Probability tolerance maps: A new statistical model for non linear tolerance analysis applied to rectangular faces

A new statistical model for the tolerance analysis based upon joint probability distribution of the trivariate normal distributed variables involved in the construction of Tolerance-maps¹ (T-Maps) for rectangular face is presented. Central to the new model is a Tolerance-Map (T-Map) (Patent No. 69638242). It is the range of points resulting from a one-to-one mapping from all the variational possibilities of a perfect-form feature, within its tolerance-zone, to a specially designed Euclidean point-space. The model is fully compatible with the ASME/ANSI/ISO Standard for geometric tolerances. In this research, 4D probability T-Maps (prob T-Maps) have been developed in which the probability value of a point in space is represented by the size of the marker and the associated color. Additionally, 3D prob T-Maps (3D cross sections of the 4D prob T-Maps at pre specified values) are used to represent the probability values of two variables at a time for a constant value of the third variable on a plane. Superposition of the probability point cloud with the T-Map clearly identifies which points are inside and which are outside the T-Map. This represents the pass percentage for parts manufactured with the statistical parameters such as mean and standard deviation as of the assumed trivariate probability distribution. The effect of refinement with form and orientation tolerance is highlighted by calculating the change in pass percentage with the pass percentage for size only. Delaunay triangulation and ray tracing algorithms have been used to automate the process of identifying the points inside and outside the T-Map. Proof of concept software has been implemented to demonstrate this model and to determine pass percentages for various cases. The model is further extended to assemblies by employing convolution algorithms on two trivariate statistical distributions to arrive at the statistical distribution of the assembly. Accumulation T-Maps generated by using Minkowski Sum techniques on the T-Maps of the individual parts is superimposed on the probability point cloud resulting from convolution. Delaunay triangulation and ray tracing algorithms are employed to determine the assemleability percentages for the assembly.