Multi-Dimensional Tolerance Analysis (Automated Method)

Transcription

1 Cpt Multi-Dimnsionl Toln Anlsis (Automtd Mtod Knnt W. Cs, P.D. Bigm Young nivsit Povo, t D. Cs s tugt mnil ngining t t Bigm Young nivsit sin 968. An dvot o omput tnolog, s svd s onsultnt to indust on numous pojts involving ngining sotw pplitions. H svd s viw o t Motool Si Sigm Pogm t its inption. H lso svd on n NSF slt pnl o vluting toln nlsis s nds. In 984, oundd t ADCATS onsotium o t dvlopmnt o CAD-sd tools o toln nlsis o mnil ssmlis. Mo tn sponsod gdut tss v n dvotd to t dvlopmnt o t toln tnolog ontind in t CATS sotw. Svl ult nd studnts untl involvd in od sptum o s pojts nd indust s studis on sttistil vition nlsis. Pst nd unt sponsos inlud Allid Signl, Boing, Cummins, FMC, Fod, GE, HP, Hugs, IBM, Motool, Sndi Ls, Ts Instumnts, nd t S Nv.. Intodution In tis pt, n ltntiv mtod to t on dsid in Cpt is psntd. Tis mtod is sd on vto loop ssml modls, ut wit t ollowing distint dins:. A st o uls is povidd to ssu vlid st o vto loops is otind. T loops inlud onl tos ontolld dimnsions tt ontiut to ssml vition. All dimnsions dtum nd.. A st o kinmti modling lmnts is intodud to ssist in idntiing t djustl dimnsions witin t ssml tt ng to ommodt dimnsionl vitions. -

2 - Cpt Titn. In ddition to dsiing vition in ssml gps, ompnsiv st o ssml toln uimnts is intodud, wi usul to dsigns s pomn uimnts. 4. Algi mnipultion to div n pliit pssion o t ssml tu is limintd. Tis mtod opts ull wll on impliit ssml utions. T loop utions solvd t sm w v tim, so it is wll suitd o omput utomtion. Tis pt distinguiss itsl om Cpt pling dintition o omplitd ssml pssion wit singl mti option, wi dtmins ll nss toln snsitivitis simultnousl. Sin t mti onl ontins sins nd osins, divtions simpl. As wit t mtod sown in Cpt, tis mtod m lso inlud ot sous o vition, su s position toln, plllism o, o poil vitions.. T Sous o Vition in Assmlis T t min sous o vition, wi must ountd o in mnil ssmlis:. Dimnsionl vitions (lngts nd ngls. Gomti om nd tu vitions (position, oundnss, ngulit, t.. Kinmti vitions (smll djustmnts twn mting pts Dimnsionl nd om vitions t sult o vitions in t mnutuing posss o w mtils usd in podution. Kinmti vitions ou t ssml tim, wnv smll djustmnts twn mting pts uid to ommodt dimnsionl o om vitions. T two-omponnt ssml sown in Figs. - nd - dmonstts t ltionsip twn dimnsionl nd om vitions in n ssml nd t smll kinmti djustmnts tt ou t ssml tim. T pts ssmld insting t lind into t goov until it mks ontt on t two sids o t goov. Fo st o pts, t distn will djust to ommodt t unt vlu o dimnsions A,, nd θ. T ssml sultnt psnts t nominl position o t lind, wil psnts t position o t lind wn t vitions A,, nd θ psnt. Tis djustilit o t ssml dsis kinmti onstint, o losu onstint on t ssml. A D A A D D A D Figu - Kinmti djustmnt du to omponnt dimnsion vitions Figu - Adjustmnt du to gomti sp vitions

3 Multi-Dimnsionl Toln Anlsis (Automtd Mtod - It is impotnt to distinguis twn omponnt nd ssml dimnsions in Fig. -. Ws A,, nd θ omponnt dimnsions, sujt to ndom poss vitions, distn is not omponnt dimnsion. It is sultnt ssml dimnsion. is not mnutuing poss vil, it is kinmti ssml vil. Vitions in n onl msud t t pts ssmld. A,, nd θ t indpndnt ndom sous o vition in tis ssml. T t inputs. is dpndnt ssml vil. It is t output. Fig. - illustts t sm ssml wit ggtd gomti tu vitions. Fo podution pts, t ontt sus not ll lt nd t lind is not ptl ound. T pttn o su wvinss will di om on pt to t nt. In tis ssml, t lind mks ontt on pk o t low ontt su, wil t nt ssml m mk ontt in vll. Simill, t low su is in ontt wit lo o t lind, wil t nt ssml m mk ontt twn los. Lol su vitions su s ts n popgt toug n ssml nd umult just s si vitions do. Tus, in omplt ssml modl ll t sous o vition must ountd o to ssu listi nd ut sults.. Empl -D Assml Stkd Bloks T ssml in Fig. - illustts t toln modling poss. It onsists o t pts: Blok, sting on Fm, is usd to position Clind, s sown. T ou dint mting su onditions tt must modld. T gp G, twn t top o t Clind nd t Fm, is t itil ssml tu w wis to ontol. Dimnsions toug,,, nd θ dimnsions o omponnt tus tt ontiut to ssml vition. Tolns stimts o t mnutuing poss vitions. Dimnsion g is utilit dimnsion usd in loting gp G. g G Dim Nominl. mm. Toln ±. mm ±. Clind Blok d ±. ±. ±. d g 75.. ±.5 ± Fm. 4. ±. ±. 7. dg ±. dg Figu - Stkd loks ssml

