Nonlinear dynamics of MEMS/NEMS resonators: analytical solution by the homotopy analysis method
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Due to various sources of nonlinearities, micro/nano-electro-mechanical-system (MEMS/NEMS) resonators present highly nonlinear behaviors including softening- or hardening-type frequency responses, bistability, chaos, etc. The general Duffing equation with quadratic and cubic nonlinearities serves as a characterizing model for a wide class of MEMS/NEMS resonators as well as lots of other engineering and physical systems. In this paper, after brief reviewing of various sources of nonlinearities in micro/nano-resonators and discussing how they contribute to the Duffing-type nonlinearities, we propose a Homotopy Analysis Method (HAM) approach for derivation of analytical solutions for the frequency response of the resonators. Toward this aim, we first apply the HAM to the proposed Duffing equation, and through this procedure, we derive the first-order and second-order HAM-based analytical solutions for the frequency response of the resonator. As the main novelty, we show that the second-order solution benefits from a tunable parameter, known as the convergence-control parameter, which is a distinguishing aspect of the HAM and plays a key role in enhancing the accuracy of the obtained analytical expressions in strongly nonlinear problems. We use the obtained analytical solutions for the study of nonlinear dynamics in two types of electrostatically actuated MEMS resonators proposing hardening, softening or mixed behaviors near their primary resonance frequency. Numerical simulations are performed to validate the analytical results.
KeywordsFrequency Response Homotopy Analysis Method Analytical Frequency Response Squeeze Film Multiple Scale Method
The authors would like to gratefully thank the Iran National Science Foundation (INSF) for their financial support.
- Kovacic I, Brennan MJ (2011) The Duffing equation, nonlinear oscillators and their behavior. Joun Wiley & SonsGoogle Scholar
- Liao S (2003) Beyond perturbation introduction to homotopy Analysis method. CRC PressGoogle Scholar
- Liao S (2012) Homotopy Analysis Method in Nonlinear Differential Equations. SpringerGoogle Scholar
- Liao S (2013) Chance and challenge : a brief review of the homotopy analysis method. In: advances of the homotopy analysis method. World Scientific PressGoogle Scholar
- Mestrom RMC, Fey RHB, Phan KL, Nijmeijer H (2009) Experimental validation of hardening and softening resonances in a clamped-clamped beam MEMS resonator. Procedia Chem 00:4–7Google Scholar
- Nayfeh AH (2004) Perturbation Methods. Wiley-VCHGoogle Scholar
- Nayfeh AH, Mook DT (1995) Nonlinear oscillations. Wiley-VCHGoogle Scholar
- Tajaddodianfar F, Yazdi MH, Pishkenari HN (2014) Dynamics of bistable initially curved shallow microbeams: effects of the electrostatic fringing fields. In: 2014 IEEE/ASME international conference on advanced intelligent mechatronics (AIM2014), 8–11 July 2014, Besacon, France. doi: 10.1109/AIM.2014.6878258