Analysis of the Jump and Bifurcation Phenomena in a Nonlinear Cantilever Beam under the Forced Vibration

In this study the vibration of a cantilever beam with geometric nonlinearity was focused with respect to the jump and bifurcation phenomena. Firstly, the partial differential equation of the cantilever beam with geometric nonlinearity was presented. Next, based on the first mode shape of the linear beam, the governing nonlinear ordinary differential equation (ODE) was achieved. This nonlinear ODE was solved with the harmonic balance method. Moreover, the super harmonic balance was utilized to analyze the super harmonic behavior of the nonlinear system. The achieved frequency response was used to investigate the jump phenomenon and instability of system’s amplitude versus the excitation frequency. Whereas the system was sensitive in one third frequency of resonance, the jump, bifurcation and chaos phenomenon were analyzed with super harmonic method. Finally the numerical forth order Runge-Kutta method was used for the evaluation. The comparison between the result of presented method and the numerical results showed a precise agreement.