Modified Gyroscope

Modified Gyroscope

Background
Benchmark ID	modifiedGyroscope_n17931m1q1
Category	misc
System-Class	AP-LTI-SOS
Parameters
nstates	17931
ninputs	1
noutputs	1
nparameters	3
components	B, C, E, K, M
Copyright
License	NA
Creator	Lihong Feng
Editor	Lihong Feng Christian Himpe Judith Schneider Jens Saak Kathryn Lund
Location
NA

Description

Figure 1: Schematic representation of the gyroscope.

The device is a MEMS gyroscope based on the butterfly gyroscope developed at the Imego institute in Gothenburg, Sweden (see also: Butterfly Gyroscope, where a non-parametrized model for the device is given). 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, see also . Without applied external rotation, the paddles vibrate in phase with the function $z(t),$ see Fig. 1. 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

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.

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 $\theta$ 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 . The system is of the following form:

$\begin{align} M(d)\ddot{x}(t) +D(d,\theta)\dot{x}(t) +T(d)x(t) &= B, \\ y(t) &=Cx(t). \end{align}$

Here,

• $M(d)=M_1 + dM_2\in \mathbb R^{n\times n}$ is the mass matrix,
• $D(d,\theta)=\theta(D_1 + d D_2)\in \mathbb R^{n\times n}$ is the damping matrix,
• $T(d)=T_1+(1/d)T_2+dT_3\in \mathbb R^{n\times n}$ is the stiffness matrix,
• $B \in \mathbb R^{n \times 1}$ is the load vector,
• $C \in \mathbb R^{1 \times n}$ is the output matrix,
• $x \in \mathbb R^{n}$ is the state vector,
• and $y \in \mathbb R$ is the output response.

The quantity of interest $y$ of the system is $\delta z(t)$, which is the difference of the displacement $z(t)$ between the two red markings on the east side of the bearing (see xx--CrossReference--dft--fig:gyro--xx).

The parameters of the system, $d$ and $\theta$, represent the width of the bearing($d$) and the rotation velocity along the $x$-axis ($\theta$), with the ranges: $\theta\in [10^{-7}, 10^{-5}]$ and $d\in [1,2]$.

The device works in the frequency range $f \in [0.025, 0.25]$MHz and the degrees of freedom are $n = 17913$.

Data

The model is generated in ANSYS. The system matrices $M_1, \, M_2, \, D_1, \, D_2, \, T_1, \, T_2, \, T_3$, and $B$ are in the MatrixMarket format, and can be downloaded here: Gyroscope_modi.tgz. The matrix $C$ defines the output, which has zeros on all the entries, except on the 2315th entry, where the value is $-1$, and on the 5806th entry, the value is $1$; in MATLAB notation, it is C(1,2315) = -1 and C(1,5806) = 1.

The matrices have been reindexed for MORB, as shown in the next section. In particular, $D_i$ is denoted instead by $E_i$, and $T_i$ by $K_i$.

Dimensions

System structure:

$\begin{array}{rcl} (M_1 + d M_2)\ddot{x}(t) + \theta(E_1 + d E_2) \dot{x}(t) + (K_1 + d^{-1} K_2 + d K_3)x(t) &=& B \\ y(t) &=& Cx(t) \end{array}$

System dimensions:

$M_{1,2}, E_{1,2}, K_{1,2,3} \in \mathbb{R}^{17931 \times 17931}$, $B \in \mathbb{R}^{17931 \times 1}$, $C \in \mathbb{R}^{1 \times 17931}$.

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

The MORwiki Community, Modified Gyroscope. MORwiki - Model Order Reduction Wiki, 2018. https://modelreduction.org/morwiki/Modified_Gyroscope

@MISC{morwiki_modgyro,
  author =       {{The MORwiki Community}},
  title =        {Modified Gyroscope},
  howpublished = {{MORwiki} -- Model Order Reduction Wiki},
  url =          {https://modelreduction.org/morwiki/Modified_Gyroscope},
  year =         2018
}

• For the background on the benchmark: morMoo07 (BibTeX)

Contact

Lihong Feng