Accuracy calculation of a trapezoidal thread obtained by turning with a tool with tangential offset

Abstract

The accuracy of the trapezoidal thread profile for lead screws determines their assemblability, load-bearing capacity, and stability of kinematic characteristics. This is especially critical for small-diameter screws with a large pitch, where the geometry of the cutting tool significantly influences the form-generation process. This work presents an analytical modeling of the impact of tangential displacement of a lathe cutter tip on the geometry of a trapezoidal thread profile, taking into account the actual spatial orientation of the cutting edge. Dependencies are derived to predict the increase in the flank angle and the corresponding displacement of the pitch diameter resulting from the tangential deviation of the tool. Particular attention is paid to the analysis of small-diameter threads with a large pitch. Calculations were performed for a Tr 22×8 trapezoidal thread manufactured according to the ISO 2904:2020 standard. For a cutter tip displacement of 200 µm, an increase in the flank angle by 6.24 arcminutes was obtained, which constitutes over 28% of the allowable deviation according to the ASME B1.5 standard. The required diametral compensation for the pitch diameter reaches 26.4 µm, corresponding to approximately 9% of the tolerance field. It is demonstrated that such deviations are systematic and cannot be eliminated solely by improving machine tool accuracy, as they are inherent to the geometry of the thread-cutting process and the technologically necessary tool setting parameters. The obtained results confirm the advisability of preliminary analytical prediction of profile errors and accounting for tangential displacement when setting up cutters to ensure the interchangeability and assemblability of critical screw pairs.