Higher-Order Complex Periodic Motions in a Nonlinear, Electromagnetically Tuned Mass Damper System

In this paper, the existence and bifurcations of higher-order complex periodic motions in a nonlinear, electromagnetically tuned mass damper system are studied through period-3, period-9, and period-12 motions. For such a nonlinear system, there exist three branches of period-3 motions, one branch of period-9 motions, and one branch of period-12 motion. The corresponding stability and bifurcations of periodic motions are determined. The frequency-amplitude characteristics for bifurcation routes of such higher-order periodic motions are presented. Numerical illustrations are presented for complex periodic motions. Higher-order unstable and stable periodic motions exist in nonlinear systems, which are difficult to control for energy harvesting and vibration reduction. The unstable higher-order periodic motions make the system responses more complicated. The higher-order stable and unstable periodic motions have been studied comprehensively. This study on periodic motions can help people better understand the dynamical behaviors of an electromagnetically tuned mass damper system under different excitation frequencies. The study of higher-order periodic motion stability and bifurcations will also benefit the new development and design of vibration reduction and energy harvesting systems.

Automated summary

This paper studies the existence, bifurcations, and stability of higher-order complex periodic motions in a nonlinear, electromagnetically tuned mass damper system. The authors present three branches of period-3 motions, one branch of period-9 motions, and one branch of period-12 motion in such a system. The frequency-amplitude characteristics for bifurcation routes of these higher-order periodic motions are determined. The study on periodic motions provides insights into the dynamical behaviors of the system and can contribute to the development and design of vibration reduction and energy harvesting systems.