Electrostatic Beam

Electrostatic Beam

Background
Benchmark ID	electrostaticBeam_n38m1q1 electrostaticBeam_n398m1q1
Category	oberwolfach
System-Class	NLTI-SOS
Parameters
nstates	38 398
ninputs	1
noutputs	1
nparameters	0
components	B, C, E, F, K, M, f
Copyright
License	NA
Creator	Christian Himpe
Editor	Christian Himpe
Location
NA

Description: Beam Actuated by Electrostatic Force

Figure 1

Figure 2: Degrees of freedom $y$ and $\theta_z$

Moving structures are an essential part for many microsystem devices, among them fluidic components like pumps and electrically controllable valves, sensing cantilevers, and optical structures.

Several actuation principles can be employed on microscopic length scales, certainly most frequent electromagnetic forces. While electrostatic actuation falls behind at the macro scale, the effect of charged bodies outperforms magnetic forces in the micro scale both in terms of performance and manufacturing expense.

While the single component can easily be simulated on a usual desktop computer, the calculation of a system of many coupled devices still presents a challenge. This challenge is raised by the fact that many of these devices show a nonlinear behavior. Especially for electrostatic structures, a further difficulty is the large reach of the electrostatic forces, leading to a strong spatial coupling of charges.

Accurate modelling of such a system typically leads to high order models. The tasks of simulation, analysis and controller design of high order nonlinear control systems can be simplified by reducing the order of the original system and approximating it by a lower order model.

An application of electrostatic moving structures are e.g. RF switches or filters. Given a simple enough shape, they often can be modelled as one-dimensional beams embedded in two or three dimensional space.

Model description

This model describes a slender beam which is actuated by a voltage between the beam and the ground electrode below (see Fig. 1).

On the beam, at least three degrees of freedom per node have to be considered:

$y$	Flexural displacement
$\theta_z$	Flexural rotation
$q$	Charge

On the ground electrode, all spatial degrees of freedom are fixed, so only charge has to be considered. The beam is supported either on the left side or on both sides. The damping matrix is calculated by a linear combination of the mass matrix $M$ and the stiffness matrix $K$.

The calculation of the electrostatic force would require a boundary element discretization, where it would be necessary to recalculate the capacity matrix for each time-step due to the motion of the charges This would require an integration over the beam's elements and could be written in analytical form by using e.g. Gauss integration; however, the complexity of the resulting system would be too high. We therefore use the method shown in , i.e. we concentrate the charges on the nodes. The capacity matrix then follows a simple $1/r$ law.

Origin

This benchmark is part of the Oberwolfach Benchmark Collection.

Data

Based on the finite element discretization presented in , an interactive matrix generator has been created using Wolfram Research's webMathematica. Models produced by this generator are in the DSIF format, which allows for nonlinear terms.

All examples are made for a silicon beam with the following properties:

Property	Value
Beam length (l)	$10^{-5} m$
Beam height (h)	$10^{-6} m$
Beam width (w)	$15 \cdot 10^{-6} m$
Distance between beams (s)	$200 \cdot 10^{-9} m$
Material density (rho)	$2330 kg/m^3$
Cross-sectional area (A)	$150 \cdot 10^{12} m^2$
Moment of inertia (I)	$1.25 \cdot 10^{-21} m^4$
Modulus of elasticity (E)	$1.31 \cdot 10^{11} Pa$
Contribution of $M$ to damping	$0$
Contribution of $K$ to damping	$10^{-6}$
Support	Both sides, $y$ DOF only

The following examples are available (all files are zipped compressed DSIF files, Units: SI):

File	Number of Nodes	Number of Equations	Compressed Size [kB]
ElectrostaticBeam-dim1e1-E10.zip	10	38	4144
ElectrostaticBeam-dim1e2-E100.zip	100	398	347679

The .m files contain matrices $M$, $E$, $K$, $B$, $F$ and $C$, the vector $f$ and initial conditions for the following system of equations:

$\begin{align} M \ddot{x}(t) + E\dot{x}(t) + K x(t) &= B u(t) + F f(x(t),u(t)) \\ y(t) &= C x(t) \end{align}$

where $B$ is a $n \times 1$ matrix with $1$ at all charge DOFs of the upper beam and $C$ is a $1 \times n$ matrix with the only nonzero entry at the $y$ DOF of the middle node.

Details of the implementation are available in a separate report (ElectrostaticBeam.pdf), see also .

A typical input to this system is a step response; periodic on/off switching is also possible. The reduced model should thus both represent the step response as well as the possible influence of higher order harmonics.

Dimensions

System structure:

$\begin{align} M \ddot{x}(t) + E\dot{x}(t) + K x(t) &= B u(t) + F f(x(t)) \\ y(t) &= C x(t) \end{align}$

System dimensions:

$M \in \mathbb{R}^{N \times N}$, $E \in \mathbb{R}^{N \times N}$, $K \in \mathbb{R}^{N \times N}$, $B \in \mathbb{R}^{N \times 1}$, $F \in \mathbb{R}^{N \times S}$, $f : \mathbb{R}^N \to \mathbb{R}^S$, $C \in \mathbb{R}^{1 \times N}$.

System variants: E10: $N = 38$, $S = 28$, E100: $N = 398$, $S = 298$.

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

The MORwiki Community, Electrostatic Beam. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Electrostatic_Beam

@MISC{morwiki_ebeam,
  author =       {{The MORwiki Community}},
  title =        {Electrostatic Beam},
  howpublished = {{MORwiki} -- Model Order Reduction Wiki},
  url =          {https://modelreduction.org/morwiki/Electrostatic_Beam},
  year =         {20XX}
}

• For the background on the benchmark:

@ARTICLE{morLieRK06,
  author =       {J. Lienemann and E.B. Rudnyi and J.G. Korvink},
  title =        {{MST} {MEMS} model order reduction: Requirements and benchmarks},
  journal =      {Journal of Biomechanics},
  volume =       {415},
  number =       {2--3},
  pages =        {469--498},
  year =         {2006},
  doi =          {10.1016/j.laa.2005.04.002}
}