Identification of energy dependent synchronization in coupled pendulums using semi-analytical method

The synchronization pattern of the coupled pendulums is vital in the design of pendulum energy harvesters. This understanding will help in optimizing the design of the harvester. Typically, the numerical solutions of these problems are initial-condition dependent and solved using time-consuming numerical integration techniques. Semi-analytical methods can effectively find system responses compared to nonlinear normal modes and then calculate the response. Semi-analytical methods are faster as one looks for periodic/quasiperiodic solutions. However, direct application in solving homogeneous equations using semi-analytical methods is complex. Hence, in the present paper, the synchronization nature of multi-degrees of freedom pendulum system was evaluated using multi-harmonic energy balance method (MHEBM), with different energy distribution between mass elements. Here, the proposed method is validated using a 2 DOF system with one pendulum and a 3 DOF system with two pendulums. The present study considers MHEBM with two independent time scales for solving quasi-periodic responses. Numerical integration results and analytical solutions for the linearized system (derived and solved using the method of characteristics) validate the proposed method. The ability of the system to get into in-phase and out-of-phase synchronization with different energy distributions is also demonstrated.

Automated summary

The synchronization pattern of coupled pendulums is crucial in the design of pendulum energy harvesters. This paper introduces a multi-harmonic energy balance method to evaluate the synchronization nature of the system and validates it in different scenarios.