Residual vibration suppression for duffing nonlinear systems with electromagnetical actuation using nonlinear command shaping techniques

Residual vibration control is crucial for numerous applications in precision machinery with negligible damping such as magnetically actuated systems. In certain magnetically actuated applications, the systems could also be highly nonlinear and conditionally stable. Although traditional command shaping techniques work well for linear and weakly nonlinear systems, they show little effects for dealing with systems with both strong structural and actuation nonlinearities. In this paper, a general input shaper design methodology for single degree of freedom systems with both Duffing spring and electromagnetic forcing nonlinearities is successfully devised using an energy approach. Following this method, two-step and three-step shapers are developed, which in the linear limit reduce to the traditional zero-vibration (ZV) and zero-vibration-and-derivative (ZVD) shapers, respectively. The robustness of these nonlinear shapers is investigated numerically through several case studies and the results show that the three-step shaper is sufficiently robust to resist significant amounts of parameter variations without exciting significant residual vibration. The two-step shaper, however, is somewhat less robust with respect to parameter variations. Meanwhile, an electromagnetically driven Duffing mechanical system is also constructed so that the performances and robustness of the nonlinear shapers in vibration suppression can be examined. It is shown that the nonlinear shapers result in a significant improvement in residual vibration suppression and settling time reduction in comparison with the traditional linearized ZV and ZVD shapers.