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## Q. 9.7

Design a pair of spur gears to be used as a part of the drive for a chipper to prepare pulpwood for use in a paper mill. Intermittent use is expected. An electric motor transmits 3.0 horsepower to the pinion at 1750 rpm and the gear must rotate between 460 and 465 rpm. A compact design is desired.

## Verified Solution

Step 1. Considering the transmitted power, $P$, the pinion speed, $n_P$, and the application, refer to Figure 9–11 to determine a trial value for the diametral pitch, $P_d$. The overload factor, $K_o$, can be determined from Table 9–1, considering both the power source and the driven machine.

For this problem, $P = 3.0$ hp and $n_P = 1750$ rpm, $K_o = 1.75$ (uniform driver, heavy shock driven machine). Then $P_{des}$ = (1.75)(3.0 hp) = 5.25 hp. Try $P_d = 12$ for the initial design.
Step 2. Specify the number of teeth in the pinion. For small size, use 17 to 20 teeth as a start.
For this problem, let’s specify $N_P = 18$.
Step 3. Compute the nominal velocity ratio from $VR = n_P/n_G$.
For this problem, use $n_G = 462.5$ rpm at the middle of the acceptable range.
$VR = n_P/n_G = 1750/462.5 = 3.78$

Step 4. Compute the approximate number of teeth in the gear from $N_G = N_P(VR)$.
For this problem, $N_G = N_P(VR) = 18(3.78) = 68.04$ Specify $N_G = 68$.
Step 5. Compute the actual velocity ratio from $VR = N_G/N_P$.
For this problem, $VR = N_G /N_P = 68/18 = 3.778$.
Step 6. Compute the actual output speed from $n_G = n_P(N_P/N_G)$.
For this problem, $n_G = n_P(N_P /N_G)$ = (1750 rpm)(18/68) = 463.2 rpm. OK.
Step 7. Compute the pitch diameters, center distance, pitch line speed, and transmitted load and judge the general acceptability of the results.
For this problem, the pitch diameters are:
$D_P = N_P /P_d = 18/12 = 1.500$ in
$D_G = N_G /P_d = 68/12 = 5.667$ in

Center distance:
$C = (N_P + N_G)/(2P_d) = (18 + 68)/(24) = 3.583$ in
Pitch line speed: $n_t = πD_Pn_P/12 = [π(1.500)(1.750)]/12 = 687$ ft/min
Transmitted load: $W_t = 33 000(P)/v_t = 33 000(3.0)/687 = 144$ lb
These values seem to be acceptable.

Step 8. Specify the face width of the pinion and the gear.
For this problem: Lower limit = $8/P_d = 8/12 = 0.667$ in.
Upper limit = $16/P_d = 16/12 = 1.333$ in
Nominal value = $12/P_d = 12/12 = 1.00$ in. Use this value.
Step 9. Specify the type of material for the gears and determine $C_p$ from Table 9–7.
For this problem, specify two steel gears. $C_p = 2300$.
Step 10. Specify the quality number, Av, using Table 9–5 as a guide. Determine the dynamic factor
from Figure 9–16.

For this problem, specify $A_v = 11$ for a wood chipper. $K_v = 1.35$.
Step 11. Specify the tooth form, the bending geometry factors for the pinion and the gear from
Figure 9–10 and the pitting geometry factor from Figure 9–17.
For this problem, specify 20° full depth teeth. $J_P = 0.325, J_G = 0.410, I = 0.104$.
Step 12. Determine the load-distribution factor, $K_m$, from Equation (9–17) ($K_m = 1.0 + C_{pf} + C_{ma}$) and Figures 9–12 and 9–13. The precision class of the gear system design must be specified. Values may be computed from equations in the figures or read from the graphs.

For this problem: $F = 1$.00 in, $D_P = 1.500. F/D_P = 0.667$. Then $C_{pf} = 0.042$.
Specify open gearing for the wood chipper, mounted to the frame. $C_{ma} = 0.264$.
Compute: $K_m = 1.0 + C_{pf} + C_{ma} + 0.042 + 0.264 = 1.31$
Step 13. Specify the size factor, $K_s$, from Table 9–2.
For this problem, $K_s = 1.00$ for $P_d = 12$.
Step 14. Specify the rim thickness factor, $K_B$, from Figure 9–14.
For this problem, specify a solid gear blank. $K_B$ = 1.00.

