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Chapter 9

Q. 9.8

A gear pair is to be designed to transmit 15.0 kilowatts (kW) of power to a large meat grinder in a commercial meat processing plant. The pinion is attached to the shaft of an electric motor rotating at 575 rpm.
The gear must operate at 270 to 280 rpm. The gear unit will be enclosed and of commercial quality.

Commercially hobbed (quality number A11), 20°, full-depth, involute gears are to be used in the metric module system. The maximum center distance is to be 200 mm. Specify the design of the gears.

Step-by-Step

Verified Solution

The nominal velocity ratio is
VR = 575/275 = 2.09
Specify an overload factor of K_o = 1.50 from Table 9–1 for a uniform power source and moderate shock for the meat grinder. Then compute design power,

P_{des} = K_{o}P = (1.50)(15 kW) = 22.5 kW
From Figure 9–11, m = 5 is a reasonable trial module. Then
N_P = 18 (design decision)
D_P = N_Pm = (18)(5) = 90 mm
N_G = N_P(VR) = (18)(2.09) = 37.6 (Use 38)
D_G = N_Gm = (38)(5) = 190 mm
Final output speed = n_G = n_P(N_P/N_G)
n_G = 575 rpm × (18/38) = 272 rpm (OK)

Center distance = C = (N_P + N_G)m/2[Table 8–1]
C = (18 + 38)(5)/2 = 140 mm (OK)
In SI units, the pitch line speed in meters per second (m/s) is
v_t = πD_Pn_P/(60 000) = [(π)(90)(575)]/(60 000) = 2.71 m/s
In SI units, the transmitted load, W_t, is in newtons (N). If the power, P, is in kW, and v_t is in m/s,
W_t = 1000(P)/v_t = (1000)(15)/(2.71) = 5536 N
Face width, F: Let’s specify the nominal F = 12m = 12(5) = 60 mm

Factors in stress analysis:

K_o = 1.50 (found earlier)
K_s = 1.00 (Table 9–2; m = 5)              K_B = 1.00 (Use solid gear blanks)
K_R = 1.00 (Table 9–11; 0.99 reliability)                  S_F = 1.00 (No unusual conditions)
K_v = 1.31 (Figure 9–16;      A_v = 11)
K_m = 1.21 (Figures 9–12 and 9–13;  F = 60 mm;  F/D_P = 60/90 = 0.67)
J_P = 0.315; J_G = 0.380 (Figure 9–10; N_P = 18, N_G = 38)
C_p = 191 (Table 9–7)      I = 0.092 (Figure 9–17)

Pinion contact stress:
(Equation 9–23)

S_{c}=C_{P} \sqrt{\frac{W_{t} K_{o} K_{s} K_{m} K_{v}}{F D_{P} l}}=191 \sqrt{\frac{(5536)(1.50)(1.0)(1.21)(1.31)}{(60)(90)(0.092)}}=983 \mathrm{MPa}
Adjustments for number of cycles, from Figures 9-21 and 9-22:
Y_{N p}=0.94 \quad Z_{N P}=0.91 \quad Y_{N G}=0.96 \quad Z_{N G}=0.92
Required s_{a C P}=s_{C}(S F)\left(K_{R}\right) / Z_{N P}=983 \mathrm{MPa}(1.0)(1.0) / 0.91=1080 \mathrm{MPa}
Using S_{a c P}=1080 \mathrm{MPa}, Figure 9-19 shows the required hardness =\mathrm{HB} 396 for through-hardened Grade 1 steel. This is acceptable but near the upper end of recommended range.

Material specification:

From Figure A4–5 (other possibilities exist),
SAE 4340 OQT 800; HB 415; s_y = 1324 MPa; s_u = 1448 MPa; 12% elongation.

Check other stresses:

The contact stress for the gear and the bending stress for the pinion and the gear are expected to require less material hardness and strength.
Required s_{acG} = s_c(SF)(K_R)/Z_{NG} = 983 MPa(1.0)(1.0)/0.92 = 1068 MPa
This is slightly lower than for the pinion (OK)

\begin{aligned}S_{t P} &=\frac{W_{t} K_{0} K_{S} K_{B} K_{m} K_{v}}{F m J_{P}}=\frac{(5536)(1.50)(1)(1)(1.21)(1.31)}{(60)(5)(0.315)}=139 \mathrm{MPa} \\\text { Required } S_{a t P} &=S_{t P}(S F)\left(K_{R}\right) / Y_{N P}=139 \mathrm{MPa}(1.0)(1.0) / 0.94=148 \mathrm{MPa}\end{aligned}
Referring to Figure 9-18, it is obvious that bending stress requires far lower hardness for the gear teeth, less than HB 180. The stress in the gear is always less than that in the pinion so it will obviously be safe as well.

