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Abstract

An analytical method for free vibration analysis of arbitrarily shaped plates with simply supported edges is presented using the non-dimensional dynamic influence function (NDIF) method, which was introduced by the author. A major difficulty in the theoretical formulation of the proposed method is to analytically measure the values of diagonal elements of the system matrix that gives the eigenvalues and mode shapes of the plate of interest. In addition, particular attentions are given to remove the spurious eigenvalues of the plate. Various case studies show that the proposed method yields very accurate eigenvalues and mode shapes compared with other analytical and numerical methods in spite of its theoretical simplicity.

Keywords

NDIF method free vibration arbitrarily shaped plate simply supported edge eigenvalue mode shape

Introduction

For vibration problems of arbitrarily shaped membranes and plates, numerical approximate methods such as the finite element and boundary element methods are usually applied.^1,2 However, these methods cannot be expected to give very accurate results because a large amount of numerical calculation is required as the number of nodes increases.^3–5 For highly accurate results, the author introduced the non-dimensional dynamic influence function (NDIF) method for the free vibration analysis of arbitrarily shaped membranes.^6,7

The NDIF method is an alternative superposition method based on the general collocation method that utilizes two-dimensional basis functions of two independent variables for the approximate solution of a plate. In order to simplify a large amount of numerical calculation that may be caused by the two-dimensional basis functions, a special one-dimensional basis function called NDIF is employed in the NDIF method.⁶ The function is a wave-type function having one independent variable and exactly satisfies the governing differential equation of a plate. For more detailed explanation on the function, refer to the author’s previous papers.⁶

Later, the author improved the NDIF method to effectively extract the eigenvalues of membranes^8,9 In addition, the author extended the NDIF method to arbitrarily shaped acoustic cavities^10,11 and plates with the clamped boundary condition,^12,13 a mixed boundary condition,¹⁴ and the free boundary condition.¹⁵ As known in the author’s previous researches,^12–15 the author developed the NDIF method for arbitrarily shaped plates with various boundary conditions except the simply supported boundary condition.

Although many researches for plates with arbitrary shapes have been performed up to recently, several papers that have recently been published are introduced for lack of space. Fantuzzi et al.¹⁶ dealt with the free vibration problem of arbitrarily shaped plates with FG-CNT material. Also, they solved the free vibration problems of arbitrarily shaped plates using three different numerical approaches in terms of accuracy and reliability.¹⁷ In addition, they considered arbitrarily shaped plates with arbitrary boundaries generated with NURBS using an analogy between fixed membranes and simply supported thin plates.¹⁸ Liu et al.¹⁹ developed a numerical technique for studying arbitrarily shaped plates with extremely accurate methodology which is able to employ a different number of points on the edges and in the domain.

On the other hand, the NDIF method for arbitrarily shaped plates with simply supported edges was formulated by Shi et al.²⁰ However, he failed to obtain the closed-form expressions for the diagonal elements of the system matrix, of which the determinant gives the eigenvalues of a plate. Instead, he employed the approximation of reducing the Bessel functions as serious expansions and omitting higher order terms. It should be noted that in the NDIF method that the diagonal elements of the system matrix for a simply supported plate become singular when the source and collocation points coincide with each other.²⁰

In the study, an improved theoretical formulation of the NDIF method is performed for a simply supported plate and particular care is given in obtaining the closed-form expressions for the diagonal elements of the system matrix to complete Shi’s work.²⁰ Finally, the proposed method is validated by comparing its results with those of the exact method, FEM (NASTRAN), and other analytical methods.

Theoretical formulation

Equation of motion and boundary conditions

It is known that the governing differential equation for small amplitude free transverse vibration of a thin plate with no in-plane force may be written as²¹

D \nabla^{4} w + ρ_{s} \frac{\partial^{2} w}{\partial t^{2}} = 0

(1)

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

\nabla^{4} W - Λ^{4} W = 0

(2)

Λ = {(ρ_{s} ω^{2} / D)}^{1 / 4}

(3)

in which $Λ$ is called a frequency parameter.

The two boundary conditions for a simply supported edge are given by²¹

W = 0

(4)

\frac{\partial^{2} W}{\partial n^{2}} + \frac{v}{R} \frac{\partial W}{\partial n} = 0

(5)

in which $n$ and $R$ represent the normal direction and the radius of curvature of the edge, respectively. For straight-line edges, the second term in equation (5) is eliminated due to $R = \infty$ and as the result, Poisson’s ratio is not included in the boundary conditions.

