Allocation of Geometric Tolerances zNew Criterion and Methodology

8/13/2019 Allocation of Geometric Tolerances zNew Criterion and Methodology

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E &=Set of feasible values for r and r' satisfyingthe functional requirements6 9 R1=Set of values of ra nd r' produced by themanufacturing process S1CZZ Rp=Set of values of r and r' produced by themanufacturing process S2~ GT=Set of feasible values of r and r' satisfyingthe geometric tolerance specificationFig.4 Feasibility Space for a Cylindrical Feature

Since it would be difficult o verify the functional requirementsafter completing the assembly, he geometry of the individualmanufactured parts is controlled with a set of geometrictolerances in order to reject bad parts during inspection. Theunderstanding of the relationship between geometrictolerances, assembly functional requirements andmanufactured parts is facilitated by drawing the feasibleareas over two radii (two points are used for demonstrationsince it is impossible to view a diagram of the whole featurein hyperspace) of a manufactured feature as shown n Fig.4.The area enclosed by the dashed lines in Fig.4 RGT)represent the set of feasible values for r and r' satisfying theimposed geometric tolerances. If the imposed geometrictolerances are allocated to capture the design intent, thenthe set of feasible values of r and r' satisfying the functionalrequirements should satisfy the geometric tolerances andthe set of values r and r' which violate the functionalrequirements should be rejected when checked for thegeometric tolerances. In other words, geometric tolerancesshould be selected to minimize the hyper-volume R, shownin Fig5 where:

Fig.5 Mismatch RegionAssume that the range of values of each radius r;. is coveredby a probability distribution q then the integral:P = ... u rl, 2 ...,r,, dr,drz ..dr,, 2)

r l . r*.... n E R,

is equal to the probability that the assembly will belong to theregion R,, where u(r1 r2,...,rn) is the joint probabilitydistribution over all radii and n is the number of radii (randomvariables).P will be denoted as the probability of mismatch.Hence, f the geometric tolerances are allocated such that allparts, are accepted during inspection, will produceassemblies which satisfy the design functionalrequirements, f m would be equal to zero.4.1 Evaluation of the Mismatch ProbabilityThe integral of equation can be evaluated using MonteCario simulation. where instances of the coordinates offeature points can be generated, using some probabilitydistributions. These instances are used to simulate both theinspection and assembly processes and a counter isincremented whenever a mismatch is encountered. Thereare two Monte Carlo methods that can be usad to evaluateP . These are the hit-and-miss method and thesample-mean method. The sample-mean method requiresthe prior knowledge of a joint probability distribution havingits highest density over the regionRm . Since the region Ris not known, the hit-and-miss Monte Carlo method, whichassumes a uniform distribution associated with eachvariable, is more appropriate. However, using thenit-and-miss method, a large error in the evaluated value ofP can develop especially with a large number of variables.This problem can be remedied by incorporating variancereduction techniques in the simulation such as Latinhypercube sampling and antithetic variates (7). Theevaluation process is described in the following algorithm:Given:

An assembly with the number of featureslsurfacesEach featurekurface is represented by a number ofpoints Np whose locations in space follow theprobability distribution of the manufacturing processwhich generate the featurelsurface. The totalnumber of random variables N is therefore equal to

NSUrf.

NsudN = , 3)

J1iii. A number of functional requirements Nreq.iv. A Monte Carlo simulation sample size N,.a. Latin hypercube sampling step

1. repeat Ns times,Assume a uniform distribution over the range ofpossible values of the random variables. Divide thecumulative probability function of each randomvariable into m equal probability intervals.Generate a random number Ui for each interval anduse the inverse uniform probability function to deliveran instance of the random variable creating thematrix r. where:

r: 1 r1.2 r13 ... r1.m...= . . . .. .

For each random variable r, rearrange the mgenerated values in a random permutation

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b. Antithetic variates stepFor every random variable 9 use the random number1-Ui to deliver another set of m instances forminganother matrix r .

c. Since each column of the r and F matrices forms aninstance of a set of manufactured parts, then for eachcolumn in the r matrix do:

i. Calculate the geometric deviations on all featurescheck them for the imposed tolerances.ii.Simulate the assembly sequence then verify theassembly functional requirements.

iii.if the assembly requirements are in spec and thetolerances are not in spec or if the assemblyrequirements are not inspec and the tolerances arein spec thenelseend if

e. end-forf. Repeat the loop of step d or the f matrix obtaining

ndm.

Increment Na = (Counter for belonging to Rm)Increment Nf (Counter for not belonging to Rm).

