Traditional Approaches to Analyzing Mechanical Tolerance Stacks

Transcription

1 P A R T 3 DESIGN

2 Chapte 9 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks Paul Dake 9. Itoducto Toleace aalyss s the pocess o takg kow toleaces ad aalyzg the combato o these toleaces at a assembly level. Ths chapte wll dee the pocess o aalyzg toleace stacks. It wll show how to set up a loop dagam to deteme a omal peomace/assembly value ad ou techques to calculate vaato om omal. The most mpotat goal o ths chapte s o the eade to udestad the assumptos ad sks that go alog wth each toleace aalyss method. 9. Aalyzg Toleace Stacks Fg. 9- descbes the toleace aalyss pocess. 9.. Establshg Peomace/Assembly Requemets The st step the pocess s to dety the equemets o the system. These ae usually equemets that deteme the peomace ad/o assembly o the system. The system equemets wll, ethe dectly, o dectly, low dow equemets to the mechacal subassembles. These equemets usually deteme what eeds to be aalyzed. I geeal, a equemet that apples o most mechacal subassembles s that pats must t togethe. Fg. 9- shows a coss secto o a moto assembly. I ths eample, thee ae seveal equemets. Requemet. The gap betwee the shat ad the e beag cap must always be geate tha zeo to esue that the oto s clamped ad the beags ae peloaded. Requemet. The gap betwee the housg cap ad the housg must always be geate tha zeo to esue that the stato s clamped. 9-

3 9- Chapte Ne. Establsh the Peomace Requemets. Daw a Loop Dagam 3. Covet All Dmesos to Mea Dmeso wth a Equal Blateal Toleace 4. Calculate the Mea Value o the Peomace Requemet 5. Deteme the Method o Aalyss 6. Calculate the Vaato o the Peomace Requemet Fgue 9- Toleace aalyss pocess Requemet 3. The moutg suaces o the oto ad stato must be wth ±.005 o the moto to opeate. Requemet 4. The beag oute ace must always potude beyod the ma housg, so that the beag stays clamped. Requemet 5. The thead o the beag cap scew must have a mmum thead egagemet o.00 ches. Requemet 6. The bottom o the beag cap scew thead must eve touch the bottom o the emale thead o the shat. Requemet 7. The oto ad stato must eve touch. The mamum adal dstace betwee the oto ad stato s.00. Othe eamples o peomace/assembly equemets ae: Themal equemets, such as cotact betwee a themal plae ad a heat sk, Amout o squeeze o a o-g Amout o peload o beags Sucet mateal o subsequet machg pocesses Aeodyamc equemets Iteeece equemets, such as whe pessg ps to holes Stuctual equemets Optcal equemets, such as algmet o optcal elemets The secod pat o Step s to covet each equemet to a assembly gap equemet. We would covet each o the pevous equemets to the ollowg. Requemet. Gap 0 Requemet. Gap 0

4 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks 9-3 Requemet 3. Gap Requemet 4. Gap 4 0 Requemet 5. Gap 5.00 Requemet 6. Gap 6 0 Requemet 7. Gap 7 0 ad Loop Dagam Fgue 9- Moto assembly The loop dagam s a gaphcal epesetato o each aalyss. Each equemet eques a sepaate loop dagam. Smple loop dagams ae usually hozotal o vetcal. Fo smple aalyses, vetcal loop dagams wll gaphcally epeset the dmesoal cotbutos o vetcal gaps. Lkewse, hozotal

5 9-4 Chapte Ne loop dagams gaphcally epeset dmesoal cotbutos o hozotal gaps. The steps o dawg the loop dagam ollow.. Fo hozotal dmeso loops, stat at the suace o the let o the gap. Follow a complete dmeso loop, to the suace o the ght. Fo vetcal dmeso loops, stat at the suace o the bottom o the gap. Follow a complete dmeso loop, to the suace o the top.. Usg vectos, ceate a closed loop dagam om the statg suace to the edg suace. Do ot clude gaps whe selectg the path o the dmeso loop. Each vecto the loop dagam epesets a dmeso. 3. Use a aow to show the decto o each vecto the dmeso loop. Idety each vecto as postve (), o egatve ( ), usg the ollowg coveto. Fo hozotal dmesos: Use a sg o dmesos ollowed om let to ght. Use a sg o dmesos ollowed om ght to let. Fo vetcal dmesos: Use a sg o dmesos ollowed om bottom to top. Use a sg o dmesos ollowed om top to bottom. 4. Assg a vaable ame to each dmeso the loop. (Fo eample, the st dmeso s assged the vaable ame A, the secod, B.) Fg. 9-3 shows a hozotal loop dagam o Requemet 6. Fgue 9-3 Hozotal loop dagam o Requemet 6 5. Recod sestvtes o each dmeso. The magtude o the sestvty s the value that the gap chages, whe the dmeso chages ut. Fo eample, the gap chages.00 whe the dmeso chages.00, the the magtude o the sestvty s (.00/.00). O the othe had, the gap chages.0005 o a.00 chage the dmeso, the the sestvty s.5 (.0005/.00). I the dmeso vecto s postve (potg to the ght o hozotal loops, o up o vetcal loops), ete a postve sestvty. I a dmeso wth a postve sestvty ceases, the gap wll also cease. I the vecto s egatve (potg to the let o hozotal loops, o dow o vetcal loops), ete a egatve sestvty. I a dmeso wth a egatve sestvty ceases, the gap wll decease. Note, Fg. 9-3, all o the sestvtes ae equal to ±.

6 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks Deteme whethe each dmeso s ed o vaable. A ed dmeso s oe whch we have o cotol, such as a vedo pat dmeso. A vaable dmeso s oe that we ca chage to luece the outcome o the toleace stack. (Ths wll become mpotat late, because we wll be able to adjust o esze the vaable dmesos ad toleaces to acheve a desed assembly peomace. We ae ot able to esze ed dmesos o toleaces.) 9..3 Covetg Dmesos to Equal Blateal Toleaces I Fg. 9-, thee wee seveal dmesos that wee toleaced usg ulateal toleaces (such as /-.03, /-.000 ad /-.05) o uequal blateal toleaces (such as /-.004 ). I we look at the legth o the shat, we see that thee ae seveal deet ways we could have appled the toleaces. Fg. 9-4 shows seveal ways we ca dmeso ad toleace the legth o the shat to acheve the same uppe ad lowe toleace lmts (3.03/3.09). Fom a desg pespectve, all o these methods peom the same ucto. They gve a bouday wth whch the dmeso s acceptable. Fgue 9-4 Methods to dmeso the legth o a shat The desge mght thk that chagg the omal dmeso has a eect o the assembly. Fo eample, a desge may dmeso the pat legth as / I dog so, the desge may alsely thk that ths wll help mmze the gap o Requemet. A dawg, howeve, does t gve peeece to ay dmeso wth the toleace age. Fg. 9-5 shows what happes to the mauactug yeld the mauactue ams o the dmeso stated o the dawg ad the pocess ollows the omal dstbuto. I ths eample, the mauactue amed o 3.09, hal o the pats would be outsde o the toleace zoe. Sce mauactug shops wat to mamze the yeld o each dmeso, they wll am o the omal that yelds the lagest umbe o good pats. Ths helps them mmze the costs. I ths eample, the mauactue would am o Ths allows them the hghest pobablty o makg good pats. I they amed o 3.09 o 3.03, hal o the mauactued pats would be outsde the toleace lmts. As the pevous eample, may mauactug pocesses ae omally dstbuted. Theeoe, we put ay ulateal, o uequal blateal toleaces o dmesos, the mauactue would covet them to a mea dmeso wth a equal blateal toleace. The steps o covetg to a equal blateal toleace ollow.

