Vibrations of a nonlinear dynamic beam between two stops. (English) Zbl 1167.74019

Andrews, K. T.; M’Bengue, M. F.; Shillor, M.

Discrete Contin. Dyn. Syst., Ser. B 12, No. 1, 23-38 (2009).

Summary: This work extends the model developed by D. Y. Gao [Mech. Res. Commun. 23, No. 1, 11–17 (1996; Zbl 0843.73042)] 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.

MSC:

74H45	Vibrations in dynamical problems in solid mechanics
74K10	Rods (beams, columns, shafts, arches, rings, etc.)
74H20	Existence of solutions of dynamical problems in solid mechanics
74H25	Uniqueness of solutions of dynamical problems in solid mechanics
74M15	Contact in solid mechanics

Keywords:

existence; uniqueness; Signorini condition; pseudomonotone operators