General solution and discrete boundary conditions

In Figure 1, the plate under consideration is depicted by the dotted line and the boundary of the plate is discretized by $N$ boundary points $P_{1}, P_{2}, \dots, P_{N}$ . $r$ , $r_{i}$ , and $r_{k}$ represent position vectors for boundary points $P$ , $P_{i}$ , and $P_{k}$ , respectively, and $n_{i}$ denotes the direction of the outward normal of the boundary at point $P_{i}$ . As the same manner as in the authors’ previous paper^12,13 that dealt with arbitrarily shaped clamped plates, a general solution of the plate is assumed by linearly superposing the NDIFs as follows

W (r) = \sum_{k = 1}^{N} A_{k} J_{0} (Λ | r - r_{k} |) + B_{k} I_{0} (Λ | r - r_{k} |)

(6)

where $J_{0}$ and $I_{0}$ denote the Bessel functions of order zero of the first and the second kind, $A_{k}$ and $B_{k}$ are unknown coefficients associated with the strengths of participation of NDIFs defined at the boundary points, and $| r - r_{k} |$ is the distance between points $P$ and $P_{k}$ (see Figure 1). Note that the NDIF is a wave-type solution that exactly satisfies the given governing differential equation and physically describes the displacement response of a point in an infinite plate due to a unit displacement excited at another point.^6,12,13

Figure 1.

Arbitrarily shaped plate depicted by the dotted contour along which $N$ boundary points are distributed.

In order to consider the simply supported boundary conditions (equations (4) and (5)) at $N$ boundary points $P_{1}, P_{2}, \dots, P_{N}$ , equations (4) and (5) are discretized as follows

W (r_{i}) = 0, i = 1, 2, \dots, N

(7)

\frac{\partial^{2} W (r_{i})}{\partial n_{i}^{2}} + \frac{v}{R_{i}} \frac{\partial W (r_{i})}{\partial n_{i}} = 0, i = 1, 2, \dots, N

(8)

where $n_{i}$ represents the normal direction from an edge at boundary point $P_{i}$ and $R_{i}$ denotes the radius of curvature of the edge.

Substituting the general solution (equation (6)) into the discrete boundary conditions (equations (7) and (8)) gives

\begin{matrix} \sum_{k = 1}^{N} {A_{k} J_{0} (Λ | r_{i} - r_{k} |) + B_{k} I_{0} (Λ | r_{i} - r_{k} |)} = 0, \\ i = 1, 2, \dots, N \end{matrix}

(9)

\begin{matrix} \frac{\partial^{2}}{\partial n_{i}^{2}} \sum_{k = 1}^{N} {A_{k} J_{0} (Λ | r_{i} - r_{k} |) + B_{k} I_{0} (Λ | r_{i} - r_{k} |)} \\ + \frac{ν}{R_{i}} \frac{\partial}{\partial n_{i}} \sum_{k = 1}^{N} {A_{k} J_{0} (Λ | r_{i} - r_{k} |) + B_{k} I_{0} (Λ | r_{i} - r_{k} |)} \\ = 0, i = 1, 2, \dots, N \end{matrix}

(10)

For simplicity, equations (9) and (10) are rewritten in matrix equations

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

(11)

{SM}_{M}^{(J)} A + {SM}_{M}^{(I)} B = 0

(12)

where the elements of matrices $S M^{(J)}$ , $S M^{(I)}$ , ${SM}_{M}^{(J)}$ , and ${SM}_{M}^{(I)}$ are given by

{SM}^{(J)} (i, k) = J_{0} (Λ | r_{i} - r_{k} |)

(13)

{SM}^{(I)} (i, k) = I_{0} (Λ | r_{i} - r_{k} |)

(14)

{SM}_{M}^{(J)} (i, k) = \frac{\partial^{2} J_{0} (Λ d_{ik})}{\partial n_{i}^{2}} + \frac{ν}{R_{i}} \frac{\partial J_{0} (Λ d_{ik})}{\partial n_{i}}

(15)

{SM}_{M}^{(I)} (i, k) = \frac{\partial^{2} I_{0} (Λ d_{ik})}{\partial n_{i}^{2}} + \frac{ν}{R_{i}} \frac{\partial I_{0} (Λ d_{ik})}{\partial n_{i}}

(16)

where $d_{ik} = | r_{i} - r_{k} |$ denotes the distance between two boundary points $P_{i}$ and $P_{k}$ .

On the other hand, it should be pointed out that ${SM}_{M}^{(J)} (i, k)$ and ${SM}_{M}^{(I)} (i, k)$ given by equations (15) and (16) cannot be evaluated in the normal manner for $i = k$ because there exist non-computable terms when they are differentiated. For this reason, a theoretical formulation of the NDIF method for simply supported plates was delayed for years. In order to overcome this problem, a close investigation is made into equations (15) and (16) and the closed-form expressions of the equations are derived without any approximation in the following section. It should be noted that Shi et al.²⁰ employed the approximation of omitting higher order terms in the process of calculating equations (15) and (16).

