Effect of nonhomogenous material properties on transverse vibrations of elastic cantilevers

In this paper, the method of the influence functions and the method of partial discretization are proposed to solve the boundary-value problem of free transverse vibrations of a nonhomogenous cantilever with a concentrated mass attached to the free end. In order to demonstrate the possibilities of the methods, the case of a cantilever in the form of a sharp cone, a frustum of a cone, and a linear wedge made of two different materials is treated in detail. The general characteristic equations which allow one to take into consideration the nonhomogeneous material properties and the cross-sectional geometry of cantilever are introduced. The expressions for the first three terms of the characteristic series are obtained in closed form using the method of Cauchy influence functions. The results of calculations of the first two frequencies of free transverse vibrations are presented for selected material combinations and various cantilever geometries. There is very good agreement between the numerical results obtained using the method of partial discretization and the analytical results obtained using the method of influence functions. The high accuracy of the proposed methods and agreement with known theoretical data and with the experimental results obtained by the authors in the homogeneous cantilever case are shown. Presented at the International Conference on the Theory of Machines and Mechanisms, Poland, 1996. Published in Prikladnaya Mekhanika, Vol. 35, No. 6, pp. 103–110, June, 1999.