Module 2 - GEARS Lecture 10 - SPUR GEAR DESIGN Contents 10.1 Problem 3 Spur gear design 10.1.1 Buckingham Approach 10.1.2 AGMA Approach 10.1 Problem 3 - Design of Spur gear A pair of gears is to be designed to transmit 30kW power from a pinion running at 960rpm to a gear running at 320rpm. Design the gears so that they can last for 10 8 cycles. Assume 20 o full depth involute spur gear for the system. Motor shaft diameter is 30mm. Data given: W = 30 kw; n 1 = 960 rpm; n 2 = 320 rpm; Life = 10 8 cycles; 20 o full depth involute spur gear. Solution: i = n 1 / n 2 = 960 / 320 = 3 In order to keep the size small and meet the centre distance, Z 1 = 17 chosen Z 2 = i Z 1 = 3 x 17 = 51 n xx 2 2 960 1 100. 48 rad/s 60 60 Torque is given by, w 30x1000 T1 298.57Nm 100.48
rom Lewis equation for pinion we have, 2T t 1 p [ ] 2 p bym byz1m (10.1) SAE 1050 hardened by OQT with permissible bending strength of 532 and hardness of 223Bhn is selected for pinion and SAE 1045 hardened by WQT with permissible bending strength of 487 and hardness of 215 Bhn is selected for the gear from Table 10.1. ace width b = 10m is chosen for both wheels. Table 10.1 Safe static stresses for use in the Lewis equation Material [σ ] MPa BHN Gray cast iron ASTM 25 ASTM 35 ASTM 50 Cast steel(low carbon) 0.2% C not heat treated 0.2% C WQT orged carbon steel SAE 1020 case hardened and WQT SAE 1030 not heat treated SAE 1035 not heat treated SAE 1040 not heat treated SAE 1045 hardened by WQT SAE 1045 hardened by WQT SAE 1050 hardened by OQT Alloy steel SAE 2320 case hardened and WQT SAE 2345 hardened by OQT SAE 3115 case hardened and OQT SAE 3145 hardened by OQT SAE 3245 hardened by OQT SAE 4340 hardened by OQT SAE 4640 hardened by OQT SAE 6145 hardened by OQT 122 183 228 304 380 274 304 350 380 456 487 532 761 761 563 806 989 989 837 1019 174 212 223 180 250 156 180 190 202 205 215 223 225 475 212 475 475 475 475 475
10.1.1 Buckingham approach: The preliminary dimensions are found from Lewis equation and then they are checked for dynamic loads by Buckingham equation. rom equation (10.1) substituting the value of b = 10m, we have, T 1 p 3 5YZ1m [ ] ( p (10.2) rom Table 10.2, for the pinion Y = 0.25808 for Z 1 = 17 or the gear, Y = 0.39872, for Z 2 = 51 or gear, Y[σ] g = 0.39872x 487 = 194.17 or pinion, Y[σ] p = 0.25808 x 542 = 139.87 Table 10.2 Values of the Lewis form factor Y Number of teeth 12 13 14 15 16 17 18 19 20 21 22 24 26 28 30 34 38 45 50 60 75 100 Φ=20 a=0.8m* b=m 0.335 12 0.348 27 0.359 85 0.370 13 0.379 31 0.387 57 0.395 02 0.401 79 0.407 97 0.413 63 0.418 83 0.428 06 0.436 01 0.442 94 0.449 20 0.459 20 0.467 40 0.478 46 0.484 58 0.493 91 0.503 45 0.513 21 Φ=20 a=m b=1.25m 0.229 60 0.243 17 0.255 30 0.266 22 0.276 10 0.285 08 0.293 27 0.300 78 0.307 69 0.314 06.0319 97 0.330 56 0.339 79 0.347 90 0.355 10 0.367 31 0.377 27 0.390 93 0.398 60 0.410 47 0.422 83 0.435 74 Φ=25 a=m b=1.25m 0.276 77 0.292 81 0.307 17 0.320 09 0.331 78 0.342 40 0.352 10 0.360 99 0.369 16 0.376 71 0.383 70 0.396 24 0.407 17 0.416 78 0.425 30 0.439 76 0.451 56 0.467 74 0.476 81 0.490 86 0.505 46 0.520 71
