Free Vibration Analysis of Rectangular Cantilever Plates Using the Hyperbolic-Trigonometric Series

Abstract

The purpose of this paper is to develop a numerical analytical method for an accurate solution to the problem on frequencies and mode shapes of a rectangular cantilever plate. The problem is reduced to an infinite system of linear algebraic equations relative to ratios of trigonometric series, which contains vibration frequency as a parameter. The core of the method is using two hyperbolic-trigonometric series by two coordinates with six undetermined ratios. Functional series are subject to the main differential equation of vibrations and undetermined ratios are obtained from boundary conditions of the problem. Symmetric and asymmetric mode shapes are considered separately. The symmetric solution required the introduction of an additional function to compensate free terms in the decomposition of hyperbolic functions into Fourier series. The infinite system relative to six successions of undetermined ratios was reduced to a homogeneous infinite system relative to one (basic) succession of ratios. The iterative process of its solution at the chosen vibration frequency was presented. A compact resolving system of homogeneous linear equations was obtained, relative to basic ratios of mode shapes of a rectangular cantilever plate. The search for natural frequencies was done with simple exhaustion of a frequency parameter up to the values, at which the basic ratios become invariable, starting with some iteration. The simplicity of the algorithm and the resolving system allows fast obtaining natural frequencies with high accuracy. The calculation accuracy is analyzed. The results in this study are well coincided with the results of the authors, who fulfilled all the problem's conditions most accurately. The obtained results can be used to do highly accurate dynamic calculations in nanoengineering. The calculation accuracy with this algorithm can be enhanced by increasing the number of terms in series, the number of iterations and the size of the mantissa.