Minute examination of ${SM}_{M}^{(J)} (i, k)$ and ${SM}_{M}^{(I)} (i, k)$ for $i = k$

First, equations (15) and (16) are intentionally expressed using the Cartesian coordinate system $(x, y)$ in Figure 1 as follows

\begin{matrix} {SM}_{M}^{(J)} (i, k) = n_{ix}^{2} \frac{\partial^{2} J_{0} (Λ d_{ik})}{\partial x^{2}} + 2 n_{ix} n_{iy} \frac{\partial^{2} J_{0} (Λ d_{ik})}{\partial x \partial y} \\ + n_{iy}^{2} \frac{\partial^{2} J_{0} (Λ d_{ik})}{\partial y^{2}} \\ + \frac{ν}{R_{i}} (n_{ix} \frac{\partial J_{0} (Λ d_{ik})}{\partial x} + n_{iy} \frac{\partial J_{0} (Λ d_{ik})}{\partial y}) \end{matrix}

(17)

\begin{matrix} {SM}_{M}^{(I)} (i, k) = n_{ix}^{2} \frac{\partial^{2} I_{0} (Λ d_{ik})}{\partial x^{2}} + 2 n_{ix} n_{iy} \frac{\partial^{2} I_{0} (Λ d_{ik})}{\partial x \partial y} \\ + n_{iy}^{2} \frac{\partial^{2} I_{0} (Λ d_{ik})}{\partial y^{2}} \\ + \frac{ν}{R_{i}} (n_{ix} \frac{\partial I_{0} (Λ d_{ik})}{\partial x} + n_{iy} \frac{\partial I_{0} (Λ d_{ik})}{\partial y}) \end{matrix}

(18)

where $n_{ix}$ and $n_{iy}$ represent components of unit normal vector $n_{i} = (n_{ix}, n_{iy})$ defined at boundary point $P_{i}$ . In addition, $d_{ik}$ in equations (17) and (18) may take the form

d_{ik} = \sqrt{{(x_{i} - x_{k})}^{2} + {(y_{i} - y_{k})}^{2}}

(19)

where $(x_{i}, y_{i})$ and $(x_{k}, y_{k})$ indicate the coordinates for boundary points $P_{i}$ and $P_{k}$ , respectively.

$\partial J_{0} (Λ d_{ik}) / \partial x$ in equation (17) can be differentiated as

\frac{\partial J_{0} (Λ d_{ik})}{\partial x} = - Λ J_{1} (Λ d_{ik}) \frac{\partial d_{ik}}{\partial x}

(20)

which, using equation (19), leads to

\frac{\partial J_{0} (Λ d_{ik})}{\partial x} = - Λ J_{1} (Λ d_{ik}) \cos α_{ik}

(21)

where $α_{ik}$ denotes the angle between the x-axis of Cartesian coordinates and the vector $r_{i} - r_{k}$ in Figure 1. Similarly, $\partial J_{0} (Λ d_{ik}) / \partial y$ in equation (17) may lead to

\frac{\partial J_{0} (Λ d_{ik})}{\partial y} = - Λ J_{1} (Λ d_{ik}) \sin α_{ik}

(22)

Especially, in the case of $i = k$ , equations (21) and (22) become

\frac{\partial J_{0} (Λ d_{ii})}{\partial x} = - Λ J_{1} (0) \cos α_{ii}

(23)

\frac{\partial J_{0} (Λ d_{ii})}{\partial y} = - Λ J_{1} (0) \sin α_{ii}

(24)

where $J_{1} (0) = 0$ but $α_{ii}$ cannot be evaluated. However, since $\cos α_{ii}$ and $\sin α_{ii}$ have finite values regardless of $α_{ii}$ , equations (23) and (24) lead to

\frac{\partial J_{0} (Λ d_{ii})}{\partial x} = 0

(25)

\frac{\partial J_{0} (Λ d_{ii})}{\partial y} = 0

(26)

In a similar way, $\partial I_{0} (Λ d_{ik}) / \partial x$ and $\partial I_{0} (Λ d_{ik}) / \partial y$ in equation (18) can lead to

\frac{\partial I_{0} (Λ d_{ii})}{\partial x} = 0

(27)

\frac{\partial I_{0} (Λ d_{ii})}{\partial y} = 0

(28)

Next, $\partial^{2} J_{0} (Λ d_{ik}) / \partial x^{2}$ in equation (17) can be differentiated as

\begin{matrix} \frac{\partial^{2} J_{0} (Λ d_{ik})}{\partial x^{2}} = - Λ \frac{J_{1} (Λ d_{ik})}{d_{ik}} \sin^{2} α_{ik} \\ - \frac{Λ^{2}}{2} [J_{0} (Λ d_{ik}) - J_{2} (Λ d_{ik})] \cos^{2} α_{ik} \end{matrix}

(29)

Especially, for $i = k$ , equation (29) becomes

\begin{matrix} \frac{\partial^{2} J_{0} (Λ d_{ii})}{\partial x^{2}} = - Λ \frac{J_{1} (0)}{0} \sin^{2} α_{ii} \\ - \frac{Λ^{2}}{2} [J_{0} (0) - J_{2} (0)] \cos^{2} α_{ii} \end{matrix}

(30)

where $J_{0} (0) = 1$ and $J_{2} (0) = 0$ but both $J_{1} (0) / 0$ and $α_{ii}$ cannot be evaluated. Accordingly, a limit value of $J_{1} (0) / 0$ is found as follows

\frac{J_{1} (0)}{0} = lim_{i \to k} \frac{J_{1} (Λ d_{ik})}{d_{ik}} = lim_{d_{ik} \to 0} \frac{J_{1} (Λ d_{ik})}{d_{ik}} = \frac{Λ}{2}

