Shock and Vibration

Volume 2018, Article ID 1071830, 8 pages

https://doi.org/10.1155/2018/1071830

## A Dynamic Coefficient Matrix Method for the Free Vibration of Thin Rectangular Isotropic Plates

Department of Aerospace Engineering, Ryerson University, Toronto, Canada

Correspondence should be addressed to Supun Jayasinghe; ac.nosreyr@nisayajh

Received 17 February 2018; Revised 17 May 2018; Accepted 3 June 2018; Published 11 July 2018

Academic Editor: Lorenzo Dozio

Copyright © 2018 Supun Jayasinghe and Seyed M. Hashemi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The free flexural vibration of thin rectangular plates is revisited. A new, quasi-exact solution to the governing differential equation is formed by following a unique method of decomposing the governing equation into two beam-like expressions. Using the proposed quasi-exact solution, a Dynamic Coefficient Matrix (DCM) method is formed and used to investigate the free lateral vibration of a rectangular thin plate, subjected to various boundary conditions. Exploiting a special code written on MATLAB®, the flexural natural frequencies of the plate are found by sweeping the frequency domain in search of specific frequencies that yield a zero determinant. Results are validated extensively both by the limited exact results available in the open literature and by numerical studies using ANSYS® and in-house conventional FEM programs using both 12- and 16-DOF plate elements. The accuracy of all methods for lateral free vibration analysis is assessed and critically examined through benchmark solutions. It is envisioned that the proposed quasi-exact solution and the DCM method will allow engineers to more conveniently investigate the vibration behaviour of two-dimensional structural components during the preliminary design stages, before a detailed design begins.

#### 1. Introduction

Many vibrating airframe structural components could be modelled as thin plates. Not only that do these structural elements transmit various internal and external loads that may affect their stiffness but they are also frequently in close proximity to vibrating components such as engines. Therefore, it is of utmost importance to device and develop solution techniques to study the vibrational characteristics of these structures during preliminary design stages. Such vibrational analyses would allow the designers to investigate the effects of various boundary conditions the structural elements would be subjected to during its operation and the vibrational characteristics of the component before progressing to advanced stages of design. Using these results designers could alter the geometry or the materials used to avoid resonance and gain a favourable outcome.

Among the many methods available for vibration analysis, the analytical and semianalytical methods yield the highest accuracy but one major hurdle in using these methods is that they require the closed form solution to the governing partial differential equation. This can be a very tedious process if at all a tractable one. To circumvent this problem, many simplifying assumptions have been incorporated into the existing exact methods and as a result they exhibit many limitations. Having lost their generality, these exact methods are then only applicable to specific plate shapes, geometries, and those subjected to certain boundary conditions.

The orthogonality, completeness, and stability of Fourier series expansions have resulted in their frequent application to plate vibration problems [1]. The Navier [2] and Levy methods [3, 4] are two of the most common analytical procedures available for plate vibration analysis that incorporate such Fourier series expansions, where the former exploits a double Fourier series to solve the governing differential equation, the latter is based on a single Fourier series. However, both methods have a common drawback in that they are only applicable to plates having at least two simply supported boundaries. In addition, the Levy method is also limited to rectangular shaped plate configurations and is incapable of taking into account the effects of bending-twisting coupling. In addition to the above weaknesses, all methods that are based on conventional Fourier series expansions consist of a convergence problem along the boundaries arising as a result of discontinuities in displacement and its derivatives [1]. Therefore, both of these methods are unsuitable for most aerospace applications as they could only tackle simple and special cases. In order to overcome the discontinuity in displacement and its derivatives along the boundaries, the Improved Fourier Series Method (IFSM) [5] was later proposed. Although IFSM possesses a higher rate of convergence and is more readily applicable to a host of plate configurations and boundary types, it is still inadequate to study problems comprising material and geometric nonlinearity.

The Rayleigh-Ritz method is another very popular exact method that has been exploited by many researchers in the past. It was first introduced by Rayleigh [6] and later improved by Ritz [7] by assuming a set of admissible trial functions, each of which had independent amplitude coefficients; thus, it is termed the Rayleigh-Ritz method or Ritz method. Young [8] and Warburton [9] used the Ritz method to study the vibration behaviour of rectangular plates. Later, Vijayakumar and Ramaiah [10] studied the vibration of clamped square plates using the Rayleigh-Ritz method. The flexural vibration of simply supported rectangular plates was investigated by Dickinson [11, 12] using Rayleigh’s method. One of the most comprehensive studies on thin isotropic rectangular plate vibration was carried out by Leissa [13, 14] using the Rayleigh-Ritz method. Warburton [15] later extended the Rayleigh-Ritz method for the response calculation of a damped rectangular plate. The vibration of rectangular plates with elastically restrained edges was studied by Warburton and Edney [16]. The Rayleigh-Ritz method was again used to study the vibration of rectangular plates using plate characteristic equations as shape function by Rajalingam et al. [17]. However, the Ritz method in general is based on the weak form of the governing equations and is only applicable to self-adjoint problems. Furthermore, the choice of test functions in formulating the weak form is restricted to the approximation functions and it is required that the test and approximation functions are defined across the full domain of the problem, which is a major disadvantage.

