Simple and Versatile Dynamic Model of Spherical Roller Bearing

Hindawi Publishing Corporation International Journal of Rotating Machinery Volume 2013, Article ID 567542, 13 pages http://dx.doi.org/10.1155/2013/567542 Research Article Simple and Versatile Dynamic Model of Spherical Roller Bearing Behnam Ghalamchi, Jussi Sopanen, and Aki Mikkola Department of Mechanical Engineering, Lappeenranta University of Technology, P.O. Box 20, 53851 Lappeenranta, Finland Correspondence should be addressed to Behnam Ghalamchi; Received 7 February 2013; Accepted 21 August 2013 Academic Editor: Paolo Pennacchi Copyright © 2013 Behnam Ghalamchi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Rolling element bearings are essential components of rotating machinery. The spherical roller bearing (SRB) is one variant witnessing increasing use because it is self-aligning and can support high loads. It is becoming increasingly important to understand how the SRB responds dynamically under a variety of conditions. This study introduces a computationally efficient, three-degree-offreedom, SRB model that was developed to predict the transient dynamic behaviors of a rotor-SRB system. In the model, bearing forces and deflections were calculated as a function of contact deformation and bearing geometry parameters according to the nonlinear Hertzian contact theory. The results reveal how some of the more important parameters, such as diametral clearance, the number of rollers, and osculation number, influence ultimate bearing performance. One pair of calculations looked at bearing displacement with respect to time for two separate arrangements of the caged side-by-side roller arrays, when they are aligned and when they are staggered. As theory suggests, significantly lower displacement variations were predicted for the staggered arrangement. Following model verification, a numerical simulation was carried out successfully for a full rotor-bearing system to demonstrate the application of this newly developed SRB model in a typical real world analysis. 1. Introduction Bearings are one of the most important components in mechanical systems, and their reliable operation is necessary to ensure the safe and efficient operation of rotating machinery [1]. For this reason, a multipurpose dynamic roller bearing model capable of predicting the dynamic vibration responses of rotor-bearing systems is important. However, bearings introduce nonlinearities, often leading to unexpected behaviors, and these behaviors are sensitive to initial conditions. For rolling element bearings, the significant sources of nonlinearity are radial clearance between the rolling elements and raceways and the nonlinear restoring forces between the various curved surfaces in contact. A special type of nonlinearity is introduced to the system if the contact surfaces have distributed defects, such as waviness, or localized defects, such as inner or outer ring defects. Goenka and Booker [2] extended the general applicability of the finite element method to include spherical roller bearings (SRBs). In their research, triangular finite elements with linear interpolation functions were used to model the lubricant film. Loading conditions for spherical roller bearings with elastohydrodynamic and hydrodynamic lubrication effects were analyzed by Kleckner and Pirvics [3]. They simulated the mechanical behavior of spherical roller bearings in isothermal conditions. Creju et al. [4, 5] improved the dynamic analysis of tapered roller bearings by improving integration of the differential equations that describe the dynamics of the rollers and bearing cage. Their study considered the effects of centrifugal forces and the gyroscopic moments of the rollers. The effects of correction parameters for roller generatrices in spherical roller bearings were discussed by Krzemiński-Freda and Warda [6]. They focused in their study on determining a proper ratio of osculation coefficients for both races to obtain self-stabilization of the barrel shaped roller and to minimize friction losses. Olofsson and Björklund [7] performed 3D surface measurements and analysis on spherical roller thrust bearings that revealed the different wear mechanisms. A theoretical model for estimating the stiffness coefficients of spherical roller bearings was developed by Royston and Basdogan [8] showing that coefficient values are complicated functions, dependent on radial and axial preloads. 2 While this work is useful for qualitative analysis, it cannot deliver the dynamic insights needed for understanding the high performance machine systems. Olofsson et al. [9] simulated the wear of boundary lubricated spherical roller thrust bearings. A wear model was developed in which the normal load distribution, tangential tractions, and sliding distances can be calculated to simulate the changes in surface profile due to wear. Taking into account internal geometry and preload impacts, Bercea et al. [10] applied a vector-and-matrix method to describe total elastic deflection between double-row bearing races. This study focused only on static analysis. It is not capable of delivering a detailed analysis of the complex dynamic behaviors of spherical roller bearing systems involving nonlinear interactions between rollers and inner/outer races. Cao and Xiao [11, 12] established and applied a comprehensive spherical roller bearing model to provide quantitative performance analyses of SRBs. In addition to the vertical and horizontal displacements considered in previous investigations, the impacts of axial displacement and load were addressed by introducing degrees-of-freedom in the axial shaft direction. The point contacts between rollers and inner/outer races were considered. These bearing models have a large number of degrees-of-freedom since there is one degree-of-freedom (DOF) for each roller and an additional 3 to 5 DOFs for the inner race. Its high complexity makes this bearing model unattractive for the analysis of complete rotorbearing systems. For example, a single gear-box can contain up to ten roller bearings. The effect of centrifugal forces on lubricant supply layer thickness in the roller bearings was considered by van Zoelen et al. [13]. In particular, this model is used to predict lubricant layer thickness on the surface of the inner and outer raceways and each of the rollers. In this extended model, it is assumed that the lubricant layers for each of the roller raceway contacts are divided equally between the diverging surfaces. Although a large number of ball bearing models exist, there has been little study of spherical roller bearing dynamics. For example, Harsha et al. [14, 15] studied the rolling element dynamics for certain imperfect configurations of single row deep-grooved ball bearings. The study revealed dynamic behaviors that are extremely sensitive t (...truncated)