7 9-6 Chapte Ne Fgue 9-5 Methods o ceteg mauactug pocesses. Covet the dmeso wth toleaces to a uppe lmt ad a lowe lmt. (Fo eample, / has a uppe lmt o 3.03 ad a lowe lmt o 3.09.). Subtact the lowe lmt om the uppe lmt to get the total toleace bad. ( ) 3. Dvde the toleace bad by two to get a equal blateal toleace. (.0/.006) 4. Add the equal blateal toleace to the lowe lmt to get the mea dmeso. ( ). Alteately, you could subtact the equal blateal toleace om the uppe lmt. ( ) As a ule, desges should use equal blateal toleaces. Sometmes, usg equal blateal toleaces may oce mauactug to use ostadad tools. I these cases, we should ot use equal blateal toleaces. Fo eample, we would ot wat to covet a dlled hole damete om.5.005/-.00 to.7 ±.003. I ths case, we wat the mauactue to use a stadad.5 dll. I the mauactue sees.7 o a dawg, he may thk he eeds to buld a specal tool. I the case o dlled holes, we would also wat to use a uequal blateal toleace because the mea o the dllg pocess s usually lage tha the stadad dll sze. These dmesos should have a lage plus toleace tha mus toleace. As we wll see late, whe we covet dmesos to equal blateal toleaces, we do t eed to keep tack o whch toleaces ae postve ad whch toleaces ae egatve because the postve toleaces ae equal to the egatve toleaces. Ths makes the aalyss ease. Table 9- covets the ecessay dmesos ad toleaces to mea dmesos wth equal blateal toleaces.

8 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks Calculatg the Mea Value (Gap) o the Requemet The st step calculatg the vaato at the gap s to calculate the mea value o the equemet. The mea value at the gap s: d g whee a d Table 9- Covetg to mea dmesos wth equal blateal toleaces Ogal Dmeso/Toleace Mea Dmeso wth Equal Blateal Toleace / / / / / / / /-.006 dg the mea value at the gap. I dg s postve, the mea gap has cleaace, ad dg s egatve, the mea gap has teeece. the umbe o depedet vaables (dmesos) the stackup a sestvty acto that dees the decto ad magtude o the th dmeso. I a oedmesoal stackup, ths value s usually o. Sometmes, a oe-dmesoal stackup, ths value may be.5 o -.5 a adus s the cotbutg acto o a damete callout o a dawg. d the mea value o the th dmeso the loop dagam. Table 9- shows the dmesos that ae mpotat to deteme the mea gap o Requemet 6. We have assged Vaable Name to each dmeso so that we ca wte a loop equato. We have also added Table 9- Dmesos ad toleaces used Requemet 6 /- Equal Vaable Mea Fed/ Blateal Descpto Name Dmeso Sestvty Vaable Toleace Scew thead legth A Fed.055 Washe legth B.030 Fed.000 Ie beag cap C.0600 Vaable.0030 tued legth Beag legth D.4305 Fed.0075 Space tued legth E.00 Vaable.0050 Roto legth F.5030 Fed.0070 Space tued legth G.00 Vaable.0050 Beag legth H.4305 Fed.0075 Pulley castg legth I.4500 Vaable.0070 Shat tued legth J Vaable.0060 Tapped hole depth K.3000 Vaable.0300 (9.)

9 9-8 Chapte Ne a colum ttled Fed/Vaable. Ths detes whch dmesos ad toleaces ae ed the aalyss, ad whch oes ae allowed to vay (vaable). Typcally, we have o cotol ove vedo tems, so we teat these dmesos as ed. As we make adjustmets to dmesos ad toleaces, we wll oly chage the vaable dmesos ad toleaces. The mea o Gap 6 s: Gap 6 a d a d a 3 d 3 a 4 d 4 a 5 d 5 a 6 d 6 a 7 d 7 a 8 d 8 a 9 d 9 a 0 d 0 a d Gap 6 (-)A ()B ()C ()D ()E ()F ()G ()H()I ( )J ()K Gap 6 (-).3595().030().0600().4305().00().5030().00 ().4305().4500(-)3.050().0300 Gap Deteme the Method o Aalyss Eq. (9.) oly calculates the omal value o the gap. The et step s to aalyze the vaato at the gap. Hstocally, mechacal egees have used two types o toleacg models to aalyze these vaatos: ) a wost case (WC) model, ad ) a statstcal model. Each appoach oes tadeos betwee pecepat toleaces ad assembly qualty. I Chaptes ad 4, we wll see that thee ae othe methods based o the optmzato o pecepat ad assembly qualty ad the optmzato o total cost. Fg. 9-6 shows how the assumptos about the pecepats aect the equemets (gaps), usg the wost case ad statstcal methods. I ths gue, the hozotal as epesets the mauactued dmeso. The vetcal as epesets the umbe o pats that ae mauactued at a patcula dmeso o the hozotal as. Fgue 9-6 Combg pecepat vaatos usg wost case ad statstcal methods

10 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks 9-9 I the Wost Case Model, we vey that the pats wll peom the teded ucto 00 pecet o the tme. Ths s otetmes a cosevatve appoach. I the statstcal modelg appoach, we assume that most o the mauactued pats ae ceteed o the mea dmeso. Ths s usually less cosevatve tha a wost case appoach, but t oes seveal beets whch we wll dscuss late. Thee ae two tadtoal statstcal methods; the Root Sum o the Squaes (RSS) Model, ad the Moded Root Sum o the Squaes (MRSS) Model Calculatg the Vaato o the Requemet Dug the desg pocess, the desg egee makes tadeos usg oe o the thee classc models. Typcally, the desge aalyzes the equemets usg wost case toleaces. I the wost case toleaces met the equed assembly peomace, the desge would stop thee. O the othe had, ths model dd ot meet the equemets, the desge ceased the pecepat toleaces (to make the pats moe mauactuable) at the sk o ocoomace at the assembly level. The desge would make tades, usg the RSS ad MRSS models. The ollowg sectos dscuss the tadtoal Wost Case, RSS, ad MRSS models. Addtoally, we dscuss the Estmated Mea Sht Model that cludes Wost Case ad RSS models as eteme cases Wost Case Toleacg Model The Wost Case Model, sometmes eeed to as the Method o Etemes, s the smplest ad most cosevatve o the tadtoal appoaches. I ths appoach, the toleace at the teace s smply the sum o the dvdual toleaces. The ollowg equato calculates the epected vaato at the gap. t wc a t whee t wc mamum epected vaato (equal blateal) usg the Wost Case Model. t equal blateal toleace o the th compoet the stackup. (9.) The vaato at the gap o Requemet 6 s: t wc (-).055 ().0030 ().0050 ().0075 ().0050 ().0070 ().0050 ().0075 ().0070 (-).0060 ().0300 t wc.0955 Usg the Wost Case Model, the mmum gap s equal to the mea value mus the wost case vaato at the gap. The mamum gap s equal to the mea value plus the wost case vaato at the gap. Mmum gap d g - t wc Mamum gap d g t wc The mamum ad mmum assembly gaps o Requemet 6 ae: Mmum Gap 6 d g - t wc Mamum Gap 6 d g t wc

