Mass-Spring-Damper

Note: This page has not been verified by our editors.

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:

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

First Order Representation

$\begin{align} \begin{pmatrix} 1 & 0 \\ 0 & M \end{pmatrix} \begin{pmatrix} \dot{p} \\ \dot{v} \end{pmatrix} &= \begin{pmatrix} 0 & 1 \\ K & D \end{pmatrix} \begin{pmatrix} p \\ v \end{pmatrix} + \begin{pmatrix} 0 \\ f(p) \end{pmatrix} + \begin{pmatrix} 0 \\ B \end{pmatrix} \\ y &= \begin{pmatrix} C & 0 \end{pmatrix} \begin{pmatrix} p \\ v \end{pmatrix} \end{align}$

$M = m \begin{pmatrix} 1 \\ & \ddots \end{pmatrix} K = k_l \begin{pmatrix} -2 & 1 \\ 1 & -2 & \ddots \\ & \ddots & \ddots \end{pmatrix}, \quad D = d \begin{pmatrix} -2 & 1 \\ 1 & -2 & \ddots \\ & \ddots & \ddots \end{pmatrix}, \quad B = \begin{pmatrix} 0 \\ \vdots \\ 0 \\ 1 \end{pmatrix}, \quad C = \begin{pmatrix} 0 & \dots & 0 & 1 \end{pmatrix}, \quad f(p) = -k_n \Big( \begin{pmatrix} 1 & -1 \\ & \ddots & \ddots \end{pmatrix} p \Big)^3 -k_n \Big( \begin{pmatrix} 1 \\ -1 & \ddots \\ & \ddots \end{pmatrix} p \Big)^3$

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

Christian Himpe