mathbf{S}{\mathbf{M}}^{(I)}\mathbf{B}=\mathbf{0}$$

(11)

$${\mathbf{SM}}_n^{(J)}\mathbf{A}+{\mathbf{SM}}_n^{(I)}\mathbf{B}=\mathbf{0}$$

(12)

where system matrices $\mathbf{S}{\mathbf{M}}^{(J)}$ and $\mathbf{S}{\mathbf{M}}^{(I)}$ are given by $S{M}^{(J)}(i,k)=J_0(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|)$ and $S{M}^{(I)}(i,k)=I_0(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|)$ , respectively, and system matrices ${\mathbf{SM}}_n^{(J)}$ and ${\mathbf{SM}}_n^{(I)}$ are given by ${\mathrm{SM}}_n^{(J)}(i,k)=(\partial /\partial n_i)J_0(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|)$ and ${\mathrm{SM}}_n^{(I)}(i,k)=(\partial /\partial n_i)I_0(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|)$ , respectively. Note that the calculation method for the diagonal elements of the above four system matrices for ${\mathbf{r}}_i-{\mathbf{r}}_k=0$ (i.e. $i=k$ ) was presented in the authors’ previous paper. ⁹

Furthermore, equations (11) and (12) can be combined into a single equation

$$\mathbf{SM}(\Lambda )\mathbf{C}=\mathbf{0}$$

(13)

where the final system matrix $\mathbf{SM}(\Lambda )$ and the unknown vector $\mathbf{C}$ are given by

$$\mathbf{SM}(\Lambda )=\left[\begin{array}{cc}\hfill \mathbf{S}{\mathbf{M}}^{(J)}\hfill & \hfill \mathbf{S}{\mathbf{M}}^{(I)}\hfill \\ \multicolumn{1}{c}{{\mathbf{SM}}_n^{(J)}}& \hfill {\mathbf{SM}}_n^{(I)}\hfill \end{array}\right]$$

(14)

$$\mathbf{C}=\left\{\begin{array}{c}\hfill \mathbf{A}\hfill \\ \multicolumn{1}{c}{\mathbf{B}}\end{array}\right\}$$

(15)

The condition that the system matrix equation (equation (13)) has a non-trivial solution gives that the determinant of the system matrix $\mathbf{SM}(\Lambda )$ is zero as follows

$$\det [\mathbf{SM}(\Lambda )]=0$$

(16)

Eigenvalues of the plate can be obtained from the values of the frequency parameter ( $\Lambda$ ) that satisfy equation (16). The mode shapes corresponding to the eigenvalues can be produced by substituting $A_k$ and $B_k$ obtained from equation (13) into equation (5). Note that a matrix manipulation process to reduce the system matrix size in equation (13) has not been described in detail here.

It may be seen in equation (13) that elements of the system matrix $\mathbf{SM}(\Lambda )$ depend on the frequency parameter $\Lambda$ . Therefore, the inefficient procedure of searching for the frequency parameter that makes the system matrix singular by sweeping the frequency parameter in the range of interest is required to extract eigenvalues in the NDIF method. The next section presents an improved theoretical formulation for the NDIF method that makes the system matrix independent of the frequency parameter.

Improved theoretical formulation

The four Bessel functions $J_0$ , $I_0$ , $J_1$ , and $I_1$ in equations (8) and (10) are expanded in a Taylor series ²¹ as follows

$$J_0\left(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|\right)\approx \sum _{s=0}^M\frac{{(-1)}^s{(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|/2)}^{2s}}{{[\Gamma (s+1)]}^2}$$

(17)

$$I_0\left(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|\right)\approx \sum _{s=0}^M\frac{{(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|/2)}^{2s}}{{[\Gamma (s+1)]}^2}$$

(18)

$$J_1\left(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|\right)\approx \sum _{s=0}^M\frac{{(-1)}^s{(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|/2)}^{1+2s}}{\Gamma (s+1)\Gamma (s+2)}$$

(19)

$$I_1\left(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|\right)\approx \sum _{s=0}^M\frac{{(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|/2)}^{1+2s}}{\Gamma (s+1)\Gamma (s+2)}$$

(20)

where M denotes the number of terms of the series and $\Gamma (\dots )$ represents the gamma function. For simplicity, equations (17)–(20) may be rewritten as

$$J_0\left(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|\right)\approx \sum _{s=0}^M{\Lambda }^{2s}{\varphi }_s^J(i,k)$$

