Available online at ScienceDirect. Procedia CIRP 27 (2015 ) th CIRP conference on Computer Aided Tolerancing

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1 Available olie at ScieceDirect Procedia CIRP 27 (205 ) th CIRP coferece o Computer Aided Toleracig Applicatio of Mote Carlo Method i Tolerace Aalysis Huiwe Ya, Xiagji Wu, Jiagxi Yag* The State Key Lab of Fluid Power Trasmissio ad Cotrol, Zhejiag Uiversity, Hagzhou, Chia * Correspodig author. Tel.: ; fax: address: yagjx@zju.edu.c. Abstract Mote-Carlo simulatio is the most popular ad simplest method for oliear statistical tolerace aalysis. Radom values for every part are got accordig to the part distributios, ad the value of the respose fuctio is computed for each set of part values. A sample of respose fuctio values is thus got, ad the momets of the sample are computed usig the stadard statistical formula. Atpreset, the researches o assembly tolerace aalysis have bee focused o dimesio tolerace ad hardly take geometric tolerace itocosideratio. I this paper, geometric tolerace is treated as dimesio tolerace whose omial value is zero, ad the Mote Carlo SimulatioMethod is applied to tolerace aalysis icludig geometric tolerace. Fially, a case of a top colum assembly verifies the validity ad accuracy of the method. 205 The Authors. Published by Elsevier B.V. This is a ope access article uder the CC BY-NC-ND licese Peer-review ( uder resposibility of the orgaizig committee of 3th CIRP coferece o Computer Aided Toleracig. Peer-review uder resposibility of the orgaizig committee of 3th CIRP coferece o Computer Aided Toleracig Keywords:Tolerace aalysis; Geometric tolerace; Mote-Carlo Simulatio Method;. Itroductio Tolerace aalysis has become more ad more importat ad has received cosiderable attetio i maufacturig idustriesad thus i may literatureswith icreasigly fierce market competitio. The fial target of tolerace aalysis is to ot oly improve the performace of product, but also reduce the cost of product. So far, a large amout of researchers have devoted themselves to mathematical modelsad methods of tolerace aalysis.may well-kow models or methods are itroduced i several review literatures for tolerace aalysis [-]. The variatioal model has its roots i parametric geometric modellig, where geometry ca be modelled by mathematical equatios that allow shape ad sizeattributes to be chaged ad cotrolled through a reduced set of parameters[5-6].the vector loop model adoptsa graph-like schematizatio where ay relevat liear dimesio i the assemblyis represeted by a vector, ad a associated tolerace is represeted as a smallvariatio of such a vector[7-].the tolerace map(t- Map) represets all possible variatios of size, form, positio, ad orietatio for a target feature[2].the small displacemet torsor (SDT) model uses six small displacemet vectors to represet the positio ad orietatio of aideal surface i relatio to aother ideal surface i a kiematic way[ 3-5].The matrix model aims at derivig the explicit mathematical represetatio of geometry of each tolerace regio through displacemet matrices[6-8]. Mote Carlo simulatio[9-20] is probably the simplest statistical tolerace aalysis method ad it takes ito cosideratio the probabilistic behaviour of the maufacturig process. Its geeral procedure is: () Use a geerator to radomly geerate sets of maufactured dimesios i a assembly with specified compoet distributios. (2) Get a sample of assembly fuctios employig the sets of maufactured dimesios. (3)Estimate the assembly performace parameters, such as mea, stadard deviatio ad reject rate of the assembly. So far few research at home ad abroad o effect of geometric tolerace o assembly quality,ad the geometric tolerace may have great effect i some cases,resultig from rigid body effects. This paper aims at icludig geometric tolerace i the assembly tolerace aalysis with the Mote Carlo simulatio method.the paper is orgaized as six parts.i the followig part,geometric tolerace is illustrated.the effect of geometric tolerace o the tolerace aalysis is discussed i Sectio 3; Sectio will discuss how to iclude geometric tolerace of compoets i Mote Carlo simulatios.i the Sectio 5, a case of a top colum assemblyverifies the validity ad accuracy of the method.the coclusios are summarized i the Sectio The Authors. Published by Elsevier B.V. This is a ope access article uder the CC BY-NC-ND licese ( Peer-review uder resposibility of the orgaizig committee of 3th CIRP coferece o Computer Aided Toleracig doi:0.06/j.procir

