Chapter 3 MATHEMATICAL MODELING OF TOLERANCE ALLOCATION AND OVERVIEW OF EVOLUTIONARY ALGORITHMS

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1 28 Chapter 3 MATHEMATICAL MODELING OF TOLERANCE ALLOCATION AND OVERVIEW OF EVOLUTIONARY ALGORITHMS Tolerace sythesis deals with the allocatio of tolerace values to various dimesios of idividual compoets of a assembly. The allotted tolerace has direct relatio with the maufacturig ad quality cost of the mechaical assembly. I this chapter, mathematical (aalytical) formulatio for simultaeous tolerace allocatio usig various methods like Worst Case Sceario (WCS), Root Sum Square (RSS) method, Spoot s Modified criteri a (SM) ad Estimated Mea Shift (EMS) criteria have bee explaied. Moreover, the importace of tolerace of positio cotrol is discussed i detail. 3.1 MATHEMATICAL FORMULATION OF THE PROBLEM The existig literature (Ye ad salustri, 2003) shows that maufacturig costs have bee icreased due to the allocatio of tight toleraces after employig quality loss ad tolerace sythesis methods. The cocurret tolerace allocatio method simultaeously assigs toleraces to strike balace betwee quality ad maufacturig cost ad optimizes cost over the life of the product. It is also importat to ote that positioal tolerace (Data, 2005) which is a geometric tolerace is also havig a impact o the total

2 29 cost of a assembly. I the preset thesis, the optimizatio of tolerace sythesis problem has bee solved after cosiderig both the maufacturig ad quality cost for various mechaical assemblies, such as, pisto-cylider, wheel-mout ad rotor-key. For rotor-key assembly, positioal cotrol is icorporated i the stackup coditios. By combiig maufacturig ad quality cost measures, this method tries to get equilibrium betwee them ad attai a miimal total cost (CT) of the assembly. I the preset model, process ad desig toleraces are cosidered to be the decisio variables. The maufacturig cost is a fuctio of process toleraces, whereas the quality loss is a fuctio of desig toleraces. Depedig o the miimizatio of total cost of assembly a optimizatio problem is formulated as follows. 3.2 FORMULATION AS AN OPTIMIZATION PROBLEM I this preset work, the objective fuctio cosidered miimizes the total cost (CT) i.e maufacturig ad quality cost of the assembly. I the objective fuctio, the weights W1 for maufacturig cost ad W2 for quality cost are icorporated. The Assumptios for Formulatig the Mathematical Model are give below: 1. The desig fuctio ca be retrieved ad formulated from a assembly cotext. This fuctio defies a relatioship betwee a set of deviatios i a dimesio chai ad a product performace

3 30 characteristic. I this work desig fuctio is assumed oedimesioal. 2. The resultat tolerace of a dimesio chai is give. This is the fuctioal tolerace, which comes from fuctioal or customer requiremets ad is available from the outset of a desig project. 3. A process pla for each dimesio is give. I a cocurret egieerig eviromet, it is commoplace to fid maufacturig egieers developig process plas while desigers are still detailig the product. Eve with icomplete or ureliable iformatio, a maufacturig egieer ca estimate a process pla. 4. Each process has a ormal distributio ad is uder statistical cotrol. This allows six-sigma theory to be applied. 5. The dimesios i a dimesio chai ad the processes for each dimesio are idepedet. below. The formulatio of the problem as optimizatio problem is give Mi. pi 2 C() T W C tpij W 2 K i j y, (3.1) Subjected to: t T i 1 id f, worst case criteria (3.2) or 2 t i 1 id T f, RSS criteria (3.3) or

4 31 Z tid tid Tf i 1 i mitid (1) mi tid Tf i 1 3 i 1 Spotts criteria (3.4) or Estimated mea shift criteria (3.5) All the above costraits are used to optimize the objective fuctio without positio cotrol (that is, geometrical tolerace). The costraits with positio cotrol (Data, 2005) are give below. t 1 id T i f T positio worst case criteria (3.6) or 2 t i 1 id T f T positio RSS criteria (3.7) or 1 2 or 2 tid tid Tf Tpositio i 1 i 1 Spotts criteria (3.8) Z 2 2 mitid (1) mi tid Tf Tpositio i 1 3 i 1 Estimated mea shift criteria (3.9) ad ad ad t ij t i j 1 mi max ij ij ij Tp ij, (3.10) tp tp tp, (3.11) tp ipi t id, (3.12)

5 32 where the costrait equatios 3.2 to 3.9 represets the desig tolerace costraits for differet cases, whereas equatio 3.10 idicates the maufacturig tolerace costrait. Moreover, the equatios 3.11 ad 3.12 represet the process tolerace costraits ad desig toleraces costraits, respectively. The sigificace of the termiology used i the above equatios is give below. The total maufacturig cost of a assembly represets the summatio of the maufacturig costs of idividual machiig process for the etire compoet s dimesio. Table 3.1 shows a few cost fuctios proposed by Dog et al. (1991, 1994) which have a extra edge over the traditioal cost fuctios i modelig the maufacturig cost tolerace characteristics precisely. The expoetial cost fuctio model proposed by Spekhart (1972) has proved to be the best amog all. Table 3.1: Cost tolerace models Cost tolerace Model Combiatio of reciprocal powers Ad expoetial Combiatio of liear ad Expoetial fuctio B splie cost model Cubic polyomial model Fourth order polyomial model Fifth order polyomial model Mathematical Represetatio c C() t a b/ t dexp() et C() t a bt cexp() dt C()()()()()() t a B t a B t a B t a B t a B t C() t a a t a t a t C() t a a t a t a t a t C() t a a t a t a t a t a t