11 9-0 Chapte Ne The equemet o Gap 6 s that the mmum gap must be geate tha 0. Theeoe, we must cease the mmum gap by.0340 to meet the mmum gap equemet. Oe way to cease the mmum gap s to mody the dmesos (d s) to cease the omal gap. Dog ths wll also cease the mamum gap o the assembly by Sometmes, we ca t do ths because the mamum equemet may ot allow t, o othe equemets (such as Requemet 5) wo t allow t. Aothe opto s to educe the toleace values (t s) the stackup. Reszg Toleaces the Wost Case Model Thee ae two ways to educe the toleaces the stackup.. The desge could adomly chage the toleaces ad aalyze the ew umbes, o. I the ogal umbes wee weghted the same, the all vaable toleaces (those ude the cotol o the desge) could be multpled by a esze acto to yeld the mmum assembly gap. Ths s the coect appoach the desge assged ogal toleaces that wee equally poducble. Reszg s a method o allocatg toleaces. (See Chaptes ad 4 o uthe dscusso o toleace allocato.) I allocato, we stat wth a desed assembly peomace ad deteme the pecepat toleaces that wll meet ths equemet. The esze acto, F wc, scales the ogal wost case toleaces up o dow to acheve the desed assembly peomace. Sce the desge has o cotol ove toleaces o puchased pats (ed toleaces), the scalg acto oly apples to vaable toleaces. Eq. (9.) becomes: t wc p a t q j j j k a t k k whee, a j sestvty acto o the j th, ed compoet the stackup a k sestvty acto o the k th, vaable compoet the stackup t j equal blateal toleace o the j th, ed compoet the stackup t kv equal blateal toleace o the k th, vaable compoet the stackup p umbe o depedet, ed dmesos the stackup q umbe o depedet, vaable dmesos the stackup F The esze acto o the Wost Case Model s: d g g m wc q k p j a t k kv a t j j whee g m mmum value at the (assembly) gap. Ths value s zeo o teeece o cleaace s allowed. The ew vaable toleaces (t kv,wc, eszed ) ae the old toleaces multpled by the acto F wc. t kv,wc,eszed F wc t kv t kv,wc,eszed equal blateal toleace o the k th, vaable compoet the stackup ate eszg usg the Wost Case Model.

12 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks 9- Fg. 9-7 shows the elatoshp betwee the pecepat toleaces ad the assembly toleace beoe ad ate eszg Ogal Toleaces K Pecepat Toleace Reszed Toleaces I J E & G C Assembly Toleace Fgue 9-7 Gaph o pecepat toleaces vesus assembly toleace beoe ad ate eszg usg the Wost Case Model The esze acto o Requemet 6 equals.399. (Fo eample,.0030 s eszed to.399* ) Table 9-3 shows the ew (eszed) toleaces that would gve a mmum gap o zeo. Table 9-3 Reszed toleaces usg the Wost Case Model Vaable Name Mea Dmeso Fed/ Vaable /- Equal Blateal Toleace Reszed /- Equal Blateal Toleace (t v,wc,eszed ) A.3595 Fed.055 B.030 Fed.000 C.0600 Vaable D.4305 Fed.0075 E.00 Vaable F.5030 Fed.0070 G.00 Vaable H.4305 Fed.0075 I.4500 Vaable J Vaable K.3000 Vaable

13 9- Chapte Ne As a check, we ca show that the ew mamum epected assembly gap o Requemet 6, usg the eszed toleaces, s: t wc,eszed t.066 wc,eszed The vaato at the gap s: Mmum Gap 6 d g - t wc,eszed Mamum Gap 6 d g t wc,eszed Assumptos ad Rsks o Usg the Wost Case Model I the wost case appoach, the desge does ot make ay assumptos about how the dvdual pecepat dmesos ae dstbuted wth the toleace ages. The oly assumpto s that all pecepats ae wth the toleace lmts. Whle ths may ot always be tue, the method s so cosevatve that pats wll pobably stll t. Ths s the method s majo advatage. The majo dsadvatage o the Wost Case Model s whe thee ae a lage umbe o compoets o a small gap (as the pevous eample). I such applcatos, the Wost Case Model yelds small toleaces, whch wll be costly RSS Model I desges caot acheve poducble pecepat toleaces o a gve equemet, they ca take advatage o pobablty theoy to cease them. Ths theoy s kow as the Root Sum o the Squaes (RSS) Model. The RSS Model s based o the pemse that t s moe lkely o pats to be mauactued ea the cete o the toleace age tha at the eds. Epeece mauactug dcates that small eos ae usually moe umeous tha lage eos. The devatos ae buched aoud the mea o the dmeso ad ae ewe at pots athe om the mea dmeso. The umbe o mauactued peces wth lage devatos om the mea, postve o egatve, may appoach zeo as the devatos om the mea cease. The RSS Model assumes that the mauactued dmesos t a statstcal dstbuto called a omal cuve. Ths model also assumes that t s ulkely that pats a assembly wll be adomly chose such a way that the wost case codtos aalyzed eale wll occu. Devato o the RSS Equato* We ll deve the RSS equato based o statstcal pcples o combatos o stadad devatos. To make ou devato as geec as possble, let s stat wth a ucto o depedet vaables such as y(,,, ). Fom ths ucto, we eed to be able to calculate the stadad devato o y, o σ y. But how do we d σ y all we have s omato about the compoets? Let s stat wth the deto o σ y. σ y ( y µ y ) *Deved by Dale Va Wyk ad epted by pemsso o Raytheo Systems Compay

14 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks 9-3 whee, µy the mea o the adom vaable y the total umbe o measuemets the populato o teest Let y y -µ y I y s small, whch s usually the case, y d... d d dy (9.3) Theeoe, dy y σ (9.4) Fom Eq. (9.3), ( ) ( ) ( ) ( )( ) k j j k k j k j d d d d d d d d dy I all the vaables ae depedet, ( )( ) 0 k j j k k j k j d d The same would hold tue o all smla tems. As a esult, ( ) ( ) ( ) ( ) d... d d dy Each patal devatve s evaluated at ts mea value, whch s chose as the omal. Thus, C whee C s a costat o each, ( ) ( ) ( ) ( ) d... d d dy (9.5)