4 -4 Cpt Titn.4 Stps in Cting n Assml Toln Modl Stp. Ct n ssml gp An ssml gp is simpliid digm psnting n ssml. All gomt nd dimnsions movd. Onl t mting onditions twn t pts sown. E pt is sown s lloon. T Clind Gp Loop Loop Fm Loop Blok Figu -4 Assml gp o t stkd loks ssml ontts o joints twn mting pts sown s s o dgs joining t osponding pts. Fig. -4 sows t ssml gp o t smpl polm. T ssml gp lts ou s t ltionsip twn t pts in t ssml. It lso vls insption ow mn loops (dimnsion ins will uid to uild t toln modl. Loops nd losd loop ssml onstints, wi lot t Blok nd Clind ltiv to t Fm. Loop is n opn loop dsiing t ssml pomn uimnt. A sstmti podu o dining t loops is illusttd in t stps tt ollow. Smols v n ddd to dg idntiing t tp o ontt twn t mting sus. Btwn t Blok nd Fm t two ontts: pln-to-pln nd dg-to-pln. Ts lld Pln nd Edg Slid joints, sptivl, t ti kinmti ountpts. Onl si kinmti joint tps uid to dsi t mting pt ontts ouing in most -D ssmlis, s sown in Fig. -5. Aows indit t dgs o dom o joint, wi pmit ltiv motion twn t mting sus. Also sown two dtum sstms dsid in t nt stion. Pln Clind Slid Edg Slid volut Plll Clinds igid tngul Dtum Cnt Dtum Figu -5 -D kinmti joint nd dtum tps

5 Multi-Dimnsionl Toln Anlsis (Automtd Mtod -5 Stp. Lot t dtum n m o pt Cting t toln modl gins wit n ssml dwing, pl dwn to sl. Elmnts o t toln modl ddd to t ssml dwing s n ovl. T ist lmnts ddd st o lol oodint sstms, lld Dtum n Fms, o DFs. E pt must v its own DF. T DF is usd to lot tus on pt. You pol will oos t dtum plns usd to din t pts. But, l to pimnt. As ou pom t toln nlsis, ou m ind dint dimnsioning sm tt dus t num o vition sous o is lss snsitiv to vition. Idntiing su ts nd ommnding ppopit dsign ngs is on o t gols o toln nlsis. In Fig. -6, t Fm nd Blok ot v tngul DFs lotd t ti low lt ons, wit s ointd long otogonl sus. T Clind s lindil DF sstm t its nt. A sond nt dtum s n usd to lot t nt o t lg on t Blok. Tis is lld tu dtum nd it is usd to lot singl tu on pt. It psnts vitul point on t Blok nd must lotd ltiv to t Blok DF. Clind DF G Blok DF Fm DF Figu -6 Pt dtums nd ssml vils Also sown in Fig. -6 t ssml vils ouing witin tis ssml.,, nd djustl dimnsions dtmind t sliding ontts twn t pts.,, nd din t djustl ottions tt ou in spons to dimnsionl vitions. E o t djustl dimnsions is ssoitd wit kinmti joint. Dimnsion G is t gp wos vition must ontolld stting ppopit tolns on t omponnt dimnsions. Stp. Lot kinmti joints nd t dtum pts In Fig. -7, t ou kinmti joints in t ssml lotd t points o ontt nd ointd su tt t joint s lign wit t djustl ssml dimnsions (lld t joint dgs o dom. Tis is don insption o t ontt sus. T simpl modling uls o joint tp. Joint is n dg slid. It psnts n dg ontting pln su. It s two dgs o dom: it n slid long t ontt pln ( nd ott ltiv to t ontt point (. O ous, it is onstind not to slid o ott ontt wit mting pts, ut ng in dimnsions,,, d, o θ will us nd to djust odingl.

6 -6 Cpt Titn Joint DF Clind DF Blok Joint DF Fm Figu -7 Dtum pts o Joints nd Joint is pln joint dsiing sliding ontt twn two plns. lots n point on t ontting su ltiv to t Blok DF. is onstind t on o t Blok sting ginst t vtil wll o t Fm. In Fig. -8, Joint lots t ontt point twn t Clind nd t Fm. A lind slid s two dgs o dom: is in t sliding pln nd is msud t t nt dtum o t Clind. Joint 4 psnts ontt twn two plll linds. T point o ontt on t Clind is lotd ; on t Blok,. Joints nd 4 simill onstind. Howv, ngs in omponnt dimnsions us djustmnts in t points o ontt om on ssml to t nt. Joint Clind DF Joint 4 Blok DF DF Fm Figu -8 Dtum pts o Joints nd 4

