Study on chaotic characteristics of heterogeneous physical double pendulum

To address the impact of materials with uneven mass distribution on the motion of double pendulum systems in engineering practice, a heterogeneous double pendulum model was established based on the homogeneous physical double pendulum model. This model incorporated variables, such as the position of the center of mass and the moment of inertia of the pendulum. To further explore the chaotic characteristics of the system, the heterogeneous double pendulum system was approximated from a Hamilton system to a quasi-Hamilton system. The dynamics equation of the double pendulum system was obtained using the Euler-Lagrange equation of the second kind. The energy threshold for Smale horseshoe chaos in the quasi-Hamiltonian system was determined using Melnikov method with two degrees of freedom, serving as the chaos condition for the Hamiltonian system. Through programming in Matlab, the correctness of the chaos condition was verified using numerical methods such as maximum Lyapunov exponential diagram, bifurcation diagram and Poincare section diagram. The influence of each parameter on the system’s motion state and action mechanism was analyzed in detail. The results show that the chaos threshold of heterogeneous double pendulums depends on various factors, including the initial energy of the double pendulums, the position of the center of mass of the first pendulums and the ratio of the moment of inertia of the two pendulums. With different values for each parameter, the system changes from a regular motion state with a chaos threshold to an irregular and complex motion state without a chaos threshold. The reasons for the system alternating between chaotic and quasi-periodic states were explained, and theoretical predictions were validated. The differences between the theoretical threshold and actual numerical simulation results were explained when the center of mass of the first pendulum, the mass ratio of the two pendulums, and the ratio of moment of inertia of the two pendulums were set to their limit values. On this basis, the relationship between parameter values and the applicability of Melnikov method were further explored, and the parameters under which Melnikov method was no longer applicable were discussed by numerical simulation classification.