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

First Order Representation

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

\begin{aligned} (\begin{matrix} 1 & 0 \\ 0 & M \end{matrix}) (\begin{matrix} \dot{p} \\ \dot{v} \end{matrix}) & = (\begin{matrix} 0 & 1 \\ K & D \end{matrix}) (\begin{matrix} p \\ v \end{matrix}) + (\begin{matrix} 0 \\ f_{p} (p) \end{matrix}) + (\begin{matrix} 0 \\ B_{v} \end{matrix}) \\ y & = (\begin{matrix} C_{p} & 0 \end{matrix}) (\begin{matrix} p \\ v \end{matrix}) \end{aligned}

with the components:

M = m (\begin{matrix} 1 \\ ⋱ \end{matrix}), K_{0} = k_{l} (\begin{matrix} - 2 & 1 \\ 1 & - 2 & ⋱ \\ ⋱ & ⋱ \end{matrix}), D = d (\begin{matrix} - 2 & 1 \\ 1 & - 2 & ⋱ \\ ⋱ & ⋱ \end{matrix}), B_{v} = (\begin{matrix} 0 \\ ⋮ \\ 0 \\ 1 \end{matrix}), C_{p} = (\begin{matrix} 0 & \dots & 0 & 1 \end{matrix}),

and the nonlinear term:

f_{p} (p) = - k_{n} ((\begin{matrix} 1 & - 1 \\ ⋱ & ⋱ \end{matrix}) p)^{3} - k_{n} ((\begin{matrix} 1 \\ - 1 & ⋱ \\ ⋱ \end{matrix}) p)^{3}

and thus yielding the classic first order components:

E = (\begin{matrix} 1 & 0 \\ 0 & M \end{matrix}), A = (\begin{matrix} 0 & 1 \\ K & D \end{matrix}), B = (\begin{matrix} 0 \\ B_{v} \end{matrix}), C = (\begin{matrix} C_{p} & 0 \end{matrix}) .

The parameters for the mass $m$ , linear spring constant $k_{l}$ , nonlinear spring constant $k_{n}$ , and damping $d$ are chosen in ^[1] 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; k_{n}) + B u (t) \\ y (t) & = & C x (t) \end{array}

System dimensions:

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

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

↑ ^1.0 ^1.1 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.

[kawano15-1] 1.0 ^1.1 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.

[1]

@@ Line 23: / Line 23: @@
 ===First Order Representation===
+The second order system can be represented as a first order system as follows:
 :<math>
@@ Line 28: / Line 30: @@
 \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} \\
+\begin{pmatrix} 0 \\ f_p(p) \end{pmatrix} + \begin{pmatrix} 0 \\ B_v \end{pmatrix} \\
-y &= \begin{pmatrix} C & 0 \end{pmatrix} \begin{pmatrix} p \\ v \end{pmatrix}
+y &= \begin{pmatrix} C_p & 0 \end{pmatrix} \begin{pmatrix} p \\ v \end{pmatrix}
 \end{align}
 </math>
+with the components:
 :<math>
-M = m \begin{pmatrix} 1 \\ & \ddots \end{pmatrix}
+M = m \begin{pmatrix} 1 \\ & \ddots \end{pmatrix}, \quad
-K = k_l \begin{pmatrix} -2 & 1 \\ 1 & -2 & \ddots \\ & \ddots & \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 = \begin{pmatrix} 0 \\ \vdots \\ 0 \\ 1 \end{pmatrix}, \quad
+B_v = \begin{pmatrix} 0 \\ \vdots \\ 0 \\ 1 \end{pmatrix}, \quad
-C = \begin{pmatrix} 0 & \dots & 0 & 1 \end{pmatrix}, \quad
+C_p = \begin{pmatrix} 0 & \dots & 0 & 1 \end{pmatrix},
-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
+</math>
+and the nonlinear term:
+:<math>
+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
 </math>
+and thus yielding the classic first order components:
+:<math>
+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}.
+</math>
+The parameters for the mass <math>m</math>, linear spring constant <math>k_l</math>, nonlinear spring constant <math>k_n</math>, and damping <math>d</math> are chosen in <ref name="kawano15"/> as <math>m=1</math>, <math>k_l=1</math>, <math>k_n=2</math>, and <math>d=1</math>.
 ==Data==
+The following Matlab code assembles the above described <math>E</math>, <math>A</math>, <math>B</math> and <math>C</math> parameter dependent matrices and the function <math>f</math> for a given number of subsystems <math>N</math>.
+<div class="thumbinner" style="width:540px;text-align:left;">
+<source lang="matlab">
+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
+</source>
+</div>
@@ Line 51: / Line 91: @@
 System structure:
+:<math>
+\begin{array}{rcl}
+E(m) \dot{x}(t) &=& A(k_l,d)x(t) + f(x;k_n) + Bu(t) \\
+y(t) &=& Cx(t)
+\end{array}
+</math>
+System dimensions:
+<math>E \in \mathbb{R}^{2N \times 2N}</math>,
+<math>A \in \mathbb{R}^{2N \times 2N}</math>,
+<math>B \in \mathbb{R}^{2N \times 1}</math>,
+<math>C \in \mathbb{R}^{1 \times 2N}</math>.
 ==Citation==
@@ Line 80: / Line 132: @@
     doi =          {10.1007/978-3-319-20988-3}
   }
 ==References==
@@ Line 89: / Line 140: @@
 </references>
-==Contact==
-[[User:Himpe]]