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.

Acknowledgement & Origin

This model was developed and investigated

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