Resonant behaviors of a nonlinear cantilever beam with tip mass subject to an axial force and electrostatic excitation

Abstract

In this study, the primary, sub- and super-harmonic resonant behaviors of a cantilever beam-type micro-scale device are analytically solved and examined. The device under study includes a tip mass and is subjected to an axial force and electrostatic excitement. An appropriate derivation of orthogonality conditions and their application enable us to properly discretize the governing nonlinear field equation along with its boundary conditions to an equation form suitable for ‘single mode approximation’. This procedure results in a Mathieu–Hill type differential equation and causes associated parametric instability problems. Using a Taylor series expansion with an electrostatic forcing term, a quadratic nonlinear term naturally appears in the resulting differential equation. This term often requires more rigorous mathematical treatment than other conventional approaches. To resolve this problem, the concept of nonlinear normal mode is introduced in this study. A perturbation technique and asymptotic expansions of modal displacement are employed to accurately solve the resulting nonlinear differential equation by applying an appropriate ordering scheme. Finally, the effects of parameters/operating conditions on the resonant characteristics of the device are thoroughly investigated, and the associated parametric instability issue is also discussed.