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

含流体动力轴承系统的转子动力学特性受到非线性轴承力的重要影响。对于快速旋转、轻负荷的转子，这会导致潜在高振幅的次同步自激振荡，从而降低部件的耐用性，导致临界噪声排放，并影响机器的能源效率。为了减少昂贵的试验台实验和产品开发过程中耗时的迭代，设计必须基于对运行行为的精确模拟分析，并考虑轴承力和轴振动之间的非线性相互作用。为此，将弹性轴的运动方程纳入时间积分方案，并与描述流体动力轴承压力产生的雷诺兹方程耦合。因此，模拟的每个时间步都包含雷诺方程的一个解，该解采用数值方法、解析近似和查找表。虽然数值方法导致相当大的计算时间，往往是不可接受的，但解析解只有在大量简化的情况下才有可能。在某种程度上，查找表方法提供了这两个极端之间的权衡，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。半解析尺度边界有限元法（SBFEM）是开发一种新的、数值上有效的解决方案而不受解析方法或查找表技术的实质性限制的有希望的基础。在初步工作中已经推导出用SBFEM求解Reynolds方程的基本原理，但该方法的潜力尚未得到开发，这是本项目的目标。为了进一步减少数值计算的工作量，需要将高阶形状函数与自动、自适应网格细化和粗化相结合，并分析了以平滑解的方式对雷诺方程进行的变换。另一个值得研究的策略是在时间积分方案中避免特征值问题的重复求解。这就要求在进行转子动力学仿真之前，将特征值问题对轴位移参数进行微分，并发展成一个级数。与前期工作相比，为了提高SBFEM解的建模深度，需要研究纳入质量守恒空化模型和竖井倾斜的策略。在最后一步中，将对所开发的方法进行有效性验证和分析。为了确保真实的背景，这是在转子动力学或MBS公式的框架内完成的，从而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.