Non-linear Free Vibration Analysis of Stepped Beams under Gravity by Transfer Matrix Method

Abstract

The non-linear free vibrations of stepped beams under gravity are analyzed by using the transfer matrix method. By using the harmonic balance method, in which a constant term and a sinusoidal harmonic term are assumed as time functions, the governing partial differential equation reduces to simultaneous ordinary non-linear differential equations for the corresponding spatial functions. The transfer matrices are derived from these equations. The numerical results for simply supported and clamped, one-stepped thickness beams with rectangular cross-section are presented and the effects of gravity, the beam geometry, the concentrated mass, and the boundary conditions on the non-linear vibration characteristics are discussed.