15 9-4 Chapte Ne Usg the esults o Eq. (9.5) ad setg to Eq. (9.4) ( ) ( ) ( ) d... d d y σ ( ) ( ) ( ) d... d d y σ (9.6)... y σ σ σ σ Now, let s apply ths statstcal pcple to toleace aalyss. We ll cosde each o the vaables to be a dmeso, D, wth a toleace, T. I the omal dmeso, D, s the same as the mea o a omal dstbuto, we ca use the deto o a stadad omal vaable, Z, as ollows. (See Chaptes 0 ad o uthe dscussos o Z.) T D USL Z σ σ Z T σ (9.7) I the pecepats ae adomly selected, ths elatoshp apples o the ucto y as well as o each T. Fo oe-dmesoal toleace stacks, a D y whee each a epesets the sestvty. I ths case, a y ad Eq. (9.6) becomes y a... a a σ σ σ σ (9.8) Whe you combe Eq. (9.7) ad Eq. (9.8), y y Z T a... Z T a Z a T Z T (9.9) I all o the dmesos ae equally poducble, o eample all ae eactly 3σ toleaces, o all ae 6σ toleaces, Z y Z Z Z. I addto, let a a a /-. Eq. (9.9) wll the educe to y T... T T T o y T... T T T (9.0) whch s the classcal RSS equato.

16 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks 9-5 Let s evew the assumptos that wet to the devato o ths equato. All the dmesos D ae statstcally depedet. The mea value o D s lage compaed to s. The ecommedato s that D /σ should be geate tha ve. The omal value s tuly the mea o D. The dstbutos o the dmesos ae Gaussa, o omal. The pecepats ae adomly assembled. Each o the dmesos s equally poducble. Each o the sestvtes has a magtude o. Z equatos assume equal blateal toleaces. The valdty o each o these assumptos wll mpact how well the RSS pedcto matches the ealty o poducto. Note that whle Eq. (9.0) s the classcal RSS equato, we should geeally wte t as ollows so that we do t lose sestvtes. t ss a t a t... a t (9.) Hstocally, Eq. (9.) assumed that all o the compoet toleaces (t ) epeset a 3σ value o the mauactug pocesses. Thus, all the compoet dstbutos ae assumed to be omal, the the pobablty that a dmeso s betwee ±t s 99.73%. I ths s tue, the the assembly gap dstbuto s omal ad the pobablty that t s ±t ss betwee s 99.73%. Although most people have assumed a value o ±3σ o pecepat toleaces, the RSS equato woks o equal σ values. I the desge assumed that the put toleaces wee ±4σ values o the pecepat mauactug pocesses, the the pobablty that the assembly s betwee ±t ss s (4σ). The 3σ pocess lmts usg the RSS Model ae smla to the Wost Case Model. The mmum gap s equal to the mea value mus the RSS vaato at the gap. The mamum gap s equal to the mea value plus the RSS vaato at the gap. Mmum 3σ pocess lmt d g - t ss Mamum 3σ pocess lmt d g t ss Usg the ogal toleaces o Requemet 6, t ss s: t ss ( ).055 ().000 ().0030 ().0075 ().0050 ().0070 ().0050 ().0075 ().0070 ( ).0060 ().0300 t ss.038 The thee sgma vaato at the gap s: Mmum 3σ pocess vaato o Gap 6 d g t ss Mamum 3σ pocess vaato o Gap 6 d g t ss

17 9-6 Chapte Ne Reszg Toleaces the RSS Model Usg the RSS Model, the mmum gap s geate tha the equemet. As the Wost Case Model, we ca esze the vaable toleaces to acheve the desed assembly peomace. As beoe, the scalg acto oly apples to vaable toleaces. The esze acto, F ss, o the RSS Model s: F ss ( d g g m ) ( a j t j ) q ( ak tkv ) k p j The ew vaable toleaces (t kv,ss, eszed ) ae the old toleaces multpled by the acto F ss. t kv,ss,eszed F ss t kv t kv,ss,eszed equal blateal toleace o the k th, vaable compoet the stackup ate eszg usg the RSS Model. Fg. 9-8 shows the elatoshp betwee the pecepat toleaces ad the assembly toleace beoe ad ate eszg. Reszed Toleaces K Ogal Toleaces Pecepat Toleace I J E & G C Assembly Toleace Fgue 9-8 Gaph o pecepat toleaces vesus assembly toleace beoe ad ate eszg usg the RSS Model The ew vaable toleaces ae the old toleaces multpled by the acto F ss. The esze acto o Requemet 6 s (Fo eample,.0030 s eszed to.7984* )

18 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks 9-7 Table 9-4 shows the ew toleaces that would gve a mmum gap o zeo. As a check, we ca show that the ew mamum epected assembly gap o Requemet 6, usg the eszed toleaces, s: t ss, eszed Vaable Name t ss,eszed.065 ( ).055 ().000 ().0054 ().0075 ().0090 ().0070 ().0090 ().0075 ().06 ( ).008 ().0540 The vaato at the gap s: Table 9-4 Reszed toleaces usg the RSS Model Mea Dmeso Fed/ Vaable Mmum 3σ pocess vaato o Gap 6 d g t ss,eszed Mamum 3σ pocess vaato o Gap 6 d g t ss,eszed Assumptos ad Rsks o Usg the RSS Model Ogal /- Equal Blateal Toleace A.3595 Fed.055 B.030 Fed.000 The RSS Model yelds lage pecepat toleaces o a gve assembly gap, but the sk o deects at assembly s hghe. The RSS Model assumes: a) Pecepat toleaces ae ted to pocess capabltes. Ths model assumes that whe the desge chages a toleace, the pocess capabltes wll also chage. b) All pocess dstbutos ae ceteed o the mdpot o the dmeso. It does ot allow o mea shts (tool wea, etc.) o o puposeul deceteg. c) All pecepat dmesos ae depedet (covaace equals zeo). Reszed /- Equal Blateal Toleace (t v,ss,eszed ) C.0600 Vaable D.4305 Fed.0075 E.00 Vaable F.5030 Fed.0070 G.00 Vaable H.4305 Fed.0075 I.4500 Vaable J Vaable K.3000 Vaable

19 9-8 Chapte Ne d) The bad pats ae thow wth the good the assembly. The RSS Model does ot take to accout pat sceeg (specto). e) The pats cluded ay assembly have bee thooughly med ad the compoets cluded ay assembly have bee selected at adom. ) The RSS devato assumes equal blateal toleaces. Remembe that by devg the RSS equato, we made the assumpto that all toleaces (t s) wee equally poducble. Ths s usually ot the case. The oly way to kow a toleace s poducble s by udestadg the pocess capablty o each dmeso. The tadtoal assumpto s that the toleace (t ) s equal to 3σ, ad the pobablty o a deect at the gap wll be about.7%. I ealty, t s vey ulkely to be a 3σ value, but athe some ukow umbe. The RSS Model s bette tha the Wost Case Model because t accouts o the tedecy o pecepats to be ceteed o a mea dmeso. I geeal, the RSS Model s ot used thee ae less tha ou dmesos the stackup Moded Root Sum o the Squaes Toleacg Model I ealty, the pobablty o a wost case assembly s vey low. At the othe eteme, empcal studes have show that the RSS Model does ot accuately pedct what s mauactued because some (o all) o the RSS assumptos ae ot vald. Theeoe, a opto desges ca use s the RSS Model wth a coecto acto. Ths model s called the Moded Root Sum o the Squaes Method. t mss C a t a t... a t whee C coecto acto used the MRSS equato. t mss epected vaato (equal blateal) usg the MRSS model. Seveal epets have suggested coecto actos (C ) the age o.4 to.8 (Reeeces,4,5 ad 6). Hstocally, the most commo acto s.5. The vaato at the gap s: Mmum gap d g - t mss Mamum gap d g t mss C C I ou eample, we wll use the coecto acto suggested Reeece. 0.5 t ( t t ) ss wc ss ( ) Ths coecto acto wll always gve a t mss value that s less tha t wc. I ou eample, C s: (.038 ) C.35 ( )

