Linear 1D Beam

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Description

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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 with four degrees of freedom per node:

x	Axial displacement
$\theta_x$	Axial rotation
$y$	Flexural displacement
$\theta_z$	Flexural rotation

See xx--CrossReference--dft--fig2--xx for Degree of Freedom $x$, xx--CrossReference--dft--fig3--xx for Degree of Freedom $\theta_x$ and xx--CrossReference--dft--fig4--xx 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 $K$.

Origin

This benchmark is part of the Oberwolfach Benchmark Collection; No. 38861.

Data

Based on the finite element discretization presented in, 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/m3
Cross-sectional area (A)	$7.854\cdot 10^{-7}$ m2
Moment of inertia (I)	$4.909\cdot 10^{-14}$ m4
Polar moment of inertia (J)	$9.817\cdot 10^{-14}$
Modulus of elasticity (E)	$2\cdot 10^{11}$ Pa
Poisson ratio (nu)	$0.29$
Contribution of $M$ to damping	$100$
Contribution of $K$ to damping	$0.01$
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 [kB]	Compressed size [kB]
LF10.zip	flexural ($y$ and $\theta_z$)	10	18	5935	2384
LF10000.zip	flexural ($y$ and $\theta_z$)	10000	19998	6640324	716807
LFAT5.zip	flexural ($y$ and $\theta_z$), axial, torsional	5	14	4045	2255
LFAT5000	flexural ($y$ and $\theta_z$), axial, torsional	50000	19994	5532532	627991

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

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

where $B$ is a $n \times 1$ matrix 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. 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. See also .

Dimensions

System structure:

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

System dimensions:

$M \in \mathbb{R}^{N \times N}$, $E \in \mathbb{R}^{N \times N}$, $K \in \mathbb{R}^{N \times N}$, $B \in \mathbb{R}^{N \times 1}$, $C \in \mathbb{R}^{1 \times N}$

System variants:

LF10: $N = 18$, LF100: $N = 19998$, LFAT5: $N = 14$, LFAT5000: $N = 19994$,