(31)

Using both $J_{1} (0) / 0 = Λ / 2$ and $\sin^{2} α_{ii} + \cos^{2} α_{ii} = 1$ , equation (30) can be reduced to

\frac{\partial^{2} J_{0} (Λ d_{ii})}{\partial x^{2}} = - \frac{Λ^{2}}{2}

(32)

Similarly, $\partial^{2} J_{0} (Λ d_{ik}) / \partial y^{2}$ , $\partial^{2} I_{0} (Λ d_{ik}) / \partial x^{2}$ , $\partial^{2} I_{0} (Λ d_{ik}) / \partial y^{2}$ , $\partial^{2} J_{0} (Λ d_{ik}) / \partial x \partial y$ , and $\partial^{2} I_{0} (Λ d_{ik}) / \partial x \partial y$ in equations (17) and (18) lead to

\frac{\partial^{2} J_{0} (Λ d_{ii})}{\partial y^{2}} = - \frac{Λ^{2}}{2}

(33)

\frac{\partial^{2} I_{0} (Λ d_{ii})}{\partial x^{2}} = \frac{Λ^{2}}{2}

(34)

\frac{\partial^{2} I_{0} (Λ d_{ii})}{\partial y^{2}} = \frac{Λ^{2}}{2}

(35)

\frac{\partial^{2} J_{0} (Λ d_{ii})}{\partial x \partial y} = 0

(36)

\frac{\partial^{2} I_{0} (Λ d_{ii})}{\partial x \partial y} = 0

(37)

Finally, by substituting equations (25)–(28), (32)–(37) into equations (17) and (18), closed-form expressions for the diagonal elements of the system matrices can be obtained as follows

\begin{matrix} {SM}_{M}^{(J)} (i, k) = n_{ix}^{2} (- \frac{Λ^{2}}{2}) \\ + n_{iy}^{2} (- \frac{Λ^{2}}{2}) = - \frac{Λ^{2}}{2} for i = k \end{matrix}

(38)

{SM}_{M}^{(I)} (i, k) = n_{ix}^{2} (\frac{Λ^{2}}{2}) + n_{iy}^{2} (\frac{Λ^{2}}{2}) = \frac{Λ^{2}}{2} for i = k

(39)

due to $n_{ix}^{2} + n_{iy}^{2} = 1$ . If using equations (38) and (39), equations (15) and (16) can be rewritten as

{SM}_{M}^{(J)} (i, k) = {\begin{matrix} - \frac{Λ^{2}}{2}, i = k \\ \frac{\partial^{2} J_{0} (Λ d_{ik})}{\partial n_{i}^{2}} + \frac{ν}{R_{i}} \frac{\partial J_{0} (Λ d_{ik})}{\partial n_{i}}, i \neq k \end{matrix}

(40)

{SM}_{M}^{(I)} (i, k) = {\begin{matrix} \frac{Λ^{2}}{2}, i = k \\ \frac{\partial^{2} I_{0} (Λ d_{ik})}{\partial n_{i}^{2}} + \frac{ν}{R_{i}} \frac{\partial I_{0} (Λ d_{ik})}{\partial n_{i}}, i \neq k \end{matrix}

(41)

System matrix and eigenvalues

To extract the system matrix that gives the eigenvalues and mode shapes, equations (11) and (12) given in section “General solution and discrete boundary conditions” are considered. Equation (11) is rewritten as

B = - S {M^{(I)}}^{- 1} S M^{(J)} A

(42)

where $S {M^{(I)}}^{- 1}$ denotes the inverse matrix of $S M^{(I)}$ . Substituting equation (42) into equation (12) yields

SM A = 0

(43)

where the system matrix $SM$ is given by

SM (Λ) = {SM}_{M}^{(J)} - {SM}_{M}^{(I)} S {M^{(I)}}^{- 1} S M^{(J)}

(44)

which is a function of the frequency parameter ( $Λ$ ).

The condition for which equation (43) has a non-trivial solution is that the determinant of the system matrix becomes zero, that is

\det (SM (Λ)) = 0

(45)

The eigenvalues may correspond to the values of the frequency parameter ( $Λ$ ) which satisfy equation (45). The mode shapes for the eigenvalues can be obtained by plotting equation (6) in which $A_{k}$ ’s and $B_{k}$ ’s are given by equations (42) and (43).

Elimination of spurious eigenvalues

As in the authors’ previous papers,^12–15 equation (45) may yield not only the eigenvalues of the plate of interest but also the eigenvalues of a similarly shaped membrane, which are called the spurious eigenvalues. To confirm these spurious eigenvalues, the proposed method is applied to a simply supported circular plate of unit radius and Poisson’s ratio of 0.3.

