Non-linear Dynamics and Chaos Control of a Physical Pendulum with Vibrating and Rotating Support

The dynamic behavior of a physical pendulum system of which the support is subjected to both rotation and vertical vibration are studied in this paper. Both analytical and computational results are employed to obtain the characteristics of the system. By using Lyapunov's direct method the conditions of stability of the relative equilibrium position can be determined. Melnikov's method is applied to identify the existence of chaotic motion. The incremental harmonic balance method is used to "nd the stable and unstable periodic solutions for the strong non-linear system. By applying various numerical results such as phase portrait, PioncareH map, time history and power spectrum analysis, a variety of the periodic solutions and the phenomena of the chaotic motion can be presented. The e!ects of the changes of parameters in the system could be found in the bifurcation and parametric diagrams. Further, chaotic motion can be veri"ed by using Lyapunov exponent and Lyapunov dimension. The global analysis of basin boundary and fractal structure are observed by the modi"ed interpolated cell mapping method. Besides, non-feedback control, delayed feedback control, adaptive control, and variable structure control are used to control the chaos e!ectively. ( 2001 Academic Press