Hence, for the same face width pinion will be weaker and consideration for the design is, T 298.57 x1000 13610 3 5YZ m 5x0.25808x17m m p 3 3 1 532MPa (10.3) m = 2.93 mm. Since motor shaft diameter is 30 mm, to get sufficiently large pinion m = 4 mm is taken. Table 10.3 Data for pinion and gear Wheel Z m b=10m d V =wrv Material Hardness Pinion 17 4mm 40 mm 68mm 3.42 m/s SAE1050 223 Gear 51 4mm 40 mm 204mm 3.42 m/s SAE1045 215 We will now use Buckingham dynamic load approach for the design. t = T 1 /r 1 = 298.57/0.034 = 8781 N Buckingham dynamic load is given by, i 9.84V(Cb+ t ) 9.84V +0.4696 Cb+t (10.4) or V=3.42 m/s permissible error is e= 0.088 mm from ig.10.1. rom Table 10.4, if we choose I class commercial cut gears, expected error is 0.050 for m=4mm. In order to keep the dynamic load low precision cut gears are chosen. So, e = 0.0125
ig. 10.1 Permissible error Table 10.4 Expected error in tooth profile Gear quality and expected error e Module irst class commercial gears Carefully cut gears Precision gears Up to 4 5 6 7 8 9 10 0.050 0.056 0.064 0.072 0.080 0.085 0.090 0.025 0.025 0.030 0.035 0.038 0.041 0.044 0.0125 0.0125 0.0150 0.0170 0.0190 0.0205 0.0220 Table 10.5 Value of C Tooth form Material of pinion and gear 14.5 o Cast iron and cast iron steel and cast iron steel and steel 20 o ull depth Cast iron and cast iron steel and cast iron steel and steel 20 o Stub tooth Cast iron and cast iron steel and cast iron steel and steel C 5720 e 7850 e 11440 e 5930 e 8150 e 11860 e 6150 e 8450 e 12300 e rom Table 10.5, if material for both gear and pinion are steel, then,
C = 11860e = 11860 x 0.0125 = 148.25 Substituting the values t = 8781 N, C = 148.25, V=3.42 m/s, b= 40mm in eqn. (10.4), Buckingham dynamic load is given by, 9.84x3.42(148.25x40 +8781) i 5464N 9.84x3.42+0.4696 148.25x40+8781 (10.5) d = t + i = 8781 + 5464 = 14245 N Beam strength of the pinion is given by, tp = bym [σ] p = 40x0.25808 x4x542 = 22381 N Since tp (22381)> d (14245) the design is safe from tooth bending failure consideration. Wear strength of the pinion is given by, [ ] H bd I C p ts 1 2 (10.6) rom Table 10. 6 for steel vs steel, pinion and gear C p = 191 MPa 0.5 and substituting i = 3, Ф=20 0 we o o sin cos i sin 20 cos20 3 I 0.1205 2 i1 2 31 Table 10.6 Elastic coefficient C p for spur gears in MPa 0.5 Pinion Material (µ=0.3 in all cases) Gear Material Steel Cast iron Al Bronze Tin Bronze Steel, E=207 GPa 191 166 162 158 Cast iron, E=131 GPa 166 149 149 145 Al Bronze, E=121 GPa 162 149 145 141 Tin Bronze, E=110 GPa 158 145 141 137
