Amplitude- and Gas Pressure-Dependent Nonlinear Damping of High-Q Oscillatory MEMS Micro Mirrors

Abstract

Silicon-based micro-electromechanical systems (MEMS) can be fabricated using bulk and surface micromachining technology. A micro mirror designed as an oscillatory MEMS constitutes a prominent example. Typically, in order to minimize energy consumption, the micro mirror is designed to have high quality factors. In addition, a phase-locked loop guarantees resonant actuation despite the occurrence of frequency shifts. In these cases, the oscillation amplitude of the micro mirror is expected to scale linearly with the actuation input power. Here however, we report on an experimental observation which clearly shows an amplitude depletion that is not in accordance with any linear behaviour. As a consequence, the actuation forces needed to reach the desired oscillation amplitude are by multiples higher than expected. We are able to explain the experimental observations accurately by introducing a single degree-of-freedom model including an amplitude-dependent nonlinear damping term. Remarkably, we find that the nonlinear damping shows a clear gas pressure dependency. We investigate the concepts and compare our findings on two different micro mirror design layouts.

Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction

“…to determine the amplitude A 0 and phase φ 0 [36]- [38] by solving the nonlinear algebraic steady-state equation…”

Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level

“…disturbances only occur above a certain threshold deflection (or critical point) of the drive mode. Below threshold, a single degree-of-freedom nonlinearly damped Duffing oscillator provides an accurate estimate of the device behaviour [20], [21]. Here, in accordance with SPDC, we can observe resonant actuation of two parasitic modes, denoted by the indices 1 and 2, in some devices, whenever the linear mode frequencies f 0,1 and f 0,2 fulfil the condition f 0,0 ≈ f 0,1 + f 0,2 (with the drive mode linear frequency denoted by f 0,0 ).…”

“…It needs to be calculated individually for each device, since process tolerances during MEMS fabrication influence the mode spectrum and thus, the scope of possible mode couplings. When the phase difference between the actuation force and the drive mode response is controlled by a phase-locked-loop (PLL) [21], [23] to ensure an actuation at the resonance frequency of the drive mode, the critical amplitude is given by…”

Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes

“…F 0 denotes the amplitude of the external periodic force that actuates the drive mode. We employ the method of averaging [13] to reduce the system to first order differential equations for amplitude and phase of each mode, as shown previously [14]. This approach is valid for resonant systems with high quality factors, where two time scales for fast and slow oscillation can be identified.…”

“…This approach is valid for resonant systems with high quality factors, where two time scales for fast and slow oscillation can be identified. Assuming steady-state, we obtain implicit analytical equations for all solution branches: The method of averaging for the additional terms is carried out as shown by [14] and yields for n = 3…”