Improvement of the numerical efficiency of rotordynamic simulations by applying the Scaled Boundary Finite Element Method to compute the hydrodynamic bearings

非线性轴承力对动压轴承系统的转子动力学特性有着重要的影响。对于快速旋转、轻负载的转子，这会导致具有潜在高振幅的次同步自激振荡，这会降低部件的耐用性，导致严重的噪声排放，并影响机器的能量效率。为了减少产品开发过程中昂贵的测试台架实验和耗时的迭代，设计必须基于对轴承力和轴振动之间的非线性相互作用下的操作行为的精确模拟分析。为此，弹性轴的运动方程被纳入一个时间积分方案，并与雷诺方程，它描述了在流体动力轴承的压力产生耦合。因此，模拟的每个时间步包括雷诺方程的解，对于该雷诺方程，采用数值方法、解析近似和查找表。虽然数值方法导致相当大的，往往是不可接受的计算时间，分析解决方案是唯一可能的结合大量的简化。查找表方法在一定程度上提供了这两个极端之间的折衷，而建模深度通常是有限的，因为插值工作随着每一个考虑的物理效应而增加。一个有前途的基础，发展一种新的，数值有效的解决方案，而没有实质性的限制解析方法或查找表技术是半解析缩放边界有限元法（SBFEM）。用SBFEM求解Reynolds方程的基本原理已经在初步工作中得到，但是该方法的潜力还没有被开发，这是本项目的目标。为了进一步减少数值工作量，高阶形状函数需要与自动的、自适应的网格细化以及粗化相结合，并以平滑解决方案的方式分析雷诺方程的变换。另一个值得研究的策略是避免在时间积分方案中重复求解特征值问题。这就要求在转子动力学模拟之前，将特征值问题与轴位移参数区分开来，并将其发展成一系列。为了提高SBFEM解决方案的建模深度相比，初步的工作，战略，将质量守恒空化模型以及轴倾斜需要进行调查。在最后一步中，开发的方法是验证和分析其效率。为了确保真实的环境，这是在转子动力学或MBS公式的框架内完成的，由此也可以模拟复杂的技术整体系统。

The rotordynamic properties of systems with hydrodynamic bearings are affected crucially by the nonlinear bearing forces. Regarding fast-rotating, lightly-loaded rotors, this causes subsynchronous self-excited oscillations with potentially high amplitudes, which can reduce the durability of the components, cause critical noise emissions, and affect the energy efficiency of the machine. To reduce expensive test bench experiments and time-consuming iterations in the product development process, the design has to be based on precise simulative analyses of the operating behavior under consideration of the nonlinear interactions between the bearing forces and the shaft vibrations. To this end, the equation of motion of the elastic shaft is incorporated into a time integration scheme and coupled with the Reynolds equation, which describes the pressure generation in hydrodynamic bearings. Hence, each time step of the simulation includes a solution of the Reynolds equation, for which numerical methods, analytical approximations, and look-up tables are employed. While numerical methods lead to considerable and often inacceptable computational times, analytical solutions are only possible in conjunction with substantial simplifications. The look-up table approach, to some extent, offers a tradeoff between these two extremes, while the modeling depth is usually limited, since the interpolation effort increases with every considered physical effect.A promising basis for the development of a novel, numerically efficient solution without the substantial limitations of analytical methods or look-up table techniques is the semi-analytical Scaled Boundary Finite Element Method (SBFEM). The fundamentals for solving the Reynolds equation with the SBFEM have been derived in preliminary work, but the potential of the approach has not been exploited yet, which is the objective of this project. In order to further reduce the numerical effort, high-order shape functions need to be employed in combination with an automatic, adaptive mesh refinement as well as coarsening and a transformation of the Reynolds equation in a manner that smoothens the solution is analyzed. Another strategy worth investigating is to avoid the repeated solution of eigenvalue problems within the time integration scheme. This requires that the eigenvalue problem is differentiated with respect to the parameters of the shaft displacement and developed into a series prior to the rotordynamic simulation. In order to improve the modeling depth of the SBFEM solution compared to the preliminary work, strategies for incorporating mass-conserving cavitation models as well as shaft tilting need to be investigated. In the last step, the developed methodology is to be verified and analyzed with regard to its efficiency. To ensure a realistic context, this is done within the framework of a rotor dynamics or MBS formulation, whereby complex technical overall systems can also be simulated.