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Description: Mass-Spring-Damper System
This benchmark is a generalization of the nonlinear mass-spring-damper system presented in [1], which is concerned with modeling the a mechanical systems consisting of chained masses, linear and nonlinear springs, and dampers. The underlying mathematical model is a second order system: \[ \begin{align} M \ddot{x}(t) + D \dot{x}(t) + K x(t) + f(\dot{x}(t)) &= B u(t), \\ y(t) &= C x(t). \end{align} \]
First Order Representation
\[ \begin{align} \begin{pmatrix} 1 & 0 \\ 0 & M \end{pmatrix} \begin{pmatrix} \dot{p} \\ \dot{v} \end{pmatrix} &= \begin{pmatrix} 0 & 1 \\ K & D \end{pmatrix} \begin{pmatrix} p \\ v \end{pmatrix} + \begin{pmatrix} 0 \\ f(p) \end{pmatrix} + \begin{pmatrix} 0 \\ B \end{pmatrix} \\ y &= \begin{pmatrix} C & 0 \end{pmatrix} \begin{pmatrix} p \\ v \end{pmatrix} \end{align} \]
\[ M = m \begin{pmatrix} 1 \\ & \ddots \end{pmatrix} K = k_l \begin{pmatrix} -2 & 1 \\ 1 & -2 & \ddots \\ & \ddots & \ddots \end{pmatrix}, \quad D = d \begin{pmatrix} -2 & 1 \\ 1 & -2 & \ddots \\ & \ddots & \ddots \end{pmatrix}, \quad B = \begin{pmatrix} 0 \\ \vdots \\ 0 \\ 1 \end{pmatrix}, \quad C = \begin{pmatrix} 0 & \dots & 0 & 1 \end{pmatrix}, \quad f(p) = -k_n \Big( \begin{pmatrix} 1 & -1 \\ & \ddots & \ddots \end{pmatrix} p \Big)^3 -k_n \Big( \begin{pmatrix} 1 \\ -1 & \ddots \\ & \ddots \end{pmatrix} p \Big)^3 \]
Data
Dimensions
System structure:
Citation
To cite this benchmark, use the following references:
- For the benchmark itself and its data:
- The MORwiki Community, Mass-Spring-Damper System. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Mass-Spring-Damper
@MISC{morwiki_msd,
author = {{The MORwiki Community}},
title = {Mass-Spring-Damper System},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = {http://modelreduction.org/index.php/Mass-Spring-Damper},
year = 2018
}
- For the background on the benchmark:
@INPROCEEDINGS{morKawS15,
title = {Model Reduction by Generalized Differential Balancing},
author = {Y. Kawano and J.M.A. Scherpen},
booktitle = {Mathematical Control Theory I: Nonlinear and Hybrid Control Systems},
series = {Lecture Notes in Control and Information Sciences},
volume = {461},
pages = {349--362},
year = {2015},
doi = {10.1007/978-3-319-20988-3}
}
References
- ↑ Y. Kawano and J.M.A. Scherpen, Model Reduction by Generalized Differential Balancing, In: Mathematical Control Theory I: Nonlinear and Hybrid Control Systems, Lecture Notes in Control and Information Sciences 461: 349--362, 2015.