CH 17: lexible Mehnil Elements lexible mehnil elements (belts, hins, ropes re use in onveying systems n to trnsmit power over long istnes (inste of using shfts n gers. The use of flexible elements simplifies the esign n reues ost. Also, sine these elements re elsti n usully long, they ply role in bsorbing shok los n reuing vibrtions. Disvntge, they hve shorter life thn gers, shfts, et. Belts There re four bsi types of belts (Tble 17-1: lt belts ~ rowne pulleys. Roun belts ~ groove pulleys. V-belts ~ groove pulleys. Timing belts ~ toothe pulleys. Chrteristis of belt rives: Pulley xis must be seprte by ertin minimum istne. Why? Cn be use for long enters istne. Exept for timing belts, there is some slipping between belt n pulley, thus ngulr veloity rtio is not onstnt or equl to the rtio of pulley imeters. A tension pulley n be use to mintin tension in the belt. There re two min onfigurtions for belt rives; open n rosse (ig 17-1 where the iretion of rottion will be reverse for the rosse belt rive. The figure shows reversing n non-reversing belt rives, lwys there is one loose sie epening on the river pulley n the iretion of rottion. ig. (17-3 shows flt belt rive for out of-plne pulleys. ig. (17-4 shows how luthing tion n be obtine by shifting the belt from loose to tight pulley. ig. (17-5 shows two types of vrible-spee belt rives. higley s Mehil Egieerig Desig, th E. Ch.17 (R1 Pge 1 of 1
lt n Roun Belt Drives lt belt rivers proue very little noise n they bsorb more vibrtion from the system thn V-belts. Also, flt belts rives hve high effiieny of bout 98 % (sme s for gers ompre to 70-96 % for V-belts. or open belt rives, the ontt ngles re: D sin sin D C 1 D C 1 Lrger ontt-ngle for the lrge pulley where: D : imeter of lrger pulley : imeter of smller pulley C : enters istne An the length of the belt is: L 4C ( D 1 ( D D or rosse belt rives, the ontt ngle is the sme for both pulleys: sin An the belt length is: D C 1 L 4C ( D 1 ( D ore Anlysis: Tight sie tension: 1 i i T / D higley s Mehil Egieerig Desig, th E. Ch.17 (R1 Pge of 1
Loose sie tension: i i T / D Note tht D refers to the imeter of the river pulley where i : initil tension, n : hoop tension ue to entrifugl fore, : tension ue to trnsmitte torque. The totl trnsmitte fore is the ifferene between 1 & T 1 D The entrifugl tension C n be foun s: = mr where is the ngulr veloity & m is the mss per unit length. It lso n be written s: w V g where g = 9.81 m/s, w : is weight per unit length, V=πDn The initil tension n be expresse s: 1 i (1 The belting eqution reltes the possible belt tension vlues with the oeffiient of frition n it is efine s: 1 e where f: oeffiient of frition, : ontt ngle. Substituting in eqn.(1 we fin the reltion between f Note tht is the smllest vlue of the ontt ngle i n T f T e 1 i f D e 1 This eqution shows tht if trnsmitte torque. Minimum vlue of i neee to trnsmit ertin vlue of torque without slipping i is zero; then T is zero (i.e. there is no higley s Mehil Egieerig Desig, th E. Ch.17 (R1 Pge 3 of 1
Substituting in 1 & equtions we get: = + + = + + Use to fin the 1 & vlues when the belt is on the verge of slipping or to fin 1 & for smll i vlues where slipping is ourring (note tht the kineti oeffiient of frition shoul be use in suh se Plotting 1 & vs. i we n see tht the initil tension nees to be suffiient so tht the ifferene between 1 & urves is T/D. Tble 17- gives the mnufturers speifitions for the llowble tension for eh type of belts. When belt is selete, the tension in the tight sie is set to be equl to the mx llowble tension for tht belt type. However, severity of flexing t the pulley, n the belt spee ffet the belt life, thus they nee to be ounte for. Therefore the mx llowble tension is foun s: where: ( 1 =b C P C V : llowble tension per unit with for speifi belt mteril (kn/m (Tble 17- b : belt with (m C P : pulley orretion ftor (for the severity of flexing, it is foun from (Tble 17-4 for the smll pulley imeter. Use C P =1 for urethne belts C V : veloity orretion ftor. (or veloities other thn 3 m/s, it is foun from ig. 17-9 for lether belts. or polymie or urethne belts use C V =1 higley s Mehil Egieerig Desig, th E. Ch.17 (R1 Pge 4 of 1
