Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators

Inthe last decadewe havewitnessed excitingtechnologicaladvancesin the fabricationand control of microelectromechanical and nanoelectromechanical systems (MEMS& NEMS) [1–5]. Such systems are being developed for a host of nanotechnologicalapplications,suchashighly-sensitivemass[6–8],spin[9],andchargedetectors[10,11],aswellasforbasicresearchinthemesoscopicphysicsofphonons[12],andthegeneralstudy of the behavior of mechanical degrees of freedom at the interface between thequantum and the classical worlds [13,14]. Surprisingly, MEMS & NEMS have alsoopenedup a whole new experimentalwindowinto the study of the nonlineardynamicsof discrete systems in the form of nonlinear micromechanical and nanomechanicaloscillators and resonators.Thepurposeofthisreviewisto providean introductionto thenonlineardynamicsofmicromechanical and nanomechanical resonators that starts from the basics, but alsotouches upon some of the advanced topics that are relevant for current experimentswith MEMS & NEMS devices. We begin in this section with a general motivation,explaining why nonlinearities are so often observed in NEMS & MEMS devices. In§ 1.2 we describe the dynamics of one of the simplest nonlinear devices—the Duffingresonator—while giving a tutorial in secular perturbation theory as we calculate itsresponse to an external drive. We continue to use the same analytical tools in § 1.3 todiscuss the dynamics of a parametrically-excited Duffing resonator, building up to thedescription of the dynamicsof an array of coupled parametrically-excitedDuffing res-onatorsin § 1.4. We conclude in § 1.5 by giving an amplitude equation description forthe array of coupled Duffing resonators, allowing us to extend our analytic capabilitiesin predicting and explaining the nature of its dynamics.