PROCESS CAP J~BILITY ANALYSIS FOR PRODUCTION TOLERANCEASSIGNMENT

Anshum Jain, Nuo Xu, Samuel H. HuangIntelligent CAM Systems Laboratory

Departmelnt of Mechanical, Industrial and Nuclear EngineeringUniversity of Cincinnati, Cincinnati, OH 45221

Y. Kevin Rong

Computer-aided Manufacturing LaboratoryDepartment of Mechanical Engineering

Worclester Polytechnic Institute, Worcester, MA 01609

Tolerance Charting has been used traditionallyto find out the working dimensions and allocateproduction tolerances. With the increasing useof computers in industry I manual tolerancecharting not only seems to be an inefficientmethod but also hinders complete automation.For this reasons several researchers havefocused their attention over the past decade onautomating the process of tolerance charting.This is mainly achieved by converting tolerancecharting into a linear or non-linear programmingproblem. The choice of objective function andconstraints differs from one researcher toanother.

ABSTRACT

Tolerance control is critical in assuring productquality in discrete part manufacturing. Thispaper deals with an important aspect oftolerance control, namely, the assignment ofproduction tolerance i3nd determination ofworking dimensions. The guiding principle isthat tolerance is used to account for theinevitable errors in manufacturing, which arecollectively reflected in process capability.Therefore, a methodology based on processcapability analysis is proposed. It assumes theuse of existing production equipment wherehistorical data is available. Algorithms aredeveloped and their use is illustrated using asimplified actual part.

~,!Ii;

~,;

i:IDong and Hu (1991) used non-linearoptimization to determine the optimal productiontolerances with least production cost. Tang et al.(1994) used a cost function with productiontolerance and stock removal as variables tosimultaneously allocate the productiontolerances and stock removal. Even thoughimportant, cost may not always be the onlydeciding factor for determining productiontolerances. Ji (1993) proposed a LP (LinearProgramming) model for allocating theproduction tolerances. Maximizing the sum of allthe production tolerances was considered asobjective function subjected to the constraints ofdesign tolerances and the process capability. Ji(1994) tried to improve on his model byproposing a "proportional smoothing approach"for allocating tolerances in a tolerance chain.

INTRODUCTION

Discrete part manufacturing constitutes the vastmajority of manufacturing processes. It is a $60Billion per annum industry in this country.Tolerance control is critical to assure quality ofdiscrete parts. However, current techniques fortolerance control (more specifically, productiontolerance assignment) used in the industry areheuristic in nature. They typically evolve fromthe experience of indivicjual machinists and lackscientific rigor.

1!i~)..\hl,.!~~"~i.;I';~m'

of NAMRI/,SME 531 Volume XXXI. 2003

Ngoi and Teck (1993) pr9Posed a method to findthe working dimension by first representingthem in a form of line r equations based ontolerance charts and then solving theseequations to get the v lues of each workingdimension. -rhe indivi ual tolerances weredetermined b;( maximizi g an objective function(W1d1 + W2dz + + WndJ where W1,Wz, ,Wn wlere the eights given to eachdimension d1, dz, , n depending on thecriticality of each proce s, machining time, costor any other (jesired fac or. Wei and Lee (1995)tried to improve upon goi's method by takinginto account the cap bility of the processproducing the! workpiec .They proposed a wayto estimate the reliabilit of the entire processand hence thl3 need to r view the manufacturingprocesses if the failur rate is found to beunacceptable. The ba is of finding out theworking dimensions wa still the tolerance chart.This was just an all mpt to automate theprocess of tolerance carting. Ji et al. (1995)developed yet another LP model for toleranceassignment. Tolerance or each dimension wasmaximized with blueprin dimension and processcapability as constraint. Mean sizes of workingdimensions ",,'ere also c Iculated by solving a setof linear equations. i (1996) proposed abackward derivation a proach for determiningworking dimensions ba ed on tolerance chart.He and Gibson (1992) ttempted to incorporatethe relation:ship be een geometrical anddimensional tolerances, ignored by previousresearchers, in form lating the toleranceallocation problem. The objective function usedwas the minimization f scrap cost and theconstraints used were t e design requirements,geometrical tolerance requirement and themachining allowances. Ngoi and Tang (1998)presented 81 com pre ensive review of themethodologie~s used by various researchers fordimensioning and t lerancing in processplanning. Most of the ethods either model thetolerancing pfoblem wi h a linear or non linearcost minimi;~ation obj ctive function or usetolerance chart as the asic tool for solving theproblem. The drawbac of using tolerance chartfor finding OlJt the wor ing dimensions is that itonly aims to 'nnd out the working dimensions thatcan satisfy the blueprin dimensions. There is noconsideratiorl given to the fact that with theseworking (jimensions geometrical anddimensional tolerance r quirements mayor maynot be met, i.e., the inimum stock removalrequired for each ma hining operation is notconsidered. This m y result in working

