12. Spur Gear Design and selection. Standard proportions. Forces on spur gear teeth. Forces on spur gear teeth. Specifications for standard gear teeth
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1 . Spur Gear Desig ad selecio Objecives Apply priciples leared i Chaper 11 o acual desig ad selecio of spur gear sysems. Calculae forces o eeh of spur gears, icludig impac forces associaed wih velociy ad clearaces. Deermie allowable force o gear eeh, icludig he facors ecessary due o agle of ivolue of ooh shape ad maerials seleced for gears. Desig acual gear sysems, icludig specifyig maerials, maufacurig accuracy, ad oher facors ecessary for complee spur gear desig. Udersad ad deermie ecessary surface hardess of gears o miimize or preve surface wear. Udersad how lubricaio ca cushio he impac o gearig sysems ad cool hem. Selec sadard gears available from sockig maufacurers or disribuors. Sadard proporios America Sadard Associaio (ASA) America Gear Maufacurers Associaio (AGMA) Brow ad Sharp 14 ½ deg; 20 deg; 25 deg pressure agle ull deph ad sub ooh sysems P N RAO 1 P N RAO 2 Specificaios for sadard gear eeh orces o spur gear eeh Iem Pressure agle Addedum (i.) Dededum (i.) ull deph & piches coarser ha /P 1.250/P /P 1.250/P ull deph & piches fier ha /P 1.2/P ½ full deph 14½ 1/P 1.157/P P N RAO 3 Trasmied force Normal force or separaig force r Resula force θ pressure agle a θ r cos θ P N RAO 4 orces o spur gear eeh Example Problem -1: orces o Spur Gear Teeh Power, P; P T 63,000 or 63,000 T P 20-ooh, pich, 1-ich-wide, 20 piio rasmis 5 hp a 1725 rpm o a 60-ooh gear. Torque, T r ad r D p /2 Combiig he above we ca wrie 2 T D p Deermie drivig force, separaig force, ad maximum force ha would ac o mouig shafs. P N RAO 5 P N RAO 6 1
2 Example Problem -1: orces o Spur Gear Teeh 20-ooh, pich, 1-ich-wide, 20 piio rasmis 5 hp a 1725 rpm o a 60-ooh gear. Deermie drivig force, separaig force, ad maximum force ha would ac o mouig shafs. (2-6) T P 63,000 T 63,000P T (63,000) i-lb Example Problem -1: orces o Spur Gear Teeh (co d.) id rasmied force: 2T Dp (2)13 i-lb 146 lb 2.5 i id separaig force: a θ 146 lb a 20 (-3) (-1) id pich circle: (11-4) Dp Np 20 eeh Dp 2.5 i eeh/i diameer P N RAO 7 id maximum force: 53 lb (-2) r cos θ 146 lb r cos 20 r 155 lb P N RAO Surface Speed Example Problem -2: Surface Speed Surface speed (V m ) is ofe referred o as pich-lie speed I previous problem, deermie he surface speed: Vm π D (-7) V m V m π Dp f/mi π Dp m/mi -- Meric uis 1000 or Vm π Dp f Vm π 2.5 i 1725 rpm i Vm 19 f/mi (-5) P N RAO 9 P N RAO 10 Sregh of Gear Teeh orces o Gear Tooh Lewis form facor mehod igure orces acig o idividual gear ooh. P N RAO McGraw-Hill, Hamrock, Jacobso ad Schmid P N RAO 2
3 Lewis equaio S Y b s P s Allowable dyamic bedig force (lb) S Allowable sress (lb/i 2 ). Use edurace limi ad accou for he fille as he sress coceraio facor b ace widh (i.) Y Lewis form facor (Table.1) P d Diameral pich d P N RAO 13 Table.1 Lewis form facors (Y) P N RAO 14 Example Problem -3: Sregh of Gear Teeh Example Problem -3: Sregh of Gear Teeh (co d.) I Example Problem -1, deermie he force allowable ( s ) o hese eeh if he piio is made from a AISI 4140 aealed seel, he maig gear is made from AISI 1137 