First, the boundary of the circular plate is discretized using 16 equally spaced points, as shown in Figure 2. Logarithmic values of $det (SM (Λ))$ in equation (45) are plotted as a function of the frequency parameter ( $Λ$ ) in Figure 3 where the values of $Λ$ corresponding to troughs $S_{1} ~ S_{12}$ indicate the singular values of the system matrix $SM$ . The singular values $S_{1} ~ S_{12}$ are summarized in Table 1 where it is confirmed that only $S_{1}$ , $S_{3}$ , $S_{5}$ , $S_{7}$ , $S_{9}$ , and $S_{11}$ correspond to the first six exact eigenvalues ( $p_{1} ~ p_{6}$ ) of the simply supported circular plate, respectively, and that the remaining singular values correspond to the first six exact eigenvalues ( $m_{1} ~ m_{6}$ ) of the similarly shaped fixed membrane.

Figure 2.

Circular plate discretized by 16 boundary points.

Figure 3.

Logarithm curve for $det (SM (Λ))$ of the simply supported circular plate using 16 boundary points.

Table 1.

Singular values of the system matrix $SM$ of the simply supported circular plate, which are compared with the exact eigenvalues of the plate²² and those of the similarly shaped fixed membrane.²³

Singular values	Simply supported circular plate		Fixed circular membrane
Singular values	Present method	Exact solution²²	Exact solution²³
$S_{1}$	2.2215	2.2215 ( $p_{1}$ )
$S_{2}$	2.4048		2.4048 ( $m_{1}$ )
$S_{3}$	3.7280	3.7280 ( $p_{2}$ )
$S_{4}$	3.8317		3.8317 ( $m_{2}$ )
$S_{5}$	5.0610	5.0610 ( $p_{3}$ )
$S_{6}$	5.1356		5.1356 ( $m_{3}$ )
$S_{7}$	5.4516	5.4516 ( $p_{4}$ )
$S_{8}$	5.5201		5.5201 ( $m_{4}$ )
$S_{9}$	6.3212	6.3212 ( $p_{5}$ )
$S_{10}$	6.3802		6.3802 ( $m_{5}$ )
$S_{11}$	6.9627	6.9627 ( $p_{6}$ )
$S_{12}$	7.0156		7.0156 ( $m_{6}$ )

From the fact that the singular values of $SM$ consist of the eigenvalues of the plate and those of the membrane, equation (45) may be expressed in the form

\det (SM) = \det (S M_{plate}) \det (S M_{membrane}) = 0

(46)

where $S M_{plate}$ denotes the system matrix of the plate that gives only the eigenvalues of the plate and $S M_{membrane}$ denotes the system matrix of the membrane that gives only the eigenvalues of the membrane. Using equation (46), one can extract

\det (SM) = \det (S M_{plate} S M_{membrane})

(47)

from which

SM = S M_{plate} S M_{membrane}

(48)

By the rearrangement of equation (48), $S M_{plate}$ can be obtained as

S M_{plate} = SM {SM}_{membrane}^{- 1}

(49)

where ${SM}_{membrane}^{- 1}$ denotes the inverse matrix of $S M_{membrane}$ . Substituting equation (44) into equation (49) yields

\begin{matrix} S M_{plate} = {SM}_{M}^{(J)} {SM}_{membrane}^{- 1} \\ - {SM}_{M}^{(I)} {SM}^{(I) - 1} S M^{(J)} {SM}_{membrane}^{- 1} \end{matrix}

(50)

Since $S M^{(J)}$ corresponds to ${SM}_{membrane}$ ,⁶ equations (50) becomes

S M_{plate} = {SM}_{M}^{(J)} {SM}_{membrane}^{- 1} - {SM}_{M}^{(I)} {SM}^{(I) - 1}

(51)

Finally, the eigenvalues of the plate may be obtained from the roots of the determinant of $S M_{plate}$ given by equation (51) as follows

det (S M_{plate} (Λ)) = 0

(52)

where note that $S M_{plate}$ is the function of the frequency parameter ( $Λ$ ).

Case studies

Numerical tests are presented to show the validity and accuracy of the proposed method for various shapes of plates with simply supported edges.

Circular plate

The present method is applied to a simply supported circular plate of unit radius with Poisson’s ratio of 0.3. The boundary of the plate is discretized with 16, 18, and 20 boundary points. For $N = 16$ , $N = 18$ , and $N = 20$ , logarithmic values of $det (S M_{plate} (Λ))$ in equation (52) are plotted as a function of $Λ$ in Figure 4 where the values of $Λ$ corresponding to troughs $p_{1} ~ p_{6}$ indicate the eigenvalues of the simply supported circular plate, and crests $m_{1} ~ m_{6}$ indicate the spurious eigenvalues, which coincide with the eigenvalues of the similarly shaped circular membrane.

Figure 4.

Logarithm curves for $det (S M_{plate} (Λ))$ of the simply supported circular plate for $N = 16$ , $N = 18$ , and $N = 20$ .

The eigenvalues of the circular plate obtained from Figure 4 are summarized in Table 2 where they are compared with those obtained by the exact solution,²² FEM (NASTRAN), and Lam’s method.²⁴ The comparison shows that the eigenvalues by the present method using 16, 18, and 20 boundary points are identical to the exact solution. On the other hand, it is found in Table 2 that except for the first eigenvalue, the eigenvalues by FEM do not fully converge to the exact solution, even though 2409 nodes are used, and that the eigenvalues by Lam et al.²⁴ are slightly higher than the exact solution. From the above comparisons, it may be said that the present method yields very accurate eigenvalues compared with other methods, although it needs a small amount of numerical computation. This high accuracy results from the fact that the present method employs the radius of curvature in its theoretical formulation as in equation (8) unlike FEM. Note that computational efficiency between the NDIF method and FEM can be clearly compared by considering the max matrix dimension (rank) of each method.