7 Multi-Dimnsionl Toln Anlsis (Automtd Mtod -7 T vtos ovlid on Figs. -7 nd -8 lld t dtum pts. A dtum pt is in o dimnsions tt lots t point o ontt t joint wit spt to pt DF. Fo mpl, Joint in Fig. -7 joins t Blok to t Fm. T point o ontt must dind om ot t Fm nd Blok DFs. T two vto pts tt lv Joint. lis on t sliding pln nd points to t Blok DF. Vtos nd point to t Fm DF. T two dtum pts o Joint : vtos nd lding to t Fm DF, nd dius nd vto, lding to t Blok DF. In Fig. -8, Joint is lotd dius pointing to t Clind DF, nd nd dining t pt to t Fm DF. T ontt point o Joint 4 is lotd sond dius pointing to t Clind DF nd dius nd lding to t Blok DF. Modling uls din t pt vto loop must ollow to oss joint. Fig. -9 sows t ot vto pts o ossing ou -D joints. T ul stts tt t loop must nt nd it joint toug t lol joint dtums. Fo t Pln nd Edg Slid joints, vto (it inoming o outgoing must li in t sliding pln. Lol Dtum psnts n point on t sliding pln, om wi t ontt point is lotd. Fo t Clindil Slid joint, t inoming vto psss toug nt dtum o t lind, ollows dius vto to t ontt point nd lvs toug vto in t sliding pln. T pt toug t plll lind joint psss om t nt dtum o on lind to t nt dtum o t ot, pssing toug t ontt point nd two olin dii in twn. om Dtum om Dtum Dtum Dtum Edg Slid Pln Dtum Dtum Dtum Dtum Clindil Slid Plll Clinds Figu -9 -D vto pt toug t joint ontt point As w td t two dtum pts om joint, w w in t ting t inoming nd outgoing vtos o joint. Altoug t w ot dwn s outgoing vto pts, wn w omin tm to om t vto loops, on o t dtum pts will vsd in dition to ospond to t vto loop dition. E joint intodus kinmti vils into t ssml, wi must inludd in t vto modl. T uls ssu tt t kinmti vils intodud joint inludd in t loop, nml, t vto in sliding pln, nd t ltiv ngl.

8 -8 Cpt Titn E dtum pt must ollow ontolld ngining dimnsions o djustl ssml dimnsions. Tis is itil tsk, s it dtmins wi dimnsions will inludd in t toln nlsis. All joint dgs o dom must lso inludd in t dtum pts. T t unknown vitions in t ssml toln nlsis. Stp 4. Ct vto loops Vto loops din t ssml onstints tt lot t pts o t ssml ltiv to ot. T vtos psnt t dimnsions tt ontiut to toln stkup in t ssml. T vtos joind tip-to-til, oming in, pssing toug pt in t ssml in sussion. A vto loop must o tin modling uls s it psss toug pt. It must: Ent toug joint Follow t dtum pt to t DF Follow sond dtum pt lding to not joint, nd Eit to t nt djnt pt in t ssml Tis is illusttd smtill in Fig. -. Tus, vto loops td simpl linking togt t dtum pts. B so doing, ll t dimnsions will dtum nd. DF Pt Inoming Joint d Dtum Pts Outgoing Joint Figu - -D vto pt oss pt Additionl modling uls o vto loops inlud: Loops must pss toug v pt nd v joint in t ssml. A singl vto loop m not pss toug t sm pt o t sm joint twi, ut it m stt nd nd in t sm pt. I vto loop inluds t t sm dimnsion twi, in opposit ditions, t dimnsion is dundnt nd must omittd. T must noug loops to solv o ll o t kinmti vils (joint dgs o dom. You will nd on loop o o t t vils. Two losd loops uid o t mpl ssml, s w sw in t ssml gp o Fig. -4. T sulting loops sown in Figs. - nd -. Noti ow simil t loops to t dtum pts o Figs. -7 nd -8. Also, noti tt som o t vtos in t dtum pts w vsd to kp ll t vtos in loop going in t sm dition.

9 Multi-Dimnsionl Toln Anlsis (Automtd Mtod -9 Clind Joint DF DF Loop Blok Joint Fm DF Figu - Assml Loop Joint Clind DF Joint 4 Loop Blok DF Joint Fm DF Figu - Assml Loop

10 - Cpt Titn Stp 5. Add gomti vitions Gomti vitions o om, ointtion, nd lotion n intodu vition into n ssml. Su vitions n umult sttistill nd popgt kinmtill t sm s si vitions. T mnn in wi gomti vition popgts oss mting sus dpnds on t ntu o t ontt. Fig. - illustts tis onpt. Nominl il Tnsltionl vition Toln on Toln on ottionl vition Toln on Clind on pln su Blok on pln su Figu - Popgtion o -D tnsltionl nd ottionl vition du to su wvinss Consid lind on pln, ot o wi sujt to su wvinss, psntd toln on. As t two pts ougt togt to ssmld, t lind ould st on t top o ill o down in vll o su wv. Tus, o tis s, t nt o t lind will iit tnsltionl vition om ssml-to-ssml in dition noml to t su. Simill, t lind ould lod, s sown in t igu, sulting in n dditionl vtil tnsltion, dpnding on wt t pt sts on lo o in twn. In ontst to t lind/pln joint, t lok on pln sown in Fig. - iits ottionl vition. In t tm s, on on o t lok ould st on wvinss pk, wil t opposit on ould t t ottom o t vll. T mgnitud o ottion would v om ssml-tossml. Wvinss on t su o t lok would v simil t. In gnl, o two mting sus, w would v two indpndnt su vitions tt intodu vition into t ssml. How it popgts dpnds on t ntu o t ontt, tt is, t tp o kinmti joint. Wil t is littl o no pulisd dt on tpil su vitions o mnutuing posss, it is still instutiv to inst stimts o vitions nd lult t mgnitud o ti possil ontiution. Fig. -4 illustts svl stimtd gomti vitions ddd to t smpl ssml modl. Onl on vition is dind t joint, sin ot mting sus v t sm snsitivit. Emining t pnt ontiution to t gp vition will nl us to dtmin wi sus sould v GD&T toln ontol. Stp 6. Din pomn uimnts Pomn uimnts ngining dsign uimnts. T ppl to ssmlis o pts. In toln nlsis, t t spiid limits o vition o t ssml tus tt itil to podut pomn, somtims lld t k tistis o itil tu tolns. Svl mpls w illusttd in Cpt 9 o n lti moto ssml. Simpl its twn ing nd st, o ing nd ousing, would onl involv two pts, wil t dil nd il ln twn t mtu nd ousing would involv toln stkup o svl pts nd dimnsions.

