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Statistical Models for the Relationship between Production Errors and the Position Tolerance of a Hole
Statistical Models for the Relationship between Production Errors and the Position Tolerance of a Hole E. A. Lehtihet, U. N. Gunasena; Penn State University/USA - Submitted by R. Weill (1) Received on January 12,1990
This paper starts with experimental observations regarding positional and size errors when drilling holes. Based on these observations, statistical models relating production crrors to position tolerance o f a hole are presented. 2-D models for RFS and MMC position tolerance specilications sre developed. Modeling is extended to a speclal 2-D case including orientation errors. Numerical applications show the effects o f variouc parameters on the probability o r a n acceptable hole. Use of a simulation package for the solution 0 1 these problcms is also discussed. These modcls and solutions may help manufacturing select equipment with the accuracy and precision required to meet design tolerance specirkation. KEY WORDS: Position tolerance. production error models, statistical tolerance models. drilled holes
INTRODUCI'ION
'I hc process plaiining Tunction and subscqucnt niaiiufiicturing act must lead 10 the rcali7atioii of' products in conrorniance with rcquircmcnts establislicd hy dcsign and devclopmcnt actiuitics. 'I'licsc rcquircmcnts arc convcycd in the forin or an cnginccring tlrawing Tor each part to hc mmiu~acturcd. For each nicclianical part, an engineering tlrawing spccifics a structured sct or clcincntap gcomctric wraccs. 'I'licsc surraccs arc rclatcd, directly or indirectly. by tolcranced diincnsions: l'licy niag also be constraincd bp gcomctric toleranccs as tlictatcd hy product fuiictionality. Kcading and comprchcnsion or these specifications must hc easy and without ambiguity. T o this end. various national II ] and international 121 standardizing Imdics have cvolvul norms that cstablisli a rational basis Tor tlic tolerancing or cnginccring design fcatures. Although problciiis with tlicsc standards havc bccn raised 131, thcrc documents rcmaiii Tor thc timc bcing. thc basis on which communication bctwccn dcsign and manuracturing is carried out. Positional tolcrancing or a rcaturc is an important part or the systcm or gcoinetrical tolcrancing. In this systcm, thc location or a holc, Tor example, is indicated by a pair or iintolcranced coordinatc valucs spccirying truc position oT thc ccntcr. and a rcaturc control rraine spccirying a cylindrical tolcrancc zonc wherc the axis or tlic manuracturcd holc must hc containcd. Variants of this rcprrsciitatiori arc shown iii figurc I. This systcni prucnts ccveral advantages relativc to the convciitional coordiiiatc tolcrancing method. 'l%e resultant circular sliapc oT llic tolcrancc 70nc is largcr and more rational than thc convcntional squwc tolcrancc 7onc. Wlierc runctioiial rcquircincnts permit it, thc iolerancc zonc may hc niaJc variiihlc by invoking the maximum material condition modificr 8. 'Iliis allows manuTacturing to takc advantagc or an additional position tolcrancc rclatcd to thc fcature sizc tolcrance. Whcn several liolcs arc involved, gcomctric position tolcrancing eliminates tlic undcsirable accuinulatioii or tolcranccs which can rcsult wlicn convcntional coordinatc c h i n dimciisionirig is uectl. Finally, this mctliod corrcsponds well to control cxcrciscd hy functional gagcs which remain, in sonic CBSCS,an important inspectioil nlcthotl. A hole such as that or Figurc I. is produccd lhrougli metal rcmoval on machining cquipmcnt. 1:quipnlfnt prccicion limits do not allow thc rcaliiation of identical gcomctric char8ctcrislics on a scflcs of workpicces. Dimensional and gcomctric cliaractcristics or manuracturcd rcaturcs will thus cxliibit dispcrsions. 'l'hcse dispcrsions may rcsult from both rystcmatic variation (tool wear elTcct on sizc) and randoin variation (positioning crrors, dcrorindtion or inachine-toolworkpicce-system. ctc.). Knowledge or tlicsc dispcrsions and evaluation or their magnitudes cnablc matiufactctring to sclect prodtlctiori cquipmcnt or mcthods compatiblc with dimensional and geometric tolcranccs appearing on the drawing. For the holc shown in Figure I., inanuhcturing would benefit from dcvelopinent or statistical modcls showing tlic rclationship hctwccn specificd position tolcrancc and global knowlcdgc oT dispcrsions inhcrent in the production or a holc. Global knowlcdgc oT dispcrsions may be. retrieved Trom technological data bases or through limited preproduction cxpcrimcntation. It is tlic piirposc or this papcr to discuss dcvclopmcnt and solutions tools for idciilized statistical motlcls Tor the rclatioiisliip hetwccn the position tolcrancc spccification oTa singlc holc and production errors. A nuniher or rcscarclicrs have looked at prohlcms associated with thc production or holcs. Scvcral papers 14.5,61 have studicd the 'walking' phcnomcnon in drilling. 'I'liesc models arc outsidc the scopc or this paper. Other authors liavc obscrvcd 17.Xl thc quality or hole inaking. Ilcaphg and Gruska 191 liavc analycd drillcd holec and used a hivariarc ncrmal distribution l o dcscribe
position errors in tlic x-y planc. In a palrr describing the application of probability in tolerance tcclinology, (iliitlman [lo] considers a statistical distribution Tor a position tolcrancc qualified by the maximum material condition modificr 0.'I'hc rorm or this distribution is intuitively appealing and uscful at tlic design stage. I#jorke 11 I J uscs the Raylcigh distribution as a rnodel for wccntricities relativc to true position. In this paper, we bcgiii with some cxpcrimcntal observations of the global errors obscrvcd on the position and s i x of a drillcd holc. Prom these observations, an idcalizcd modcl Tor production errors is assumed. 'This modcl is used to develop statistical modcls Tor tlic rclationship hctwccn productiorl crrors and the position tolerance or a hole at RFS (Kcgardlcss or I'caturc Size) and MMC (at Maximum Material Condition). 'the paper concludes with the discussion or a simulation tool developed to cnablc manuracturing to obtain quick and reliable solutions Tor tlicsc problcnu. EXPERIMENTAL OBSERVATIONS A 1 4 i l A1 platc ( 203.2~203.2~25.4miii ) was carcrully prepared Tor positioning and drilling on a numcrically coiitrollcd machining center. A standard I ligh Speed Stccl drill or 6.35 mm dianictcr and I10 dcg. point angle was trtcd ror drilling. Drilling conditions wcrr set at a surrace speed 01-46 m/min and a rccd oTO.18 mm/rcv. Forty nine (49) holcs with 25.4 mm spacing bctwccn
