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A common problem in transient rotordynamic simulations is the numerical effort necessary for the computation of hydrodynamic bearing forces. Due to the nonlinear interaction between the rotordynamic and hydrodynamic systems, an adequate prediction of shaft oscillations requires a solution of the Reynolds equation at every time step. Since closed-form analytical solutions are only known for highly simplified models, numerical methods or look-up table techniques are usually employed. Numerical solutions provide excellent accuracy and allow a consideration of various physical influences that may affect the pressure generation in the bearing (e.g., cavitation or shaft tilting), but they are computationally expensive. Look-up tables are less universal because the interpolation effort and the database size increase significantly with every considered physical effect that introduces additional independent variables. In recent studies, the Reynolds equation was solved semianalytically by means of the scaled boundary finite element method (SBFEM). Compared to the finite element method (FEM), this solution is relatively fast if a small discretization error is desired or if the slenderness ratio of the bearing is large. The accuracy and efficiency of this approach, which have already been investigated for single calls of the Reynolds equation, are now examined in the context of rotordynamic simulations. For comparison of the simulation results and the computational effort, two numerical reference solutions based on the FEM and the finite volume method (FVM) are also analyzed. [ABSTRACT FROM AUTHOR]

The numerical effort of rotordynamic simulations with hydrodynamic bearings is often dominated by the solution of the Reynolds equation. Because of the nonlinear fluid–structure interaction, this equation needs to be solved in every time step. Although computationally efficient analytical approximations and look-up table techniques exist, these approaches often do not provide the necessary modeling depth and accuracy. Thus, in many cases, numerical methods are preferred despite their computational cost. In recent studies, a semi-analytical solution of the Reynolds equation has been developed based on the scaled boundary finite element method (SBFEM) with the objective of improving the numerical efficiency. The developed approach assumed Gümbel conditions, which means that cavitation was handled in a highly simplified manner. In this work, the SBFEM solution is for the first time combined with a more complex, nonlinear cavitation model based on the Elrod algorithm. Because of the semi-analytical technique, this requires a simplification of the switch function that defines the locations of the pressure and cavitation zones. However, as demonstrated in this paper, the resulting bearing forces are still in good agreement with a standard numerical reference solution based on the finite volume method (FVM). Since the SBFEM model uses only a one-dimensional discretization, its high-order formulation is very straightforward, even in case of a non-equidistant grid with varying shape functions that take the smooth and non-smooth regions of the solution into account. As a result, the numerical effort is reduced significantly in comparison to the standard FVM. [Display omitted] • An SBFEM solution of the Reynolds equation is combined with an Elrod cavitation model. • The accuracy of the hydrodynamic forces computed with this approach is adequate. • The developed method is numerically efficient compared to the standard FVM. [ABSTRACT FROM AUTHOR]

• A semi-analytical solution of the Reynolds equation for hydrodynamic journal bearings is derived. • The solution is based on the scaled boundary finite element method (SBFEM). • The SBFEM solution is numerically efficient for bearings with large slenderness ratios. • For simplified oil supply pressure boundary conditions, the numerical effort is further reduced. [Display omitted] The behavior of rotor systems with hydrodynamic bearings can be affected crucially by the nonlinear bearing forces. These forces are incorporated in rotordynamic simulations by solving the Reynolds equation in every step of a time integration scheme. Since closed-form analytical solutions of the Reynolds equation are known only for simplified boundary conditions, numerical methods and look-up table approaches are applied. To obtain accurate and convergent results, numerical solutions require a fine discretization, while the look-up tables require a fine grid of interpolation points. In both cases, the numerical effort of the rotordynamic simulation is dominated by the computation of the bearing forces. In this study, a semi-analytical solution of the Reynolds equation based on the Scaled Boundary Finite Element Method (SBFEM) is developed to reduce the numerical effort of the overall simulation. In contrast to a purely numerical solution, the two-dimensional pressure field is discretized in only one coordinate direction, whereas in the other direction, an analytical formulation is used. This results in a system of non-homogeneous ordinary differential equations, which is solved under consideration of the boundary conditions. Since the complexity of the SBFEM model and the resulting numerical effort depend on the modeling of the oil supply groove, three different models are analyzed in this paper. The equations are derived, the solutions are verified, and their numerical efficiency is investigated in comparison to an FEM solution, which is one of the standard numerical approaches. [ABSTRACT FROM AUTHOR]