Date of Award


Degree Type


Degree Name

Doctor of Philosophy (PhD)


Mechanical Engineering


Dr. Hoda A. ElMaraghy


Manufacturing processes generate surfaces with variable dimensions and geometries. The produced surfaces deviate from their nominal geometry and consequently, critical dimensions and clearaces deviate from their designed values when the parts are assembled. Since it is impossible to check whether the functional requirements fall within their allowable range after the assembly precess, geometric tolerances are specified on individual features during the design stage such that, parts whose geometric deviations would cause a violation of the functional requirements when assembled to other parts, would be rejected during the inspection process. Therefore, wrong choice of geometric tolerances could lead to either rejecting good parts, or the acceptance of bad pans leading to an assembly that violates the functional requirements. Furthermore, tolerance selection is not limited to the magnitudes of the geometric tolerances. Each feature in the assembly has four geometric variations that need to be controlled. These are size, position, form and orientation. Each of these variations, except variation in size, can be controlled by several types of tolerances and the selection of tolerance types affects the percentage of accepted or rejected parts during inspection.

This dissertation presents a novel computer-aided method for the synthesis of magnitudes and types of geometric tolerances in mechanical assemblies. Tolerance selection is formulated as a combinatorial optimization problem, where each feature has seven variables. These are the types of tolerances controling the orientation, form and position variations, the magnitudes of these tolerances as well as the magnitude size tolerance. A new criterion was developed to allocate geometric tolerances which is the minimum mismatch probability defined as the probability of rejecting a part during the inspection process which satisfy the functional requirements when assembled to other parts, or accepting one violating the functional requirements when assembled to other parts.

In order to evaluate the objective function, a probabilistic analysis method, based on Monte Carlo simulation, was developed to calculate the rejection probabilities of assemblies with geometric tolerances. The surface of each feature in the assembly is represented with a number of points where the coordinates of these points are the random variables. In each simulation cycle, the coordinates of the surface points are generated using the probability distribution associated with the manufacturing process. The inspection process is then simulated where the geometric deviations on each feature are checked against the specified tolerances. Finally, the functional requirements are checked. Several methods for parts joining were examined and a new genetic algorithms based method is developed to evaluate the maximum and minimum values of critical clearances. The use of genetic algorithms ensures the arrival to the global minimum and maximum values of the clearance. Due to the large number of random variables, and since the probabilistic analysis is used in every optimization step in the tolerance allocation algorithm, two variance reduction techniques are incorporated with the standard Monte Carlo simulation to reduce the sample size.

A number of genetic algorithms based routines are used for checking of geometric deviations on the generated parts. In many cases the evaluation of geometric deviations involve optimization. The use of global optimizers ensures the correct evaluation of deviations and avoids the unnecessary rejection of good pans. The advantage of using genetic algorithms is demonstrated with several examples used by previous researchers. Furthermore, a new parametric surface interpolation method is developed to approximate the actual surface of the manufactured parts and help in the evaluation of some geometric deviations that cannot be evaluated directly using the generated points.

The new tolerance allocation method, presented in this dissertation, attempts to fill a void area in the tolerancing research, which is the selection of the "types" in addition to magnitudes of geometric tolerances. Although the skiil of a tolerancing practitioner is still needed to specify candidate tolerance types for each geometric control, the developed robust mathematical formulation of the problem avoids the random human factors in the selection. The proposed methodology can be extended for incorporation within computer-aided tolerancing systems to assist designers in selecting geometric tolerances.

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