A new Galerkin Reduction approach for the analysis of a fully coupled foil air bearing rotor system with bilinear foil model

The foil air bearing (FAB) is a key enabler of the rapidly expanding technology of oil-free turbomachinery. However, it presents a formidable computational challenge due to the intricacy of the mathematical modelling of the nonlinear interaction between the foil structure, air film, and rotor. Galerkin Reduction (GR) allows the integration of the compressible Reynolds Equation within a FAB-rotor model without the need for spatial discretisation, resulting in a considerable gain in computational efficiency relative to grid/mesh-based methods like Finite Difference (FD) and Finite Element (FE). GR has so far been limited to a simple model that ignores the detachment of the top foil from the underlying foil. A new GR approach is developed to overcome this limitation. Following transient and static equilibrium analyses of the nonlinear system, Jacobian-based linearization is used to extract the full mode set, Campbell diagram and linear stability map. The use of GR instead of FD reduced the computational time by over 50% and required 20-60 times less memory for saving the results, without compromising accuracy and reliability. In particular, Jacobian-based linearization using GR retains the ability to detect instabilities (like top foil flutter) that are beyond the capability of the traditional linear force coefficients method.