dimensions that are impossible to maintain in

machining.

Most of these algorithms for toleranceassignment use tolerance chart as basis and theeffect of manufacturing error sources ontolerance assignment is overlooked. Ourproposed methodology not only take intoconsideration the errors due to tolerancestackup but also other sources of errors presentin a manufacturing environment i.e. work holdingerror, machine tool error, cutting tool error andmeasurement error. Since process capability isa good representation of all the errors involvedin manufacturing, its use for the purpose oftolerance allocation is justified. This is a morerealistic method of tolerance assignment ascompared to the traditional method of assigningtolerances by trial and error method andprogressively checking it using tolerance chart.

It is important to note that although the processcapability index is defined as Cp = (USL -LSL)/

6a, it is the 6a which will be used for toleranceallocation in the proposed methodology. Thereason is, though the process capability indexindicates the ability of a given process to meetthe drawing specifications for a given part, it isthe process dispersion (6a) which governs theability of the process to meet the requirementsof any generic part.

BACKGROUND

In metal removal manufacturing, the finalselection of basic processes for machining anypart is dictated by the part accuracy specified inthe design, which comprises of mainly threecomponents, roughness, geometrical toleranceand dimensional tolerance. To make a feasibleprocess plan, a production engineer needs toknow the process capability of each chosenprocess, and compare them with the designedtolerances; hence assuring each specificationcould be met.

Consider a machined surface having the

following specifications: roughness -201.Jm,flatness -O.O15mm, dimension requirement inrelation to bottom surface -20:f:O.1 mm andparallelism in relation to bottom surface -

O.O20mm. Then it is easy to see that at leastfour capability indexes need to be developed forthis process, namely, roughness, flatness,dimension tolerance and parallelism.

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As discussed above. the deviating angle insetting up any similar workpiece will remainsame. Thus for the workpiece 2 having therepresentative length Lr2 (different fromworkpiece 1) and deviating angle cP 112 = cP III I

expected parallelism tolerance can be calculatedusing the following equation:

.-I ( £5 2 )CPl12 = sm -

£'2

It is reasonable to assume that these ndices forroughness, dimension tolerance an flatnesscan be obtained from some datab se in theindustry because the capability of aintainingthe roughness, dimension and fI tness ofmachined surfac;e is an inht~rent chara teristic ofthe milling process. But thi~; is not the case withparallelism since the resultant paralle ism is notonly determined by the milling oper tion itself,the positioning and clamping conditi ns of themachined part in machine tool is a majorcontributor as well. Actually all datu -requiredgeometric tolerance share this commonattribute; furthermore, onc:e the po ition errorcomes into the scope, the size of machinedsurface and datum surface need to b taken into

consideration.

Our general process capability a alysis fortolerance assignment will not be co plete if wecould not address pro,cess cap bility formaintaining the datum-required geometrictolerance. The concept of tolerancenormalization proposed by Liu a d Huang(2000) can be used to solve this roblem. Anormalized tolerance is an angle r presentingthe maximum permissible! rotation rror whenlocating a component. It i:s calculate based onrigorous analysis of rnanufactur ng errorsinvolved in component setups. Tab e 1 showshow to calculate normalized tol rance fordifferent datum-required geometric tolerance.Consider the same millin~1 example 0 see howthe tolerance normalization conce t can beapplied to develop a methodology f r modelingdatum-required geometrilc toleran e processcapability. Assuming that Iprocess ca ability hascertain stability, i.e. the ability to setup andmachine a part and hlence the workpiecelocating errors, tool errors and t e machineerrors remain same, then the expectedparallelism tolerance for a given art can beestimated if the parallelism value a hieved in asimilar part of other dimensions i known byapplying the concept of tolerance no alization.

tp //1 : Angle between ideal plane (So) and deviated

plane (S)Lr1: Line segment of maximum length lying on

surface So

1. PARALLELISM ERROR.

From equation (1). we have

.-1 ( °1 ) ._1 ( °2 )sm -= sm -Or

Lrl Lr2 .

