MiniHex

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1 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.

2 Description

Thermal FE-modeling with segmented spindle surface to enter position-dependent heat flows

The mobile demonstrator-machine MiniHex consists of six struts of variable length, which are driven by ball screws, each with a servo-motor . Following one of the feedaxes is considered. It is divided into four stationary subassemblies: belt housing with motor, spindle, spindle nut and feed tube. The thermal finite element (FE) model was generated in ANSYS and afterwards the FE system matrices were exported for post-proceessing, such as MOR, to MATLAB. The evolution of the temperature field is modelled with the heat equation

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

with

$\begin{align} f=q_{fric}+\kappa_c(T_{c_i}-T_{c_k}), \quad i,k=1,\dots,4, \quad i\neq k, \end{align}$

and

$k \in \mathbb{R}$	- number of the subassembly, $k=1,\dots, 4$
$T$	- temperature
$c_p$	- specific heat capacity
$\rho$	- density
$\lambda$	- heat conductivity
$\Omega_k$	- domain of the k-th subassembly
$\Gamma_{c_k}$	- contact boundary of the k-th subassembly (partly time varying, moves with the position of the nut)
$\kappa_{ext}$	- heat transfer coefficient between a subassembly and the ambient air
$T_{ext}$	- external temperature
$\Gamma_{ext_k}$	- contact boundary with the ambience
$q_{fric}$	- friction driven heat flow induced by the movement
$\kappa_c$	- heat transfer coefficient between two subassemblies
$T_{c_k}$	- temperature of the contact area of subassembly k.

The finite element discretization of the heat conduction models leads to the four systems

$\begin{align} E_k\dot{T}_k&=(A_{0_k}-\sum_{i=1}^{l_k}\kappa_{ext_i}(t)E^\Gamma_{k_i})T_k+B_k u_k(t),\quad k=1,\dots,4\\ y_k(t)&=C_k T_k,\\ \end{align}$

with

$E_k \in \mathbb{R}^{n\times n}$	- FE mass matrix
$A_{0_k} \in \mathbb{R}^{n\times n}$	- basic stiffness matrix (discrete Laplacian)
$l \in \mathbb{R}$	- number of boundary conditions with the ambience (Robin boundary conditions (RBC))
$E^{\Gamma}_k \in \mathbb{R}^{n\times n}$	- boundary mass matrix (part of stiffness matrix arising from RBC)
$T_k \in \mathbb{R}^n$	- state vector (discrete temperature)
$B_k \in \mathbb{R}^{n \times m}$	- input map
$u_k \in \mathbb{R}^m$	- input vector
$y_k \in \mathbb{R}^p$	- output vector
$C_k \in \mathbb{R}^{p \times n}$	- output map.

The interaction between the subassemblies is considered as position-dependent heat flows in the input $u(t)$. The movement of the MiniHex is characterized by the position $d$ of the nut on the spindle with $d \in [-0.26,0.26]\text{ }m$. This movement induces a variability of the contact areas between the subassemblies. To include this variability in the model the boundaries of the subassemblies are divided into smaller parts and the average temperature of these parts is used for computing the input $u(t)$. To be more precise, the spindle is divided into 32 parts and the feed tube into 17 parts . In the simulation the calculated average temperature of an area in the current time step serves as input temperature of this area in the next time step and is also used for calculating the thermally dependent heat transfer coefficients. Thus the average temperatures of all contact areas are needed as outputs $y(t)$. Further, the temperatures in certain nodes of interest complete the output $y(t)$. For more information concerning MOR for systems with moving loads see e.g. and the references therein.

3 Data

The system matrices are given in the Matrix Market format and can be found in MiniHex.tar.gz. There is a set of matrices for each subassembly. The numbering of the subassemblies is as follows: 1 - belt housing with motor, 2 - spindle, 3 - spindle nut and 4 - feed tube. Further the archive contains some MATLAB files with the description and computation of the 18 contacts between the subassemblies as well as the computation of the temperature-dependent heat transfer coefficients. The overall system dimension is $n=116892$ with $n_1=44902$ , $n_2=49590$, $n_3=1336$ and $n_4=21064$.

4 Origin

The MiniHex is used as a demonstrator in the CRC/TR 96 Project-ID 174223256 financed by the German Research Foundation DFG.

6 Contact

Jens Saak