Shigley's Mechanical Engineering Design Tutorial 3-16: Press and Shrink Fits

Transcription

1 Shgley's Mechancal Engneerng Desgn Tutral 3-16: Press and Shrnk Fts APPLICATION OF THICK-WALL CYLINDRICAL PRESSURE VESSEL THEORY RELATING TO STRESSES DEVELOPED FROM INTERFERENCE FITS: 1. Desgn applcatn whch uses the cylndrcal pressure vessel Thck-Wall Thery. 2. Stresses develp between cylnders due t the cntact pressure generated by an nterference ft. The nterference ft s acheved by pressng a larger nsde member nt the smaller penng f an utsde member. In the specfc case f a shaft press ft nt the hub f a gear, the utsde dameter (OD) f the shaft s slghtly larger than the nsde hle dameter (ID) f the hub. The dametral dfference between the shaft OD and the ID f the hub hle s referred t as the nterference ft. The radal defrmatn requred by the nterference ft causes an nterfacal pressure, p, t develp at the nmnal radus, at r = R. Cnsequently, radal and tangental stresses, r and t, are prduced. Assumng uncapped ends ( l = 0), a baxal state f stress exsts fr whch tw nn-zer prncpal stresses must be cnsdered. Frm the cylndrcal pressure vessel thery, the radal and tangental stresses represent prncpal stresses. The length f the uter member s assumed t be equal t the length f the nner member. Outer Member r R δ R r r Inner Member r (a) End vew f nner and uter members, press ft tgether. (b) Crss-sectn f cylnders shwng nternal utsde radus larger than external nsde radus by a small amunt f δ. FIGURE T Interference ft f tw cylnders f fnte length and equal lengths. Text Eq. refers t Shgley's Mechancal Engneerng Desgn, 8 th edtn text by Rchard G. Budynas and J. Keth Nsbett; equatns wth the prefx T refer t the present tutral.

2 3. Referrng t Fg. T3-16-1, the gemetrc features f the cylndrcal parts are defned as: r = the nsde radus f the nner cylnder R = nmnal radus f nternal utsde radus and external nsde radus after assembly r = utsde radus f the uter cylnder δ = radal nterference INSIDE CYLINDER Inner member experences an external pressure, p = p, resultng n cmpressve tangental and radal stresses. Thck-Wall Thery may be appled wth r = R: (T-1) R + r t r= R = p pc = t R r ( ) ( ) = p = p r r= R OUTSIDE CYLINDER Outer member nly experences nternal pressure, p = p, resultng n tensle tangental stress and cmpressve radal stress. Thck-Wall Thery s, as always, applcable wth r = R: (T-2) r + R t r= R = p pc = t r R ( ) ( ) = p = p r r= R DEFINITION OF INTERFACIAL PRESSURE We presently have tw equatns and three unknwns fr bth the nsde and utsde cylnder analyses. A thrd equatn whch relates the cntact pressure and the nterference can be derved by examnng the defrmatn f the members. Deflectn Equatn The ttal radal nterference may be defned as: δ ttal = δ + δ where, Budynas & Nsbett Machne Desgn Tutral 3-16: Press and Shrnk Fts 2/11

3 δ = decrease n radus f nner cylnder δ = ncrease n radus f hle The defrmatn may als be expressed as: δ = prk + prk (Mdfed Text Eq. 3-56) ttal where the utsde member cnstant, K, s defned as, K 1 r + R 1 = + ν = + E r R E [ C ν ] Usng the radus rat frm defned fr the cylndrcal pressure vessel frmulatn, ζ = r / R, we can defne C 2 2 ζ + 1 = ζ 1 Nte that the C term s a functn f gemetry nly, whle the member cnstant term K s a functn f bth gemetry and materal parameters. Smlarly fr the nsde member cnstant K, K 1 R + r 1 = ν = E R r E wth ζ = R / r, C s defned as: 2 C = ζ 2 ζ [ C ν ] Fr the case f a sld shaft, r = 0, ζ = R/ r = and C = 1. We can nw slve fr the defrmatn fr a gven class f nterference fts, δ = [ K K ] pr ttal + Or, rearrangng, the cntact pressure, p, may be expressed as a functn f the nterference wthut assumptns regardng materal prperty values: 1 δ ttal p = (Mdfed Text Eq. 3-56) K + K R Budynas & Nsbett Machne Desgn Tutral 3-16: Press and Shrnk Fts 3/11

