CMM-based determination of the thread pitch diameter using a three-dimensional dual-flank tangency model

Abstract Accurate assessment of thread pitch diameter is essential to ensure the functional reliability of screw threads. Conventional calibration typically relies on mechanical probing with wires (or balls) and is often restricted to sectional evaluation, planar approximations, and manual implementation. This paper presents a coordinate measuring machines (CMMs)-based method that explicitly preserves the dual-flank condition in a full three-dimensional formulation. The approach models the exact spatial tangency of a spherical probing element to opposing helicoidal thread flanks and determines the pitch diameter directly from probe-centre coordinates by numerically solving the resulting nonlinear system. The proposed framework was experimentally validated on an ISO M27 × 3 external thread plug gauge, demonstrated numerical stability and yielded a well-defined global pitch diameter with a standard uncertainty of 0.3 µm. A Monte Carlo-based sensitivity analysis reveals that the uncertainty budget is dominated by the probe diameter calibration (accounting for the majority of variance), confirming that the global fit effectively averages out CMM coordinate noise. Additionally, comparison with a three-wire reference demonstrated agreement within the combined expanded uncertainty, yielding a normalized error of E n = 0.32 , indicating metrological consistency of the proposed CMM-based dual-flank tangency evaluation with established mechanical probing reference practice. By reconciling functional two-flank contact concepts with modern coordinate metrology, the framework offers a robust complementary approach to conventional wire-based calibration, enabling richer spatial sampling of thread geometry. This work validates that maintaining rigorous three-dimensional tangency constraints in CMM-based evaluation preserves the essential functional definitions while leveraging the advantages of digital coordinate measurement, providing a reliable bridge between traditional tactile metrology and advanced coordinate-based calibration.