Coventor's MEMS Design Blog

Nonlinear dynamics in MEMS, aka Frequency Hysteresis, Hard Spring Effect, or Duffing Effect

  

  

  

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Introduction

The frequency response of a MEMS device under some desired exciation is a key tool in understanding device performance.  Typically it can be assumed that the excitation is sufficiently small that the device remains within its linear regime and therefore the amplitude of the response is proportional to the excitation, regardless of the magnitude of excitation. This is the assumption when using Harmonic Analysis is CoventorWare's Analyzer tool, or AC Analysis in CoventorWare's Architect tool. However, in several MEMS applications, the amplitude of motion is large enough that this assumption is violated either by the strong nonlinearity of the electrostatic force, which varies like 1/(g -x)^2 where g is the gap and x is the displacement, or by the nonlinearity in beam bending.  The figure below shows how the frequency response can vary for large amplitude motion. This figure was taken from a paper by Tilman's [1] for resonant strain gauges.  But similar curves can be found in applications for scanning mirrors [2] published by Schenk and [3] Ataman as well as RF resonators [4]

As the amplitude grows, two effects occur:

1. the device appears to stiffen since the peak frequency gets larger
2. the curve starts to look like a breaking ocean wave.  This describes a hysteresis effect that shows that for frequencies underneath the breaking wave, there are two possible amplitudes of oscillation.  Which one the device oscillates at depends on the path to get there:  One for slowly increasing the frequency and one for decreasing the frequency.   This is not unlike a pull-in/lift-off curve.

Simulation using CoventorWare ARCHITECT 

Since AC analysis is a small signal analysis, it cannot be used to compute a frequency hysteresis curve.  Instead, one must use a transient analysis with a sinusoidal drive frequency of appropriate large amplitude.

An application note and schematics for using Architect for this purpose are available from Coventor's support.

Good example of this type of simulation in CoventorWare ARCHITECT on a MEMS micro-mirror has just been presented at MEMS2010 in Hong Kong by ASTRI.

[1] JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 7, NO. 1, MARCH 1998. Nonlinearity and Hysteresis of Resonant Strain Gauges, Chengqun Gui, Rob Legtenberg, Harrie A. C. Tilmans, Jan H. J. Fluitman, and Miko Elwenspoek

[2] IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 6, NO. 5, SEPTEMBER/OCTOBER 2000. Large Deflection Micromechanical Scanning Mirrors for Linear Scans and Pattern Generation. Harald Schenk, Peter Dürr, Thomas Haase, Detlef Kunze, Udo Sobe, Hubert Lakner, and Heinz Kück.

[3] Proceedings of SPIE Vol. 5348 (SPIE, Bellingham, WA, 2004). Nonlinear Frequency Response of Comb-Driven Microscanners. Caglar Ataman, Hakan Urey