Periodic and chaotic behaviors of a compound pendulum driven by a horizontal periodic external force

Abstract Based on the oscillation experiments of a compound pendulum driven by a horizontal periodic external force, a nonlinear dynamical model is established and studied numerically. The periodic, quasiperiodic and chaotic orbits are found in the motions of the compound pendulum in response to several driving forces. The numerical results are in qualitative agreement with those in the experiments. It is found that the chaotic attractor in the three-dimensional phase space displays a two-dimensional torus structure. On a Poincare section, the chaotic attractor consists of the relative rotating core and its trailing tails. The fractal dimension of the chaotic attractor on the Poincare section is determined by using the box-counting method. The bifurcation diagram shows that the transition of the system from periodic motion to chaos is realized by the period-doubling bifurcation. The first Feigenbaum universal constant is approximately determined in the period-doubling bifurcation process. The competition between the inherent vibration and the external driving vibration of the system is thus thought as the physical mechanism for leading to the complex phenomena such as the period-doubling bifurcation and chaos. The numerical results combining with the experimental ones can provide an intuitive and in-depth understanding of chaotic phenomena, which is of great significance for the optimization design and the stability control in the engineering technology.