centers were drilled in thc platc as shown in Figurc 2. 'l'he plate was tlicn taken to a coordinate inensurcincnt machine and tlic rollowing mcasurcnients wcrc made (see Figure 3.). a. d,, : llole diainetcr at top of thc platc h. dW : I lolc diameter at hottom or the plate c. X,,, : llolc ccnter dcviation Trom ~ N C position, in thc X direction, at thc top or thc plate d. Yr,,, : Ilolc ccnter dpviation from true position, in thc Y dircction, at the top oTtlic.platc c. X P . h : llolc ccntcr deviation rrom true position, in tlic X direction, at the bottom orthe plate r. Y p . h : llolc centcr dcviation rrom true position, in thc Y direction, at the bottom or the platc g. X , : Deviation between top and bottoni ccnters, X direction : Deviation hctwccn top and bottom ccntcrs, Y direction h. Y, Mcasurcments (9) and (h) wcrc dcrivcd rrom mcasurcments (c.0. A series of statistical tests wcrc conducted and rclcvant rcsults arc summarizcd below. Distrihutinn Test.. A standard xz tcst was used to chcck if measured variables could bc adcquatcly dcscribed by normal distributions. With the exception of tlic hole diam measured at the hottoin or thc plate ( d b ) , all variables could be rcasonably well approxirnatcd by nonnal distributions. Somc or the resultant histograms arc shown in Figure 4. Tests on Means: t-Tcstr wcrc irsed 011 individual means and within sclcctcd pairs or nlcans to chcck for quality or m a n s and check il dcviations wcrc centered about true position. Without cxccption, ~ncansor variables (c,Q were round to bc statictically dilTercnt from 7er0, indicating that the corrcsponding nonnal distributions wcrc displaced relative to true position. Similarly, the means Tor variables (g,li) wcrc also round to be dilTcrciit from 7cr0, indicating that orientation or pcrpcndicularity errors relative lo the ccnter of' the top plate wcre also displaced. lcsts within pairs or means 4iowcd that mwns for (c) and (d) could he considered cqual; this however was not true for incans OT (e,l) and (g.h). BOTTOM O F TOP O F HOLE HOLE
a' b
Y
t
TRUE POSITION
I:igure I : I'ositioii 'l'olcrancc Spccifioition il) KI.3 b)MM(' Figurc 2 : Ilrillctl I'latc
Annals of the ClRP Vd. 3/1/lSUl
Figurc 3 : I lole Mcasurcments
569
10
Pr(A.1l.S.) = Pr( p+-t0/2
< q5 < p,+t+/2
-1
) = 2@(1,/20+)
(4 (3)
Assuming that holc size q5 and position crror R arc indcpcndcnt, thcn the prohability o f a n acccptable hole is givcn hy
5
Pr(A.11.) = Pr(A.ll.S.)~Pr(A.P.) = 0 0
(4)
l h c expression Tor Pr(A.P.) will now IJC figures 5.a and 5.b.
0
dtop
Adjwd8bk System For an adjustable system, the probability of an acceptable position is given hy:
Pr( R S 142 )
Pr(A.P.)
-
&(r) dr
(5)
When X and Y arc indepcndcntly and idcntically distributed normal variables with mean zero and common variancc 0'. the density $(r) or R = is the well known Raylcigh distribution having cquatioii .2505
rRw =
.2521
(5)
ev{--$]
(6)
It rollows that I:igore 4 : Example Ilistogra~tts
5
Pr(A.P.) =
$}-xe-p{-
= 1
- cxp{-$}
Tesfs on V 8 r i a ~ I;-tests . were run to test Tor homogeneity or variances within pairs or random variahlcs. Without exception, variances nT (c,d). (e,f), and (8.h) could bc considcrcd to hc statistically similar. llowcvcr vnriancc or (a) was statisticallv diKcrent than that oT(h). ~ o m ~ r t i oTCS~S: n Correlation coehcicnts wcrc calculated lor pairs or selcctd variablw. No correlation was round betwccn (c,d). Tlic samc was trUC Tor (d,c). Some correlation was round hetween (c,c) and the samc was truc Tor (d.0. No significant correlation was round between (g,h).
and
IDEALIZED 2-DMODEL FOR PRODUCTION ERRORS Based on the previous obscrvations. idealired 2-11 models Tor production position errors can Ix: Tormulated. By a 2-Dmodcl. it is meant a model which considers position errors in the machinc tool tahlc planc only (x-y directions) and ignores oricntation errors in the verticnl direction. 'Ihis niodcl is suitahlc Tor holes produced through thin niatcrials such as print& circuit hoards or sheet metal where pcrpcndicularity errors may be insignificant. Within this 2-D model. we wiU distinguish hctwccn two diKcrcnt systems: an -Adjustable System' and a "Fixed System". If a hole is produced on a numcrically controllcd machinc tool, for example, adjustments and compensation Tor possihlc displaccmcnts or the means or errors in the x-y directions arc always possible in principle. Such systems will he rererrcd to as Adjustahlc Systcms and a modcl Tor them is shown in Figurc 5a. The produced hole is thus assumed to have errors or position in the xy directions which arc Identically and lndepcndently Distributed Normal (IIDN) variables with 7,ro mean and standard deviation u. These assumptions arc consistent with experimental observations. Howcver, i r a holc is produced say on o manual machinc and tool guidirig and positioning is achieved with a jig, thcn we know that a n offset or the means of errors in the x-y directions relative to true position cannot be adjusted. Thesc cases will be reTcrred to as Fixed systems and a positioning model for thcm is shown in Figurc 5h. x-y position errors are thus assumed to be utdcpcndently distributed normal variables with means pi and p2 relative to truc 'position and common standard deviation u.
