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Vibration analysis of vertically standing plates subject to gravitational forces is crucial for designing walls, panels, and windows in building structures. This paper investigates the fundamental frequency of a vertically oriented, heavy plate with simplysupported edges. A pseudospectral method, a robust numerical technique, is employed to solve the governing differential equation incorporating various boundary conditions. The study presents frequency parameter values for a range of weight and width parameters. Additionally, the numerical technique is extended to determine the frequencies of a plate on a Winkler foundation. Comparative assessments are conducted to validate the accuracy and reliability of the proposed method.
free vibration, gravity, pseudospectral, standing plate, Winkler
Verticallyoriented plated structures are extensively employed in various engineering applications, including curtain facades, walls, panels, and windows of buildings, as well as in propelling missiles and rockets [1]. In these cases, the standing plate is subjected to its own weight, and accelerating mobile structures can generate body forces equivalent to gravity [2]. The body forces developed in the midplane of the plate, due to its weight or acceleration in its plane, influence the plate's stability and natural frequencies. Consequently, accounting for the effects of gravity on the vibration characteristics of these plates is crucial for designing such structures.
Free vibration analysis of rectangular plates represents a classical structural mechanics problem that holds particular interest for professionals in mechanical, civil, and aerospace engineering fields. The free vibration characteristics of thin, uniformthickness rectangular plates have been comprehensively reported by Leissa [3]. Nonetheless, the introduction of complicating effects and additional geometric or material parameters renders the vibration characteristics data virtually limitless [4]. While vibrations of rectangular plates with varying properties are wellstudied, the vibration analysis of standing vertical plates remains relatively scarce [1, 5].
Large standing plates with simply supported vertical sides are frequently utilized in walls and windows of buildings [5]. For heavy plates, the effect of gravity is a significant factor in their design. Moreover, accelerating mobile structures can generate body forces equivalent to gravity [6]. Therefore, investigating the vibration characteristics of heavy standing plates under their own weight can enhance the understanding and applicability of these structures in practical engineering designs [1].
Herrmann [7] is among the early references that employed an energy method to study specific cases related to the vibration of standing rectangular plates. Yu and Wang [5] used the semianalytical Levy interpolation method to determine the fundamental frequency of standing plates with vertical simply supported edges. They applied the Levy separation method to reduce the governing equation to an ordinary differential equation with nonconstant coefficients. The twopoint boundary value problem was subsequently transformed into two deterministic initial value problems. This method was further employed in study [8] to determine the fundamental frequencies of a standing plate with simply supported edges and weakened by a horizontal internal hinge. Lai and Xiang [1] adopted the Discrete Singular Convolution (DSC) method to examine the influence of body forces on the buckling and vibration behavior of elastically restrained vertical plates. Recently, Guguloth et al. [9] presented a free vibration analysis of simply supported rectangular plates using ANSYS software, although weight was neglected.
This paper aims to apply the pseudospectral method for studying the vibration characteristics of a standing plate with simply supported vertical sides. Conventional spectral collocation methods that utilize differentiation matrices based on Lagrange interpolating polynomials encounter difficulties when imposing two boundary conditions at one node. Several approaches have been proposed by researchers to address this issue [10]. Numerous recursion codes and packages implementing spectral collocation methods exist [1113]; however, incorporating general boundary conditions into the formulation remains challenging [14]. The proposed formulation overcomes the problem associated with the imposition of two boundary conditions at one node. The technique is first applied to study the free vibration of a heavy standing plate and subsequently employed to determine the frequencies of a plate on a Winkler foundation. The obtained results are compared with the semianalytical method [5] and differential transform method [15]. The motivation for utilizing this approach lies in its numerical stability and flexible implementation for vibration analysis.
Consider an isotropic standing rectangular plate of height L and width aL and uniform thickness h, simply supported on the vertical sides with horizontal and vertical sides parallel to x,y axis respectively in the xy plane as shown in Figure 1.
Figure 1. A rectangular plate with simply supported vertical edges