2 282 Huiwe Ya et al. / Procedia CIRP 27 ( 205 ) Geometric tolerace The geometric tolerace ca be defied ito four mai groups, i.e., form tolerace, orietatio tolerace, locatio tolerace, ad ruout tolerace, accordig to GB/T [2]: ()A form tolerace states how far a actual surface or feature is permitted to vary from the desired form implied by the drawig. It icludes straightess, flatess, circularity, profile of a lie ad profile of a surface. (2)A orietatio tolerace states how far a actual surface or feature is permitted to vary relative to a datum or datums. It cosists of parallelism, perpedicularity, agularity, profile of a lie ad profile of a surface. (3)A locatio tolerace states how far a actual size feature is permitted to vary from the perfect locatio implied by the drawig as related to a datum, ordatums, or other features. This category icludes positio, cocetricity, Symmetry, profile of a lie ad profile of a surface. ()A ruout tolerace states how far a actual surface or feature is permitted to vary from the desired form implied by the drawig durig full (360 ) rotatio of the part o a datum axis. A ruout ca be either a circular ruout or a total ruout. Tolerace zoe is the space limited by oe or several geometrically perfect lies or surfaces, ad characterized by a liear dimesio, called a tolerace. Accordig to the characteristic to be toleraced ad the maer i which it is dimesioed, thetolerace zoe is oe of the followig: () the space withi a circle; (2) the space betwee two cocetric circles; (3) the space betwee two equidistat lies or twoparallel straight lies; () the space withi a cylider; (5) the space betwee two coaxial cyliders (6) the space betwee two equidistat surfaces or two parallel plaes; (7) the space withi a sphere. 3. The Effect of Geometric Tolerace o the Tolerace Aalysis There are three mai sources of variatios i a mechaical assembly.two of them are the result of the atural variatio i maufacturig processes ad the third is from assembly processes ad procedures.these three sources are ) dimesioal variatio, 2)geometric feature variatio ad 3)variatio due to kiematic adjustmets at assembly time. The geometric feature tolerace ca be treated as dimesio tolerace whose omial value is zero ad added to the related cotact surfaces. Ad its directio is maily determied by the kiematic joit type ad the geometric characteristic. The effect of the geometric feature toleraces associated with each of the jois may result i traslatioal variatio or rotatioal variatio. Fig. illustrates how a flatess tolerace zoe of the low plae ca affect two matig parts differetly whe viewed i 2-D. The cylider o the left illustrates a traslatioal variatio, while the block o the right exhibits the rotatioal variatio, due to the same geometric feature variatio. The traslatioal variatio for the cylider i Fig. is related to the flatess tolerace, while the rotatioal variatio for the block is determied by ot oly the flatess tolerace but the cotact legth of the block, i this case, the horizotal dimesio of the block. 2 T () T L ta ( ) (2) where T is the flatess tolerace, L is the cotact legth of the block, is the traslatioal variatio caused by the flatess i a plaar joit, ad is the rotatioal variatio resulted from the flatess i a plaar joit ad the cotact legth of the block. Fig.. The effects of geometric feature tolerace. If the traslatioal variatio caused by the geometric feature tolerace is represeted by T ad rotatioal variatio by R, all the possible combiatios of the geometric feature toleraces with the kiematic joit types ca be summarized i Table.The empty cells i the table mea that the correspodig geometric feature tolerace ad kiematic joit combiatio does ot apply. Table.Rotatioal ad traslatioal variatios associated with correspodig geometric feature tolerace-kiematic joit combiatio i 2-D. Plaar R R R R R R RT RT Cyl Slider T T T T T T T T T Edge Slide T T T T T T T T T T T Revolute T T Par Cylid T T T T T. Compoets withgeometric Tolerace i Mote Carlo Simulatios The effect of geometric feature toleraces ca be treated as dimesio toleraces whose value ca be got by theformula () or (2).So the Mote Carlo Simulatio ca be used to estimate the effect of both the dimesioal toleraces ad geometric feature toleraces.