(21)

$$I_0\left(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|\right)\approx \sum _{s=0}^M{\Lambda }^{2s}{\varphi }_s^I(i,k)$$

(22)

$$J_1\left(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|\right)\approx \sum _{s=0}^M{\Lambda }^{1+2s}{\phi }_s^J(i,k)$$

(23)

$$I_1\left(\Lambda |{\mathbf{r}}_i-{\mathbf{r}}_k|\right)\approx \sum _{s=0}^M{\Lambda }^{1+2s}{\phi }_s^I(i,k)$$

(24)

where ${\varphi }_s^J(i,k)$ , ${\varphi }_s^I(i,k)$ , ${\phi }_s^J(i,k)$ , and ${\phi }_s^I(i,k)$ are given by

$${\varphi }_s^J(i,k)=\frac{{(-1)}^s{(|{\mathbf{r}}_i-{\mathbf{r}}_k|/2)}^{2s}}{{[\Gamma (s+1)]}^2}$$

(25)

$${\varphi }_s^I(i,k)=\frac{{(|{\mathbf{r}}_i-{\mathbf{r}}_k|/2)}^{2s}}{{[\Gamma (s+1)]}^2}$$

(26)

$${\phi }_s^J(i,k)=\frac{{(-1)}^s{(|{\mathbf{r}}_i-{\mathbf{r}}_k|/2)}^{1+2s}}{\Gamma (s+1)\Gamma (s+2)}$$

(27)

$${\phi }_s^I(i,k)=\frac{{(|{\mathbf{r}}_i-{\mathbf{r}}_k|/2)}^{1+2s}}{\Gamma (s+1)\Gamma (s+2)}$$

(28)

Substituting equations (21) and (22) into equation (8) yields

$${\displaystyle \sum _{k=1}^N{A}_k\left({\displaystyle \sum _{s=0}^M{\Lambda }^{2s}{\varphi }_s^J(i,k)}\right)}+{\displaystyle \sum _{k=1}^N{B}_k\left({\displaystyle \sum _{s=0}^M{\Lambda }^{2s}{\varphi }_s^I(i,k)}\right)}=0,i=1,2,\dots ,N$$

(29)

Similarly, substituting equations (23) and (24) into equation (10) yields

$${\displaystyle \sum _{k=1}^N{A}_k\Lambda \left({\displaystyle \sum _{s=0}^M{\Lambda }^{1+2s}{\phi }_s^J(i,k)}\right)\frac{\partial }{\partial n_i}\left|r_i-r_k\right|}-{\displaystyle \sum _{k=1}^N{B}_k\Lambda \left({\displaystyle \sum _{s=0}^M{\Lambda }^{1+2s}{\phi }_s^I(i,k)}\right)\frac{\partial }{\partial n_i}\left|r_i-r_k\right|}=0,i=1,2,\dots ,N$$

(30)

Equation (30) is simplified further as

$${\displaystyle \sum _{k=1}^N{A}_k\Lambda \left({\displaystyle \sum _{s=0}^M{\Lambda }^{1+2s}{\psi }_s^J(i,k)}\right)}-{\displaystyle \sum _{k=1}^N{B}_k\Lambda \left({\displaystyle \sum _{s=0}^M{\Lambda }^{1+2s}{\psi }_s^I(i,k)}\right)}=0,i=1,2,\dots ,N$$

(31)

where ${\psi }_s^{(J)}(i,k)$ and ${\psi }_s^{(I)}(i,k)$ are

$${\psi }_s^{(J)}(i,k)={\phi }_s^{(J)}(i,k)\frac{\partial }{\partial n_i}|{\mathbf{r}}_i-{\mathbf{r}}_k|$$

(32)

$${\psi }_s^{(I)}(i,k)={\phi }_s^{(I)}(i,k)\frac{\partial }{\partial n_i}|{\mathbf{r}}_i-{\mathbf{r}}_k|$$

(33)

