Pseudospectral Method for Free Vibration Analysis of Vertically Standing Plates

Vertically-oriented 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 mid-plane 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, uniform-thickness 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 well-studied, 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 semi-analytical 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 non-constant coefficients. The two-point 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 [11-13]; 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 semi-analytical method [5] and differential transform method [15]. The motivation for utilizing this approach lies in its numerical stability and flexible implementation for vibration analysis.

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=2y-1. 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

$\begin{array}{r}T_k(t)=\cos \left(k \cos ^{-1} t\right), \quad-1 \leq t \leq 1 \\ T_k(t)=\cos (k \theta) \text { where } \theta=\cos ^{-1} t\end{array}$

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 Chebyshev-Gauss-Lobatto (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

$\frac{d^n}{d t^n} T_k(t)=( \pm 1)^{k+n} \prod_{p=0}^{n-1} \frac{k^2-p^2}{2 p+1}$, at $\mathrm{t}= \pm 1$     (12)

In the collocation framework, we select N-3 points in (0, π) and require Y(θ) to satisfy Eq. (2) at these N-3 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

$t_i=\cos \left(\frac{(N-i) \pi}{N}\right), i=1,2, \ldots, N-3$     (13)

that corresponds to the points

$\theta_i=\frac{(N-i) \pi}{N}, i=1,2, \ldots, N-3$ 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 N-3 collocation points given by Eq. (14) yielding N-4 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 ill-conditioned matrices and is the main algorithm in the eigenvalue solver.