U -sequence in electrostatic microelectromechanical systems (MEMS)

Abstract

In this paper, the presence of U (Universal)-sequence (a sequence of periodic windows that appear beyond the period doubling (PD) route to chaos) in electrostatic microelectromechanical systems (MEMS) is reported. The MEM system is first brought to a nonlinear steady state by the application of a large dc bias close to the dynamic pull-in voltage of the device. An ac voltage (the bifurcation parameter) is next applied to the system and increased gradually. A sequence of PD bifurcations leading to chaos is observed for resonant and superharmonic excitations (frequency of the ac voltage). On further increase in the ac voltage (beyond where chaos sets in), U -sequence is observed in the system. Under superharmonic excitation, the sequence is found to be a modified form of the U -sequence referred to as the ‘ UM -sequence’ in this paper. The appearance of a periodic window with K oscillations per period or K -cycles in the normal U -sequence is replaced by a corresponding periodic window with KM -cycles in the UM -sequence. M stands for the M th superharmonic frequency of excitation. The formation of the periodic windows from a chaotic state in the UM -sequence takes place through intermittent chaos as the ac voltage is gradually increased. On the other hand, the periodic states/cycles formed through intermittent chaos transform back into a chaotic state through the period doubling route. A sequence of period doubling bifurcations of the UM -sequence cycles result in the formation of -cycles in electrostatic MEMS. n corresponds to the n th period doubling bifurcation in the sequence. A simplified mass–spring–damper (MSD) model for MEMS is used to understand the physical mechanism that gives rise to these nonlinear dynamic properties in MEMS. The nonlinear nature of the electrostatic force acting on the MEM device is found to be responsible for the reported observations.