Question 11.6: Power Transmitted by a Gear Based on Bending Strength and Us...
Power Transmitted by a Gear Based on Bending Strength and Using the Lewis Formula
A 25° pressure angle, 25-tooth spur gear having a module of 2 mm, and a 45 mm face width are to operate at 900 rpm. Determine
a. The allowable bending load applying the Lewis formula
b. The maximum tangential load and power that the gear can transmit
Design Decisions: The gear is made of SAE 1040 steel. A fatigue stress concentration factor of 1.5 is used.
We have Y = 0.402 for 25 teeth (Table 11.2) and \sigma _{o} = 172 MPa (Table 11.3). The pitch diameter is d = mN = 2(25) = 50 mm and V = \pi dn = \pi (0.05)(15) = 2.356 m/s = 463.7 fpm.
a. Using Equation 11.33 with 1/P = m, we have
F_b=\frac{\sigma_o b}{K_f} \frac{Y}{P} (11.33)
F_b=\frac{\sigma_o b \Upsilon m}{K_f}=\frac{1}{1.5}(172 \times 45 \times 0.402 \times 2)=4.149 kN
b. From Equation 11.24a, the dynamic load is
F_d=\frac{600+V}{600} F_t \quad(\text { for } 0<V \leq 2000 fpm ) (11.24a)
F_d=\frac{600+463.7}{600} F_t=1.77 F_t
The limiting value of the transmitted load, applying Equation 11.34, is
F_{b} ≥ F_{d} (11.34)
4.149=1.77 F_t \quad \text { or } \quad F_t=2.344 kN
The corresponding gear power, by Equation 1.15, is
kW =\frac{F_{t} V}{1000}=\frac{T n}{9549} (1.15)
\begin{aligned} kW &=\frac{F_t \pi d n}{60} \\ &=\frac{(2.344) \pi(0.05) 900}{60}=5.52 \end{aligned}
| Table 11.2 Values of the Lewis Form Factor for Some Common Full-Depth Teeth |
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| No. of Teeth | 20° Y | 25° Y | No. of Teeth | 20° Y | 25° Y |
| 12 | 0.245 | 0.277 | 26 | 0.344 | 0.407 |
| 13 | 0.264 | 0.293 | 28 | 0.352 | 0.417 |
| 14 | 0.276 | 0.307 | 30 | 0.358 | 0.425 |
| 15 | 0.289 | 0.320 | 35 | 0.373 | 0.443 |
| 16 | 0.295 | 0.332 | 40 | 0.389 | 0.457 |
| 17 | 0.302 | 0.342 | 50 | 0.408 | 0.477 |
| 18 | 0.308 | 0.352 | 60 | 0.421 | 0.491 |
| 19 | 0.314 | 0.361 | 75 | 0.433 | 0.506 |
| 20 | 0.320 | 0.369 | 100 | 0.446 | 0.521 |
| 21 | 0.326 | 0.377 | 150 | 0.458 | 0.537 |
| 22 | 0.330 | 0.384 | 200 | 0.463 | 0.545 |
| 24 | 0.337 | 0.396 | 300 | 0.471 | 0.554 |
| 25 | 0.340 | 0.402 | Rack | 0.484 | 0.566 |
| Table 11.3 Allowable Static Bending Stresses for Use in the Lewis Equation |
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| \sigma _{o} | ||||
| Material | Treatment | ksi | (MPa) | Average Bhn |
| Cast iron ASTM 35 ASTM 50 |
12 15 |
(82.7) (103) |
210 220 |
|
| Cast steel 0.20% C 0.20% C |
WQ&T | 20 25 |
(138) (172) |
180 250 |
| Forged steel SAE 1020 SAE 1030 SAE 1040 SAE 1045 SAE 1050 |
WQ&T | 18 20 25 32 35 |
(124) (138) (172) (221) (241) |
155 180 200 205 220 |
| Alloy steels SAE 2345 SAE 4340 SAE 6145 |
OQ&T OQ&T OQ&T |
50 65 67 |
(345) (448) (462) |
475 475 475 |
| SAE 65 (phosphor bronze) | 12 | (82.7) | 100 | |
| Note: WQ&T, water-quenched and tempered; OQ&T, oil-quenched and tempered. | ||||