Changing the order of $\sum _{k=1}^N$ and $\sum _{s=0}^M$ in equations (29) and (31) leads to

$${\displaystyle \sum _{s=0}^M{\Lambda }^{2s}\left({\displaystyle \sum _{k=1}^N{A}_k{\varphi }_s^J(i,k)}\right)}+{\displaystyle \sum _{s=0}^M{\Lambda }^{2s}\left({\displaystyle \sum _{k=1}^N{B}_k{\varphi }_s^I(i,k)}\right)}=0i=1,2,\dots ,N$$

(34)

$${\displaystyle \sum _{s=0}^M{\Lambda }^{2(1+s)}\left({\displaystyle \sum _{k=1}^N{A}_k{\psi }_s^J(i,k)}\right)}-{\displaystyle \sum _{s=0}^M{\Lambda }^{2(1+s)}\left({\displaystyle \sum _{k=1}^N{B}_k{\psi }_s^I(i,k)}\right)}=0,i=1,2,\dots ,N$$

(35)

After defining $\lambda ={\Lambda }^2$ , equations (34) and (35) are rewritten as the forms of polynomial equations with respect to $\lambda$ as follows

$$\begin{array}{c}{\lambda }^0\sum _{k=1}^N{A}_k{\varphi }_0^J(i,k)+{\lambda }^1\sum _{k=1}^N{A}_k{\varphi }_1^J(i,k)\hfill \\ +\cdots +{\lambda }^M\sum _{k=1}^N{A}_k{\varphi }_M^J(i,k)\hfill \\ +{\lambda }^0\sum _{k=1}^N{B}_k{\varphi }_0^I(i,k)+{\lambda }^1\sum _{k=1}^N{B}_k{\varphi }_1^I(i,k)\hfill \\ +\cdots +{\lambda }^M\sum _{k=1}^N{B}_k{\varphi }_M^I(i,k)=0\hfill \end{array}$$

(36)

$$\begin{array}{c}{\lambda }^0\sum _{k=1}^N{A}_k{\psi }_0^J(i,k)+{\lambda }^1\sum _{k=1}^N{A}_k{\psi }_1^J(i,k)\hfill \\ +\cdots +{\lambda }^M\sum _{k=1}^N{A}_k{\psi }_M^J(i,k)\hfill \\ -{\lambda }^0\sum _{k=1}^N{B}_k{\psi }_0^I(i,k)+{\lambda }^1\sum _{k=1}^N{B}_k{\psi }_1^I(i,k)\hfill \\ +\cdots +{\lambda }^M\sum _{k=1}^N{B}_k{\psi }_M^I(i,k)=0\hfill \end{array}$$

(37)

Next, equations (36) and (37) may be expressed in matrix equations as follows

$$\left({\lambda }^0{\mathbf{\Phi }}_0^J+{\lambda }^1{\mathbf{\Phi }}_1^J+\dots +{\lambda }^M{\mathbf{\Phi }}_M^J\right)A+\left({\lambda }^0{\mathbf{\Phi }}_0^I+{\lambda }^1{\mathbf{\Phi }}_1^I+\dots +{\lambda }^M{\mathbf{\Phi }}_M^I\right)B=0$$

(38)

$$\left({\lambda }^0{\Psi }_0^J+{\lambda }^1{\Psi }_1^J+\cdots +{\lambda }^M{\Psi }_M^J\right)A-\left({\lambda }^0{\Psi }_0^I+{\lambda }^1{\Psi }_1^I+\cdots +{\lambda }^M{\Psi }_M^I\right)B=0$$

(39)

where the elements of matrices ${\mathbf{\Phi }}_s^{(\circ )}$ and ${\mathbf{\Psi }}_s^{(\circ )}$ are given by

$${\mathbf{\Phi }}_s^{(\circ )}(i,k)={\varphi }_j^{(\circ )}(i,k)$$

(40)

$${\mathbf{\Psi }}_s^{(\circ )}(i,k)={\psi }_j^{(\circ )}(i,k)$$

(41)

and unknown vectors $\mathbf{A}$ and $\mathbf{B}$ are

$$\mathbf{A}={\left\{\begin{array}{cccc}\hfill A_1\hfill & \hfill A_2\hfill & \hfill \cdots \hfill & \hfill A_N\hfill \end{array}\right\}}^T$$

(42)

$$\mathbf{B}={\left\{\begin{array}{cccc}\hfill B_1\hfill & \hfill B_2\hfill & \hfill \cdots \hfill & \hfill B_N\hfill \end{array}\right\}}^T$$

