A rod-beam system with dynamic contact and thermal exchange condition

摘要

We study a mathematical hybrid dynamic system that models the setting, in which one end of a nonlinear viscoelastic Timoshenko beam is clamped while the another end is jointed with the bottom of a thermoviscoelastic rod. The top of the rod may come in contact with a rigid support. Two conditions are applied to the contacting end; Signorini’s contact condition and Barber’s heat exchange condition. We formulate a partial differential equation (PDE) system with the relevant boundary conditions that describes the motion of the combined rod-beam, taking into account the dynamic contact and thermal interaction. A variational formulation to the model is obtained in an abstract setting. Then, a hybrid of several numerical methods is employed to abstract formulations. It guarantees that all the conditions at each time step are satisfied. Convergence of numerical trajectories is shown, by passing to the limits as time step sizes approach zero. We derive a new energy balance form that plays an important role in establishing the numerical stability. The fully discrete numerical schemes are then used to compute numerical solutions and some representative numerical simulations are presented.