20 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks 9-9 Usg the ogal toleaces o Requemet 6, t mss s: ( ).055 ().000 ().0030 ().0075 ().0050 ().0070 t mss.35 ().0050 ().0075 ().0070 ( ).0060 ().0300 t mss.0505 The vaato at the gap s: Mmum Gap 6 d g - t mss Mamum Gap 6 d g t mss Reszg Toleaces the RSS Model Smla to the RSS Model, the mmum gap usg the MRSS Model s geate tha the equemet. Lke the othe models, we ca esze the vaable toleaces to acheve the desed assembly peomace. The equato o the esze acto, F mss, s much moe comple o ths model. The value o F mss s a oot o the ollowg quadatc equato. af mss bf mss c 0 whee q a 0.5 a t b 0.5 k k kv k p c 0.5 a jt j q p aj t j.5 ( a t ) 3 ( a t ) ( a t ) p q q ( a t ) ( a t ) a t ( d g ) a t ( d g ) j Theeoe, k k kv k p ( d g ) ( d g ) ( d g ) a t ( d g ) p p p ( ) ( ) ( ) ( ) d g.5 a t 3 a t a t j k kv j g j j m g q F mss -b - b - 4ac a k kv m j j j g k g q m k kv m j k k kv k j g j q k kv m g j j j m j j j g m

21 9-0 Chapte Ne Fg. 9-9 shows the elatoshp betwee the pecepat toleaces ad the assembly toleace beoe ad ate eszg. The ew vaable toleaces (t kv,mss, eszed ) ae the old toleaces multpled by the acto F mss Ogal Toleace Reszed Toleaces K Pecepat Toleace I J E & G C Assembly Toleace Fgue 9-9 Gaph o pecepat toleaces vesus assembly toleace beoe ad ate eszg usg the MRSS Model t kv,mss,eszed F mss t kv t kv,mss,eszed equal blateal toleace o the k th, vaable compoet the stackup ate eszg usg the MRSS Model. The esze acto o Requemet 6 s.309. (Fo eample,.0030 s eszed to.309* ) Table 9-5 shows the ew toleaces that would gve a mmum gap o zeo. Vaable Name Table 9-5 Reszed toleaces usg the MRSS Model Mea Dmeso Fed/ Vaable Ogal /- Equal Blateal Toleace A.3595 Fed.055 B.030 Fed.000 Reszed /- Equal Blateal Toleace (t v,mss,eszed ) C.0600 Vaable D.4305 Fed.0075 E.00 Vaable F.5030 Fed.0070 G.00 Vaable H.4305 Fed.0075 I.4500 Vaable J Vaable K.3000 Vaable

22 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks 9- As a check, we show the ollowg calculatos o the eszed toleaces. t wc, eszed t wc, eszed.34 t ss, eszed t ss, eszed.047 ( ).055 ().000 ().0040 ().0075 ().0066 ().0070 ().0066 ().0075 ().009 ( ).0079 () ( ).047 ( ) C, eszed C, eszed.303 ( ).055 ().000 ().0040 ().0075 ().0066 ().0070 t mss, eszed.303 ().0066 ().0075 ().009 ( ).0079 ().0396 t mss, eszed.065 As a check, we ca show that the epected assembly gap o Requemet 6, usg the eszed toleaces, s: Mmum Gap 6 d g t mss,eszed Mamum Gap 6 d g t mss,eszed Assumptos ad Rsks o Usg the MRSS Model The ucetaty assocated wth the MRSS Model s that thee s o mathematcal easo o the acto C. The coecto acto ca be thought o as a saety acto. The moe the RSS assumptos depat om ealty, the hghe the saety acto should be. The MRSS Model also has othe poblems. a) It apples the same saety acto to all the toleaces, eve though they do t devate om the RSS assumptos equally. b) I ed coecto actos poposed the lteatue ae used, the MRSS toleace ca be lage tha the wost case stackup. Ths poblem s elmated wth the use o the calculated C show hee. c) I the toleaces ae equal ad thee ae oly two o them, the MRSS assembly toleace wll always be lage tha the wost case assembly toleace whe usg the calculated coecto acto. The MRSS Model s geeally cosdeed bette tha the RSS ad Wost Case models because t tes to model what has bee measued the eal wold.

23 9- Chapte Ne Compaso o Vaato Models Table 9-6 summazes the Wost Case, RSS, ad MRSS models o Requemet 6. The Reszed colums show the toleaces that wll gve a mmum epected gap value o zeo, ad a mamum epected gap value o.30 ch. As epected, the wost case toleace values ae the smallest. I ths eample, the eszed RSS toleace values ae appomately thee tmes geate tha the wost case toleaces. It s obvous that the RSS toleaces wll yeld moe pecepats. The MRSS eszed toleace values all betwee the wost case (most cosevatve) ad RSS (most sk o assembly deects) values. Table 9-6 Compaso o esults usg the Wost Case, RSS, ad MRSS models Toleace Aalyss Dm. Wost Case RSS MRSS Mea Dm. Ses. Type Ogal Reszed Ogal Reszed Ogal Reszed Vaable Fed Vaable Fed Vaable Fed Vaable Fed Vaable Vaable Vaable Nomal Gap Mmum Gap Epected Vaato Table 9-7 summazes the tadeos o the thee models. All the models have deet degees o sk o deects. The wost case toleaces have the least amout o sk (.e. lagest umbe o assembles wth the epected assembly equemets). Because o the tght toleaces we wll eject moe pecepats. Wost case also mples that we ae dog 00% specto. Sce we have to tghte up the toleaces to meet the assembly speccato, the umbe o ejected pecepats ceases. Theeoe, ths model has the hghest costs assocated wth t. The RSS toleaces wll yeld the least pecepat cost at the epese o a lowe pobablty o assembly coomace. The MRSS Model tes to take the best o both o these models. It gves a hghe pobablty o assembly coomace tha the RSS Model, ad lowe pecepat costs tha the Wost Case Model. Wth the lmtatos, the tadtoal toleacg models have woked the past. The desg egee, howeve, could ot quaty how well they woked. He also could ot quaty how cost eectve the toleace values wee. Obvously, these methods caot cosstetly acheve qualty goals. Oe way to acheve qualty goals s to elmate the assumptos that go alog wth the classcal toleacg models. By dog so, we ca quaty (sgma level, deects pe mllo oppotutes (dpmo)) the toleaces ad optmze toleaces o mamum poducblty. These ssues ae dscussed Chapte, Pedctg Assembly Qualty.

