Translational Thermo-elastic Timoshenko Beams

Of concern is a beam subject to thermal effects and attached to a base in translational movement at one end. A dynamics tip mass is attached to its free end. This kind of beams arises in robot arms in industry. The model takes into account the deflection due to transverse shear strains and is known as the Timoshenko model. This model is therefore coupled with an ordinary differential equation describing the dynamic of the translational base. Although dampened by a thermal effect through the coupling with a heat equation, the structure cannot be driven to a desired position. Therefore there is a need for an extra damping through a control force acting not on the body but rather on the base. We propose to study the stabilization of this problem. After deriving the model, we shall use the multiplier technique to find an appropriate (reasonable and economic) control which is able to drive the system to a desired position. This will be established for two types of damping, namely the viscous damping and the viscoelastic damping separately. Obviously, the classical energy of the system will not be a useful tool to work with. There will be a need to come up with new Lyapunov-type functionals carefully selected to deal with all the undesirable terms appearing in the computation of the rate of change of the energy. This should allow us to choose an appropriate boundary control force. We expect to face a considerable challenge as the translational displacement has, in general, a destabilizing effect on the structure. It will compete with whatever damping we use. Therefore, it is interesting to have an insight on how strong is a thermal damping in such a situation.