~ = ( Lr21 L'1 ) 01 (2)

Equation (1) was derived to account for onlyworkpiece locating error In a given setup. Theeffect of other errors in machining a part,namely, tool wear, motion errors, vibration,thermal errors etc. can be accounted for if 01(achievable parallelism tolerance for workpiece1) is derived from the process capability valuefor parallelism. A total process dispersion of 60"1(for achieving parallelism while machiningworkpiece 1), indicates the ability of the processto achieve a parallelism tolerance of 01 with99.7% probability (assuming Cp = 1). Thus of allpractical purpose, 60"1 = 01

Let the parallelism tolerance aphieved in

workpiece 1 (Figure 1.) with re ~ resentative length Lr1 be 81. Then the deviating ngle rp III in

setting the workpiece can be written as (Liu and

Huang, 2000)'.c

.-I ( 01 )£PIli = sm (1)

LrlNow equation (2) can be written as

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product quality. The setup plan containsinformation only about the best combination ofdatum and machined surfaces for every setup.The total number of setups and manufacturingprocess to be used for each setup is not knownat this point. Manufacturing process and the totalnumber of setups needs to be finalized beforedetermining working dimensions and productiontolerance assignment.

I~ = 6{ Lr2/Ln )(U1) or U2 = (Lr2/ Ln )(U1)

Similar equcltions can ~e derived for other typeof tolerances;, r

ImcrCV9

~

etennine Wor1ting Dimension

ToleranceAssignment

I/l~v-

Verification

f-Unsatisfactory

METHODOLOGY

2. TOLERANCE ASSIGNMENT METHODOLOGY.The proposed methodology for productiontolerance assignment is I shown in Figure 2. Agiven setup pl,an will hav- the information on themachines used, cutting t<l>°ls used, fixtures usedand measuring instrume~ts used for each metalcutting operation. The ~orking dimensions foreach metal cutting oper~.ron can be determinedbased on the minim9m material removalrequired for each <l>peration. Productiontolerances cain then be assigned based onoverall proce~is varianc~. Once the workingdimensions and produption tolerances aredetermined, it needs to ~ be verified if the final

design dimensions an tolerances can be

maintained. If not, set p plan needs to beimproved by selecting machines, tools, orfixtures of higher accu~acy. The process iscontinued until the tolerance assignment isfound satisfactory upon vQrification.

Process dispersion is a good measure of theaccuracy of a manufacturing process. Anapproximate finishing process can be selectedby comparing the design tolerance of the givenpart with the process dispersion of the given setof machines. For example, if the designtolerance of a given part is :t 0.01 and none ofthe turning machines have process dispersion(:t3u) less then 0.01, we can conclude that amachining operation with process dispersionless then 0.01 (finish turning, grinding etc.)needs to be performed after turning operation.

Once an approximate process is selected,number of setups required is also known. Afterthis, working dimensions and productiontolerances for each machining cut needs to bedetermined. The main reason for leaving themachining stock in the pre-finishing operation isto have enough material for finishing operationwhich results in a closer dimensional andgeometrical control of the features. This ideacan be used for determining the workingdimensions for all the setups as outlined in

Determination of Workina Dimensions andProduction Tolerance Assianment

The purpose of setup planning is to arrangemanufacturing features! into an appropriatesequence of ~Iroups in prder to assure best

Transactions of NAMRI/SME 534 Volume XXXI,

Figure 3. Machining stock required f or an operation can be calculated based on quired

precision and the process (jispersion of pre-

finishing operation.

\f(fg) a constant, \f(fg) = 1 if fg is an

external feature; otherwise, \f(fg) = -1

r,{dg) a constant ,r,{dg) = 1, if dg is altered

in setup i; otherwise r,{dg) = 0

<1>/{dg) nominal dimension between f;and fgin setup i

S(dg, a) stock removal affecting dij in setup a

setup = i i = nulmber of set~ps

Mi = number of madhiningCUlts in setup i I

cut = M!

