Softening Effect of the Nonlocality against the Hardening Effect of the Stretching in a Capacitive Micro-beam

This paper investigates the nonlinear resonant behavior of a capacitive micro-beam based on the nonlocal theory of elasticity. The micro-beam is deflected by a DC voltage, where it acts as a micro-resonator by superimposing an AC voltage. Taking into account stretching effects, the Galerkin projection method is used to discretize the partial differential equations into a set of nonlinear, ordinary differential equations. Multiple-scales method is used to obtain an approximate analytical solution to construct the nonlinear resonant curves of the transverse vibration amplitude. Taking into account the classical and nonlocal elasticity theories, the frequency response curves are plotted for different values of DC voltage. Effects of mid-plane stretching on the resonant curves are also examined. The hardening behavior of the system is shown to decrease due to the presence of the nonlocality as well as the DC voltage. However, mid-plane stretching increases the hardening effects. The results show that, in spite of the existence of nonlinearity in the system, this conflict effect can result in a linear frequency response curve for some values of the nonlocal parameter.