Stiffness matrix formulation for double row angular contact ball bearings: Analytical development and validation

Abstract Though double row angular contact ball bearings are widely used in industrial, automotive, and aircraft applications, the scientific literature on double row bearings is sparse. It is also shown that the stiffness matrices of two single row bearings may not be simply superposed to obtain the stiffness matrix of a double row bearing. To overcome the deficiency in the literature, a new, comprehensive, analytical approach is proposed based on the Hertzian theory for back-to-back, face-to-face, and tandem arrangements. The elements of the five-dimensional stiffness matrix for double row angular contact ball bearings are computed given either the mean bearing displacement or the mean load vector. The diagonal elements of the proposed stiffness matrix are verified with a commercial code for all arrangements under three loading scenarios. Some changes in stiffness coefficients are investigated by varying critical kinematic and geometric parameters to provide more insight. Finally, the calculated natural frequencies of a shaft-bearing experiment are successfully compared with measurements, thus validating the proposed stiffness formulation. For double row angular contact ball bearings, the moment stiffness and cross-coupling stiffness terms are significant, and the contact angle changes under loads. The proposed formulation is also valid for paired (duplex) bearings which behave as an integrated double row unit when the surrounding structural elements are sufficiently rigid.

[1]  B. J. Stone.  The State of the Art in the Measurement of the Stiffness and Damping of Rolling Element Bearings , 1982 .

[2]  J. J. Blech,et al.  In Situ Determination of Rolling Bearing Stiffness and Damping by Modal Analysis , 1987 .

[3]  M. Sartor,et al.  Calculation of the Stiffness Matrix of Angular Contact Ball Bearings by Using the Analytical Approach , 2000 .

[4]  Rajendra Singh,et al.  Effect of bearing preloads on the modal characteristics of a shaft-bearing assembly: Experiments on double row angular contact ball bearings , 2012 .

[5]  R T.J..  VIBRATION TRANSMISSION THROUGH SELF-ALIGNING (SPHERICAL) ROLLING ELEMENT BEARINGS: THEORY AND EXPERIMENT , 1998 .

[6]  Nalinaksh S. Vyas,et al.  ESTIMATION OF NON-LINEAR STIFFNESS PARAMETERS OF ROLLING ELEMENT BEARINGS FROM RANDOM RESPONSE OF ROTOR-BEARING SYSTEMS , 1995 .

[7]  B. Stone,et al.  The Measurement of the Radial Stiffness of Rolling Element Bearings under Oscillating Conditions , 1980 .

[8]  Eric R. Marsh,et al.  EXPERIMENTAL MEASUREMENT OF PRECISION BEARING DYNAMIC STIFFNESS , 1997 .

[9]  A. B. Jones.  A General Theory for Elastically Constrained Ball and Radial Roller Bearings Under Arbitrary Load and Speed Conditions , 1960 .

[10]  Rajendra Singh,et al.  Vibration transmission through rolling element bearings, part IV: statistical energy analysis , 1992 .

[11]  M. Cao.  A Refined Double-row Spherical Roller Bearing Model and its Application in Performance Assessment of Moving Race Shaft Misalignments , 2007 .

[12]  Aydin Gunduz,et al.  Multi-dimensional stiffness characteristics of double row angular contact ball bearings and their role in influencing vibration modes , 2012 .

[13]  Rajendra Singh,et al.  Vibration Transmission through Rolling Element Bearings, Part V: Effect of Distributed Contact Load on Roller Bearing Stiffness Matrix , 1994 .

[14]  Rajendra Singh,et al.  Vibration transmission through rolling element bearings, part I: Bearing stiffness formulation , 1990 .

[15]  Rajendra Singh,et al.  VIBRATION TRANSMISSION THROUGH ROLLING ELEMENT BEARINGS, PART II: SYSTEM STUDIES , 1990 .

[16]  Daniel Nelias,et al.  A unified and simplified treatment of the non-linear equilibrium problem of double-row rolling bearings. Part 1: Rolling bearing model , 2003 .

[17]  A. Palmgren.  Ball and roller bearing engineering , 1945 .

[18]  Teik C. Lim,et al.  Vibration transmission through rolling element bearings. III - Geared rotor system studies , 1991 .

[19]  Juhn-Shyue Lin.  Experimental analysis of dynamic force transmissibility through bearings , 1989 .

[20]  Yusuf Altintas,et al.  A General Method for the Modeling of Spindle-Bearing Systems , 2004 .

[21]  Mohammad Dawod Rajab.  Modeling of the transmissibility through rolling-element bearings under radial and moment loads / , 1982 .

[22]  Daniel Nelias,et al.  A unified and simplified treatment of the non-linear equilibrium problem of double-row rolling bearings. Part 2: Application to taper rolling bearings supporting a flexible shaft , 2003 .

[23]  M. Cao,et al.  A comprehensive dynamic model of double-row spherical roller bearing—Model development and case studies on surface defects, preloads, and radial clearance , 2008 .

[24]  M. While.  Rolling Element Bearing Vibration Transfer Characteristics: Effect of Stiffness , 1979 .

[25]  T. A. Harris,et al.  Rolling Bearing Analysis , 1967 .

[26]  Dong-Hoon Choi,et al.  A Design Method of an Automotive Wheel-Bearing Unit With Discrete Design Variables Using Genetic Algorithms , 2001 .

[27]  Wei Xu,et al.  A secant method for nonlinear least-squares minimization , 2012, Comput. Optim. Appl..

[28]  Johannes Brändlein,et al.  Ball and roller bearings: Theory, design, and application , 1985 .

[29]  Robert G. Parker,et al.  Stiffness matrix calculation of rolling element bearings using a finite element/contact mechanics model , 2012 .

[30]  Miha Boltežar,et al.  An indirect approach to investigating the dynamics of a structure containing ball bearings , 2004 .

[31]  J. M. de Mul,et al.  Equilibrium and Associated Load Distribution in Ball and Roller Bearings Loaded in Five Degrees of Freedom While Neglecting Friction—Part I: General Theory and Application to Ball Bearings , 1989 .

[32]  S. J. Lacey,et al.  Dynamic and static characteristics of a wide speed range machine tool spindle , 1983 .

[33]  Jan Swevers,et al.  A piezo-based bearing for the active structural acoustic control of rotating machinery , 2010 .