Form Tolerance Evaluation of Toroidal Surfaces Using Particle Swarm Optimization

Toroidal surfaces are used in many machinery parts such as ball bearing races, bull nose and indexable endmills, continuously-variable transmissions, and enveloping worm gears. Torisity, the form tolerance of toroidal surfaces, plays an important role in mechanical design and quality control such as component functionality, assemblability, and interchangeability. Compared with simple forms, such as planes, spheres, cylinders, and cones, toroidal surfaces are more complex and the evaluation of torisity has not been thoroughly performed. This paper proposes a nonlinear approach to evaluate torisity information using particle swarm optimization (PSO). Three sets of PSO parameters are compared based on the least-squares objective function and minimum-zone objective function using toroidal geometry characteristics. The effectiveness and robustness of the PSO approaches are validated by experiments on simulated entire, outer, inner, thrust, and endmill toroidal surfaces, which appear in various machinery components. Compared with previous studies in torisity evaluation, the proposed approach is more efficient in the ability to provide the form tolerance and the surface profile information simultaneously. The proposed approach is also more consistent with the tolerance zone definition of ASME standard Y14.5M on dimensioning and tolerancing.