Table 2.

Eigenvalues of the simply supported circular plate obtained by the present method, the exact solution,²² FEM (NASTRAN), and Lam et al.²⁴

Source of results	Eigenvalues
Source of results	$p_{1}$	$p_{2}$	$p_{3}$	$p_{4}$	$p_{5}$	$p_{6}$
Present (16 points)	2.2215	3.7280	5.0610	5.4516	6.3212	6.9627
Present (18 points)	2.2215	3.7280	5.0610	5.4516	6.322	6.9627
Present (20 points)	2.2215	3.7280	5.0610	5.4516	6.322	6.9627
Exact solution²²	2.2215	3.7280	5.0610	5.4516	6.3212	6.9627
FEM (2409 nodes)	2.2215	3.7282	5.0613	5.4519	6.3217	6.9635
FEM (1527 nodes)	2.2216	3.7283	5.0615	5.4522	6.3221	6.9643
FEM (785 nodes)	2.2217	3.7288	5.0623	5.4530	6.3232	6.9648
Lam et al.²⁴ (40 terms)	2.2216	3.7282	5.0611	5.4528	N/P	N/P

Figures 5 and 6 show the mode shapes produced by the present method and FEM, respectively. It may be said from the comparison of the mode shapes that although only 16 boundary points are employed, the mode shapes by the present method agree well with those by FEM using 1527 nodes.

Figure 5.

First six mode shapes of the circular plate obtained by the present method using 16 boundary points: (a) first mode, (b) second mode, (c) third mode, (d) fourth mode, (e) fifth mode, and (f) sixth mode.

Figure 6.

First six mode shapes of the circular plate obtained by FEM (NASTRAN) using 1527 nodes: (a) first mode, (b) second mode, (c) third mode, (d) fourth mode, (e) fifth mode, and (f) sixth mode.

Elliptical plate

A simply supported elliptical plate of which the lengths of the major and minor axes are given by 1.2 and 1.0 m (Poisson’s ratio $ν = 0.25$ ) is considered as another verification example. The boundary of the plate is discretized using 16 boundary points, as shown in Figure 7. A logarithm curve for $det [S M_{plate} (Λ)]$ is shown in Figure 8 where the values of the frequency parameter corresponding to troughs $p_{1} ~ p_{6}$ indicate the eigenvalues of the plate, which are tabulated in Table 3.

Figure 7.

Elliptical plate discretized by 16 boundary points.

Figure 8.

Logarithm curve for $det (S M_{plate} (Λ))$ of the simply supported elliptical plate using 16 boundary points (its major and minor axes are given by 1.2 and 1.0 m; Poisson’s ratio $ν = 0.25$ ).

Table 3.

Eigenvalues of the simply supported elliptical plate obtained by the present method, FEM (NASTRAN), Singh and Chakraverty,²⁵ and Leissa²⁶ (its major and minor axes are given by 1.2 and 1.0 m; Poisson’s ratio $ν = 0.25$ ).

Source of results	Eigenvalues
Source of results	$p_{1}$	$p_{2}$	$p_{3}$	$p_{4}$	$p_{5}$	$p_{6}$
Present (16 points)	2.0378	3.2453	3.5956	4.4516	4.6510	5.1894
FEM (2871 nodes)	2.0378	3.2454	3.5956	4.4515	4.6514	5.1891
Singh and Chakraverty²⁵	2.0378	–	–	–	–	–
Leissa²⁶	2.0389	–	–	–	–	–

It may be said in Table 3 that the eigenvalues obtained by the present method using only 16 boundary points agree well with those calculated by FEM using 2871 nodes. In addition, it may be seen that the fundamental eigenvalue obtained by the present method is identical to the value presented by FEM and Singh and Chakraverty²⁵ but is slightly higher than the value presented by Leissa.²⁶ The difference is 0.054% in value. As shown in Figure 9, the present method successfully gives the first six mode shapes for only 16 boundary points. It is revealed that these mode shapes are good agreement with those produced from FEM, which are given in Figure 10.

Figure 9.

First six mode shapes of the elliptical plate obtained by the present method using 16 boundary points (its major and minor axes are given by 1.2 and 1.0 m; Poisson’s ratio $ν = 0.25$ ): (a) first mode, (b) second mode, (c) third mode, (d) fourth mode, (e) fifth mode, and (f) sixth mode.

Figure 10.

First six mode shapes of the elliptic plate obtained by FEM (NASTRAN) using 2871 nodes (its major and minor axes are given by 1.2 and 1.0 m; Poisson’s ratio $ν = 0.25$ ): (a) first mode, (b) second mode, (c) third mode, (d) fourth mode, (e) fifth mode, and (f) sixth mode.

