A complete model describing the non linear dynamics of Mathieu resonators is presented in order to study the stability of resonant MEM gyroscopes

Baguet, Sébastien

Dufour, Régis

Hentz, S.

Kacem, Najib

English

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Non linear dynamics of Mathieu resonators for resonant
gyroscope applications
Najib Kacem, S. Hentz, Sébastien Baguet, Régis Dufour
To cite this version:
Najib Kacem, S. Hentz, Sébastien Baguet, Régis Dufour. Non linear dynamics of Mathieu resonators
for resonant gyroscope applications. 12th conference on Nonlinear Vibrations, Dynamics, and Multi-
body Systems, Jun 2008, Blacksburg, Virginia, United States. ￿hal-00506166￿
  
Non linear dynamics of Mathieu resonators for resonant gyroscope applications 
 
N.Kacem
12
, S.Hentz
1
, S.Baguet
2
 and R. Dufour
2 
1
Microsystems Components Laboratory, CEA/LETI - MINATEC, Grenoble, France 
2
LaMCoS, INSA-Lyon, CNRS-UMR5259, 569621, France 
 
For the first time, a complete model describing the non linear dynamics of Mathieu resonators is presented in 
order to study the stability of resonant MEM gyroscopes.  
 
Introduction: 
The resonant sensing technique [1] is highly sensitive, has the potential for large dynamic range, good linearity, low 
noise and potentially low power. However, when scaling sensors down to NEMS, nonlinearities occur sooner [2] 
restricting the benefits of the resonant sensors. Moreover, for resonant gyroscopes, the frequency of the coriolis forces 
becomes closer to the resonator frequency. The idea is to investigate the dynamic behaviour of nonlinear Mathieu 
resonators [3] in order to find the optimum physical conditions for gyroscope designers to maximise the sensors 
performances. 
Device and equations: 
The resonant output gyroscope [4] shown in Figure (1), as its name implies, utilizes resonant sensing as the basis for 
Coriolis force detection. In its simplest form, the device consists of three resonating elements, a proof mass and two 
resonating sense elements. The dynamics of the device can be described by a series of coupled differential equations. 
The proof mass dynamics can be described for most part by a classical spring-mass-damper equation (1). The 
dynamics of the resonators subjected to an axial time-varying Coriolis force is described by a nonlinear partial 
differential equation (2) with boundary conditions (3) as follows: 
 
( ) ( ) ( )
M
F
tX
dt
tdX
Qdt
tXd e
=++ 2
2
2
δδ     
( ) [ ] ( ) ( ) ( ) ( ) ( )[ ]2
2
02
2
2
2
0
2
4
4
,2
1,,,,,
txwg
bV
t
txw
s
t
txw
c
x
txw
dx
x
txw
l
ES
tCosFN
x
txw
EI R
l
c
−
=
∂
∂
+
∂
∂
+
∂
∂












∂
∂
++−
∂
∂
∫ ερδ
    
( ) ( ) 0),(,),0(,0 =
∂
∂
==
∂
∂
= tl
t
w
tlwt
t
w
tw     
Model: 
The Galerkin discretization procedure was used in order to represent the solution of Equation 2 in terms of a linearly 
independent set of basis functions ( )xnφ  where each basis function satisfies the boundary conditions described in 
Equation 3. This procedure permits transforming the nonlinear partial differential Equation 2 into a finite system of 
nonlinear Mathieu equations. 
Since we are interested in the response of the resonator at resonance when the first mode is dominant, we reduce the 
system of ordinary differential equations given by Galerkin discretization to a nonlinear Mathieu equation. 
A perturbation technique [5] was used in order to obtain two first order non-linear ordinary-differential equations 
which describe the amplitude and phase modulation of the response and permits the computation of its stability. 
Results: 
The effect of the Coriolis force frequency in the resonator frequency response at its primary resonance is shown in the 
Figure (2). We observe the separation of the curve branches when the Coriolis frequency is closed to the resonator 
frequency. This particular case is treated in Figure (3) for different kinds of resonator behavior. It appears that the 
quality factor decreases when the angular rate of the microgyroscope increases. Figure (4) shows the effect of the 
parametric and the nonlinear terms in the periodicity of the resonator response and its amplitude. 
Conclusions: 
Another limitation for the full scale of a resonant microgyroscope is underlined: the frequency ratio between the proof 
mass actuation and the resonator sensing. It appears in Figure (5) that the symmetry can be broken between negative 
and positive Coriolis stress effect when the resonator lose the stability for high coriolis forces amplitude and 
frequency. The maximum of sensitivity is situated at 0.6 of frequency ratio. 
 
