Nonlinear analysis of a parametrically excited cantilever beam - Effect of the tip mass on stationary response

For a parametrically excited cantilever beam the effect of the tip mass on the nonlinear characteristics of the frequency-response is theoretically presented. The equation of motion governing the system is formulated by Hamilton's principle, taking into account the inertia and curvature nonlinearities and a quadratic damping effect of the beam. Using the method of multiple scales and center manifold theory, the bifurcation points of the frequency-response curve are analyzed. It follows that there are two transcritical bifurcations, and in addition to these bifurcations there are two saddle-node bifurcations, in the cases when the tip mass is relatively light and heavy, respectively. Experiments are also performed and the results show good qualitative agreement with the theoretical ones.