Step 15. Specify a service factor, $SF$, typically from 1.00 to 1.50, based on uncertainty of data.
For this problem, there is no unusual uncertainty. Let $SF$ = 1.00.
Step 16. Specify a reliability factor using Table 9–11 as a guideline.
For this problem, specify a reliability of 0.99. $K_R$ = 1.00.

Step 17. Specify a design life. Compute the number of loading cycles for the pinion and the gear.
Determine the stress cycle factors for bending ($Y_N$) and pitting ($Z_N$) for the pinion and the gear.
For this problem, intermittent use is expected. Specify the design life to be 3000 hours, similar to agricultural machinery. The numbers of loading cycles are:
$N_{cP} = (60)(3000$ hr$)(1750$ rpm$)(1) = 3.15 × 10^8$ cycles
$N_{cG} = (60)(3000$ hr$)(463.2$ rpm$)(1) = 8.34 × 10^7$ cycles
Then, from Figure 9–21, $Y_{NP} = 0.96, Y_{NG} = 0.98$. From Figure 9–22, $Z_{NP} = 0.92, Z_{NG} = 0.95$.

Step 18. Compute the expected bending stresses in the pinion and the gear using Equation (9-16).
\begin{aligned}&s_{t P}=\frac{W_{t} P_{d}}{F J_{P}} K_{o} K_{s} K_{m} K_{B} K_{v}=\frac{(144)(12)}{(1.00)(0.325)}(1.75)(1.0)(1.31)(1.0)(1.35)=16455 \mathrm{psi} \\&s_{t G}=s_{t P}\left(J_{P} / J_{G}\right)=(16455)(0.325 / 0.410)=13044 \mathrm{psi}\end{aligned}
Step 19. Adjust the bending stresses using Equation 9-30.
For this problem, for the pinion:
$s_{a t P}>s_{t P} \frac{K_{R}(S F)}{Y_{(N P)}}=(16455) \frac{(1.00)(1.00)}{0.96}=17141 \mathrm{psi}$
For the gear:
$s_{\text {atG }}>s_{t G} \frac{K_{R}(S F)}{Y_{(N G)}}=(13044) \frac{(1.00)(1.00)}{0.98}=13310 \mathrm{psi}$
Step 20. Compute the expected contact stress in the pinion and the gear from Equation (9-23). lote that this value will be the same for both the pinion and the gear.
$s_{c}=C_{P} \sqrt{\frac{W_{t} K_{o} K_{s} K_{m} K_{v}}{F D_{P} I}}=2300 \sqrt{\frac{(144)(1.75)(1.0)(1.31)(1.35)}{(1.00)(1.50)(0.104)}}=122933 \mathrm{psi}$
Step 21. Adjust the contact stresses for the pinion and the gear using Equation (9-31).
$s_{a c P}>s_{C P} \frac{K_{R}(S F)}{Z_{N P}}=(122933) \frac{(1.00)(1.00)}{(0.92)}=133623 \mathrm{psi}$
For the gear:
$s_{a c G}>s_{C G} \frac{K_{R}(S F)}{Z_{N G}}=(122933) \frac{(1.00)(1.00)}{(0.95)}=129403 \mathrm{psi}$

Step 22. Specify materials for the pinion and the gear that will have suitable through hardening or case hardening to provide allowable bending and contact stresses greater than those required from Steps 19 and 21. Typically the contact stress in the pinion is the controlling factor. Refer to Figures 9–18 and 9–19 and Tables 9–9 and 9–10 for data on required hardness. Refer to Appendices 3 to 5 for properties of steel to specify a particular alloy and heat treatment.

For this problem, the contact stress for the pinion is the controlling factor, as is often the case.
A steel must be specified that is rated to handle approximately $s_{ac} = 133.6$ ksi. First check Figure 9–19 to explore whether or not through-hardened steel is practical. We can use the equation for Grade 1 steel in U.S. units to determine the required Brinell hardness number, $HB$.

Reqd. $HB = (s_{ac} – 29.10)/0.322$= (133.6 ksi – 29.10)/0.322 = 324
This value is well within the recommended hardness for through-hardened steels. Using Appendix 4, we can specify SAE 4140 OQT 1000 steel having HB = 341 and 18% elongation indicating good ductility.
We can also check the required hardness for the gear that has a required $s_{ac} = 129.4$ ksi.