6 33 A way to evaluate ecoomic loss from a quality deviatio elimiatig the eed to create the uique fuctio for every quality characteristic is provided by Taguchi quality loss fuctio (Taguchi, 1989). These losses comprise of customer satisfactio, performace, ad maufacturig ad supplier efficiecy. These are real i ature ad ca be estimated i moey terms. Figure 3.1 depicts Taguchi s Loss Fuctio ad is defied as follows: Moetary loss is a fuctio of each product feature (x), ad its differece from the best (target) value. Figure 3.1: Taguchi s Loss Fuctio ad ormal distributio x = measuremet of a compoet characteristic. T = X target value. a = extet of loss whe the quatity of x deviates from the quatity of T b = extet that x deviates from T. I this illustratio, T = mea of the sample of X s. The Nomial characteristic is pictorially depicted i the Figure 3.2. The loss ( L) icreases as the output value ( y) departs from the

7 34 target value ( m) risig the mea squared deviatio. No loss is show eve whe the output value equal the target value (y = m). Figure 3.2: Graphical represetatio of the Nomial Characteristic Quality loss is quatified, as a quadratic expressio represetig the loss to the deviatio of a product property L(Y) = k(y-m) 2, (3.13) where k=a/tf, A is the replacemet cost. Whe the dimesio does ot match the tolerace requiremets, Tf is the fuctioal tolerace requiremet, m is the fuctioal dimesio target value, ad y represets the desig characteristic. A ormal distributio havig the mea at the target value is assumed to be the fuctioal dimesio. The stadard deviatio of the fuctioal dimesio represets the quality loss. The loss fuctio the ca be give as follows QL = E(L(Y)) = k( (μ m) 2 +σy 2 ), (3.14) where µ is the mea ad σ 2 y is the variace of Y. The distace of the mea of Y i.e µ from the target value m ad the variace of Y are

8 35 liearly combied i the equatio. Durig parameter desig a quality egieer alters µ to reduce the quality loss. The value of process variability σy will ot be affected by these adjustmets. Fially the quality loss may be show as: QL = E(L(Y))= k σy 2, (3.15) Based o the desig fuctio, a set of idividual quality characteristics ca be used to estimate the overall quality characteristic. It is to be oted that Taylor s series expressios ca be used to foud these approximate fuctios. The idividual quality characteristic i terms of the resultat variace is expressed as fallows: 2 2 f 2 y i 1 x xi, (3.16) i Toleraces are i geeral related ad desiged i combiatio with the applicatio of a particular maufacturig process. The the process capability idex Cp, is writte as: t Cp id, (3.17) 3 xi The above expressio ca be modified as t id xi, (3.18) 3C pi Substitutig the Equatio 3.18 i Equatio 3.16

9 f t id y, (3.19) i 1 x i 3C p ad the total quality loss is 3.3 CONSTRAINTS 2 QL t d k y, (3.20) The optimizatio problem discussed above has costraits pertaiig to both desig ad maufacturig toleraces Desig Tolerace Costraits The fudametal cosideratios of various dimesios of the assembly are based o the desig tolerace. The desig tolerace costrait is formulated with a coditio that i the dimesioal chai, accumulated tolerace should ot go over the specified tolerace o the fuctioal dimesio. Several approaches had bee proposed to estimate the accumulated toleraces, which are valid uder variable coditios. I additio to regular stack up approaches, like RSS ad worst case (Paul, 1999), other otraditioal approaches, such as Spoot s criteria ad Estimated Mea Shift criteria have also bee used. The relatioships related to these cases are give i equatios 3.2 to 3.9. Worst case sceario (WCS): The ame worst case sceario implies that it adds or subtracts all the upper or lower toleraces coected with the omial set poit for idividual compoet. I this case, laws of probability are ot

10 37 cosidered ad it keeps the assembly at the highest or lowest possible dimesio. All the part dimesios are summed up at the worst case miimum or maximum tolerace values (paul, 1999). Based o the worst possible coditio, the simplest of all is the method of extremes (equatio 3.2). The outcome of this method allocates very precise toleraces, which i tur lead to higher maufacturig costs of a assembly Root Sum of Squares (RSS) method: RSS is a kid of statistical toleracig method. This method is helpful to take care of the rare case of all dimesios occurrig at their ed limits oe after the other. The said method is a mathematical treatmet method of the data to assist the tolerable additio of measures of variability. The mere assumptio cosidered i the Root-sum-square (RSS) criteria (Paul, 1999) (equatio 3.3) is a ormal process distributio with the mea cetered at the omial value. It is also called as simple statistical model ad depeds o impractical assumptio. The applicatio of this method yields low maufacturig cost of assembly due to very loose toleraces. Spotts Modified (SM) criteria: This method is used due to its simplicity. Some modificatios have bee proposed to accout for the realistic distributios of process. Based o mea results of WC ad RSS stack-up coditios, Spott s developed a ew method called Spotts modified (SM) criteria (Spott s, 1978). This model is represeted i equatio 3.4.

