Quasi-static Analysis of Thrust-loaded Angular Contact Ball Bearings Part I: Theoretical Formulation

CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol. 25, No. 1, 2012 ·71· DOI: 10.3901/CJME.2012.01.071, available online at www.springerlink.com; www.cjmenet.com; www.cjmenet.com.cn Quasi-static Analysis of Thrust-loaded Angular Contact Ball Bearings Part I: Theoretical Formulation CHEN Wenhua 1, 2, * , MA Zikui 1 , GAO Liang 1 , LI Xinglin 3 , and PAN Jun 2 1 Institute of Mechanical Design, Zhejiang University, Hangzhou 310027, China 2 Institute of Mechanical Design and Manufacturing, Zhejiang Sci-Tech University, Hangzhou 310018, China 3 State Testing Laboratory of Hangzhou Bearing Test & Research Center, Hangzhou 310022, China Received October 10, 2010; revised March 22, 2011; accepted May 26, 2011 Abstract: Ball bearings play an important role in various rotating machineries, but the complicated kinematic and tribological features of ball bearings make many aspects of their operating behaviors still inconclusive. Most theoretical analyses of ball bearings up to date are based on either the hypothesis of race control or other empirical models to determine the ball motion of ball bearings, but none of these strategies can reveal and consequently employ the intrinsic coupling mechanism between the spin and the tangential traction of contacting bodies rolling upon one another. To remedy the deficiency of current analytical models for ball bearing analysis, the rolling contact theory is employed to establish an explicit link between motions and interactions within ball bearings. A differential slip model is established to precisely define the slip component due to the significant curvature of the common contact patches between the ball and inner/outer raceways. The creepage and the spin ratio are formulated to accurately define the relative rigid motion between the ball and the inner/outer raceway. Then a quasi-static analytical model is established that can accurately determine the motions of the balls and races of the ball bearing. It can also give a vivid description of the slip and traction distributions within the contact area. The analytical model can be effectively used to analyze the operational conditions and tribological features of solid-lubricated ball bearings. It can also be used optimize the construction of ball bearings for specific applications. Key words: ball bearing, quasi-static, creepage, differential slip, tangential elastic compliance Notations , ab —Semi-major/minor axes of contact ellipse, C —Definition of contact area/patch, d —Diameter of raceway, diameter of pitch circle, D —Diameter of the ball, E —Modulus of elasticity, E ¢ —Reduced modulus of elasticity, f —Conformity, i.e., , f rd = F —Concentrated load G —Shear modulus of elasticity, i.e., 2(1 ), G E = + I —Mass moment of inertia, k —Ellipticity of contact ellipse, i.e., , k ab = L — Flexibility parameter for Kalker’s rolling contact theory, M—Moment, torque, o —Origin of coordinate system, p —Normal stress or tangential traction, d P —Diametral clearance of ball bearing, r — Radius of ball, radius of curvature of raceway, radius of pitch circle of the ball bearing, R — Radius of curvature of deformed contact patch, radius of curvature of the ball and groove, * Corresponding author. E-mail: [email protected] This project is supported by National Natural Science Foundation of China (Grant No. 50935002, Grant No. 51105342) © Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2012 R¢ —Curvature sum of contacting bodies, s —Slip component, i.e., relative velocity of contacting particles, t —Time, T —Resultant traction, u —Tangential elastic displacement/compliance, normal elastic deformation, V —Rolling velocity of contacting bodies, w —Difference of tangential elastic displacement or compliance, x —x-axis, longitudinal/rolling direction, y —y-axis, lateral direction, z —z-axis, normal direction, Z —Number of balls of the ball bearing, 0 —Free contact angle of the ball bearing, —Contact angle, pitch angle of the angular velocity vector of the ball, —Yaw angle of angular velocity vector of the ball, —Elastic approach between contacting bodies, —Coefficient of friction, —Poisson ratio, —Creepage in longitudinal or lateral direction, —Density of the ball, —Complete elliptic integral of the 2nd kind, —Spin ratio,

Quasi-static Analysis of Thrust-loaded Angular Contact Ball Bearings Part I: Theoretical Formulation

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