24 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks 9-3 Table 9-7 Compaso o aalyss models Wost Case Cosdeato Model RSS Model MRSS Model Rsk o Deect Lowest Hghest Mddle Cost Hghest Lowest Mddle Assumptos about Noe The pocess ollows a The pocess ollows compoet omal dstbuto. The a omal dstbuto. pocesses mea o the pocess s The mea o the pocess equal to the omal dstbuto s ot dmeso. Pocesses ecessaly equal to the ae depedet. omal dmeso. Assumptos about Dmesos The toleace s elated The toleace s elated dawg toleaces outsde the to a mauactug to a mauactug toleace age pocess capablty. pocess capablty. ae sceeed Usually the toleace Usually the toleace out. age s assumed to be age s assumed to be the /- 3 sgma lmt the /- 3 sgma lmt o the pocess. o the pocess. Assumptos about 00% o the The assembly 99.73% o the assembles epected assembly pats ae wth dstbuto s omal. wll be betwee the vaato the mamum Depedg o the mmum ad mamum ad mmum pecepat assumptos, gap. The coecto peomace a pecetage o the acto (C ) s a saety age. assembles wll be acto. betwee the mmum ad mamum gap. Hstocally, ths has bee 99.73%. Some out o speccato pats each assembly Estmated Mea Sht Model Geeally, we do t have kowledge about the pocesses o mauactug a pat, such as a vedo pat, we ae moe cled to use the Wost Case Model. O the othe had, we have kowledge about the pocesses that make the pat, we ae moe cled to use a statstcal model. Chase ad Geewood poposed a toleacg model that bleds the Wost Case ad RSS models. (Reeece 6) Ths Estmated Mea Sht Model s: t ems m a t (( m ) a t ) whee m the mea sht acto o the th compoet

25 9-4 Chapte Ne I ths model, the mea sht acto s a umbe betwee 0 ad.0 ad epesets the amout that the mdpot s estmated to sht as a acto o the toleace age. I a pocess wee closely cotolled, we would use a small mea sht, such as.. I we kow less about the pocess, we would use hghe mea sht actos. Usg a mea sht acto o. o the vaable compoets ad.8 o the ed compoets, the epected vaato o Requemet 6 s: t ems.8( ).055.8().000.() ().0075.() ().0070.() ().0075.().0070.( ).0060.().0300 (.( ).055 ) (. ().000 ) (.8 ().0030 ) (. ().0075 ) (.8 ().0050 ) (. ().0070 ) (.8 ().0050 ) (. ().0075 ) (.8 ().0070 ) (.8 ( ).0060 ) (.8 ().0300 ) t ems.0690 The st pat o the Estmated Mea Sht Model s the sum o the mea shts ad s smla to the Wost Case Model. Notce we set the mea sht acto to.0 o all the compoets, t ems s equal to.0955, whch s the same as t wc. The secod pat o the model s the sum o the statstcal compoets. Notce we used a mea sht acto o zeo o all o the compoets, t ems s equal to.038, whch s the same as t ss. The two majo advatages o the Estmated Mea Sht Model ae: It allows leblty the desg. Some compoets may be modeled lke wost case, ad some may be modeled statstcally. The model ca be used to estmate desgs (usg cosevatve sht actos), o t ca accept mauactug data ( t s avalable). 9.3 Aalyzg Geometc Toleaces The pevous dscussos have oly cluded toleaces assocated wth dmesos the toleace aalyss. We have ot yet addessed how to model geometc toleaces the loop dagam. Geeally, geometc cotols wll esta oe o seveal o the ollowg attbutes: Locato o the eatue Oetato o the eatue Fom o the eatue The most dcult task whe modelg geometc toleaces s detemg whch o the geometc cotols cotbute to the equemet ad how these cotols should be modeled the loop dagam. Because the geometc cotols ae teelated, thee ae o had ad ast ules that tell us how to clude geometc cotols toleace aalyses. Sce thee ae seveal modelg methods, sometmes we clude GD&T the model, ad sometmes we do ot. Geeally, howeve, a eatue s cotolled wth geometc toleaces, the ollowg apply. I thee s a locato cotol o a eatue the loop dagam, we wll usually clude t the aalyss. I thee s a oetato cotol o a eatue the loop dagam, we may clude t the aalyss as log as the locato o the eatue s ot a cotbuto to the equemet.

26 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks 9-5 I thee s a om cotol o a eatue the loop dagam, we may clude t the aalyss as log as the locato, oetato, o sze o the eatue s ot a cotbuto to the equemet. Ay tme pats come togethe, howeve, we have suace vaatos that toduce vaatos the model. Geometc om ad oetato cotols o datum eatues ae usually ot cluded loop dagams. Sce datums ae the statg pots o measuemets, ad ae deed as the geometc coutepats (hgh pots) o the datum eatue, the vaatos the datum eatues usually do t cotbute to the vaato aalyss. Thee s a deece betwee a GD&T cotol (such as a om cotol) ad a eatue vaato (such as om vaato). I we add a GD&T cotol to a stack, we add to the output. Theeoe, we should oly clude the GD&T cotols that add to the output. GD&T cotols ae geeally used oly wost case aalyses. Pevously we sad that the Wost Case Model assumes 00% specto. Sce GD&T cotols ae the speccato lmts o specto, t makes sese to use them ths type o aalyss. I a statstcal aalyss, howeve, we ethe make assumptos about the mauactug pocesses (as show pevously), o use eal data om the mauactug pocesses (as show Chapte ). Sce the mauactug pocesses ae souces o vaato, they should be puts to the statstcal aalyses. Sce GD&T cotols ae ot souces o vaato, they should ot be used a statstcal aalyss. The ollowg sectos show eamples o how to model geometc toleaces. The eamples ae sgle pat stacks, but the cocepts ca be appled to stacks wth multple compoets Fom Cotols Fom cotols should seldom be cluded a vaato aalyss. Fo osze eatues, the locato, o oetato toleace usually cotols the etet o the vaato o the eatue. The om toleace s typcally a eemet o oe o these cotols. I a om cotol s appled to a sze eatue (ad the Idvdual Featue o Sze Rule apples om ASME Y4.5), the sze toleace s usually cluded the vaato aalyss. I these cases, the om toleace bouday s sde the sze toleace bouday, the locato toleace bouday, o the oetato toleace bouday, so the om cotol s ot modeled. I om toleaces ae used the loop dagam, they ae modeled wth a omal dmeso equal to zeo, ad a equal blateal toleace equal to the om toleace. (Depedg o the applcato, sometmes the equal blateal toleace s equal to hal the om toleace.) Fg. 9-0 shows a assembly wth ou pats. I ths eample, the equemet s o the Gap to be geate tha zeo. Fo ths equemet, the ollowg apples to the om cotols. Flatess o.00 o the substate s ot cluded the loop dagam because t s a datum. Flatess o.00 o the heatsk s cluded the loop dagam. Flatess o.00 o the housg s ot cluded the loop dagam because t s a eemet o the locato toleace. Flatess o.004 o the housg s ot cluded the loop dagam because t s a datum. Flatess o.006 o the housg s ot cluded the loop dagam because t s a eemet o the locato.

