A Modified Fourier–Ritz Formulation for Vibration Analysis of Arbitrarily Restrained Rectangular Plate with Cutouts

Yiming Liu,Guoyong Jin,Zhuang Lin,Hu Ding,Sensen Yan

doi:10.1155/2018/8759074

Abstract

A modified Fourier–Ritz method is developed for the flexural and in-plane vibration analysis of plates with two rectangular cutouts with arbitrary boundary conditions, aiming to provide a unified solving process for cases that the plate has various locations or sizes of cutout, and different kinds of boundary conditions. Under the current framework, modifying the position of the cutout or the boundary conditions of the plate is just as changing the geometric parameters of the plate, and there is no need to change the solution procedures. The arbitrary boundary conditions can be obtained by setting the stiffness constant of the boundary springs which are fixed uniformly along the edges of the plate at proper values. The strain and kinetic energy functions of a plate with rectangular cutout are derived in detail. The convergence and accuracy of the present method are demonstrated by comparing the present results with those obtained from the FEM software. In this paper, free in-plane and flexural vibration characteristics of the plate with rectangular cutout under general boundary conditions are studied. From the results, it can be found that the geometric parameters and positions of the cutout and the boundary conditions of the plate will obviously influence the natural vibration characteristics of the structures.

Highlights

As one of the most commonly used structures, the rectangular plate has been in study for a long time
It is well known that, in a vibrating thin plate, there exist three types of modes: bending mode, longitudinal mode, and shear mode. e bending mode is referred to the out-of-plane vibration, while the longitudinal mode and shear mode are referred to the in-plane vibration [1]. e flexural vibration of thin plates with various boundary conditions has been attracting researchers for a long time
For the plates under arbitrary boundary conditions, the displacement functions expressed in the form of standard Fourier cosine or sine series can avoid this problem, but discontinuities of the displacement functions and their derivatives will potentially exist along the boundaries of the rectangular plates. is kind of problem is demonstrated in detail [27,28,29,30,31,32,33]

Summary

Introduction

As one of the most commonly used structures, the rectangular plate has been in study for a long time. E flexural vibration of thin plates with various boundary conditions has been attracting researchers for a long time. Gorman [4, 5] and Mochida [8] used the superposition-Galerkin method to study the vibration characteristics of rectangular plates under free boundary condition. Gorman [13, 14] analyzed the free in-plane vibration characteristics of plates with different kinds of boundary conditions by the superposition method. Laura et al [23] studied the out-of-plane vibration of the plate with rectangular cutout with simple supported boundary condition. Shufrin and Esienberger [25] used the multiterm extended Kantorovich method to study the in-plane vibration of the plate with rectangular cutouts. E natural frequencies and mode results of plates with rectangular cutouts are obtained by using the Rayleigh–Ritz procedure

Theoretical Formulations

B61 B62 B66 D61 D62 D66

Validation and Discussion

Conclusions