The bottom edge, bearing the total weight is clamped or simply supported and the top edge bearing no load is either free or simply supported. By simply supported, we mean a plate boundary that is prevented from deflecting but free to rotate about a line along the boundary edge. In other words, for a simply supported edge, the displacement is zero and the moment perpendicular to edge is also zero. A clamped edge in a plate is an edge wherein both the deflection and its slope are absent normally at an edge of a plate, a twisting moment, a bending moment and transverse shear force act. An edge for which all of these stress resultants vanish is considered to be a free edge. Adopting the classical plate theory and normalizing all lengths by the plate height L, the governing equation [16] is
$\nabla^4 w+\gamma \frac{\partial}{\partial y}\left[(1y) \frac{\partial w}{\partial y}\right]K^4 w=0$ (1)
where, $\nabla^4$ is the biharmonic differential operator. w is the lateral deflection, ρ the mass per unit area, g the gravitational acceleration, γ the weight parameter, K the frequency parameter.
$\gamma=\frac{\rho g L^3}{D}, K^4=\frac{\rho \Omega^2 L^4}{D}, D=\frac{E h^3}{12\left(1v^2\right)}$
where, D is the flexural rigidity and Ω the frequency and υ is the Poisson’s ratio. As the vertical sides are simply supported, Levy separation of variables is used wherein $w(x, y)=\sin \alpha x Y(y)$.
where $\alpha=\frac{n \pi}{a}, n$ an integer. Eq. (1) becomes
$Y^{\prime \prime \prime \prime}+Y^{\prime \prime}\left(2 \alpha^2+\gamma(1y)\right)\gamma Y^{\prime}+\left(\alpha^4K^4\right) Y=0$ (2)
where, the primes denote derivatives with respect to y. The boundary conditions to be satisfied at the horizontal edges are:
$Y=\frac{d Y}{d y}=0$ for clamped edge (3)
$\begin{aligned} \begin{aligned} & \frac{d^2 Y}{d y^2}\alpha^2 v Y=0 \\& \frac{d^3 Y}{d y^3}\alpha^2(2v) \frac{d Y}{d y}=0\end{aligned} & \text { for free edge } \end{aligned}$ (4)
$Y=\frac{d^2 Y}{d y^2}=0$ for simply supported edge (5)
The boundary value problem is difficult to solve, even numerically [5]. Here a novel pseudospectral method is employed to numerically solve the problem. Eq. (2) is to be solved with the boundary conditions at y=0 and y=1. When the weight is absent and the plate is resting on a Winkler foundation with K_{w} being the foundation modulus parameter, Eq. (2) becomes
$Y^{\prime \prime \prime \prime}2 \alpha^2 Y^{\prime \prime}+\alpha^4 Y+K_w Y=K^4 Y$ (6)
Plates on an elastic foundation are common structural elements that are widely employed in many civil engineering applications and Winkler model is supposed to be simplest model for an elastic foundation. The model assumed that the vertical displacement and pressure underneath it is linearly related to each other. The foundation reaction is included in the governing differential equation of the plate through the foundation parameter (K_{w}). The boundary value problems given by Eq. (2) and Eq. (6) are to be solved subject to the boundary conditions Eqns. (3)(5) as required.
The methodology of solving the boundary value problems using the proposed pseudospectral method is outlined in the next section.
The pseudospectral method can be implemented using several approaches. In the literature, the approach of Fornberg [11] and the differentiation matrices approach [12, 13] are generally employed. An analysis of the numerical instabilities that may occur as the order of the derivative and the number of nodes are increased has been presented by Sadiq and Viswanath [17]. In addition, the difficulty in the incorporation of different boundary conditions in these approaches have led us to develop an approach of pseudospectral method that is simple to use and efficient in implementation. The methodology developed is particularly useful in solving vibration problems of rods, beams and plates of different geometries and configurations. The general steps in the proposed method can be outlined as follows: The physical domain 0≤y≤1 is first transformed to the computational domain 1≤t≤1 using the transformation t=2y1. With this transformation, $D_y^{(m)}=2^m D_t^{(m)}, m=1,2,3,4$ where $D_y^{(m)}$ is the differential operator with the subscript denoting the differentiation variable and super script in bracket denoting the order of differentiation. To apply the collocation technique, we assume
$Y=\sum_{k=0}^N a_k T_k(t)$ (7)
where, $T_k(t),(k=0,1,2, \ldots)$ are Chebyshev polynomials that can be described as
$\begin{array}{r}T_k(t)=\cos \left(k \cos ^{1} t\right), \quad1 \leq t \leq 1 \\ T_k(t)=\cos (k \theta) \text { where } \theta=\cos ^{1} t\end{array}$
The transformation t=cosθ is exploited here to evaluate the derivative directly in terms of the cosine and sine. This change of coordinates trick can reduce the error for a given number of grid points [18]. The grid points chosen in the paper are the ChebyshevGaussLobatto (CGL) nodes whose Lebesgue constant is very close to the optimal. In solving boundary value problems, it is opined [19] that CGL nodes often yield the best results. In addition, in the implementation of derivatives of Chebyshev polynomials using recurrence relations, the trigonometric derivative formulas are simpler to use and require fewer loops. This is used in the present work.
The derivatives are given by:
For $1<t<1$;
$\frac{d}{d t} T_k(t)=\frac{k \sin (k \theta)}{\sin \theta}$ (8)
$\frac{d^2}{d t^2} T_k(t)=\frac{k^2 \cos (k \theta)}{\sin ^2 \theta}+\frac{k \cos \theta}{\sin ^3 \theta} \sin k \theta$ (9)
