Vibrations of a nonlinear dynamic beam between two stops

Abstract

This work extends the model developed by Gao (1996) for the vibrations ofa nonlinear beam to the case when one of its ends is constrained to move between two reactiveor rigid stops. Contact is modeled with the normal compliance condition for the deformablestops, and with the Signorini condition for the rigid stops. The existence of weak solutions to theproblem with reactive stops is shown by using truncation and an abstract existence theorem involving pseudomonotoneoperators. The solution of the Signorini-type problem with rigid stops is obtainedby passing to the limit when the normal compliance coefficient approaches infinity. This requiresa continuity property for the beam operator similar to a continuity property for the wave operatorthat is a consequence of the so-called div-curl lemma of compensated compactness.