For the modcl assumcd in figure 5.b. the joint density of X and Y is givcn by /u)' (10) rx.,(x.y) = 1 / ( ~ ~ ~ 3 c x r ~ - f ( ( x - p ~ )-+((Y-PJ/u)'} and
t
Pr(A.11.) = C2@(t,/20+) - I ][I F h d Sydem:In a similar a fixed is given by, -
Pr(A.P.)
= Pr(
- cxp(t$3a2):] the probability
t)
t
R-Jx'+y's
=
oran acccptahle position for rr
Pr( (XJ) E D) = J] Tx,,(x,y) dydx (9)
"
-
witho = - , @ 4 T a n d p J94-x' The integral givcn by equation ( I I) does not have a closed rorm solution and needs to bc cvaluatcd numcrically. Sctting it cqual to I. ivc thcn have Pr(A.1i.J
-
r2q&)
- I:JI
(12)
POSITION TOLERANCE AT M M C I r a position tolcrancc is spccificd at MMC, as in figure I.b, holc center position fnuR be contained within a variable tolcrancc zone depcndent on the spccified tolcrancc zone t, and the deviation of actual holc size from thc maximum material condition. Ilole size requircmcnt is exprcsscd by:
-4
and d, = p +& d, < q5
-
Adjustable System: a. Normallv Distributed IIole Sizc: h t thc actual holc be normally distributed with mean p+ and standard deviation u+. Since q5 and 2 2R arc independent. it follows that
-
:R&E
Figure 5 :
(a) Model for a n
POSITION
Rcrcrring to figurc 6, tcc prohahility of an acccptahlc holc is given hy
Adjustable System
(b) Model for a Fixed System
Pr(A.11.) =
fa(+, z) dz dq5 R.
POSITION TOLERANCE AT RFS Ixt X = x direction positioning error rclativc to truc position Y = y dircction positioning error relative to truc position R =f,(i) = probability density function o f variahle I t+ = hole size tolerance zone 5 = position tokrance zone prohability of an acceptable holc size Pr(A.H.S.) probability o f an acceptable position Pr(h.P.) = probability oTan acccptahlc hole (position and sim) Pr(h.11.) IT a position tolcrancc is spccificd Regardless or I'cature Sim (RFS), as in Fig. I.a, then the requirement Ior an acceptable position is given by
--
t
R < +
(1)
Ifthe hole diamcter q5 is normally distributcd with mean p, and standard
deviation uc, then the probability oTan acceptable hole sire is given by:
570
P
Figure 6 : ( q5, Z ) Rcgion orhcccptahle lloler
2
Figurc 7 : Integration Region 0
for Fixed System at MMC
The integral can be expressed in a form containing the normal probability The result is given by distribution function
-
Pr(A.H.)
w).
24%)
- I - -.[‘WAd,-B)
- @(Adl-B)J
(17)
1 .o
0.9
and a(.) is given by equation (3). b. Unirormly Distributed Hole Sue: Let the actual hole size be uniformly distributed with density given by
-:
0.7
1 0 elsewhere We define d,’ and d,’ such that QI’ max(a. dl), and d,’ max(b, d,). The probability o r a n acceptable hole is given by
-
-
0.8
v
0 p/a
* p/a
L4
* p/u - 2
p/o = 1
0.5
-
cp d l d t h -cp d‘-d+t (d ‘-dl’) b-a L-%[ b-a
( ’ 2i ) ( ’ 2i .)]
(19)
Fixed System A closed form solution for this case cannot be obtained and the probability or an acceptable hole needs to be evaluated by numerical integration. Let d, and d, represent the MMC and LMC conditions of the hole size. If zo represents the allowable tolerance zone corresponding to diameter &, then zo is given by (20) z* $ + W-dJ The position tolerance requireMnt is expressed by (21) 2R < z* The region oracceptable holes in this case corresponds to the 3-D volume represented by the h s t r u m of the cone given in figure 7. For a normally distributed hole size, the trivariate density of the X, Y position errors and hole
-
The probability o r a n akeptable hole is then given by
NUMERICAL APPLICATION In order to illustrate the influence of different models and variables on the probability of producing an acceptable hole, a set of selected numerical values was used to compute and plot the probability values. The variahility of hole size was fixed such that the ratio rl of hole size range to hole size standard t&, 6.0. The ratio r, of positioning deviation is constant and equal to r, variability u to hole size variability u+ was restrictcd to r, = u/a* = 1. Both variabilities are thus considered at equal magnitudcs. The accuracy requirement wax expressed by the ratio or specified position tolerance t, to positioning variability u and this ratio r3 $ / a was varied From one to six in order to observe its effect on the fraction of acceptable holes. l h e influence of hole size density is illustrated by considering both rectangular and truncated normal densities of the same range for hole size. As expected, the probability of an acccptable hole was round to be dependent on the magnitude p rather than individual combinations of pI and p, values and plots are thus presented in function of p values. The plots are shown in figures 8 and 9.
- -
I
-
= 0 Nom. D l s t r . Hole Size = 0 Rect. D l s t r . Bole Size Norm. D l s t r . Hole Size Nora. D l s t r . Hole Size
0.4
0
3
2
1
tp/a
4
5
6
Figure 9 : Probability of an Acceptable llolc for Position Tolerance Spccificd at MMC
The plots clearly show that performance increases as the magnitude 0 1 the tolerance zone $ increases. The plots also show the deterioration in performance which results when distributions are no longer centered about true position (p/o 0). The considerable advantage afforded by the spccilication of a position tolerance with the maximum material condition modifier 0 can be clearly seen hy comparing the corresponding curves in figures 8 and 9 at p/a-O. Finally, the influcnce or hole size density shape when the maximum matcrial condition modifier @ is invoked can bc clearly seen by comparing curves in figure 9 Tor p/a = . 0.