11 Multi-Dimnsionl Toln Anlsis (Automtd Mtod -.4. A. A Clind DF Blok DF Fm -A- DF. Figu -4 Applid gomti vitions t ontt points Componnt tolns st s sult o nling toln stkup in n ssml nd dtmining ow omponnt dimnsion ontiuts to ssml vition. Posss nd tooling sltd to mt t uid omponnt tolns. Insption nd gging uipmnt nd podus lso dtmind t sulting omponnt tolns. Tus, w s tt t pomn uimnts v pvsiv inlun on t nti mnutuing ntpis. It is t dsign s tsk to tnsom pomn uimnt into ssml tolns nd osponding omponnt tolns. T svl ssml tus tt ommonl is in podut dsign. A il ompnsiv st n dvlopd mining gomti dimnsioning nd tolning tu ontols nd oming osponding st o ssmlis. Fig. -5 sows si st tt n ppl to wid ng o ssmlis. Not tt wn pplid to n ssml tu, plllism pplis to two sus on two dint pts, wil GD&T stndds onl ontol plllism twn two sus on t sm pt. T sm n sid out t ot ssml ontols, wit t ption o position. Position toln in GD&T lts ssmlis o two pts, wil t position toln in Fig. -5 ould involv wol in o intmdit pts ontiuting vition to t position o mting tus on t two nd pts. An mpl o t pplition o ssml toln ontols is t lignmnt uimnts in doo ssml. T gp twn t dg o t doo nd t doo m must uniom nd lus (plll in two plns. T doo stik must lin up wit t doo lok mnism (position. E ssml tu, su s gp o plllism, uis n opn loop to dsi t vition. You n v n num o opn loops in n ssml toln modl, on p itil tu. Closd loops, on t ot nd, limitd to t num o loops uid to lot ll o t pts in t ssml. It is uniu num dtmind t num o pts nd joints in t ssml. L J P w L is t uid num o loops, J is t num o joints, nd P is t num o pts. Fo t mpl polm: L 4 wi is t num w dtmind insption o t ssml gp.

12 - Cpt Titn Assml Lngt u ± du Ppndiulit & Angulit A Assml Gp u±du d Conntiit & unout A Assml Angl A d -A- -A- -A- Position A B Plllism Pt A Pt T mpl ssml s spiid gp toln twn lindil su nd pln, s sown in Fig. -6. T vto loop dsiing t gp is sown in Fig. -6. It gins wit vto g, on on sid o t gp, pods om pt-to-pt, nd nds t t top o t lind, on t opposit sid o t gp. Not tt vto, t t DF o t Fm, pps twi in t sm loop in opposit ditions. It is to dundnt nd ot vtos must limintd. Vto lso pps twi in t lind; owv, t two vtos not in opposit ditions, so t must ot inludd in t loop. Vto g, inidntll, is not mnutud dimnsion. It is ll kinmti vil, wi djusts to lot t point on t gp opposit t igst point on t lind. It ws givn o toln, us it dos not ontiut to t vition o t gp. T stps illusttd ov dsi ompnsiv sstm o ting ssml modls o toln nlsis. Wit just w si lmnts, wid vit o ssmlis m psntd. Nt, w will illustt t stps in poming vitionl nlsis o n ssml modl..5 Stps in Anling n Assml Toln Modl Figu -5 Assml toln ontols In -D o -D ssml, omponnt dimnsions n ontiut to ssml vition in mo tn on dition. T mgnitud o t omponnt ontiutions to t vition in itil ssml tu is dtmind t podut o t poss vition nd t toln snsitivit, summd wost s

13 Multi-Dimnsionl Toln Anlsis (Automtd Mtod - Loop g Gp Clind DF Blok DF DF Fm Figu -6 Opn loop dsiing itil ssml gp o oot Sum Sud (SS. I t ssml is in podution, tul poss pilit dt m usd to pdit ssml vition. I podution s not t gun, t poss vition is ppoimtd sustituting t spiid tolns o t dimnsions, s dsid li. T toln snsitivitis m otind numill om n pliit ssml untion, s illusttd in Cpt. An ltntiv podu will dmonsttd, wi dos not ui t divtion o n pliit ssml untion. It is sstmti mtod, wi m pplid to n vto loop ssml modl. Stp. Gnt ssml utions om vto loops T ist stp in n nlsis is to gnt t ssml utions om t vto loops. T sl utions dsi losd vto loop. T divd summing t vto omponnts in t nd ditions, nd summing t vto ottions s ou t t loop. Fo losd loops, t omponnts sum to o. Fo opn, t sum to nono gp o ngl. T utions dsiing t stkd lok ssml sown low. Fo Closd Loops nd,,, nd θ t sums o t,, nd ottion omponnts, sptivl. S Es. (. nd (.. Bot loops stt t t low lt on, wit vto. Fo Opn Loop, onl on sl ution (E. (.6 is ndd, sin t gp s onl vtil omponnt. Opn loops stt t on sid o t gp nd nd t t opposit sid. Closd Loop os( os(9 os(9 os(9 8 os(θ os(9 os(8 sin( sin(9 sin(9 sin(9 8 sin(θ (. sin(9 sin(8 θ θ 9 9 8