$\frac{d^3 T_k(t)}{d t^3}=\frac{1}{\sin ^5 \theta}\left(\begin{array}{l}\sin ^2 \theta \\ \left(k \sin (k \theta)k^3 \sin (k \theta)\right) \\ +3 k \sin (k \theta) \cos ^2 \theta \\ 3 k^2 \cos (k \theta) \cos \theta \sin \theta\end{array}\right)$ (10)
$\frac{d^4 T_k(t)}{d t^4}=\frac{\left[\begin{array}{l}\sin ^2 \theta\left(\begin{array}{l}9 k \sin (k \theta) \cos \theta \\ 6 k^3 \sin (k \theta) \cos \theta\end{array}\right) \\ \sin ^3 \theta\left(\begin{array}{l}4 k^2 \cos (k \theta) \\ k^4 \cos (k \theta)\end{array}\right) \\ +15 k \sin (k \theta) \cos ^3 \theta \\ 15 k^2 \cos (k \theta) \cos ^2 \theta \sin \theta\end{array}\right]}{\sin ^7 \theta}$ (11)
and
$\frac{d^n}{d t^n} T_k(t)=( \pm 1)^{k+n} \prod_{p=0}^{n1} \frac{k^2p^2}{2 p+1}$, at $\mathrm{t}= \pm 1$ (12)
In the collocation framework, we select N3 points in (0, π) and require Y(θ) to satisfy Eq. (2) at these N3 points in addition to satisfying the boundary conditions at the end points. The internal points which are the extrema of T_{N}(t) are given by
$t_i=\cos \left(\frac{(Ni) \pi}{N}\right), i=1,2, \ldots, N3$ (13)
that corresponds to the points
$\theta_i=\frac{(Ni) \pi}{N}, i=1,2, \ldots, N3$ in $(0, \pi)$ (14)
Substituting Eq. (7) in Eq. (2) and using Eqns. (8)(11), an equivalent differential equation on $\theta \in[0, \pi]$ is obtained that is collocated at N3 collocation points given by Eq. (14) yielding N4 equations in N+1 unknowns a_{k.} Imposing the boundary conditions, we get a system of four equations in N+1 unknowns for each of the boundary conditions given by Eqns. (3)(5). The resulting N+1 by N+1 system of equations is expressed as a matrix eigenvalue problem and solved using a standard eigensolver. There is different eigenvalue solver in different software that can be utilized to solve the generalized eigenvalue problem. The algorithms used for computing the eigenvalues are the Cholesky factorization method or the QZ algorithm which is based on the generalized Schur decomposition. In general, the two algorithms return the result. The QZ algorithm is found to be stable in problems involving illconditioned matrices and is the main algorithm in the eigenvalue solver.
With the plate having vertical simply supported edges, we denote the boundary conditions using two letters with the first letter denoting the bottom condition and the second letter denoting the top condition. For example, if the letters C, S and F are used to denote clamped, simply supported and free conditions then a plate with clamped bottom and simply supported top is denoted by CS. In all the cases, $\alpha=\frac{\pi}{a}(n=1)$ and the Poisson's ration υ is 0.3. The proposed PS method is first applied to obtain the fundamental frequency of vibration of a rectangular plate under selfweight. The algebraic eigenvalue problem obtained is solved for the eigenvalues using the eigensolver of MATLAB. The program was run for different values of N until we get the frequency parameter values correct to six decimal places. The results of the PS method obtained for N=25 (26 collocation points) are presented. The frequency parameter (K) for the CF, CS and SF plates are presented in Tables 13 respectively. The results obtained using the PS method are compared with the results obtained using the semianalytic Levyintegration method [5]. In the tables γ=0 corresponds to the case when selfweight is absent.
The weight parameter γ is varied over the values 0, 7, 20 and 100 in Table 1; 0, 10, 50, 100, 200 in Table 2 and 0, 10, 50 and 100 in Table 3 with the width parameter a varying over the values 0.2, 0.5, 1 and 2 in Table 1, Table 2 and an additional value of a=10 in Table 3. The variation in the frequency parameter with variation in a and $\gamma$ is reflected in the tables.
It is observed that for fixed width as the weight increases, the frequency decreases until the plate buckles statically. In the tables, the frequency parameter values marked "" means that the plate has already buckled. The results obtained using the PS method are almost same as those obtained using the semi analytic method. In the semi analytic method [5] the method of solution is tedious as twopoint boundary value problem is first converted into two initial value problems and then the bisection algorithm was utilized to solve the resulting nonlinear equations. In the present work, the PS method efficiently obtains the frequency values with relatively good accuracy that is comparable with the results of reference [5]. In the second instance, the PS method is used to analyze the free transverse vibration of a rectangular plate resting on a Winkler foundation. Here the weight is absent and the foundation parameter (K_{w}) takes the values 100, 300 with the ratio (a) of width to height taking the values 1.0 and 2.0. The values of the frequency parameter (K^{2}) obtained using the PS method with N=20 (21 collocation points) are given in Table 4 for SC and SS plates. It is observed that the frequency parameter values are greater in SC plate case than in the SF plate case. Further it increases with the increasing value of the foundation modulus parameter and aspects ratio.
The results obtained are compared with those obtained using the differential transform method [15]. A comparison of the results obtained using the PS method show that the formulation can provide highly accurate results in a simple and efficient manner.
Table 1. Frequency parameter K for the CF plate
γ 


