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.
Highlights
The problem of free vibrations of rectangular cantilever plates does not have an exact closed-form solution
This paper offers an effective method of determining natural frequencies and mode shapes of a rectangular cantilever Kirchhoff plate using two trigonometric series, which contain hyperbolic functions by another coordinate
The problem of finding natural frequencies is brought to simple exhaustion of frequencies in a resolution homogeneous reduced system of linear algebraic equations relative to one sequence of trigonometric series ratios
Summary
The problem of free vibrations of rectangular cantilever plates does not have an exact closed-form solution. Some authors use methods that allow finding an approximate solution using a simple computational procedure, others tend to satisfy all conditions of the problem more precisely, which causes considerable difficulties of analytical and computational nature. The numerical results of such computations often differ from each other. A rectangular cantilever plate is a computational scheme for many elements of different constructions, devices and plants. Requires highprecision dynamic computations of these elements. The goal of this work is to find an accurate solution to the problem, which is an infinite system of linear algebraic equations relative to the ratios of hyperbolictrigonometric function series satisfying all conditions of the problem. Selection of frequencies that give nontrivial solutions of a reduced system and an increase in the dimension of this system allow obtaining an accurate solution within the limit
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