Products
Rewards
from HOLOOLY

We are determined to provide the latest solutions related to all subjects FREE of charge!

Enjoy Limited offers, deals & Discounts by signing up to Holooly Rewards Program

HOLOOLY

HOLOOLY
TABLES

All the data tables that you may search for.

HOLOOLY
ARABIA

For Arabic Users, find a teacher/tutor in your City or country in the Middle East.

HOLOOLY
TEXTBOOKS

Find the Source, Textbook, Solution Manual that you are looking for in 1 click.

HOLOOLY
HELP DESK

Need Help? We got you covered.

## Q. 9.5

Consider three different materials and calculate the safety factor for the pinion and gear using the results from Example Problem 9–3 for the computed bending stresses. Design for a reliability of 0.999, fewer than one failure in 1000. The application is a drive for an industrial saw that will be fully utilized on a normal, one-shift, five-day per-week operation. The materials proposed are:
a. Ductile iron ASTM A536 type 80-55-06 quenched and tempered.
b. Though hardened steel pinion and gear: SAE 4340 OQT 1000.
c. SAE 8620 steel pinion and gear case hardened by carburizing to a hardness of Rockwell C 60-64 with a minimum effective case depth of 0.025 in.

## Verified Solution

The results required from Example Problem 9–3 are summarized here:

$\begin{array}{lllc}N_{P}=20 & N_{G}=70 & P_{d}=6 & F=2.00 \mathrm{in} \\n_{P}=1750 \mathrm{rpm} &n_{G}=500 \mathrm{rpm} & D_{P}=3.333 \mathrm{in} & D_{G}=11.667 \mathrm{in} \\S_{t P}=12376\mathrm{psi} & S_{t G}=9871 \mathrm{psi} & &\end{array}$

The equation relating the bending stress number with the adjusted allowable bending stress number is:

$s_{t}

Solving this equation for the safety factor, SF:

$S F=\frac{s_{a t}}{s_{t}} \cdot \frac{Y_{N}}{K_{R}}$

Design decisions will be made on the reliability and design life. From Table 9-11, we find $K_{R}=1.25$ for the desired reliability of 0.999 as specified in the problem. Because the saw will be fully utilized in an industrial environment, we choose a life of $L=20000$ hours, using Table 9-12 as a guide.

Compute the number of load cycles for the pinion and the gear using Equation (9-27) ($N_c = (60)(L)(n)(q)$). Each tooth sees one load cycle per revolution, $q=1$.

\begin{aligned}&N_{c P}=(60) \cdot L \cdot n_{p} \cdot q=60 \cdot 20000 \cdot 1750 \cdot 1=2.10\times 10^{9} \text { cycles } \\&N_{C G}=(60) \cdot L \cdot n_{G} \cdot q=60 \cdot 20000\cdot 500 \cdot 1=6.00 \times 10^{8} \text { cycles }\end{aligned}

From Figure 9-21, the bending strength stress cycle factor for the pinion and gear is:

\begin{aligned}&Y_{N P}=0.93 \\&Y_{N G}=0.95\end{aligned}

Part (a): Let’s look first at using ductile iron ASTM type A536 80-55-06. Using Table 9-10, the allowable bending stress number for this material is

$s_{a t}=22000 \mathrm{psi}$

Solving for the safety factor for the pinion:
$S F=\frac{S_{a t}}{S_{tP}} \cdot \frac{Y_{N P}}{K_{R}}=\frac{22000 \mathrm{psi}}{12376\mathrm{psi}} \cdot \frac{0.93}{1.25}=1.3$
Using the same equation for the safety factor of the gear:
$S F=\frac{S_{a t}}{S_{t G}} \cdot \frac{Y_{N G}}{K_{R}}=\frac{22000 \mathrm{psi}}{9871 \mathrm{psi}} \cdot \frac{0.95}{1.25}=1.7$
The minimum safety factor is for the pinion and it is within the range of 1.0 to 1.5, as recommended by AGMA, so this design is considered to be satisfactory for bending stress.

Part (b) considers a through hardened steel material SAE 4340 OQT 1000 for the pinion and gear. Using Appendix 3 the properties of the material are:
$s_{u}=171000 \text { psi, } s_{y}=158000 \text { psi, HB }=363,16 \% \text { elongation }$
Using Figure 9-18, along with the material Brinell hardness of $\mathrm{HB}=363$, the allowable bending stress number for the though-hardened steel gear is:
$S_{a t}=41000 \mathrm{psi}$
Solving for the safety factor SF of the pinion and the gear:
\begin{aligned}&S F_{p}=\frac{S_{a t}}{S_{t P}} \cdot \frac{Y_{N P}}{K_{R}}=\frac{41000\mathrm{psi}}{12376 \mathrm{psi}} \cdot \frac{0.93}{1.25}=2.5 \\&S F_{G}=\frac{S_{\mathrm{AT}}}{S_{\mathrm{tG}}} \cdot \frac{Y_{\mathrm{NG}}}{K_{R}}=\frac{41000 \mathrm{psi}}{9871 \mathrm{psi}} \cdot \frac{0.95}{1.25}=3.2\end{aligned}
The safety factor is well over 1.5 for both the pinion and the gear for bending stress with this through-hardened steel.

Part (c) considers a gear material, SAE 8620 case hardened by carburizing to Rockwell $\mathrm{C} 60-64$ with a minimum effective case depth of 0.025 in.
Using Table 9-9, the allowable bending stress number for both the pinion and the gear is:
$s_{\text {at }}=54000 \mathrm{psi}$
Solving for the safety factor of the pinion and the gear:
\begin{aligned}&S F_{p}=\frac{S_{a t}}{S_{t P}} \cdot \frac{Y_{N P}}{K_{R}}=\frac{54000 \mathrm{psi}}{12376 \mathrm{psi}} \cdot \frac{0.93}{1.25}=3.2 \\&S F_{G}=\frac{S_{a t}}{s_{t G}}\cdot \frac{Y_{N G}}{K_{R}}=\frac{54000 \mathrm{psi}}{9871 \mathrm{psi}} \cdot \frac{0.95}{1.25}=4.2\end{aligned}
The safety factor is well over 1.5 for both the pinion and the gear for bending stress with a carburized, case hardened steel material.

 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 9–12 Recommended Design Life Domestic appliances 1000–2000 Aircraft engines 1000–4000 Automotive 1500–5000 Agricultural equipment 3000–6000 Elevators, industrial fans, multipurpose gearing 8000–15 000 Electric motors, industrial blowers, general industrial machines 20 000–30 000 Pumps and compressors 40 000–60 000 Critical equipment in continuous 24-h operation 100 000–200 000
 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