Framework for efficient and accurate simulation of the dynamics of machine tools

Abstract

The aim of this dissertation is to formulate a comprehensive framework of methods, algorithms, and techniques for efficient and accurate simulation of the dynamics of machine tools, to implement these methods in a software package, and to give advise of how to use simulation analyses effectively dur-ing development of new machine tool products. Relevant properties of machine tool structures are derived and a thorough literature study reveals that there is a lack of methods for efficient model order reduction, reduction error estimation, and model-ling of moving interfaces which are compatible with reduced-order models. Furthermore, there are no complete frameworks for the simulation of moving and rotating axes with reduced-order models. A new model order reduction method which is based on Krylov and modal subspace projection is pre-sented. An in-depth analysis of the quality of approximation of them both shows that a combination of them leads to a beneficial combination of their properties. Krylov subspace based reduction allows matching the frequency response function at specific frequencies and modal subspace projection leads to accurately matching poles of the frequency response function. The frequency response functions of the resulting reduced models allow the estimation of an upper bound for the relative error. The result is an a-priori error estimation which depends on the number of Krylov iterations per expansion point, the expansion points, and the number of eigenvectors used for reduction only. The quality of the estima-tion is analysed using randomly generated systems as well as using a real-world example. Modelling of moving interfaces in combination with reduced-order models is challenging because changing the location of action of an interface on a flexible body involves changing the finite element nodes involved. Using all potentially loaded nodes as independent inputs and outputs to a system is not feasible for accurate model order reduction methods, because every input and output to a system enhances the number of degrees of freedom for the reduced system. Therefore, a new method for the approximation of a density function along a path by means of trigonometric interpolation is presented. This method leads to a minimum number of independent interfaces which represent the harmonics of a Fourier series. After having reduced the models, the harmonics can be superposed in order to achieve the desired density function, e.g. a trapezoidal force at the location of a linear guide carriage. Creation of system matrices for rotated bodies is addressed using a floating frame of reference formu-lation. Therefore, an application-oriented derivation of the required information for the assembly of system matrices of the bodies in any orientation in space is presented. The methods above are embedded in a comprehensive software package that interfaces with ANSYS Mechanical for the creation of finite element models and with MATLAB and Simulink for modelling of the complete mechatronic system. Special attention is paid to an effective work-flow. Changes in any part of a model affect this part only. Neither a time-consuming computation nor a cumbersome manual task have to be repeated, if it is not logically required. In order to speed up the process of designing or redesigning a machine tool structure, the Design to Specifications approach is introduced. This approach describes the derivation of requirements on the structure of a machine tool used for the fulfilment of the specifications on productivity and accuracy. Modelling with the presented methods is started as soon as a first design is finalised and for an effi-cient evaluation of the actual dynamic performance, a method based on weighted error transfer func-tions is presented. For verification, a test bench is modelled and compared with measurements. Despite not having fitted any parameter for optimising the accordance between measurement and simulation, the results are accurate in qualitative and quantitative terms. The methods for derivation of requirements as well as the analysis of the dynamic performance are applied to the model of the test bench and are proved valid. To summarise, this thesis gives a complete framework consisting of simulation methods, algorithms, software, and application techniques, leading to an exceptionally efficient work-flow for the simula-tion and design of machine tools.