ME311 Machine Design

Transcription

1 ME311 Machne Desgn Lectue 8: Cylnes W Dnfel Nv017 Fafel Unvesty Schl f Engneeng Thn-Walle Cylnes (Yu aleay cvee ths n Bee & Jhnstn.) A essuze cylne s cnsee t be Thn-Walle f ts wall thckness s less than.5% (1/40th) f ts nse amete. Une these cntns: 1. We assume the stess stbutn s unfm thughut the wall thckness bth n the h (ccumfeental) ectn an n the lngtunal (axal) ectn.. We assume that the aal stess s neglgble. Then: H θ t Axal z t an θ Hamck Page 46

2 Thn-Walle Cylne Quz Knwng all that yu abut essuze cylnes (.e., that the h stess s twce the lngtunal stess), whch ectn wul yu ect that these essuze cylnes wll factue? A. Lengthwse ue t h stess. B. Csswse ue t axal stess. C. On a 45 angle ue t shea stess tque. A B C Stesses n Thck-Walle Cylnes Thck-Walle cylnes have a wall thckness geate than 1/0 th f the aveage aus. They ae essuze ntenally an/ extenally. The ncal stesses ae ccumfeental (h) c, aal, an lngtunal (axal) l. l c Hamck Sectn 10.3.

3 ( ) Ccumfeental & aal Stesses F the geneal case f bth ntenal an extenal essue, the ccumfeental an aal stesses at aus n the wall ae: Whee the ± s: ± ( ) f ccumfeental, an - f aal stess. / F the secal case f nly ntenal essue, 0, an the stesses at aus ae: l ( ) 1 ± c The sgn cnventn s the same. Eqns 10.0/10. Eqns 10.3/10.4 Lngtunal Stesses The lngtunal stess s smly gven by a Fce/Aea, whee the Fce s tmes the ccula nse aea π, an the Aea s the annula aea f the cylne css sectn, π( - ), : l Ths s geneally nly cnsee f the case f ntenal essuzatn ( 0). Un-numbee Equatn just belw Eqn l c

4 Stesses vs. aus Fst, the easy bsevatn: aal stesses at the nne an ute sufaces ae equal t mnus the essuzatn. If a suface s unessuze, the aal stess thee s ze. If a suface s essuze, the aal stess thee -, because t s n cmessn. Nw let s lk at an ntenally essuze cylne, an hw the aal an ccumfeental stesses vay acss the wall thckness at aus. l c ( ) 1 ± ( s ccumfeental, - s aal ) Eqns 10.3/10.4 Stesses f Intenal Pessuzatn H Stess (KSI) 16 1 Thck-Walle Cylne wth ntenal essue f 5300 s aal aus (n.) HOOP STESS ADIAL STESS 1 ± ( s h, - s aal )

5 Stesses vs. aus - Intenal Pessue aal stess s as ecte: s at the nne, essuze suface. 0 at the unessuze ute suface. H stess s: Maxmum at the nne suface, 13.9 ks. Lwe, but nt ze, at the unessuze ute suface, 8.5 ks. Lage n magntue than the aal stess Lngtunal stess s (tust me): 4.3 ks, cnsee as a unfm, aveage stess acss the thckness f the wall. Nw let s lk at an extenally essuze cylne. H aal Stesses f Extenal Pessuzatn Stess (KSI) H aal Thck-Walle Cylne wth extenal essue f 5300 s. ADIAL STESS HOOP STESS aus (n.) 1 ± ( s h, - s aal )

6 Stesses vs. aus - Extenal Pessue aal stess s as ecte: 0 at the unessuze nne suface s at the ute, essuze suface. aal H stess s: Mnmum at the ute suface, -8.9 ks. Maxmum at the (unessuze) nne suface, -14. ks. Lage than the aal stess Lngtunal stess s: Nt usually cnsee f extenal essuzatn. H Bust Tubng Analyss ID ; OD ; 16,000s What was the h stess n the tube? Analyze t as bth thn-wall an thck-wall. Whch s t?

7 Stess (PSI) 80,000 70,000 60,000 50,000 40,000 30,000 0,000 10, ,000-0,000-30,000 Stesses vs f Tube Ccumfeental (H) aal aus (n.) 1 ± tatng ngs Stesses (aal & tangental) ae smla t thse n thck-walle cylnes. The fces cme fm centfugal las n all f the ng atcles nstea f fm the ntenal essue. Cntns: t. t s cnstant 3 ν h ρω 8 whee 1 3ν 3 ν 3 Lb n ρ Mass Densty, ω aans sec, ν Pssn' s 386n s aal 3 ν ρω 8 Eqn at Eqn ω t