(43)

Furthermore, equations (38) and (39) may lead to a single matrix equation

$${\lambda }^0\left[\begin{array}{cc}{\Phi }_0^J& {\Phi }_0^I\\ {\Psi }_0^J& -{\Psi }_0^I\end{array}\right]\left\{\begin{array}{c}A\\ B\end{array}\right\}+{\lambda }^1\left[\begin{array}{cc}{\Phi }_1^J& {\Phi }_1^I\\ {\Psi }_1^J& -{\Psi }_1^I\end{array}\right]\left\{\begin{array}{c}A\\ B\end{array}\right\}+\cdots +{\lambda }^M\left[\begin{array}{cc}{\Phi }_M^J& {\Phi }_M^I\\ {\Psi }_M^J& -{\Psi }_M^I\end{array}\right]\left\{\begin{array}{c}A\\ B\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\end{array}\right\}$$

(44)

which is expressed as a simple form as follows

$${\lambda }^0\mathbf{S}{\mathbf{M}}_0\mathbf{C}+{\lambda }^1\mathbf{S}{\mathbf{M}}_1\mathbf{C}+\cdots +{\lambda }^M\mathbf{S}{\mathbf{M}}_M\mathbf{C}=\mathbf{0}$$

(45)

where $\mathbf{C}=\{\begin{array}{cc}\hfill \mathbf{A}\hfill & \hfill \mathbf{B}{\}}^T\hfill \end{array}$ , and $\mathbf{S}{\mathbf{M}}_s$ for $s=0,1,\dots ,M$ is given by

$$\mathbf{S}{\mathbf{M}}_s=\left[\begin{array}{cc}\hfill {\mathbf{\Phi }}_s^J\hfill & \hfill {\mathbf{\Phi }}_s^I\hfill \\ \multicolumn{1}{c}{{\mathbf{\Psi }}_s^J}& \hfill -{\mathbf{\Psi }}_s^I\hfill \end{array}\right]$$

(46)

Finally, equation (45) may be changed into the algebraic eigenvalue problem ²² as follows

$$\mathbf{S}{\mathbf{M}}_L\mathbf{D}=\lambda \mathbf{S}{\mathbf{M}}_R\mathbf{D}$$

(47)

where the system matrices $\mathbf{S}{\mathbf{M}}_L$ and $\mathbf{S}{\mathbf{M}}_R$ are given using the diagonal matrix $\mathbf{I}$ by

$$\mathbf{S}{\mathbf{M}}_L=\left[\begin{array}{ccccc}\hfill \mathbf{0}\hfill & \hfill \mathbf{I}\hfill & \hfill \mathbf{0}\hfill & \hfill \cdots \hfill & \hfill \mathbf{0}\hfill \\ \multicolumn{1}{c}{\mathbf{0}}& \hfill \mathbf{0}\hfill & \hfill \mathbf{I}\hfill & \hfill \cdots \hfill & \hfill \mathbf{0}\hfill \\ \multicolumn{1}{c}{⋮}& \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \multicolumn{1}{c}{⋮}& \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill \mathbf{I}\hfill \\ \multicolumn{1}{c}{-\mathbf{S}{\mathbf{M}}_{\mathbf{0}}}& \hfill -\mathbf{S}{\mathbf{M}}_1\hfill & \hfill -\mathbf{S}{\mathbf{M}}_2\hfill & \hfill \cdots \hfill & \hfill -\mathbf{S}{\mathbf{M}}_{M-1}\hfill \end{array}\right]$$

(48)

$$\mathbf{S}{\mathbf{M}}_R=\left[\begin{array}{ccccc}\hfill \mathbf{I}\hfill & \hfill \mathbf{0}\hfill & \hfill \mathbf{0}\hfill & \hfill \cdots \hfill & \hfill \mathbf{0}\hfill \\ \multicolumn{1}{c}{\mathbf{0}}& \hfill \mathbf{I}\hfill & \hfill \mathbf{0}\hfill & \hfill \cdots \hfill & \hfill \mathbf{0}\hfill \\ \multicolumn{1}{c}{\mathbf{0}}& \hfill \mathbf{0}\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \multicolumn{1}{c}{⋮}& \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill \mathbf{I}\hfill & \hfill \mathbf{0}\hfill \\ \multicolumn{1}{c}{\mathbf{0}}& \hfill \mathbf{0}\hfill & \hfill \cdots \hfill & \hfill \mathbf{0}\hfill & \hfill \mathbf{S}{\mathbf{M}}_M\hfill \end{array}\right]$$

