Dynamic contact of two Gao beams. (English) Zbl 1302.74116

Ahn, Jeongho; Kuttler, Kenneth L.; Shillor, Meir

Electron. J. Differ. Equ. 2012, Paper No. 194, 43 p. (2012).

Summary: The dynamic contact of two nonlinear Gao beams that are connected with a joint is modeled, analyzed, and numerically simulated. Contact is modeled with either (i) the normal compliance condition, or (ii) the unilateral Signorini condition. The model is in the form of a variational equality in case (i) and a variational inequality in case (ii). The existence of the unique variational solution is established for the problem with normal compliance and the existence of a weak solution is proved in case (ii). The solution in the second case is obtained as a limit of the solutions of the first case when the normal compliance stiffness tends to infinity. A numerical algorithm for the problem is constructed using finite elements and a mixed time discretization. Simulation results, based on the implementation of the algorithm, of the two cases when the horizontal traction vanishes or when it is sufficiently large to cause buckling, are presented. The spectrum of the vibrations, using the FFT, shows a rather noisy system.

MSC:

74M15	Contact in solid mechanics
74K10	Rods (beams, columns, shafts, arches, rings, etc.)
35Q74	PDEs in connection with mechanics of deformable solids
35L86	Unilateral problems for nonlinear hyperbolic equations and variational inequalities with nonlinear hyperbolic operators
35L70	Second-order nonlinear hyperbolic equations

Keywords:

dynamic contact; Gao beam; mechanical joint; normal compliance; Signorini condition; weak solutions; numerical scheme