ho-rolled seel, ad log life is desired. Piio: S.5 Su.5 (95 ksi) 47.5 ksi (-9) s S b Y id Lewis form facor (Y) from Table -1, assumig full-deph eeh: Gear: S.5 ( ksi) 44 ksi Y ,000 (1).421 s s 2316 lb (Table -1) Y ,500 (1).320 s Use s 1900 lb for desig purposes. s 1900 lb P N RAO 15 P N RAO 16 Classes of Gears Trasmied load depeds o he accuracy of he gears Gear Maufacure Casig Machiig ormig Hobbig Shapig ad Plaig ormig sampig P N RAO 17 orce Trasmied Trasmied load depeds o he accuracy of he gears A dyamic load facor is added o ake care of his. Trasmied force d Dyamic force Commercial Vm 600 P N RAO 1 3
4 Precisio Hobbed or shaved Classes of Gears 00 + Vm Carefully cu V m 50 + V m Desig Mehods Sregh of gear ooh should be greaer ha he dyamic force; s d You should also iclude he facor of safey, s N sf d P N RAO 19 P N RAO 20 Table.2 Service acors ace widh of Gears 1 < < < < < < < < 2 Uiform load wihou shock Medium shock, freque sars Moderaely heavy shock Heavy shock Relaio bewee he widh of gears ad he diameral pich.5 < b < P d P d P N RAO 21 P N RAO 22 Example Problem -4: Desig Mehods If, i Example Problem -1, he gears are commercial grade, deermie dyamic load ad, based o force allowable from Example Problem -3, would his be a accepable desig if a facor of safey of 2 were desired? Use surface speed ad force rasmied from Example Problems -2 ad -3. Dyamic load: Vm (-10) (146 lb) lb Comparig o force allowable: s 1900 lb lb > 421 lb This desig mees he crieria. P N RAO 23 Example Problem -5: Desig Mehods Spur gears from he caalog page show i igure -3 are made from a.2% carbo seel wih o special hea reame. Wha facor of safey do hey appear o use i his caalog? Try a 24-ooh a 100 rpm gear for example purposes. rom Appedix 4, a AISI 1020 ho-rolled seel would have.2% carbo wih a S y 30 ksi ad S u 55 ksi. Therefore: id Dp : S.5 Su S.5 (55 ksi) 27.5 ksi Dp Np Dp 24 2 i (11-4) P N RAO 24 4
5 Example Problem -5: Desig Mehods (co d) id Vm : Example Problem -5: Desig Mehods (co d) Se s ad solve for : Vm π Dp f Vm π 2 i (100 rpm) i (-5) s Vm lb lb id s : Vm 942 f/mi T Dp 2 (-3) s S b Y (-9) T 202 lb 2 i 2 T 202 i-lb Y ,500 (¾).302 s (from Table -1) s 519 lb P N RAO 25 (2-6) T 202 (100) P 5. hp 63,000 63,000 or (-6) Vm 202 (942) P 33,000 P N RAO 5. hp 33, Example Problem -5: Desig Mehods (co d) Example Problem -6: Desig Mehods Compared o caalog: Nsf hp - calculaed hp - caalog 5. Nsf Pair of commercial-grade spur gears is o rasmi 2 hp a a speed of 900 rpm of he piio ad 300 rpm for gear. If class 30 cas iro is o be used, specify a possible desig for his problem. The followig variables are ukow: P d, D p, b, N,, θ. As i is impossible o solve for all simulaeously, ry he followig: P d, N 4, θ 14½, 2 Solve for face widh b. Miscellaeous properies: Appears o be reasoable value. Maufacurer may also have reduced is raig for wear purposes as hese are o hardeed gears. Dp Np 4 4 i Ng Np Vr 4(3) 144 eeh (11-4) P N RAO 27 P N RAO 2 Example Problem -6: Desig Mehods (co d.) Surface speed: Vm π Dp 4 i 900 rpm Vm π i/f Vm 943 f/mi idig force o eeh: 33,000 hp Vm 33,000 (2) 943 (-5) (-6) 70 lb P N RAO 29 Example Problem -6: Desig Mehods (co d.) Dyamic force Sice widh b is he ukow: Vm lb s ad s S b Y S b Y N sf (-10) (-) Class 30 CI; S u 30 ksi; S.4 S u (.4 is used because cas iro): S ksi Y.344 (Table -1) P N RAO 30 5
