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July  2009, 12(1): 23-38. doi: 10.3934/dcdsb.2009.12.23

Vibrations of a nonlinear dynamic beam between two stops

K. T. Andrews 1, , M. F. M'Bengue 1, and  Meir Shillor 1,

1. 

Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, United States, United States, United States

Received  September 2008 Revised  February 2009 Published  May 2009

This work extends the model developed by Gao (1996) for the vibrations of a nonlinear beam to the case when one of its ends is constrained to move between two reactive or rigid stops. Contact is modeled with the normal compliance condition for the deformable stops, and with the Signorini condition for the rigid stops. The existence of weak solutions to the problem with reactive stops is shown by using truncation and an abstract existence theorem involving pseudomonotone operators. The solution of the Signorini-type problem with rigid stops is obtained by passing to the limit when the normal compliance coefficient approaches infinity. This requires a continuity property for the beam operator similar to a continuity property for the wave operator that is a consequence of the so-called div-curl lemma of compensated compactness.
Keywords: Gao beam, dynamic nonlinear beam, Signorini condition., existence and uniqueness, contact.
Mathematics Subject Classification: Primary: 74M15; Secondary: 74H20, 74H25, 74K10, 35L7.
Citation: K. T. Andrews, M. F. M'Bengue, Meir Shillor. Vibrations of a nonlinear dynamic beam between two stops. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 23-38. doi: 10.3934/dcdsb.2009.12.23
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