Reqd. $HB = (s_{ac} – 29.10)/0.322$ = (129.4 ksi – 29.10)/0.322 = 311
This value can be met by SAE 4140 OQT 1100 steel having $HB$ = 311 and 20% elongation. However, because both the pinion and the gear experience nearly the same contact stress, it may be prudent to specify the same heat treatment for both to permit them to be produced by the same process.

 TABLE 9–1 Suggested Overload Factors, $K_o$ Driven Machine Power source Uniform Light shock Moderate shock Heavy shock Uniform 1.00 1.25 1.50 1.75 Light shock 1.20 1.40 1.75 2.25 Moderate shock 1.30 1.70 2.00 2.75

 TABLE 9–7 Elastic Coefficient, $C_p$ Gear material and modulus of elasticity, $E_G, lb/in^2 (MPa)$ Pinion material Modulus of elasticity, $E_P,lb/in^2$ (MPa) Steel $30×10^6$$(2×10^5)$ Malleable iron $25×10^6$$(1.7×10^5)$ Nodular iron $24×10^6$$(1.7×10^5)$ Cast iron $22×10^6$$(1.5×10^5)$ Aluminum bronze $17.5×10^6$$(1.2×10^5)$ Tin bronze $16×10^6$$(1.1×10^5)$ Steel $30×10^6$ 2300 2180 2160 2100 1950 1900 $(2×10^5)$ 191 181 179 174 162 158 Mall. Iron $25×10^6$ 2180 2090 2070 2020 1900 1850 $(1.7×10^5)$ 181 174 172 168 158 154 Nod. Iron $24×10^6$ 2160 2070 2050 2000 1880 1830 $(1.7×10^5)$ 179 172 170 166 156 152 Cast iron $22×10^6$ 2100 2020 2000 1960 1850 1800 $(1.5×10^5)$ 174 168 166 163 154 149 Al. bronze $1.75×10^6$ 1950 1900 1880 1850 1750 1700 $(1.2×10^5)$ 162 158 156 154 145 141 Tin bronze $16×10^6$ 1900 1850 1830 1800 1700 1650 $(1.1×10^5)$ 158 154 152 149 141 137

 TABLE 9–5 Recommended AGMA Quality Numbers Application Quality number Application Quality number Cement mixer drum drive A11 Small power drill A9 Cement kiln A11 Clothes washing machine A8 Steel mill drives A11 Printing press A7 Grain harvester A10 Computing mechanism A6 Cranes A10 Automotive transmission A6 Punch press A10 Radar antenna drive A5 Mining conveyor A10 Marine propulsion drive A5 Paper-box-making machine A9 Aircraft engine drive A4 Gas meter mechanism A9 Gyroscope A2 Machine tool drives and drives for other high-quality mechanical systems Pitch line speed (fpm) Quality number Pitch line speed (m/s) 0–800 A10 0-4 800–2000 A8 4–11 2000–4000 A6 11–22 Over 4000 A4 Over 22

 TABLE 9–2 Suggested Size Factors, $K_s$ Diametral pitch, $P_d$ Metric module, m Size factor, $K_s$ ≥ 5 ≤ 5 1.00 4 6 1.05 3 8 1.15 2 12 1.25 1.25 20 1.40

 TABLE 9–11 Reliability Factor, $K_R$ Reliability $K_R$ 0.90, one failure in 10 0.85 0.99, one failure in 100 1.00 0.999, one failure in 1000 1.25 0.9999, one failure in 10 000 1.5
 TABLE 9–10 Allowable Stress Numbers for Iron and Bronze Gears Material designation Minimum hardness at surface (HB) Allowable bending stress number, $s_{at}$ Allowable contact stress number, $s_{ac}$ Gray cast iron, ASTM A48, as cast Class 20 5 35 50 345 Class 30 174 8.5 59 65 448 Class 40 201 13 90 75 517 Ductile (nodular) iron, ASTM A536 60-40-18 annealed 140 22 152 77 530 80-55-06 quenched and tempered 179 22 152 77 530 100-70-03 quenched and tempered 229 27 186 92 634 120-90-02 quenched and tempered 269 31 214 103 710 Bronze, sand-cast, $s_{u min}$ = 40 ksi (275 MPa) 5.7 39 30 207 Bronze, heat-treated,  $s_{u min}$ = 90 ksi (620 MPa) 23.6 163 65 448