27 9-6 Chapte Ne Fgue 9-0 Substate package 9.3. Oetato Cotols Lke om cotols, we do ot ote clude oetato cotols a vaato aalyss. Typcally we deteme the eatue s wost-case toleace bouday usg the locato o sze toleace. I oetato toleaces ae used the loop dagam, they ae modeled lke om toleaces. They have a omal dmeso equal to zeo, ad a equal blateal toleace equal to the oetato toleace. (Depedg o the applcato, sometmes the equal blateal toleace s equal to hal the oetato toleace.) I Fg. 9-0, the ollowg descbes the applcato o the oetato cotols to the Gap aalyss. Paallelsm o.004 to datum A o the Substate s ot cluded the loop dagam because t s a eemet o the sze dmeso (.040 ±.003). Paallelsm o.004 to datum A o the Housg s ot cluded the loop dagam because t s a eemet o the locato toleace. Paallelsm o.004 to datum A o the Wdow s cluded the loop dagam.

28 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks 9-7 Theeoe, the equato o the Gap Fg. 9-0 s: Gap -AB-CDE whee A.040 ±.003 B 0 ±.00 C.5 ±.005 D.85 ±.008 E 0 ± Posto Thee ae seveal ways to model a posto geometc costat. Whe we use posto at egadless o eatue sze (RFS), the sze o the eatue, ad the locato o the eatue ae teated depedetly. Whe we use posto at mamum mateal codto (MMC) o at least mateal codto (LMC), the sze ad locato dmesos caot be teated depedetly. The ollowg sectos show how to aalyze these stuatos Posto at RFS Fg. 9- shows a hole postoed at RFS. The equato o the Gap Fg. 9- s: Gap A/B whee A.065 ±.000 B.50 ± Posto at MMC o LMC Fgue 9- Posto at RFS As stated eale, whe we use posto at MMC o LMC, the sze ad locato dmesos should be combed to oe compoet the loop dagam. We ca do ths usg the ollowg method. ) Calculate the lagest oute bouday allowed by the dmesos ad toleaces. ) Calculate the smallest e bouday allowed by the dmesos ad toleaces. 3) Covet the e ad oute bouday to a omal damete wth a equal blateal toleace.

29 9-8 Chapte Ne Vtual ad Resultat Codtos Whe calculatg the teal ad eteal boudaes o eatues o sze, t s helpul to udestad the ollowg detos om ASME Y4.5M-994. Vtual Codto: A costat bouday geeated by the collectve eects o a sze eatue s speced MMC o LMC ad the geometc toleace o that mateal codto. The vtual codto (oute bouday) o a eteal eatue, called out at MMC, s equal to ts mamum mateal codto plus ts toleace at mamum mateal codto. The vtual codto (e bouday) o a teal eatue, called out at MMC, s equal to ts mamum mateal codto mus ts toleace at mamum mateal codto. The vtual codto (e bouday) o a eteal eatue, called out at LMC, s equal to ts least mateal codto mus ts toleace at least mateal codto. The vtual codto (oute bouday) o a teal eatue, called out at LMC, s equal to ts least mateal codto plus ts toleace at least mateal codto. Resultat Codto: The vaable bouday geeated by the collectve eects o a sze eatue s speced MMC o LMC, the geometc toleace o that mateal codto, the sze toleace, ad the addtoal geometc toleace deved om ts speced mateal codto. The smallest esultat codto (e bouday) o a eteal eatue, called out at MMC, s equal to ts least mateal codto mus ts toleace at least mateal codto. The lagest esultat codto (oute bouday) o a teal eatue, called out at MMC, s equal to ts least mateal codto plus ts toleace at least mateal codto. The lagest esultat codto (oute bouday) o a eteal eatue, called out at LMC, s equal to ts mamum mateal codto plus ts toleace at mamum mateal codto. The smallest esultat codto (e bouday) o a teal eatue, called out at LMC, s equal to ts mamum mateal codto mus ts toleace at mamum mateal codto Equatos We ca use the ollowg equatos to calculate the e ad oute boudaes. Fo a eteal eatue at MMC oute bouday VC MMC Geometc Toleace at MMC e bouday (smallest) RC LMC Toleace at LMC Fo a teal eatue at MMC e bouday VC MMC - Geometc Toleace at MMC oute bouday (lagest) RC LMC Toleace at LMC Fo a eteal eatue at LMC e bouday VC LMC - Geometc Toleace at LMC oute bouday (lagest) RC MMC Toleace at MMC Fo a teal eatue at LMC oute bouday VC LMC Geometc Toleace at LMC e bouday (smallest) RC MMC Toleace at MMC

30 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks 9-9 Covetg a Iteal Featue at MMC to a Nomal Value wth a Equal Blateal Toleace Fg. 9- shows a hole that s postoed at MMC. The value o B the loop dagam s: Lagest oute bouday Smallest e bouday Nomal damete (.65.5)/.45 Equal blateal toleace.00 Fo posto at MMC, a ease way to covet ths s: LMC ± (total sze toleace toleace the eatue cotol ame).45 ± ( ).45±.00 The equato o the Gap Fg. 9- s: Gap A-B/ whee A.3 ±0 ad B.45 ±.00 Fgue 9- Posto at MMC teal eatue

31 9-30 Chapte Ne Covetg a Eteal Featue at MMC to a Nomal Value wth a Equal Blateal Toleace Fg. 9-3 shows a p postoed at MMC. Fgue 9-3 Posto at MMC eteal eatue The value o B the loop dagam s: Lagest oute bouday Smallest e bouday Nomal damete ( )/.064 Equal blateal toleace.004 As show eale, the ease coveso o posto at MMC, s: LMC ±(total sze toleace toleace the eatue cotol ame).064 ±( ).064/-.004 The equato o the Gap Fg. 9-3 s: Gap -A/B whee A.064 ±.004 B.50 ±0 Covetg a Iteal Featue at LMC to a Nomal Value wth a Equal Blateal Toleace Fg. 9-4 shows a hole that s postoed at LMC. The value o B the loop dagam s: Lagest oute bouday Smallest e bouday Nomal damete (.55.4)/.48 Equal blateal toleace.07

32 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks 9-3 Fgue 9-4 Posto at LMC teal eatue Fo posto at LMC, a ease way to covet ths s: MMC ±(total sze toleace toleace the eatue cotol ame).48 ± (04.03).48 ±.07 The equato o the Gap Fg. 9-4 s: Gap A B/ whee A.70 ±0 B.48 ±.07 Covetg a Eteal Featue at LMC to a Nomal Value wth a Equal Blateal Toleace Fg. 9-5 shows a boss that s postoed at LMC. The value o B the loop dagam s: Lagest oute bouday Smallest e bouday Nomal damete (.3.93)/.03 Equal blateal toleace.0 Fgue 9-5 Posto at LMC eteal eatue

