Vertical Stand

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Description

CAD Geometry

The vertical stand represents a structural part of a machine tool. On one of its surfaces there are guide rails located. On these rails a tool slide is moving due to the machining process the slide has to perform by the machine tool on top. The machining process produces a certain amount of heat which is transported through the structure into the vertical stand. This heat source is considered to be a temperature input at the guide rails. This transfered heat amount leads to deformations within the device induced by the prevailed temperature field denoted by

$x$

. The evolution of this field is modeled by the heat equation

$c_p\rho \frac{\partial x}{\partial t}=\mathrm{\nabla }.(\lambda \mathrm{\nabla }x)=0$

with the boundary conditions

$\lambda \frac{\partial x}{\partial n}=q$ on ${\mathrm{\Gamma }}_{slide}$ (surface where the tool slide is moving on the guide rails),

describing the heat transfer between the tool slide and the vertical stand and

$\lambda \frac{\partial x}{\partial n}=\kappa (x-x_{ext})$ on ${\mathrm{\Gamma }}_{surf}$ (remaining boundaries),

which describes the heat transfer to the ambience.

The heat load $q$ induced by the slide and the external temperature $x_{ext}$ serve as the input $u$ of the corresponding state-space system.

The position of the moving slide has been included into the system as a parameter dependency $\mu$. Due to this motion the boundary ${\mathrm{\Gamma }}_{Slide}$, where the input acts on the system is varying. Thus, the system matrix $A$ and the input matrix $B$ become parameter dependent.

Finally, the system describing the heat evolution induced by the moving heat source is given by:

$\begin{array}{ll}E\dot{x}& =A(\mu )x+B_{surf}{x}_{ext}+B_{slide}(\mu )q,\\ & =A(\mu )x+B(\mu )u,\\ \end{array}$

where

$\begin{array}{ll}B(\mu )& =[B_{surf},B_{slide}(\mu )],\\ u& =[x_{ext}^T,q{]}^T.\\ \end{array}$

The quantity $x_{ext}$ can be assumed as a vector including different ambient temperatures corresponding to different locations of the geometry.

Data

The data file Data_VertStand.tar.gz contains a MAT_File matrices.mat which consists of the matrices

$E,A\in {\mathbb{ℝ}}^{n\times n},B_{slide}\in {\mathbb{ℝ}}^{n\times 1},B_{surf}\in {\mathbb{ℝ}}^{n\times 5},n=16626$

in sparse format and a file with the coordinates of the mesh nodes called coord.txt.

Here $B_{slide}$ consists of all nodes located on the guide rails.

In order to get a parameter dependent matrix $B_{slide}(\mu )$ one has to pick the "active" nodes (nodes hit by tool carriage) at vertical position $\mu$. The "active" nodes are in the interval of $[\mu -\frac{d}{2},\mu +\frac{d}{2}]$, where $d$ is the heigth of the slide. Therefore the file coord.txt is provided in Data_VertStand.tar.gz.

This file includes a column with indices followed by three additional columns containing the spatial coordinates $x,y,z$ of the corresponding nodes.

The matrix $B_{surf}$ describes the locations where the external temperatures act on. The first column is responsible for the input of the temperature at the clamped bottom slice of the structure. Column 2 describes the ... part of the stand. Columns 3 to 5 describe different thresholds with respect to the height of ambient air temperature. The third column includes the nodes of the lower third $(y\in [0,670]mm)$ of the stand. In column 4 all nodes of the middle third $(y\in [670,1340]mm)$ of the geometry are contained and the fifth column of $B_{surf}$ includes the missing upper $(y\in [1340,2010]mm)$ part.

To clarify this partitioning the geometrical data is given as follows:

Width ($x$ direction): $519mm$, Height ($y$ direction): $2010mm$, Depth ($z$ direction): $480mm$

Contact

User:Lnor