Development of a model order reduction method for the direct generation of time-discrete, low-dimensional models of machine tools

现代机床的发展面临着巨大的创新、时间和成本压力。随着生产力需求的不断增长，机床的复杂性也随之增加。为了应对这些挑战，对开发和测试的创新方法的需求很高。它可以防止昂贵的物理原型设计或开发失败，并减少以后的调试时间。因此，需要时间确定性模型，一方面具有高模型质量，另一方面具有预定时间步长的解。由于典型有限元模型的模型大小，这些时间步长不适合实时计算。因此，需要自适应的模型降阶来解决实时问题。目前，还不存在适用于复杂有限元模型的用于创建实时可计算系统的自动化程序。本研究项目的主题是开发一种用于直接生成时间离散的机床低维模型的模型降阶方法。该项目的主要目标是自动化和定制化的特定应用程序，提供离散时间、确定性、简化系统到具有实时能力的机床动态行为模型。为此，将开发一种创新的模型降阶方法，该方法提供对原始模型的固定时间步长的低维近似以及相关的误差估计，并保持系统固有的属性(如无源性、稳态精度)。该方法基于机床或机床模块的有限元模型所提供的高维常微分方程组和微分代数方程。此外，该方法还考虑了用于实时计算的时间步长。此外，误差估计作为中止准则，允许模型的自动降阶。通过生产工程和数学的跨学科合作，支持这一创新过程，并缩短机电系统和机床的开发。这种方法是在开发过程的早期阶段使用的，直到虚拟调试。该方法对结构模型的建立和模型精度的提高做出了重要贡献。

The development of modern machine tools is under great innovation, time and cost pressure. With continuous growth of productivity demands, also the complexity of machine tools increases. There is a high demand for innovative methods of developing and testing, to handle with these challenges. It results in preventing expensive design- or development-failures of a physical prototype as well as reducing commissioning times at a later point in time. Therefore time-deterministic models are needed, on the one hand having a high model quality and on the other hand a solution at a predetermined time step. Due to the model size of typical finite element models, these time steps are not suitable for real-time computations. Consequently, there is a demand for adapted model order reduction to solve real time problems. Currently, no such automated procedure for the creation of real-time computable systems exists applicable for complex finite-element models. The subject of this research project is to develop a model order reduction method for the direct generation of time-discrete, low-dimensional models of machine tools. The main goal of the project is the automated and customized application specific provision of discrete-time, deterministic, reduced systems to real-time capable models of the dynamic behavior of machine tools. For this purpose an innovative model order reduction method will be developed, which provides a low-dimensional approximation for a fixed time step of an original model along with the associated error estimate and retaining inherent system properties (such as passivity, steady-state accuracy). The method is based on high-dimensional ordinary differential equations and differential-algebraic equations, which are provided by finite element models of machine tools or rather machine modules. Furthermore, the method allows for the consideration of the time step for real-time calculations. In addition, the error estimation serves as abort criterion allowing the automatic model order reduction. Through interdisciplinary cooperation of production engineering and mathematics to support this innovative process and shorten the development of mechatronic systems and as machine tool is provided. This method is used in the early stage of the development process as far as to the point of virtual commissioning. The method makes a major contribution to the creation of structural models and to the increase of the accuracy of these models.