Suppression of overshoots and undershoots in nonlinear structural and mechanical systems

In this work, we consider the problem of having a transit in a nonlinear SDOF system from a given rest position to another one, both associated to given and constant values of the external force, without spurious undershoots or overshoots of the solution after the final state is reached. It extends to the nonlinear regime the same problem considered in Udwadia (Acta Mech 231:3157-3182, 2020) in the linear realm. Two different approaches are considered. In the first, which is the more general, the free dynamics of the nonlinear system is not considered, and a very simple solution is obtained, showing how it is robust with respect to perturbations. In the second case, on the other hand, the free dynamics of the system is exploited during the transits, thereby allowing less control parameters to be determined. However, aiming at having a closed form solution, we limit this study to the Duffing equation, although the ideas are general and can be applied to other systems even when closed form solutions are not available.