Research and concepts Inertial tolerancing

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1 Research and concets Inertial tolerancing The author is a Professor in the Deartment of Industrial Engineering at the Université de Savoie, Annecy le Vieux, France Keywords Taguchi methods, Statistical tolerance levels, Production costs Abstract Traditionally, tolerances are defined by an interval [LSL; USL] which can lead to several ambiguous interretations of conformity This aer examines an alternative method for setting secifications: inertial tolerancing Inertial tolerancing consists of tolerancing the mean square deviation from the target rather than the distance This alternative has numerous advantages over the traditional aroach, articularly in the case of roduct assembly, mixed batches and conformity analysis Couled with a caability index Ci, this alternative method leads to minimizing roduction costs for a secified level of quality We roose to comare both aroaches: traditional and inertial tolerancing Electronic access The Emerald Research Register for this journal is available at wwwemeraldinsightcom/researchregister The current issue and full text archive of this journal is available at wwwemeraldinsightcom/ xhtm 1 Main tolerancing methods in the case of assembled roducts 11 Conformity and tolerancing In the case of assembled roducts or, more generally, in the case where a functional characteristic is the result of a linear combination of several elementary characteristics, we have the roblem of tolerancing the elementary characteristics With no variation in roduction, it would be useless to determine tolerances and define the target The urose of tolerancing is to hel maximize organizational rofitability by minimizing the total cost of assembly, test, and design while simultaneously maximizing roduct quality and reliability Minimizing the total cost of assembly leads to increasing the tolerance whilst maximizing roduct quality, which leads to a reduction in the tolerances Tolerancing consists in obtaining the interval that will give the best comromise economically Many studies have been carried out on this subject and was standardized (ANSI, 1982; ASME, 1994a, b; ISO 286, 1988; ISO 5458, 1998) Tolerancing is linked to conformity The quality of the roduct is judged in terms of whether it meets secification or design requirements Failure of a secific characteristic to meet the secification reresents a defect or non-conformance However, if all roduction is very close to the limit, it will have serious roblems in assembly It is generally viewed that the conformity is acceted when the roducts are included in the bi-oint [USL; LSL] Several caability indexes are founded on this definition as Ck (Ford, 1991) The caability index Cm (Chan et al, 1988) is based on another definition, the Taguchi s loss function (Taguchi, 1987) In the case of assembled roducts with the traditional bi-oint, one must share the final functional condition Y of all elementary characteristics X i Several aroaches were roosed Let us remind ourselves of the main results in the case of a linear relationshi between Y and x i (Chase and Prakinson, 1991; Grant, 1946; Graves, 1997, 2001; Graves and Bisgaard, 2000; Shewhart, 1931): Y ¼ a 0 þ Xn i¼1 a i x i ð1þ Volume 16 Number q Emerald Grou Publishing Limited ISSN X DOI / Worst case tolerancing In this case, one considers that the final condition Y will be resected in all cases of assembling In worst case tolerancing, the elementary tolerances are calculated by the equations: t y ¼ X a i t xi ð2þ

2 D y ¼ X ja i jd xi ð3þ Notation: t y : Y target D Y : USL[Y ]2LSL[Y ] The tolerance sharing can be carried out using different methods (Graves, 2001): uniform tolerances sharing; consideration of standard or design rules; roortional to the square root of the nominal dimension; and consideration of the caability exerience The well known disadvantage of worst case tolerancing is the high cost associated with this method Indeed, it leads to very tight tolerances which are very difficult to obtain in roduction Consequently, the cost of insection, rejects and reworking increases as does the choice of a more sohisticated roduct method The main advantage is the guarantee of the secification at the final level of assembly 13 Statistical