 TABLE 9–9 Allowable Stress Numbers for Case-Hardened Grade 1 Steel Materials Allowable bending stress number, $s_{at}$ Allowable contact stress number, $s_{ac}$ Hardness at surface (ksi) (Mpa) (ksi) (Mpa) Flame- or induction-hardened 50 HRC 45 170 170 1172 54 HRC 45 175 175 1207 Carburized and case-hardened 55–64 HRC 55 379 180 1241

 APPENDIX 3 Design Properties of Carbon and Alloy Steel Material designation (SAE number) Condition Tensile strength Yield strength Ductility (percent elongation in 2 in) Brinell hardness (HB) (ksi) (MPa) (ksi) (MPa) 1020 Hot-rolled 55 379 30 207 25 111 1020 Cold-drawn 61 420 51 352 15 122 1020 Annealed 60 414 43 296 38 121 1040 Hot-rolled 72 496 42 290 18 144 1040 Cold-drawn 80 552 71 490 12 160 1040 OQT 1300 88 607 61 421 33 183 1040 OQT 400 113 779 87 600 19 262 1050 Hot-rolled 90 620 49 338 15 180 1050 Cold-drawn 100 690 84 579 10 200 1050 OQT 1300 96 662 61 421 30 192 1050 OQT 400 143 986 110 758 10 321 1117 Hot-rolled 65 448 40 276 33 124 1117 Cold-drawn 80 552 65 448 20 138 1117 WQT 350 89 614 50 345 22 178 1137 Hot-rolled 88 607 48 331 15 176 1137 Cold-drawn 98 676 82 565 10 196 1137 OQT 1300 87 600 60 414 28 174 1137 OQT 400 157 1083 136 938 5 352 1144 Hot-rolled 94 648 51 352 15 188 1144 Cold-drawn 100 690 90 621 10 200 1144 OQT 1300 96 662 68 496 25 200 1144 OQT 400 127 876 91 627 16 277 1213 Hot-rolled 55 379 33 228 25 110 1213 Cold-drawn 75 517 58 340 10 150 12L13 Hot-rolled 57 393 34 234 22 114 12L13 Cold-drawn 70 483 60 414 10 140 1340 Annealed 102 703 63 434 26 207 1340 OQT 1300 100 690 75 517 25 235 1340 OQT 1000 144 993 132 910 17 363 1340 OQT 700 221 1520 197 1360 10 444 1340 OQT 400 285 1960 234 1610 8 578 3140 Annealed 95 655 67 462 25 187 3140 OQT 1300 115 792 94 648 23 233 3140 OQT 1000 152 1050 133 920 17 311 3140 OQT 700 220 1520 200 1380 13 461 3140 OQT 400 280 1930 248 1710 11 555 4130 Annealed 81 558 52 359 28 156 4130 WQT 1300 98 676 89 614 28 202 4130 WQT 1000 143 986 132 910 16 302 4130 WQT 700 208 1430 180 1240 13 415 4130 WQT 400 234 1610 197 1360 12 461 4140 Annealed 95 655 54 372 26 197 4140 OQT 1300 117 807 100 690 23 235 4140 OQT 1000 168 1160 152 1050 17 341 4140 OQT 700 231 1590 212 1460 13 461 4140 OQT 400 290 2000 251 1730 11 578 4150 Annealed 106 731 55 379 20 197 4150 OQT 1300 127 880 116 800 20 262 4150 OQT 1000 197 1360 181 1250 11 401 4150 OQT 700 247 1700 229 1580 10 495 4150 OQT 400 300 2070 248 1710 10 578 4340 Annealed 108 745 68 469 22 217 4340 OQT 1300 140 965 120 827 23 280 4340 OQT 1000 171 1180 158 1090 16 363 4340 OQT 700 230 1590 206 1420 12 461 4340 OQT 400 283 1950 228 1570 11 555 5140 Annealed 83 572 42 290 29 167 5140 OQT 1300 104 717 83 572 27 207 5140 OQT 1000 145 1000 130 896 18 302 5140 OQT 700 220 1520 200 1380 11 429 5140 OQT 400 276 1900 226 1560 7 534 5150 Annealed 98 676 52 359 22 197 5150 OQT 1300 116 800 102 700 22 241 5150 OQT 1000 160 1100 149 1030 15 321 5150 OQT 700 240 1650 220 1520 10 461 5150 OQT 400 312 2150 250 1720 8 601 5160 Annealed 105 724 40 276 17 197 5160 OQT 1300 115 793 100 690 23 229 5160 OQT 1000 170 1170 151 1040 14 341 5160 OQT 700 263 1810 237 1630 9 514 5160 OQT 400 322 2220 260 1790 4 627 6150 Annealed 96 662 59 407 23 197 6150 OQT 1300 118 814 107 738 21 241 6150 OQT 1000 183 1260 173 1190 12 375 6150 OQT 700 247 1700 223 1540 10 495 6150 OQT 400 315 2170 270 1860 7 601 8650 Annealed 104 717 56 386 22 212 8650 OQT 1300 122 841 113 779 21 255 8650 OQT 1000 176 1210 155 1070 14 363 8650 OQT 700 240 1650 222 1530 12 495 8650 OQT 400 282 1940 250 1720 11 555 8740 Annealed 100 690 60 414 22 201 8740 OQT 1300 119 820 100 690 25 241 8740 OQT 1000 175 1210 167 1150 15 363 8740 OQT 700 228 1570 212 1460 12 461 8740 OQT 400 290 2000 240 1650 10 578 9255 Annealed 113 780 71 490 22 229 9255 O&T 1300 130 896 102 703 21 262 9255 O&T 1000 181 1250 160 1100 14 352 9255 O&T 700 260 1790 240 1650 5 534 9255 O&T 400 310 2140 287 1980 2 601

