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## Q. 9.10

Design a pair of plastic gears for a paper shredder to transmit 0.25 horsepower at a pinion speed of 1160 rpm. The pinion will be mounted on the shaft of an electric motor that has a diameter of 0.625 in with a keyway for a 3/16 × 3/16 in key. The gear is to rotate approximately 300 rpm.

Given Data:

$P = 0.25$ hp, $n_P = 1160$ rpm,
Shaft diameter = $D_s = 0.625$ in, Keyway for a 3/16 × 3/16 in key.
Approximate gear speed = $n_G = 300$ rpm

## Verified Solution

Use the design procedure outlined in this section.
Step 1. Consider the given data.
Step 2. Specify $N_P = 18$ and $P_d = 16$ (Design decisions)
Step 3. $D_P = N_P/P_d = 18/16 = 1.125$ in. This seems reasonable for mounting on the 0.625 in motor shaft.
Step 4. Compute the transmitted load,
$W_t = (126 000)(P)/(n_PD_P) = (126 000)(0.25)/[(1160)(1.125)] = 24.1$ lb

Step 5. Specify 20° full-depth teeth. Then $Y = 0.521$ for 18 teeth from Table 9–15.
Step 6. Specify a safety factor, $SF$. The shredder will likely experience light shock; the preference is to operate the gears without lubrication. Specify SF = 1.50 from Table 9–1.
Step 7. Specify unfilled nylon. From Table 9–14, $s_{at} = 6000$ psi.
Step 8. Compute the required face width using Equation (9–42).

$F=\frac{W_tP_d(SF)}{S_{at}Y}=\frac{(24.1)(16)(1.50)}{(6000)(0.521)}=0.185$ in

Step 9. The dimensions seem reasonable.
Step 10.Appendix 2 lists a preferred size for the face width of 0.200 in.

Comment:

In summary, the proposed pinion has the following features:
$P_d = 16, N_P = 18$ teeth, $D_P = 1.125$ in, $F = 0.200$ in, Bore = 0.625 in,
Keyway for a 3/16 × 3/16 in key. Unfilled nylon material.

Step 11. Gear design: Specify $F = 0.200$ in, $P_d = 16$. Compute the number of teeth in the gear.
$N_G = N_P(n_P/n_G) = 18(1160/300) = 69.6$ teeth
Specify $N_G = 70$ teeth
Pitch diameter of gear = $D_G = N_G/P_d = 70/16 = 4.375$ in
From Table 9–15, $Y_G = 0.728$ by interpolation.
Stress in gear teeth using Equation (9–41):

$σ_t=\frac{W_tP_d(SF)}{FY}=\frac{(24.1)(16)(1.50)}{(0.200)(0.728)}=3973$ psi

Comment:

This stress level is safe for nylon. The gear could also be made from acetal to achieve better wear performance.