Corresponding author: N. Kacem, CEA/LETI /DRT/DIHS/LCMS, 17 rue des Martyrs 38054 Grenoble, France 
Tel: +33-4-38-78-01-27; Fax: +33-4-38-78-51-69; E-mail: najib.kacem@cea.fr  
 
References: 
[1] T. Roessig, “IntegratedMEMStuning fork oscillators for sensor applications,” Ph.D. dissertation, Department of Mechanical Engineering, University of 
California, Berkeley, 1998.  
[2] Gui, C., Legrenberg, R., Tilmans, H.A., Fluitman, J.H.J., Elwenspoek, M.: “Nonlinearity and hysteresis of resonant strain gauges”, J. Microelectromech. Syst. 
7, 122–127 (1998)    
[3] Barry E. DeMartini, Holly E. Butterfield,, Jeff Moehlis, and Kimberly L. Turner, Member, “Chaos for a Microelectromechanical Oscillator Governed by the 
Nonlinear Mathieu Equation”, J. Microelectromech. Syst., Vol. 16, No. 6, December 2007.                 
[4] Seshia, A.A. ; Howe, R.T.; Montague, S, “An integrated microelectromechanical resonant output gyroscope,” Technical Digest Fifteenth IEEE International 
Conference on Micro Electro Mechanical Systems (MEMS 2002), Las Vegas, NV, p.722-726. 
[5] Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981). 
 
 
 
(1) 
(2) 
(3) 
  
  
    
  
 
 
 
10 20 30 40
t
-0.10
-0.05
0.05
0.10
w
 
 
0.2 0.4 0.6 0.8 1.0
D
0.010
0.012
0.014
0.016
wmax
q = 250 ±ês
q = 100 ±ês
q = -100 ±ês
q = -250 ±ês
 
 
 
 
Fe
 
Fc
 
M 
Fc
 
Fc
 
Fc
 
Fc
 
Fig.3: Predicted forced frequency responses for different angular rates of the resonant microgyrosope at a frequency equal to the 
resonator frequency with different kinds of behaviour. Wmax is the normalized displacement with respect to the gap. 
. 
 Black curve:  0 °/s 
 
 Blue curve:  250 °/s 
 
 Green curve:  500 °/s 
 
 Red curve:  1000 °/s 
 Black curve:  0 °/s 
 
 Blue curve:  100 °/s 
 
 Green curve:  200 °/s 
 
 Red curve:  500 °/s 
 Black curve:  0 °/s 
 
 Blue curve:  100 °/s 
 
 Green curve:  200 °/s 
 
 Red curve:  500 °/s 
Fig.3.a: Softening behaviour 
. 
Fig.3.b: Mixed behaviour 
. 
Fig.3.c: Hardening behaviour 
. 
Resonator design: length =100µm, width = 5µm, thickness = 2µm. 
Resonator design: length =50µm,  
width = 0.5µm, thickness = 2µm. 
Fig.1: Schema of the resonant microgyroscope. 
. 
Fig.2: Predicted forced frequency responses including Coriolis force 
at different frequencies. Microgyroscope angular rate = 150 °/s. 
 Black curve: 
Coriolis frequency = 0.25 resonator frequency 
 Blue curve:   
Coriolis frequency = 0.5 resonator frequency 
 Green curve:   
Coriolis frequency = 0.75 resonator frequency 
 Red curve:   
Coriolis frequency = resonator frequency 
  
Resonator design:  
Length =50µm 
Width = 0.5µm 
Thickness = 2µm 
Fig.5 Variation of Wmax with the frequency ratio 
between the coriolis force and the sensing resonance. 
θ is the angular rate of the microgyroscope. 
Fig.4.a: Phase plane 
. 
Fig.4.b: Time history 
. 
Coriolis frequency = resonator frequency 
 
 Green curve :  
Coriolis frequency = 0.1 resonator frequency 
 Black curve : 
Coriolis frequency = resonator frequency 
 
Fig.4: Long time integration 
Resonator design: length =50µm,  
width = 0.5µm, thickness = 2µm 
Microgyroscope angular rate = 1000 °/s 
 
Y 
X 
Z 
θ 
Fc
 
Fe
 


Non linear dynamics of Mathieu resonators for resonant gyroscope applications

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Non linear dynamics of Mathieu resonators for resonant gyroscope applications

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