 APPENDIX 4 Properties of Heat-Treated Steel

 APPENDIX 5 Properties of Carburized Steel Core properties Material designation (SAE number) Condition Tensile strength Yield strength Ductility (percent elongation in 2 in) Brinell hardness (HB) Case hardness (HRC) (ksi) (MPa) (ksi) (MPa) 1015 SWQT 350 106 731 60 414 15 217 62 1020 SWQT 350 129 889 72 496 11 255 62 1022 SWQT 350 135 931 75 517 14 262 62 1117 SWQT 350 125 862 66 455 10 235 65 1118 SWQT 350 144 993 90 621 13 285 61 4118 SOQT 300 143 986 93 641 17 293 62 4118 DOQT 300 126 869 63 434 21 241 62 4118 SOQT 450 138 952 89 614 17 277 56 4118 DOQT 450 120 827 63 434 22 229 56 4320 SOQT 300 218 1500 178 1230 13 429 62 4320 DOQT 300 151 1040 97 669 19 302 62 4320 SOQT 450 211 1450 173 1190 12 415 59 4320 DOQT 450 145 1000 94 648 21 293 59 4620 SOQT 300 119 820 83 572 19 277 62 4620 DOQT 300 122 841 77 531 22 248 62 4620 SOQT 450 115 793 80 552 20 248 59 4620 DOQT 450 115 793 77 531 22 235 59 4820 SOQT 300 207 1430 167 1150 13 415 61 4820 DOQT 300 204 1405 165 1140 13 415 60 4820 SOQT 450 205 1410 184 1270 13 415 57 4820 DOQT 450 196 1350 171 1180 13 401 56 8620 SOQT 300 188 1300 149 1030 11 388 64 8620 DOQT 300 133 917 83 572 20 269 64 8620 SOQT 450 167 1150 120 827 14 341 61 8620 DOQT 450 130 896 77 531 22 262 61 E9310 SOQT 300 173 1190 135 931 15 363 62 E9310 DOQT 300 174 1200 139 958 15 363 60 E9310 SOQT 450 168 1160 137 945 15 341 59 E9310 DOQT 450 169 1170 138 952 15 352 58 Notes: Properties given are for a single set of tests on 1/2-in round bars. SWQT: single water-quenched and tempered. SOQT: single oil-quenched and tempered. DOQT: double oil-quenched and tempered. 300 and 450 are the tempering temperatures in °F. Steel was carburized for 8 h. Case depth ranged from 0.045 to 0.075 in.