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Q. 10.1

A helical gear has a normal diametral pitch, $P_{nd}$, of 8, a normal pressure angle of 20°, 32 teeth, a face width of 3.00 in, and a helix angle of 15°. Compute the diametral pitch, the transverse pressure angle, and the pitch diameter. If the gear is rotating at 650 rpm while transmitting 7.50 hp, compute the pitch line speed, the tangential force, the axial force, and the radial force.

Verified Solution

Diametral Pitch:
$P_{d}=P_{n d} \cos \psi=8 \cos \left(15^{\circ}\right)=7.727$
Transverse Pressure Angle: [Equation (10-1)]
\begin{aligned}\phi_{t} &=\tan ^{-1}\left(\tan \phi_{n} / \cos \psi\right) \\\phi_{t} &=\tan ^{-1}\left[\tan \left(20^{\circ}\right) / \cos \left(15^{\circ}\right)\right]=20.65^{\circ}\end{aligned}
Pitch Diameter:
$D=N / P_{d}=32 / 7.727=4.141 \text { in }$
Pitch Line Speed, $v_{t}$ : [Equation (10-5)]
$v_{t}=\pi D n / 12=\pi(4.141)(650) / 12=704.7 \mathrm{ft} / \mathrm{min}$
Tangential Force, $W_{t}$ : [Equation (10-7)]
$W_{t}=33000(P) / v_{t}=33000(7.5) / 704.7=351 \mathrm{lb}$
Axial Force, $W_{x}$ : [Equation (10-9)]
$W_{x}=W_{t} \tan \psi=351 \tan \left(15^{\circ}\right)=94 \mathrm{lb}$
Radial Force, $W_{r}$ : [Equation (10-8)]
$W_{r}=W_{t} \tan \phi_{t}=351 \tan \left(20.65^{\circ}\right)=132 \mathrm{lb}$