Power Transmission Shafting

Power Transmission Shafting
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  Polver ransmission shafting Continuous mechanical power s usually fransmitted along andbetween otating shafts. The ransfer betlveen shafu is accomplished ygears, belts, chains or other similar means or matching thetorque/speed characteristics f the interconnected hafts eg. a cNneeds gears between the engine crankshaft and drive wheel hal f-shafts.shafu rotafing only at constant speed n (rev/s) are consid.ered ere,and as shafts are usually statically determinate they may be examinedby the techniques of elementary statics. AIso, since -power = force ( i.l) .linear velocity m/s)in translational applicatioru and?ower = torque ( Nm) * angular velocity (= 2nn rad/s) in rotationalappiications,. . . . then it follows that torque s a major load. component n power bansmittingrotating shafbs. Torque maybe ransferr.a o or from the end ofone shaft by u second oaxial haft this s a puretorque, a twist about he shaft axis. The ransferis accomplished y a shaft couoling wh_ichconsists of two often identical half-couplirgr which are dath mounted.on one of the shafu before being connected ogether non-perrnanently.The mounting of a half-coupting on a shaft must prevent relafiverotation when torque is applied - this is achieved by a positivelyinierlocking element exempiified by a key (or a derivative of this suchas a spline) or by friction. Shown here s the free body of a shaft keyedto a coupling half - the torque keyon the shaft is equilibrated by sh#r an equal artd opposite orqueon the half-coupling. Freebodies of the individual eiements larify how the key,s aces ,ve rise oT .qoitibrating couple on both he shaft and the hjf-couplt g,The ypical friction-based mounting below comprises split taperedbush which is forced by screws nto a mating aper machined n thebore of the half-coupling. Thebush confractsonto the shaft ,causing high {T \-t halF.z\co upling FBD fshalt r,ffi FBD ofEhaltFBD f half+oupling  force n pulley piessllres and coffesponding high surface riction i,vhjch equilibratesthe torque - provided dghterirg is sufficient.Torque may be transferred also at any point along a shaft by u gear.,belt :,?:iflil, pulley, or chain sprocket or example, mounted shait +F on the shaft as described above, These common .&.f' . g elements pply orces offset r-om he shaft axis,r=F' and therefore sefui orque T must beaccompanied y a conesponding adial oad withconsequential ending this s why the generalised haft design model- ,ffi.incorporate both orque and bending moment oad components.A spur gear and a belt pulley are sketched, ach ubjected o loadingtangential o its effectiae r pitch cylinder of diameter D. The oad on the spur geat arises rom inter-tooth contact .. ..ipursearptch with ts mating ger and comprises wo ',,.i;i4l^ds components the useful angential omponent ffilffiF, and he unwanted ut unavoidable adial \ry cofnponent { (commonly 0.36 F, ). Gegr orms other han spur -SVe rise also o a loadcomponent parallel to the shaft axis - but for all gears, shifting the offsetforceasabove, =F.D/2. .-:A beIt, being flexible, cannot withstand compression the puIley istherefore subjected o two sfand tensions F ,* and F * both of whichmust exceed zero. The net torque T = (F-u* F ) D/2is clockwisehere. The tension ratio F* ,,1F,* must iie between a lower iimit of unity(corresponding o no torque transfer) and an upper limit dictated byincipient belt slip around the pulley. (rhis may be demonstrated byequally pulling the ends of a bit of rope vwapped around a posf thenprogressively ncreasing he puil on one end.) A chain sprocket ssimilar though the minimum tension rl:ray top to zero due to thepositive drive not relying on friction.The mounting of a half-coupling and the like on a shaft is, localIpintrinsically indeterminate. This is exemplified by the pulley bossillustrated which extends rom A to C along the stepped shaft. Theactual distributions of bending and torque n the shaft 4 qare shown. To the left of A and to the right of C there rt,*tff', -\are no immed,iate ransverse oads, so the bending ^ T\4':- to the shaft's tiffness nd also distributes he pulleyload graduaily nto the shaft, o he bending momentvariation s non-linear. imilarly he orque s someconstant alue zero ere) n the shaft o the eft of A T . ,0. aFDlAl a are no immed,iate ransverse oads, o he bending ^ T\fi'l-moment s linear. But within AC the boss contributes rtvvrc i r,,,,,t and another onstant alue T to the right of C -  l,.trpr1s f.a -If.