14 -4 Cpt Titn Closd Loop os( os(9 os( os( os( 8 os( os(θ os( 9 os( 8 sin( sin(9 sin( sin( sin( 8 sin( sin(θ sin( 9 sin( 8 (. θ θ Opn Loop Gp sin( 9 sin(8 sin( 9 sin(9 g sin( (. T loop utions lt t ssml vils:,,,,,, nd Gp to t omponnt dimnsions:,,,,, g,,, nd θ. W onnd wit t t o smll ngs in t omponnt vils on t vition in t ssml vils. Not t uniomit o t utions. All omponnts in tms o t osin o t ngl t vto mks wit t -is. All in tms o t sin. In t, just pl t osins in t ution wit sins to gt t ution. T loop utions lws v tis om. Tis mks t utions v s to div. In CAD implmnttion, ution gntion m utomtd. T θ utions t sum o ltiv ottions om on vto to t nt s ou pod ound t loop. Countlokwis ottions positiv. Fig. -7 ts t ltiv ottions o Loop. A inl ottion o 8 is ddd to ing t ottions to losu. Wil t gumnts o t sins nd osins in t nd utions psnt t solut ngl om t -is, t ngls pssd s t sum o ltiv ottions up to tt point in t loop. sing ltiv ottions is itil to t ot ssml modl vio. It llows ottionl vitions to popgt otl toug t ssml. -8 ltiv ottions θ θ Loop is -9 Figu -7 ltiv ottions o Loop

15 Multi-Dimnsionl Toln Anlsis (Automtd Mtod -5 A sotut ws usd o t gumnts o vtos,, nd. T sum o ltiv ottions ws pld wit ti known solut ditions. T sum o ltiv ngls o is ( θ θ 9, ut it must lign wit t ngld pln o t m (θ. Simill, vtos nd will lws vtil nd oiontl, sptivl, gdlss o t pding ottionl vitions in t loop. pling t ngls o, C, nd is uivlnt to solving t θ ution o θ nd sustituting in t gumnts to limint som o t ngl vils. I ou t it ot ws, ou will s tt ou gt t sm sults o t pditd vitions. T sults lso indpndnt o t stting point o t loop. W ould v sttd wit n vto in t loop. Stp. Clult divtivs nd om mti utions T loop utions nonlin nd impliit. T ontin poduts nd tigonomti untions o t vils. To solv o t ssml vils in tis sstm o utions would ui nonlin ution solv. Fotuntl, w onl intstd in t ng in ssml vils o smll ngs in t omponnts. Tis is dil omplisd liniing t utions ist-od Tlo s sis pnsion. E. (.4 sows t linid utions o Loop. θ θ θ θ θ θ (.4 w psnts smll ng in dimnsion, nd so on. Not tt t tms v n ngd, gouping t omponnt vils,,,,,, nd θ togt nd ssml vils,,,,, nd togt. T Loop nd Loop utions m pssd simill. Poming t ptil dintition o t sptiv,, nd θ utions ilds t oiints o t lin sstm o utions. T ptils s to pom us t onl sins nd osins to dl wit. E. (.5 sows t ptils o t Loop ution.

16 -6 Cpt Titn Componnt Vils Assml Vils os( os( 8 os( 9 os( 7 os( 9 sin( sin( 9 sin( 7 os( 9 os( (.5 E ptil is vlutd t t nominl vlu o ll dimnsions. T nominl omponnt dimnsions known om t ngining dwings o CAD modl. T nominl ssml vlus m otind uing t CAD modl. T ptil divtivs ov not t toln snsitivitis w sk, ut t n usd to otin tm. Stp. Solv o ssml toln snsitivitis T linid loop utions m wittn in mti om nd solvd o t toln snsitivitis mti lg. T si losd loop sl utions n pssd in mti om s ollows: [A]{X} [B]{} {} w: [A] is t mti o ptil divtivs wit spt to t omponnt vils, [B] is t mti o ptil divtivs wit spt to t ssml vils, {X} is t vto o smll vitions in t omponnt dimnsions, nd {} is t vto o osponding losd loop ssml vitions. W n solv o t losd loop ssml vitions in tms o t omponnt vitions mti lg: {} [B A]{X} (.6 T mti [B - A] is t mti o toln snsitivitis o t losd loop ssml vils. Poming t invs o t mti [B] nd multipling [B - A] m id out using spdst o ot mt utilit pogm on dsktop omput o pogmml lulto.