 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 9–14 Approximate Allowable Tooth Bending Stress, $s_{at}$, in Plastic Gears Approximate allowable bending stress, $s_{at}$, psi (MPa) Material Unfilled Glass-filled ABS 3000 (21) 6000 (41) Acetal 5000 (34) 7000 (48) Nylon 6000 (41) 12 000 (83) Polycarbonate 6000 (41) 9000 (62) Polyester 3500 (24) 8000 (55) Polyurethane 2500 (17)
 Table 9–15 Lewis Tooth Form Factor, Y, for Load Near the Pitch Point Tooth form Number teeth 14 1/2° Full depth 20° Full depth 20° Stub 14 – – 0.54 15 – – 0.566 16 – – 0.578 17 – 0.512 0.587 18 – 0.521 0.603 19 – 0.534 0.616 20 – 0.544 0.628 22 – 0.559 0.648 24 0.509 0.572 0.664 26 0.522 0.588 0.678 28 0.535 0.597 0.688 30 0.54 0.606 0.698 34 0.553 0.628 0.714 38 0.566 0.651 0.729 43 0.575 0.672 0.739 50 0.588 0.694 0.758 60 0.604 0.713 0.774 75 0.613 0.735 0.792 100 0.622 0.757 0.808
 TABLE A2–1 Preferred Basic Sizes Fractional (in) Decimal (in) SI metric (mm) $\begin{array}{ll}1 / 64 & 0.015625 \\1 / 32 & 0.03125 \\1 / 16 & 0.0625 \\3 / 32 & 0.09375\\1 / 8 & 0.1250\end{array}$ $\begin{array}{ll}5 & 5.000 \\5 \frac{1}{4} & 5.250 \\5 \frac{1}{2} & 5.500 \\5 \frac{3}{4}& 5.750 \\6 & 6.000\end{array}$ $\begin{array}{rrr}0.010 & 2.00 & 8.50 \\0.012 & 2.20 & 9.00 \\0.016 & 2.40 & 9.50 \\0.020&2.60 & 10.00 \\0.025 & 2.80 & 10.50\end{array}$ $\begin{array}{ll}1.0 & 40 \\1.1 & 45 \\1.2 & 50 \\1.4 & 55 \\1.6 & 60\end{array}$ $\begin{array}{cc}5 / 32 & 0.15625 \\3 / 16 & 0.1875 \\1 / 4 & 0.2500 \\5 / 16 & 0.3125 \\3/ 8 & 0.3750\end{array}$ $\begin{array}{ll}6 \frac{1}{2} & 6.500 \\7 & 7.000 \\7 \frac{1}{2} & 7.500 \\8 & 8.000 \\8\frac{1}{2} & 8.500\end{array}$ $\begin{array}{lll}0.032 & 3.00 & 11.00 \\0.040 & 3.20 & 11.50 \\0.05 & 3.40 & 12.00 \\0.06& 3.60 & 12.50 \\0.08 & 3.80 & 13.00\end{array}$ $\begin{array}{ll}1.8 & 70 \\2.0 & 80 \\2.2 & 90 \\2.5 & 100 \\2.8 & 110\end{array}$ $\begin{array}{cc}7 / 16 & 0.4375 \\1 / 2 & 0.5000 \\9 / 16 & 0.5625 \\5 / 8 & 0.6250\end{array}$ $\begin{array}{cc}9 & 9.000 \\9 \frac{1}{2} & 9.500 \\10 & 10.000 \\10 \frac{1}{2}&10.500\end{array}$ $\begin{array}{lll}0.10 & 4.00 & 13.50 \\0.12 & 4.20 & 14.00 \\0.16 & 4.40 & 14.50 \\0.20 &4.60 & 15.00\end{array}$ $\begin{array}{ll}3.0 & 120 \\3.5 & 140 \\4.0 & 160 \\4.5 & 180\end{array}$ $\begin{array}{cc}11 / 16 & 0.6875 \\3 / 4 & 0.7500 \\7 / 8 & 0.8750\end{array}$ $\begin{array}{ll}11 & 11.000 \\11 \frac{1}{2} & 11.500 \\12 & 12.000\end{array}$ $\begin{array}{lll}0.24 & 4.80 & 15.50 \\0.30 & 5.00 & 16.00 \\0.40 & 5.20 & 16.50\end{array}$ $\begin{array}{ll}5.0 & 200 \\5.5 & 220 \\6 & 250\end{array}$ $\begin{array}{cc}1 & 1.000 \\1 \frac{1}{4} & 1.250 \\1 \frac{1}{2} & 1.500 \\1 \frac{3}{4}& 1.750\end{array}$ $\begin{array}{ll}12 \frac{1}{2} & 12.500 \\13 & 13.000 \\13 \frac{1}{2} & 13.500 \\14 &14.000\end{array}$ $\begin{array}{lll}0.50 & 5.40 & 17.00 \\0.60 & 5.60 & 17.50 \\0.80 & 5.80 & 18.00 \\1.00 &6.00 & 18.50\end{array}$ $\begin{array}{rr}7 & 280 \\8 & 300 \\9 & 350 \\10 & 400\end{array}$ $\begin{array}{cc}2 & 2.000 \\2 \frac{1}{4} & 2.250 \\2 \frac{1}{2} & 2.500 \\2 \frac{3}{4}& 2.750\end{array}$ $\begin{array}{ll}14 \frac{1}{2} & 14.500 \\15 & 15.000 \\15 \frac{1}{2} & 15.500 \\16 &16.000\end{array}$ $\begin{array}{lll}1.20 & 6.50 & 19.00 \\1.40 & 7.00 & 19.50 \\1.60 & 7.50 & 20.00 \\1.80 &8.00 &\end{array}$ $\begin{array}{ll}11 & 450 \\12 & 500 \\14 & 550 \\16 & 600\end{array}$ $\begin{array}{ll}3 & 3.000 \\3 \frac{1}{4} & 3.250 \\3 \frac{1}{2} & 3.500 \\3 \frac{3}{4}& 3.750\end{array}$ $\begin{array}{ll}16 \frac{1}{2} & 16.500 \\17 & 17.000 \\17 \frac{1}{2} & 17.500 \\18 &18.000\end{array}$ $\begin{array}{cc}18 & 700 \\20 & 800 \\22 & 900 \\25 & 1000\end{array}$ $\begin{array}{cc}4 & 4.000 \\4 \frac{1}{4} & 4.250 \\4 \frac{1}{2} & 4.500 \\4 \frac{3}{4}& 4.750\end{array}$ $\begin{array}{ll}18 \frac{1}{2} & 18.500 \\19 & 19.000 \\19 \frac{1}{2} & 19.500 \\20 &20.000\end{array}$ \begin{aligned}&28 \\&30 \\&35\end{aligned}