Asymptotic decay rate of non-classical strain gradient Timoshenko micro-cantilevers by boundary feedback

In non-classical micro-beams, the strain energy of the system is obtained based on the non-classical continuum mechanics. This paper presents the problem of boundary control of a vibrating non-classical micro-cantilever Timoshenko beam to achieve the asymptotic decay rate of the closed loop system. For this aim, we need to establish the well-posedness of the governing partial differential equations (PDEs) of motion in presence of boundary feedbacks. A linear control law is constructed to suppress the system vibration. The control forces and moments consist of feedbacks of the velocities and spatial derivatives of them at tip of the micro-beam. To verify the effectiveness of the proposed boundary controllers, numerical simulations of the open loop and closed loop PDE models of the system are worked out using finite element method (FEM). New Timoshenko beam element stiffness and mass matrices are derived based on the strain gradient theory and verification of this new beam element is accomplished.