Original article
Machine tool simulation based on reduced order FE models

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Abstract

Numerical simulations of the behavior of machine tools are usually based on a finite element (FE) discretization of their mechanical structure. After linearization one obtains a second-order system of ordinary differential equations. In order to capture all necessary details the system that inevitable arises is too complex to meet the expediency requirements of real time simulation and control. In commercial FE simulation software often modal reduction is used to obtain a model of lower order which allows for faster simulation. In recent years new methods to reduce large and sparse dynamical systems emerged. This work concentrates on the reduction of certain FE systems arising in machine tool simulation with Krylov subspace methods. The main goal of this work is to discuss whether these methods are suitable for the type of application considered here. Several Krylov subspace methods for first or second-order systems were tested. Numerical examples comparing our results to modal reduction are presented.

Introduction

The integrated simulation of machine tools consists of two major parts: the structural model of the machine tool representing its reaction on certain control inputs on the one hand and the control loop generating those inputs on the other hand. The reaction of the mechanical structure on control inputs is described by a system of FE semi-discretized partial differential equations. After linearization one obtains a system of ordinary differential equations of second orderMx¨(t)+Dx˙(t)+Kx(t)=Fu(t),y(t)=Cvx˙(t)+Cpx(t),where M,D,KRn×n, FRn×m, Cv,CpRq×n, x(t)Rn, u(t)Rm, y(t)Rq. Here Rayleigh damping is considered, that is, the damping matrix D is proportional to the mass matrix M and the stiffness matrix K: D = α · M + β · K, where α and β are real parameters which are chosen by the experience of the design engineer and lie between 0 and 0.1. The system matrices are large, sparse and non-symmetric. All of this accounts for unacceptable computational and resource demands in simulation and control of these models. In order to reduce these demands to acceptable computational times, usually model order reduction techniques are employed which generate a reduced order model that captures the essential dynamics of the system, preserves its important properties, and has nearly the same response characteristic. Model order reduction methods are methods to find a second-order system of reduced dimension r  nM˜x˜¨(t)+D˜x˜˙(t)+K˜x˜(t)=F˜u(t),y˜(t)=C˜vx˜˙(t)+C˜px˜(t),where M˜,D˜,K˜Rr×r, F˜Rr×m, C˜v,C˜pRq×r, x˜(t)Rr, u(t)Rm, y˜(t)Rq, which approximates the original system in some sense.

Any second-order model can be transformed into a first order systemK00MEx˙(t)x¨(t)z˙(t)=0KKDAx(t)x˙(t)z(t)+0FBu(t),y(t)=[CpCv]Cx(t)x˙(t)z(t),where E,AR2n×2n, BR2n×m, CRq×2n, z(t)R2n, u(t)Rm, y(t)Rq. Various other linearizations have been proposed in the literature, see, e.g. [14]. The linearization (3) is usually preferred as it is symmetry preserving in case K, M, D are symmetric. The system considered here is non-symmetric, so one of the various other possible linearizations could be used instead. Note that by the transformation process the dimension of the system doubles. The corresponding reduced order system is of the formE˜z˜˙(t)=Ãz˜(t)+B˜u(t),y˜(t)=C˜z˜(t),where E˜,ÃRr×r, B˜Rr×m, C˜Rq×r, z˜(t)Rr, u(t)Rm, y˜(t)Rq.

In engineering modal reduction [2] is most common. This method has the disadvantage that the reduced system only contains information of the modes chosen to generate the reduced system. Moreover, the choice of the essential modes is usually based on a heuristic knowledge of the design engineer and cannot be fully automated. Therefore, there is a need for alternative reduction methods which can be fully automated. In the last years new reduction methods to reduce large and sparse dynamical systems were presented, see [1] for an overview of methods for linear systems. The two most famous methods are balanced truncation approximation (BTA) and Krylov subspace methods. Here we consider the reduction of large structural mechanical FE models by Krylov subspace methods.

Section snippets

Model reduction via Krylov subspace methods

The Laplace transform of the impulse response H (that is, the transfer function) is, for the systems considered here, a rational function. One way to approximate a system is to approximate its transfer function by a rational function of lower degree. This can be done by matching some terms of the Laurent series expansion of H at various points of the complex plane. This problem can be solved in a numerically efficient way by employing Krylov subspace projection based model reduction methods.

Numerical results

Our test model is a simplified, abstract mechanical structure of a machine tool designed using the CAD environment Nastran© (see Fig. 1 with courtesy of iwb,1 here TCP denotes the tool center point). The test model is of order n = 4983, it has four inputs (m = 4) and eight outputs. Due to the modeling, the output vectors do not span an 8-dimensional space. Here only

Conclusions

We propose a modified reduction method based on IRKA adapted for the special properties of reduced models occurring in the simulation of machine tools, called MIRKA. Several Krylov methods to reduce first- or second-order systems were implemented to test the method.

For the problem at hand the methods with more than one expansion point should be used, they give better results at a wider range of frequencies. In time response we obtained similar approximation results whether by using reduction

Acknowledgments

This research has been supported by the research project WAZorFEM: Integrierte Simulation des Systems “Werkzeugmaschine-Antrieb-Zerspanprozess” auf der Grundlage ordnungsreduzierter FEM-Strukturmodelle funded by the German Science Foundation (DFG) under grant FA 276/12-1.

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