×

On the stability of periodic solutions with defined sign in MEMS via lower and upper solutions. (English) Zbl 1464.34062

Summary: This article considers the existence and linear stability of positive periodic solutions for the Nathanson’s model and the Comb-drive finger model of Micro-Electro-Mechanical-Systems (MEMS). Our results contribute to a better understanding of these models and therefore, they add up to the set of tools to decide the range of usefulness in sensing and acting of these two models. Both models are considered without damping and having an AC-DC input voltage \(\mathcal{V}(t) > 0\) of period \(T^\ast > 0\). For the Nathanson’s model, we prove the existence of two \(T^\ast\)-periodic solutions \(0 < \varphi(t) < \psi(t)\) under the assumption that the voltage is less than the pull-in voltage and greater than a certain positive constant depending on the temporal frequency of \(\mathcal{V}\) and physical parameters. The solution \(\varphi\) is elliptic and, if the parameter \(\lambda := \frac{\min \mathcal{V}}{\max \mathcal{V}}\) is greater than a certain critical value, the solution \(\psi\) is hyperbolic. We point out that if \(\mathcal{V}\) is greater than the pull-in voltage, there are not \(T^\ast\)-periodic solutions. Under the same conditions that guarantee the existence, we prove the uniqueness of these two solutions and we classify their linear stability. We can view this existence/uniqueness/linear stability classification theorem as an extension of the known results for the autonomous case when the voltage \(\mathcal{V}\) is constant. The Comb-drive model is studied under a cubic stiffness with coefficient \(\alpha > 0\), and its dynamics depends on whether \(\alpha \leq 2\) or \(\alpha > 2\). We will prove that for an adequate range of \(\mathcal{V}\) the equilibrium (the zero solution) is elliptic. This suggests the existence of periodic solutions with defined sign. In this document we prove the existence of periodic positive (negative) solutions for the Comb-drive model for which the number and the linear stability depend on \(\alpha\) and \(\lambda\) respectively. When \(0 < \alpha \leq 2\), we prove the existence of an unstable periodic solution for \(\lambda\) greater than a certain critical value and \(\mathcal{V}\) below the pull in voltage. For the case \(\alpha > 2\), similar to the Nathanson case, we prove the existence of two positive periodic solutions one of which is linearized stable and the other is unstable. The methodology uses the Lower and Upper Solution Method. Some numerical examples are provided to illustrate the results.

MSC:

34C25 Periodic solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
70K40 Forced motions for nonlinear problems in mechanics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bernstein, D. H.; Pelesko, J. A., Modeling MEMS and NEMS, (2002), Chapman and Hall/CRC Press Boca Raton, FL · Zbl 1031.74003
[2] Zhang, W. M.; Yan, H.; Peng, Z. K.; Meng, G., Electrostatic pull-in instability in MEMS/NEMS: A review, Sensors Actuators A, 214, 719-732, (2014)
[3] Nathanson, H. C.; Newell, W. E.; Wickstrom, R. A.; Davis, J. R., The resonant gate transistor, IEEE Trans. Electron. Dev., 14, 3, 117-133, (1967)
[4] Luo, A. C.; Wang, F. Y., Nonlinear dynamics of a micro-electro-mechanical system with time-varying capacitors, J. Vib. Acoust., 126, 77-83, (2004)
[5] Meng, G.; Zhang, W., Nonlinear dynamical system of micro-cantilever under combined parametric and forcing excitations in MEMS, Sensors Actuators A, 119, 291-299, (2005)
[6] Ai, S.; Pelesko, J. A., Dynamics of a canonical electrostatic MEMS/NEMS system, J. Dyn. Differ. Equ., 20, 609-641, (2007) · Zbl 1155.34027
[7] Gutiérrez, A.; Torres, P. J., Non-autonomous saddle-node bifurcation in a canonical electrostatic MEMS, Int. J. Bifurc. Chaos Appl. Sci. Eng., 23, 5, (2013), 1350088 (9p) · Zbl 1270.34125
[8] Gutiérrez, A.; Núñez, D.; Rivera, A., Effects of voltage change on the dynamics in a comb-drive finger of an electrostatic actuator, Int. J. Non-Linear Mech., 95, 224-232, (2017)
[9] Yang, Y.; Zhang, R., And le zhao dynamics of electrostatic microelectromechanical systems actuators, J. Math. Phys., 53, 022703, (2012) · Zbl 1274.74050
[10] Younis, M. I., MEMS linear and nonlinear statics and dynamics, (2011), Springer
[11] Hartman, P., Ordinary differential equations, (1982), Birkhäuser Boston, Basel, Stuttgart · Zbl 0125.32102
[12] J. Lei, X. Li, P. Yan, M. Zhang, Twist character of the last amplitude periodic solution of the forced pendulum, SIAM J. Math. Anal. 35(4) 844-867. · Zbl 1189.37064
[13] Magnus, W.; Winkler, S., Hill’s equation, (1979), Dover New York · Zbl 0158.09604
[14] Perdomo, O., A bifurcation in the family of periodic orbits for the spatial isosceles 3 body problem, Qual. Theory Dyn. Syst, 77-83, (2017)
[15] De Coster, C.; Habets, P., (Two-Point Boundary Value Problems: Lowet and Upper Solutions, Mathematics in Science and Engineering, vol. 205, (2006), Elsevier Amsterdam)
[16] Núñez, D., The method of lower and upper solutions and the stability of periodic oscillations, Nonlinear Anal. Theory Methods Appl., 51, 1207-1222, (2002) · Zbl 1043.34044
[17] Núñez, D.; Torres, P. J., Periodic solutions of twist type of an Earth satellite equation, Discrete Contin. Dyn. Syst., 7, 2, 303-306, (2001) · Zbl 1068.70027
[18] Núñez, D.; Torres, P. J., Stable odd solutions of some periodic equations modelling satellite motion, J. Math. Anal. Appl., 279, 2, 700-709, (2003) · Zbl 1034.34051
[19] Núñez, D.; Rivera, A., Twist periodic solutions in the relativistic driven harmonic oscillator, Abstr. Appl. Anal., (2016), ID 6084082. http://dx.doi.org/101155/2016/6084082 · Zbl 1470.34121
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.