Vertical Stand

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

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

describing the heat transfer to the ambience and

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

which describes the heat transfer between tool slide and vertical stand.

The heat load $q$ induced by the slide and the external temperature $x_{ext}$ serves 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 region there the input acts on the system is varying and thus one obtains a parameter dependent input matrix $B(\mu )$.

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

$\begin{array}{l}E\dot{x}=Ax+B_{surf}{x}_{ext}+B_{slide}(\mu )q,\\ =Ax+B(\mu )u,\\ y=Cx\end{array}$

Data

File:Matrices VertStand.tar.gz

The data file consists of the matrices: $E,A,B_{slide},B_{surf}\in {\mathbb{ℝ}}^{n\times n}$

and some

Origins

Contact information:

User:Lnor