MODES OF NON-HOMOGENEOUS DAMPED BEAMS ON A WINKLER-TYPE ELASTIC LAYER

Abstract

The stepped beam considered here can be used to model many parts in mechanical assemblies, such as crankshafts, gearboxes, etc., where a beam is submerged into some medium. In such cases, the beam consists of two parts with different material properties (mass, stiffness). It is externally non-homogenously damped and it rests on a Winkler elastic layer. The elastic layer is represented by continuously distributed springs along the stepped beam. Using Newton’s second law and classical elasticity theory, a system of partial differential equations of motion is derived. Different boundary conditions are applied. In order to determine eigenvalues and eigenvectors, we exploit the finite difference method for solving vibration problems of stepped beams. Results obtained using the finite difference method are compared with the analytical results, which are obtained using the Bernoulli-Fourier method. It is determined that the difference of the values obtained using two different methods is negligibly slight. For the analytical solution, the complete derivation of the characteristic equation for the clamped-clamped boundary conditions is given. For the other boundary conditions, (pinned-pinned, clamped-free, free-free) characteristic equations are given without derivation since the procedure is similar. Overdamped and underdamped vibration is investigated. The influence of the stiffness of the Winkler layer on Eigen characteristics is discussed.