Tolerancespecifications of the

finishinQ cut

,

proc 1ss I disper ion

datDetermine

minimum stockremoval required

for the cut

Calculatdlmenslo

i orevio

The algorithm is as follow:

Assign values to all 'P(fij)

Assign values to all [,{dij)for i = N to 1 step-1

for j = 1 to M;

if i= Nw,{dij) = <1>,{dij)

elsea = min{t I [t(dij) ~ O}, "if t> i

w,{dij) = <1>,{dij) + 'P(fij).S(dij. a)

endend

For the purpose of calculating the minimummachining stock requirements in setup a, S(dij,a), interrelationship between various tolerancesneeds to be examined.

~L<::1~-:"-M"::>

~--r-:;::I setup ~ setup I

-1

Icut=C

The following rules can then be used todetermine the machining stock:

,L _-:~~~:setup ~

Yes

Stop

3. METHOD OF DETERMINING WORKINGIDIMENSION AND TOLERANCE ASSIGN~ENT.

Alaorithm for Determining WorkingDimensions

Determination of workin!J dimens; ons for

rotational parts can be f rmulated

mathematically using the following nota ions:

1 .Any two tolerances can be compared only ifthey control the same feature and have the

same datum(s).2. For the purpose of calculating machining

stock, any two tolerances are dependent ifthe improvement in one also improves

another.3. For the tolerances which are dependent,

required stock removal will be the maximumof the stock removal required for individual

tolerances.4. For the tolerances which are independent,

required stock removal will be the sum of thestock removal required for individual

tolerances.5. For a surface feature having more then two

tolerance requirements, minimum machiningstock can be calculated using the following

notations:

N

M;

number of different sets of dependent

tolerance specificationsnumber of tolerance specificationsinseti; i=1,2, ,Nmachining stock required for set imachining stock required for

s;

Sij

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tolerance j in s~t i value and hence can be specified only asThe setup plan of the part is as shown in8.N

S(dfjl a) = Is;

{=I

Where S; = max (So-); ~ = 1 ,2,. . .M;

Relationship! of the v rious tolerances can bedetermined 'from the m trix in Figure 4. where '0'represents tolerances i dependent of each otherand '1' represents t lerances dependent oneach other. Sfj can be determined from table inFigure 5. where a is th process dispersion.

1-lalol.b'J//1 ""LIIr\IOI"-IYI"'I@I-=-I-~Lr\O"- Y@-=--~::::1:J

1

-t..

~I~

4. TOLERANCE RELATlpNSHIP MATRIX.

f)roduction Tolerance AssiQnment

Once the working dim nsions for every setup isdetermined, producti n tolerance can beassigned to, each wor ing dimension based onprocess dislpersion (:t u). Tolerance assignmentbased on process disp rsion is a realistic one asthere is no point in assigning the toleranceswhich can riot be ac ieved with the availablemachines.

5. MINIMUM STOCK REMOVAL FOR DIFFERENTTOLERANCES

I:;ASE STUDY 84 76~O20

45.89~O.20

The cylindrical part s own in Figure 6. is usedas an example to illustrate the algorithm fordetermining! working imensions and allocatingproduction tolerances The capability values ofthe turning machine T1 and T2 available tomachine th'e part in t rms of process dispersion(6a) for various types f tolerances are shown inFigure 7. 6a and 3a values are indicatedseparately as the to al process dispersion isgiven by 6a, but th bilateral tolerances areallocated based on :t a. Moreover, geometricaltolerances like flatn ss do no have a mean

:?ci..0;.,:~

".....,...

6. EXAMPLE PART

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tolerance j in set value and hence can be specified only asThe setup plan of the part is as shown in8.

N

S(dgt a) = Is;;=\

Where S; = max (Sij); j = 1,2"

.M;

Relationship of the various tolerances can bede1:ermined from the matrix in Figure 4. where '0're~lresents tolerances independent of each otheran(j '1' represents tolerances dependent onea(~h other. Sfj can be determined from table inFigure 5. where a is the process dispersion.

