Load-Displacement Relationships for Ball and Spherical Roller Bearings

Analytical relationships for calculating three rolling element bearing loads (F_x, F_y, and F_z) and two tilting moments (M_y and M_z) as a function of three relative race translations (dx, dy, and dz) and two relative race tilting angles (dθ_y and dθ_z) have been given in a previous paper. The previous approach was suggested for any rolling element bearing type, although it has been recognized that the assumption of a constant rolling element-race contact angle is not well supported by deep groove ball bearings (DGBB) or angular contact ball bearings (ACBB). The new approach described in this paper addresses the latter weaknesses by accounting for the variation of the contact angle on the most loaded ball and also shows that misalignment effects on spherical roller bearing (SRB) loads are negligible. Comparisons between the simplified approach (option 1) and the “enhanced” numerical approach (option 2, which requires a summation of the load components on each ball with the appropriate contact angle included) is made, showing a good correlation as long as the relative misalignment remains reasonable or occurs in the plane corresponding to maximum radial displacement. Option 2 can, however, be recommended since it is easy to program and quite accurate at any misalignment level. Other pros and cons of both options are described. As in the previous paper, a full coupling between all displacements and forces, as well as roller and raceway crown radii, are considered, meaning that Hertzian point contact stiffness is used in roller bearings at low load with a smooth transition toward Hertzian line contact as the load increases. This approach is particularly recommended for programming the rolling element bearing behavior in any finite element analysis or multibody system dynamic tool, since only two nodes are considered: one for the inner race (IR) center, usually connected to a shaft, and another node for the outer race (OR) center, connected to the housing.