# Eletrostatic beam (38882)

### Beam Actuated by Electrostatic Force (38882)

Jan Lienemann <lieneman (at) imtek (dot) de>, Andreas Greiner <greiner (at) imtek (dot) de>, and Jan G. Korvink <korvink (at) imtek (dot) de>

## Introduction

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, the most frequent certainly the 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 fabrication 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 approximate it by a lower order model.

A 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 beem and the ground electrode below (see figure 1).

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

**y**- Flexural displacement
**thetaz**- Flexural rotation
**q**- Charge

(Figure 2: degrees of freedom y and theta z).

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 timestep due to the motion of the charges. This would require an integration over the beam's elements. This 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 [1], i.e. we concentrate the charges on the nodes; The capacity matrix then follows a simple 1/r law.

## Benchmark examples

Based on the finite element
discretization presented in [2], 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) | 100e-6 m |

Beam height (h) | 10e-6 m |

Beam width (w) | 15e-6 m |

Distance between beams (s) | 200e-9 m |

Material density (rho) | 2330 kg/m^{3} |

Cross-sectional area (A) | 150e-12 m^{2} |

Moment of inertia (I) | 1.25e-21 m^{4} |

Modulus of elasticity (E) | 1.31e11 Pa |

Contribution of M to damping | 0 |

Contribution of K to damping | 1e-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 | File size/Compressed size |
---|---|---|---|

(File 1) |
10 | 38 | 47537 / 4144 |

(File 2) |
100 | 398 | 5007860 / 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:

M x" + E x' + K x | = B u + F f(x,u) |

y | = C x |

where B is a n×1 matrices with 1 at all charge DOFs of the upper beam and C is a 1×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 (File 3). A full paper will soon appear in
*Linear Algebra and its Applications*. 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.

[1] L. Silverberg, and L. Weaver, Jr. "Dynamics and Control
of Electrostatic Structures", *Journal of Applied
Mechanics*, Vol. 63, p. 383--391 (June 1996)

[2] William Weaver, Jr., Stephen P. Timoshenko, and Donovan
H. Young, *Vibration problems in engineering*,
5th ed., Wiley (1990)

[3] Lienemann Jan, Rudnyi Evgenii B, Korvink Jan G: MST MEMS model order reduction: Requirements and benchmarks, Linear Algebra and its Applications Linear Algebra Appl, 2006; 415 (2-3): 469-498 http://dx.doi.org/10.1016/j.laa.2005.04.002