Machine tool MAX - Revision history

Schuetz2 at 07:39, 3 March 2026

2026-03-03T07:39:44Z

← Older revision		Revision as of 07:39, 3 March 2026
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	}}		}}

	~~__NUMBEREDHEADINGS__~~

	==Motivation==		==Motivation==

Vettermann at 12:56, 26 October 2023

2023-10-26T12:56:53Z

← Older revision		Revision as of 12:56, 26 October 2023
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	* For the background on the benchmark:		* For the background on the benchmark:

	@article{~~SauVNetal21~~,		@article{morVetSNetal21,
	author = {Vettermann, J. and Sauerzapf, S. and Naumann, A. and Beitelschmidt, M. and Herzog, R. and Benner, P. and Saak, J.},		author = {Vettermann, J. and Sauerzapf, S. and Naumann, A. and Beitelschmidt, M. and Herzog, R. and Benner, P. and Saak, J.},
	title = {Model order reduction methods for coupled machine tool models},		title = {Model order reduction methods for coupled machine tool models},

Vettermann: add citation

2023-10-26T12:55:53Z

add citation

@@ Line 141: / Line 141: @@
 The '''MAX''' is used as an experimental machine tool in the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Category:CRC-TR-96 CRC/TR 96] Project-ID 174223256 financed by the German Research Foundation DFG.
 ==References==
@@ Line 149: / Line 177: @@
 <ref name="vettermann2021">
-J. Vettermann, S. Sauerzapf, A. Naumann, J. Saak, P. Benner, M. Beitel-
+J. Vettermann, S. Sauerzapf, A. Naumann, M. Beitelschmidt, R. Herzog, P. Benner and J. Saak, "<span class="plainlinks">[https://doi.org/10.17973/MMSJ.2021_7_2021072 Model order reduction methods for coupled machine tool models]</span>", MM Science Journal, Special Issue ICTIMT 2021, 2021.
 </ref>

Vettermann at 11:02, 26 October 2023

2023-10-26T11:02:21Z

← Older revision		Revision as of 11:02, 26 October 2023
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	{{Infobox		{{Infobox
	\|Title = MAX ~~stand~~		\|Title = Machine tool MAX
	\|Benchmark ID =		\|Benchmark ID =
	* machineToolMAX_feCoupled_n1265497m69q11		* machineToolMAX_feCoupled_n1265497m69q11

Vettermann: Add Infobox and Zenodo link

2023-10-26T11:01:42Z

Add Infobox and Zenodo link

@@ Line 8: / Line 8: @@
 [[Category:MIMO]]
 [[Category:CRC-TR-96]]
 __NUMBEREDHEADINGS__
@@ Line 107: / Line 134: @@
 </figure>
-The system matrices are available in the .mat file format and can be downloaded under the following link [https://tuc.cloud/index.php/s/diyz2JNzjkkt9RA MAX.tar.gz]. The model has <math> n=1\,265\,497 </math> degrees of freedom and is available for two diffenrent coupling approaches, which are explained in more detail in <ref name="vettermann2021"/>, see also Fig. 3:
+The system matrices are available in the .mat file format and can be downloaded from [https://zenodo.org/records/10041041 Zenodo]. The model has <math> n=1\,265\,497 </math> degrees of freedom and is available for two different coupling approaches, which are explained in more detail in <ref name="vettermann2021"/>, see also Fig. 3:
-* A so called output-coupled model: In this case, an input-output coupling was used, i.e., the model is block diagonal and thus the subassemly models can be reduced separately for structure preserving MOR. There is a set of matrices for each subassembly with a total number of <math> m=287 </math> inputs and <math> p=224 </math> outputs.
+* A so called output-coupled model: In this case, an input-output coupling was used, i.e., the model is block diagonal with a total number of <math> m=287 </math> inputs and <math> p=224 </math> outputs. Thus the subassemlby models can be reduced separately for structure preserving MOR. The block diagonal system can be split into a set of matrices for each subassembly by the script "generate_subsystems.m".
 * A so called FE-coupled model: In this case, the subassemblies are coupled on FE-level, i.e., the conductivity matrix A of the overall system has additional (off-diagonal) coupling blocks. Thus the subassembly models cannot be reduced separately anymore. There is one set of matrices for the overall model with <math> m=69 </math> inputs and <math> p=11 </math> outputs.