6 Example Problem -6: Desig Mehods (co d.) Subsiuig: S b Y Nsf,000 b () b 1.0 iches Check raio of widh o pich: < b <.5 < 1 <.5.66 < 1 < 1.04 This is a accepable desig. May oher desigs are also possible. P N RAO 31 (-9) (-14) To icrease he dyamic beam sregh of he gear Icrease ooh size by decreasig he diameral pich Icrease face widh upo he pich diameer of he piio Selec maerial of greaer edurace limi Machie ooh profiles more precisely Mou gears more precisely Use proper lubrica ad reduce coamiaio P N RAO 32 Buckigham Mehod of Gear Desig ig..4 Expeced error i ooh profiles I offers greaer flexibiliy Expeced error is based o differe-pich eeh More coservaive desig P N RAO 33 P N RAO 34 Table.3 Values of C for e ich Buckigham Mehod of Gear Desig Maerial Gray iro ad Gray iro Gray iro ad seel Seel ad seel 14 ½ deg 00 1,100 1, deg 30 1,140 1,660 d Vm (bc + ) 0.05 V + (bc + ) m 0.5 P N RAO 35 P N RAO 36 6
7 ig..5 Recommeded maximum error i gear eeh Example Problem -7: Buckigham Mehod of Gear Desig ad Expeced Error A pair of seel gears is made from aealed AISI Each gear has a surface hardess of BHN 350. The piio is a precisio, 16 pich, 20 ivolue, wih 24 eeh 1 ich wide. The gear has 42 eeh. To rasmi 3 hp a a speed of 3450 rpm for a safey facor of 1.4, is his a suiable desig? Su 95 ksi S.5 Su.5(95 ksi) 47.5 ksi (Appedix 4) (11-4) Dp Np i 16 P N RAO 37 P N RAO 3 Example Problem -7: Buckigham Mehod of Gear Desig ad Expeced Error (co d.) id orque: Example Problem -7: Buckigham Mehod of Gear Desig ad Expeced Error (co d.) id surface speed: T P 63,000 T P (63,000) (2-6) Vm π Dp π 1.5 i 3450 rpm Vm f/i (-5) T 3 (63,000) 3450 T 55 i-lb id force rasmied: 2T Dp 2(55 i-lb) 1.5 i (-3) 73 lb P N RAO 39 Vm 1355 f/mi id force allowable (s): s S b Y Y ,500 (1).337 s 16 (-9) (from Table -1) s 1000 P N RAO 40 Example Problem -7: Buckigham Mehod of Gear Desig ad Expeced Error (co d.) Expeced error from igure -4: e.0005 The value of C from Table -3 for seel ad seel ad 20 ivolue agle gears is 1660: C.0005 (1000) 1660 C 30 Solve for dyamic load usig Buckigham s equaio:.05vm ( bc + ) d + 1/ 2 (-15).05 Vm + ( bc + ).05 (1355) ( ) / 2.05 (1355) + ( ) 699 lb s > 699 Wear sregh (Buckigham) D W P 2 D g b K g Dp + Dg w ooh wear sregh (lb) D p diameral pich of piio (i.) D g diameral pich of gear (i.) b face widh (i.) K g load sress facor (Appedix 13) This mees he crieria. P N RAO 41 P N RAO 42 7
8 Example Problem -: Wear of Gears I prior example problem, verify he surface is suiable for wear cosideraios. or wear use N ss 1.2 Wear formula: Dp b Q Kg (-16) Example Problem -: Wear of Gears (co d.) Subsiuig io equaio -16: 1.5 (1) 1.27 (270) 514 id Q : (-17) 2 Ng Q Ng + Np 2 (42) Q Q 1.27 (from Appedix 13) Kg 270 P N RAO 43 This would o be suiable. Try if surfaces each had a BHN 450. K g 470 (from Appedix 13) 1.5 (1) (1.27) (470) > > 699 This would ow be accepable if he gear eeh were hardeed o a BHN of 450. P N RAO 44
A = p 4 (0.05)2 = 1.9635(10-3 ) m 2. J = p 2 (0.025)4 = 0.61359(10-4 ) m 4. s = P A = 2(10 3 ) 1.9635(10-3 = 1.019 MPa. t = Tc J = 500(0.
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