A SYSTEMATIC APPROACH FOR COST OPTIMAL TOLERANCE DESIGN

Transcription

1 INTERNATIONAL CONFERENCE ON ENGINEERING DESIGN, ICED AUGUST 007, CITE DES SCIENCES ET DE L'INDUSTRIE, PARIS, FRANCE A SYSTEMATIC APPROACH FOR COST OPTIMAL TOLERANCE DESIGN Sotiria Ch. Dimitrellou, Stefaos C. Diplaris ad Michael M. Sfatsikopoulos Natioal Techical Uiversity of Athes (NTUA, Greece ABSTRACT This paper presets a ew method ad a prototype system for cost optimal tolerace desig i mechaical assemblies. The system overcomes iefficiecies ad/or hadicaps met i curret methods ad tools that require expert cost-tolerace iput refereced to the participatig i the assembly chai dimesios or the availability of case-drive experimetal cost-tolerace data. This is accomplished by itroducig the cocept of the Tolerace Elemet (TE, a machiig process related geometric etity with attributes associated with the accuracy cost ad idetifiable i coformace with the stadard idustrial uderstadig. It becomes thus possible to assig cost optimum ad ratioal compoet toleraces through a algorithmic mode that ca be directly itegrated i a CAD eviromet. Withi this -based o the TE approach- frame, machiig capability ad cost per TEclass of a particular machie shop are appropriately recorded ad processed ad cost-tolerace fuctios automatically established ad stored i the system database to be used for tolerace optimizatio. The latter oce created, eeds the updatig oly whe chages take place i the machie shop resources ad/or expertise. Tolerace allocatio ca be thus achieved with miimum maufacturig cost withi the machie shop accuracy gettig capability i a effort ad time efficiet way through realistic ad machie shop focused cost-tolerace data. A applicatio example demostrates the method ad its positive evaluatio i compariso with alterative problem solutios. Keywords: Toleracig, Tolerace allocatio, Cost-Tolerace Fuctio, Optimum Toleracig, Toleraces Chais 1 INTRODUCTION Withi the frame of further developmet of CAD tools emphasis is give to computer-aided toleracig techiques for cost optimal allocatio of toleraces i mechaical assemblies [1-4]. I egieerig desig as optimal toleracig is meat the assigmet of toleraces to the compoets of a mechaical assembly i terms ot oly of fuctioality but also of miimum maufacturig cost. The fuctioal performace of a assembly, as a result of the deviatios of its critical dimesios, is decisively affected by the compoet toleraces accordig to the tolerace chais. Tight toleraces geerally result to high dimesioal accuracy, quality ad modularity of the product. High accuracy is maily based, o the other had, o machiig processes ad always imposes additioal effort hece higher maufacturig costs. Give that accuracy costs costitute a vital issue i the idustrial productio, various cost-tolerace relatioships have bee proposed ad tried to this date, [3,5-8]. Critical characteristic of these relatioships however, is that their expoets ad coefficiets are obtaied experimetally uder particular each time experimetal coditios ad caot therefore be broadly used. CAD software for cost optimal tolerace allocatio does ot make use of such case-drive costtolerace fuctios as they may well ot be represetative of the machie shop where the compoets will be maufactured. Istead, they require from the user to feed the system with cost-tolerace data based o his ow judgemet ad cost-tolerace kowledge. The procedure is, apparetly, ot systematic ad subjective to the way the particular user uderstads the accuracy cost of a machiig tolerace. It ca also be time-cosumig i case the assembly has a large umber of membertoleraces. Cosiderable depedecy of the produced results o user iput, misiterpretatios, mistakes ad uecessary repetitios are uavoidable. ICED 07/170 1

