Mass-Spring-Damper - Revision history

Saak: /* Data */

2026-01-23T17:26:10Z

Data

← Older revision		Revision as of 17:26, 23 January 2026
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	<div class="thumbinner" style="width:540px;text-align:left;">		<div class="thumbinner" style="width:540px;text-align:left;">
	<~~source~~ lang="matlab">		<syntaxhighlight lang="matlab">
	function [E,A,B,C,f] = msd(N)		function [E,A,B,C,f] = msd(N)

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	f = @(x,kn) [z;-kn( (H'x(N+1:end)).^3 - (H*x(N+1:end)).^3)];		f = @(x,kn) [z;-kn( (H'x(N+1:end)).^3 - (H*x(N+1:end)).^3)];
	end		end
	</~~source~~>		</syntaxhighlight>
	</div>		</div>


	==Dimensions==		==Dimensions==

Schuetz2 at 05:34, 17 June 2025

2025-06-17T05:34:19Z

← Older revision		Revision as of 05:34, 17 June 2025
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	title = {Mass-Spring-Damper System},		title = {Mass-Spring-Damper System},
	howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},		howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
	url = <nowiki>{~~http~~://modelreduction.org/~~index.php~~/Mass-Spring-Damper}</nowiki>,		url = <nowiki>{https://modelreduction.org/morwiki/Mass-Spring-Damper}</nowiki>,
	year = 2018		year = 2018
	}		}

Mitchell: x(t) instead of x instead f

2022-03-24T13:54:28Z

x(t) instead of x instead f

← Older revision		Revision as of 13:54, 24 March 2022
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	:<math>		:<math>
	\begin{array}{rcl}		\begin{array}{rcl}
	E(m) \dot{x}(t) &=& A(k_l,d)x(t) + f(x;k_n) + Bu(t) \\		E(m) \dot{x}(t) &=& A(k_l,d)x(t) + f(x(t);k_n) + Bu(t) \\
	y(t) &=& Cx(t)		y(t) &=& Cx(t)
	\end{array}		\end{array}

Himpe: fix 2nd order form and doi.

2019-06-25T14:15:42Z

fix 2nd order form and doi.

← Older revision		Revision as of 14:15, 25 June 2019
Line 17:		Line 17:
	:<math>		:<math>
	\begin{align}		\begin{align}
	M \ddot{x}(t) + D \dot{x}(t) + K x(t) + f(~~\dot{~~x}(t)) &= B u(t), \\		M \ddot{x}(t) + D \dot{x}(t) + K x(t) + f(x(t)) &= B u(t), \\
	y(t) &= C x(t).		y(t) &= C x(t).
	\end{align}		\end{align}
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	<references>		<references>

	<ref name="kawano15">Y. Kawano and J.M.A. Scherpen, <span class="plainlinks">[https://doi.org/10.1007/978-3-319-20988-3 Model Reduction by Generalized Differential Balancing]</span>, In: Mathematical Control Theory I: Nonlinear and Hybrid Control Systems, Lecture Notes in Control and Information Sciences 461: 349--362, 2015.</ref>		<ref name="kawano15">Y. Kawano and J.M.A. Scherpen, <span class="plainlinks">[https://doi.org/10.1007/978-3-319-20988-3_19 Model Reduction by Generalized Differential Balancing]</span>, In: Mathematical Control Theory I: Nonlinear and Hybrid Control Systems, Lecture Notes in Control and Information Sciences 461: 349--362, 2015.</ref>

	</references>		</references>

Himpe: more content

2018-12-11T13:14:08Z

Himpe: init

2018-12-10T15:10:11Z

init

New page

{{preliminary}} 

[[Category:benchmark]]
[[Category:nonlinear]]
[[Category:ODE]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:second differential order]]
[[Category:Procedural]]
[[Category:SISO]]

==Description: Mass-Spring-Damper System==

This benchmark is a generalization of the nonlinear [[wikipedia:Mass-spring-damper_model|mass-spring-damper]] system presented in <ref name="kawano15"/>,
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:
:<math>
\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}
</math>

===First Order Representation===

:<math>
\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}
</math>

:<math>
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
</math>

==Data==

==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 = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Mass-Spring-Damper System},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Mass-Spring-Damper}</nowiki>,
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==

<references>

<ref name="kawano15">Y. Kawano and J.M.A. Scherpen, <span class="plainlinks">[https://doi.org/10.1007/978-3-319-20988-3 Model Reduction by Generalized Differential Balancing]</span>, In: Mathematical Control Theory I: Nonlinear and Hybrid Control Systems, Lecture Notes in Control and Information Sciences 461: 349--362, 2015.</ref>

</references>

==Contact==

[[User:Himpe]]

@@ Line 23: / Line 23: @@
 ===First Order Representation===
 :<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>
 :<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>
 </math>
 ==Data==
@@ Line 51: / Line 91: @@
 System structure:
 ==Citation==
@@ Line 80: / Line 132: @@
     doi =          {10.1007/978-3-319-20988-3}
+  }
 ==References==
@@ Line 89: / Line 140: @@
 </references>