8 Nonlinear Dynamics of Nanomechanical Resonators

In the last decade we have witnessed exciting technological advances in the fabri-cation and control of microelectromechanical and nanoelectromechanical systems (MEMS & NEMS) [16, 19, 26, 54, 55]. Such systems are being developed for a host of nanotechnological applications, such as highly sensitive mass [25, 34, 67], spin [56], and charge detectors [17, 18], as well as for basic research in the mesoscopic physics of phonons [63], and the general study of the behavior of mechanical degrees of freedom at the interface between the quantum and the classical worlds [5, 64]. Sur-prisingly, MEMS & NEMS have also opened up a whole new experimental window into the study of the nonlinear dynamics of discrete systems in the form of nonlin-ear micromechanical and nanomechanical oscillators and resonators. The purpose of this review is to provide an introduction to the nonlinear dynam-ics of micromechanical and nanomechanical resonators that starts from the basics, but also touches upon some of the advanced topics that are relevant for current ex-periments with MEMS & NEMS devices. We begin in this section with a general motivation, explaining why nonlinearities are so often observed in NEMS & MEMS devices. In Section 8.2 we describe the dynamics of one of the simplest nonlinear devices, the Duffing resonator, while giving a tutorial in secular perturbation the-ory as we calculate its response to an external drive. We continue to use the same analytical tools in Section 8.3 to discuss the dynamics of a parametrically-excited Duffing resonator, building up to the description of the dynamics of an array of coupled parametrically-excited Duffing resonators in Section 8.4. We conclude in Section 8.5 by giving an amplitude equation description for the array of coupled Duffing resonators, allowing us to extend our analytic capabilities in predicting and explaining the nature of its dynamics