Coupled Flexural-Torsional Nonlinear Vibrations of Piezoelectrically Actuated Microcantilevers With Application to Friction Force Microscopy

The problem of vibrations of microcantilevers has recently received considerable attention due to its application in several nanotechnological instruments, such as atomic force microscopy, nanomechanical cantilever sensors, and friction force microscopy. Along this line, this paper undertakes the problem of coupled flexural-torsional nonlinear vibrations of a piezoelectrically actuated microcantilever beam as a typical configuration utilized in these applications. The actuation and sensing are both facilitated through bonding a piezoelectric layer (here, ZnO) on the microcantilever surface. The beam is considered to have simultaneous flexural, torsional, and longitudinal vibrations. The piezoelectric properties combined with nonlinear geometry of the beam introduce both linear and nonlinear couplings between flexural vibration as well as longitudinal and torsional vibrations. Of particular interest is the inextensibility configuration, for which the governing equations reduce to coupled flexural-torsional nonlinear equations 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. Both linear and nonlinear simulation results are compared to the experimental results and it is observed that nonlinear modeling response matches the experimental findings very closely. More specifically, the softening phenomenon in fundamental flexural frequency, which is due to nonlinearity of the system, is analytically and experimentally verified. It is also disclosed that the initial twisting in the microcantilever can influence the value of the flexural vibration resonance. The experimental results from a macroscale beam are utilized to demonstrate such twist-flexure coupling. This unique coupling effect may lead to the possibility of indirect measurement of small torsional vibration without the need for any angular displacement sensor. This observation could significantly extend the application of friction force microscopy to measure the friction of a surface indirectly.