Dynamic stability of parametrically-excited linear resonant beams under periodic axial force

Abstract

The parametric dynamic stability of resonant beams with various parameters under periodic axial force is studied. It is assumed that the theoretical formulations are based on Euler—Bernoulli beam theory. The governing equations of motion are derived by using the Rayleigh—Ritz method and transformed into Mathieu equations, which are formed to determine the stability criterion and stability regions for parametrically-excited linear resonant beams. An improved stability criterion is obtained using periodic Lyapunov functions. The boundary points on the stable regions are determined by using a small parameter perturbation method. Numerical results and discussion are presented to highlight the effects of beam length, axial force and damped coefficient on the stability criterion and stability regions. While some stability rules are easy to anticipate, we draw some conclusions: with the increase of damped coefficient, stable regions arise; with the decrease of beam length, the conditions of the damped coefficient arise instead. These conclusions can provide a reference for the robust design of parametrically-excited linear resonant sensors.