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:

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

First Order Representation

The second order system can be represented as a first order system as follows:

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

with the components:

$M=m\left(\begin{array}{c}1\\ & \ddots \end{array}\right),K_0=k_l\left(\begin{array}{cc}-2& 1\\ 1& -2& \ddots \\ & \ddots & \ddots \end{array}\right),D=d\left(\begin{array}{cc}-2& 1\\ 1& -2& \ddots \\ & \ddots & \ddots \end{array}\right),B_v=\left(\begin{array}{c}0\\ ⋮\\ 0\\ 1\end{array}\right),C_p=\left(\begin{array}{cccc}0& \dots & 0& 1\end{array}\right),$

and the nonlinear term:

$f_p(p)=-k_n(\left(\begin{array}{cc}1& -1\\ & \ddots & \ddots \end{array}\right)p{)}^3-k_n(\left(\begin{array}{c}1\\ -1& \ddots \\ & \ddots \end{array}\right)p{)}^3$

and thus yielding the classic first order components:

$E=\left(\begin{array}{cc}1& 0\\ 0& M\end{array}\right),A=\left(\begin{array}{cc}0& 1\\ K& D\end{array}\right),B=\left(\begin{array}{c}0\\ B_v\end{array}\right),C=\left(\begin{array}{cc}{C}_p& 0\end{array}\right).$

The parameters for the mass $m$, linear spring constant $k_l$, nonlinear spring constant $k_n$, and damping $d$ are chosen in as $m=1$, $k_l=1$, $k_n=2$, and $d=1$.

Data

The following Matlab code assembles the above described $E$, $A$, $B$ and $C$ parameter dependent matrices and the function $f$ for a given number of subsystems $N$.

function [E,A,B,C,f] = msd(N)

    U = speye(N);                     % Sparse unit matrix
    T = gallery('tridiag',N,-1,2,-1); % Sparse tridiagonal matrix
    H = gallery('tridiag',N,-1,1,0);  % Sparse transport matrix
    Z = sparse(N,N);                  % Sparse all zero matrix
    z = sparse(N,1);                  % Sparse all zero vector

    E = @(m) [U,Z;Z,m*U];             % Handle to parametric E matrix
    A = @(kl,d) [Z,U;kl*T,d*T];       % Handle to parametric A matrix
    B = sparse(2*N,1,1,2*N,1);
    C = sparse(N,1,1,2*N,1);
    f = @(x,kn) [z;-kn*( (H'*x(N+1:end)).^3 - (H*x(N+1:end)).^3)];
end

Dimensions

System structure:

$\begin{array}{rcl}E(m)\dot{x}(t)& =& A(k_l,d)x(t)+f(x(t);k_n)+Bu(t)\\ y(t)& =& Cx(t)\end{array}$

System dimensions:

$E\in {\mathbb{ℝ}}^{2N\times 2N}$, $A\in {\mathbb{ℝ}}^{2N\times 2N}$, $B\in {\mathbb{ℝ}}^{2N\times 1}$, $C\in {\mathbb{ℝ}}^{1\times 2N}$.

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 =          {https://modelreduction.org/morwiki/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}
}