Mathematical Problems in Engineering

Volume 2019, Article ID 4687082, 20 pages

https://doi.org/10.1155/2019/4687082

## Free In-Plane Vibrations of Orthotropic Rectangular Plates by Using an Accurate Solution

^{1}Naval Research Academy, Beijing 100161, China^{2}State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha 410083, China^{3}Beijing Aerospace Technology Institute, China Aerospace Science & Industry Corp, Beijing 100074, China

Correspondence should be addressed to Dong Shao; moc.kooltuo@natasoahsgnod

Received 29 May 2019; Revised 3 August 2019; Accepted 16 October 2019; Published 11 November 2019

Academic Editor: Arkadiusz Zak

Copyright © 2019 Yuan Cao et al. 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

Many numerical methods have been developed for in-plane vibration of orthotropic rectangular plates with various boundary conditions; however, the exact results for such structures with elastic boundary conditions are very scarce. Therefore, the object of this paper is to present an accurate solution for free in-plane vibration of orthotropic rectangular plates with various boundary conditions by the method of reverberation ray matrix (MRRM) and improved golden section search (IGSS) algorithm. The boundary condition studied in this paper is defined as that a set of opposite edges is with one kind of simply supported boundary conditions, while the other set is with any kind of classical and general elastic boundary conditions or their combination. Its accuracy, reliability, and efficiency are verified by some numerical examples where the results are compared with other exact solutions in the published literature and the FEA results based on the ABAQUS software. Finally, some new accurate results for free in-plane vibration of orthotropic rectangular plates with elastic boundary conditions are examined and further can be treated as the reference data for other approximate methods or accurate solutions.

#### 1. Introduction

The orthotropic rectangular plates, as the fundamental components, have achieved a widespread use in aerospace, military, and marine industries and various engineering fields, and their structural stability and safety performance rely on the vibration characteristics. Thus, in the past decades, a lot of studies for free vibrations of rectangular plates have been carried out. Among the existing jobs, most of the researchers focus on the transverse vibrations of plates while the in-plane vibration research is very scarce. For example, Li et al. [1–4] proposed the analytical symplectic superposition method to solve free vibration solutions of rectangular plates. Zhang et al. [5, 6] presented a finite integral transform solution to determine the vibration behavior of rectangular orthotropic thin plates with different boundary conditions. Ullah et al. [7, 8] extended the integral transform method to study the buckling behavior of rectangular plates. Of course, there are many other excellent achievements, and there is no detailed discussion here. In theory, the aforementioned methods for studying transverse vibration characteristics of rectangular plates also have potential to conduct the analysis of in-plane vibration characteristics. According to the current research situation, there are relatively few related literatures on in-plane vibration characteristics of rectangular plates; the importance of which, however, has been shown in the complicated plate structures and sandwich plates [9–16]. As a matter of fact, the publications pertaining to in-plane vibrations have been increasing. In recent years, the study on in-plane vibrations has increased greatly and a lot of research results have been published.

Bardell et al. [9] studied in-plane vibrations of isotropic rectangular plates due to the lack of the related valid results at that time and provided some useful benchmark data of the in-plane frequencies of rectangular plates with simply supported, clamped, and free boundary conditions, where the Rayleigh–Ritz method is adopted. Du et al. [12] studied the in-plane vibration of isotropic rectangular plates with elastically restrained edges by the Rayleigh–Ritz method in conjunction with the Fourier series method. Gorman et al. [17] studied in-plane vibration of rectangular plates subject to completely free boundaries by extending the superposition method. Applying the variational approximation method, Seok et al. [18] studied free in-plane vibration of a cantilevered rectangular plate. Dozio [19] studied the accurate in-plane modal properties of rectangular plates using the Ritz method, where arbitrary nonuniform elastic edge restraints were considered. Singh and Muhammad [20] investigated free in-plane vibration of isotropic nonrectangular plates subjected to classical boundary conditions using the variational method. Du et al. [21] applied a generalized Fourier method to study the free vibration of the rectangular plate with different point-supported boundary conditions, which enriched the in-plane vibration of the plate. Based on the dynamic stiffness method, Nefovska-Danilovic and Petronijevic [22] carried out the in-plane free vibration and response analysis of isotropic rectangular plates, in which the numerical results were very consistent with FEM.

