Design of command shapers for residual vibration suppression in Duffing nonlinear systems

Residual vibrations generated from rest-to-rest maneuvers are crucial for applications in precision engineering, active structural control, space engineering, and other mechatronics applications. In certain applications, the structures to be controlled could be highly nonlinear yet lightly damped. Although the traditional input shaping techniques, which utilize destructive interference, work well for linear and weakly nonlinear systems, they show little effects on systems with strong nonlinearity. In this paper, a general input shaper design methodology for single-degree-of-freedom systems with Duffing nonlinearity is developed by an energy approach. Following this approach, two-step and three-step shapers, as well as their design procedures, are developed, which in the linear limit reduce to the traditional zero-vibration and zero-vibration-and-derivative shapers, respectively. The robustness of these nonlinear shapers is investigated numerically through case studies. The results show that the three-step shapers are sufficiently robust to resist certain level of parameter variations (from their designed values) without exciting significant residual vibrations. The two-step shapers, however, are less robust in comparison. Meanwhile, it is also found that the presence of damping effectively disturbs the energy flow and thus induces residual vibrations. For the less robust two-step shapers, an effective ‘online tuning’ scheme is also proposed here to further improve its performance in a damped nonlinear system. These shaping schemes, as well as their practical adjustment routing, could be applied to the particular structural or mechatronics systems with Duffing or other nonlinearities for vibration suppression to enhance the performance of mechatronics systems.