Surface fatigue strength of the pinion material is σ sf = σ sf K L K R K T σ sf = 2.8(Bhn) 69MPa = 2.8 x 223-69 = 555.4MPa K L = 0.9 for 10 8 cycles life from graph1 K R = 1.0 taken for 99 reliability K T = 1.0 for operating temperature <120 C (assumed) o Table 10.7 Surface fatigue strength σ Pa) for metallic spur gears (10 7 sf (M cycle life 99% reliability and temperature <120 o C) Material Steel Nodular iron σ sf (MPa) 2.8 (Bhn)-69MPa 0.95(2.8(Bhn)-69MPa) Cast iron, grade 20 379 Cast iron, grade 30 482 Cast iron, grade 40 551 Tin Bronze, AGMA 2C (11% Sn) 207 Aluminium Bronze (ASTM 148 52) (Alloy 9C H.T.) 448 ig.10.2 Life factor K L
Table 10.8 Reliability factor K R Reliability (%) K R 50 1.25 99 1.00 99.9 0.80 Surface fatigue strength of the pinion material is σ sf = σ sf K L K R K T = 555.4x0.9x1x1 = 500MPa Assuming, factor of safety, s = 1.1 [σ H ] = σ sf /s = 500/1.1 = 455MPa Wear strength of the pinion is: 2 2 [σ H] 455 ts = bd1 I = 40x68x0.1205 =1860 N C p 191 Since ts (1860) << d (14245), the design is not safe. Revision is necessary. As the SAE1050 can attain a hardness of 800 VPN(~750 Bhn) after oil quenching, increase the hardness to 475 Bhn and increase the b to 13m = 13 x 4 = 52 mm. rom Table 10.7, we know that, σ sf = 2.8(Bhn) 69MPa = 2.8 x 475-69 = 1261MPa K L = 0.9 for 10 8 cycles life from ig.10.1 K R = 1.0 taken for 99 reliability K T = 1.0 for operating temperature <120 o C Assumed. Surface fatigue strength of the pinion material is σ sf = σ sf K L K R K T = 1261x0.9x1x1 = 1135MPa
ig.10.3 Effect of carbon content on the hardness of fully hardened steel Assuming factor of safety, s = 1.1 [σh] = σ sf /s = 1135 /1.1 = 1032MPa 2 2 [σ H ] 1032 ts = bd1i =52x68x0.1205 =12439 N C p 191 Since ts (12439)< d (14245), still it is not safe. Hence increase the module to 5mm. Table 10.9 Properties of pinion an d gear Wheel Z m b=13m d V =wrv Material Hardness Pinion 17 5mm 65 mm 85mm 4.27 m/s C 50 475 Gear 51 5mm 65 mm 255mm 4.27 m/s C 45 450
With new dimensions d = 16098 N ts = 19436 N. Since ts > d, the revised design is safe from surface fatigue (pitting) considerations. If b = 50 mm, d = 13186 N ts = 14951 N, ace width of 50 mm is adequate 10.1.2 AGMA Approach Data given: W = 30 kw; n 1 = 960 rpm; n 2 = 320 rpm; Life = 10 8 cycles; 20 o full depth involute spur gear. Solution: i = n 1 / n 2 = 960 / 320 = 3 In order to keep the size of gears small and avoid interference, Z 1 = 17 is chosen. Z2 = i Z 1 = 3 x 17 = 51 2n 2x960 60 60 1 1 = = 100.48rad / s 1000W 1000x30 T1 298.57Nm 1 100.48 AGMA equation for tooth bending stress is, t bmj K v Ko Km d 1 = m Z 1 2T1 2 bz m J 1 K K K [ ] v o m ace width, b= 10 to 13 m.
