The simulation of thermal effects in hydrodynamic bearings requires high-precision recording of the temperature distributions in the shaft, lubricating film and bearing shell due to the non-linear interactions with the pressure build-up and the resulting bearing forces. In order to limit the simulation time, the bearing shell in particular is often modelled as an idealized hollow cylinder, which means that some essential geometric details are lost and relevant thermal gradients are not taken into account.
In order to counteract these deficits and still ensure a moderate simulation time, a high-order finite element method for the entire thermal simulation is being developed as part of this project, which enables significantly increased accuracy and efficient calculation at the same time thanks to higher-order approach functions.
The first step is to develop a numerical algorithm for solving the heat conduction equation based on high-order FEM with variable polynomial orders for 2D rotationally symmetric and 3D problems. The use of higher polynomial orders allows a detailed mapping of local temperature gradients in the bearing shell (3D) and shaft (2D) without having to increase the mesh refinement disproportionately.
Based on this, the energy equation of the lubricating film (3D) is also formulated as a high-order FEM model in the second step. In contrast to the heat conduction equation, there are additional convective terms in the energy equation, which means that numerical solutions tend to be unstable. The approaches developed in the first step are adapted to the extended differential equation. At the same time, stabilization methods are investigated and implemented to ensure that a robust and convergent solution can be determined despite the high approximation order.
The subsequent coupling of the models of shaft, lubricating film and bearing shell results in a consistent, global system of equations that can be solved efficiently. The increased numerical precision of the high-order FEM enables a significantly improved overall prediction of the temperature distribution in the plain bearing, while the efficiency of the method noticeably reduces the calculation time compared to classic approaches with low polynomial order. The developed method thus forms a reliable and at the same time efficient basis for future thermal analyses of plain bearings.
This text was translated with DeepL on 18/12/2025

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In the area of high-speed turbomachinery, oil-free bearing concepts are coming to the front due to the desired mobility and energy transition, which also requires charged fuel cell systems in particular.
Air-lubricated bearings are suitable here due to their many advantages: No need for an external lubricant supply; low losses due to friction power; inert lubricant, which is why expensive seals can also be dispensed with in fuel cells. On the other hand, there are disadvantages such as the low load-bearing capacity and the excitation of subsynchronous vibrations specific to journal bearings.
The load capacity can be increased by various concepts, but mainly by a flexible bearing shell in the form of a foil structure, whereby the focus of the research project is on bump-type bearings.
The prediction of the subsynchronous vibrations mentioned above is of great importance in the development process of rotordynamic systems, as these vibrations can have negative effects such as acoustic irregularities, rubbing of the rotor on the housing or contact in the bearing and must therefore be avoided or at least minimized in amplitude.
The research project aims to develop, implement and experimentally validate an efficient simulation method for rotors with air bearings that allows a valid prediction of the subsynchronous vibrations and the associated stability, taking into account the aerodynamics of the air gap, the deformation of the foil structure and the thermodynamic effects in the radial air bearings. The integration into an overall rotordynamic simulation also makes it possible to take into account the time-dependent rotor misalignment and its influence on the bearing damping and thus to predict the vibration amplitudes and the occurrence of potential instabilities with high quality.

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