Design of a Speed Reducer for Bending Strength by the AGMA Method A conveyor drive involving heavy shock torsional loading is to be operated by an electric motor turning at a speed of n, as shown schematically in Figure 11.17. The speed ratio of the spur gears connecting the motor and conveyor or

Question

Design of a Speed Reducer for Bending Strength by the AGMA Method

A conveyor drive involving heavy shock torsional loading is to be operated by an electric motor turning at a speed of [latex]n[/latex], as shown schematically in Figure 11.17. The speed ratio of the spur gears connecting the motor and conveyor or speed reducer is to be [latex]r_{s}[/latex] = 1:2. Determine the maximum horsepower that the gearset can transmit, based on bending strength and applying the AGMA formulas.

Given: Both gears are of the same 300 Bhn steel and have a face width of [latex]b[/latex] = 1.5 in. Pinion rotates at [latex]n[/latex] = 1600 rpm. [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

The pinion pitch diameter and number of teeth of the gear are

[latex] d_p=\frac{N_p}{P}=\frac{18}{10}=1.8 ext { in., } \quad N_g=N_p\left(\frac{1}{r_s} ight)=18(2)=36 [/latex]

The pitch-line velocity, using Equation 11.20, is

[latex] V=\frac{\pi d_p n_p}{12}=\frac{\pi(1.8)(1600)}{12}=754 fpm [/latex]

The allowable bending stress is estimated from Equation 11.36:

[latex] \sigma_{ ext {all }}=\frac{S_{ t } K_L}{K_T K_R} [/latex] (a)

where

[latex]S_{t}[/latex] = 41.5 ksi (from Table 11.6, for average strength)

[latex]K_{L}[/latex] = 1.0 (from Table 11.7, for indefinite life)

[latex]K_{T}[/latex] = 1 (oil temperature should be <160°F)

[latex]K_{R}[/latex] = 1.25 (by Table 11.8, for 99.9% reliability)

Carrying the foregoing values into Equation (a) results in

[latex] \sigma_{ ext {all }}=\frac{41.5(1)}{1(1.25)}=33.2 ksi [/latex]

The maximum allowable transmitted load is now obtained, from Equation 11.35 by setting [latex]\sigma _{all} = \sigma[/latex], as

[latex] \sigma=F_t K_o K_{\upsilon} \frac{P}{b} \frac{K_s K_m}{J} [/latex] (11.35)

[latex] F_t=\frac{33,200}{K_o K_{\upsilon}} \frac{b}{P} \frac{J}{K_s K_m} [/latex] (b)

In the foregoing, we have

[latex]P[/latex] = 10

[latex]b[/latex] = 1.5 in.

[latex]K_{\upsilon}[/latex] = 1.55 (from curve [latex]C[/latex] of Figure 11.15)

[latex]J[/latex] = 0.235 (from Figure 11.16a, the load acts at the tip of the tooth, [latex]N_{p}[/latex] = 18)

[latex]K_{o}[/latex] = 1.75 (by Table 11.4)

[latex]K_{s}[/latex] = 1.0 (for standard gears)

[latex]K_{m}[/latex] = 1.6 (from Table 11.5)

Equation (b) yields

[latex] F_t=\frac{33,200(1.5)(0.235)}{(1.75)(1.55)(10)(1.0)(1.6)}=270 lb [/latex]

The allowable power is then, by Equation 11.22,

[latex] F_t=\frac{33,000 hp }{V} [/latex] (11.22)

[latex] hp =\frac{F_{ t } V}{33,000}=\frac{270(754)}{33,000}=6.2 [/latex]

Table 11.4 Overload Correction Factor [latex]K_{o}[/latex]
	Load on Driven Machine
Source of Power	Uniform	Moderate Shock	Heavy Shock
Uniform	1.00	1.25	1.75
Light shock	1.25	1.50	2.00
Medium shock	1.50	1.75	2.25

Table 11.5 Mounting Correction Factor [latex]K_{m}[/latex]
	Face Width (in.)
Condition of Support	0–2	6	9	16 up
Accurate mounting, low bearing clearances, maximum deflection, precision gears	1.3	1.4	1.5	1.8
Less rigid mountings, less accurate gears, contact across the full face	1.6	1.7	1.8	2.2
Accuracy and mounting such that less than full-face contact exists	Over 2.2

Table 11.6 Bending Strength [latex]S_{t}[/latex] of Spur, Helical, and Bevel Gear Teeth
			[latex]S_{t}[/latex]
Material	Heat Treatment	Minimum Hardness or Tensile Strength	ksi	(MPa)
Steel	Normalized Q&T Q&T Q&T Case carburized Nitrided AISI-4140	140 Bhn 180 Bhn 300 Bhn 400 Bhn 55 [latex]R_{C}[/latex] 60 [latex]R_{C}[/latex] 48 [latex]R_{C}[/latex] case 300 Bhn core	19–25 25–33 36–47 42–56 55–65 55–70 34–45	(131–172) (172–223) (248–324) (290–386) (380–448) (379–483) (234–310)
Cast iron AGMA grade 30 AGMA grade 40		175 Bhn 200 Bhn	8.5 13	(58.6) (89.6)
Nodular iron ASTM grade 60-40-18 80-55-06 100-70-18 120-90-02	Annealed Normalized Q&T		15 20 26 30	(103) (138) (179) (207)
Bronze, AGMA 2C	Sand cast	40 ksi (276 MPa)	5.7	(39.3)
Source: ANSI/AGMA Standard 218.01. Note: Q&T, Quenched and tempered.

Table 11.7 Life Factor [latex]K_{L}[/latex] for Spur and Helical Steel Gears
Number of Cycles	160 Bhn	250 Bhn	450 Bhn	Case Carburized (55–63 [latex]R_{C}[/latex])
[latex]10^{3}[/latex]	1.6	2.4	3.4	2.7–4.6
[latex]10^{4}[/latex]	1.4	1.9	2.4	2.0–3.1
[latex]10^{5}[/latex]	1.2	1.4	1.7	1.5–2.1
[latex]10^{6}[/latex]	1.1	1.1	1.2	1.1–1.4
[latex]10^{7}[/latex]	1.0	1.0	1.0	1.0
Source: ANSI/AGMA Standard 218.01.

Table 11.8 Reliability Factor [latex]K_{R}[/latex]
Reliability (%)	50	90	99	99.9	99.99
Factor [latex]K_{R}[/latex]	0.7	0.85	1.00	1.25	1.50
Source: From ANSI/AGMA Standard 2001–C95.

Question 11.7: Design of a Speed Reducer for Bending Strength by the AGMA M...

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