0

II

.£.1

("\r"'J"I

4. TOLERANCE RELATIONSHIP MATRIX.

~)duction Tolerance AssiQnment

Once the working dimensions for every setup isdetermined, production tolerance can beassigned to each working dimension based onprocess dispersion (j;3a). Tolerance assignmentbased on process dispersion is a realistic one asthere is no point in assigning the toleranceswhich cannot be achieved with the availablemachines.

5. MINIMUM STOCK REMOVAL FOR DIFFERENTTOLERANCES

CASE STUDY 84.76:0.20

45.89:0.20

The cylindrical part shown in Figure 6. is usedas an example to illustrate the algorithm fordetermining working dimensions and allocatingproduction tolerances. The capability values ofthe turning machine T1 and T2 available tomachine the part in terms of process dispersion(6a) for various types of tolerances are shown inFigure 7. 6a and :t3a values are indicatedseparately as the total process dispersion isgiven by 6a, but the bilateral tolerances areallocated based on :t3a. Moreover, geometricaltolerances like flatness do no have a mean

~'";!!00""00

11ci..~

6. EXAMPLE PART

Volume XXXI.536Transactions of NAMRI/SME

~m--

44.75%0.05

9.33zo.osHe, J. R., and P. R. Gibson, (1992),Aided Geometrical DimensioningTolerancing for Process-OperationQuality Control", International JournalAdvanced Manufacturing Technology, Vol. 711-20.

~

~Iil..Ji, P., (1993), "A linear programmingtolerance assignment in a toleranceInternational journal of productionVol. 31, No.3, pp. 739-751.S~TUP 3

Ji, P., (1994), "An AutomaticAssignment Approach for Tolerance

84.76,00.07

11.24,00.10

9.23,00.10

~ Technology, Vol. 9, pp. 362-368.

Ji, P., M. Ke and R. S. Ahluwalia, (1995),

"Computer-aided operational dimensioning forprocess planning", International journal ofmachine tools and manufacture, Vol. 35, No. 10,

pp.1353-1362.

V7

-L~118: 1 ~

.00

~ ~'--1,.- Ji, P., (1996), "Determining Dimensions forProcess Planning: A Backward DerivationApproach, International Journal of AdvancedManufacturing Technology, Vol. 11, pp. 52-58.

SI$TUP4

9. FINAL SETUP PLAN ~CONTINUED).Liu, Q., and S. H. Huang, (2000), "GeometricTolerance Normalization and Its Application,"Transactions of the North AmericanManufacturing Research Institution/SME. Vol.28, pp. 305-310.

CONCLUSION

Ngoi, B. K. A., and C. T. Ong, (1993), "Acomplete tolerance charting system",International journal of production research,Vol. 31, No.2, pp. 453-469.

Ngoi, B. K. A., and C. T. Tang, (1998), "Productand Process Dimensioning and TolerancingTechniques. A State-of-the-Art Review",International Journal of Advanced ManufacturingTechnology, Vol. 14, pp. 910-917.

Computer-aided desi n (CAD) and computer-aided manufacturing (CAM) systems are nowstandard engineering tools used inmanufacturing compa ies. Although these toolsallow genleration of cutting tool path andvisualizatioln of mach ning, they cannot assurethe quality of finishe parts due to the lack oftolerance c:ontrol cap bility. In this paper. Weproposed a methodol gy to address productiontolerance assignme t problem in tolerancecontrol. It is appro riate for a plant whereexisting production quipment is used andhistorical ~Irocess ca ability data is available.Future research will a dress the issue of dealingwith new plroduction ines where historical datais unavailable.

Tang, G. R., R. Kung and J. Y. Chen, (1994),"Optimal allocation of process tolerances andstock removals", International journal ofproduction research, Vol. 32, No.1, pp. 25-35.

REFERENCESWei, C.-C., and Y.-C. Lee, (1995), "Determiningthe Process Tolerances Based on theManufacturing Process Capability", InternationalJournal of Advanced Manufacturing Technology,Vol. 10, pp. 416-421.

Dong, Z., and W. H ~ (1991)' "Optimal process sequence identificati n and optimal process

tolerance assignment in computer-aided process

planning", Computer. in Industry, Vol. 17, pp.

19-32.

Volume XXXI, 2003Transactions of NAMRWSME 538