At the end of this lesson, the students should be able to understand:

Transcription

1 Instructional Objctivs: At th nd of this lsson, th studnts should b abl to undrstand: Dsign thod for variabl load Equivalnt strss on shaft Dsign basd on stiffnss and torsional rigidit Critical spd of shaft 8.2. Dsign of Shaft for variabl load Dsign of shaft for strngth involvs crtain changs whn it is actd upon b variabl load. It is rquird to calculat th an strss and strss aplitud for all th loads, nal, axial, bnding and torsion. Thraftr, an of th dsign thods for variabl load, that is, Sodrbrg, Goodan or Grbr critria is utilizd. Onc again, th failiar dsign diagra for variabl load in trs of th strss aplitud and th an strss is rproducd blow. Grbr Strss Aplitud a A (dsign point) Goodan Sodrbrg Man Strss ut Fig Diagra for dsign undr variabl load

2 A is th dsign point, for which, th strss aplitud is a and an strss is. In th Sodrbrg critrion th an strss atrial proprt is th ild point, whras in th Grbr and th Goodan critria th atrial proprt is th ultiat strss ut For th fatigu loading, atrial proprt is th nduranc liit, in rvrs bnding. Th corrsponding quations for all th thr abov critria ar givn as, Goodan critrion: Sodrbrg critrion: a a ut FS FS Grbr (8.2.) critrion: FS a FS + ut 2 = Whr, a = Strss aplitud; = Enduranc liit; = Man strss; = Yild point; ut = Ultiat strss and FS= factor of saft. Siilar quation (8.2.) also can b writtn for th shar strss. For th dsign of shaft, it is ost coon to us th Sodrbrg critrion. Hnc, w shall liit our discussion onl to Sodrbrg critrion. Noral strss quation is givn as, (8.2.2) Kfa FS k ultipling b, = f a FS q Siilarl, shar strss quation is givn as Kfsτa τ τ τ FS

3 (8.2.3) τ K τ τ ultipling b τ, + τ = = τ fs a τ fs In quations (8.2.2) and (8.2.3), to considr th ffct of variabl load, th noral strss aplitud, a is ultiplid b th fatigu strss concntration factor, K f and th corrsponding tr, shar strss aplitud is ultiplid b a fatigu strss concntration factor in shar, K fs. Th phsical significanc of quations (8.2.2) and (8.2.3) is that, th ffct of variabl strss on th achin br (lft hand sid of th quations), has bn ffctivl dfind as an quivalnt static strss. Thrfor, th probl is tratd as a dsign for static loads. Hr, or τ ar quivalnt to allowabl τ strss, or. Hraftr, convntional failur thoris can b usd to FS FS coplt th dsign. q Maxiu shar strss thor It stats that a achin br fails whn th axiu shar strss at a point xcds th axiu allowabl shar strss for th shaft atrial. Thrfor, (8.2.4) τ =τ = +τ 2 q 2 ax allowabl q 2 substitution of q and τ q fro (8.2.3) will giv th rquird shaft diatr Dsign basd on Stiffnss In addition to th strngth, dsign a b basd on stiffnss. In th contxt of shaft, dsign for stiffnss ans that th latral dflction of th shaft and/or angl of twist of th shaft should b within so prscribd liit. Thrfor, dsign for stiffnss is basd on latral stiffnss and torsional rigidit.

4 Latral stiffnss Lt us considr a ba loadd as shown in Fig Th ba dflcts b du to th load P. So th rquirnt for th dsign is that whr, on has to liit th dflction. Hnc, th dsign procdur is as follows, Dtrin th axiu shaft dflction, using an of th following thods, Intgration thod Mont-ara thod, or Enrg thod (Thor of Castigliano) P L Fig Dflction of a ba Now, th dflction, = f (applid load, atrial proprt, ont of inrtia and givn dinsion of th ba). Fro th xprssion of ont of inrtia, and known dsign paratrs, including, shaft dinsion is obtaind Torsional rigidit To dsign a shaft basd on torsional rigidit, th liit of angl of twist should b known. Th angl of twist is givn as follows, TL θ rad = GI p or, 584TL θ = Gd ( k ) dg d = 584ΤL G(-k ) θ O 4 4 dg (8.2.5) Whr, θ = angl of twist L = lngth of th shaft

5 2 G = shar odulus of lasticit I p = Polar ont of inrtia Th liiting valu of θ varis fro 0.3 dg/ to 3 dg/ for achin tool shaft to lin shaft rspctivl. With th knowldg of dsign paratrs, th shaft dinsion can b obtaind fro (8.2.5) A not on critical spd of rotating shaft Critical spd of a rotating shaft is th spd whr it bcos dnaicall unstabl. It can b shown that th frqunc of fr vibration of a non-rotating shaft is sa as its critical spd. Th quation of fundantal or lowst critical spd of a shaft on two supports is, g(w + W W ) f critical = 2 π (W + W W ) (8.2.6) 2 2 n n n Whr, W, W 2. : wights of th rotating bodis, 2. : dflctions of th rspctiv bodis This particular quation (8.2.6) has bn drivd using th following assuption. Assuptions: Th shaft is wightlss Th wights ar concntratd and Barings/supports ar not flxibl Whr, W,W 2 : Wights of th rotating bodis and 2 : Dflctions of th rspctiv bodis

6 Th oprating spd of th shaft should b wll abov or blow a critical spd valu. Thr ar nubr of critical spds dpnding upon nubr of rotating bodis. Sapl probl Dsign a solid shaft of lngth, carring a load of 5 kn at th cntr and is sipl supportd as shown in figur. Th axiu shaft dflction is. E=200GPa. P Solution Th axiu dflction of th shaft is givn as, L 3 PL ax = 48EI πd whr, for a solid shaft, I = 64 4 O 4PL πE 3 3 π d 4 4 O = = ax = 57 fro standard shaft siz, d O = 58 This probl is not a coplt on. Th agnitud of torqu on th shaft is not spcifid. Th dsign calculations should b first basd on strngth, whr, both bnding ont and torsion ar rquird. With th givn liits of latral dflction and angular twist, th dsign should b chckd. Qustions and answrs Q. What is an quivalnt strss? A. Whn a shaft is subjctd to variabl load, both th strss aplitud and an strss can b convnintl rprsntd as quivalnt strss. Th quivalnt strss is concptuall an quivalnt static strss. Q2. What ar th liiting valus of th angl of twist of a shaft?

7 A2. Th liiting valu of angl of twist of a shaft varis fro 0.3 dg/ to 3 dg/ for achin tool shaft to lin shaft rspctivl. Q3. What ar th assuptions ad to driv th quation for critical frqunc? Wh critical frqunc is iportant in shaft dsign? A3. Th assuptions ad to driv th quation for critical frqunc ar, Th shaft is wightlss, th wights ar concntratd and barings/supports ar not flxibl. Th critical spd valu hlps a dsignr to st th liit of shaft spd. To avoid rsonanc, th shaft spd should b uch highr or lowr than th critical spd. Rfrncs. J.E Shigl and C.R Mischk, Mchanical Enginring Dsign, McGraw Hill Publication, 5 th Edition M.F Spotts, Dsign of Machin Elnts, Prntic Hall India Pvt. Liitd, 6 th Edition, Khuri, R.S. and Gupta J.K., Txt book on Machin Dsign, Eurasia Publishing Hous, Nw Dlhi. 4. Shara, C.S. and Purohit Kaalsh, Dsign of Machin Elnts, Prntic Hall of India, Nw Dlhi, 2003.