|
Article information
2018 , Volume 23, ¹ 2, p.63-75
Pimanov D.O.
Study of the nonlinear oscillations in a mathematical model of microresonator
Purpose. Study of a second order non-autonomous non-linear differential equation that describes oscillations of a material point under the action of linear elastic force, frictional force and the force of electrostatic attraction which varies in time with a given frequency. The problem appears in applications of a mathematical model to MEMS resonator where a non-deformable platform with given mass attached to a spring is considered as a material point. Methodology. In connection with the periodicity of external influence, the nonlinear boundary value problem with periodicity conditions is formulated and investigated numerically by the parameter continuation on the basis of the multiple shooting method. The stability of the obtained periodic solutions is determined using the spectrum of the monodromy matrix. Findings. The regions of existence for model parameters in which the periodic solutions of the equation with the period of external forcing are defined taking into account their multiplicity and stability. Numerical integration has shown that the nonlinear oscillations can also exist as stable periodic oscillations with a multiple period. Furthermore, for appropriate values of parameters the chaotic oscillations appear by the Feigenbaum scenario. Originality. The proposed method of numerical investigation allows one to effectively study nonlinear properties of the model, such as multistability, nonlinear resonance and the appearance of chaos.
[full text]
Keywords: nonlinear vibration, boundary value problem, multiple shooting method, parameter continuation, electrostatic attraction, period doubling bifurcation, Feigenbaum cascade
doi: 10.25743/ICT.2018.23.12759
Author(s):
Pimanov Daniil Olegovich
Position: Student
Office: Novosibirsk state university
Address: 630090, Russia, Novosibirsk, 2 Pirogov st.
E-mail: pimanov-daniil@yandex.ru
References:
[1] Kostsov, E.G., Fadeev, S.I. New microelectromechanical cavities for gigahertz frequencies. Optoelectronics, Instrumentation and Data Processing. 2013; 49(2):204–210.
[2] Takamatsu, H., Sugiura, T. Nonlinear vibration of electrostatic MEMS under DC and AC applied voltage. Proc. of the 2005 Intern. Conf. on MEMS, NANO and Smart Systems, ICMENS 2005. Banff, Alberta, Canada; 2005:423–424.
[3] Younis, M.I. MEMS Linear and Nonlinear Statics and Dynamics. Springer US. Series:Microsystems. 2011; 20(XVI): 456.
[4] Fadeev, S.I., Kogay, V.V. Kraevye zadachi dlya sistem obyknovennykh differentsial'nykh uravneniy [Boundary value problems for systems of ordinary differential equations]. Novosibirsk: NGU; 2012: 278. (In Russ.)
[5] Kogai, V.V., Fadeev, S.I. Application of parameter continuation based on the multiple shooting method for the numerical investigation of nonlinear boundary value problems. Sibirskiy zhurnal industrial'noy matematiki. 2001; 4(1):83–101. (In Russ.)
[6] Metody analiza nelineynykh dinamicheskikh modeley [Methods of the analysis for nonlinear dynamic models] / M. Holodniok, A. Klich, M. Kubichek, M. Marek. Moscow: Mir; 1991: 368. (In Russ.)
[7] Demidovich, B.P. Lektsii po matematicheskoy teorii ustoychivosti [Lectures on the mathematical theory of stability]. Moscow: Nauka; 1967: 472 p. (In Russ.)
[8] Merkin, D.R. Introduction to the Theory of Stability. N.Y.: Springer; 1997: 315.
[9] Moon, F. C. Chaotic Vibrations: An Introduction for Applied Scientists and Engineers. N.Y.:Wiley-VCH; 2004: 309.
Bibliography link:
Pimanov D.O. Study of the nonlinear oscillations in a mathematical model of microresonator // Computational technologies. 2018. V. 23. ¹ 2. P. 63-75
|
