NONLINEAR VIBRATION AND FREQUENCY RESPONSE ANALYSIS OF NANOMECHANICAL CANTILEVER BEAMS

Microcantilevers are one the most commonly utilized microelectromechanical systems in a variant of the ultra-precise scanning and characterization application. Of particular interest, 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. For applications such as these, the problem of coupled flexural-torsional nonlinear vibrations of a piezoelectrically-actuated microcantilever beam is considered in this dissertation. 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 coupling between flexural vibration as well as longitudinal and torsional vibrations. Considering different geometrical configurations for the beam, it is demonstrated how beam geometry or piezoelectric properties bring different nonlinear coupling terms into the equations. Of particular interest is the inextensibility assumption, for which the governing equations reduce to coupled flexural-torsional nonlinear equations with piezoelectric nonlinearity appearing in quadratic form while inertia and stiffness nonlinearities appear as cubic. An extensive analytical stability analysis is done to investigate the frequency response in various conditions. The stability analysis for fundamental mode is performed in presence and absence of piezoelectric quadratic nonlinear term to indicate the effect of this term on the system. In addition, the one-to-three subharmonic resonance is identified in the system due to presence of inertia, stiffness and piezoelectric nonlinearities. Then, the stability of the system at this subharmonic is investigated. An experimental setup consisting of a commercial piezoelectric microcantilever installed on the stand of a modern laser-based microsystem analyzer is designed and utilized to verify the theoretical developments. First and second flexural natural frequencies are experimentally obtained, which are shown to be in good agreement with their theoretical values. Both linear and nonlinear numerical simulation results are compared with experimental results and it is observed that nonlinear modeling response matches the experimental findings very closely. More importantly, it is disclosed that the initial twisting in the microcantilever influence the value of the flexural vibration resonance. This unique and unexplored coupling effect can be utilized to indirectly measure 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