(49)

and the vector $\mathbf{D}$ is

$$\mathbf{D}=\left\{\begin{array}{ccccc}\hfill {\mathbf{C}}^T\hfill & \hfill \lambda {\mathbf{C}}^T\hfill & \hfill {\lambda }^2{\mathbf{C}}^T\hfill & \hfill \cdots \hfill & \hfill {\lambda }^{M-1}{\mathbf{C}}^T\hfill \end{array}\right\}$$

(50)

Note that unlike $\mathbf{SM}(\Lambda )$ in equation (13), the system matrices $\mathbf{S}{\mathbf{M}}_L$ and $\mathbf{S}{\mathbf{M}}_R$ in equation (47) are independent of the frequency parameter. Therefore, eigenvalues can simply be extracted from equation (47) without the inefficient procedure required in the original NDIF method. ⁹ The ith eigenvector ${\mathbf{D}}_i$ can be obtained from equation (47) for mode shapes as follows ²³

$${\mathbf{D}}_i={\left\{\begin{array}{ccccc}\hfill {\mathbf{C}}^T\hfill & \hfill \lambda {\mathbf{C}}^T\hfill & \hfill {\lambda }^2{\mathbf{C}}^T\hfill & \hfill \cdots \hfill & \hfill {\lambda }^{M-1}{\mathbf{C}}^T\hfill \end{array}\right\}}_i$$

(51)

The ith mode shape can be obtained by plotting equation (5), where the unknown coefficients $A_k$ and $B_k$ are given by ${\mathbf{C}}^T$ ( ${\mathbf{C}}^T=\{\begin{array}{cc}\hfill \mathbf{A}\hfill & \hfill \mathbf{B}\}\hfill \end{array}$ ) in equation (51).

Verification examples

We performed numerical tests for a clamped circular plate having an exact solution and an arbitrarily shaped plate with clamped edges to examine the validity and accuracy of the proposed method. The eigenvalues and mode shapes of the plates are obtained through the numerical testes. Since the eigenvalues have the same dimension as the frequency parameter in equation (2), the natural frequencies of the plates can be calculated from equation (3).

Clamped circular plate

The proposed method is first applied to a clamped circular plate of unit radius (Figure 2). The boundary of the plate is discretized with 16 nodes for the proposed method. The eigenvalues obtained by the proposed method using $M=20$ and $M=25$ are presented in Table 1, which also shows the eigenvalues given by the exact method, ²⁴ NDIF method, ⁹ and FEM (ANSYS). It can be observed that the eigenvalues obtained by the proposed method in the case of $M=25$ converge rapidly and accurately to those obtained by the exact method. Also, the errors with respect to the exact method do not decrease proportionally as M increases for $M<20$ , and a meaningful trend of eigenvalue change is not observed. Note that the errors in Table 1 are calculated by

$$\frac{\mathrm{Proposed\; method}-\mathrm{exact\; method}}{\mathrm{Exact\; method}}\times 100$$

Figure 2.

Clamped circular plate discretized with 16 nodes.

Table 1.

Eigenvalues of the circular plate obtained by the proposed method, the exact method, the NDIF method, and FEM (parenthesized values denote errors (%) with respect to the exact method).

No.	Proposed method (16 nodes)		Exact method 24	NDIF method 9 (16 nodes)	FEM (1835 nodes)
	$M=20$	$M=25$
1	3.196 (0.00)	3.196 (0.00)	3.196	3.196 (0.00)	3.201 (0.16)
2	4.611 (0.00)	4.611 (0.00)	4.611	4.611 (0.00)	4.610 (0.02)
3	5.906 (0.00)	5.906 (0.00)	5.906	5.906 (0.00)	5.906 (0.00)
4	6.306 (0.00)	6.306 (0.00)	6.306	6.306 (0.00)	6.323 (0.27)
5	7.144 (0.00)	7.144 (0.00)	7.144	7.144 (0.00)	7.143 (0.01)
6	7.820 (0.27)	7.799 (0.00)	7.799	7.799 (0.00)	7.811 (0.15)

NDIF: non-dimensional dynamic influence function; FEM: finite element method.

