Design of a Speed Reducer for Wear by the AGMA Method Determine the maximum horsepower that the speed reducer gearset in Example 11.7 can transmit, based on wear strength and applying the AGMA method. Given: Both gears are made of the same 300 Bhn steel of E = 30 × 10^6 psi, ν = 0.3, and have a

Question

Design of a Speed Reducer for Wear by the AGMA Method

Determine the maximum horsepower that the speed reducer gearset in Example 11.7 can transmit, based on wear strength and applying the AGMA method.

Given: Both gears are made of the same 300 Bhn steel of [latex]E = 30 imes 10^{6}[/latex] psi, [latex] u[/latex] = 0.3, and have a face width of [latex]b[/latex] = 1.5 in., [latex]P[/latex] = 10 in.[latex]^{−1}[/latex], and [latex]N_{p}[/latex] = 18.

Design Decisions: Rational values of the factors are chosen, as indicated in the parentheses in the solution.

Accepted Answer

Allowable contact stress is estimated from Equation 11.44 as

[latex] \sigma_{c, ext { all }}=\frac{S_C C_L C_H}{K_T K_R} [/latex] (a)

In the preceding,

[latex]S_{c}[/latex] = 127.5 ksi (from Table 11.11, for average strength)

[latex]C_{L}[/latex] = 1.0 (from Figure 11.19, for indefinite life)

[latex] C_H=1.0+A\left(\frac{N_g}{N_p}-1.0 ight)=1.0 [/latex] (by Equation 11.46)

[latex]K_{T}[/latex] = 1.0 and [latex]K_{R}[/latex] = 1.25 (both from Example 11.7)

Hence,

[latex] \sigma_{c, ext { all }}=\frac{127,500(1.0)(1.0)}{(1.0)(1.25)}=102 ksi [/latex]

The maximum allowable transmitted load is now determined, from Equation 11.42 setting [latex] \sigma_{c, ext { all }}=\sigma_{c^{\prime}} [/latex] as

[latex] \sigma_c=C_p\left(F_t K_o K_v \frac{K_s}{b d} \frac{K_m C_f}{I} ight)^{1 / 2} [/latex] (11.42)

[latex] F_t=\left(\frac{102,000}{C_p} ight)^2 \frac{1.0}{K_o K_v} \frac{b d}{K_s} \frac{I}{K_m C_f} [/latex]

where

[latex] C_p=2300 \sqrt{ psi } [/latex] (by Table 11.10)

[latex]b[/latex] = 1.5 in., [latex]d_{p}[/latex] = 1.8 in

[latex]K_{\upsilon} = 1.55, K_{o} = 1.75, K_{s} = 1, K_{m}[/latex] = 1.6 (all from Example 11.7)

[latex]C_{f}[/latex] = 1.0 (for smooth surface finish)

[latex] I=\frac{\sin \phi \cos \phi}{2 m_N} \frac{m_{ G }}{m_{ G }+1}=0.107 [/latex] (using Equation 11.43b)

Therefore,

[latex] F_t=\left(\frac{102,000}{2300} ight)^2 \frac{1.0}{(1.75)(1.55)} \frac{1.5(1.8)}{1.0} \frac{0.107}{1.6(1.0)}=131 lb [/latex]

This value applies to both mating gear tooth surfaces. The corresponding power, using Equation 11.22 with [latex]V[/latex] = 754 fpm (from Example 11.7), is

[latex] hp =\frac{F_t V}{33,000}=\frac{131(754)}{33,000}=2.99 [/latex]

Table 11.10 AGMA Elastic Coefficients [latex]C_{p}[/latex] for Spur Gears, in [latex] \sqrt{p s i} [/latex] and [latex] \sqrt{MPa} [/latex]
		Gear Material
Pinion Material	[latex]E[/latex], ksi (GPa)	Steel	Cast Iron	Aluminum Bronze	Tin Bronze
Steel	30,000	2300	2000	1950	1900
	(207)	(191)	(166)	(162)	(158)
Cast iron	19,000	2000	1800	1800	1750
	(131)	(166)	(149)	(149)	(145)
Aluminum bronze	17,500	1950	1800	1750	1700
	(121)	(162)	(149)	(145)	(141)
Tin bronze	16,000	1900	1750	1700	1650
	(110)	(158)	(145)	(141)	(137)

Table 11.11 Surface Fatigue Strength or Allowable Contact Stress [latex]S_{c}[/latex]
		[latex]S_{c}[/latex]
Material	Minimum Hardness or Tensile Strength	ksi	(MPa)
Steel	Through hardened
	180 Bhn	85–95	(586–655)
	240 Bhn	105–115	(724–793)
	300 Bhn	120–135	(827–931)
	360 Bhn	145–160	(1000–1103)
	400 Bhn	155–170	(1069–1172)
	Case carburized
	55 [latex]R_{C}[/latex]	180–200	(1241–1379)
	60 [latex]R_{C}[/latex]	200–225	(1379–1551)
	Flame or induction hardened
	50 [latex]R_{C}[/latex]	170–190	(1172–1310)
Cast iron
AGMA grade 20		50–60	(345–414)
AGMA grade 30	175 Bhn	65–75	(448–517)
AGMA grade 40	200 Bhn	75–85	(517–586)
Nodular (ductile) iron
Annealed	165 Bhn	90%–100% of the [latex]S_{c}[/latex] Value of steel with the same hardness
Normalized	210 Bhn
OQ&T	255 Bhn
Tin bronze
AGMA 2C (10%–12% tin)	40 ksi (276 MPa)	30	(207)
Aluminum bronze
ASTM B 148–52 (alloy 9C-HT)	90 ksi (621 MPa)	65	(448)
Source: ANSI/AGMA Standard 218.01. Note: OQ&T, Oil-quenched and tempered; HT, heat-treated.

Question 11.9: Design of a Speed Reducer for Wear by the AGMA Method Determ...

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