An elliptical plate with different dimensions and Poisson’s ratio is again considered to confirm the accuracy of the proposed method. The lengths of the major and minor axes are 1.0 and 0.5 m (Poisson’s ratio $ν = 0.3$ ). The eigenvalues are obtained from troughs $p_{1} ~ p_{4}$ observed in Figure 11 and are compared with those obtained by Lam et al.,²⁴ Singh and Chakraverty,²⁵ and FEM in Table 4 where the first four eigenvalues are tabulated because Singh presents only four eigenvalues. The comparison indicates that the eigenvalues obtained by the present method using 16 boundary points are slightly different from the FEM result using 2971 nodes and the differences do not exceed 0.01% in value. It is also found in Table 4 that the present method yields as accurate eigenvalues as Lam and Singh’s methods.^24,25 It should be noted that the present method can be applied to arbitrarily shaped plates unlike the two analytical methods, which can solve only elliptical plates. Although the mode shapes of the elliptical plate are not presented here, they agree well with mode shapes obtained by FEM.

Figure 11.

Logarithm curve for $det (S M_{plate} (Λ))$ of the simply supported elliptical plate using 16 boundary points (its major and minor axes are given by 1.0 and 0.5 m; Poisson’s ratio $ν = 0.3$ ).

Table 4.

Eigenvalues of the simply supported elliptical plate obtained by the present method, Lam et al.,²⁴ Singh and Chakraverty,²⁵ and FEM (NASTRAN) (its major and minor axes are given by 1.0 and 0.5 m; Poisson’s ratio $ν = 0.3$ ).

Source of results	Eigenvalues
Source of results	$p_{1}$	$p_{2}$	$p_{3}$	$p_{4}$
Present (16 points)	3.6350	4.8622	6.1907	6.7934
Lam et al.²⁴ (80 terms)	3.6350	4.8622	6.1908	6.7933
Singh and Chakraverty²⁵ (10 terms)	3.6350	4.8622	6.1931	6.7935
FEM (2971 nodes)	3.6350	4.8624	6.1913	6.7935

Arbitrarily shaped plate

In this section, an arbitrarily shaped plate is considered and its geometry is shown in Figure 12 where the plate is discretized with 20 boundary points. The normal directions at the corners having the 1st, 6th, and 16th boundary points are approximately determined by the sum of the two normal vectors for the edges adjacent to each corner.

Figure 12.

Arbitrarily shaped plate discretized by 20 boundary points at which $n_{i}$ represents the normal direction.

The eigenvalues obtained from the logarithm curve of $det [S M_{plate} (Λ)]$ shown in Figure 13 are summarized in Table 5. It may be seen in Table 5 that the present method yields accurate eigenvalues, which are slightly lower than those obtained FEM using 2212 nodes and the differences between them do not exceed 0.88% in value. The current differences are relatively larger than those for the circular and elliptical plates considered in the previous sections. This reason may be that the arbitrarily shaped plate has the three corners of which the normal directions are approximately determined. An additional research is required to reduce the differences further.

Figure 13.

Logarithm curve for $det (S M_{plate} (Λ))$ of the simply supported arbitrarily shaped plate using 20 boundary points (Poisson’s ratio $ν = 0.3$ ).

Table 5.

Eigenvalues of the simply supported, arbitrarily shaped plate obtained by the present method and FEM (NASTRAN) (Poisson’s ratio $ν = 0.3$ ).

Source of results	Eigenvalues
Source of results	$p_{1}$	$p_{2}$	$p_{3}$	$p_{4}$	$p_{5}$	$p_{6}$
Present (20 points)	2.596 (0.88%)	4.134 (0.84%)	4.308 (0.02%)	5.492 (0.53%)	5.894 (0.02%)	6.069 (0.13%)
FEM (2212 nodes)	2.619	4.169	4.309	5.521	5.895	6.077

Figures 14 and 15 show the first six mode shapes given by the present method and those obtained by FEM, respectively. From the comparison between the figures, it may be said that mode shapes by the two methods are in good agreement.

Figure 14.

First six mode shapes of the arbitrarily shaped plate obtained by the present method using 20 boundary points: (a) first mode, (b) second mode, (c) third mode, (d) fourth mode, (e) fifth mode, and (f) sixth mode.

Figure 15.

First six mode shapes of the arbitrarily shaped plate obtained by FEM (NASTRAN) using 2212 nodes: (a) first mode, (b) second mode, (c) third mode, (d) fourth mode, (e) fifth mode, and (f) sixth mode.

Conclusion

The NDIF method has been extended to the free vibration of arbitrarily shaped plates with simply supported edges. Deep consideration has been given to measure the diagonal elements of the system matrix and thus, the closed-form expressions for the diagonal elements have been achieved. It was revealed from case studies that the eigenvalues and mode shapes obtained by the proposed method are very accurate in spite of the small number of boundary points compared with other analytical or numerical methods. Although the proposed method gives very accurate eigenvalues compared with other methods, it cannot be applicable directly to laminated composite plates at this stage. Thus, an improved NDIF method effective for laminated composite plate will be developed in the another paper following this article. In addition, it will be investigated the possibility that the NDIF method can be extended to other theories such as the first-order shear deformation theory.

