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

First Order Representation

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

\[ \begin{align} \begin{pmatrix} 1 & 0 \\ 0 & M \end{pmatrix} \begin{pmatrix} \dot{p} \\ \dot{v} \end{pmatrix} &= \begin{pmatrix} 0 & 1 \\ K & D \end{pmatrix} \begin{pmatrix} p \\ v \end{pmatrix} + \begin{pmatrix} 0 \\ f_p(p) \end{pmatrix} + \begin{pmatrix} 0 \\ B_v \end{pmatrix} \\ y &= \begin{pmatrix} C_p & 0 \end{pmatrix} \begin{pmatrix} p \\ v \end{pmatrix} \end{align} \]

with the components:

\[ M = m \begin{pmatrix} 1 \\ & \ddots \end{pmatrix}, \quad K_0 = k_l \begin{pmatrix} -2 & 1 \\ 1 & -2 & \ddots \\ & \ddots & \ddots \end{pmatrix}, \quad D = d \begin{pmatrix} -2 & 1 \\ 1 & -2 & \ddots \\ & \ddots & \ddots \end{pmatrix}, \quad B_v = \begin{pmatrix} 0 \\ \vdots \\ 0 \\ 1 \end{pmatrix}, \quad C_p = \begin{pmatrix} 0 & \dots & 0 & 1 \end{pmatrix}, \]

and the nonlinear term:

\[ f_p(p) = -k_n \Big( \begin{pmatrix} 1 & -1 \\ & \ddots & \ddots \end{pmatrix} p \Big)^3 -k_n \Big( \begin{pmatrix} 1 \\ -1 & \ddots \\ & \ddots \end{pmatrix} p \Big)^3 \]

and thus yielding the classic first order components:

\[ E = \begin{pmatrix} 1 & 0 \\ 0 & M \end{pmatrix}, \quad A = \begin{pmatrix} 0 & 1 \\ K & D \end{pmatrix}, \quad B = \begin{pmatrix} 0 \\ B_v \end{pmatrix}, \quad C = \begin{pmatrix} C_p & 0 \end{pmatrix}. \]

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}
}