Mass-Spring-Damper

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Description: Mass-Spring-Damper System

This benchmark is a generalization of the nonlinear mass-spring-damper system presented in , 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:

$\begin{array}{rl}M\ddot{x}(t)+D\dot{x}(t)+Kx(t)+f(\dot{x}(t))& =Bu(t),\\ y(t)& =Cx(t).\end{array}$

First Order Representation

$\begin{array}{rl}\left(\begin{array}{cc}1& 0\\ 0& M\end{array}\right)\left(\begin{array}{c}\dot{p}\\ \dot{v}\end{array}\right)& =\left(\begin{array}{cc}0& 1\\ K& D\end{array}\right)\left(\begin{array}{c}p\\ v\end{array}\right)+\left(\begin{array}{c}0\\ f(p)\end{array}\right)+\left(\begin{array}{c}0\\ B\end{array}\right)\\ y& =\left(\begin{array}{cc}C& 0\end{array}\right)\left(\begin{array}{c}p\\ v\end{array}\right)\end{array}$

$M=m\left(\begin{array}{c}1\\ & \ddots \end{array}\right)K=k_l\left(\begin{array}{cc}-2& 1\\ 1& -2& \ddots \\ & \ddots & \ddots \end{array}\right),D=d\left(\begin{array}{cc}-2& 1\\ 1& -2& \ddots \\ & \ddots & \ddots \end{array}\right),B=\left(\begin{array}{c}0\\ ⋮\\ 0\\ 1\end{array}\right),C=\left(\begin{array}{cccc}0& \dots & 0& 1\end{array}\right),f(p)=-k_n(\left(\begin{array}{cc}1& -1\\ & \ddots & \ddots \end{array}\right)p{)}^3-k_n(\left(\begin{array}{c}1\\ -1& \ddots \\ & \ddots \end{array}\right)p{)}^3$

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}
}

Contact

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