Nonsmooth dynamic analysis of sticking impacts in rocking structures

Abstract

This paper revisits from a nonsmooth dynamics perspective the contact process encountered in two archetypal rocking structures: the rigid rocking block and the flexible rocking oscillator. The analysis assumes impact is an instantaneous (unilateral) contact, and models the behavior along the normal direction with a set-valued Newton’s law. In the tangential direction it assumes that sliding is prevented. The study formulates both impact and uplifting phenomena as linear complementarity problems (LCPs). The proposed LCP formulation encapsulates all post-impact states and liberates from the need for additional ad-hoc assumptions. The results show that the proposed nonsmooth approach verifies the corresponding results of other studies with reference to the rigid rocking block. Focusing on the flexible rocking oscillator, the proposed model is compared with pertinent analytical, numerical and experimental results from literature. The present study offers original analytical solutions describing all physically feasible post-impact states and confirms, as a special case, existing solutions of rocking initiation. Importantly, the analysis unveils that a given rocking oscillator might choose a different post-impact state (e.g. bouncing, full contact, or immediate rocking) depending on its flexural deformation at the time of impact. Further, it shows that in the case of immediate rocking after impact, the post-impact flexural velocity of the rocking oscillator is not equal with its pre-impact value. The results also reveal the dominant role of the assumed value of the Newton’s coefficient of restitution on the response-history of the rocking oscillator.