Finite element error analysis of a viscoelastic Timoshenko beam with thermodiffusion effects

In this paper, a thermomechanical problem involving a viscoelastic Timoshenko beam is analyzed from a numerical point of view. The so-called thermodiffusion effects are also included in the model. The problem is written as a linear system composed of two second-order-in-time partial differential equations for the transverse displacement and the rotational movement, and two first-order-in-time partial differential equations for the temperature and the chemical potential. The corresponding variational formulation leads to a coupled system of first-order linear variational equations written in terms of the transverse velocity, the rotation speed, the temperature and the chemical potential. The existence and uniqueness of solutions, as well as the energy decay property, are stated. Then, we focus on the numerical approximation of this weak problem by using the implicit Euler scheme to discretize the time derivatives and the classical finite element method to approximate the spatial variable. A discrete stability property and some a priori error estimates are shown, from which we can conclude the linear convergence of the approximations under suitable additional regularity conditions. Finally, some numerical simulations are performed to demonstrate the accuracy of the scheme, the behavior of the discrete energy decay and the dependence of the solution with respect to some parameters

Comparison of sliding mode and state-feedback control applied to a partially treated actively constrained layer damped (ACLD) beam

In this research, a sliding mode control (SMC) was utilized in the control of a partially treated, actively constrained layer damped (ACLD), Timoshenko beam model. The resulting vibration control was compared to the vibration control achieved by a state-feedback linear quadratic regulator (LQR) for several loading conditions. An observer was designed and model order reduction (MOR) was performed to achieve a simplified, efficient, and more controllable finite element system model. As a result of model simplification, modeling errors in the form of unstructured uncertainties were introduced into the system. It was determined that the SMC and LQR achieved similar vibration control for all loading conditions when saturation limits were imposed. The saturation limits were enforced to replicate realistic voltage constraints. Saturation limits were then removed to investigate the ideal control action of the SMC and LQR. The ideal case revealed that the SMC achieved a significant reduction in the maximum deflection and settling time (as much as 37.44% and 16.61%, respectively) for all loading conditions when compared to the LQR. The improvement in response was due to the increase in control activity and the utilization of a robust control scheme in the presence of unstructured uncertainties