tolerancing Statistical tolerancing was develoed in order to consider the low robability of having several characteristics in limit of tolerance at the same time (Evans, 1975; Shewhart, 1931) From equation (1), in the case where the variables x i are indeendent of a standard deviation s i, one has the relationshi: qxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s Y ¼ a 2i s 2i ð4þ With tolerances roortional to the standard deviation, we obtain (Chase and Prakinson, 1991) the equation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X n T Y ¼ a 2 i T 2 i ð5þ i¼1 In this tolerancing method, the basic assumtion is the centering of all the elementary characteristics on the target 14 Inflated statistical tolerancing Several methods were roosed in order to thwart the negative asects of statistical tolerancing The main one is inflated statistical tolerancing (Graves, 1997, 2001; Graves and Bisgaard, 2000) The tolerance is defined by the relationshi: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X n T Y ¼ f a 2 i T 2 i ð6þ i¼1 where f reresents the inflated coefficient We often choose this coefficient arbitrarily around 15 In the case where the a i coefficients are equal, f ¼ 1is 203 the statistical tolerancing and f ¼ ffiffiffi n is the worst case tolerancing Graves and Bisgaard (2000) discuss different situations which lead to different choices for f They roose an interesting aroach to otimize the f coefficient in relationshi to the caability indexes But, desite the great quality of the tolerancing method, it is always ossible to find a situation in which the method will not be aroriate 2 Inertial tolerancing 21 Definition The goal of tolerancing consists in determining a criterion of accetance for the elementary characteristic x i guaranteeing the accetance of the resulting characteristic Y i The tolerance limits the cost of non-quality generated by a variation comared to this target In the case of a good design, when Y is laced on the target, quality is robust comared to the oerating and environmental conditions When Y moves away from the target, the quality will be increasingly sensitive to the conditions, and could lead to the customer being dissatisfied Taguchi (1987) has showed that the financial loss associated (L) with a shift to the target was roortional (coefficient K) to the square of the variation: L ¼ KðY i 2 targetþ 2 ð7þ In the case of a batch with an average m, the associated loss is: L ¼ K s 2 Y þðm2targetþ2 ¼ K s 2 Y þ d 2 Y ð8þ where K is a constant, and I Y ¼ s 2 Y þ d2 Y is the mean square deviation (MSD) of the target The MSD behaves like inertia arround a oint : I ¼ MassðI 2 Þ 2 : In order to have the minimal inertia, we must have the roducts as near to the target as ossible As with inertia, the MSD has the very interesting roerty of additivity in the case of the linear relationshi between Y and x i To take advantage of this roerty, we roose to relace the traditional tolerance Y ^ DY by the tolerance Y(I Y ) in which I Y reresents the maximum MSD acceted on variable Y By analogy of the MSD with inertia, we have chosen to name this alternative method inertial tolerancing This way of determining the tolerances has interesting roerties that we roose to develo 22 An adated caability index (Ci) In the case of inertial tolerancing, conformity is declared if the inertia I Y of the characteristic or the batch is lower than the fixed maximum inertia I YMax I YMax is a tolerance on I Y Itis

3 ossible to define a caability indicator Ci by the equation: Ci ¼ I YMax I Y ð9þ A manufacturing rocess must obtain Ci higher than 1 (I Y,I YMax ) Later in this aer, we will show that this indicator has many roerties in articular, in the case of mixed batches The Ci of two mixed batches is equal to the average of both Ci 23 Extreme situations of accetance in the case of inertial tolerance Let us study the extreme situations of accetance in the case of inertial tolerancing Inertia can increase under the influence of two arameters: a shift of the average comared to the target (d Y )or an increase in sread (s Y ) and this indeendently of the distribution form Centered extreme situation (d Y ¼ 0): I Y ¼ s 2 Y þ d2 Y : Then IY ¼ s 2 Y ð10þ One thus obtains: s YMax ¼ ffiffiffiffiffiffiffiffiffiffiffiffi I YMax ð11þ Extreme situation with a