11 38 Estimated Mea Shift (EMS) Criteria: A itegrated approach where we ca cotrol the amout of mea shift is proposed by Greewood ad Chase (1987). It cotais the RSS ad the WC as the ed cases, ad it ca reproduce ay itermediate value. I this model, the rage of mea shift factor is cosidered as (0.0, 1.0) ad represets a amout by which the midpoit is estimated to shift, as a fractio of the tolerace rage. If a process is closely cotrolled, the a small value of mea shift, such as 0.25 is used. If the amout of iformatio kow about the process is less, a high value of mea shift factor is cosidered. If the value of mea shift factor is equal to 1.0 for all the compoets, it represets the Worst Case model ad if the mea shift factor is zero for all of the compoets, the approach is similar to Root Sum Square Model. I the preset study, the value of mea shift (m i) is to be cosidered to be 0.25 for close cotrol i all the cases. This model is represeted i equatio 3.5. Geometric Toleraces - Tolerace of Positio Cotrol: I geometric tolerace methods, tolerace of positio cotrol (TOP cotrol) is oe of the importat methods. It defies the locatio tolerace of a feature of size from its origial positio. There are two ways of specifyig the TOP cotrol boudary. The first method is to specify TOP tolerace zoe o RFS basis, where as the secod method employs either MMC or LMC. I both the methods, the FOS ceter plae should be i the idetified tolerace zoe of the cotrol frame.

12 39 Advatages of Tolerace of Positio: While comparig the Tolerace of Positio ad Coordiate Toleracig, the former is foud to have distict advatages over the latter. To ame a few, the advatages are: 1. TOP allows datum shift ad additioal toleraces-bous. 2. It gives 57% larger cylidrical tolerace zoes tha square zoes. 3. TOP gives protectio to the part fuctio. 4. Maufacturig costs are reduced. 5. Accumulatio of tolerace is reduced by TOP cotrol. TOP cotrols geerally employ cylidrical tolerace zoes. Figure 3.3 shows the coversio of a liear tolerace ito cylidrical tolerace zoe. It is importat to ote that a additioal tolerace of 57% will be gaied usig a cylidrical tolerace zoe. I additio to this, additioal toleraces of 50 to 100% are goig to be added i the form of bous ad datum shift, which leads to the lower maufacturig cost. Figure 3.3: Coversio of square zoe to cylidrical positioal tolerace zoe.

13 40 Trasformatio of square zoe ito cylidrical zoe: The true positio of a feature is described geerally by the positio tolerace, as it chooses a datum ad the omial locatio of a poit, a lie, or a plae of the feature. Let us cosider the positio tolerace to be x, the the correspodig liear dimesioal tolerace is give by dividig x with 2 2 (Yua, 1999). The expressio for this model usig GLi, (basic dimesio) ad GPTi (positioal tolerace) is i the followig form: GLi ± GPT i Maufacturig Tolerace Costraits Process capability ad the amout of machiig allowace for each operatio are importat factors to be kept i mid durig machiig tolerace allocatio. The actual amout of material removed may vary because of process wise maufacturig errors. Hece, cumulative sum of maufacturig toleraces of all the processes is to be take ito accout while fixig the rage for metal removal. I practice, typical levels of material removal are set o the basis of each process. The costraits o maufacturig tolerace formulated are show i equatio The term tij represets the maufacturig tolerace of j th operatio for producig i th dimesio, while Tpij deotes the deformatio allowace of j th operatio.

14 Process Tolerace Costrait Every process or operatio may have certai accuracy ad must be performed withi its process capability. The process tolerace costraits are formulated i equatio Simultaeous tolerace sythesis model cosiders two costs workig had i had; Maufacturig Cost ad Quality Loss. While maufacturig cost is expressed as a fuctio of process toleraces, Quality loss is expressed i terms of desig toleraces. Quality loss eed ot be cosidered for the itermediate process toleraces as they are ot the fial toleraces. Sice quality loss is associated with desig toleraces, the toleraces of the last processes will be sufficiet. By computig Quality loss i this maer, we ca add the maufacturig cost to it. The desig tolerace costraits are formulated i equatio SUMMARY I the preset chapter, a aalytical model for simultaeous allocatio of toleraces for mechaical assemblies after applyig various methods, such as WCS, RSS, SM ad EMS have bee discussed i detail. Moreover, the ifluece of tolerace of positio cotrol has bee discussed. Fially, a itroductio to the evolutioary algorithms has also bee explaied. The applicability of above models to various mechaical assemblies, such as pisto cylider, wheel- mout ad rotor-key assemblies, are cosidered as

15 42 a optimizatio problem i the followig chapter. It is to be oted that the above optimizatio problems have bee solved usig three evolutioary algorithms, such as GA, DE ad PSO.