An Analysis of Misaligned Single-Row Angular-Contact Ball Bearing

Abstract

Contributed by the Mechanisms and Robotics Committee for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received April 2003. Associate Editor: J. S. Rastegar. The analytical method taking the centrifugal force and the angular misalignment into account is developed for the contact mechanisms of a ball bearing. In this method, the misalignment, the deformations created in the axial and radial directions of a bearing system are taken as three knowns in the analysis. Several parameters such as the contact angles, the normal forces created at contact points, the deformations between ball and raceways varying with bearing position angles, and the applied loads in two directions can be evaluated by differing the angular misalignment and the elastic deformations in two directions. The radius of curvature for either the inner raceway or the outer raceway of a ball bearing is r and the curvature entry center is at point i and o, respectively (see Figure 1). Two tori can be formed for the inner and outer raceways 1. In the case of a ball bearing before applying a load, the geometric center of the torus for the outer raceway is located at the point of the coordinates 0,0,ζo, whereas the torus of the inner raceway has the coordinates of 0,0,ζi. As the loads are applied to the bearing, the geometric center of the torus for the inner raceway remains unchanged. However, the geometric center of the torus corresponding to the outer raceway is now moved to ξo,0,ζo. The coordinates for any point on the surface of the inner raceway can be written as 1(1)xi=gi+r cos θcos ψyi=gi+r cos θsin ψzi=r sin θ+ζiwhere gi=di/2+rζi=−r−D/2sin α0In Eq. (1) α0 is the ball’s contact angle under no load, as shown in Fig. 1. As the misalignment occurs in a ball bearing with a rotational angle β with respect to the y-axis as shown in Fig. 2, the contact mechanisms at the inner and outer races are changed too. The β angle is positive if it rotates counterclockwise. The bearing position angle ψ is measured from the x-axis. Then, the surface profile of the inner race after misalignment can be described as: (2)xi′yi′zi′=cos β0−sin β010sin β0cos βxiyizi=xi cos β−zi sin βyixi sin β+zi cos βSimilarly, the coordinates for any one point on the outer-raceway surface are given as 1: (3)xo=go+r cos θcos ψ−δryo=go+r cos θsin ψzo=r sin θ+ζowhere go=do/2−rζo=r−D/2sin α0+δaDue to the misalignment with an angle of β, the torus created by the profile of the inner race is rotated such that the centers of the inner and outer rings created by any section are still at the same plane although the center of the inner ring is now moved to the point i′. The relative positions of i and i′ before and after the misalignment are shown in Fig. 2. The two tori which have point i′ and o as the center of two circles and r as the radius of these two circles for the inner and outer raceways would intersect at two points, c1 and c2. At these points, xo,yo,zo must be equal to xi′,yi′,zi′. Then the equivalence of Eqs. (2) and (3) gives (4)(go+r2−−xi sin β+zi cos β−ζo2)2=xi cos β+zi sin β+δr2+yi2The solutions of θ in Eq. (4) vary with position angle ψ of the ball bearing. The above nonlinear expression is obtained for the variable contact, which can be solved by such as the Newton’s iterative scheme if the bearing elastic deformation in radial and axial direction, δr and δa, are available. If the angle θ is obtained, the contact angle α′ is thus calculated by Eq. (5). The contact angle α′ can be written as 1: (5)α′=π−θ−cos−1A/2rAccording to Eq. (5), the contact angle α′ can be obtained only when the angle θ is available. This contact angle is the same in the inner and the outer raceways if the centrifugal force is ignored. The distance between two ring centers, i′ and o, is thus calculated: (6)A=‖g⃗i−g⃗o‖where g⃗i=cos β0−sin β010sin β0cos βs˙gi cos ψgi sin ψζig⃗o=go cos ψgo sin ψζoThe contact angle at the inner and the outer raceways is variable instead of constant. It varies depending upon th