4 Example T3.16.1: Shaft & Hub Shrnk Ft Prblem Statement: A carbn-steel gear hub havng a nmnal hle dameter f 1 nch s t be shrnk-ftted t a carbn-steel shaft usng a class FN4 ft. The hub has a nmnal thckness f ½ nch. Fnd: 1. the maxmum tangental and radal stresses n the hub and shaft when the lsest ft s btaned; 2. the same as (a), except usng the tghtest ft as a cndtn. Slutn Methdlgy: Schematc: 1. Usng Table A-1-2 f ths dcument, dentfy the dametral sze ranges fr the shaft and the hle based n an ANSI US Custmary Standard class FN4 ft whch s dscussed belw. 2. Calculate the nterferences fr the lsest and tghtest fts. 3. Cmpute the nterfacal pressure fr the lsest and tghtest fts. 4. Fr each ft, calculate the radal and tangental stresses fr bth the hub and the shaft. Shaft Slutn: Hub 1. Shaft and Hub Sze Ranges: Ths prblem specfes a class FN4 ft. The class FN4 ft s a specfcatn defned by the Amercan Natnal Standards Insttute (ANSI), Amercan Standard Lmts fr Cylndrcal Parts ANSI B The FN4 ft s a Frce ft sutable fr parts whch can be hghly stressed r fr shrnk fts where the heavy pressng frces requred are mpractcal. Excerpts frm the ANSI B standard are prvded n Table A-1 f ths tutral. Frm Table A-1-2, fr a nmnal shaft/hle dameter f 1 n., the apprprate sze range s n. The allwable tlerances, n thusandths f an nch, are: Budynas & Nsbett Machne Desgn Tutral 3-16: Press and Shrnk Fts 4/11

5 Largest Tlerance Smallest Tlerance (10-3 n.) (10-3 n.) Hub hle Shaft The shaft and hub sze ranges fr a nmnal 1 n. dameter are: Largest Dameter (n.) Smallest Dameter (n.) Hub hle Shaft Dametral and Radal Interference Calculatns: Lsest Ft Tghtest Ft Hub hle Shaft The radal nterferences are therefre δ = n. (lsest ft) and δ = n. (tghtest ft). The nterference s taken as a pstve number by cnventn. Cnsequently, the radal nterferences fr the tw cases are: 3. Interfacal Pressure δ = n. (lsest ft) δ = n. (tghtest ft) The nterface pressure can be cmputed frm Mdfed Eq. (3-56): δ p = K + K R 1 ttal Fr ths prblem, the gemetrc features have been defned as: d 1n. d r = + t = n. = 1.0 n. r = 0.0 R = = 0.5 n. 2 Thus, fr the hub (utsde member), Budynas & Nsbett Machne Desgn Tutral 3-16: Press and Shrnk Fts 5/11

