Dynamics and Global Stability of Beam-based Electrostatic Microactuators

Abstract

We investigate the dynamics and global stability of a beam-based electrostatic microactuator, which is modeled as a first-order approximation of a reduced-order model (ROM) derived using the differential quadrature method (DQM). We show that the ROM dynamics is qualitatively similar to that of a higher-order approximation. We simulate the occurrence of dynamic pull-in for excitations near the first primary resonance using the finite difference method (FDM) and long-time integration. Limit-cycle solutions are obtained using the FDM, the generated frequency- and force-response curves exhibit cyclic-fold, saddle-node, and period-doubling bifurcations. We verify that symmetry breaking is not likely to occur because the orbit is already asymmetric. We identify the basin of attraction of bounded motions using various approximation levels. The simulations reveal that the erosion of the basin of attraction depends heavily on the amplitude and frequency of the AC voltage. We show that smoothness of the boundary of the basin of attraction can be lost and replaced by fractal tongues, which dramatically increase the sensitivity of the microbeam to initial conditions. According to these simulations, the locations of the two fixed points are likely to be disturbed.

A double microbeam MEMS ohmic switch for RF-applications with low actuation voltage

“…They showed that dynamic pull-in occurs under voltages lower than the static pull-in voltage, as low as 25%, when the frequency of the AC component is in the neighborhood of a resonant frequency. On the other hand, Najar et al [21] and Lenci and Rega [22] studied the basin of attraction of bounded motions and showed that the erosion of the basin of attraction is the principle reason for the occurrence of dynamic pull-in by homoclinic bifurcation. They showed that smoothness of the boundary of the basin of attraction of bounded motions can be lost and replaced with fractal tongues as the excitation amplitude is increased.…”

Dynamics of a close-loop controlled MEMS resonator

“…Only the branch of period-two orbits, subsequent to the first period-doubling bifurcation, is shown due to the small span of the higherorder orbits. A homoclinic bifurcation was reported by Najar et al [17] to interrupt an incomplete cascade of period-doubling bifurcations and to result in dynamic pull-in of an open-loop electrostatic resonator. In our case, the presence of the close-loop controller allows the cascade of period-doubling bifurcation to continue culminating in chaos due to a homoclinic entanglement.…”

“…In fact, the electrostatic force grows and approaches infinity as the resonator displacement grows and approaches the size of the capacitor gap d → g. On the other hand, the beam stiffness k is finite, which causes the static equilibrium of the resonator to lose stability at about one-third of the gap g in a phenomenon known as static pull-in. The static and dynamic response of uncontrolled electrostatic MEMS resonators have been studied extensively by many researchers, for example Najar et al in [17,23].…”

“…Najar et al [17] reported an incomplete cascade of period-doubling bifurcations in a similar model just before pull-in. They showed that chaos cannot occur in open-loop electrostatic actuators due to the occurrence of a homoclinic bifurcation which leads to rapid dynamic pull-in.…”

Vibration analysis of three-layered nanobeams based on nonlocal elasticity theory

“…In the last few years, scientific researchers have been focusing on nanotechnology and the resulting nano-materials which play key roles in many engineering devices at the nano-scale used in several applications including microactuators, microswtiches, biosensors, nanowires, nanoprobes, ultra-thin films and micro-and nano-electromechanical systems (MEMS and NEMS) studied in works of (Hung and Senturia, 1999;Li et al, 2003;Moser and Gijs, 2007;Pei et al, 2004;Najar et al, 2010). In fact, nano-materials have special mechanical, chemical, electrical, optical and electronic properties.…”