Bifurcation and chaotic behavior of micro-plate with size-dependency
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TECHNICAL PAPER
Bifurcation and chaotic behavior of micro-plate with size-dependency Jihai Yuan1,2 • Xiangmin Zhang2 • Changping Chen1,2 Received: 22 June 2020 / Accepted: 5 September 2020 Ó Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper investigates bifurcation and chaotic response of a size-dependent micro-plate under electrostatic force applied on its both sides. The size dependency of the micro-plate is taken into account using modified couple stress theory (MCST). Via Hamilton’s principle, the dynamic equations of motion for the micro-plate are obtained. With application of Galerkin procedure and Runge–Kutta (RK) method, the nonlinear partial differential equations are solved numerically. Numerical examples are carried out and the influence of size effect, damping effect and initial gap on chaotic characteristics of the micro-plate system are displayed utilizing Poincare map and bifurcation diagram. The results reveal that complex nonlinear behaviors including chaotic and multiple periodic motions occur in the micro-plate system and the route to chaotic motion is period-doubling. The findings of this research may provide a theoretical guidance for the dynamic stability of electrostatically actuated micro-plate system.
1 Introduction Micro-electro-mechanical systems (MEMS) are complex micro-systems integrated with micro-sensors, micro-structures, micro-actuators and corresponding circuits. Micro scale effects can be generated by these systems which control, sense and actuate on a micro scale level. Flexible micro-plates are usually used in MEMS devices including micro-pump, micro-mirror and micro-phone (2002) and they are commonly electrically actuated and deflected by electrical excitation (Chuang et al. 2010; Batra et al. 2007). Due to mid-plane stretching, squeeze-film damping and electrostatic forces nonlinearity can be introduced in MEMS either geometrically or physically (Maani et al. 2014). These nonlinear characteristics of MEMS may lead to chaotic behavior (Tajaddodianfar et al. 2015). Chaos can be desirable or undesirable in MEMS applications (Park et al. 2008; Alemansour et al. 2017). In either case, making a better understanding of chaotic behavior of these microstructures can not only understand their mechanical
& Changping Chen [email protected] 1
Department of Civil Engineering, Xiamen University, Xiamen 361005, Fujian, China
2
School of Civil Engineering and Architecture, Xiamen University of Technology, Xiamen 361024, Fujian, China
characteristics, but also assist their design and application in MEMS. Numerous investigations on chaotic behavior of MEMS devices are carried out in recent years. For example, Chen et al. (2013) employed periodicity-ratio approach to analyses and characterizes chaos of electrostatically actuated piezoelectric laminated micro-beam system. With consideration of chaotic motion, sub and supercritical global dynamics of a micro-beam under axial load which varies with time were presented
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