6 r (1 n.) ζ = = = 2 R (0.5 n.) C K ζ + 1 (2) = = = = ζ 1 (2) = [ C + ν ] = + = E (1/ps) ( ) [ ] ps Smlarly fr the shaft (nsde member) R ζ = = r 2 2 ζ + 1 C = = 1.0 ζ 1 K 1 1 = [ C ν ] = = E Fr the lsest ft case: p ( ) [ ] ps (1/ps) 1 δ = = R = ps (lsest ft) lsest ft lsest ft 8 8 K + K + Fr the tghtest ft case: p 1 δ R = ps (tghtest ft) tghtest ft tghtest ft = = 8 8 K + K + 4. Radal & Tangental Stress Calculatns fr Shaft and Hub: Lsest Ft (p = ps) Hub: Shaft: r(r = R) = p = ps r + R t( r = R) = p = pc (11 250ps)(1.6667) = = ps r R (r = R) = p = ps r R + r t( r = R) = p = pc = ( 11250ps)(1) = ps R r Tghtest Ft (p = ps) Budynas & Nsbett Machne Desgn Tutral 3-16: Press and Shrnk Fts 6/11

7 Hub: Shaft: r(r = R) = p = ps r + R t( r = R) = p = pc (25 875ps)(1.6667) = = ps r R (r = R) = p = ps r R + r t( r = R) = p = pc = ( 25875ps)(1) = ps R r Thus the shaft has equal radal and tangental stress fr each tghtness cndtn, whereas, the hub s tangental stress s cnsstently hgher than ts radal stress fr bth the lsest and the tghtest cndtn. Nte: Snce the shaft length s greater than the hub length, whch s typcal n practce, ths desgn case vlates ne f the assumptns f the nterfacal pressure develpment. In ths case, there wuld be an ncrease n the nterfacal pressure at each end f the hub. Ths cndtn f ncreased nterfacal pressure wuld typcally be accunted fr by applyng a stress cncentratn factr, K t, fr stresses calculated at pnts at the end f the hub such as, = K t actual t, tangental t = K r actual t, radal r Budynas & Nsbett Machne Desgn Tutral 3-16: Press and Shrnk Fts 7/11

8 Table A-1 LIMITS AND FITS FOR CYLINDRICAL PARTS The lmts shwn n the accmpanyng tabulatns are n thusandths f an nch. The sze ranges nclude all szes ver the smallest sze n the range, up t and ncludng the largest sze n the range. The letter symbls are defned as fllws: RC Runnng and sldng fts are ntended t prvde a smlar runnng perfrmance, wth sutable lubrcatn allwance, thrughut the range f szes. The clearance fr the frst tw classes, used chefly as slde fts, ncreases mre slwly wth dameter than the ther classes, s that accurate lcatn s mantaned even at the expense f free relatve mtn. RC1 Clse sldng fts are ntended fr the accurate lcatn f parts whch must assemble wthut perceptble play. RC2 Sldng fts are ntended fr accurate lcatn but wth greater maxmum clearance than class RC1. Parts made t ths ft mve and turn easly, but are nt ntended t run freely, and n the larger szes may seze wth small temperature changes. RC3 Precsn runnng fts are abut the clsest fts whch can be expected t run freely and are ntended fr precsn wrk at slw speeds and lght jurnal pressures, but are nt sutable where apprecable temperature dfferences are lkely t be encuntered. RC4 Clse runnng fts are ntended chefly fr runnng fts n accurate machnery wth mderate surface speeds and jurnal pressures, where accurate lcatn and mnmum play s desred. RC5 RC6 Medum runnng fts are ntended fr hgher runnng speeds, heavy jurnal pressures, r bth. RC7 Free runnng fts are ntended fr use where accuracy s nt essental r where large temperature varatns are lkely t be encuntered, r under bth f these cndtns. RC8 RC9 Lse runnng fts are ntended fr use where wde cmmercal tlerances may be necessary, tgether wth an allwance, n the external member. L Lcatnal fts are fts ntended t determne nly the lcatn f the matng parts; they may prvde rgd r accurate lcatn, as wth nterference fts, r prvde sme freedm f lcatn, as wth clearance fts. Accrdngly, they are dvded nt three grups: clearance fts, transtn fts, and nterference fts. LC Lcatnal clearance fts are ntended fr parts whch are nrmally statnary but whch can be freely assembled r dsassembled. They run frm snug fts fr parts requrng accuracy f lcatn, thrugh the medum clearance fts fr parts such as ball, race, and husng, t the lser fastener fts where freedm f assembly s f prme mprtance. LT Lcatnal transtn fts are a cmprmse between clearance and nterference fts, fr applcatn where accuracy f lcatn s mprtant, but ether a small amunt f clearance r nterference s permssble. LN Lcatnal nterference fts are used where accuracy f lcatn s f prme mprtance and fr parts requrng rgdty and algnment wth n specal requrements fr bre pressure. Such fts are nt ntended fr parts desgned t transmt frctnal lads frm ne part t anther by vrtue f the tghtness f ft, snce these cndtns are cvered by frce fts. FN Frce and shrnk fts cnsttute a specal type f nterference ft, nrmally characterzed by mantenance f cnstant bre pressure thrughut the range f szes. The nterference therefre vares almst drectly wth dameter, and the dfference between ts mnmum and maxmum value s small s as t mantan the resultng pressures wthn reasnable lmts. FN1 Lght drve fts are thse requrng lght assembly pressures and prducng mre r less permanent assembles. They are sutable fr thn sectns r lng fts r n cast-rn external members. FN2 Medum drve fts are sutable fr rdnary steel parts r fr shrnk fts n lght sectns. They are abut the tghtest fts that can be used wth hgh-grade cast-rn external members. FN3 Heavy drve fts are sutable fr heaver steel parts r fr shrnk fts n medum sectns. FN4 FN5 Frce fts are sutable fr parts whch can be hghly stressed r fr shrnk fts where the heavy pressng frces requred are mpractcal. Extracted frm Amercan Standard Lmts fr Cylndrcal Parts ANSI B , wth the permssn f the publshers, The Amercan Scety f Mechancal Engneers, Unted Engneerng Center, 345 East 47 th Street, New Yrk Lmt dmensns are tabulated n ths standard fr nmnal szes up t and ncludng 200 n. An SI versn s als avalable. Budynas & Nsbett Machne Desgn Tutral 3-16: Press and Shrnk Fts 8/11