17 Multi-Dimnsionl Toln Anlsis (Automtd Mtod -7 Fo t mpl ssml, t sulting mtis nd vtos o t losd loop solution : { } θ X { } [ ] A os( 8 sin( sin( sin( sin( 8 os( os( os( os( sin(9 sin(7 sin( os(9 os(7 θ θ θ θ

18 -8 Cpt Titn [ ] B ( ( ( ( ( ( ( ( ( ( os os 8 os os sin( sin(9 sin sin 8 sin sin os( os(9 sin( sin(9 os(7 9 os sin(7 9 sin os( os(9 θ θ θ θ [ ] B

19 Multi-Dimnsionl Toln Anlsis (Automtd Mtod -9 {} -[B - A]{X} ( θ Estimts o vition o t ssml pomn uimnts otind liniing t opn loop utions podu simil to t losd loop utions. In gnl, t will sstm o nonlin sl utions wi m linid Tlo s sis pnsion. Gouping tms s o, w n pss t linid utions in mti om: {V} [C]{X} [E]{} (.8 w {V} is t vto o vitions in t ssml pomn uimnts, [C] is t mti o ptil divtivs wit spt to t omponnt vils, [E] is t mti o ptil divtivs wit spt to t ssml vils, {X} is t vto o smll vitions in t omponnt dimnsions, nd {} is t vto o osponding losd loop ssml vitions. W n solv o t opn loop ssml vitions in tms o t omponnt vitions mti lg, sustituting t sults o t losd loop solution. Sustituting o {}: {V} [C]{X} [E][B A]{X} [CΕ B A]{X} T mti [CE B - A] is t mti o toln snsitivitis o t opn loop ssml vils. T B - A tms om om t losd loop onstints on t ssml. T B - A tms psnt t t o smll intnl kinmti djustmnts ouing t ssml tim in spons to dimnsionl vitions. T intnl djustmnts t t {V} s wll s t {}. It is impotnt to not tt ou nnot simpl solv o t vlus o {} in E. (.6 nd sustitut tm ditl into E. (.8, s toug {} w just not omponnt vition. I ou do, ou tting {} s toug it is indpndnt o {X}. But {} dpnds on {X} toug t losd loop onstints. You must vlut t ull mti [CE B - A] to otin t toln snsitivitis. Allowing t B - A tms to intt wit C nd E is nss to dtmin t t o t kinmti djustmnts on {V}. Tting tm sptl is simil to tking t solut vlu o tm, tn summing o Wost Cs, t tn summing lik tms o tking t solut vlu. T sm is tu o SS nlsis. It is simil to suing tm, tn summing, t tn summing lik tms o suing. Fo t mpl ssml, t ution o {V} dus to singl sl ution o t Gp vil.

20 - Cpt Titn Gp Gp Gp Gp Gp Gp Gp g g Gp Gp Gp Gp Gp Gp θ θ Gp Gp Gp Gp [sin(9sin(8] sin(9 sin( g sin(9 Sustituting o om t losd loop sults (E. (.7 nd gouping tms: Gp ( θ ( θ Wil E. (.9 psss t ssml vition Gp in tms o t omponnt vitions X, it is not n stimt o t toln umultion. To stimt umultion, ou must us modl, su s Wost Cs o oot Sum Sus. Stp 4. Fom Wost Cs nd SS pssions As s n sown li, stimts o toln umultion o o V m lultd summing t poduts o t toln snsitivitis nd omponnt vitions: Wost Cs SS o V Σ Sij j o V ( S ij j S ij is t toln snsitivitis o ssml tus to omponnt vitions. I t ssml vil o intst is losd loop vil i, S ij is otind om t ppopit ow o t B - A mti. I V i is wntd, S ij oms om t [C-E B - A] mti. I msud vition dt vill, j is t ±σ poss vition. I podution o pts s not gun, j is usull tkn to ul to t ±σ dsign tolns on t omponnts. In t mpl ssml, lngt is losd loop ssml vil. dtmins t lotion o t ontt point twn t Clind nd t Fm. To stimt t vition in, w would multipl t ist ow o [B - A] wit {X} nd sum Wost Cs o SS. Wost Cs: S S S S 4 S 5 S 6 S 7 θ ±.69 mm

21 Multi-Dimnsionl Toln Anlsis (Automtd Mtod - SS: [(S (S (S (S 4 (S 5 (S 6 (S 7 θ ].5 [(.57. (.57. (. (.457. ( (.. ( ]. 5 ±.665 mm Not tt t toln on θ s n onvtd to ±.745 dins sin t snsitivit is lultd p din. Fo t vition in t Gp, w would multipl t ist ow o [C-EB - A] wit {X} nd sum Wost Cs o SS. Vto {X} is tndd to inlud nd g. Wost Cs: Gp S S S S 4 S 5 S 6 S 7 θ S 8 S 9 g ±.9 mm SS: Gp [(S (S (S (S 4 (S 5 (S 6 (S 7 θ (S 8 (S 9 g ]. 5 [(.57. (.57. (. ( ( (.. ( (.5 ( ].5 ±.8675 mm B oming simil pssions, w m otin stimts o ll t ssml vils (Tl -. Tl - Estimtd vition in opn nd losd loop ssml tus Assml Mn o WC SS Vil Nominl d d 59.6 mm.69 mm.665 mm mm.589 mm.644 mm 6.79 mm.9855 mm.494 mm Gp mm.9 mm.8675 mm Stp 5. Evlution nd dsign ittion T sults o t vition nlsis vlutd omping t pditd vition wit t spiid dsign uimnt. I t vition is gt o lss tn t spiid ssml toln, t pssions n usd to lp did wi tolns to tigtn o loosn Pnt jts T pnt jts m stimtd om Stndd Noml tls lulting t num o stndd dvitions om t mn to t upp nd low limits (L nd LL.