=-
A MODEL WITH ORIENTATION ERROR The procedure used Tor the 2-D model can be extended to a 3-11 model incorporating perpendicularity errors. An idealized 3-D model is presented in figure 10. The hole making tool enters the plate a t point A with position errors XI,Y, assumed to be identically and independently distributed normal random variables with zero mean and common variance a: as in the case of an adjustable system. As the tool progresses through the plate, it d r i h and emerges at point B with deviations X,, Y, relative to A. A simplifying assumption is introduced concerning these deviations and they are assumed to be identically and independently distributed normal variables with zero mean (relative to A) and common variance a:. Thc axis orthe hole is then represented by the segment A-B. Letting rl Oh, r, = AD, and r3 OB tlie random variables involved have the following densities. rl 4 Rayleigh distributed with parameter uI r, + Rayleigh distributed with parameter a, 0 + Uniformly distributed with bounds (-x, x ) a -t Uniformly distributed with bounds (-x, x ) Bacause or symmctry, a can be simply ignored. IF thc position tolerance $ is specified at RFS. the hole is acceptable from a position tolerance point orview irr, < $/2 and r3 5 tJ2. This is quivalent to the pair of constraints
-
(0
< r, < tJ2
) and ( 0
<
-
r,
<
r,cos0
+
-
The probability of an acceptable position is then given by Pr(A.P.) v~ c x p { - ~ ~ } j ~ ~ j ~ ~ dr, e d0 x pdr, { - (25) ~ ~ }
1 .o
4 q4 - rtsin’8
where a = r,cos8 i
A similar approach can be taken for a position tolerance spccilicd at MMC. With the addition of variable accounting ror thcr influence of hole si7.q
coinputations become very tedious and do not sccm to he practicable in a manuracturing cnvironmcnt in necd of quick solutions. The use of simulation in thcse cases seems to be more appropriate.
0.8
0.6 h
5 0.4 D4
0.2
t
EXIT P INT
0 0
1
2
3
4
5
6
tp/a
Figure 8 : Probahility of an Acceptable llole Tor Position Tolerance Spccificd at R W
I
&RUE
POSITION
x
Figure 10 : 1-1) Position Error Model (Adjustable Systeiii)
571
USING SIMULATION 'The problems associated with dcveloping solutions such as the one described previously have led us to develop a simulation tool for tolerance analysis in design and manufacturing. This soltwarc package TOLCON is described in more detail elsewhere 1121. The system is n m u driven, require no simulation code to be written by the user and runs on a personal computer. A detailed flowchart can be seen in figure I I . Thc system provides a graphical output which can be interrogatd to estimate the desired probabilities. The 'orientation' problem described by equation (25) can be simulated with TOLCQN by simply defining the problem: xI. yI + IIDN(0,u:) x2.y2 -t IlDN(0,o:) J n QA
(USER I
INPUT^
SPECIFY DENSITY
-
OB = J (xI + x2)' + (yI + YJ' Z = max (OA, OB) Thc position tolerance requirement at RFS is then given by t Z < f The MMC case can be simulated equally easily by realizing that the position requirement in this case is given by t O-dl max (Oh. OR) < 9 2 2 Defining a new random variable 2 max (OA, OD) - (4 - d,)/2, the position requirement for the MMC case is then given hy t
+-
-
ZS:f
A numerical application was developd to generate results using equation
Y
.
11
EDITOR Modify, Generate. Export, Histogmn. Statistics. Edit. Compile, Rm, Conelation, syslcm. change, End.
lB SPECIFY OUTPUT
sw COMPILER
1
Y S v n a tx Errors
T
Logic Errors
RUN
(25) and using the simulation models run with TOLCON. The results are summarized in table 1. It can be seen that the simulated results are very
ITOLCON COMPILER^
I
adequate. Simulation orers the flexibility of a wide array of densities and very easy problem formulation, modifications, and solutions. CONCLUSION In this paper, idealired statistical models for the relationship between production errors and the position tolerance of a hole have been developed. Starting with observations on the global errors of size and position of drilled holes, 2-D models were sct up and solved. Numerical applications clearly show the influence of tolerance specification (RPS vs MMC), size of the tolerance zone. hole si7x density at MMC and production errors on the probability of producing an acceptable hole. Given global knowledge relative to production errors, obtained from historical data or by cxperimcntation, production should find these models useful in assessing the suitability o f various equipment used to produce parts in compliance with design specifications. Under certain assumptions, closal form solutions for some models were possible. Mom gcncral solutions however required the use of numerical intcgration routines. 'Iliis docs not appear to be a convenient solution procedure in a manufacturing environment in nccd of quick solutions. 'Ilie use of special simulation soltware such as TOLCON appears to be more appropriate under tlicsc conditions. It would stein interesting to extend this typc of modeling to multiple hole cases where thc position tolerancc o f a hole is specified relative to another hole taken as datum or where a compositc position tolerance is specified for a pattern of holm. REFERENCES [ I [ American National Standards Institute, 1982, Dimensioning and lolerancing for Engineering Drawings, ANSI Y l4.5M-1982, American Society of Mechanical Engineers, New York, NY. 121 International Organization Tor Standardization, 1983, Technical Drawings Geometrical Tolerancing IS0 1101, Geneva, Switzerland. 131 Requicha, A.A.O., 1983, Towards a Theory of Geometric Tolerancing, International Journal of Robotics Research, Vol.2, No.4, pp. 45-60. [4] Galloway, D.P., 1957, Some Experiments on the Influence of Various Factors on Droll Performance, Trans. ASME, Vo1.79, pp. 191-231. [S) Tueda, M. et al., 1961, On Walking Phenomcnon of Drill, Trans. Japan Society of Mechanical Engineering, Vol. 27, No.178, pp. 816-825. [6] Lee, SJ., Eman, K.F., Wu, S.M., An Analysis of the Drill Wandering Motion, Journal of Engineering for Industry, V01.109, pp. 297-304. 171 langy, J.C, Geslot, R., 1975, Precision des Trous Realises en Percage en Ponction de la Geometry dArutage du Foret, CIRP Annals, VoL24, No. I . 181 Kahng, C.H., Ilam, I., 1975, A Study on Scquential Quality Improvement in llole Making Processes. CIRP AMak, Vo1.24. No.1, pp.27-32. [9l I leaphy, M.S., Gruska, G.F.,1983,
-
572
Figure I I : TOLCON Flowchart
Table I
-
- Numerical
: Model With orientation Errors Results Vs Simulation Results ( u,
$X 103
Pr( A.P. ) using Equation (25)
.04454 30148 ,59200 .79886 .91913 .97350 39292 9859 99975
Integration
.0005 u2 = .001 )
Pr( A.P. ) obtained through simulation of max(OA,OB) i p/2 using TOLCON
.04384 29944 3999 .7979a ,91791 .97268 .99255 .99834 .99970
Simulated valuw obtained from tuns totalling 32000 observations.