9 Table A-1-1 RUNNING AND SLIDING FITS Dameter Sze Range (n.) Class RC1 Hle Shaft RC2 Hle Shaft RC3 Hle Shaft RC4 Hle Shaft RC5 Hle Shaft RC6 Hle Shaft RC7 Hle Shaft RC8 Hle Shaft RC9 Hle Shaft Dameter Sze Range (n.) Class RC1 Hle Shaft RC2 Hle Shaft RC3 Hle Shaft RC4 Hle Shaft RC5 Hle Shaft RC6 Hle Shaft RC7 Hle Shaft RC8 Hle Shaft RC9 Hle Shaft Budynas & Nsbett Machne Desgn Tutral 3-16: Press and Shrnk Fts 9/11

10 Table A-1-2 FORCE AND SHRINK FITS Dameter Sze Range (n.) Class FN1 Hle Shaft FN2 Hle Shaft FN3 Hle Shaft FN4 Hle Shaft FN5 Hle Shaft Dameter Sze Range (n.) Class FN1 Hle Shaft FN2 Hle Shaft FN3 Hle Shaft FN4 Hle Shaft FN5 Hle Shaft Dameter Sze Range (n.) Class FN1 Hle Shaft FN2 Hle Shaft FN3 Hle Shaft FN4 Hle Shaft FN5 Hle Shaft Budynas & Nsbett Machne Desgn Tutral 3-16: Press and Shrnk Fts 10/11

11 Table A-1-2 FORCE AND SHRINK FITS (CONTINUED) Dameter Sze Range (n.) Class FN1 Hle Shaft FN2 Hle Shaft FN3 Hle Shaft FN4 Hle Shaft FN5 Hle Shaft Budynas & Nsbett Machne Desgn Tutral 3-16: Press and Shrnk Fts 11/11