0 
7 
20 
100 

a 
Ref. [5] 
PS method 
Ref. [5] 
PS method 
Ref. [5] 
PS method 
Ref. [5] 
PS method 
0.2 
15.740 
15.739725 
15.739 
15.739034 
15.738 
15.737751 
15.729 
15.729818 
0.5 
6.458 
6.457703 
6.448 
6.447648 
6.429 
6.428792 
6.306 
6.306385 
1 
3.562 
3.561932 
3.501 
3.501093 
3.378 
3.378442 
1.388 
1.387723 
2 
2.388 
2.388276 
2.158 
2.158045 
1.092 
1.091664 
 
 
Table 2. Frequency parameter K for the CS plate
γ 


0 
7 
20 
100 

a 
Ref. [5] 
PS method 
Ref. [5] 
PS method 
Ref. [5] 
PS method 
Ref. [5] 
PS method 
0.2 
16.048 
16.048178 
16.045 
16.045315 
16.045 
16.033734 
16.014 
16.018945 
0.5 
7.189 
7.188482 
7.158 
7.157805 
7.027 
7.026795 
6.839 
6.838520 
1 
4.863 
4.862748 
4.764 
4.764093 
4.265 
4.264998 
2.901 
2.901079 
2 
4.163 
4.163142 
4.004 
4.003524 
2.933 
2.932533 
 
 
10 
3.936 
3.935991 
3.745 
3.745056 
1.944 
1.944364 
 
 
Table 3. Frequency parameter K for the SF plate
$\boldsymbol{\gamma}$ 


0 
7 
20 
100 

a 
Ref. [5] 
PS method 
Ref. [5] 
PS method 
Ref. [5] 
PS method 
Ref. [5] 
PS method 
0.2 
15.735 
15.734484 
15.734 
15.733494 
15.730 
15.729524 
15.714 
15.714396 
0.5 
6.419 
6.418461 
6.404 
6.403488 
6.341 
6.340647 
4.491 
4.491206 
1 
3.148 
3.418265 
3.316 
3.315907 
2.623 
2.623214 
 
 
2 
2.008 
2.008405 
0.958 
0.958176 
 
 
 
 
Table 4. The first three frequency parameter values (K^{2}) for plates on Winkler foundation

SC Plate 
SF Plate 

Kw 
a=1 
a=2 
a=1 
a=2 


Ref. [15] 
PSM 
Ref. [15] 
PSM 
Ref. [15] 
PSM 
Ref. [15] 
PSM 
100 
25.6739 
25.673886 
52.6330 
52.632980 
15.3795 
15.379479 
42.3930 
42.392972 
59.4928 
59.492822 
86.7130 
86.71300 
29.5028 
29.502790 
59.9061 
59.906048 

113.6690 
113.668826 
141.2000 
141.200112 
62.6637 
62.663668 
95.0114 
9.011427 

300 
29.3112 
29.311233 
54.4998 
54.499822 
20.8933 
20.893262 
44.6897 
44.689641 
61.1506 
61.150600 
87.8586 
87.8586700 
32.7172 
32.717192 
61.5527 
61.552698 

114.5450 
114.545196 
141.9070 
141.906559 
64.2397 
64.239671 
96.0581 
96.058166 

500 
32.5446 62.7646 115.4150 
32.544561 62.764607 115.414913 
56.3048 88.9896 142.6100 
56.304801 88.989583 142.609507 
25.2295 35.6429 65.7779 
25.229514 35.642876 65.777924 
46.8739 63.1564 97.0936 
46.873917 63.15643 97.09362 
The pseudospectral method is employed to analyze the vibration of a rectangular plate under selfweight and is also further used to analyze free transverse vibration of rectangular plates of uniform thickness resting on a Winkler foundation. The two opposite edges of the plate are assumed to be simply supported and different combinations of clamped, free and simply supported conditions are taken on the other two edges. The accuracy of the method is confirmed via comparison studies and the results obtained show the effectiveness of the method for free vibration studies.
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