22 - Cpt Titn T onl ssml tu wit pomn uimnt is t Gp. T ptl ng o pop pomn is: Gp 6. ±. mm. Clulting t distn om t mn Gp to L nd LL in units ul to t stndd dvition o t Gp: L µ Gp ZL.467σ L 6 ppm σ.89 Z LL Gp LL µ σ Gp Gp σ T totl pditd jts 544 ppm Pnt Contiution Cts LL 8 ppm T pnt ontiution t tlls t dsign ow dimnsion ontiuts to t totl Gp vition. T ontiution inluds t t o ot t snsitivit nd t toln. T lultion is dint o Wost Cs o SS vition stimts. Wost Cs SS Gp j j % Cont Gp i i Gp j j % Cont Gp i i It is ommon pti to psnt t sults s t, sotd oding to mgnitud. T sults o t smpl ssml sown in Fig % Contiution Figu -8 Pnt ontiution t o t smpl ssml

23 Multi-Dimnsionl Toln Anlsis (Automtd Mtod - It is l tt t outsid dimnsion o t Gp,, is t pinipl ontiuto, ollowd t dius. Tis plot sows t dsign w to ous dsign modiition ots. Simpl nging t tolns on w dimnsions n ng t t dmtill. Suppos w tigtn t toln on, sin it is ltivl s to ontol, nd loosn t tolns on nd, sin t mo diiult to lot nd min wit pision. W will s t Clind is vndo-supplid, so it nnot modiid. Tl - sows t nw tolns. Tl - Modiid dimnsionl toln spiitions Dimnsion ±Toln Oiginl Modiid. mm. mm. mm. mm. mm. mm. mm.4 mm. mm. mm. mm.4 mm θ...5 mm.4 mm Now, nd t lding ontiutos, wil s doppd to tid. O ous, nging t tolns uis modiition o t posss. S Fig. -9. Tigtning t toln on, o mpl, migt ui nging t d o spd o num o inis psss on mill. Sin it is t podut o t snsitivit tims t toln tt dtmins t pnt ontiution, t snsitivit is lso n impotnt vition vlution id % Contiution Figu -9 Pnt ontiution t o t smpl ssml wit modiid tolns

24 -4 Cpt Titn.5.5. Snsitivit Anlsis T toln snsitivitis tll ow t ngmnt o t pts nd t gomt ontiut to ssml vition. W n ln gt dl out t ol pld dimnsion mining t snsitivitis. Fo t smpl ssml, Tl - sows t lultd Gp snsitivitis. Tl - Clultd snsitivitis o t Gp Dimnsion Snsitivit θ Not tt t snsitivit o θ is lultd p din. Fo. mm ng in o, t Gp will ng.57 mm. T ngtiv sign o mns t Gp will ds s inss. Fo mm ins in, t Gp dss n ul mount. Tis vio oms l on mining Fig. -. As inss. mm, t Blok is pusd up t inlind pln, ising t Blok nd Clind t tn(7 o.57 nd dsing t Gp. As inss. mm, t pln is pusd out om und t Blok, using it to low t sm mount. Insing. mm, uss vting to slid stigt up, dsing t Gp. Dimnsions,,, nd θ mo ompl us svl djustmnts ou simultnousl. As inss, t Clind gows, using it to slid up t wll, wil mintining ontt wit t onv su o t Blok. As t Clind iss, t Gp dss. As inss, t onv su movs dp into t lok, using t Clind to dop, wi inss t Gp. Insing uss t Blok to tikn, oing t ont on up t wll nd pusing t Blok up t pln. T nt t is to is t onv su, dsing t Gp. Insing θ uss t Blok to ott out t ont dg o t inlind pln, wil t ont on slids down t wll. T wdg ngl twn t onv su nd t wll dss, suing t Clind upwd nd dsing t Gp. T lg snsitivitis o nd θ ost ti smll osponding tolns Modiing Gomt T most ommon gomt modiition is to ng t nominl vlus o on o mo dimnsions to nt t nominl vlu o gp twn its L nd LL. Fo mpl, i w wntd to ng t Gp spiitions to 5. ±. mm, w ould simpl ins t nominl vlu o. mm. Sin t snsitivit o t Gp to is., t Gp will ds. mm. Simill, t snsitivitis m modiid nging t gomt. Sin t snsitivitis ptil divtivs, wi vlutd t t nominl vlus o t omponnt dimnsions, t n onl ngd nging t nominl vlus. An intsting is is to modi t gomt o t mpl ssml to mk t Gp insnsitiv to vition in θ ; tt is, to mk t snsitivit o θ go to o. You will nd nonlin ution solv sotw to solv t oiginl loop utions (Es. (-4, (-5, nd (-6, o nw st o nominl ssml vlus. Solv o t kinmti ssml vils:,,,,, nd, osponding to ou nw nominl dimnsions:,,,,,, θ,, nd Gp.

