Nonlinear characteristics of an autoparametric vibration system

The nonlinear characteristics of an autoparametric vibration system are investigated. This system consists of a base structure and a cantilever beam with a tip mass. The dynamic equations for the system are derived using the extended Hamilton's principle. The method of multiple scales (MMS) is used to determine an approximate analytical solution of the nonlinear governing equations and, hence, analyze the stability and bifurcation of the system. Compared with the numerical simulation, the first-order MMS is not sufficient. A Lagrangian-based approach is proposed to perform a second-order analysis, which is applicable to a large class of nonlinear systems. The effects of the amplitude and frequency of the external force, damping and frequency of the attached cantilever beam, and the tip mass on the nonlinear responses of the autoparametric vibration system are determined. The results show that this system exhibits many interesting nonlinear phenomena including saturation, jumps, hysteresis and different kinds of bifurcations, such as saddle-node, supercritical pitchfork and subcritical pitchfork bifurcations. Power spectra, phase portraits and Poincare maps are employed to analyze the unstable behavior and the associated Hopf bifurcation and chaos. Depending on the application of such a system, its dynamical behaviors could be exploited or avoided.