Boundary stabilization of non-classical micro-scale beams

In this paper, the problem of boundary stabilization of a vibrating non-classical micro-scale Euler-Bernoulli beam is considered. In non-classical micro-beams, the governing Partial Differential Equation (PDE) of motion is obtained based on the non-classical continuum mechanics which introduces material length scale parameters. In this research, linear boundary control laws are constructed to stabilize the free vibration of non-classical micro-beams which its governing PDE is derived based on the modified strain gradient theory as one of the most inclusive non-classical continuum theories. Well-posedness and asymptotic stabilization of the closed loop system are investigated for both cases of complete and incomplete boundary control inputs. To illustrate the performance of the designed controllers, the closed loop PDE model of the system is simulated via Finite Element Method (FEM). To this end, new strain gradient beam element stiffness and mass matrices are derived in this work.