Vettermann at 06:24, 5 September 2023

2023-09-05T06:24:22Z

← Older revision		Revision as of 06:24, 5 September 2023
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	c_p\rho\frac{\partial T}{\partial t}&=\lambda \Delta T, & &\text{ on } \Omega_k , \text{ } k=1,\dots,50, \\		c_p\rho\frac{\partial T}{\partial t}&=\lambda \Delta T, & &\text{ on } \Omega_k , \text{ } k=1,\dots,50, \\
	\lambda\frac{\partial T}{\partial n}&=f, & &\text{ on } \Gamma_{c_k} \subset\partial\Omega_k , \text{ } k=1,\dots,50, \\		\lambda\frac{\partial T}{\partial n}&=f, & &\text{ on } \Gamma_{c_k} \subset\partial\Omega_k , \text{ } k=1,\dots,50, \\
	\lambda\frac{\partial T}{\partial n}&=\alpha_{ext}(T_{ext}-T), & &\text{ on } \Gamma_{ext_k} \subset\partial\Omega_k , \text{ } k=1,\dots,4, \\		\lambda\frac{\partial T}{\partial n}&=\alpha_{ext}(T_{ext}-T), & &\text{ on } \Gamma_{ext_k} \subset\partial\Omega_k , \text{ } k=1,\dots,50, \\
	T(0)&=T_0, & &\text{ at } t=0		T(0)&=T_0, & &\text{ at } t=0
	\end{align}		\end{align}
Line 46:		Line 46:

	:{\| style="text-align:left"		:{\| style="text-align:left"
	\| <math> k </math> \|\| - number of the subassembly, <math> k=1,\dots, 4 </math>		\| <math> k </math> \|\| - number of the subassembly, <math> k=1,\dots, 50 </math>
	\|-		\|-
	\| <math> T </math> \|\| - temperature		\| <math> T </math> \|\| - temperature

Vettermann: Created page with "{{preliminary}} Category:benchmark Category:Sparse Category:linear Category:first differential order Category:ODE Category:MIMO..."

2023-09-04T12:17:17Z

Created page with "{{preliminary}}  Category:benchmark Category:Sparse Category:linear Category:first differential order Category:ODE Category:MIMO..."

New page

{{preliminary}} 

[[Category:benchmark]]
[[Category:Sparse]]
[[Category:linear]]
[[Category:first differential order]]
[[Category:ODE]]
[[Category:MIMO]]
[[Category:CRC-TR-96]]

__NUMBEREDHEADINGS__

==Motivation==

Due to the increasing interest in manufacturing accuracy without an additional energy demand for cooling, knowledge about the thermo-elastic behavior of entire machine tools becomes crucial. Methods to correct the thermally induced position error between the tool-center-point (TCP) and the workpiece in real-time are needed. Therefore, reduced-order models that enable a fast simulation of entire machine tools are applied in the design and production process.

==Description==
<figure id="fig:plot1">
[[File:MAX_real.jpg|480px|thumb|right|<caption>Experimental machine tool MAX.</caption>]]
</figure>

<figure id="fig:plot2">
[[File:MAX_FE.pdf|480px|thumb|right|<caption>Subassemblies and FE-model of the benchmark MAX.</caption>]]
</figure>

The '''MAX''' benchmark is a linear time-invariant thermal model of the demonstrator machine tool '''[https://tu-dresden.de/ing/maschinenwesen/imd/lwm/forschung/werkzeugmaschinen-versuchsfeld/versuchstraeger-max MAX]'''(Fig. 1). Due to its lightweight construction based on aluminium structures, low heat capacities and high thermal coefficients of expansion are to be expected <ref name="grossmann2015"/>. It is divided into 50 stationary subassemblies (SA), see Fig. 2. The thermal finite element (FE) model was generated in ANSYS and afterwards the model was exported for post-processing as described in <ref name="vettermann2021"/>. The interaction between the subassemblies is modeled by contact boundary conditions. The evolution of the temperature field is modeled with the heat equation