The trnsmitte horsepower n be foun s: H V ( 1 Tn However, when esigning belt rive, esign ftor n nees to be inlue to ount for unquntifible effets. Also nother orretion ftor K S is inlue to ount for lo evitions from the nominl vlue (i.e., over los. Thus the esign horsepower is: H H nom K S n Steps for nlyzing flt belts inlue: 1. in for the smllest pulley from geometry (fin e f if neee.. rom belt mteril n spee fin C. 3. in the trnsmitte torque. T H n ( H nomk Sn n 4. rom torque T, fin the trnsmitte lo. C w V g ( 1 T 5. rom belt mteril, rive geometry & spee, fin ( 1. D 6. in ( 1 b C P C v ( 1 (( 1 7. rom ( 1, & C fin i. Note tht must be lrger thn zero i ( 1 C 8. Chek if the frition of the belt mteril is suffiient to trnsmit the torque. higley s Mehil Egieerig Desig, th E. Ch.17 (R1 Pge 5 of 1
where > 1 ( f ln 1 9. in the ftor of sfety n fs H H nomk S C C Minimum frition neee to trnsmit the lo without slipping Alterntively, the omprison n be me between the lulte i n the minimum require vlue of i See Exmple 17-1 from text V-Belts The ross setionl imensions of V-belts re stnrize. Eh letter esigntes ertin ross setion (see Tble 17-9. A V-belt n be speifie by the ross setion letter followe by the insie irumferene length. Tble 17-10 gives the stnr lengths for V-belts. However, lultions involving the belt length re usully bse on pith length for stnr belts. Tble 17-11 gives the quntity to be e to the insie length. Exmple: Pith length of C-1500 belt is: 1500 + 7 = 157 mm. The stnr ngle for the V-belts ross setion is 40 ; however the sheve ngle is slightly smller using the belt to wege itself insie the sheve to inrese frition. The operting spee for V-belts nees to be high n the reommene spee rnge is from 5 to 5 m/s. Best performne is obtine t spee of 0 m/s. or V-belts, the pith length L P, n enter-to-enter istne re foun s: L P C ( D / ( D /(4C n C 0.5 LP ( D LP ( D ( D higley s Mehil Egieerig Desig, th E. Ch.17 (R1 Pge 6 of 1
While there re no limittions on the enter-to-enter istne for flt belts, for V-belts the enter-to-enter istne shoul not exee 3(D+ beuse the exessive vibrtions of the loose sie will shorten the belt life. why? Also the enters istne shoul not be less thn D. Horsepower: Tble 17-1 gives the horsepower rting for eh belt ross-setion (oring to sheve pith imeter n belt spee. The llowble horsepower per-belt, H is foun s: H K1K H tb Power tht n be trnsmitte by eh belt where, from Tble 17-1 K 1 : ontt ngle orretion ftor (Tble 17-13. Note: the ontt ngles for V-belts re foun using the sme equtions use for flt belts. K : belt length orretion ftor (Tble 17-14. The esign horsepower is foun s: H H K n where, nom S Power tht nees to be trnsmitte from the power soure to the riven mhine H nom : nominl horsepower of the power soure. K S : servie ftor for overlos (Tble 17-15. n : esign ftor of sfety. The number of belts neee to trnsmit the esign horsepower is foun s: H N b where N H b is n integer higley s Mehil Egieerig Desig, th E. Ch.17 (R1 Pge 7 of 1
The belting eqution for V-belts is the sme eqution use for flt belts. The effetive oeffiient of frition for Gtes Rubber Compny belts is 0.513 Thus, 1 0.513 e Where the entrifugl tension is foun s: V.4 KC K C : ounts for mss of the belt (Tble 17-16. The trnsmitte fore per belt ( 1 - is foun s: H Nb 1 n( where n (r/s & re for the river pulley. f e Thus, 1 n be foun s: 1 1 f e 1 Then n be foun from: 1 1 An i is foun s: i 1 In flt-belt fore nlysis, the tension inue from bening the belt ws not ounte for expliitly (sine belt thikness is not tht lrge, however, in V-belts the effet of flexurl stress is more pronoune, n thus it ffets the urbility (life of the belt. The figure shows the two tension peks T 1 & T resulting from belt flexure. The vlues of tension peks re foun s: higley s Mehil Egieerig Desig, th E. Ch.17 (R1 Pge 8 of 1
