| In the geometric dimensioning and tolerancing (GD&T) field, the use of coordinate measuring machine (CMM) became widely common. It consists in probing the surfaces in a finaite number of points and to fit a perfect geometric form according to chosen criteria (least squares, mini-max...). Then, the design tolerances validation test is performed using the tolerancing interpretation standards.; However, the probing procedure still depends mainly on the programmer's experience and knowledge. He actually estimates the ideal probing densities considering two criteria: the manufacturing process capabilities and the design tolerances.; In this present work, we propose an original numerical method that allows estimating the probing point densities taking into account these two criteria.; This method consists in generating numerically a finite number of point measured on an imperfect feature randomly deviating them in the local normal vector direction of a perfectly shaped generating feature, in order to simulate manufacturing deviation such as vibrations, tool elasticity....; The generating feature is derived from a random deviation of the nominal feature, simulating tool/fixture position variation....; Then, we fit the perfectly shaped substitute feature to these points using the least squares criteria since it's the most commonly used by CMMs. A design tolerances validation test is performed on this feature using the tolerancing interpretation standards.; Concurrently, we perform the same test on the generating feature. This last diagnostic is the one that we would have had if the substitute feature was derived from in infinite number of points since the least-square substitute feature tends to the generating feature when the number of points tends toward infinity.; If both diagnostics are identical, the number of points is judged sufficient. In order to consolidate the result, we perform the same procedure several times (Monte-Carlo simulations) and we count how many times the two diagnostics are identical and we derive the Good Diagnostic R ate (GDR) relatively to the total number of inspected pieces.; The final step is to set a GDRthreshold and to increase the probing densities in each Monte-Carlo simulation till we reach it and thus we get the ideal probing density.; Moreover, the complexity of the method and the importance of the least-square feature substitution step, we took a particular car to implement it using two different approaches: Bourdet's geometrical approach and the Forbes mathematical approach, to compare their performances in terms of precision and convergence speed and to test them in various conditions to confirm their reliability.; Through the probing densities prediction method application to simple examples, we developed custom strategies to find solutions in specific cases: datum probing and surface features (other than lines and circles), then we applied it to more complete examples: concentric cylinders and plan orientation tolerance.; To end, the proposed method offer a simple solution to the probing densities prediction issue, but requires a numerical simulation for each case that might be demanding in terms of calculations. |