2 I this paper a prototype system for cost optimal tolerace desig i mechaical assemblies is preseted. The system overcomes iefficiecies met i existig CAD tools that require from the user expert cost-tolerace iput or the availability of case-drive experimetal cost-tolerace data. The developed methodology itroduces the cocept of the Tolerace Elemets that are geometric etities with attributes associated with the accuracy cost of the specific machiig eviromet where the compoets will be maufactured. Oce established for this eviromet, the system automatically creates ad makes use of appropriate cost-tolerace fuctios for the assembly chai members uder cosideratio. Cost optimal tolerace desig ca thus be achieved i a systematic, cost ad time efficiet way by meas of realistic ad machie shop focused cost-tolerace data. The beefits of this system are demostrated ad discussed through a applicatio example. THE METHOD.1 Tolerace Elemets Accuracy cost is the cost required to produce a give dimesio withi its specified tolerace limits. It depeds, apparetly, o the processes ad resources eeded for the part productio. Give the workpiece material ad toleraces, the part geometrical characteristics such as size, shape, feature details, iteral surfaces, are take ito cosideratio for plaig the machiig operatios, programmig the machie tools, specifyig fixtures, etc. These geometrical characteristics have thus a direct impact o the machiig cost of the required accuracy by determiig, idirectly, its magitude [9]. A Tolerace Elemet (TE is defied as a 3D form feature of particular shape, size ad tolerace. It icorporate attributes associated with its shape, size, positio, the presece of additioal feature details ad the ratio of the pricipal dimesios of the part to which it belogs. To each TE correspods oe cost-tolerace fuctio that stads for its accuracy cost ad is directly related with the particular machie shop where the TE will be produced (machie tools, ispectio equipmet, supportig facilities, expertise. Tolerace Elemets are classified through a five level class hierarchy system, Figure 1. Class level attributes are all machiig process related, geeric ad straightforwardly idetifiable i coformace with the existig idustrial uderstadig. Figure 1. TE classificatio ICED 07/170

3 I first level, TEs are classified accordig to the basic geometry of the part to which they belog, i.e. rotatioal TEs ad prismatic TEs. Rotatioal TEs belog to rotatioal parts maufactured maily by turig ad borig, while prismatic TEs belog to prismatic parts maily maufactured by millig. I secod level, TEs are classified accordig to the size ratio of the pricipal dimesios of the part to which they belog, cosiderig the required resources for part machiig ad/or the elastic deformatios due to cuttig forces or part ow weight. I this way TEs are classified as short [L/D 3] ad log [L/D >3] TEs, followig a typical way of classificatio [10-11]. For a rotatioal part, D represets its largest diameter, L the part legth. For a prismatic part L represets its largest dimesio ad D the largest oe i the directio of the two remaiig Cartesia axes. I third level TEs are classified to exteral ad iteral oes as the achievemet of tight toleraces i iteral TEs usually results to higher accuracy cost. The fourth TE classificatio level distiguishes betwee plai ad complex TEs depedig o the absece or presece of additioal feature details. Such details iclude grooves, wedges, ribs, threads or otches. They do ot chage the pricipal TE geometry but they idirectly cotribute to the icrease of the accuracy cost. I the fial fifth level, the ivolvemet of the TE size to the accuracy cost is cosidered. TEs are classified, accordig to the omial size of their participatig i the chai dimesio, ito six groups by itegratig two sequetial ISO 86-1 size rages. The five-digit TE classificatio code is show i Table 1. Figure illustrates a rotatioal part with rotatioal TEs ad their codes. Table 1. TE codig scheme DIGIT 1 DIGIT DIGIT 3 DIGIT 4 1 Rotatioal TE 1 Short TE 1 Exteral TE 1 Plai TE 0 Prismatic TE 0 Log TE 0 Iteral TE 0 Complex TE DIGIT Dimesio 10 mm 10< Dimesio 30 mm 3 30< Dimesio 80 mm 4 80< Dimesio 180 mm 5 180< Dimesio 315 mm 6 315< Dimesio 500 mm Figure. Rotatioal TEs (assembly chai i the axis directio ICED 07/170 3

4 . Modellig of cost-tolerace fuctios Based o the TE-method the actual machiig accuracy capabilities ad the relative cost per TE-class of a particular machie shop are recorded through the Database Feedback Form (DFF of Figure 3. The latter icludes the accuracy cost for all the 6=96 TE-classes i the size rage mm ad toleraces rage IT6-IT10. The relative cost cocept represets, o a comparative basis, the expoetial icrease of costs required for expertise, effort ad resources as the toleraces become tighter, [3,6,8,11]. The 10-grade relative cost scale is show i Table. A DFF is filled oce, whe the system is commissioed, by the expert egieers of the machie shop where the assembly compoets will be maufactured. It ca the be updated each time chages occur i the shop machies, facilities ad/or expertise. Processig of the database data is performed per TE-class through the least-squares approximatio i order to costruct the cost-tolerace relatioship of the power fuctio type [1,7], k ( t = A + B t C / (1 I equatio (1 C(t is the relative cost for the productio of the machiig tolerace ±t ad A, B, k are costats. Costat A represets the cost for producig the TE dimesio with highly relaxed accuracy. Withi the frame of this work this cost compoet is costraied to the relative cost of IT10. The secod part of equatio (1 costitutes, therefore, the accuracy cost for the productio of the TEdimesio with icreased accuracy. Cost-tolerace fuctios established with the described way for all the TE-classes are stored i the system database ad become thus available for machie shop focused cost optimum toleracig. Figure 3. Database Feedback Form ICED 07/170 4