In the aforementioned works, the solutions by the related solving methods like the Rayleigh–Ritz method are classified as approximate solutions. Although the methods for approximate solutions have the merits of the simple solving process and high precision, considering that the in-plane natural frequencies of plates are generally in high-frequency regions, the minor errors of results may also cause the poor structure stability and safety performance. Thus, to achieve higher precision of in-plane natural frequencies is the ultimate goal and the corresponding solution is also called the exact solution. Gorman [23] presented the exact solutions for free in-plane vibration of rectangular plates whose one set of opposite edges is simply supported and the other set both clamped or both free. Xing et al. [24, 25] obtained exact solutions for free in-plane vibrations of rectangular plates employing the direct separation of variables, where SS1-SS2-SS1-SS2, SS1-SS1-SS1-SS1, SS2-SS2-SS2-SS2, SS1-C-SS1-C, SS1-F-SS1-F, SS1-SS1-SS1-C, SS1-SS2-SS1-C, SS1-SS1-SS1-F, SS1-SS2-SS1-F, SS1-C-SS1-F, and other possible classical boundary conditions were considered. Later, Xing’s [26] group obtained the exact solutions of the free in-plane vibration of orthotropic rectangular plates with SS1-C-SS1-C, SS2-F-SS2-F, SS1-C-SS2-F, SS2-SS1-SS2-C, SS2-SS2-SS2-F, and SS2-SS1-SS1-F boundary conditions. The symbols S, C, F, and E represent simply supported boundary, clamped boundary, free boundary, and elastically restrained boundary conditions, respectively. Two forms of simply supported boundary conditions are as follows [23]: SS1—displacement parallel to the edge and normal stress (*N*_{xx}) perpendicular to the edge are forbidden; SS2—shear stress (*N*_{xy}) along the edge and displacement normal to the edge are forbidden. Take boundary *x* = 0 as an example, the symbol SS1 indicates that *N*_{xx} *=* 0 and and SS2 means that *u* *=* 0 and *N*_{xy} *=* 0.

Through the above review, it can be known that the exact solutions for in-plane vibration analysis of orthotropic rectangular plates are deficient. By investigating the existing literature, it is found that researchers have been pursuing more accurate in-plane vibration results because it can better promote the development of numerical algorithms. As far as authors know, only “*comprehensive exact solutions for free in-plane vibrations of orthotropic rectangular plates*” [26] presents the exact solutions for the titled problem. However, in this paper, only the classical boundary conditions are considered. In the existing engineering applications, there are still a large number of elastic boundaries except for the classical boundary conditions. At present, numerical methods can solve many problems well, but researchers have been pursuing the exact solution. Thus, it is necessary and urgent to establish an accurate analytical model to conduct in-plane vibration analysis of orthotropic rectangular plates with elastic boundary conditions. Before that, the members of the author’s team have done some innovative work around the method of reverberation ray matrix (MRRM) in solving the exact solutions of structural vibration, such as the exact solutions of isotropic rectangular plates with elastic constraints [27]. However, it should be pointed out that the mechanical properties of orthotropic rectangular plates are superior to those of isotropic rectangular plates due to the difference of material properties. Therefore, it is necessary to study the in-plane exact solutions of orthotropic rectangular plates. This paper can be regarded as an extension of previous work.

In this paper, the method of reverberation ray matrix (MRRM) and the improved golden section search (IGSS) algorithm are adopted to solve accurate solutions for free in-plane vibration of orthotropic rectangular plates. MRRM is used to obtain the natural frequency characteristic equation and IGSS algorithm to acquire in-plane natural frequencies and modal shapes. The results are compared with existing published results and FEA results, by which the accuracy, reliability, and efficiency can be validated. On this basis, some new results for the orthotropic rectangular plates with elastic boundary conditions are presented for the first time, which may be worked as the benchmark data.

#### 2. Theoretical Formulations of Accurate Solutions

##### 2.1. Differential Equations and Boundary Conditions

Figure 1 shows an orthotropic rectangular plate characterized by length *L*_{x}, width *L*_{y,} and thickness *h* in *x*-, *y*-, and *z-*directions, respectively. According to the small deformation, the stresses are written asin whichwhere *E*_{x}(*E*_{y}) and *μ*_{x}(*μ*_{y}) are Young’s modulus and Poisson’s ratio in *x*-direction (*y*-direction), respectively, and *G*_{xy} is the shear modulus.