Boundary exponential stabilization of non-classical micro/nano beams subjected to nonlinear distributed forces

In this paper, the vibration suppression of micro- or nano-scale cantilever beams used in M/NEMS devices is studied. The beam is subjected to some nonlinear distributed forces, namely electrostatics force with first order fringing field correction, Casimir, and van der Waals forces. For the sake of precision, the beam is modeled by strain gradient elasticity theory capable of predicting the size effects in mechanical behavior of small-scale flexible structures. Since the governing partial differential equation of motion is nonlinear, the linearization approach is adopted to tackle the control problem. A novel control law is proposed that guarantees the exponential stability of the linearized closed-loop system and also the local stability of original nonlinear closed-loop system. To prepare the model for computer simulations, the continuous model is truncated to a set of nonlinear ordinary differential equations by using Kantorovich method. Simulation results show that the proposed controller not only suppresses the forced vibration of the beam before crossing dynamic pull-in threshold, but also it extends the dynamic pull-in criterion.