Vertical Stand

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1 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

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

with the boundary conditions

$\lambda\frac{\partial x}{\partial n}=q \qquad\qquad\qquad$ on $\Gamma_{slide}$ (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 $\Gamma_{surf}$ (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 $\Gamma_{Slide}$, 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:

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

where

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

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

2 Acknowledgement & Origin

This model was developed and investigated, in the Collaborative Research Centre Transregio 96 Thermo-Energetic Design of Machine Tools funded by the Deutsche Forschungsgemeinschaft .

3 Data

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

$E,A\in\mathbb{R}^{n\times n},B_{slide}\in\mathbb{R}^{n\times 1},B_{surf}\in\mathbb{R}^{n\times 5}, n=16\,626$

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.

The file coord.txt provided in Data_VertStand.tar.gz 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 $(y\in[670,1\,340]mm)$ of the geometry are contained and the fifth column of $B_{surf}$ includes the missing upper $(y\in[1\,340,2\,010]mm)$ part.

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

Width ($x$ direction): $519mm$,

Height ($y$ direction): $2\,010mm$,

Depth ($z$ direction): $480mm$

5 Contact

Jens Saak