The Galerkin method is also an analytical method which falls under the category of indirect classical variational methods. The Galerkin method has also been extensively used by researchers around the world. Although being somewhat similar in nature to the Rayleigh-Ritz method and belonging to the wide group of weighted residual methods, there are some distinct differences between the two techniques. Unlike the Rayleigh-Ritz method the Galerkin method commences with the weighted integral equations that are not comprised of boundary conditions. Thus, comparatively, the Galerkin method demands higher order approximation functions. Secondly, the Galerkin method does not require the system to be self-adjoint. But both methods take the test and approximation functions to be equivalent. Among many who exploited the Galerkin method for plate vibration analysis purposes, the transverse vibration of a rectangular plate was studies by Galin [18]. Munakata [19] used the Galerkin method to investigate the vibration and elastic stability of a rectangular plate clamped at its four edges. Aynola [20] and Stanisic [21] also studied the vibration behaviour of rectangular plates using the Galerkin method. Laura and Saffell [22] investigated the small-amplitude vibration of clamped rectangular plates. Later Laura and Duran [23] applied the Galerkin method to determine the vibration characteristics of a clamped rectangular plate subjected to forced vibration. Nevertheless, one of the biggest drawbacks associated with classical variation methods in general such as Rayleigh-Ritz and Galerkin methods is the difficulty involved in accurately developing the approximating functions for arbitrary domains. This difficulty associated with constructing the arbitrary test and approximate functions that should satisfy essential edge conditions, smoothness levels, linear independence, and completeness and continuity conditions is a massive limiting factor and the complicatedness of the problem becomes even more severe in magnitude for difficult geometries commonly found in most aerospace structures. Therefore, the lack of a credible method to formulate proper approximation functions for a specific geometry drastically reduces the convergence quality and applicability of classical variation methods.

The method of superposition is also a very powerful approximate analytical method that has been used extensively by many researchers in the past to obtain highly accurate results for problems involving plate vibrations. It was developed by Gorman [24] who utilised it to analyse the vibrational behaviour of thin isotropic rectangular plates. In this method, the plate is divided into a number of subsystems, termed building blocks, under different boundary conditions compared to the global system, and subjected to a distributed force, moment, rotation, and translation [24]. The steady-state response of each subsystem is then superimposed. Unlike most other exact methods, this method is applicable to a variety of plate types, which include orthotropic, hybrid, and laminated plates. The superposition technique also allows for the application of various classical and nonclassical boundary conditions as well as loading configurations and is readily applicable to thin plates, thick Mindlin plates, transverse shear deformable laminated plates, and open cylindrical shells. Furthermore, throughout the entire domain of the plate, the governing differential equations are satisfied exactly by all the solutions [24]. Gorman and Sharma [25] used the superposition method to conduct a free vibration analysis of rectangular plates. A free vibration analysis of cantilevered plates was also carried out by Gorman [26] using the superposition method. Later, Gorman [27] also conducted a study on the free vibration analysis of completely free rectangular plates using the superposition-Galerkin method. However, the main problem with the method of superposition is that, for mixed boundary types, it has not been verified yet if the results yielded are an upper bound or a lower bound. Thus, this uncertainty may well be a problem when trying to estimate the error of the results.

Among the exact methods commonly used for the vibration analysis of plates is the dynamic stiffness method (DSM), which was first presented by Kolousek [28] in the forties. Later Boscolo and Banerjee [29] applied DSM to study the vibration behaviour of plates using both the Classical Plate Theory (CPT) and the First-Order Shear Deformation Theory (FSDT). Banerjee and Papkov [30] also presented a DSM solution of a rectangular plate for the most general case. Subsequently, the free vibration of plates subjected to arbitrary boundary conditions was investigated by Liu and Banerjee [31] using a novel spectral dynamic stiffness method. However, the DSM method is cumbersome to use when applied to complex, real-life plate configurations consisting of material and geometric nonlinearity.

Thus, the objective of this work is twofold. Firstly, the authors wish to develop a new quasi-exact solution to the plate governing equation by treating the governing equation as a sum of two beam-like expressions, an approach that does not incorporate any simplifying assumptions, thus, preserving the generality of the solution and which, to the best of the authors’ knowledge, has not been explored before. The second objective will be to develop a new Dynamic Coefficient Matrix (DCM) method for the modal analyses of thin rectangular plates, having any aspect ratio, based on the new quasi-exact solution. To the best of the authors’ knowledge, the new DCM method built upon a quasi-exact Dynamic Coefficient Matrix has also not been developed and presented in the open literature. What distinguishes the DCM method from other classical exact methods is the frequency-dependent nature of the resulting system’s matrix and most importantly the fact that its generality is not compensated by any simplifying assumptions. Together, the new DCM method and the quasi-exact solution would, upon further development in the future, provide researchers with the flexibility to study the vibration of thin rectangular plates of any dimension or thin isotropic plate assemblies modelled using rectangular elements, subjected to any boundary condition.

#### 2. Theoretical Background

Consider a linearly elastic, homogeneous, isotropic, thin rectangular plate, as shown in Figure 1, having length , width , and thickness . The thickness is assumed to be much smaller compared to the other characteristic dimensions as well as the wavelength. Thus, Classical Plate Theory is used for the purpose of this study. As a result, during vibration only small deflections are assumed and the rotary inertia and shear effects are neglected.