Unbalance Responses of Rotor/Stator Systems with Nonlinear Bearings by the Time Finite Element Method

Abstract

Since the early 1970s, major works in rotordynamics were oriented toward the calculation of critical speeds and unbalance responses. The current trend is to take into account many kinds of non-linearities in order to obtain more realistic predictions. The use of algorithms based on nonlinear methods is therefore needed. This article first describes the time finite element method. The method is then applied to nonlinear rotor/stator systems where bearings present a radial clearance.Keywords Bearing clearance, Hertz contact, Nonlinear dynamics, Stability, Time finite element Many investigations have taken place concerning the calculation of critical speeds and unbalance responses in rotor dynamics and research now tends to get more realistic predictions (Ehrich, 1992). This new realism is actually due to the trend for high-performance rotating machinery, claiming reduced sizes and higher efficiency.Engineers now have to take into account non-linearities in their models. In the aircraft engine domain, those non-linearities come from components, such as bearings or squeeze film dampers (Vance, 1988), and contact phenomena, such as friction in joints or rubbing between rotor and stator (Choi and Bae, 2001). Special algorithms are needed to deal with such nonlinearities. Although those based on temporal integration are able to track all kinds of non-linearities, they are not well suited for rotordynamics. Indeed, the steady-state solution caused by unbalance is often of interest; the use of a temporal integration is therefore not appropriate since one has to wait until after the transient phase has died which can be very time-consuming.Thus, frequential methods like the incremental harmonic balance (Kim et al., 1991) or trigonometric collocation methods (Nataraj, 1989) are of great use thanks to their time efficiency. However, these methods become less attractive as the frequency content of the solution includes a high number of harmonics. On the other hand, methods in the time domain, such as the shooting method (Sundararajan and Noah, 1997), can also be used to find out periodic solutions however this one becomes time-consuming when dealing with large systems.In this article, the time finite element method is described. This time-based method enables us to get steady-state solutions and to assess their stability. To prove its reliability in rotordynamics problems, two examples are addressed. Both consist of rotor/stator systems with radial clearance in the bearings. THE TIME FINITE ELEMENT METHOD Description of the MethodThe aim of the time finite element method is to find out the periodic solutions of forced systems. It is based on Hamilton's Law of Varying Action (Bayley, 1975;Baruch and Riff, 1982):i.e.,The principle of this method is to interpolate the displacement of all spatial degrees of freedom between given instants t i and t i+1 by polynomials (Wang, 1995(Wang, , 1997. In this article, the Lagrange polynomials of order k are used, however all kinds of polynomials may be used (Park, 1996). The displacement ...