= + = + = + = + Where, ( b 1 & ( b re the e omponents of tension ue to the flexure of the belt on the smller n lrger pulleys. K b is use to ount for belt flexure n it is foun from Tble 17-16. The life of V-belts is efine s the number of psses the belt n o (N P, n it is foun s: = [( + ( ] where K & b re foun from Tble 17-17. Then, life time (in hours is foun s: = 6 Note: K & b vlues given in Tble 17-17 re vli only for the inite rnge. Thus, if N P is foun to be lrger thn 10 9 it is reporte s N P =10 9 life tie i hours t is fou usig N P =10 9. Also, if it is foun to be less thn 10 8, the belt life is onsiere to be short n inpproprite. Steps for nlyzing V-belts inlue: in V, L P, C, n e 0.513 in H, H then the number of belts N b in,, 1, & i in T 1, T n then belt life N P & t See Exmple 17-4 from text higley s Mehil Egieerig Desig, th E. Ch.17 (R1 Pge 9 of 1
Wire Ropes Wire ropes re me out of steel wires n re use in mny pplitions (suh s hoisting, hulge, irrft, et... There re two bsi wys of wining of wire ropes: Regulr ly: wires n strns re twiste in opposite iretions (o not kink or untwist. Lng ly: wires n strns re twiste in the sme iretion (more resistne to wer n ftigue. Ropes re esignte by size n onfigurtion, for exmple: 5-mm 6x7 hulge rope mens: imeter is 5 mm n hs 6 strns eh hving 7 wires. Tble 17-4 lists some of the stnr ropes long with their properties. Also see Tble 17-7. When rope psses roun sheve, bening stress evelops (espeilly in the outer wires ue to flexing. Using mehnis priniples, the stress in one of the wires of the rope n be foun s: = where, E r : moules of elstiity of the rope w : imeter of the wire D: sheve imeter This eqution shows the importne of using lrge imeter sheves (where it reues the stress evelope in the outer wires. The reommene rtio is 400 n up. Tension in the rope using the sme stress use by bening is lle the equivlent bening lo, b, n it is foun s: = = where, A m : is the re of the metl in the rope, n it is usully A m =0.38 (or from Tble 17-7 w : imeter of the wire & D: sheve imeter higley s Mehil Egieerig Desig, th E. Ch.17 (R1 Pge 10 of 1
Wire ropes re selete oring to two onsiertions: Stti onsiertions: the bility of the rope to rry the los. Wer life (ftigue onsiertions: the bility of the rope to live for ertin number of loing yles. Stti onsiertions: irst step is to etermine the tension use in the rope by the los (this inlues the e weight n tension use by elertion n shok los or exmple, the tension use in hoisting rope ue to lo n elertion /eelertion is: = ( + ( + where, W: totl weight of the lo m: number of ropes supporting the lo w: weight per unit length of the rope l: suspene length of the rope : mximum elertion/eelertion experiene : grvity elertion The tension ue to los is then ompre to the ultimte tensile lo of the rope to fin the stti ftor of sfety. u = strength of the rope nominl re of the rope Mximum lo tht n be supporte Thus, the stti ftor of sfety is: = However, the ultimte tensile lo must be reue ue to the inrese tension use by flexing the rope over the sheve n thus the ftor of sfety n be foun s: = *more urte* Tble 17-5 gives the minimum rope ftors of sfety for ifferent pplitions. higley s Mehil Egieerig Desig, th E. Ch.17 (R1 Pge 11 of 1
tigue onsiertions: The mount of wer tht ours in ropes epens on the bering pressure on the rope use by the sheve n by the number of bens (number of the psses of the rope over the sheve of the rope uring opertion. The llowble ftigue tension (ftigue strength for rope is foun s: Where, = Mximum tension tht n be supporte uner ertin bering pressure (P for ertin number of bens. (P/S u : bering pressure to ultimte strength rtio. It is foun oring to the speifie life from ig 17-1. S u : ultimte tensile strength of the wires. : imeter of the rope & D: imeter of the sheve. It shoul be note tht S u is the ultimte strength of the wires not the strength of the rope. (it usully not liste in the tbles but it n be etermine from hrness test. o Approximte rnges of the ultimte strength of wires S u for ifferent wire mterils re liste below: Improve plow steel (monitor Plow steel Mil plow steel 1655 < S u < 1930 MP 1448 < S u < 1665 MP 141 < S u < 1448 MP Thus, the ftigue ftor of sfety n be foun s: = It shoul be unerstoo tht the ftigue filure in wire ropes is not suen, s in soli boies, but rther progressive. It shows s breking of the outsie wires (sine they re subjete to the highest stress. Therefore, it n be etete by perioi inspetion. See Exmple 17-6 from text higley s Mehil Egieerig Desig, th E. Ch.17 (R1 Pge 1 of 1