< 1s(au*indeterminate fashion.lsrcgen Se E l.r.r A Qgx^.eeex No attempt is madeto resolve his bending/torsional\u arternPt 15 maqe to resolve hls bendlng/ torsional ndeterminacy -the actual distributions are nstead eplace dby approximatedistributions based on external oads concentrated at the mid-point B.This leads o a bending moment diagram characterised. y sh.aightsegments and a torque diagram which increases tepwise at B. Theapproximation is justified in view of unknown stiffness details and.stress concentration due to shoulders used or positive longifudinallocation) and mountings (square keys, shrink fits etc.) n thii example across-section between B and C is critical since the d.iameter s thesmaller Do (neglecting he blending radii which reduce stressconcentration at the shoulder), he torque s the fully developed T, andthe bending moment is to all intents and purposes the peak MB.Shafts are suPported in two bearings (sliding or rolling) which allow theshafts to turn freely - there is no appreciable orque exlrted by thebearings. A sliding bearing needs a lubricant film in the clearince spacebetween shaft and bearing bush; n the fully hydrodynamic bearing illustrated the oil is dragged into ctearance ress oitrirm,thickness the wedge-shaped up ursi.g u 'tk u' ,,,ffin'*'ohe wedge-shaped up u,rri.g 'tk u' ,,,ffin'*'o tr^,H:,Tll;;iJffiffi*?i ffi the shaft without metal-to-metal llifi BEA*N. 'nlo,*ul3 contact and little friction. tr lslling bearing (of which the ball bearing llustrated. s but oneexample) omprises wo hardened. aces between which a number ofidentical hardened rements o[. Not shown are various necessarydetails such as means or preventing axial movement of races, or dustsealing and for 1ubricant etention.There are many different embodiments f these wo bearing classes. bearing .tpport, a shaft over 1ftrytu axial ength and so s Int insicallyindeterminate no matter what ts form - howeier it is approximated ikethe boss ABC above by a concentrated eaction at its *jai_point.The ransverse orces on a shaft are serdom oplanar, o t is usuanynecessary o resolve orces nto two convenient rthogonal ongifudinalplanes and to separatery onsider ending n each. oi exampre hehorizontal shaft ABCD, igure (a) is simpiy supported n bearings t A,D and carries gears or pulleys at B, C. SLIDING EAFING  The offset gear/belt forces are shifted to the shaft axis as above, givingrise to the shaft bending loads, igure (a). The coffesponding orquediagram figure @) shows that torque is transmitted between B and C(there s no torque sink outside BC as he bearings can offer tuIr4';1hIno appreciable orsional resistance). he bending oads are ;. $+qresolved nto vertical and horizontal planes, igure (c). [:;.',1iii1#.,,MtTreating each plane separately, he bearing reactions are \ /computed and the bending moment diagram completed,resulting in two component BMDs figure (d). Since moments arevectors, he resultant bendiog moment at any cross-section along theshaft can be found easily from the two components by Pythagoras asM = {( M*' +Mu2 ), recalling that it is the moment magnitude only whichis relevant,Finallp some practical tips for shaft analysis -. Equilibrium in the axial view is usually examined ust, to figureout the magnitude of the torque(s) and the ocations betweenw$ch torque is transmitted. Thus the shaft, figure (e) similar toABCD above, s freely supported in bearings and equipped witho a 150 mm diameter belt pulleywith strand tensions given as2100 and 500 No d spur gear of 100 mm pitchdiameter with unknowntangential and radial forces. Each orce s shifted f necessary o pass hrough he shaft axiswhilst infroducing he corresponding orque igure (f). Thusstarting urith the two known belt forces, he net clockwise orquedue o their shift s (2100 500 x 0.075 120 Nm.This s the torque T transmitted between elt and gear. Theanticlockwise orque ntroduced when he gear's angential orce sshifted o the axis s F, x 0.05 Nm, assuming F units are N. Forrotational quilibrium herefore 0.05 , = \20, ie. F, = 2400 N.Finally F, may be taken as 0.36 F, = 864 N (though note hat hree rtr zpyFr 2rooJ4,qqLEsoor,rfffiosr, rtt+uit11.ttt1t1ttvlttt*t:iiwsntf sltttst
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