3 Huiwe Ya et al. / Procedia CIRP 27 ( 205 ) The traditioal relatioship betwee the iput parameters ad output variables i a mechaical assembly ca be rewritte as: critical feature is the magitude of the variatio o dimesio r 3 whe the toleraces of the compoets icludig geometric feature variatios are cosidered. Y f( X,, X, X,, X ) (3) m m m where the X,, Xm m m,the maufactured dimesio, are radom variables, typically derived from the drawig dimesios, the X,, X represetig the effect of geometric feature variatios are also radom variables ad Y, the output variable,is the assembly fuctio to be cotrolled. I this paper, all the X i are assumed to be idepedet ad follow Gaussia distributios. After a sample ofy is got, the first momets of Y ca be determied by the followig equatios: Y= Y i () i= 2 2 Y= ( Yi Y) i (5) 3= ( ) Yi Y 3 (6) i = ( ) 3 3 Yi Y (7) i = (8) 2 = where the ad 2 are the coefficiets of skewess ad kurtosis respectively. Oce the momets have bee got, we may compute a tolerace rage for the respose fuctio Y that ecompasses a give fractio of the assembly yield, or we may compute the acceptace fractio for a give tolerace rage. 5. Case Study As a example[22]to demostrate how to iclude geometric feature toleraces i the variatio estimatio of kiematic or assembly variables ad predictio of assembly rejects employig the Mote Carlo Simulatio Method, a top colum assembly is studied. Fig. 2 shows the assembly with geometric feature toleraces applied. I this case study, effort will be focused o the etire top colum assembly model which is used to mout. The assembly cosists of a hadle which press o the colum. The desig requiremet for the assembly is to have adequate iterferece betwee the colum ad the box. So the (9) Fig. 2. Top colum assembly model icludig geometric feature tolerace. Table 2 shows the geometric iformatio for the assembly. Dimesios with a give tolerace are the maufactured variables. Table 3 lists all the geometric feature toleraces applied. Table 2.Dimesioal data of the top colum assembly. Symbol Nomial Dim Tolerace r 30mm 0.05mm r2 8mm 0.mm r5 55mm 0.0mm r6 5mm 0.02mm r7 0mm 0.5mm θ 55 0 Table 3.Geometric feature data of the top colum assembly. Symbol Name Nomial Dim Tolerace Bad α Profile α 2 Plaar α 3 Roudess α Cocetricity The assembly fuctio ca be writte as below accordig to Fig. 2: r 3 r5 r ( + r7)cos( ) r r2 (0) The adopt the simulatio method, ad its basic steps are illustrated i thefollowig: () Set the sample size. (2) Geerate sets of maufactured dimesiosad geometric dimesios. For each maufactured dimesio or geometric dimesio followig Gaussia distributio N(mea,sigma), mea is its omial dimesio ad sigma is oe sixth of its tolerace. (3) Get a sample of assembly fuctios. For each set of maufactured dimesioad geometric dimesio, a assembly fuctio ca be obtaied usig the Formula(0). () Estimate the assembly performace parameters, such as mea, stadard deviatio, employig the Formula (-9). (5) Chage the sample size, ad repeat the step (2-5) util the sample size is large eough to guaratee the aalysis accuracy. So The relatioships betwee first four order momets ad sample size ca be obtaied as show i the Fig. 3.Accordig to the Fig. 3, each relatioship is periodically oscillatig cetered at a value. So we ca set first four order momets equal to their mea values. The result is listed as below: Mea=3.93 mm Stadard deviatio=0.069 mm Coefficiet of Skewess=0.007

4 28 Huiwe Ya et al. / Procedia CIRP 27 ( 205 ) Coefficiet of Kurtosis=3.007 (6) Obtai the results of tolerace aalysis.because the coefficiet of skewess of r3is close to zero ad the coefficiet of kurtosis of r3is close to three, the distributio of r3 ca be regarded as ormal distributio ad the tolerace of r3ca be set to be 6 stadard deviatio. Fig. 3. The relatioships betwee first four order momets ad sample size. Table.The result of tolerace aalysis. Symbol Nomial Upper Deviatioviatio Low De- Tolerace Dimmesio r3 35mm mm +0.5mm 0.6mm 6. Coclusio I this paper, assembly tolerace aalysis icludig geometric tolerace is studied.there are three mai sources of variatios i a mechaical assembly.two of them are the result of the atural variatio i maufacturig processes ad the third is from assembly processed ad procedures.these three sources are ) dimesioal variatio,2)geometric feature variatio ad 3)variatio due to kiematic adjustmets at assembly time. Geometric tolerace is treated as dimesio tolerace whose omial value is zero, ad the Mote Carlo Method is applied to tolerace aalysis icludig geometric tolerace. Ackowledgemets This research was supported by the Natioal Social Sciece Foudatio of Chia (No. 2&ZD 206), the Natioal Basic Research Program of Chia (973 Program, No.20CB706505), ad Natioal Sciece ad techology support program of Chia (No. 203BAC6B02). Refereces [] Marziale M, Polii W. A review of two models for tolerace aalysis of aassembly: Jacobia ad Torsor. It J Comput Itegr Mauf 20;2():786. [2] Laperrière L, Desrochers A. Modelig assembly quality requiremets usig Jacobia or screw trasforms: a compariso[c]//assembly ad Task Plaig, 200, Proceedigs of the IEEE Iteratioal Symposium o. IEEE, 200: [3]Ameta G, Serge S, Giordao M. Compariso of spatial math models for tolerace aalysis: tolerace-maps, deviatio domai, ad TTRS[J]. Joural of Computig ad Iformatio Sciece i Egieerig, 20, (2): [] Che H, Ji S, Li Z, et al. A comprehesive study of three dimesioal tolerace aalysis methods[j]. Computer-Aided Desig, 20, 53: -3. [5] Gupta S, Turer J U. Variatioal solid modelig for tolerace aalysis[j]. Computer Graphics ad Applicatios, IEEE, 993, 3(3): 6-7. [6] Li B, Roy U. Relative positioig of toleraced polyhedral parts i a assembly[j]. IIE Trasactios, 200, 33(): [7] Gao J, Chase K W, Magleby S P. Geeralized 3-D tolerace aalysis of mechaical assemblies with small kiematic adjustmets[j]. IIE trasactios, 998, 30(): [8] Chase K W, Gao J, Magleby S P. Geeral 2-D tolerace aalysis of mechaical assemblies with small kiematic adjustmets[j]. Joural of Desig ad Maufacturig, 995, 5:

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