Harmonic balance analysis of magnetically coupled two-degree-of-freedom bistable energy harvesters

Because a magnetically coupled two-degree-of-freedom bistable energy harvester (2-DOF MCBEH) shows the rich, complicated nonlinear behaviors caused by its coupled cubic nonlinearities, understanding the dynamics remains challenging. This paper reports and investigates the important nonlinear dynamical phenomena of the 2-DOF MCBEHs by performing the harmonic balance analysis (HBA). All periodic solution branches are identified in order to study and comprehend the complicated dynamics of the 2-DOF MCBEHs. This end requires care when truncating the harmonic balance solution. For a 1-DOF MCBEH, which is the conventional type, the fundamental harmonic is able to approximately describe the steady-state periodic response. However, high-order harmonics are significant for the 2-DOF MCBEH. This paper demonstrates that the harmonic balance solution should involve the high-order terms instead of using the oversimplified single-harmonic solution. By performing the proposed HBA, important solution branches are reported, and their dynamical behaviors are studied. Moreover, the complete architecture of the frequency response of the 2-DOF MCBEH is disclosed across the entire frequency range. The HBA also reveals the underlying physics of building a bridge between the first and second primary resonant areas under a strong excitation. In the future, the findings in the present report can be utilized in the design process of the 2-DOF MCBEHs.