Furthermore, note that the proposed method using only 16 nodes yields more accurate eigenvalues than FEM (ANSYS) using 1835 nodes. Although the computational speed (CPU time) of the proposed method is not presented here, it has improved vastly compared with that of the NDIF method.

In addition, mode shapes produced by the proposed method using 16 nodes for $M=25$ are presented in Figure 3, and they agree well with those given by the exact method. ²⁴

Figure 3.

Mode shapes of the circular plate produced by the proposed method using 16 nodes for $M=25$ : (a) 1st mode, (b) 2nd mode, (c) 3rd mode, (d) 4th mode, (e) 5th mode, and (f) 6th mode.

The accuracy of an eigenvalue obtained by the proposed method can be verified by plotting its mode shape. If the plotted mode shape does not exactly satisfy the given boundary condition (the clamped boundary condition), it may be said that the eigenvalue is not accurate, and increasing the number of nodes or series functions is required to improve its accuracy.

Arbitrarily shaped plate with clamped edges

An arbitrarily shaped plate whose boundary is composed of a semicircle of unit radius and two equilateral edges of $\sqrt{2}\mathrm{m}$ in length is shown in Figure 4. The normal directions at the three corner points $P_1$ , $P_6$ , and $P_{16}$ are approximately determined by the sum of the two normal vectors of the edges adjacent to each corner. The eigenvalues obtained by the proposed method, NDIF method, ⁹ and FEM (ANSYS) are summarized in Table 2. Since the current plate has no exact solution, errors of the proposed method with respect to FEM using 961 nodes are calculated. It may be observed that the proposed method for $M=25$ has very small errors within 0.2% despite using only 20 nodes. Note that the errors with respect to FEM (961 nodes) do not decrease proportionally as M increases, and a meaningful trend of eigenvalue change is not observed in the same manner as the clamped circular plate.

Figure 4.

Arbitrarily shaped plate discretized by 20 boundary nodes (the arrows denote the normal directions of the corresponding nodes).

Table 2.

Eigenvalues of the arbitrarily shaped plate obtained by the proposed method, the NDIF method, and FEM (parenthesized values denote errors (%) with respect to FEM using 961 nodes).

No.	Proposed method (20 nodes)		NDIF method 9 (16 nodes)	FEM
	$M=20$	$M=25$		961 nodes	576 nodes
1	3.634 (0.03)	3.634 (0.03)	3.634 (0.03)	3.633	3.631
2	5.121 (0.04)	5.121 (0.04)	5.121 (0.04)	5.119	5.116
3	5.283 (0.08)	5.283 (0.08)	5.283 (0.08)	5.279	5.274
4	6.457 (0.16)	6.457 (0.16)	6.457 (0.16)	6.447	6.439
5	6.844 (0.10)	6.844 (0.10)	6.844 (0.10)	6.837	6.830
6	7.013 (0.21)	7.014 (0.20)	7.014 (0.20)	7.028	7.019

NDIF: non-dimensional dynamic influence function; FEM: finite element method.

Figure 5 shows the mode shapes obtained by the proposed method, which are in good agreement with those obtained by FEM (ANSYS), which are shown in Figure 6.

Figure 5.

Mode shapes of the arbitrarily shaped plate obtained by the proposed method using 20 nodes for $M=25$ : (a) 1st mode, (b) 2nd mode, (c) 3rd mode, (d) 4th mode, (e) 5th mode, and (f) 6th mode.

Figure 6.

Mode shapes of the arbitrarily shaped plate obtained by FEM (ANSYS): (a) 1st mode, (b) 2nd mode, (c) 3rd mode, (d) 4th mode, (e) 5th mode, and (f) 6th mode.

Conclusion

An improved NDIF method is proposed to more efficiently extract eigenvalues and mode shapes of arbitrarily shaped plates with clamped edges. It is revealed that the proposed method yields highly accurate eigenvalues, which converge to the exact solution. Owing to its concise formulation, the proposed method gives eigenvalues that are much more accurate than those obtained by FEM, which uses a large number of nodes. The proposed method should be extended to accurately analyze concave plates because it gives accurate results for only convex plates. To overcome this limitation, a sub-domain method of dividing the concave region of interest into several convex regions will be developed in future research.