This paper starts with experimental observations regarding positional and size errors when drilling holes. Based on these observations, statistical models relating production crrors to position tolerance o f a hole are presented. 2-D models for RFS and MMC position tolerance specilications sre developed. Modeling is extended to a speclal 2-D case including orientation errors. Numerical applications show the effects o f variouc parameters on the probability o r a n acceptable hole. Use of a simulation package for the solution 0 1 these problcms is also discussed. These modcls and solutions may help manufacturing select equipment with the accuracy and precision required to meet design tolerance specirkation. KEY WORDS: Position tolerance. production error models, statistical tolerance models. drilled holes
INTRODUCI'ION
'I hc process plaiining Tunction and subscqucnt niaiiufiicturing act must lead 10 the rcali7atioii of' products in conrorniance with rcquircmcnts establislicd hy dcsign and devclopmcnt actiuitics. 'I'licsc rcquircmcnts arc convcycd in the forin or an cnginccring tlrawing Tor each part to hc mmiu~acturcd. For each nicclianical part, an engineering tlrawing spccifics a structured sct or clcincntap gcomctric wraccs. 'I'licsc surraccs arc rclatcd, directly or indirectly. by tolcranced diincnsions: l'licy niag also be constraincd bp gcomctric toleranccs as tlictatcd hy product fuiictionality. Kcading and comprchcnsion or these specifications must hc easy and without ambiguity. T o this end. various national II ] and international 121 standardizing Imdics have cvolvul norms that cstablisli a rational basis Tor tlic tolerancing or cnginccring design fcatures. Although problciiis with tlicsc standards havc bccn raised 131, thcrc documents rcmaiii Tor thc timc bcing. thc basis on which communication bctwccn dcsign and manuracturing is carried out. Positional tolcrancing or a rcaturc is an important part or the systcm or gcoinetrical tolcrancing. In this systcm, thc location or a holc, Tor example, is indicated by a pair or iintolcranced coordinatc valucs spccirying truc position oT thc ccntcr. and a rcaturc control rraine spccirying a cylindrical tolcrancc zonc wherc the axis or tlic manuracturcd holc must hc containcd. Variants of this rcprrsciitatiori arc shown iii figurc I. This systcni prucnts ccveral advantages relativc to the convciitional coordiiiatc tolcrancing method. 'l%e resultant circular sliapc oT llic tolcrancc 70nc is largcr and more rational than thc convcntional squwc tolcrancc 7onc. Wlierc runctioiial rcquircincnts permit it, thc iolerancc zonc may hc niaJc variiihlc by invoking the maximum material condition modificr 8. 'Iliis allows manuTacturing to takc advantagc or an additional position tolcrancc rclatcd to thc fcature sizc tolcrance. Whcn several liolcs arc involved, gcomctric position tolcrancing eliminates tlic undcsirable accuinulatioii or tolcranccs which can rcsult wlicn convcntional coordinatc c h i n dimciisionirig is uectl. Finally, this mctliod corrcsponds well to control cxcrciscd hy functional gagcs which remain, in sonic CBSCS,an important inspectioil nlcthotl. A hole such as that or Figurc I. is produccd lhrougli metal rcmoval on machining cquipmcnt. 1:quipnlfnt prccicion limits do not allow thc rcaliiation of identical gcomctric char8ctcrislics on a scflcs of workpicces. Dimensional and gcomctric cliaractcristics or manuracturcd rcaturcs will thus cxliibit dispcrsions. 'l'hcse dispcrsions may rcsult from both rystcmatic variation (tool wear elTcct on sizc) and randoin variation (positioning crrors, dcrorindtion or inachine-toolworkpicce-system. ctc.). Knowledge or tlicsc dispcrsions and evaluation or their magnitudes cnablc matiufactctring to sclect prodtlctiori cquipmcnt or mcthods compatiblc with dimensional and geometric tolcranccs appearing on the drawing. For the holc shown in Figure I., inanuhcturing would benefit from dcvelopinent or statistical modcls showing tlic rclationship hctwccn specificd position tolcrancc and global knowlcdgc oT dispcrsions inhcrent in the production or a holc. Global knowlcdgc oT dispcrsions may be. retrieved Trom technological data bases or through limited preproduction cxpcrimcntation. It is tlic piirposc or this papcr to discuss dcvclopmcnt and solutions tools for idciilized statistical motlcls Tor the rclatioiisliip hetwccn the position tolcrancc spccification oTa singlc holc and production errors. A nuniher or rcscarclicrs have looked at prohlcms associated with thc production or holcs. Scvcral papers 14.5,61 have studicd the 'walking' phcnomcnon in drilling. 'I'liesc models arc outsidc the scopc or this paper. Other authors liavc obscrvcd 17.Xl thc quality or hole inaking. Ilcaphg and Gruska 191 liavc analycd drillcd holec and used a hivariarc ncrmal distribution l o dcscribe
position errors in tlic x-y planc. In a palrr describing the application of probability in tolerance tcclinology, (iliitlman [lo] considers a statistical distribution Tor a position tolcrancc qualified by the maximum material condition modificr 0.'I'hc rorm or this distribution is intuitively appealing and uscful at tlic design stage. I#jorke 11 I J uscs the Raylcigh distribution as a rnodel for wccntricities relativc to true position. In this paper, we bcgiii with some cxpcrimcntal observations of the global errors obscrvcd on the position and s i x of a drillcd holc. Prom these observations, an idcalizcd modcl Tor production errors is assumed. 'This modcl is used to develop statistical modcls Tor tlic rclationship hctwccn productiorl crrors and the position tolerance or a hole at RFS (Kcgardlcss or I'caturc Size) and MMC (at Maximum Material Condition). 'the paper concludes with the discussion or a simulation tool developed to cnablc manuracturing to obtain quick and reliable solutions Tor tlicsc problcnu. EXPERIMENTAL OBSERVATIONS A 1 4 i l A1 platc ( 203.2~203.2~25.4miii ) was carcrully prepared Tor positioning and drilling on a numcrically coiitrollcd machining center. A standard I ligh Speed Stccl drill or 6.35 mm dianictcr and I10 dcg. point angle was trtcd ror drilling. Drilling conditions wcrr set at a surrace speed 01-46 m/min and a rccd oTO.18 mm/rcv. Forty nine (49) holcs with 25.4 mm spacing bctwccn
