Efficient simulation of hydrodynamic bearings using the SBFEM with eigenvalue problem derivatives

Abstract This work presents a semi-analytical solution of the Reynolds equation under Gümbel conditions for hydrodynamic bearings in rotordynamic simulations. The algorithm is based on the SBFEM (scaled boundary finite element method) in combination with eigenvalue problem derivatives. The pressure field is discretized by standard finite elements along the circumferential coordinate, while an exact analytical formulation is used in the axial direction. This transforms the partial differential equation into a system of ordinary differential equations and leads to an eigenvalue problem. The computation of the continuously changing eigenvalues and eigenvectors is achieved by accurate Taylor approximations so that the repeated call of a numerically expensive eigensolver can be avoided. The algorithm is incorporated into a time integration scheme for the dynamic analysis of a simple system, consisting of a rotor with two degrees of freedom in a hydrodynamic bearing. Thereby, the method is verified, and its numerical efficiency is demonstrated. For a bearing with a slenderness ratio (length-to-diameter ratio) of 1, the computational time is reduced to $$8.8\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>8.8</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> compared to a standard FVM (finite volume method) solution.