Footnotes

Handling Editor: Daxu Zhang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was financially supported by Hansung University.

ORCID iD

Sangwook Kang

References

Hughes

TJR

. The finite element method. Upper Saddle River, NJ: Prentice Hall, 1987.

Brebbia

Telles

JCF

Wrobel

. Boundary element techniques. New York: Springer-Verlag, 1984.

Dawe

. A finite element approach to plate vibration problems. J Mech Eng Sci 1965; 7: 28–32.

Guminiak

. Free vibrations analysis of thin plates by the boundary element method in non-singular approach. Sci Res Inst Math Comput Sci 2007; 6: 75–90.

Ramu

Mohanty

. Study on free vibration analysis of rectangular plate structures using finite element method. Procedia Engineer 2012; 38: 2758–2766.

Kang

Lee

Kang

. Vibration analysis of arbitrarily shaped membranes using non-dimensional dynamic influence function. J Sound Vib 1999; 221: 117–132.

Kang

Lee

. Application of free vibration analysis of membranes using the non-dimensional dynamic influence function. J Sound Vib 2000; 234: 455–470.

Kang

Kim

Atluri

. Application of the nondimensional dynamic influence function method for free vibration analysis of arbitrarily shaped membranes. J Vib Acoust 2012; 134: 041008.

Kang

Atluri

. Improved non-dimensional dynamic influence function method based on tow-domain method for vibration analysis of membranes. Adv Mech Eng 2015; 7: 1–8.

10.

Kang

Lee

. Eigenmode analysis of arbitrarily shaped two-dimensional cavities by the method of point-matching. J Acoust Soc Am 2000; 107: 1153–1160.

11.

Kang

Atluri

. Application of non-dimensional dynamic influence function method for eigenmode analysis of two-dimensional acoustic cavities. Adv Mech Eng 2014; 6: 1–9.

12.

Kang

Lee

. Free vibration analysis of arbitrarily shaped plates with clamped edges using wave-type functions. J Sound Vib 2001; 242: 9–26.

13.

Kang

Atluri

. Improved non-dimensional dynamic influence function method for vibration analysis of arbitrarily shaped plates with clamped edges. Adv Mech Eng 2016; 8: 1–8.

14.

Kang

. Free vibration analysis of arbitrarily shaped plates with a mixed boundary condition using non-dimensional dynamic influence functions. J Sound Vib 2002; 256: 533–549.

15.

Kang

Kim

Lee

. Free vibration analysis of arbitrarily shaped plates with smoothly varying free edges using NDIF method. J Vib Acoust 2008; 130: 041010.

16.

Fantuzzi

Tornabene

Bacciocchi

et al . Free vibration analysis of arbitrarily shaped functionally graded carbon nanotube-reinforced plates. Compos Part B: Eng 2017; 115: 384–408.

17.

Fantuzzi

Tornabene

Bacciocchi

et al . Stability and accuracy of three Fourier expansion-based strong form finite elements for the free vibration analysis of laminated composite plates. Int J Numer Meth Eng 2017; 111: 354–382.

18.

Fantuzzi

Puppa

Tornabene

et al . Strong formulation isogeometric analysis for the vibration of thin membranes of general shape. Int J Mech Sci 2017; 120: 322–340.

19.

Liu

Zhao

et al . A differential quadrature hierarchical finite element method and its applications to vibration and bending of Mindlin plates with curvilinear domains. Int J Numer Meth Eng 2017; 109: 174–197.

20.

Shi

Chen

Wang

. Free vibration analysis of arbitrary shaped plates by boundary knot method. Acta Mech Solida Sin 2009; 22: 328–336.

21.

Meirovitch

. Analytical methods in vibrations. New York: Macmillan Publishers, 1967.

22.

Prescott

. Applied elasticity. New York: Dover Publications, 1961.

23.

Blevins

. Formulas for natural frequency and mode shape. New York: Litton Educational Publishing, 1979.

24.

Lam

Liew

Chow

. Use of two-dimensional orthogonal polynomials for vibration analysis of circular and elliptical plate. J Sound Vib 1992; 154: 261–269.

25.

Singh

Chakraverty

. Transverse vibration of simply supported elliptical and circular plates using boundary characteristic orthogonal polynomials in two variables. J Sound Vib 1992; 152: 149–155.

26.

Leissa

. Vibrations of a simply-supported elliptical plate. J Sound Vib 1967; 6: 145–148.

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

Abstract

Keywords

Introduction

Theoretical formulation

Equation of motion and boundary conditions

General solution and discrete boundary conditions

Minute examination of SM M ( J ) ( i , k ) and SM M ( I ) ( i , k ) for i = k

System matrix and eigenvalues

Elimination of spurious eigenvalues

Case studies

Circular plate

Elliptical plate

Arbitrarily shaped plate

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Minute examination of ${SM}_{M}^{(J)} (i, k)$ and ${SM}_{M}^{(I)} (i, k)$ for $i = k$