centers were drilled in thc platc as shown in Figurc 2. 'l'he plate was tlicn taken to a coordinate inensurcincnt machine and tlic rollowing mcasurcnients wcrc made (see Figure 3.). a. d,, : llole diainetcr at top of thc platc h. dW : I lolc diameter at hottom or the plate c. X,,, : llolc ccnter dcviation Trom ~ N C position, in thc X direction, at thc top or thc plate d. Yr,,, : Ilolc ccnter dpviation from true position, in thc Y dircction, at the top oTtlic.platc c. X P . h : llolc ccntcr deviation rrom true position, in tlic X direction, at the bottom orthe plate r. Y p . h : llolc centcr dcviation rrom true position, in thc Y direction, at the bottom or the platc g. X , : Deviation between top and bottoni ccnters, X direction : Deviation hctwccn top and bottom ccntcrs, Y direction h. Y, Mcasurcments (9) and (h) wcrc dcrivcd rrom mcasurcments (c.0. A series of statistical tests wcrc conducted and rclcvant rcsults arc summarizcd below. Distrihutinn Test.. A standard xz tcst was used to chcck if measured variables could bc adcquatcly dcscribed by normal distributions. With the exception of tlic hole diam measured at the hottoin or thc plate ( d b ) , all variables could be rcasonably well approxirnatcd by nonnal distributions. Somc or the resultant histograms arc shown in Figure 4. Tests on Means: t-Tcstr wcrc irsed 011 individual means and within sclcctcd pairs or nlcans to chcck for quality or m a n s and check il dcviations wcrc centered about true position. Without cxccption, ~ncansor variables (c,Q were round to bc statictically dilTercnt from 7er0, indicating that the corrcsponding nonnal distributions wcrc displaced relative to true position. Similarly, the means Tor variables (g,li) wcrc also round to be dilTcrciit from 7cr0, indicating that orientation or pcrpcndicularity errors relative lo the ccnter of' the top plate wcre also displaced. lcsts within pairs or means 4iowcd that mwns for (c) and (d) could he considered cqual; this however was not true for incans OT (e,l) and (g.h). BOTTOM O F TOP O F HOLE HOLE
a' b
Y
t
TRUE POSITION
I:igure I : I'ositioii 'l'olcrancc Spccifioition il) KI.3 b)MM(' Figurc 2 : Ilrillctl I'latc
Annals of the ClRP Vd. 3/1/lSUl
Figurc 3 : I lole Mcasurcments
569
10
Pr(A.1l.S.) = Pr( p+-t0/2
< q5 < p,+t+/2
-1
) = 2@(1,/20+)
(4 (3)
Assuming that holc size q5 and position crror R arc indcpcndcnt, thcn the prohability o f a n acccptable hole is givcn hy
5
Pr(A.11.) = Pr(A.ll.S.)~Pr(A.P.) = 0 0
(4)
l h c expression Tor Pr(A.P.) will now IJC figures 5.a and 5.b.
0
dtop
Adjwd8bk System For an adjustable system, the probability of an acceptable position is given hy:
Pr( R S 142 )
Pr(A.P.)
-
&(r) dr
(5)
When X and Y arc indepcndcntly and idcntically distributed normal variables with mean zero and common variancc 0'. the density $(r) or R = is the well known Raylcigh distribution having cquatioii .2505
rRw =
.2521
(5)
ev{--$]
(6)
It rollows that I:igore 4 : Example Ilistogra~tts
5
Pr(A.P.) =
$}-xe-p{-
= 1
- cxp{-$}
Tesfs on V 8 r i a ~ I;-tests . were run to test Tor homogeneity or variances within pairs or random variahlcs. Without exception, variances nT (c,d). (e,f), and (8.h) could bc considcrcd to hc statistically similar. llowcvcr vnriancc or (a) was statisticallv diKcrent than that oT(h). ~ o m ~ r t i oTCS~S: n Correlation coehcicnts wcrc calculated lor pairs or selcctd variablw. No correlation was round betwccn (c,d). Tlic samc was trUC Tor (d,c). Some correlation was round hetween (c,c) and the samc was truc Tor (d.0. No significant correlation was round between (g,h).
and
IDEALIZED 2-DMODEL FOR PRODUCTION ERRORS Based on the previous obscrvations. idealired 2-11 models Tor production position errors can Ix: Tormulated. By a 2-Dmodcl. it is meant a model which considers position errors in the machinc tool tahlc planc only (x-y directions) and ignores oricntation errors in the verticnl direction. 'Ihis niodcl is suitahlc Tor holes produced through thin niatcrials such as print& circuit hoards or sheet metal where pcrpcndicularity errors may be insignificant. Within this 2-D model. we wiU distinguish hctwccn two diKcrcnt systems: an -Adjustable System' and a "Fixed System". If a hole is produced on a numcrically controllcd machinc tool, for example, adjustments and compensation Tor possihlc displaccmcnts or the means or errors in the x-y directions arc always possible in principle. Such systems will he rererrcd to as Adjustahlc Systcms and a modcl Tor them is shown in Figurc 5a. The produced hole is thus assumed to have errors or position in the xy directions which arc Identically and lndepcndently Distributed Normal (IIDN) variables with 7,ro mean and standard deviation u. These assumptions arc consistent with experimental observations. Howcver, i r a holc is produced say on o manual machinc and tool guidirig and positioning is achieved with a jig, thcn we know that a n offset or the means of errors in the x-y directions relative to true position cannot be adjusted. Thesc cases will be reTcrred to as Fixed systems and a positioning model for thcm is shown in Figurc 5h. x-y position errors are thus assumed to be utdcpcndently distributed normal variables with means pi and p2 relative to truc 'position and common standard deviation u.
For the modcl assumcd in figure 5.b. the joint density of X and Y is givcn by /u)' (10) rx.,(x.y) = 1 / ( ~ ~ ~ 3 c x r ~ - f ( ( x - p ~ )-+((Y-PJ/u)'} and
t
Pr(A.11.) = C2@(t,/20+) - I ][I F h d Sydem:In a similar a fixed is given by, -
Pr(A.P.)
