# Linear 1D Beam Model (38861)

### Linear 1D Beam Model (38861)

Jan Lienemann <lieneman (at) imtek (dot) de>, Andreas Greiner <greiner (at) imtek (dot) de>, 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

(see Figure 1)

This model describes a slender beam with four degrees of freedom per node:

**x**- Axial displacement
**thetax**- Axial rotation
**y**- Flexural displacement
**thetaz**- Flexural rotation

See figure 2 for Degree of Freedom x, figure 3 for Degree of Freedom theta x and figure 4 for Degrees of freedom y and theta z.

The beam is supported either on the left side or on both
sides. For the left side (fixed) support, the force is applied on
the rightmost node in *y* direction, whereas for the
support on both sides (simply supported), a node in the middle is
loaded. The damping matrix is calculated by a linear combination
of the mass matrix *M* and the stiffness matrix
*K*.

## Benchmark examples

Based on the finite element
discretization presented in [1], an interactive matrix generator
has been created using Wolfram
Research's *webMathematica*. However, models produced by this
generator are in the DSIF format, which allows for nonlinear
terms. For the purpose of the benchmark collection, we have
precomputed four systems and converted them to the Matrix market
format which is easier to import in standard computer algebra
packages.

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

Property | Value |
---|---|

Beam length (l) | 0.1 m |

Material density (rho) | 8000 kg/m^{3} |

Cross-sectional area (A) | 7.854e-7 m^{2} |

Moment of inertia (I) | 4.909e-14 m^{4} |

Polar moment of inertia (J) | 9.817e-14 |

Modulus of elasticity (E) | 2e11 Pa |

Poisson ratio (nu) | 0.29 |

Contribution of M to damping | 1e2 |

Contribution of K to damping | 1e-2 |

Support | Simple, both sides |

The following examples are available (all files are compressed .zip archives, Units: SI):

File | Degrees of freedom | Number of nodes | Number of equations | File size/Compressed size |
---|---|---|---|---|

File 1 |
flexural (y and thetaz) | 10 | 18 | 5935 / 2384 |

File 2 |
flexural (y and thetaz) | 10000 | 19998 | 6640324 / 716807 |

File 3 |
flexural (y and thetaz), axial, torsional | 5 | 14 | 4045 / 2255 |

File 4 |
flexural (y and thetaz), axial, torsional | 50000 | 19994 | 5532532 / 627991 |

The zip files contain matrices M, E, K, B and E for the following system of equations:

M x" + E x' + K x | = B u |

y | = C x |

where B is a n×1 matrix 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 5). 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] William Weaver, Jr., Stephen P. Timoshenko, and Donovan
H. Young, *Vibration problems in engineering*,
5th ed., Wiley (1990)

[2] 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

Last modified: Wed Sep 8 16:53:01 CEST 2004