Coaxiality evaluation based on double material condition

Abstract

To ensure the assemblability of a workpiece, Coaxiality-DM can be applied to two coaxiality cylinders. According to this specification, the relative orientation and location between an actual datum feature and the derived datum axis are variable. Thus, the Coaxiality-DM tolerance is mainly verified by real gauges, which is costly for small batch production. To develop a proper tool for that production, a evaluation model for Coaxiality-DM tolerance was investigated in this paper. According to the International Organization for Standardization (ISO), the geometry of the real gauge was analyzed, and a virtual gauge model was established. Based on the model, an evaluation method was proposed. Furthermore, an example of a stepped shaft was provided to illustrate the applicability of the proposed method. Finally, a comparative experiment was carried out, and the results revealed that the proposed model could produce more accurate results than the existing mathematical methods.However, MRDR causes a big difference between the evaluation of Coaxiality-DM tolerance and other datum-fixed tolerances. In this case, five major mathematical evaluation approaches are applicable: direct calculation, least squares method (LSM), convex-hull method, geometrical search method, and optimization method. 1. When MRDR of the actual considered feature is allowed, any location error of the considered feature (such as Coaxiality-DM error) varies and cannot be calculated directly. 2. LSM creates an error function for the considered feature before the least squares of measurements are calculated [3,4].However, when MRDR is allowed, a function for location error cannot be created directly. Thus, LSM cannot be applied to Coaxiality-DM directly. In addition, an LSM error is not exact according to the International Organization for Standardization (ISO). 3. The convex-hull method establishes the convex-hull of the considered feature before assessing the minimum error by enumeration, thus reducing the data size significantly [5]. However, as Coaxiality-DM error cannot be directly evaluated, the convex-hull method can only be employed to reduce the data size during the evaluation of Coaxiality-DM error. 4. A geometric search method is generally proposed for the error evaluation of one or a few types of features, and it gets the search direction by developed geometric rules. Hence, the geometric search method performs well for certain types of features, whereas the performance is unsatisfactory for others. Currently, it is mainly applied to single simple features, such as the straightness of straight lines [6], sphericity of balls [7], and flatness of planes [6,8]. However, it is difficult to develop a geometric search method for Coaxiality-DM, which is more complex. 5. If an optimization method is used to evaluate an error as a maximum or a minimum, an error objective function should be established. Currently, a plethora of optimization methods are available, such as the Quasi-Newton method for unconstrained problems; repetitive bracketing method...

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