b = 10 m is assumed for the first trial. J = 0.34404 for pinion Z 1 = 17 mating with gear Z 2 =51 or gear J = 0.40808 These values are obtained from the table Table 10.10 AGMA geometry factor J for teeth having Ø = 20 o, a=1m, b=1.25m and r f =0.300m Number Number of teeth in mating gear of teeth 1 17 25 35 50 85 300 1000 18 0.244 86 0.324 04 0.332 12 0.338 40 0.344 04 0.350 50 0.355 94 0.361 12 19 0.247 94 0.330 29 0.338 78 0.345 37 0.351 34 0.358 22 0.364 05 0.369 63 20 0.250 72 0.336 00 0.344 85 0.351 76 0.358 04 0.365 32 0.371 51 0.377 49 21 0.253 23 0.341 24 0.350 44 0.357 64 0.364 22 0.371 86 0.378 41 0.384 75 22 0.255 52 0.346 07 0.355 59 0.363 06 0.369 92 0.377 92 0.384 79 0.391 48 24 0.259 51 0.354 68 0.364 77 0.372 75 0.380 12 0.388 77 0.396 26 0.403 60 26 0.262 89 0.362 11 0.372 72 0.381 15 0.388 97 0.398 21 0.406 25 0.414 18 28 0.265 80 0.368 60 0.379 67 0.388 51 0.396 73 0.406 50 0.415 04 0.423 51 30 0.268 31 0.374 62 0.385 80 0.395 00 0.403 59 0.413 83 0.422 83 0.431 79 34 0.272 47 0.383 94 0.396 71 0.405 94 0.415 17 0.426 24 0.436 04 0.445 86 38 0.275 75 0.391 70 0.404 46 0.414 80 0.424 56 0.436 33 0.446 80 0.457 35 45 0.280 13 0.402 23 0.415 79 0.426 85 0.437 35 0.450 10 0.461 52 0.473 10 50 0.282 52 0.408 08 0.422 08 0.435 55 0.444 48 0.457 78 0.469 75 0.481 93 60 0.286 13 0.417 02 0.431 73 0.443 83 0.455 40 0.469 60 0.482 43 0.495 57 75 0.289 79 0.426 20 0.441 63 0.454 40 0.466 68 0.481 79 0.495 54 0.509 70 100 0.293 13 0.435 61 0.451 80 0.465 27 0.478 27 0.494 37 0.509 09 0.524 35 150 0.297 38 0.445 30 0.462 26 0.476 45 0.490 23 0.507 36 0.523 12 0.539 54 300 0.301 41 0.455 26 0.473 04 0.487 98 0.502 78 0.520 78 0.537 65 0.555 33 Rack 0.305 75 0.465 54 0.484 15 0.499 88 0.534 67 0.534 67 0.552 72 0.571 73 The tooth bending stress is given by, 0.5 0.5 78 (200V) Kv 1.15 78 is assumed.
K o = 1.25 is taken assuming uniform power source and moderate shock load from the table 7 K m = 1.3 assuming accurat e mounting and precision cut gears for face width of about 50mm. Table 10.11 -Ov erload factor K o Driven Machinery Source of power Uniform Moderate Shock Heavy Shock Uniform 1.00 1.25 1.75 Light shock 1.25 1.50 2.00 Medium shock 1.50 1.75 2.25 Table 10.12 Load distribution factor K m ace width ( mm) Characteristics of Support 0-50 150 225 400 up Accurate mountings, small bearing 1.3 1.4 1.5 1.8 clearances, minimum deflection, precision gears Less rigid mountings, less accurate gears, 1.6 1.7 1.8 2.2 contact across the full face Accuracy and mounting such that less than Over Over Over Over full-face contact exists 2.2 2.2 2.2 2.2 Substituting these values in the equation, 2T K K K 298570 x1.15 x1.25 x1.3 10m x17xm x 0.34404 1 1 2 v o m 2 bz1 m J 9539 3 m
σ e = σ e k L k v k s k r k T k f k m The pinion is of steel C50 OQT with 223Bhn hardness and tensile strength of 660MPa and the gear is of C45 OQT with hardness 210Bhn and tensile strength of 465MPa. or pinion σ e = 0.5 σ ut = 0.5 x 660 = 330MPa k L = 1 for bending, k V = 1 assumed expecting m to be <5mm; k S = 0.73 from the ig.10.4 for σut= 660MPa, k r = 0.897 for 90% reliability ig. 10. 4 Surface factor k s Table 10.13 Reliabili ty factor Kr Reliability factor R 0.50 0.90 0.95 0.99 0.999 0.9999 actor K r 1.000 0.897 0.868 0.814 0.753 0.702 k T =1 assumed based on operating temperature <120 o C k f = 1.- and k m = 1.33 for σ ut = 660MPa ( Ultimate tensile strength = 660 MPa for SAE 1050 OQT condition) σ e = σ e k L k v k s k r k T k f k m = 330x1x1x0.73x0.897x1x1x1.33 = 287.4MPa actor of safety on bending of 1.5 assumed
