The method of partial discretization in free vibration problems of circular plates with variable distribution of parameters

Abstract

This paper presents the results of applying the partial discretization method to study thin circular plates of varying thickness carrying concentrated inclusions. In this method, the plate with distributed or discrete-distributed mass is reduced to a discrete K-step degree system with the same rigidity function as that of the original plate. The most important task in this method is to form the influence matrix using Cauchy's influence function. This matrix is further used to obtain a few first terms of the characteristic series in the frequency parameter. The use of this method together with Bernstein's double estimators and the first three terms retained in the characteristic series reveals its rapid convergence to the exact solutions obtained by Conway and other authors. This demonstrates the rationality and efficiency of the method Keywords: axial symmetric vibrations, fixed circular plates, varying thickness, partial discretization method, characteristic equation, Cauchy influence function, Bernstein's double estimators 1. Introduction. The boundary-value problem of transverse vibrations for a fixed plane circular plate is one of the most frequently considered issues in engineering practice [4,20]. The necessity of considering the discrete masses and variable thickness of the plate leads to particular problems. Many authors showed that ignoring varying thickness, even in the case of thin plates, leads to significant errors in the calculation of natural frequencies [4,13]. In his papers [12, 13] Conway found characteristic equations using Bessel functions for particular cases where the thickness of the plate varies as a power function. This dilemma is described by means of ordinary differential equations with variable coefficients [7,9,13]. The exact solution to the vibration problems for a circular constant-thickness plate fixed around its circumference and having an additional mass at its center of symmetry was provided by Roberson [19]. The most frequent way of solving these problems is to apply analytic and numerical approximation methods such as finite elements and finite differences, as well as transfer matrix [2]. However, the lack of the exact solution does not allow us to estimate the accuracy of approximate solutions [4,20]. That is why the characteristic series method, based on Cauchy's influence function, proposed in this paper may appear attractive. It ought to be made clear that the method of spectral functions proposed by Bernstein in 1960 was used solely for analysis of systems with constant parameters, and no consideration was given to friction [1]. The authors successfully applied this method to solve the problem of critical Euler load for a vertical cantilever tapered beam [16]. In [8], we applied the characteristic series method to solve the boundary-value problem of free transverse vibrations for an elastic supported beam with variable flexural rigidity, distributed mass, elastic base, axial load, external friction, and discrete inclusions in distributed characteristics. Moreover, we pres...

Green's function in frequency analysis of circular thin plates of variable thickness

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