Chin S. Chu Graduate Research Assistant, Kristin L. Wood Associate Professor. Ilene J. Busch-Vishniac Professor. Department of Mecfianical Engineering, Ttie University of Texas, Austin, TX A Nonlinear Dynamic Model With Confidence Bounds for Hydrodynamic Bearings In conventional rotordynamic modeling, hydrodynamic bearings are often char- acterized by a set of linear stiffness and damping coefficients obtained from a first-order Taylor series expansion of bearing reactions. Theoretically, these coefficients are only valid for small amplitude motion about an equilibrium position. In this paper, a nonlinear dynamic model that overcomes the small amplitude assumption in the conventional linear analysis is described. By includ- ing higher-order terms in the bearing reaction expansion, nonlinearity in the oil film forces for large amplitude motion can be captured and represented by a set of nonlinear stiffness and damping coefficients. These coefficients are functions of static bearing displacement. A finite difference approach is de- scribed and is used to solve for these coefficients. The stated model is applied to a conventional slider bearing and a mechanical smart slider bearing that experiences large variations in load. Error assessment is performed numerically on the higher-order solutions to determine an acceptable displacement bound for the higher order coefficients. 1 Introduction There are generally two approaches to representing fluid film bearings in rotordynamic modeling: namely, a linearized model and a time-transient model (Shapiro and Rumbarger, 1971). The linearized model represents bearing systems by constant stiffness and damping coefficients evaluated about a nominal operating position. Dynamic response and stability conditions can be evaluated using a time-invariant linear dynamic model with these stiffness and damping coefficients. The time-transient model, on the other hand, produces a time history of the bearing displacement in all degrees of freedom without any linearization assumptions. The latter approach provides the closest simulation of the actual system performance; however, repetitive solution of the governing fluid equation is required over a time span, resulting in costly computational requirements. It is therefore a common practice to use the linearized approach unless the linearized assumption deteriorates, i.e., when the bearing motion amplitude becomes large. In some cases, the linearized model may be shown to be valid over a range of bearing parameters and operating condi- tions. Shapiro and Rumbarger (1971) suggest that displace- ments over 10 percent of the operating film thickness would require appropriate modification of linear spring and damping constants. Lund (1987), who introduced the linearized bear- ing analysis, states that the linear coefficients maybe valid for amplitudes up to 40 percent of bearing minimum clearance. However, there is no justification of his criterion in terms of coefficient accuracy at such high displacement amplitudes. Obviously, such a linearized approach would require an evalu- ation of the linearized error to set forth an acceptable bound for the linear solutions. Hashi and Sankar (1984) have at- tempted to estimate the linearized error. Deviations of linear- ized journal bearing coefficients from the nonlinear systems are calculated. Error estimation charts for some important cases of plain journal bearings under unbalance loading are provided. Contributed by the Tribology Division for publication in the JOURNAL OF TRIBOLOGY. Manuscript received by the Tribology Division June 7, 1995; revised manuscript received June 16, 1997. Associate Technical Editor: M. J. Braun. When the bearing displacement amplitude exceeds a certain limit, bearing reactions become nonlinear. There are limited studies which have been performed on the nonlinearity within the bearing oil film. Hattori (1993) shows that for a rotary compressor under large dynamic loads (displacement ampli- tude varies more than 50 percent of the radial clearance per revolution), journal bearing stiffness and damping coeffi- cients can vary by more than one order of magnitude, demon- strating that oil film nonlinearity seriously influences rotor motion. The linearized model is therefore not acceptable for such cases and some form of nonlinear analysis is required. Choy, et al. (1991) use a nonlinear approach in modefing the nonlinearity within the oil film forces. They expand the bear- ing reactions using a power series and retain the higher order terms. They show that at displacements far away from the equilibrium, nonlinearity in the oil film is significant and can be modeled closely by higher order stiffness and damping coefficients. Their results indicate that the range of accuracy for the higher-order coefficients can be extended by including more terms in the expansion. In this paper, a quasi-static nonlinear dynamic model for hydrodynamic bearings is described. There are two features to this model: (j) a nonlinear dynamic model capable of capturing the nonlinearity in oil film forces under large amplitude excur- sions, and («) an error scheme that evaluates higher-order trun- cation error in the oil film force expansion. The first feature enables the designer to calculate the nonlinear oil film forces. The second feature provides a useful way for the designer to set a confidence bound on the nonlinear solutions. Depending on the application requirements, additional terms can be added to the bearing load expansion. A finite difference scheme with successive-over-relaxation (SOR) is employed to solve the stated nonlinear model. This model is first applied to a conventional slider bearing to bench- mark the new model and then to a new Micro-Electro-Mechani- cal Systems (MEMs) smart bearing concept (Hearn et al., 1995). The MEMs smart bearing concept involves active con- trol of the film thickness profile to induce desirable bearing performance. A typical MEMs smart bearing consists of an actively deformable surface (actuators) and pressure sensors (Hearn et al., 1995). Preliminary results from a linear model Journal of Tribology Copyright © 1998 by ASME JULY 1998, Vol. 120 / 595 Downloaded 24 Aug 2012 to 128.83.63.20. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm