SIMULATION-BASED TOLERANCE STACKUP ANALYSIS IN~'ACHINING

Rami A. Musia and Samuel H. HuangIntelligent CAM Systems Laboratory

Department of Mechanical, Industrial and Nuclear EngineeringUniversity of Cincinnati, Cincinnati, OH

Y. Kevin Rong

Computer-aided Manufacturing LaboratoryDepartment of Mechanical Engineering

Worcester Polytec;hnic Institute, Worcester, MA

KEYWORDS INTRODUCTION

Monte Carlo Simulation, Tolerance Chain,Tolerance Chart, Tolerance Allocation

Tolerance stackup can be defined as theaccumulation of errors when machining afeature using different operational datums thanthe ones specified in the blueprints. Analysis oftolerance stackup is critical to ensure accuracyof the machined component. The two traditionalmethods used to analyze tolerance stackup inmachining are worst-case analysis andstatistical analysis. These methods are based onassumptions that are too restrictive and haveseveral drawbacks:

ABSTRACT

Tolerance stackup in machining results fromusing operational datums that are different fromdesign datums. It is inevitable due to economicconsiderations of the machining process.Conventional methods used for tolerancestackup analysis include worst-case andstatistical analysis. These methods are basedon strong assumptions and have certaindrawbacks, the most critical one being theinability to analyze geometric tolerances. Thispaper presents a novel method based on featurediscretization, manufacturing error analysis,Monte Carlo simulation, and virtual inspection. Itis generally applicable to stackup analysis ofvarious types of tolerances and produces moreaccurate and less conservative results. Thetrade off is longer computational time.

(1) Worst-case analysis assumes that alltolerances simultaneously occur at theirworst limit; and thus, is exaggeratedlypessimistic in calculating tolerance

stackup.(2) Statistical analysis assumes individual

tolerances to be independent and havea normal distribution, which allows theuse of root sum squares for stackupcalculation. This will lead toconservative results since individualtolerances are more or less correlated in

machining.(3) The analysis is restricted to dimensional

tolerances. In other words, tolerancestack between features is preformed in

Transactions ofNAMRI/SME 533 Volume 32, 2004

surface. This method is generally applicableand is particularly useful for the analysis oftolerances specified on a surface. It is moreaccurate and less conservative compared totraditional analysis methods. In other words,using the proposed method for stackupevaluation will result in much less expectedrejects per million parts when using the samemanufacturing resources.

one dimension, which does notrepresent the actual three-dimensionalfeatures of interest.

(4) The root cause of tolerances, namely,manufacturing errors, are not taken intoaccount.

The need to analyze geometric tolerancestackup became apparent in the mid 1990swhen the new ANSI standard [ANSI (1995)] ispublished with emphasis on geometrictolerancing for improved quality control.According to the ANSI standard, there are twotypes of dimensional tolerances and fourteentypes of geometric tolerances. Dimensionaltolerances include "Iimit-of-size" tolerances thatare applied to only one surface (e.g., thediameter of a hole), and those that are related totwo surfaces (e.g., the length of a shaft).Geometric tolerances can be divided into fivesubcategories: (1) form tolerances that includeStraightness, Flatness, Roundness, andCylindricity, (2) orientation tolerances thatinclude Parallelism, Angularity, andPerpendicularity, (3) location tolerances thatinclude Concentricity, Symmetry, and Position,(4) runout tolerances that include CircularRunout and Total Runout, and (5) profiletolerances that include Profile of a Line andProfile of a Surface. Form tolerances are notsubject to stackup because there are not relatedto any datums. Some researchers have studiedthe stackup of position tolerance [Ngoi et al.(1999), Shan et al. (1999)]. A position toleranceis usually specified on a hole. The axis of thehole is projected to its primary datum and theproblem is converted into a dimensionaltolerance stackup problem. Unfortunately, thisapproach cannot be extended to deal withorientation tolerances since parallelism,angularity and perpendicularity are specified ona surface.

The paper is divided into six sections. It startswith the introduction that explains the researchmotivation. The next section reviews traditionalanalytical methods in the tolerance stackupliterature. Then, the proposed simulation-basedanalysis method is explained. Afterwards.manufacturing error models, categories and theirevaluation methods are outlined. This is followedby an illustrative example. Finally, the utility andapplicability of the proposed method isdiscussed.

TRADITIONAL EVALUATION METHODS FORTOLERANCESTACKUP

Traditional tolerance stackup analysis is basedon the dimension chain (sometimes calledtolerance chain) model. A dimension chain is aclosed loop of interrelated dimensions. Itconsists of increasing, decreasing links and asingle concluding link. The concluding link is theone whose tolerance is of interest and which isproduced indirectly. Increasing and decreasinglinks (both called contributing links) are the onesthat by increasing them, the concluding linkincreases and decreases, respectively.Dimension chains for machining processes areidentified from a tolerance chart, which is agraphical representation of the underlyingprocess plan [Whybrew and Britton (1997)].

Although researchers have recognized theimportant role of manufacturing errors inmachining tolerance stackup [Lin and Zhang(2001), Huang and Zhang (1996)], no systematicanalysis method is available. This paperpresents a simulation-based method driven byfeature discretization and manufacturing erroranalysis. The basic idea is to represent thesurface of interest with a set of discrete points.The effect of various manufacturing errors onthe spatial location of these points is thensimulated. Finally, virtual inspection isperformed to evaluate geometric accuracy of the

I m

c=}:ij-}:dkj=1 k=1

Where:D': The summation of the increasing linkdimensions.Ed: The summation of the decreasing link

dimensions.j: increasing links index.k: decreasing links index.I: number of increasing links.m: number of decreasing links.

I,

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The equation for evaluating the concludinglink's nominal dimension is:

Iff

The concluding link for the dimension chainshown in Figure 1 can be found as:

c=:(i1+i2)-(d1+d2)

This problem can be resolved using Monte Carlosimulation. In fact, Monte Carlo simulation is apopular tool for assembly tolerance stackupanalysis [Chase and Parkinson (1991)]. Notethat Monte Carlo simulation is a general tool thatcan be applied to many engineering or non-engineering problems. The key to its successfulapplication is an accurate model to describe theunderlying problem. For example, the vector-loop-based model [Gao et al. (1998)] is used todescribe assembly in 2 and 3-D, wheretolerance stackup analysis can be achievedthrough Monte Carlo simulation or directlinearization. The only model that is currentlyavailable for machining tolerance stackupanalysis is the dimension chain model, whichcannot deal with geometric tolerances and doesnot take manufacturing errors into account.

Apparently, the tolerance of concluding link isdetermined by the tolerances of the increasingand decreasing links. There are differentmethods to determine this resultant tolerance. Inworst-case method, the concluding link'stolerance L\c is determined as follows:

I oc ..ocL1c = LI-:-I L1i j + LI-I L1dk

j=1 oz j k=1 od k

.~

t ',~SIMULATION-BASED TOLERANCEST ACKUP ANALYSIS

:... d2 .; C i..

.Operational datwn

..Machined mlrface

FIGURE 1. A DIMENSION CHAIN

Referring to Figure 1, the tolerance of theconcluding link is:

l1c = l1il + l1i2 + l1dl + l1d2

In order to overcome the limitations of thedimension chain model, we propose asimulation-based analysis method that utilizesthe following strategies:

.A set of discrete points is used torepresent the surface whose tolerancesare involved in the analysis (Figure 2).

.Monte Carlo simulation is used to studythe effect of various manufacturingerrors on the spatial locations of thesepoints.

.Virtual inspection can then conductedbased on the coordinates of thesepoints, which allows the analysis of anytypes of tolerances (geometric as wellas dimensional).

Referring to Figure 1, the tolerance of theconcluding link cis:

~c = ~(~4)2 +(~4)2 +(~d1)2 +(~d2)2

The worst-case method is clearly tooconservative. The statistical method issupposed to reflect the stochastic nature ofmachining and hence produces realistic results.The problem is that it assumes normaldistribution, while a machined dimension usuallyhas a flat top distribution [Gladman (1980)].

FIGURE 2. PART REPRESENTATION BY SAMPLEPOINTS.

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In statistical method, the concluding link's

tolerance t::.c is determined as follows:

g

ENHACLNo~

Virtu~1 YESInspection

FIGURE 3. SIMULATION-BASED TOLERANCE STACKUP ANALYSIS METHOD.

The simulation-based analysis procedure isshown in Figure 3 as a flowchart. Thecomponents of the flowchart are explained asfollows:

(5) Stopping criteria.The stopping criteria deal with the number of

simulation runs (iterations). Statistically [Kelton,Sadowski and Sadowski (2002)], the minimumnumber of iterations can be calculated by usingthe following equation (valid for n > 30):

S2-2n = Z ]-a/2

(1) Setup plan.Setup plan is a high-level process plan. It

includes the number of setups, the datums usedin each setup, and the setup sequence. Withthe same manufacturing resources, tolerancestackup is solely dependent on the specified

setup plan.

-h2

Where:n: number of iterationss: sample standard deviationh: desired half width of confidence interval.

Alternatively, number of iterations can be foundempirically by benchmarking the results at verybig sample size (e.g. 1 billion) and thenchoosing a proper tolerance band to determinethe sample size [Cvetko et al. (1998)].

(2) Sample plan.The sample plan deals with the discretization

of a surface into a set of points. It specifies thenumber of sample points (sample size) and theposition of these points (sample pointdistribution). In general, the sample plandepends on the surface dimension and itsinaccuracy, the desired precision, and thecapability of the machining process. More detailscan be found in [Woo et al. (1995)], where theauthors concluded that the use of Hammersleysequence and Halton-Zaremba sequence led toeffective sample plans.

(6) Virtual inspection.Since the surface of interested is represented

using a set of discrete sample points. itstolerances can be evaluated using standardCMM (coordinate measuring machine)inspection methods. Note that the inspectionresults can be fed back to improve the setupplan if necessary.

(3) Error modeling.Since our simulation is driven by

manufacturing error analysis, we need to identifythe contributing error sources that shape up thefeatures and cause the stack up. More detailsare given in the next section.

MANUFACTURING ERROR MODELS

There is a great amount of papers in theliterature pertinent to manufacturing errormodeling and its compensation. The recenttrend is moving towards virtual machining. Forexample, [Yao et al. (2002)] uses a desktopvirtual-reality approach to represent themachining and measurement processes byincluding some machining error sources in themodel. [Huang et al. (2002)] studied the sameproblem analytically to determine root-causes ofmachined part inaccuracy. Depending on the

(4) Virtual machining.Simulation starts from here by considering a

virtual part, shaping its form and orientation inthe space by the sample points and keepingtrack of the changes of feature representationdue to material removals and manufacturingerrors.

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~

type of applications, manufacturing errors areclassified based on different factors. Thesefactors are: (1) ~, which accounts for errorvariation with time; (2) randomness, errors arecategorized as deterministic and random; (3)sources of errors, including geometric errors thatrepresent the inaccuracy of surfaces movingrelative to each others, thermal errors thataccount for thermal deformation in the toolbecause of heat provided by cutting process,machine, people. thermal memory (fromprevious environments) and cooling effect fromcoolant, and cutting-force induced errors thatcome from the dynamic stiffness of allcomponents of the machine tool; (4) ~influence on aeometric accurac~, which includeslocating error that accounts for the variationbetween the ideal datum and the actual oneafter locating and clamping, and machining errorthat accounts for the variation between the idealposition of the machine tool and the actual one[Un and Zhang (2001). Un et al. (1997)].

that are numbered from 1 to 6. Milling is theprocess used to machine the surfaces. Holedrilling is not included in the simulation since itdoes not have an effect on the tolerance chainof concern. Figure 7 shows a tolerance chart ofthe part in order to predict tolerance stackupusing worst-case and statistical methods. Weare interested in the dimension shown in line 8as a concluding link. The contributing links areshown in lines 7, 4 and 1.

FIGURE 4. ABS PART (ANTI-BLOCK SYSTEMHOUSING).Since we are dealing with tolerance stackup

analysis that focuses on the geometric accuracyof features, it is logical to adopt the lastclassification method. All manufacturing errorsare lumped into two categories: (1) setup errordue to locating and clamping an imperfectworkpiece using an imperfect fixture, and (2)cutting tool deviation that accounts for all othererror sources. Setup error can be determined byfirst measuring the accuracy of the underlyingfixture (3 rotation and 3 translation components)and then synthesizing it with workpiece accuracy(measured using flatness). Cutting tool deviationcan be determined by machining a surface andthen measure surface point variations in z-coordinate. More details can be found in [Musa

(2003)].

c 4~,1~,. 100 .,

=~

.Ih-"I I II III"I r II I"

-ISJ

FIGURE 5. SIMPLIFIED PART DRAWINGS.

IllUSTRATIVE EXAMPLE The simulation is run under the followingconditions:

.Raw material surface flatness: 0.05 mm.Cutting tool deviation: N (0, 0.00751..Rotational setup error around three

axes: U (-0.002, 0.005) degrees..Translational setup error in three axes:

U (-0.0015, 0.005) mm.

In this section, an illustrative example isprovided to compare tolerance stackup analysisresults using worst-case, statistical andsimulation based methods. The comparison ismade over a dimensional tolerance, since bothworst-case and statistical analysis cannot dealwith geometric tolerances. A housing part foranti-block system is used (Figure 4). Asimplified drawing of the finished partdimensional requirements is shown in Figure 5.The setup plan for machining the part is shownin Figure 6. The part has six surfaces of interest

As was mentioned earlier, determining thenumber of iterations for the simulation is acrucial task in Monte Carlo simulation. Here, webenchmarked the results of the simulation at100,000 to calculate the errors in the first two

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moments when having less sample size. Note A comparison of the results of worst-case,that 10,000 iterations are considered large statistical, and simulation methods in findingenough by most Monte Carlo simulation concluding link tolerance stackup is shown inresearchers [Cvetko (1998)]. The first two Table 2. Monte Carlo simulation results weremoments (mean and variance) at 100,000 found to be less than both the traditionaliterations are shown in Table 1 for dimensions in methods. The ratio of simulation tolerancelines 1, 4, 7 and 8. stackup to the worst case tolerance stackup was

found to be 0.34 and the ratio of simulationtolerance stackup to the statistical methodtolerance stackup was found to be 0.59.

I~

I;

~I

i i".l ,

"I",' ,

:';

;

,TABLE 2. TOLERANCE EVALUATION USING THE ,;;,'

1THREE APPROACHES. ;',

Standard Deviation T olerance=6a

~ L1

L4 0.010698664

L7 0.010872206

La 0.010890703

WorstCase

0.010668288

0.064191982

0.065233235

0.065344216

0.193434944

0.064009727

, ~ l-'~-,"-~ ~ "-"-"---b\; ,t;c~DO [DIJ! [JIIJ: Ii

! ! Ii

,, FIGURE 6. SETUP PLAN. ..

..,Statistical 0.111683618

'4r:-o).. I Simulation

Figure 8 shows the dimension histograms ofthe concluding link (La) and the contributing links(L1, L4 and Lr). Number of iterations required toachieve certain accuracy can be predicted fromFigure 8. This figure shows means and standarddeviations predicted using the simulation interms of number of iterations (X axis). Clearly,4,000 iterations seem to have very close resultsto the 100,000 iterations. Therefore, 4000iterations can be considered as a proper choicefor the sample size of virtually machined parts

(iterations.)

I

Simulation 0.065344216

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{ t

~

~

1..;

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IWak..,.DmI I BaJob:,~ ,.";~~1-..I

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~~

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ii:1:11111 mlli 1:1 18 1J~ Jj---~

~

-~+1 10

4+~

'+'

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'---""'-

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FIGURE 7. TOLERANCE CHART.

TABLE 1.ITERATIONS.

SIMULATION RESULTS AT 100,000

Mean Variance~

~

0.00011381236394345459.994325575444784.9983246659704

0.0001144614033031970.0001182048612291290.000118607406206314

49.9926265647646

~

FIGURE 8. PROGRESS OF RESULTS WITHSAMPLE SIZE INCREASE.

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Suppose that we need to maintain 0.066 mmfor the dimension shown in line 8 01' thetolerance chart (Figure 7). Then, we will need toallocate proper tolerances for the concludinglinks (dimensions shown in lines 1, 4 and 7).According to our simulation, assigning 0.060 mmfor each of the three contributing links will begood enough. However, if we make theallocation decision based on the worst case andstatistical methods, 0.022 mm and 0.038 mm willbe needed for each contributing link,respectively. Apparently, traditional methodsoverestimate the severity of tolerance stackupand hence allocate tighter than necessaryproduction tolerances.

TABLE 3. PART PER MILLION (PPM) REJECTCOMPARISON WHEN ALLOCATING PRODUCTIONTOLERANCE USING WORST CASE, STATISTICALAND SIMULATION METHODS.

Worst .Case Statistical Simulation

Tolerance 2.02a 3.5a 6aC-p-- 0.37 0.64 1

PPM rejects 312,500 80,100 2,700

CONCLUSIONS AND FUTURE WORKRECOMMENDATIONS

A method for evaluating tolerance stackupusing Monte Carlo simulation driven by featurediscretization, manufacturing error analysis, andvirtual inspection is proposed in this work.Cutting tool deviation and setup error weresimulated along with machining and inspectingprocesses for virtual parts. This method isgenerally applicable to geometric as well asdimensional tolerances. It also gives lessconservative results compared to the traditionalones (worst case and statistical methods).Overestimating tolerance stackup could result inprecluding good process plans that should beaccepted. Therefore, accurate evaluation oftolerance stackup can lead to cost-effective

process plans.

Process capability indicates how capablemanufacturing process is to maintain designrequirements in terms of number of rejectsexpected out of a million. Usually. it isrepresented by the Cp index, which is the ratio ofdesign specifications (tolerance T) to the

process variability (6a) , i.e., C = ~. If wep 60-

choose processes that are capable to achieveour simulation requirements, then we will besatisfied with 2,700 parts per million (PPM)rejects when having the process capability indexequals to 1. However, if we use the sameprocess considering the traditional methods,much more rejects per million will beerroneously expected (as shown in Table 3 andFigure 9) since the allocated productiontolerances are tighter than necessary. In otherwords, simulation results suggest that thechosen processes are capable to do the job,whereas the traditional methods do not. Thisshows the importance of having a lessconservative method for tolerance allocation.

f>)

1

Validation is used to ensure that the simulationmodel matches accurately enough with the realworld behavior. Our future work will focus onconducting experiments to validate thesimulation results. In our illustrative example, itwas assumed that the probability distributionfunctions (pdt) of cutting tool deviation and setuperror are available and follow normal distribution.In reality, this may not be true, especially whendealing with new products. We have developedexperimental procedures to estimate error pdt'sand preliminary results showed that these errorsdo not follow normal distribution. Therefore,general probability density estimation will be animportant future research topic.

5)~s

~

~t,,~.J.l~)~PI'-ir ~TCI.sERojOCW.J,4)OO-r STA11511:AI.

Ro~.hI3~WM(SlMJLAna4

~--~~

~,/

Finally, we would like to mention that MonteCarlo simulation is computationally expensive,while analytical methods are more efficient.Therefore, when dealing with dimensionaltolerances, it should be beneficial to developstatistical methods that consider errorcorrelations and different error pdt's. This willsolve the overestimation problem in tolerancestackup analysis while preserving computational

~ 2.02a, ~

1:-. -3.Sa

~---~ J- = 6a

FIGURE 9. REJECTION AREAS COMPARISONWHEN ALLOCATING PRODUCTION TOLERANCESUSING WORST CASE, STATISTICAL ANDSIMULATION METHODS.

H

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efficiency. When dealing with geometrictolerances, analytical methods may not befeasible because of the need for featurediscretization. To improve computationalefficiency, modified Monte Carlo simulationmaybe the only way.

Huang, S. H., and H.-C. Zhang, 1996, "Use ofTolerance Chart for NC Machining," Journal ofEngineering Design and Automation, Vol. 2, No.1,pp.91-104.

Kelton, W. D., R. P. Sadowski, and D. A.Sadowski, 2002, Simulation with ArenaMcGraw-Hili, Second Edition. '

ACKNOWLEDGEMENTUn, E. and H.-C. Zhang, 2001, "TheoreticalTolerance Stackup Analysis Based on ToleranceZone Analysis," International Journal ofAdvanced Manufacturing Technology, Vol. 17,

pp.257-262.

This material is based upon work supported bythe National Science Foundation under GrantNo. 0099735. Any opinions, findings, andconclusions or recommendations expressed inthis material are those of the authors and do notnecessarily reflect the views of the NationalScience Foundation.

Un, S., H. P. Wang, and C. Zhang, 1997,"Statistical Tolerance Analysis Based on BetaDistribution," Journal of Manufacturing Systems,Vol. 16, No.2, pp. 150-158.

REFERENCESMusa, R., 2003, Simulation-based ToleranceStackup Analysis in Machining, M.S. Thesis,Department of Mechanical, Industrial, andNuclear Engineering, University of Cincinnati.

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Cvetko, R., K. W. Chase, and S. P. Magleby,1998, "New Metrics for Evaluating Monte CarloTolerance Analysis of Assemblies," Proceedingsof the ASME International MechanicalEngineering Conference and Exposition,Anaheim, CA, Nov. 15-20.

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Yuan, 2002, "VMMC: a Test-Bed for Machining,"Computers in Industry, Vol. 47, pp. 255-268.

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Volume 32, 2004Transactions ofNAMRI/SME 540

C.

C. Hsieh, and N. K.Sampling for Surfaceal

of Manufacturingpp. 345-354.