Steady State Response and Stability of an Elastically Restrained Tapered Beam

An analytical method for the study of the nonlinear forced vibrations and their stabilities of an elastically restrained tapered cantilever beam due to a direct periodic excitation is developed. The method of harmonic balance is used to study the steady state frequency response of the beam system for different values of physical parameters such as the root translational and rotational stiffness and the beam taper ratio. Results are presented for the first three modes of vibration. The stability of the frequency response for some selected values of the physical parameters is investigated, i.e. the regions on the frequency response curves at which the solution may bifurcate and then culminate into chaos. The qualitative features of the solutions are studied and identified using phase plane, Poincare maps and Fast Fourier Transform. The results are presented, discussed and conclusions on the elastically restrained tapered beam nonlinear dynamics are drawn.