:<math>
\begin{align}
c_p\rho\frac{\partial T}{\partial t}&=\lambda \Delta T, & &\text{ on } \Omega_k , \text{ } k=1,\dots,50, \\
\lambda\frac{\partial T}{\partial n}&=f, & &\text{ on } \Gamma_{c_k} \subset\partial\Omega_k , \text{ } k=1,\dots,50, \\
\lambda\frac{\partial T}{\partial n}&=\alpha_{ext}(T_{ext}-T), & &\text{ on } \Gamma_{ext_k} \subset\partial\Omega_k , \text{ } k=1,\dots,4, \\
T(0)&=T_0, & &\text{ at } t=0
\end{align}
</math>

with

:<math>
\begin{align}
f=\kappa_c(T_{c_i}-T_{c_k}), \quad i\neq k,
\end{align}
</math>

and

:{| style="text-align:left"
| <math> k </math> || - number of the subassembly, <math> k=1,\dots, 4 </math>
|-
| <math> T </math> || - temperature
|-
| <math> c_p </math> || - specific heat capacity
|-
| <math> \rho </math> || - density
|-
| <math> \lambda </math> || - heat conductivity
|-
| <math> \Omega_k </math> || - domain of the k-th subassembly
|-
| <math> \Gamma_{c_k} </math> || - contact boundary of the k-th subassembly (partly time varying, moves with the position of the z-slide (considered as heat flow))
|-
| <math> \kappa_{ext} </math> || - heat transfer coefficient between a subassembly and the ambient air
|-
| <math> T_{ext} </math> || - external temperature
|-
| <math> \Gamma_{ext_k} </math> || - contact boundary with the ambience
|-
| <math> \kappa_c </math> || - heat transfer coefficient between two subassemblies
|-
| <math> T_{c_k} </math> || - temperature of the contact area of subassembly k.
|-
|}

The finite element discretization of the heat conduction models leads to the system

:<math>
\begin{align}
E\dot{T}&=A T+B u(t),\\
y(t)&=C T,\\
\end{align}
</math>

with

:{| style="text-align:left"
| <math> E \in \mathbb{R}^{n\times n} </math> || - capacity matrix
|-
| <math> A \in \mathbb{R}^{n\times n} </math> || - conductivity matrix
|-
| <math> T \in \mathbb{R}^n </math> || - state vector (discrete temperature)
|-
| <math> B \in \mathbb{R}^{n \times m} </math> || - input map
|-
| <math> u \in \mathbb{R}^m </math> || - input vector
|-
| <math> y \in \mathbb{R}^p </math> || - output vector
|-
| <math> C \in \mathbb{R}^{p \times n} </math> || - output map.
|-
|}

==Data==

<figure id="fig:plot3">
[[File:coupled_systems.pdf|480px|thumb|right|<caption>Sketch of the two coupling approaches using the example of a model with 2 subassemblies.</caption>]]
</figure>

The system matrices are available in the .mat file format and can be downloaded under the following link [https://tuc.cloud/index.php/s/diyz2JNzjkkt9RA MAX.tar.gz]. The model has <math> n=1\,265\,497 </math> degrees of freedom and is available for two diffenrent coupling approaches, which are explained in more detail in <ref name="vettermann2021"/>, see also Fig. 3:
* A so called output-coupled model: In this case, an input-output coupling was used, i.e., the model is block diagonal and thus the subassemly models can be reduced separately for structure preserving MOR. There is a set of matrices for each subassembly with a total number of <math> m=287 </math> inputs and <math> p=224 </math> outputs.
* A so called FE-coupled model: In this case, the subassemblies are coupled on FE-level, i.e., the conductivity matrix A of the overall system has additional (off-diagonal) coupling blocks. Thus the subassembly models cannot be reduced separately anymore. There is one set of matrices for the overall model with <math> m=69 </math> inputs and <math> p=11 </math> outputs.

==Origin==

The '''MAX''' is used as an experimental machine tool in the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Category:CRC-TR-96 CRC/TR 96] Project-ID 174223256 financed by the German Research Foundation DFG.

==References==
<references>
<ref name="grossmann2015">
K. Großmann, ed., "<span class="plainlinks">[https://doi.org/10.1007/978-3-319-12625-8 Thermo-energetic Design of Machine Tools]</span>", Springer International Publishing, Switzerland, pp. 9-10, 2015.
</ref>

<ref name="vettermann2021">
J. Vettermann, S. Sauerzapf, A. Naumann, J. Saak, P. Benner, M. Beitel-
schmidt and R. Herzog, "<span class="plainlinks">[https://doi.org/10.17973/MMSJ.2021_7_2021072 Model order reduction methods for coupled machine tool models]</span>", MM Science Journal, Special Issue ICTIMT 2021, 2021.
</ref>

</references>

==Contact==

''[[User:Saak]]''