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

Question

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.

Accepted Answer

We have [latex]Y[/latex] = 0.402 for 25 teeth (Table 11.2) and [latex]\sigma _{o}[/latex] = 172 MPa (Table 11.3). The pitch diameter is [latex]d = mN[/latex] = 2(25) = 50 mm and [latex]V = \pi dn = \pi (0.05)(15)[/latex] = 2.356 m/s = 463.7 fpm.

a. Using Equation 11.33 with [latex]1/P = m,[/latex] we have

[latex] F_b=\frac{\sigma_o b}{K_f} \frac{Y}{P} [/latex] (11.33)

[latex] F_b=\frac{\sigma_o b \Upsilon m}{K_f}=\frac{1}{1.5}(172 imes 45 imes 0.402 imes 2)=4.149 kN [/latex]

b. From Equation 11.24a, the dynamic load is

[latex] F_d=\frac{600+V}{600} F_t \quad( ext { for } 0

[latex] F_d=\frac{600+463.7}{600} F_t=1.77 F_t [/latex]

The limiting value of the transmitted load, applying Equation 11.34, is

[latex]F_{b} ≥ F_{d}[/latex] (11.34)

[latex] 4.149=1.77 F_t \quad ext { or } \quad F_t=2.344 kN [/latex]

The corresponding gear power, by Equation 1.15, is

[latex] kW =\frac{F_{t} V}{1000}=\frac{T n}{9549} [/latex] (1.15)

[latex] \begin{aligned} kW &=\frac{F_t \pi d n}{60} \ &=\frac{(2.344) \pi(0.05) 900}{60}=5.52 \end{aligned} [/latex]

Table 11.2 Values of the Lewis Form Factor for Some Common Full-Depth Teeth
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
	[latex]\sigma _{o}[/latex]
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.

Question 11.6: Power Transmitted by a Gear Based on Bending Strength and Us...

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