5. end-repeat6. Calculate the probability that the assembly nstances will

belong to Rm.f , = 0 . 5 x a_.=_a(Na + Nf Na + Nf

4.2 Example

1500.0I ]

Fig.6 Simplified punching machineIn order to show how the mismatch probability can be usedto identify the effect of poor tolerance choices, consider thesimplified diagram of a punching machine shown inFig.6 which is composed of four main parts (base, stand,head and punch). The base has two holesbl and &,whosenominal diameters are 210 and 325 mm respectively. Thefunctional requirement is to keep the clearance between thepunch and the holebl between 8.6 and 12 mm. Assume thatall dimensions are produced at their nominal values exceptfor the two base holes. and that only the form and orientationof hole b l and the orientation of hole b2 are to be controlled.For form control, cylindricity would constrain the hole's formthan circularity assuming that both types have the sametolerance magnitude. The orientation of the punch in spaceis determined by the orientation of hole &, and hence thecontrol of orientation of hole l should relate to a datumestablished by hole D). he probability of mismatch wasevaluated for form control and then for orientation control.The results are shown in Table 3 .

~~ ~~Control Tolerance Tvpes lMaqnitude I f 1~~Orientation w.r.t. A for bl and bz -2 p.032

w.r.t. A for bz and 2 10.023I // w.r.t. D for blI- -

II

0.059Form of . Circularity~ - _ _ _ - - - - - - - - - - - - -1

bl Cvlindricitv 1Table 3 Probabilityof mismatch for different tolerancesThe above results indicate that higher mismatch

probabilities were associated with the poor choices oftolerance types.5. TOLERANCE ALLOCATION PROCEDURETolerance allocation consists of two optimizationprocedures. The first is the selection of manufacturingprocesses and sequences which minimize the productioncost and constrain the rejection rate. The second, allocatesthe types and magnitudes of the size, form and orientationtolerances in addition to the magnitudes of positiontolerances (The datum reference frames for positiontolerances are directly deduced from the manufacturingsequences). Since both procedures are of a combinatorialnature, genetic algorithms (3) are used as an optimizationmethod for both procedures.5.1 Selection of Processes and SequencesGiven an assembly of Np parts and a set of Nf features.Each feature has one discrete independent variable X(The manufacturing process by which it is produced) andeach part p; has another discrete independent variable Y;(The manufacturing sequence by which it is produced).Hence the total number of independent variables is equal toNp+Nf. Since the datum reference frame of the positiontolerancing scheme of each part must follow themanufacturing sequence of the part, another formulationcan be devised where each feature can have a variableXwhich stands for the manufacturing process and anothervariable Z which is the datum reference frame used in itsposition tolerance. Since some reference frames cannot beused with others, each candidate reference frame can havea set of constraints on its usage as an attribute. The objectivefunction to be minimized is the total production cost. Theprobability that the manufactured parts and features willviolate the assembly functional requirements Prej) isconstrained to a specified value v. Therefore, the objectivefunction is given by the following equation.

min(Cost(X,Z)) , subject to Prei(X,Z) 5 v (6)x.z5.2 Minimization of the Mismatch ProbabilityGiven an assembly with a set of Nf features, each featurein the assembly has two discrete-valued independentvariables (Oi and Fi) for the types of tolerance controlling itsorientation and form deviations respectively and fourreal-valued independent variables X i , Y, Zi and Ui) for themagnitudes of the tolerances of size, position, orientationand form respectively. The datum reference frame for theposition tolerance is already determined by themanufacturing sequence and controlling the size of a feature

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needs only one type of tolerance (size tolerance). Theobjective function to be minimized is the mismatchprobability P,.

min P JO , F,X, Y, 2 U 7)0.F.X.Y.Z.U

( 1 4.02 01 oc.5

6. EXAMPLE

- .1 (Features C1, C4 and C53 2I .S3 s4

m

Dimensions 3003 en mm

C2

150 f.075

F

+ A-B-F B-D*Cl frame can be

\ o 1

c2

Cdc

150 100Fig.7 Speed Reducer

The speed reducer shown in Fig.7 has a functionalrequirement, requiring the center-distance between theinput and output shafts to bel50f 0.075 mm, otherwise,backlash between the gear teeth will increase, leading inturn to an unacceptable level of gear noise. Tolerances areto be allocated to the casing holes: C1, C2, C3, C4 and C5,(Fig.2 ) and to the bushings holes 81 and 8 2 (Fig.8 ).

Fig.8 Bushing

Figure 9 shows the cost functions used for the selection ofmanufacturing processes. All processes were assumed togenerate manufactured surfaces between concentric offsetcylinders, whose offset distance s equal to the process imit,and according to a normal distribution. Table 4 shows thecandidate datum reference frames for the selection ofmanufacturing sequences. Table 5 shows the candidatetolerance types for controlling orientation. The proceduresdescribed in sections 5.1 and 5.2 are used to select themanufacturing processes and the geometric tolerances.The value v (equation6) constraining the rejection rate is setto 0.0063.