standard deviation equal to zero (s Y ¼ 0) One thus obtains: Inertial tolerancing d YMax ¼ I Y ¼ d 2 Y ffiffiffiffiffiffiffiffiffiffiffiffi I YMax ð12þ ð13þ Extreme m shift acceted according to the standard deviation (Figure 1) In the case of inertial tolerancing, acceted extreme shift will be a function of the standard deviation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d YMax ¼ I YMax 2 s 2 Y qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 2 s 2 Y =I ffiffiffiffiffiffiffiffiffiffiffiffi YMax I YMax ð14þ d= ffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I YMax ¼ 1 2 s Y = ffiffiffiffiffiffiffiffiffiffiffiffi 2 I YMax Figure 1 Limit of the m shift acceted according to the sigma for an inertia I YMax ¼1 Equation (14) is reresented in Figure 1 As long as s, 0:6 ffiffiffiffiffiffiffiffiffiffiffiffi I YMax ; one can have relatively significant shift d, 0:8 ffiffiffiffiffiffiffiffiffiffiffiffi I YMax : Similarly, as long as the shift is low, one can accet a relatively significant sread 24 Tolerance determination in the case of a mechanical assembly The roerties of inertial tolerancing come from the roerties of the inertia s additivity To illustrate this roerty, we will take the case of a resulting characteristic Y deendent on a linear function of several indeendent elementary characteristics x i (equation (1)) In this case, it is easy to show that we have: s 2 Y ¼ X a 2 i s2 i d Y ¼ X a i d i Let us calculate the inertia obtained on the characteristic Y: I Y ¼ s 2 Y þ d2 Y ¼ X a 2 i s2 i þ X a i d 2 i I Y ¼ X a 2 i s2 i þ X a 2 i d2 i þ 2 X a i a j d i d j which is written as: I Y ¼ X a 2 i I X i þ 2 X a i a j d i d j ð15þ ð16þ ð17þ The first art of the equation corresonds to the additivity of different inertia The double roduct corresonds to the case where all shifts are on the same side In the case of random distribution of the averages when the comonent count is significant, one can consider that this double roduct is equal to zero We have a hyothesis here that is close to the assumtion of traditional statistical tolerancing On the other hand, in cases where the number of comonents is not significant or when the average distribution is not random, the worst case assumtion makes it ossible to integrate the double roduct in the distribution of inertias (equation (18)) From equation (17), we can calculate the distribution of inertias of each elementary characteristic according to the inertia desired on the resulting characteristic This equation makes it ossible to integrate the risk of unfavorable systematic decentring of all the elements This equation easily makes it ossible to determine the tolerances on the elementary characteristics according to the following three hyotheses Assumtion 1 Worst case Under these conditions, we saw (equation (13)) that maximum decentring was d YMax ¼ ffiffiffiffiffi I Y

4 Equation (17) becomes: Figure 2 Academic case of a mechanical assembly I YMax ¼ X a 2 i I imax þ 2 X ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a i a j I imax I jmax ð18þ Equation (18) is simlified in the case where a i ¼ 1 and in the uniform distribution of inertia (I imax ¼ I Max ) on each characteristic: I YMax ¼ ni Max þ nðn 2 1ÞI Max ¼ n 2 I Max I Max ¼ I YMax =n 2 ð19þ 242 Assumtion 2 Random distribution of the averages In this hyothesis, the double roduct is equal to zero Equation (17) becomes: I YMax ¼ X a 2 i I imax ð20þ Equation (20) is simlified in the case where a i ¼ 1 and in the uniform distribution of inertia (I imax ¼ I Max ) on each characteristic I Y ¼ ni Max I Max ¼ I Y =n ð21þ 243 Assumtion 3 Unfavorable average shifts of k sigma for all the characteristics roduction cost We use an academic case very close to the Bisgaar s ten washers examle (Bisgaard and Graves, 1997) We will retain a uniform distribution of the tolerances on x i In traditional tolerancing, the clearance Y is 1 ^ 0:8: In the case of a centered oulation, a Ck ¼ 1:33 situation corresonds to the standard deviation s ¼ 0:2: In inertial tolerancing, the condition will be I Y ¼ s 2 Y þ d2 Y ¼ 0:2 2 ¼ 0:04 that one writes 1 (I 004) The targets are fixed in the following way: X 1 ¼ 9; X 225 ¼ Traditional tolerancing In the case of the uniform distribution of the tolerances we have (equations (2) and (5)): worst case assumtion X i ¼ target ^ 0:08; and statistical assumtion X i ¼ Target ^ 0:35 I Y ¼ X a 2 i I X i þ 2 X a i a j d i d j I Y ¼ X a 2 i I imax þ 2 X a i a j ð1 2 1=ð1 þ k 2 ÞÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I imax I jmax ð22þ Equation (20) is simlified in the case where a i ¼ 1 and in the uniform distribution of inertia (I imax ¼ I Max ) on each characteristic: I YMax ¼ ni X þ nðn 2 1Þ þ k 2 I X ¼ I YMax n nk2 þ1 1þk 2 I X ð23þ We will show that assumtion 2 (random distribution of averages) in the case of inertial tolerancing is very robust even in hard unfavorable conditions 25 Comarison with traditional tolerancing on an academic case (Figure 2) In inertial tolerancing, we only retain the assumtion: random distribution of the averages This assumtion is the best comromise between the quality of the finished roduct and the Inertial tolerancing In the case of the uniform distribution of the tolerances based on the assumtion of the random distribution of the averages we have (equation (21)): I X ¼ I Y =5 ¼ 0:008; which corresonds to a standard deviation s ¼ 0:089 in the case of a centered oulation 253 Study of the extreme situations In order to comare inertial and traditional tolerancing, we will consider two extreme situations: (1) Centered situation (d X i ¼ 0) with Ck ¼ 1:33 for the traditional tolerancing and Ci ¼ 1 for the inertial tolerancing of all the characteristics (2) s X i < 0 and the rocess is in limit of accetance (Ck ¼ 1:33 and Ci ¼ 1) Figure 3 shows the limits of the traditional tolerancing method In the worst case, when all elementary characteristics are centered, the tolerances are too tight In the statistical method, when all elementary characteristics are decentered in an unfavorable manner, the deendant variable Y is outside of the tolerances As we see in Figure 3, when the Ci index is higher than 1 in both situations (d ¼ 0 and s ¼ 0)

5 Figure 3 Extreme situations the deendant variable Y is accetable Inertial tolerancing leads to minimizing roduction costs (when the rocess is d ¼ 0, the situation is similar to classical statistical tolerancing) for a secified level of quality (when the rocess s ¼ 0; d Max is limited by the Ci, and Y is more accetable than the classical worst case) It is easy to generalize this academic case when we have the following relationshi: Y ¼ Xn x i i¼1 ð24þ Case 1: d ¼ 0 In this case, the acceted sread for the elementary characteristics is very wide, it is the same for classical statistical tolerancing Case 2: s ¼ 0 In this case, d XMax ¼ ffiffiffiffiffiffiffiffiffiffiffiffi I XMax on each elementary characteristic x i The resulting shift on Y is equal to d YMax ¼ nd XMax : 3 Conformity and inertial tolerancing 31 Case of mixed batches One of the disadvantages of the traditional caability indicator Ck aears when two batches are mixed By mixing two batches with an accetable caability (Ck 1:33), one can obtain a non-accetable caability for the mixed batch (Ck, 1:33) This disadvantage is eliminated in the case of inertial tolerance with the Ci indicator Indeed, by mixing k batches of the same size n and from 206 caability Ci k, one obtains a resulting batch such as: Demonstration: I ¼ 1 X X 2 kn i ¼ 1 X k kn Ci ¼ð1=kÞ X Ci j¼1 ðx i 2 X j þ X j Þ 2 ð25þ I ¼ 1 X n X k h i ðx i 2 X j Þ 2 þ X 2 j þ 2 X j ðx i 2 X j Þ kn i¼1 j¼1 I ¼ 1 X s 2 k k þ X 2 j ¼ 1 X Ij k ð26þ One changes immediately from equation (26) to (25) by using equation (9) This roerty is articularly interesting in the case of multigenerator rocesses such as molding machines with 64 cavities Each cavity has a articular mean, and it is imossible to adjust all cavities on the target If each cavity has an accetable Ci value, then when all the roduction is mixed, total Ci will also be accetable It is not the case with caability indicators such as Ck or Cm 32 Inertial tolerancing and lot size In the case of assembly, the decision about whether a art conforms or not is not easy Indeed, although the art is assembled, the quality of the assembled roduct will be the combination of several characteristics If one considers the case of a characteristic near the secification limit, the decision about conformity must two several known or unknown elements into account:

