Tolerance envelopes of planar mechanical parts with parametric tolerances Yaron Ostrovsky-Berman and Leo Joskowicz School of Engineering and Computer Science The Hebrew University of Jerusalem, Jerusalem 91904, Israel Email: yaronber@cs.huji.ac.il Abstract We present a framework for the systematic study of parametric variation in planar me- chanical parts and for efficiently computing approximations of their tolerance envelopes. Part features are specified by explicit functions defining their position and shape as a func- tion of parameters whose nominal values vary along tolerance intervals. Their tolerance envelopes model perfect form Least and Most Material Conditions (LMC/MMC). Tolerance envelopes are useful in many design tasks such as quantifying functional errors, identify- ing unexpected part collisions, and determining device assemblability. We derive geometric properties of the tolerance envelopes and describe four efficient algorithms for computing first-order linear approximations with successive accuracy. The results from experiments on 14 realistic part models demonstrate that on average, our algorithms are an order of magnitude faster and more accurate than the commonly used Monte Carlo simulation, and produce better results than the computationally expensive Taguchi method. Keywords: tolerancing, parametric models, tolerance zones, mechanical design. 1 Introduction Manufacturing and assembly processes are inherently imprecise, producing parts that vary in size and form. The need to control the quality of the production and to manufacture parts interchangeably led to the development of tolerance specifications. Tolerance specifications are the critical link between the designer and the manufacturer. Designers prefer tight tolerances to ensure that the part will fit in the assembly and perform its function. Manufacturers, on the other hand, prefer loose tolerances to lower the production cost and decrease the need for quality machine tools and precision measurement machines. Tolerance analysis methods play a key role in bridging between the two. Tolerance allocation is difficult even to the most skilled of designers because it requires identifying the critical interactions of toleranced dimensions, which often have complex depen- dencies. Tolerancing methods have been developed and incorporated into most modern CAD software. Given a tolerance allocation, tolerance analysis consists of predicting the effect of the allowed variations on the design functions. Tolerance synthesis consists of finding tolerance intervals that meet the functional requirements at the lowest cost. A key problem in tolerance analysis is computing the tolerance envelope of a part from its tolerance specification. Tolerance specifications define a family of parts consisting of all valid instances of the part. The tolerance zone of a part is the difference between the smallest volume containing all part instances and the largest volume contained in all part instances. Its boundaries, called the part tolerance envelope, define the worst-case variability of the part features, and thus model perfect form Most and Least Material Conditions (MMC/LMC). Part tolerance zones are useful in design tasks such as quantifying functional errors, identifying unexpected part collisions, and determining device assemblability. 1