Vibration of Circular Plate with Multiple Eccentric Circular Perforations by the Rayleigh Ritz Method

Abstract

The free vibration of a circular plate with multiple perforations is analyzed by using the Rayleigh Ritz Method. Admissible functions are assumed to be separable functions of radial and tangential coordinates. Trigonometric functions are assumed in the circumferential direction. The radial shape functions are the Boundary Characteristic Orthogonal Polynomials generated following the Gram-Schmidt recurrence scheme. The assumed functions are used to estimate the kinetic and the potential energies of the plate depending on the number and the position of the perforations. The eigen-values, representing the dimensionless natural frequencies, are compared with the results obtained using Bessel Functions, where the exact solution is available. Moreover, the eigen-vectors, which are the unknown coefficients of the Rayleigh Ritz Method, are used to present the mode shapes of the plate. In order to validate the analytical results of the plates with multiple perforations, experimental investigations are also performed. Two unique case studies that are not addressed in the existing literature are considered. The results of the Rayleigh Ritz Method are found to be in good agreement with those from the experiments. Although the method presented can be employed in the vibration analysis of plates with different boundary conditions and shapes of the perforations, circular perforations that are free on the edges are studied in this paper. The results are presented in terms of dimensionless frequencies and mode shapes.

1. INTRODUCTION

There are many applications where there is a need for circular plates with multiple perforations. One such example would be the turbo screen used to protect the turbo charger in locomotives. The turbo screen has an important role in protecting the turbo charger from broken engine parts. Potential hazard could be caused by parts of piston rings or broken valves. The perforations in the turbo screen allow the flow of exhaust gas from the engine to the turbine section of the turbo charger, thereby running the compressor of the turbo charger. Circular plates could be classified into three groups according to the location and size of the perforations: plates may have no perforations (full), one centered perforation (annular), or plates with multiple perforations. Indeed, there are many other engineering examples of perforated diaphragms such as automobile wheels, which are perforated not only for aesthetics and weight reduction but also for the air cooling of brake shoes. [1,2]. Of these three types of plates, full and annular plates have been extensively studied by many researchers. These two have been frequently used in the construction of aeronautical structures, ship structures, and in several other industrial areas [3,4].

Compared with the studies of annular plates, the studies on the plates with multiple eccentric perforations are rare. There are some studies where Finite Element Method [4] and Bessel Functions satisfying the boundary conditions [5,6] have been used for the free vibration analysis of plates with eccentric perforations. The Finite Element Method has been used [7] to study the effects of the eccentricity, the size of the perforation, and the boundary conditions on the natural frequencies and the mode shapes of a circular plate with eccentric perforation. Polynomials are employed as radial admissible functions in the Rayleigh Ritz Method in order to determine the frequencies of circular plates with circular [8,9] and rectangular [10] perforations. The Boundary Element Method (BEM) has been used to study the natural vibration of circular plates with various boundary conditions such as clamped and simply-supported [11,12]. Also the Boundary Integral Equation Method (BIEM) has been proposed as a semi-analytical solution to study the free vibration of a circular plate with circular perforations [13,14,15,16].

The Rayleigh Ritz Method has extensively gained the attention of researchers [17,18] because the assumed deflection functions are required to satisfy the geometrical boundary conditions only. The Boundary Characteristic Orthogonal Polynomials, which have been proposed by Bhat [19] and used by Rajalingham et al. [20,21] and Chakraverthy et al. [22], not only provide diagonalized mass matrix, but also satisfy the geometrical boundary conditions. In the present study, the natural frequencies and mode shapes of plates with multiple circular perforations are calculated, using the product of the Boundary Characteristic Orthogonal Polynomials and the trigonometric functions as the assumed deflection functions in the Rayleigh Ritz Method. The results are first validated for clamped circular and clamped annular plates without perforations by comparing the results with the Bessel functions solution. The results for the perforated plates are experimentally validated. Two cases that are not found elsewhere in the literature are studied:

i) A clamped circular plate with one circular perforation at the center and 8 equally spaced circular perforations close to the outer edge.

ii) A clamped circular plate with one circular perforation at the center, 8 equally spaced circular perforations close to the center and sixteen 16 equally spaced circular perforations close to the outer edge.

2. METHODOLOGY

Denoting W(r,θ) as the deflection shape expressed in the polar coordinates and ω as the natural frequency of the plate, the maximum kinetic energy Tmax can be written as:

(1)

where, ρ is the mass density per unit volume, h is the thickness of the plate, and A represents the area over the plate. Also the maximum strain energy of the deformed circular plate Umax is given by:

(2)

where ν is Poisson's ratio, and D the flexural rigidity is:

(3)

In the equation above, E is Young's modulus. Assuming the deflection function as:

(4)

where the mth Boundary Characteristic Orthogonal Polynomial in the radial direction, and substituting it into the preceding expressions for maximum kinetic and potential energies results in:

(5)

(6)

where

(7)

The Rayleigh quotient is given by

(8)

where . Applying the condition of stationarity of the natural frequencies with respect to the arbitrary constants Aij and Bij, we have:

(9)