6 (1) The distribution of the characteristics The decision should not be the same if the art nearest to the secification limit is alone or if 100 ercent of the roduction is also near the secification limit (2) The distribution of the other imortant characteristics of the assembly In general, these elements are unknown at the time the decision is made Inertial tolerancing makes it ossible to take these various elements artly into account Contrary to the traditional methods, the goal is not to obtain a level of quality measured by a ercentage outside tolerance, but to guarantee a low inertia around the target (Figure 4) Fuzzy tolerance is a articularly interesting oint of inertial tolerance In the case of unit roduction, it guarantees the erfect conformity of the roduction in the worst cases The inertial tolerancing leads to fuzzy tolerances that vary according to the quantity of the arts roduced To illustrate this oint, let us imagine the characteristic 9 (I 0008) of our academic case (Figure 2): (1) Case of one art measured at 9088 In this case, the inertia for a art is equal to 0:088 2 ¼ 0:0077: The art is only just acceted (2) Now let us take the case of three arts: 900; 9088; 9125 Inertia is then I ¼ 0:0078; the three arts are acceted One art has an individual inertia higher than 0008 (3) Finally, let us take the case of a batch of arts averaging 905 and with a standard deviation of 007 In this case I ¼ 0:0074; the batch is also acceted even though it contains 286 ercent of the roducts whose dimension is higher than which is the aroval limit in the case of unit roduction This fuzzy tolerance is a articularly interesting oint of inertial tolerance In the case of unit roduction, it guarantees the erfect conformity of the roduction in the worst cases In the case of Figure 4 Conformity and inertial tolerancing series roduction, the inertial tolerance takes account of the weak robability that at both extremes of the tolerance limits will be assembled together Thus, with only one tolerance, one is able to resond as well to cases of unit roduction as to those in series roduction 4 Inertial tolerancing in the case of unilateral tolerance 41 Case where the ideal is zero Traditional tolerancing in the higher unilateral case such as the geometry or form defect consists in fixing an uer limit secification that one should not exceed This method leads to illogical results (Pillet, 1999, 2000) which rivilege a batch having a good Ck, but with many arts near the secification limits rather than a roduction having a more modest Ck, but with a greater concentration of arts with a value close to zero (ideal value) In this case, the inertial tolerancing consists of fixing the maximum inertia acceted comared to zero The batch will be acceted if its inertia is lower than the inertia limit A machined element has a critical circularity (I 001) (Figure 5) In the case of a measurement of only one measured art with 012, the art is refused because inertia is higher than 010 On the other hand, if we consider a distribution comrising the values with 014 but whose distribution of the arts has a high density of robability close to zero, one will be able to accet the batch The third batch will be refused, although not including any value with 014 However, as the distribution of the values will be very offset comared to zero, it will lead to unaccetable inertia 42 Case where the ideal is a finished value One can quote, for examle, an outut that has an ideal value of 100 ercent or a temerature having to be the coldest ossible and whose ideal should be absolute zero In this case, one is Figure 5 Case where the ideal is zero 207

7 reduced to the receding case by defining inertia as the difference between the measured value and the ideal value 43 Case where the ideal is the infinite One can quote, for examle, a traction force which one wishes to control One is reduced to the case where the ideal is zero, by defining a maximum inertia on x 0 ¼ 1=x; with x the measured characteristic 431 Alication on a traction force after sticking The sticking of shoes has a required inertia 003 (I 003) (Figure 6) As the examle in Figure 6 shows, inertial tolerancing makes it ossible to interret the will of the designer better by considering the roduct function rather than just the values themselves In examle #1, one art measured to 4 is refused ði ¼ð1=4Þ 2 ¼ 0:0625 0:03Þ: For one