[σ] = σ e / s = 287.4 / 1.5 =191.6MPa ig.10.5 - Miscellaneous effects factor k m rom tooth bending fatigue considerations, 9539 1 3 m [ ] 191.6 Solving the equation we get m = 3.68mm Now take m=4 mm as the next standard value. rom this module, the dimensions calculated are given in Table 10.14. Table 10.14 Dimensions of pinion and gear Wheel Z m b=10m d V =wrv Material Hardness Pinion 17 4mm 40 mm 68mm 3.42 m/s C 50 223 Gear 51 4mm 40 mm 204mm 3.42 m/s C 45 205 t = T 1 / r 1 = 29857/34 = 8781N The tooth has to be checked from surface durability considerations now. The contact stress equation of AGMA is given below: C K K K t H p V o m bd1i C p = 191 MPa 0.5 from the table for steel vs steel
0 Substituting i =3, Ф=20 we get I= 0.1205 o o sin cos i sin 20 cos20 3 I 0.1205 2 i1 2 31 0.5 0.5 0.5 0.5 78 (200V) 78 (200x3.42) Kv 1 78 78.15 rom Table 10.11 and 10.12, Ko = 1.25 and K m = 1.3 assumed as in the case of bending stress calculation C K K K 191 t H p V bd1i o m 8781x1.15x1.25x1.3 40x68x0.1205 σh = 1209MPa The surface fatigue strength of the pinion material is given by, σ sf = σ sf K L K R K T Where σ sf = 2.8(Bhn) 69MPa = 2.8 x 223-69 = 555.4MPa K L = 0.9 for 10 8 cycles life from graph1 K R = 1.0 taken for 99% reliability K o T = 1.0 for operating temperature <120 C assumed. Substituting the values in the equation, σ sf = σ sf K L K R K T = 555.4x0.9x1x1 = 500MPa Assuming a factor of safety, s = 1.1 rom ig. 10.2 and Table 10.8, we get, [σ H ] = σ sf /s = 500/1.1 = 455MPa σ H = 1209MPa
Since σ H (1209) >> [σ H ] (455), the design is not safe and surface fatigue failure will occur. Solution: Increase the surface hardness of the material to 475Bhn and also increase the b to 13m = 13 x 4 = 52 mm rom ig. 3 we get, Surface fatigue strength of the pinion material as σsf = σ sf K L K R K T where σ sf = 2.8(Bhn) 69MPa = 2.8 x 475-69 = 1261MPa K L = 0.9 for 10 8 cycles life from graph1 KR = 1.0 taken for 99% reliability K <120 o T = 1.0 for operating temperature C Assumed. Substituting these values we get, σ sf = σ sf K L K R K T = 1261x0.9x1x1 = 1135MPa Assuming a factor of safety s = 1.1 [σh] = σ sf /s = 1135 /1.1 = 1032MPa C K K K 191 t H p V o m bd1 I 8781x1.15x1.25x1.3 52x68x0.1205 As σ H (1185) > [σ H ] (1032) the design is not safe from surface durability considerations. Hence increase the module to 5mm and take b=10m C K K K 191 t H p V o m bd1 I 7025x1.17x1.25x1.3 50x85x0.1205
σ H =975MPa < [σ H ] (1032MPa). Hence the design is safe from surface durability consideration. inal specification of the pinion and gear are given in the Table 10.20 and 10.21. Table 10.20 Values for gear and pinion Wheel Z m b=10m d Pinion 17 5mm 50 mm 85mm Gear 51 5mm 50 mm 255mm Table 10.21 Specification of gear and pinion Wheel Material Steel Hardness Manufacturing quality Pinion SAE1050 OQT 475Bhn Precision cut Gear SAE 1045 OQT 450Bhn Precision cut -------------------