25 Multi-Dimnsionl Toln Anlsis (Automtd Mtod -5 T snsitivit o θ will ds to nl o i w ins to vlu o 4 mm. W must lso ins to 5 mm to du t nominl Gp k to 6. mm. T [A], [B], [C], nd [E] mtis will ll nd to -vlutd nd solvd o t vitions. T modiid sults sown in Tl -4. Tl -4 Clultd snsitivitis o t Gp t modiing gomt Dimnsion Nominl ±Toln Snsitivit mm mm mm mm mm mm.4. θ mm.4. Noti tt t onl snsitivit to ng ws θ (p din. Tis is du to t lk o oupling o nd wit t ot vils. T lultd vitions sown in Tl -5. Tl -5 Vition sults o modiid nominl gomt Assml Mn o WC SS Vil Nominl ± ± mm.6497 mm.7659 mm 4.55 mm.988 mm.84 mm mm.999 mm.498 mm Gp mm.497 mm.898 mm T nw pnt ontiution t is sown in Fig. -. Bsd on t low snsitivit, ou ould now ins t toln on θ witout ting t Gp vition. Stp 6. pot sults nd doumnt ngs T inl stp in t ssml toln nlsis podu is to pp t inl pot. Figus, gps, nd tls pd. Compison tls nd gps will lp to justi dsign disions. I ou v svl ittions, it is wis to dopt s numing sm to idnti tl nd gp wit its osponding s. A list o s nums wit onis summ o t distinguising tu o would ppitd t d.

26 -6 Cpt Titn % Contiution Figu - Modiid gomt ilds o θ ontiution.6 Summ T pding stions v psntd sstmti podu o modling nd nling ssml vition. Som o t dvntgs o t modling sstm inlud: T t min sous o vition m inludd: dimnsions; gomti om, lotion, nd ointtion; nd kinmti djustmnts. Assml modls onstutd o vtos nd kinmti joints, lmnts wit wi most dsigns mili. A vit o ssml onigutions m psntd wit w si lmnts. Modling uls guid t dsign nd ssist in t tion o vlid modls. It n utomtd nd intgtd wit CAD sstm to iv ull gpil modl tion. Advntgs o t nlsis sstm inlud: T ssml untions dil divd om t gpil modl. Nonlin, impliit sstms o utions dil onvtd to lin sstm. Toln snsitivitis dtmind singl, stndd, mti lg option. Sttistil lgoitms stimt toln stkup utl nd iintl witout uiing ptd simultions. On pssions o t vition in ssml tus v n divd, t m usd o toln llotion o wt-i studis witout pting t ssml nlsis. Vition pmts usul o vlution nd dsign sil otind, su s: t mn nd stndd dvition o itil ssml tus, snsitivit nd pnt ontiution o omponnt dimnsion nd gomti om vition, pnt jts, nd ulit lvl. Toln nlsis modls omin dsign uimnts wit poss pilitis to ost opn ommunition twn dsign nd mnutuing nd sond, untittiv disions. It n utomtd to totll limint mnul divtion o utions o ution tping.

27 Multi-Dimnsionl Toln Anlsis (Automtd Mtod -7 A CAD-sd toln nlsis sstm sd on t podus dmonsttd pviousl s n dvlopd. T si ognition o t Comput-Aidd Tolning Sstm (CATS is sown smtill in Fig. -. T sstm s n intgtd wit ommil -D CAD sstm, so it looks nd ls lik t dsign s own sstm. Mn o t mnul tsks o modling nd nlsis dsid ov v n onvtd to gpil untions o utomtd. -D CAD Sstm CATS Applition Int CATS Modl CATS Anl CAD Dts Mg Poss Dts Figu - T CATS Sstm Toln nlsis s om mtu ngining dsign tool. It is untittiv tool o onunt ngining. Powul sttistil lgoitms v n omind wit gpil modling nd vlution ids to ssist dsigns inging mnutuing onsidtions into ti dsign disions. Poss sltion, tooling, nd insption uimnts m dtmind l in t podut dvlopmnt l. Poming toln nlsis on t CAD modl ts vitul pototp o idntiing vition polms o pts podud. Dsigns n mu mo tiv dsigning ssmlis tt wok in spit o mnutuing poss vitions. Costl dsign ngs to ommodt mnutuing n dud. Podut ulit nd ustom stistion n insd. Toln nlsis ould om k to in mintining omptitivnss in tod s intntionl mkts..7 ns. C, Cls D. 99. A Compnsiv Mtod o Spiing Toln uimnts o Assmlis. Mst s tsis. Bigm Young nivsit.. Cs, K. W. nd A.. Pkinson. 99. A Suv o s in t Applition o Toln Anlsis to t Dsign o Mnil Assmlis. s in Engining Dsign. (: -7.. Cs, K. W. nd Angl Tgo AutoCATS Comput-Aidd Tolning Sstm - Modl s Guid. ADCATS pot, Bigm Young nivsit. 4. Cs, K. W., J. Go nd S. P. Mgl Gnl -D Toln Anlsis o Mnil Assmlis wit Smll Kinmti Adjustmnts. Jounl o Dsign nd Mnutuing. 5(4: Cs, K. W., J. Go nd S. P. Mgl Toln Anlsis o -D nd -D Mnil Assmlis wit Smll Kinmti Adjustmnts. In Advnd Tolning Tnius. pp. -7. Nw Yok: Jon Wil. 6. Cs, K. W., J. Go, S. P. Mgl nd C. D. Sonson Inluding Gomti Ftu Vitions in Toln Anlsis o Mnil Assmlis. IIE Tnstions. 8(: Fotini, E.T Dimnsioning o Intngl Mnutu. Nw Yok, Nw Yok: Industil Pss. 8. T Amin Soit o Mnil Engins ASME Y4.5M-994, Dimnsioning nd Tolning. Nw Yok, Nw Yok: T Amin Soit o Mnil Engins.