Bifurcation in a Nonlinear Autoparametric System Using Experimental and Numerical Investigations

Experimental and numerical investigations are carried out on an autoparametric system consisting of a composite pendulum attached to a harmonically base excited mass-spring subsystem. The dynamic behavior of such a mechanical system is governedby a set of coupled nonlinear equations with periodic parameters. Particular attention is paid to the dynamic behavior of the pendulum. The periodic doubling bifurcation of the pendulum is determined from the semi-trivial solution of the linearized equations using two methods: a trigonometric approximation of the solution and a symbolic computation of the Floquet transition matrix based on Chebyshev polynominal expansions. The set of nonlinear differential equations is also integrated with respect to time using a finite difference scheme and the motion of the pendulum is analyzed via phase-plane portraits and Poincare maps. The predicted results are experimentally validated through an experimental set-up equipped with an optoelectronic set sensor that is used to measure the angular displacementof the pendulum. Period doublingand chaotic motions are observed.