Numerical Modeling and Some Optimal Control Problems of Dynamic Systems Describing Contact Problems with Friction in Elasticity

The purpose of the article is to make use of difference equations to solve the dynamic visco-elastic contact problems with friction. First of all, we propose the discretization in space and time of the equilibrium equations and the use of regularization methods to approximate the non-differentiable terms that model the contact friction, Coulomb type. For the discretization in space of the problem of dynamic elastic contact with friction, the finite element method is used and a system of ordinary differential equations of the second order will result, which will be solved using the finite difference method, the Newmark method and the Newton-Raphson iterative method of solving the linearized dynamic system. The model predictive control (MPC) and the linear quadratic regulator (LQR), described by the linear discrete-time difference equations derived from the system of nonlinear differential equations, have been used to solve some optimal control problems. This problem is of great importance in engineering applications, for example for the control of deformations, velocities and accelerations in areas of interest, and vibration mitigation. Areas of interest could be: the tool tip of a machine tool or the end efector of a robotic arm with links and joints etc.