5 Table. Relative cost scale Machiig Relative Cost Ufeasible 10 Extremely difficult 9 Quite difficult 8 Difficult 7 Feasible with some effort 6 Feasible 5 Easily feasible 4 Easy 3 Quite easy Extremely easy 1 3 TOLERANCE DESIGN I a -member dimesioal chai the toleraces of the idividual dimesios D i, i=1,,,, cotrol the variatio of a critical ed-dimesio D 0, accordig to the chai, D 0 = f ( D1, D,..., D where f (D i ca be either a liear or oliear fuctio. To esure that the ed-dimesio will be kept withi its specified tolerace zoe, two approaches are usually employed for tolerace allocatio. The worst-case costrai, that provides for 100% iterchageability, results usually to tight toleraces ad hece to high accuracy costs, f f t1 + t D D 1 f t t D 0 I case that a certai rejectio percetage ca be cosidered as acceptable, the the root-sum-squares costrai that implies the ormal distributio for the dimesioal deviatios is used istead, ( (3 f D 1 t 1 f + D t + f... + t t0 D (4 I above equatios t 0 ad t i are the toleraces of the ed-dimesio D 0 ad the dimesio D i respectively. A additioal costrai that arises from the capability of the process ad machie tools to be used is, ti Tm m = X, Y, Z with T m represetig the available accuracies alog the machiig axes X, Y, Z. Cost optimum toleracig of the assembly chai dimesios ca thus be pursued through the miimizatio of the oliear objective fuctio, C total ki ( t t = Ci ( ti = [ Ai + Bi / ti ] mi, ki 1,.., Bi / ti mi (6 i= 1 i= 1 subject to the costrais (3 or (4 ad (5. 4 METHOD IMPLEMENTATION The prototype software has bee developed usig Microsoft Excel ad Matlab tools. It cosists of two major modules, amely the database module ad the tolerace allocatio module. For the developmet of the database module the add-i Excel Lik is used that allows for data exchage betwee the two eviromets. A expert egieer of the machie shop fills the Database Feedback Form at the system commissioig stage ad sets the curret machie shop accuracy capability. The database module rus oly at this stage ad ay time sice DFF eeds updatig. Cost-tolerace fuctios for the TE-classes are thus automatically computed ad stored i the database. i= 1 (5 ICED 07/170 5

6 The secod module that deals with tolerace allocatio cosists of three submodules: (i module for the requiremets defiitio, (ii module for TE ad cost-tolerace fuctio idetificatio, ad (iii module providig for the optimum solutio through the miimizatio of the automatically established objective fuctio. A user-friedly iterface has bee developed to provide the user with a iteractive widows eviromet for easy ad efficiet system operatio. Sequetial Quadratic Programmig (SQP algorithm of the Matlab Optimizatio Toolbox for costraied oliear optimizatio is used. Startig poit of the algorithm is take the tolerace grade IT10. The flow charts of the database ad the tolerace allocatio modules are show i Figures 4 ad 5 respectively. Figure 4. Database module Figure 5. Tolerace allocatio module ICED 07/170 6

7 5 APPLICATION EXAMPLE AND DISCUSSION I the assembly of compoets A-B-C of Figure 6 below the dimesio D 4 = 75±0.5mm is cotrolled through the two dimesioal chais (D1,D,D3,D4,D5,D6,D7,D8, (D8,D9,D10,D11. Worst case cost optimum toleracig is required for all the chai dimesios. The machie shop where the assembly compoets will be maufactured has a IT6 best capability ad its DFF processed ad results stored. Chai equatios ivolved are, Figure 6. Applicatio example [ D1 ( D6 D7 D cot D3 D5 D8 ] D3 D4 = si (7 D 8 = D11 D10 D9 Case TEs ad cost-tolerace fuctios retrieved by the algorithm from the database are show i Table 3. The total cost model was costructed i accordace with the relatioships (3, (5 ad (6 ad the optimizatio process fially produced the results quoted i Table 4. For system performace evaluatio are also give, i the same table, the results produced by two alterative approaches, a covetioal (ot optimized mode that applies same IT grade, here IT8, for all the chai dimesios startig from the highest IT that satisfies the costrais ad a commercially available toleracig tool. Users A ad B of the commercial tool followed User Istructios ad worked idepedetly usig their ow cost-tolerace kowledge to drive the tool ad produce the results. The total accuracy cost of all three approaches was calculated usig the cost-tolerace fuctios of the preset method, Table 3. Table 3. Cost tolerace fuctios of the applicatio example Dimesio TE Cost tolerace fuctio C ( t = t D /( D 0111 C ( t = / ( t D C 3 ( t3 = t D C 5 ( t5 = t D C 6 ( t6 = /( t D C ( t = ( t 7 7 / 7 (8 ICED 07/170 7

