Periodic motions on bifurcation trees in an inverted pendulum with a periodically moving base

Abstract

This paper studies periodic motions of an inverted pendulum with a periodically moving base. A bifurcation tree of period-1 to period-8 motions is predicted through a discrete implicit mapping method. The stable and unstable periodic motions on the bifurcation tree are obtained semi-analytically, and the corresponding stability and bifurcation of the periodic motions are determined by eigenvalue analysis. Numerical simulations are completed for illustration of motion complexity in the inverted pendulum with a periodically moving base. The numerical and analytical results are presented for comparison. Frequency–amplitude characteristics of the bifurcation tree are presented through the finite Fourier series analysis. Through such results, one can better understand the building vibration during earthquake. In addition, stabilization of the inverted pendulum can be based on the periodically moving base.