Vibrations in Nonlinear Rotordynamics - Modelling, Simulation, and Analysis

Mechanical vibrations of rotor-bearing systems are an ubiquitous problem in mechanical engineering and the prediction of response frequencies and amplitudes with the help of mathematical models is of major importance for the design of more efficient and reliable machinery. In the present work a model for the dynamics of fast rotating, elastic beams supported in hydrodynamic bearings is derived and its vibration behavior analyzed. Special focus is put on the influence of the nonlinear bearing reaction forces on the dynamics. The continuous rotor is modeled using Euler-Bernoulli beam theory under the inclusion of rotatory inertia and gyroscopic effects. For a general class of support functions the existence of weak solutions to the equations of motion is proved. The pressure distribution in the oil-lubricated simple journal bearings is modeled by the well known Reynolds’ equation. Its derivation from the Navier-Stokes equations by an asymptotic expansion in the film thickness is reviewed and new correction terms for fluid inertia effects are derived. Additional correction terms for the short bearing approximation to Reynolds’ equation are also derived by making additional assumptions on the bearings’ width-to-radius ratio. Furthermore, the pressure distribution and the bearing reaction forces are computed numerically in dependence of the position and the velocity of the rotor inside the bearings. The finite element method is applied to discretize the beam equation and the bearing forces are included into the model as point forces in the bearing nodes. While the classical lubrication theory leads to explicit equations of motion, the inertia corrections lead to implicit equations of motion for the rotor-bearing system. The model is evaluated by comparing numerical simulations with experimental results obtained for a passenger car turbocharger. For this example it is shown that the model equations describe the dynamics well, capturing most experimentally observed phenomena, such as unbalance oscillation and self-excited oil whirl. Some differences between model and experiment can be seen in the response frequency of the subharmonic oil whirl. The inertia corrections yield a small improvement compared to the classical models. A new phenomenological correction of the short bearing approximation based on the adaptation of the average circumferential lubricant velocity is proposed and shown to influence the whirl frequency ratio strongly. Continuation methods for periodic and quasiperiodic solutions are shown to be more efficient and are hence more appropriate tools for the examination of the vibration response behavior than direct numerical simulation. It is shown numerically, that the static gravity load can be neglected for higher rotational frequencies. Combined with a transformation to a co-rotating frame of coordinates this leads to a significant simplification, since the resulting ODE becomes autonomous, and the vibration response can be computed by the continuation of periodic orbits instead of invariant tori. This is applied successfully to study the parameter range where the inertia correction for the short bearing is valid. All in all, the presented model and its variations prove to be useful for future industrial application in the design of more efficient turbomachinery. Parts of the presented research are already actively used for turbocharger design by the Toyota Central Research and Development Laboratories