Modified Gyroscope

Description of the device

The device is a MEMS gyroscope based on the butterfly gyroscope [1] developed at the Imego institute in Gothenburg, Sweden (see also: http://simulation.uni-freiburg.de/downloads/benchmark/The Butterfly Gyro (35889)). A gyroscope is a device used to measure angular rates in up to three axes.

The basic working principle of the gyroscope can be described as follows [2]. Without applied external rotation, the paddles vibrate in phase with the function z(t). Under the influence of an external rotation about the x-axis (drawn in red), an additional force due to the Coriolis acceleration acts upon the paddles. This force leads to an additional small out-of-phase vibration between two paddles on the same side of the bearing. This out-of phase vibration is measured as the difference of the z-displacement of the nodes with the red dots. Thus, measuring the displacement of two adjacent paddles, the rotation velocity can be ascertained.

Motivation of MOR

When planning for and making decisions on future improvements of the Butterfly Gyroscope, it is of importance to improve the efficiency of the gyro simulations. Repeated analysis of the sensor structure have to be conducted with respect to a number of important issues. Examples of such are sensitivity to shock, linear and angular vibration sensitivity, reaction to large rates and/or acceleration, different types of excitation load cases and the effect of force-feedback. The use of model order reduction indeed decreases run time for repeated simulations.

Description of the parametrized model

Two parameters are of special interest for the model. The first one is the quantity that is to be sensed, the rotation velocity around the $x$-axes. The second parameter is the width of the bearing, d. The parametrized system below is obtained by finite element discretization of the parametrized model (in the form of partial differential equations) for the gyroscope. (The details of constructing the parametrized system can be found in [2]. The system is of the following form: $M(d)\ddot{x}+D(\theta)\dot{x}+T(d)x=B y=Cx.$

Here, $M(d)=(M_1+dM_2)\in \mathbb R^{n\times n}$is the mass matrix,

$D(\theta)=\theta(D_1+dD_2)\in \mathbb R^{n\times n}$ is the damping matrix,

$T(d)=T_1+(1/d)T_2+dT_3\in \mathbb R^{n\times n}$ is the stiffness matrix,

$B \in \mathbb R^{n\times m_I}$ is the load vector, $C \in \mathbb R^{m_O\times n}$ is the output matrix, $x \in \mathbb R^{n}$ is the state vector, and $y \in \mathbb R^{m_O}$ is the output response.

The variables $d$ and $\theta$ are the parameters of the system, where $d$ is the width of the bearing and $\theta$ is the rotation velocity along the $x$-axis.

The interesting output $y$ of the system is $\delta$ which is the difference of the displacement $z(t)$ between the two red dots on the same side of the bearing (see Fig.1). The number of degrees of freedom is $n=17913$.

The interesting range for the parameters are: $\theta\in [10^{-7}, 10^{-5}]$ and $d\in [1,2]$. The device works in the frequency range $f\in [0.025, 0.25]$MHz. Typical values of the damping variables $\alpha$ and $\beta$ are $0$ and $10^{-6}$, respectively. It has been proven by Salimbahrami [3] that the matrices associated with the damping variables do not have to be taken into account during the process of model reduction.

Data information

The systems matrices $M_1, \, M_2, \, D_1, \, D_2, \, T_1, \, T_2$, and $T_3$ are in the MatrixMarket format (http://math.nist.gov/MatrixMarket/), and can be downloaded here File:Matrices.tgz. $M_1$ corresponds to the file Bmass1.dat, $M_2$ corresponds to the file Bmass2.dat., $D_1, \, D_2$ correspond to the files Bdamp1.dat, Bdamp2.dat, respectively. $T_1, \, T_2, \, T_3$ correspond to the files Bstiff1.dat, Bstiff2.dat, Bstiff3.dat, respectively. The load vector $B$ can be obtained from the file Bload.dat.

References

[1] J. Lienemann, D. Billger, E. B. Rudnyi, A. Greiner, J. G. Korvink, "MEMS Compact Modeling Meets Model Order Reduction: Examples of the Application of Arnoldi Methods to Microsystem Devices," Nanotech, 2004, pp. 303–306.

[2] C. Moosmann, "ParaMOR---- Model Order Reduction for parameterized MEMS applications," PhD thesis, Department of Microsystems Engineering, University of Freiburg, 2007.

[3] B. Salimbahrami, R. Eid, B. Lohmann, "Model Reduction by Second Order Krylov Subspaces: Extensions, Stability and Proportional Damping," IEEE International Symposium on Intelligent Control, 2006, pp. 2997–3002.

Fig.1