= Pr(
- cxp(t$3a2):] the probability
t)
t
R-Jx'+y's
=
oran acccptahle position for rr
Pr( (XJ) E D) = J] Tx,,(x,y) dydx (9)
"
-
witho = - , @ 4 T a n d p J94-x' The integral givcn by equation ( I I) does not have a closed rorm solution and needs to bc cvaluatcd numcrically. Sctting it cqual to I. ivc thcn have Pr(A.1i.J
-
r2q&)
- I:JI
(12)
POSITION TOLERANCE AT M M C I r a position tolcrancc is spccificd at MMC, as in figure I.b, holc center position fnuR be contained within a variable tolcrancc zone depcndent on the spccified tolcrancc zone t, and the deviation of actual holc size from thc maximum material condition. Ilole size requircmcnt is exprcsscd by:
-4
and d, = p +& d, < q5
-
Adjustable System: a. Normallv Distributed IIole Sizc: h t thc actual holc be normally distributed with mean p+ and standard deviation u+. Since q5 and 2 2R arc independent. it follows that
-
:R&E
Figure 5 :
(a) Model for a n
POSITION
Rcrcrring to figurc 6, tcc prohahility of an acccptahlc holc is given hy
Adjustable System
(b) Model for a Fixed System
Pr(A.11.) =
fa(+, z) dz dq5 R.
POSITION TOLERANCE AT RFS Ixt X = x direction positioning error rclativc to truc position Y = y dircction positioning error relative to truc position R =f,(i) = probability density function o f variahle I t+ = hole size tolerance zone 5 = position tokrance zone prohability of an acceptable holc size Pr(A.H.S.) probability o f an acceptable position Pr(h.P.) = probability oTan acccptahlc hole (position and sim) Pr(h.11.) IT a position tolcrancc is spccificd Regardless or I'cature Sim (RFS), as in Fig. I.a, then the requirement Ior an acceptable position is given by
--
t
R < +
(1)
Ifthe hole diamcter q5 is normally distributcd with mean p, and standard
deviation uc, then the probability oTan acceptable hole sire is given by:
570
P
Figure 6 : ( q5, Z ) Rcgion orhcccptahle lloler
2
Figurc 7 : Integration Region 0
for Fixed System at MMC
The integral can be expressed in a form containing the normal probability The result is given by distribution function
-
Pr(A.H.)
w).
24%)
- I - -.[‘WAd,-B)
- @(Adl-B)J
(17)
1 .o
0.9
and a(.) is given by equation (3). b. Unirormly Distributed Hole Sue: Let the actual hole size be uniformly distributed with density given by
-:
0.7
1 0 elsewhere We define d,’ and d,’ such that QI’ max(a. dl), and d,’ max(b, d,). The probability o r a n acceptable hole is given by
-
-
0.8
v
0 p/a
* p/a
L4
* p/u - 2
p/o = 1
0.5
-
cp d l d t h -cp d‘-d+t (d ‘-dl’) b-a L-%[ b-a
( ’ 2i ) ( ’ 2i .)]
(19)
Fixed System A closed form solution for this case cannot be obtained and the probability or an acceptable hole needs to be evaluated by numerical integration. Let d, and d, represent the MMC and LMC conditions of the hole size. If zo represents the allowable tolerance zone corresponding to diameter &, then zo is given by (20) z* $ + W-dJ The position tolerance requireMnt is expressed by (21) 2R < z* The region oracceptable holes in this case corresponds to the 3-D volume represented by the h s t r u m of the cone given in figure 7. For a normally distributed hole size, the trivariate density of the X, Y position errors and hole
-
The probability o r a n akeptable hole is then given by
NUMERICAL APPLICATION In order to illustrate the influence of different models and variables on the probability of producing an acceptable hole, a set of selected numerical values was used to compute and plot the probability values. The variahility of hole size was fixed such that the ratio rl of hole size range to hole size standard t&, 6.0. The ratio r, of positioning deviation is constant and equal to r, variability u to hole size variability u+ was restrictcd to r, = u/a* = 1. Both variabilities are thus considered at equal magnitudcs. The accuracy requirement wax expressed by the ratio or specified position tolerance t, to positioning variability u and this ratio r3 $ / a was varied From one to six in order to observe its effect on the fraction of acceptable holes. l h e influence of hole size density is illustrated by considering both rectangular and truncated normal densities of the same range for hole size. As expected, the probability of an acccptable hole was round to be dependent on the magnitude p rather than individual combinations of pI and p, values and plots are thus presented in function of p values. The plots are shown in figures 8 and 9.
- -
I
-
= 0 Nom. D l s t r . Hole Size = 0 Rect. D l s t r . Bole Size Norm. D l s t r . Hole Size Nora. D l s t r . Hole Size
0.4
0
3
2
1
tp/a
4
5
6
Figure 9 : Probability of an Acceptable llolc for Position Tolerance Spccificd at MMC
The plots clearly show that performance increases as the magnitude 0 1 the tolerance zone $ increases. The plots also show the deterioration in performance which results when distributions are no longer centered about true position (p/o 0). The considerable advantage afforded by the spccilication of a position tolerance with the maximum material condition modifier 0 can be clearly seen hy comparing the corresponding curves in figures 8 and 9 at p/a-O. Finally, the influcnce or hole size density shape when the maximum matcrial condition modifier @ is invoked can bc clearly seen by comparing curves in figure 9 Tor p/a = . 0.