8 tatng ngs: Effect f Cente Be aus n Stesses tatng ng vs Cente Be aus (0.15 -> ") Stesses f a nch thck steel sk tatng at 5000 PM. Stess (PSI) 5,000 0,000 15,000 1,037 10, Tangental (H) Stess t 5, aal Stess ω aus (nches) At what aus s the eak aal stess? emembe Dffeentatn? 3 ν ρω 8 aal 4 3 ν ρω 8 ( ) 3 ()(1) 0 at eak ( ) x x n nx n1 3

9 Pess Fts In a ess ft, the shaft s cmesse an the hub s exane. Befe Hub aal ntefeence, δ Hub Afte Shaft Shaft Pess Fts Pess fts, ntefeence fts, ae smla t essuze cylnes n that the lacement f an vesze shaft n an unesze hub esults n a aal essue at the nteface. Hub

10 Chaactestcs f Pess Fts 1) The shaft s cmesse an the hub s exane. ) Thee ae equal an ste essues at the matng sufaces. 3) The elatve amunt f cmessn an exansn eens n the stffness (elastcty an gemety) f the tw eces. Hub Shaft 4) The sum f the cmessn an the exansn equals the ntefeence ntuce. 5) The ctcal stess lcatn s usually the nne amete f the hub, whee max tensle h stess ccus. Analyss f Pess Fts Stat by fnng the nteface essue. If shaft an hub ae f the same mateal: Eδ ( ) ( ) ( ) Whee δ s the ADIAL ntefeence f hub an shaft f the same mateal, wth mulus f elastcty, E. Eqn 10.5, eaange If the shaft s sl, 0 an Eδ 1 Eqn 10.53, eaange 0 nly f the shaft has a hle n t. s whee the shaft an hub cntact.

11 Analyss f Pess Fts If the shaft an hub ae f ffeent mateals E ν δ Once we have the essue, we can use the cylne equatns t cmute the h stesses at the nteface. A) The ID f the hub s tensle: B) The OD f the shaft s cmessve: c E c ν Eqn 10.51, eaange E,ν Eqn Eqn E,ν ν, Pssn - f shaft s sl F Inf Only Stan Analyss f Pess Fts The ess ft has n axal essue, s l 0, an t s a baxal stess cntn. The ccumfeental stan Eqn c ν ε c E E whch equals the aal stan (because C π). Because the aal change δ ε, we get the ncease n Inne aus f the ute membe (hub): ν E δ An the ecease n Oute aus f the nne membe (shaft): E δ ν Eqn Eqn These tw eflectns a u t the aal Intefeence: δ δ δ

12 Summay f Pess Fts 1. Select amunt f ntefeence. Be caeful abut aal ametal. Intefeence s eally small maybe 1 t tenths f a ecent f amete.. Cmute the essue at the matng suface. If same mateals, use Eqn an f shaft s sl ( 0): If ffeent mateals, use Eqn (fle) 3. Cmute the tensle h stess n the ute ece. Eqn ( ) ( ) ( ) E δ 1 E δ E E ν ν δ t Pess Ft Pblem A 1 nch amete shaft s t be esse nt a 3 nch amete hub wth a aal ntefeence f nch. Bth ae AISI 1080 steel, Q&T 800 C. What s the esultng suface essue an h stess n the hub? Cautn: Mn yu a an ametes.

13 Pess Fts: Fce & Tque F The assembly fce eque wll be F max πlµ whee the nteface essue µ the ceffcent f fctn L The tque caacty avalable s T F πlµ whee the nteface aus, as befe. T We knw hw t cmute the nteface essue f these equatns! Hamck Sectn Pess Ft Pblem F the 1 n. amete steel shaft that we just calculate, f the fctn ceffcent s 0.15 an the hub s 1 n. thck, what ae: 1. The fce neee t ess the ats tgethe?. The maxmum tque the jnt cul wthstan?

14 Shnk Fts If heatng clng a at t acheve a shnk ft, the eque aal ntefeence s: δ α T whee s the nteface aus α s the ceffcent f themal exansn T s the temeatue change T select an amunt f ntefeence see ANSI/ASME tables f class FN1 (lght) t FN5 (Heavy-ve) fts. They gve ntefeence n 0.001" n amete f a ange f ametes Ex: FN4 f 0.95 t 1.19" amete, ntefeence 1 t.3 mls n amete. Nte: Cnvesn f Temeatue Change s NOT the same as Cnvetng Temeatue C 5/9 ( F 3 ) C F Blng F 9/5 C C 180 F C 5/9 F F 9/5 C 0 Feezng 3

15 Pess Ft Pblem 3 F the 1 nch amete steel shaft that we just calculate, hw many egees F wul the hub nee t heate t be able t assemble the ats wthut fcng? If we chse t cl the shaft nstea, hw many egees C wul the shaft nee t be cle t the same thng?