ar, minimum required is 578: (1/ 578) 2,003 In examles #2 and #3, the measured minimal value is 4, but it is easy to note that the overall distribution #2 will give much better satisfaction to the customers Indeed in the searation of a sole, it is necessary to consider the combination between the force necessary to wrench and the conditions of use imosed by the owner The robability of having the overla of a weak sticking and a severe use is much stronger in case #3 than in case #2 A traditional tolerance with a minimal tolerance of 3 dan/cm 2 would lead to the conclusion of identical quality for both batches (#2: Ck ¼ 1:10; #3 Ck ¼ 1:13) 5 Conclusion Inertial tolerancing In this aer, we have detailed the rincile of inertial tolerancing This aroach makes it ossible to view tolerancing differently This method has many advantages as follows: In the case of an assembled roduct, the inertial tolerancing leads to a better comromise between the cost of roduction and quality as oosed to the traditional aroaches The additivity roerty of inertia in the case of mixed batches The inertia of the mixed batch will be the weighted average of elementary inertias The ease of tolerancing bilateral, unilateral or geometrical characteristics with the same logic In all cases, there is a unilateral criterion: inertia The accetance of a art takes account of the batch size This new way of considering the tolerance marks an evolution of the traditional way It falls under a logic that rivileges centering on the target to the detriment of counting not-in conformity characteristics The accent is on the finished roduct, and not on the characteristic References ANSI (1982), Dimensioning and tolerancing, ANSI Y145M ASME (1994a), Y145M-1994: Dimensioning and tolerancing: revision of ANSI Y145M-1982, The American Society of Mechanical Engineers, New York, NY ASME (1994b), Y145M-1994: Mathematical definition of dimensioning and tolerancing, The American Society of Mechanical Engineers, New York, NY Bisgaard, S and Graves, S (1997), Quality quandaries A negative rocess caability index from assembling good comonents?, CQPI Reort No 160, Aril Chan, LJ, Cheng, SW and Siring, FA (1988), A new measure of rocess caability: CPM, Journal of Quality Technology, Vol 20 No 3, Chase, KW and Prakinson, AR (1991), A survey of research in the alication of tolerance analysis to the design of mechanical assemblies, Research in Engineering Design, Vol 3, Evans, DH (1975), Statistical tolerancing: the state-of-the-art, Journal of Quality Technology, Vol 7 No 1 Ford (1991), Quality Procedures SPC Q1, Ford Motor Comany, Dearborn, MI Grant, EL (1946), Statistical Quality Control, McGraw-Hill, New York, NY Graves, S (1997), How to reduce costs using a tolerance analysis formula tailored to your organisation, Reort No 157, CQPI, Madison, WI Figure 6 Case where the ideal is the infinite 208

8 Graves, S (2001), Tolerance analysis tailored to your organization, Journal of Quality Technology, Vol 33 No 3, Graves, S and Bisgaard, S (2000), Five ways statistical tolerancing can fail and what to do about them, Quality Engineering, Vol 13, ISO 286 (1988), ISO System of Limits and Fits, ISO, Geneva ISO 5458 (1998), Geometrical Product Secifications (GPS), ISO, Geneva Pillet, M (1999), Qualité des roduits et qualité des caractéristiques élémentaires, l objectif cible de la concetion à la fabrication, Congrès Qualité, March, Paris Pillet, M (2000), Intégrer l objectif cible dans une stratégie qualité, Congrès International de Statistique de Fès, Fès, Morocco, 12 Shewhart, WA (1931), Economic Control of Quality of Manufactured Products, Van Nostrand, New York, NY Taguchi, G (1987), System of exerimental design, Engineering Methods to Otimize Quality and Minimize Costs, Vol 1 and 2, American Sulier Institute, Inc, Livonia, MI Further reading Bisgaard, S, Graves, S and Shin, G (2000), Tolerancing mechanical assemblies with CAD and DOE, Journal of Quality Technology, Vol 32 No 3, Farnum, N (1997), Using Johnson curves to describe non-normal rocess data, Quality Engineering, Vol 9 No 2, Pillet, M, Duret, D and Sergent, A (2001), L objectif cible et la détermination statistique des tolérances, Actes, Congrès qualité, Annecy, March, Wheeler, D (2000), Normality and the rocess, Behavior Chart, Quality Press, Milwaukee, WI 209