8 D C 9 ( t9 = /( t D C 10 ( t10 = /( t D C ( t = ( t / 11 Dimesio Table 4. Applicatio example results Toleracig Tool IT8 User A User B Preset system D 1 = 190mm ± ± ± ± D = 15 mm ± ± ± ± 0.06 D 3 = 45 ± 0.07 ± ± ± 0.05 D 5 = 14 mm ± ± ± 0.06 ± 0.07 D 6 = 95 mm ± 0.07 ± ± ± D 7 = 0 mm ± ± ± ± D 9 = 75 mm ± 0.03 ± 0.01 ± ± D 10 = 1 mm ± ± ± ± 0.07 D 11 = 97 mm ± 0.07 ± ± 0.00 ± 0.06 Accuracy cost: As it ca be see from the results, miimum total cost was achieved by the preseted system. The system leads to the same toleraces whatever is the IT grade the algorithm is asked to start the accuracy cost optimizatio. That does ot happe with the commercial tool whose results are also strogly user iput depedet. Time eeded to operate such a system is also much loger tha that of the preset method. The covetioal approach, o the other had, although very simple caot guaratee a optimum solutio as it does ot take ito cosideratio cost ifluecig parameters other tha the dimesioal size. Similar results from other case studies cofirm these observatios. 6 CONCLUSIONS Assigmet of proper toleraces to the compoets of a mechaical product is of major importace for the product quality, fuctioality, modularity ad maufacturig cost. This paper itroduced the Tolerace Elemet cocept ad preseted the developmet of a prototype system for the establishmet ad processig of ratioal ad machie shop focused cost-tolerace iformatio. The ew methodology overcomes iefficiecies ad hadicaps of the curretly available toleracig tools ad ca be easily itegrated ito a CAD eviromet. The system database oce created eeds updatig oly whe chages occur i the machie shop resources ad/or expertise. The developed system was used for cost optimal tolerace desig i mechaical assemblies ad produced realistic results i a small fractio of time otherwise required by alterative -computer aided or ot- problem solutios. ACKNOWLEGEMENTS This project is co-fuded by the Europea Uio Europea Social Fud & Natioal Resources EPEAEK II. REFERENCES [1] Zhag G. Simultaeous toleracig for desig ad maufacturig. I Advaced Toleracig Techiques, 1997, pp (Joh Wiley & Sos, New York. [] Huag M.F., Zhog Y.R. ad Xu Z.G. Cocurret process tolerace desig based o miimum product maufacturig cost ad quality loss. The Iteratioal Joural of Advaced Maufacturig Techology, 005, 5, [3] Hu J. ad Xiog G. Dimesioal ad geometric tolerace desig based o costraits. The Iteratioal Joural of Advaced Maufacturig Techology, 005, 6, [4] Sigh P.K., Jai S.C. ad Jai P.K. Advaced optimal tolerace desig of mechaical assemblies with iterrelated dimesio chais ad process precisio limits. Computers i ICED 07/170 8

9 Idustry, 005, 56, [5] Sfatsikopoulos M.M. A cost-tolerace aalytical approach for desig ad maufacturig. The Iteratioal Joural of Advaced Maufacturig Techology, 1990, 5, [6] Yeo S.H., Ngoi B.K.A. ad Che H. A cost-tolerace model for process sequece optimizatio. The Iteratioal Joural of Advaced Maufacturig Techology, 1996, 1, [7] Chase K.W. Tolerace allocatio methods for desigers. ADCATS Report No. 99 6, [8] Kulkari U.A. ad Bao H.P. Close-loop cost equatio for objects maufactured by millig. Joural of Maufacturig Sciece ad Egieerig, ASME, 003, 15, [9] Ullma D.G. The Mechaical Desig Process, 199 (McGraw-Hill, New York. [10] Opitz H. A Classificatio System to Describe Workpieces, 1970 (Pergamo Press, Oxford. [11] Ehrlespiel K. Kostegüstig Kostruiere, 1985 (Spriger-Verlag, Berli. Cotact: S. Ch. Dimitrellou Natioal Techical Uiversity of Athes (NTUA Departmet of Mechaical Desig & Cotrol Systems 9 Iroo Polytechiou Str , Athes Greece ( ( sdimitre@cetral.tua.gr ICED 07/170 9