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 speed reducer is to be r_s =1:2. Determine the maximum horsepower that the gear set 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 b =1.5 in. The pinion rotates at n =1600 rpm. P=10 in.^{−1} and N_p =18.
Design Decisions: Rational values of the factors are chosen, as indicated in the parentheses in the solution.
The pinion pitch diameter and number of teeth of the gear are
d_p=\frac{N_p}{P}=\frac{18}{10}=1.8 \text { in., } \quad N_g=N_p\left(\frac{1}{r_s}\right)=18(2)=36
The pitch-line velocity, using Equation (11.20), is
V=\frac{\pi d n}{12} (11.20)
V=\frac{\pi d_p n_p}{12}=\frac{\pi(1.8)(1600)}{12}=754 fpm
The allowable bending stress is estimated from Equation (11.36):
\sigma_{\text {all }}=\frac{S_t K_L}{K_T K_R} (a)
where
S_t=41.5 ksi \text { (from Table 11.6, for average strength) }
K_L=1.0 \text { (from Table 11.7, for indefinite life) }
K_T=1 \text { (oil temperature should be }<160^{\circ} F \text { ) }
K_R=1.25 \text { (by Table } 11.8 \text {, for } 99.9 \% \text { reliability) }
Carrying the foregoing values into Equation (a) results in
\sigma_{ all }=\frac{41.5(1)}{1(1.25)}=33.2 ksi
The maximum allowable transmitted load is now obtained, from Equation (11.35) by setting \sigma_{ all }=\sigma , as
\begin{aligned} & \sigma=F_t K_o K_{ \upsilon } \frac{P}{b} \frac{K_s K_m}{J} \quad \text { (US customary units) } \\ & \sigma=F_t K_o K_{ \upsilon } \frac{1.0}{b m} \frac{K_s K_m}{J} \quad \text { (SI units) } \end{aligned} (11.35)
F_t=\frac{33,200}{K_o K_\upsilon } \frac{b}{P} \frac{J}{K_s K_m} (b)
In the foregoing, we have
P = 10
b = 1.5 in.
K_{\upsilon } = 1.55 (from curve C of Figure 11.15)
J = 0.235 (from Figure 11.16(a), the load acts at the tip of the tooth, N_p =18)
K_o = 1.75 (by Table 11.4)
K_s = 1.0 (for standard gears)
K_m = 1.6 (from Table 11.5)
Equation (b) yields
F_t=\frac{33,200(1.5)(0.235)}{(1.75)(1.55)(10)(1.0)(1.6)}=270 lb
The allowable power is then, by Equation (11.22),
F_t=\frac{33,000 hp }{V} (11.22)
hp =\frac{F_t V}{33,000}=\frac{270(754)}{33,000}=6.2
TABLE 11.6 Bending Strength S_t of Spur, Helical, and Bevel Gear Teeth |
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S_t | ||||
Material | Heat Treatment | Minimum Hardness or Tensile Strength | ksi | (MPa) |
Steel | Normalized | 140 Bhn | 19−25 | (131−172) |
Q&T | 180 Bhn | 25−33 | (172−223) | |
Q&T | 300 Bhn | 36−47 | (248−324) | |
Q&T | 400 Bhn | 42−56 | (290−386) | |
Case carburized | 55 R_C | 55−65 | (380−448) | |
60 R_C | 55−70 | (379−483) | ||
Nitrided AISI-4140 | 48 R_C case | 34−45 | (234−310) | |
300 Bhn core | ||||
Cast iron | ||||
AGMA grade 30 | 175 Bhn | 8.5 | (58.6) | |
AGMA grade 40 | 200 Bhn | 13 | (89.6) | |
Nodular iron ASTM grade | ||||
60–40–18 | 15 | (103) | ||
80–55–06 | Annealed | 20 | (138) | |
100–70–18 | Normalized | 26 | (179) | |
120–90–02 | Q&T | 30 | (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 K_L for Spur and Helical Steel Gears |
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Number of Cycles | 160 Bhn | 250 Bhn | 450 Bhn | Case Carburized (55–63 R_C ) |
10^3 | 1.6 | 2.4 | 3.4 | 2.7–4.6 |
10^4 | 1.4 | 1.9 | 2.4 | 2.0–3.1 |
10^5 | 1.2 | 1.4 | 1.7 | 1.5–2.1 |
10^6 | 1.1 | 1.1 | 1.2 | 1.1–1.4 |
10^7 | 1.0 | 1.0 | 1.0 | 1.0 |
Source: ANSI/AGMA Standard 218.01. |
TABLE 11.8 Reliability Factor K_R |
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Reliability (%) | 50 | 90 | 99 | 99.9 | 99.99 |
Factor K_R | 0.70 | 0.85 | 1.00 | 1.25 | 1.50 |
Source: From ANSI/AGMA Standard 2001-C95. |
TABLE 11.4 Overload Correction Factor K_o |
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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 K_m |
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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 Accuracy and mounting such that less than full-face contact exists |
1.6 Over 2. 2 |
1.7 | 1.8 | 2.2 |