Summary of the Design:
P=15.0 \mathrm{~kW} from an electric motor to a large meat grinder
Pinion speed: n_{P}=575 \mathrm{rpm}
Gear speed: n_{G}=272 \mathrm{rpm}
Number of teeth: N_{P}=18 ; N_{G}=38 Center distance: C=140.00 \mathrm{~mm}
Module: m=5 \mathrm{~mm}
Diameters: D_{P}=90 \mathrm{~mm} ; D_{G}=190 \mathrm{~mm}
Material: Steel-SAE 4340 OQT 800

Comment:

A redesign may be considered with several possible approaches:
1. Increase the face width, F, to lower the stresses and permit the choice of a material with more moderate required hardness and better ductility. The recommended upper limit of face width is 16m = 16(5) = 80 mm.
2. Increase the size of pinion and its number of teeth (same module) to lower stresses.
Possible trial: Module: m = 5 mm Number of teeth: N_P = 22; N_G = 46
Center distance: C = 170.00 mm Diameters: D_P = 110[latex] mm; [latex]D_G = 230 mm
3. Consider case-hardened steel for the initial design, rather than through-hardened steel. A smaller design is possible.

 

TABLE 9–1 Suggested Overload Factors, K_o
Driven Machine
Power source Uniform Light shock Moderate shock Heavy shock
Uniform 1.00 1.25 1.50 1.75
Light shock 1.20 1.40 1.75 2.25
Moderate shock 1.30 1.70 2.00 2.75
TABLE 9–11 Reliability Factor, K_R
Reliability K_R
0.90, one failure in 10 0.85
0.99, one failure in 100 1.00
0.999, one failure in 1000 1.25
0.9999, one failure in 10 000 1.5
TABLE 9–2 Suggested Size Factors, K_s
Diametral pitch, P_d Metric module, m Size factor, K_s
≥ 5 ≤ 5 1.00
4 6 1.05
3 8 1.15
2 12 1.25
1.25 20 1.40
TABLE 9–7 Elastic Coefficient, C_p
Gear material and modulus of elasticity, E_G, lb/in^2 (MPa)
Pinion
material
Modulus of
elasticity,
E_P,lb/in^2
(MPa)
Steel
30×10^6(2×10^5)
Malleable
iron
25×10^6(1.7×10^5)
Nodular
iron
24×10^6(1.7×10^5)
Cast iron
22×10^6(1.5×10^5)
Aluminum
bronze
17.5×10^6(1.2×10^5)
Tin bronze
16×10^6(1.1×10^5)
Steel 30×10^6 2300 2180 2160 2100 1950 1900
(2×10^5) 191 181 179 174 162 158
Mall. Iron 25×10^6 2180 2090 2070 2020 1900 1850
(1.7×10^5) 181 174 172 168 158 154
Nod. Iron 24×10^6 2160 2070 2050 2000 1880 1830
(1.7×10^5) 179 172 170 166 156 152
Cast iron 22×10^6 2100 2020 2000 1960 1850 1800
(1.5×10^5) 174 168 166 163 154 149
Al. bronze 1.75×10^6 1950 1900 1880 1850 1750 1700
(1.2×10^5) 162 158 156 154 145 141
Tin bronze 16×10^6 1900 1850 1830 1800 1700 1650
(1.1×10^5) 158 154 152 149 141 137

 

TABLE 8–1 Gear and Tooth Features, Diameters, Center Distance for a Gear Pair
Formulas
U.S. Full-depth involute system Metric
module
system
(mm)
Number of teeth and Pitches Symbol Definition Typical unit General formula Coarse pitch P_d < 20 (in) Fine pitch P_d ≥ 20 (in)
Number of teeth N Integer count of teeth on a gear
Circular pitch P Arc distance between corresponding points on adjacent teeth in or mm p = πD/N p = π/P_d p = πm
Diametral pitch P_d Number of teeth per inch of pitch diameter in^{-1} P_d = N/D
Module m Pitch diameter divided by number of teeth mm m = D/N m = 25.4/P_d
Diameters
Pitch diameter D Kinematic characteristic diameter for a gear; Diameter of the pitch circle in or mm D = N/P_d D = mN
Outside diameter D_o Diameter to the outside surface of the gear teeth in or mm D_o = (N + 2)/P_d D_o = m(N + 2)
Root diameter D_R Diameter to the root circle of the gear at the base of the teeth in or mm D_R = D - 2b
Gear Tooth Features
Addendum a Radial distance from pitch circle to outside of tooth in or mm a = 1.00/P_d a = 1.00m
Dedendum b Radial distance from pitch circle to bottom of tooth space in or mm b = 1.25/P_d b = 1.20/P_d + 0.002 b = 1.25m^1
Clearance c Radial distance from top of fully engaged tooth of mating gear to bottom of tooth space in or mm c = 0.25/P_d c = 0.20/P_d + 0.002 c = 0.25m^1
Whole depth h_t Radial distance from top of a tooth to bottom of tooth space in or mm h_t = a + b h_t = 2.00/P_d h_t = 2.20/P_d + 0.002 h_t = 2.25m^1
Working depth h_k Radial distance a gear tooth projects into tooth space of mating gear in or mm h_k = a + a = 2a h_k = 2.25/P_d h_k = 2.25/P_d h_k = 2.00m^1
Tooth thickness Tensile strength Theoretical arc distance equal to 1/2 of circular pitch in or mm t = p/2 t = π/[2(P_d)] t = πm/2
Face width F Width of tooth parallel to axis of gear in or mm Design decision Approximately 12/P_d
Pressure angle \phi Angle between the tangent to the pitch circle and the perpendicular to the gear tooth surface degrees Design decision Most common value = 20°
Others: 14 1/2°, 25°
Center Distance C Distance from between centerlines of mating gears in or mm C = (D_P + D_G)/2 C = (N_P + N_G)/2P_d C = m(N_P + N_G)/2