Coupled flexural-torsional nonlinear vibrations of microcantilever beams

Abstract

In this paper the problem of coupled flexural-torsional nonlinear vibrations of a piezoelectrically-actuated microcantilever beam is investigated considering beam's simultaneous flexural, torsional and longitudinal vibrations. Application of such problem is utilized in several nanotechnological instruments such as atomic force microscopy, nanomechanical cantilever sensors and friction force microscopy. The actuation and sensing are both facilitated through bonding a piezoelectric layer (here, ZnO) on the microcantilever surface. The piezoelectric properties combined with nonlinear geometry of the beam introduce both linear and nonlinear coupling between flexural vibration as well as longitudinal and torsional vibrations. The governing equations of motion are obtained with piezoelectric nonlinearity appearing in quadratic form while inertia and stiffness nonlinearities as cubic. An experimental setup consisting of a commercial piezoelectric microcantilever installed on the stand of an ultramodern laser-based microsystem analyzer is designed and utilized to verify the theoretical developments. First and second flexural natural frequencies are both experimentally and numerically obtained and are shown to be in good agreement. Both linear and nonlinear simulation results are compared with experimental results and it is observed that nonlinear modeling response matches the experimental findings very closely.