=-
A MODEL WITH ORIENTATION ERROR The procedure used Tor the 2-D model can be extended to a 3-11 model incorporating perpendicularity errors. An idealized 3-D model is presented in figure 10. The hole making tool enters the plate a t point A with position errors XI,Y, assumed to be identically and independently distributed normal random variables with zero mean and common variance a: as in the case of an adjustable system. As the tool progresses through the plate, it d r i h and emerges at point B with deviations X,, Y, relative to A. A simplifying assumption is introduced concerning these deviations and they are assumed to be identically and independently distributed normal variables with zero mean (relative to A) and common variance a:. Thc axis orthe hole is then represented by the segment A-B. Letting rl Oh, r, = AD, and r3 OB tlie random variables involved have the following densities. rl 4 Rayleigh distributed with parameter uI r, + Rayleigh distributed with parameter a, 0 + Uniformly distributed with bounds (-x, x ) a -t Uniformly distributed with bounds (-x, x ) Bacause or symmctry, a can be simply ignored. IF thc position tolerance $ is specified at RFS. the hole is acceptable from a position tolerance point orview irr, < $/2 and r3 5 tJ2. This is quivalent to the pair of constraints
-
(0
< r, < tJ2
) and ( 0
<
-
r,
<
r,cos0
+
-
The probability of an acceptable position is then given by Pr(A.P.) v~ c x p { - ~ ~ } j ~ ~ j ~ ~ dr, e d0 x pdr, { - (25) ~ ~ }
1 .o
4 q4 - rtsin’8
where a = r,cos8 i
A similar approach can be taken for a position tolerance spccilicd at MMC. With the addition of variable accounting ror thcr influence of hole si7.q
coinputations become very tedious and do not sccm to he practicable in a manuracturing cnvironmcnt in necd of quick solutions. The use of simulation in thcse cases seems to be more appropriate.
0.8
0.6 h
5 0.4 D4
0.2
t
EXIT P INT
0 0
1
2
3
4
5
6
tp/a
Figure 8 : Probahility of an Acceptable llole Tor Position Tolerance Spccificd at R W
I
&RUE
POSITION
x
Figure 10 : 1-1) Position Error Model (Adjustable Systeiii)
571
USING SIMULATION 'The problems associated with dcveloping solutions such as the one described previously have led us to develop a simulation tool for tolerance analysis in design and manufacturing. This soltwarc package TOLCON is described in more detail elsewhere 1121. The system is n m u driven, require no simulation code to be written by the user and runs on a personal computer. A detailed flowchart can be seen in figure I I . Thc system provides a graphical output which can be interrogatd to estimate the desired probabilities. The 'orientation' problem described by equation (25) can be simulated with TOLCQN by simply defining the problem: xI. yI + IIDN(0,u:) x2.y2 -t IlDN(0,o:) J n QA
(USER I
INPUT^
SPECIFY DENSITY
-
OB = J (xI + x2)' + (yI + YJ' Z = max (OA, OB) Thc position tolerance requirement at RFS is then given by t Z < f The MMC case can be simulated equally easily by realizing that the position requirement in this case is given by t O-dl max (Oh. OR) < 9 2 2 Defining a new random variable 2 max (OA, OD) - (4 - d,)/2, the position requirement for the MMC case is then given hy t
+-
-
ZS:f
A numerical application was developd to generate results using equation
Y
.
11
EDITOR Modify, Generate. Export, Histogmn. Statistics. Edit. Compile, Rm, Conelation, syslcm. change, End.
lB SPECIFY OUTPUT
sw COMPILER
1
Y S v n a tx Errors
T
Logic Errors
RUN
(25) and using the simulation models run with TOLCON. The results are summarized in table 1. It can be seen that the simulated results are very
ITOLCON COMPILER^
I
adequate. Simulation orers the flexibility of a wide array of densities and very easy problem formulation, modifications, and solutions. CONCLUSION In this paper, idealired statistical models for the relationship between production errors and the position tolerance of a hole have been developed. Starting with observations on the global errors of size and position of drilled holes, 2-D models were sct up and solved. Numerical applications clearly show the influence of tolerance specification (RPS vs MMC), size of the tolerance zone. hole si7x density at MMC and production errors on the probability of producing an acceptable hole. Given global knowledge relative to production errors, obtained from historical data or by cxperimcntation, production should find these models useful in assessing the suitability o f various equipment used to produce parts in compliance with design specifications. Under certain assumptions, closal form solutions for some models were possible. Mom gcncral solutions however required the use of numerical intcgration routines. 'Iliis docs not appear to be a convenient solution procedure in a manufacturing environment in nccd of quick solutions. 'Ilie use of special simulation soltware such as TOLCON appears to be more appropriate under tlicsc conditions. It would stein interesting to extend this typc of modeling to multiple hole cases where thc position tolerancc o f a hole is specified relative to another hole taken as datum or where a compositc position tolerance is specified for a pattern of holm. REFERENCES [ I [ American National Standards Institute, 1982, Dimensioning and lolerancing for Engineering Drawings, ANSI Y l4.5M-1982, American Society of Mechanical Engineers, New York, NY. 121 International Organization Tor Standardization, 1983, Technical Drawings Geometrical Tolerancing IS0 1101, Geneva, Switzerland. 131 Requicha, A.A.O., 1983, Towards a Theory of Geometric Tolerancing, International Journal of Robotics Research, Vol.2, No.4, pp. 45-60. [4] Galloway, D.P., 1957, Some Experiments on the Influence of Various Factors on Droll Performance, Trans. ASME, Vo1.79, pp. 191-231. [S) Tueda, M. et al., 1961, On Walking Phenomcnon of Drill, Trans. Japan Society of Mechanical Engineering, Vol. 27, No.178, pp. 816-825. [6] Lee, SJ., Eman, K.F., Wu, S.M., An Analysis of the Drill Wandering Motion, Journal of Engineering for Industry, V01.109, pp. 297-304. 171 langy, J.C, Geslot, R., 1975, Precision des Trous Realises en Percage en Ponction de la Geometry dArutage du Foret, CIRP Annals, VoL24, No. I . 181 Kahng, C.H., Ilam, I., 1975, A Study on Scquential Quality Improvement in llole Making Processes. CIRP AMak, Vo1.24. No.1, pp.27-32. [9l I leaphy, M.S., Gruska, G.F.,1983,
-
572
Figure I I : TOLCON Flowchart
Table I
-
- Numerical
: Model With orientation Errors Results Vs Simulation Results ( u,
$X 103
Pr( A.P. ) using Equation (25)
.04454 30148 ,59200 .79886 .91913 .97350 39292 9859 99975
Integration
.0005 u2 = .001 )
Pr( A.P. ) obtained through simulation of max(OA,OB) i p/2 using TOLCON
.04384 29944 3999 .7979a ,91791 .97268 .99255 .99834 .99970
Simulated valuw obtained from tuns totalling 32000 observations.