A unified and simplified treatment of the non-linear equilibrium problem of double-row rolling bearings. Part 1: Rolling bearing model

Abstract

To increase the rigidity of bearings±shaft systems, the shaft is often supported by a series of double-row rolling bearings. Little work on multiple-row rolling bearings has been published. A comprehensive model is proposed which permits the determination of the internal interactions. The total elastic de¯ection between bearing rings is described using a vector-and-matrix method. A variety of double-row rolling bearing types are analysed, such as tapered roller bearing, cylindrical roller bearing, spherical roller bearing, self-aligning ball bearings and angular contact ball bearing. The basic internal geometry (including the internal clearance) and the effect of the initial preload (in term of the initial axial compression) are considered. Keywords: double-row rolling bearings, ®ve degrees of freedom, load distribution, vector-andmatrix method NOTATION A distance between the centre of curvature of the inner and outer raceways (m) ACBB angular contact ball bearing CRB cylindrical roller bearing d m pitch circle diameter (m) D rolling element diameter (m) DB back-to-back arrangement DF face-to-face arrangement e axial distance between the roller inertial frame origins of the bearing rows (m) E 0 effective elastic modulus (N/m 2 ) F x , F y , F z bearing loads applied along the x, y and z directions respectively (N) K load±de¯ection constant for a point contact (N/m 3/2 ) and for a line contact (de®ned per unit length) (N/m 19/9 ) L roller effective length (m) M bearing tilting moment vector n exponent depending on the contact typeˆ3/2 for point contact andˆ10/9 for line contact n unit vector q contact load intensity (N/m) q contact load intensity vector Q contact load (N) Q contact load vector r i inner raceway curvature radius (m) r nominal position vector in the inertial frame r 0 displaced position vector in the inertial frame R distance from the rolling element origin frame to bearing axis (m) SABB self-aligning ball bearing SRB spherical roller bearing [T] transformation matrix from roller to inertial frame TRB tapered roller bearing u clearance (m) x roller axial coordinate (m) …x, y, z † rolling element frame …X, Y, Z † bearing inertial frame a contact angle (rad) The MS was Downloaded from d p axial clearance/compression of the bearing row (m) d x , d y , d z relative linear displacements of the bearing along the x, y and z directions respectively (m) d bearing displacement vector D total approach between bearing rings (m) ehalf tapered angle of roller (rad) y y , y z relative tilting angles of bearing around the y and z directions respectively (rad) ‰ Š outer displaced to outer ®xed transformation matrix q position vector for outer raceway in roller frame P r curvature sum (m ¡1 ) c angular position of a rolling element (rad)

Simple and Versatile Dynamic Model of Spherical Roller Bearing

“…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.…”

Effects of Preloads on Vibration Transmission Through Double Row Angular Contact Ball Bearings

“…Although single row bearings have been well and extensively studied, publications specific to double row bearings are sparse. Bercea et al [17] formulated the relative displacement between the bearing rings (also termed as the 'ring approach') for various double row bearing types such as tapered, spherical, cylindrical roller and angular contact ball bearings. Their study was limited to a formulation of bearing deflections and did not include any stiffness formulation or dynamic analysis.…”

A Method to Solve the Stiffness of Double‐Row Tapered Roller Bearing

“…However, inertial loading due to the rotational speed effect was not considered in their bearing model. Bercea et al [24,25] presented the relationship of bearing stiffness given by Harris [14] in a DRTRBs example, and formulated the relative displacement between the bearing rings for various double-row bearing types such as DRTRBs [26]. Fernandez et al [2] optimized presetting loads to improve working conditions for a doublerow TRB.…”