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)
This study is focused on a bearing arrangement made of two tapered roller bearings and a shaft, supporting a combination of axial and radial loads. The paper is divided in two parts. This part presents first a numerical procedure to obtain the internal load distribution within tapered roller bearings. The internal load distribution is required to estimate the bearing fatigue life. Afterwards, a numerical method and the related algorithm for a load support system constituted of a shaft and two tapered roller bearings are described. The effect of the initial axial preload (or total axial compression) is taken into account. In consequence the method proposed can be used to select an adequate axial preload in order to increase the system survivability. An example of model implementation, i.e. application to the transfer shaft of an automobile automatic transaxle, is presented in Part 2 of the paper.
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