Bound on amplitude of a MEMS resonator by approximating the derivative of the Lyapunov function in finite time. (English) Zbl 1535.70081
Summary: We propose a methodology to obtain the amplitude of a nonlinear differential equation that may not satisfy Lyapunov’s global stability criterion. This theory is applied to the MEMS resonator which has a high-quality factor. The derivative of the Lyapunov function approximated for a finite time and an optimization problem was formulated. The local optima were obtained using the Karush-Kuhn-Tucker conditions, for which the amplitude was analytically formulated. The obtained amplitude, when compared with that by the numerical method, showed the validity of the analytical approximation for a useful range of the nonlinearity, but accurate only at an excitation frequency \(\Omega=0.913\). This methodology will be useful to approximate the damping in a system if one obtains the amplitude from the experimental data near this excitation frequency.
MSC:
| 70K28 | Parametric resonances for nonlinear problems in mechanics |
References:
| [1] | Lifshitz, R. and Cross, M. C., Nonlinear dynamics of nanomechanical resonators, in Nonlinear Dynamics of Nanosystems (Wiley-VCH, Weinheim, 2010). · Zbl 1160.37035 |
| [2] | www.crossgroup.caltech.edu. |
| [3] | Ramanan, A., Teoh, Y. X., Ma, W. and Ye, W., Characterization of a laterally oscillating microresonator operating in the nonlinear region, Micromachines7 (2016) 132, https://doi.org/10.3390/mi7080132. |
| [4] | Antonio, D., Zanette, D. H. and López, D., Frequency stabilization in nonlinear micromechanical oscillators, Nat. Commun.3 (2012) 806, https://doi.org/10.1038/ncomms1813. |
| [5] | Almog, R., Zaitsev, S., Shtempluck, O. and Buks, E., High intermodulation gain in a micromechanical Duffing resonator, Appl. Phys. Lett.88 (2006) 213509, https://doi.org/10.1063/1.2207490. |
| [6] | Villanueva, L. G., Kenig, E., Karabalin, R. B., Matheny, M. H., Lifshitz, R., Cross, M. C. and Roukes, M. L., Surpassing fundamental limits of oscillators using nonlinear resonators, Phys. Rev. Lett.110(17) (2013) 177208, https://doi.org/10.1103/PhysRevLett.110.177208. |
| [7] | Lai, S. K., Yang, X., Wang, C. and Liu, W. J., An analytical study for nonlinear free and forced vibration of electrostatically actuated MEMS resonators, Int. J. Struct. Stab. Dyn.19 (2019) 1950072, https://doi.org/10.1142/S021945541950072X. · Zbl 1535.74165 |
| [8] | Andrzejewski, R. and Awrejcewicz, J., Nonlinear Dynamics of a Wheeled Vehicle (Springer, New York, 2005). · Zbl 1070.70001 |
| [9] | Bogusz, 1972, Technical stability, PWN. |
| [10] | Loop, B. P., Sudhoff, S. D., Zak, S. H. and Zivi, E. L., An optimization approach to estimating stability regions using genetic algorithms, in Proc. American Control Conf., Portland, OR, USA, , 2005, https://doi.org/10.1109/acc.2005.1469937. |
| [11] | A. Chatterjee, A brief introduction to nonlinear vibrations. Available at http://home.iitk.ac.in/\( \sim\) anindya/NLVnotes.pdf. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
