A rack is driven by pinion shown in Figure 8–46. The pinion rotates at 125 rpm, has 24 teeth, and a diametral pitch of 6. a. Determine the pitch diameter of the pinion. b. Determine the dimension from the pitch line to the back of the rack, B.

Question

A rack is driven by pinion shown in Figure 8–46. The pinion rotates at 125 rpm, has 24 teeth, and a diametral pitch of 6.

a. Determine the pitch diameter of the pinion.
b. Determine the dimension from the pitch line to the back of the rack, [latex]B[/latex].
c. Calculate the distance of the pinion center line to the back of the rack.
d. Determine the linear velocity of the rack.
e. How long would it take to move a rack that has a length of 20 ft?
f. How many revolutions would the pinion turn in moving the rack 20 ft?

Accepted Answer

Given: [latex]n_{pinion} = 125[/latex] rpm; Pinion [latex]P_d = 6; N_{pinion} = 24[/latex]; Length of rack [latex]= L = 20[/latex] ft
a. [latex]D_p = N_p/P_d = 24/6 = 4.000[/latex] in
b. Distance from the pitch line to the back of the rack: [latex]B[/latex] = 1.333 in [From Table 8–10]

c. Distance from back of the rack to the pinion centerline: [Call this value [latex]B-C[/latex]]
[latex]B-C = B[/latex] + Pinion radius = [latex]B + D_p/2[/latex] = 1.333 in + (4.000 in)/2 = 3.333 in

d. Linear velocity of the rack = [latex]V_{rack} = rω = (D_p/2)(n_p)[/latex]

[latex]V_{rack} = (2.000 in)(125[/latex] rev/min[latex])(2π [/latex]rad/rev[latex])(1.0[/latex] ft/12 in[latex]) = 130.9 [/latex]ft/min = 130.9 fpm

e. Time to move rack 20 ft: Using [latex]V = s/t[/latex], then [latex]t = s/V[/latex]

[latex]t=\frac{s}{V}=\frac{20ft}{130.9ft/min}.\frac{60sec}{min}=9.167[/latex] sec

f. Number of pinion revolutions, [latex]θ_P[/latex], to move rack 20 ft:
From Equation 8–37:

[latex]S_{RACK}=\frac {D_P}{2} θ_P[/latex]

Then

[latex]θ_P=\frac {S_{RACK}}{D_P/2}=\frac{20ft}{2.0 in}.\frac{12 in}{ft}=120 rad.\frac{1 rev}{2k}= 19.09rev[/latex]

TABLE 8–10 Example rack specifications
Diametral pitch	Pitch line to back (B)	Overall thickness	Face width	Nominal length [ft]
64	0.109	0.125	0.125	2
48	0.104	0.125	0.125	2
32	0.156	0.187	0.187	4
24	0.208	0.250	0.25	4
20	0.450	0.500	0.5	6
16	0.688	0.750	0.75	6
12	0.917	1.000	1	6
10	1.150	1.250	1.25	6
8	1.375	1.500	1.5	6
6	1.333	1.500	2	6
5	1.300	1.500	2.5	6
4	1.750	2.000	3.5	6

Question 8.9: A rack is driven by pinion shown in Figure 8–46. The pinion ...

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