33 9-3 Chapte Ne As show eale, the ease coveso o posto at LMC s: MMC ±(total sze toleace toleace the eatue cotol ame).03 ±(.06.04).03 /-.0 The equato o the Gap Fg. 9-5 s: Gap A-B/ whee A.70 ± 0 B.03 ± Composte Posto Fg. 9-6 shows a eample o composte postoal toleacg. Fgue 9-6 Composte posto ad composte pole Composte postoal toleacg toduces a uque elemet to the vaato aalyss; a udestadg o whch toleace to use. I a equemet oly cludes the patte o eatues ad othg else o the pat, we use the toleace the lowe segmet o the eatue cotol ame. Sce Gap Fg. 9-6 s cotolled by two eatues wth the patte, we use the toleace o.04 to calculate the vaato o Gap. Gap, howeve, cludes vaatos o the eatues back to the datum eeece ame. I ths stuato, we use the toleace the uppe segmet o the eatue cotol ame (.050) to calculate the vaato o Gap.

34 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks Ruout Aalyzg uout cotols toleace stacks s smla to aalyzg posto at RFS. Sce uout s always RFS, we ca teat the sze ad locato o the eatue depedetly. We aalyze total uout the same as ccula uout, because the wost-case bouday s the same o both cotols. Fg. 9-7 shows a hole that s postoed usg uout. Fgue 9-7 Ccula ad total uout We model the uout toleace wth a omal dmeso equal to zeo, ad a equal blateal toleace equal to hal the uout toleace. The equato o the Gap Fg. 9-7 s: Gap A/ B C/ whee A.5 ±.008 B 0 ±.003 C.06 ± Cocetcty/Symmety Aalyzg cocetcty ad symmety cotols toleace stacks s smla to aalyzg posto at RFS ad uout. Fg. 9-8 s smla to Fg. 9-7, ecept that a cocetcty toleace s used to cotol the.06 eatue to datum A. Fgue 9-8 Cocetcty

35 9-34 Chapte Ne The loop dagam o ths gap s the same as o uout. The equato o the Gap Fg. 9-8 s: Gap A/ B C/ whee A.5 ±.008 B 0 ±.003 C.06 ±.005 Symmety s aalogous to cocetcty, ecept that t s appled to plaa eatues. A loop dagam o symmety would be smla to cocetcty Pole Pole toleaces have a basc dmeso locatg the tue pole. The toleace s depcted ethe equal blateally, ulateally, o uequal blateally. Fo equal blateal toleace zoes, the pole compoet s eteed as a omal value. The compoet s equal to the basc dmeso, wth a equal blateal toleace that s hal the toleace the eatue cotol ame Pole Toleacg wth a Equal Blateal Toleace Zoe Fg. 9-9 shows a applcato o pole toleacg wth a equal blateal toleace zoe. Fgue 9-9 Equal blateal toleace pole The equato o the Gap Fg. 9-9 s: Gap -AB whee A.55 ±.003 B.755 ±.003

36 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks Pole Toleacg wth a Ulateal Toleace Zoe Fg. 9-0 shows a gue smla to Fg. 9-9 ecept the equal blateal toleace was chaged to a ulateal toleace zoe. The equato o the Gap s the same as Fg. 9-9: Gap A B Fgue 9-0 Ulateal toleace pole I ths eample, howeve, we eed to chage the basc dmesos ad ulateal toleaces to mea dmesos ad equal blateal toleaces. Theeoe, A.58 ±.003 B.758 ± Pole Toleacg wth a Uequal Blateal Toleace Zoe Fg. 9- shows a gue smla to Fg. 9-9 ecept the equal blateal toleace was chaged to a uequal blateal toleace zoe. The equato o the Gap s the same as Fg. 9-9: Gap A B Fgue 9- Uequal blateal toleace pole

37 9-36 Chapte Ne As we dd Fg. 9-0, we eed to chage the basc dmesos ad uequal blateal toleaces to mea dmesos ad equal blateal toleaces. Theeoe, A.54 ±.003 B.754 ± Composte Pole Composte pole s smla to composte posto. I a equemet oly cludes eatues wth the pole, we use the toleace the lowe segmet o the eatue cotol ame. I the equemet cludes vaatos o the pole back to the datum eeece ame, we use the toleace the uppe segmet o the eatue cotol ame. Fg. 9-6 shows a eample o composte pole toleacg. Gap 3 s cotolled by eatues wth the pole, so we would use the toleace the lowe segmet o the pole eatue cotol ame (.008) to calculate the vaato o Gap 3. Gap 4, howeve, cludes vaatos o the poled eatues back to the datum eeece ame. I ths stuato, we would use the toleace the uppe segmet o the pole eatue cotol ame (.040) to calculate the vaato o Gap Sze Datums Fg. 9- shows a eample o a patte o eatues cotolled to a secoday datum that s a eatue o sze. Fgue 9- Sze datum I ths eample, ASME Y4.5 states that the datum eatue apples at ts vtual codto, eve though t s eeeced ts eatue cotol ame at MMC. (Note, ths agumet also apples o secoday ad tetay datums voked at LMC.) I the toleace stack, ths meas that we wll get a addtoal shtg o the datum that we eed to clude the loop dagam. The way we hadle ths the loop dagam s the same way we hadled eatues cotolled wth posto at MMC o LMC. We calculate the vtual ad esultat codtos, ad covet these boudaes to a omal value wth a equal blateal toleace.

38 The value o A the loop dagam s: Lagest oute bouday Smallest e bouday Nomal damete (.54.49)/.503 Equal blateal toleace.0 Tadtoal Appoaches to Aalyzg Mechacal Toleace Stacks 9-37 A ease way to covet to ths adal value s: LMC ±(total sze toleace toleace the eatue cotol ame).503 ±( ).503±.0 The value o C the loop dagam s: Lagest oute bouday Smallest e bouday Nomal damete (.65.5)/.45 Equal blateal toleace.00 A ease way to covet to ths adal value s: LMC ±(total sze toleace toleace the eatue cotol ame).45 ±( ).45 ±.00 The equato o the Gap Fg. 9- s: Gap A/ B/ C/ whee A.503 ±.0 B.750 ±0 C.45 ± Abbevatos Vaable Deto a a j a k C C,eszed sestvty acto that dees the decto ad magtude o the th dmeso. I a oe-dmesoal stackup, ths value s usually o -. Sometmes, a oe-dmesoal stackup, ths value may be.5 o -.5 a adus s the cotbutg acto o a damete callout o a dawg. sestvty acto o the jth, ed compoet the stackup sestvty acto o the kth, vaable compoet the stackup coecto acto used the MRSS equato coecto acto used the MRSS equato, usg eszed toleaces d g d patal devatve o ucto y wth espect to the mea value at the gap. I d g s postve, the mea gap has cleaace, ad d g s egatve, the mea gap has teeece the mea value o the th dmeso the loop dagam