-,011 .012 .013 ,014Process Limits (mm)

2.01 .o .012 ,013 .014 .015Process Limits (mm)Fig.9 Cost vs. Process Limits

Feature Candidate 1 andidate 2 CONSTRAINTC2 means that

+ D-A82 4 D-A

used only iffeature C2 is+ C-E manufactured

D-C B1Table 4 Candidate Datum Reference Frames

Type? Range of ToleranceMagnitudes forPosition andOrientation ControlC1 -LB I E

Table 5 Candidate Orientation Tolerances

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ReferenceFrame IProbability of rejection due to the violation offunctional requirements= 0.0057

Table 6 Optimum Processes and SequencesFeature I C1 C2 C3 C4 I C5 I B1 j 82 I

I I

P, = 0.0067Table 7 Optimum Tolerance Types and Magnitudes

Results of the optimization processes are shown in Tables.6 nd 7, and the convergence curves of the cost vs. geneticiterations are shown in figures 10 and 11. The rejectionprobability obtained by using the manufacturing processesand datum reference frames, shown in table 6, equals0.0057, and the probability of mismatch achieved by thegenetic algorithms is equal to 0.0067. The reason that therejection probability is not exactly equal to the desired valueof 0.0063 and that the probability of mismatch is not equal tozero is that the problem is of a discrete nature, where thesearch can only find solutions near to the desired ones.

10.00

9.00ri. . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................... . . . . . . ...................... .8.001 10 20 30 40Fig.10 Cost vs Iterations0.025 . . . . .:.. . . . . :. . . . . . : . . . . . .0.0200.015

. . . . . .

Fig.11 Probability of mismatch vs Iterations

7. CONCLUSIONThe cost vs. tolerance functions which are traditionally usedto allocate dimensional tolerance cannot be used to allocate

both types and magnitudes of geometric tolerances. Theinput data to the allocation process will be cumbersome,unreliable and difficult to obtain. A new criterion, mismatchprobability, was developed to allocate geometric tolerances.The criterion was to illustrate the benefit of choosing sometolerance types over others. An algorithm was developed,based on genetic algorithms and using the proposedcriterion. to allocate tolerance types in addition to tolerancemagnitudes. The developed algorithms demonstrate amethodology which can be used to convert the geometrictolerance allocation problem from a human-based trial anderror process into a mathematically formulated problem tosearch for the best solution. The proposed algorithms do notcompletely exclude the use of the expertise of tolerancedesigners. The choice of the set of possible tolerance typesfor each feature is still made by the designer. The algorithmtook a relatively long time (six hours on an SGI workstation),because of the complexity of the objective function, and theuse of genetic algorithms which use a large population ofinstances of the independent variables in its search.However, the tolerance allocation problem is solved onlyonce at the detailed design stage. The benefit of selectingtolerance types and magnitudes outweighs the initialcomputational effort. The computational time can bereduced using a hybrid optimization model that incorporatesgenetic algorithms and direct search, where the searchspace is coursely discretized and genetic algorithms can beused to roughly estimate the location of the global minimum,and the extra search needed can be conducted using directsearch.8. REFERENCES(1 ASME. 1995, Mathematical Definition of Dimensioning(2) Farmer, L.E.. Gladman, C.A., 1986, Tolerance

Technology - Computer-Based Analysis, Annals of(3) Goldberg, D, 1989. GeLetS Alaorithms in Search.

Qpthization and Machine Learning, Addison Wesley.(4) Kanai, S., Onozuka, M. and Takahashi, H., 1995,

Optimal Tolerance Synthesis by Genetic Algorithmunder the Machining and Assembling Constraints,Proceedings of the 4th CIRP Seminars on ComputerAided Tolerancing, Tokyo, Japan.

( 5 ) Lee, W.4. and Woo, T.C., 1989, Optimum Selection ofDiscrete Tolerances, Trans. of ASME, Journal ofMechanisms, Transmissions and Automation inDesign, 111:243-251.

(6) Nassef, A.O. and EIMaraghy, H.A., 1993, Allocation ofTolerance Types and Values Using GeneticAlgorithms, Proceedings of the 3rd CIRP Seminars onComputer Aided Tolerancing, CACHAN, France.

(7) Rubinstein, R., 1981, Simulation and the Monte CarlQMethod, John Wiley and Sons, New York.(8) Zhang. C., Wang, H.P., Li, J.K., 1992, SimultaneousOptimization of Design and Manufacturing.

Tolerances with Process (Machine) Selection,Annals of the CIRP, 41569-572.

a n d a s , NSI Y14.5.1M-1994.

CIRP. 35:7-10.

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Allocation of Geometric Tolerances zNew Criterion and Methodology

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