| The kinematics of bearings has been derived and used for a hundred years, although it is conceptually incorrect. The reason is that the arms of rotation of the ball and the raceways are different, so that the angular velocities of different elements of bearings are not additive as previously assumed. The kinematics of the ball screw mechanism has been treated similar to that of bearings. This makes the previous analysis of such aspects as efficiency questionable. The efficiency equations for the ball screw mechanism, which were derived by Levit (2) and used by many books for years, are also proven to be incorrect.; This research provides a theoretical method to explore the kinematics of the ball screw mechanism through the use of Frenet-Serret Coordinates. The use of these coordinates allows a convenient way to visualize the relative motion inside the mechanism. By reducing the helix angle of the screw to zero, the theory is additionally valid for the analysis of ball bearings. It is shown that there must be slip between the balls and the nut or screw at all times and thus, the no-slip condition is unattainable. Therefore, the conventional manner of modeling friction in these machine elements needs to be reinvestigated.; In summary, this thesis takes a fundamental approach in examining the kinematics of the ball screw mechanism as well as ball bearings. This exact theory is approximated in order to better develop methodologies for designing these machine elements. Throughout the thesis comments are made as to the desirable design parameter values. It is envisioned that the closed-form solution would be used for initial determination of feasible and optimal designs and that the exact theory would then be used to further develop and understand the final design. |
