Experimental investigation and theoretical modelling on nonlinear dynamics of cantilevered microbeams

Abstract

The nonlinear vibration of cantilever nickel microbeams with thickness in micron level is investigated through experiment and theory. A non-contact dynamic vibration measurement system is built to test the fundamental linear frequency and the nonlinear amplitude-frequency curves near the primary resonance. Experimental result reveals the nonlinear amplitude-frequency curves exhibit a marked weak softening-type behavior as well as hysteresis behavior. To elucidate the experimental observations, a modified couple stress-based Euler-Bernoulli beam model incorporating geometric and inertial nonlinearities is established. The derived partial differential equation of motion is converted into a set of ordinary differential equations (ODEs) by employing the Galerkin method. And then, the multi-dimensional Lindstedt-Poincaré (L-P) method is applied to solve these ODEs. By comparing these analytical and experimental results, a good agreement is observed. Besides, a p-version Ritz method is combined with the iteration algorithm to reveal the close similarity of the beam and plate models and to further demonstrate the size-dependency in the nonlinear regime. Meanwhile the size-dependency in nonlinear regime is demonstrated. The effects of different order nonlinear terms, length scale parameter and damping coefficients on the system are investigated. It is illustrated that the period-doubling bifurcation occurs when imposing sufficiently large excitations, and it turns into chaos with increasing the excitations continuously.