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Chapter 13: Problem 32

A 79-tooth, \(30^{\circ}\) helix angle, helical gear is in mesh with a 20 -tooth pinion. The \(p_{d}=6\) and \(\phi=20^{\circ}\). Find the transverse and axial contact ratios.

Short Answer

Expert verified
The transverse contact ratio is approximately \(8.994\) and the axial contact ratio is approximately \(1.552\).

Step by step solution

01

Identify the basic parameters

The given parameters are the number of teeth on the gear \(N_g = 79\), the number of teeth on the pinion \(N_p = 20\), the helix angle of the gear \(\psi = 30^{\circ}\), the pressure angle \(\phi = 20^{\circ}\), and the diametral pitch \(p_d = 6\).
02

Calculate the Normal Diametral Pitch

The normal diametral pitch, \(p_{nd}\), is given by \(p_{nd} = p_{d} \cdot \cos{\psi}\). Substituting the given values, \(p_{nd} = 6 \cdot \cos{30^{\circ}}\). After calculating, we get \(p_{nd} = 5.196\).
03

Compute the Transverse Contact Ratio

The transverse contact ratio, \(m_t\), is given by \(m_t = \frac{\pi}{2} \cdot (\frac{N_g}{p_{nd}}) \cdot \tan{\phi}\). Substituting the given values, \(m_t = \frac{\pi}{2} \cdot (\frac{79}{5.196}) \cdot \tan{20^{\circ}}\). After calculating, we get \(m_t = 8.994\).
04

Compute the Axial Contact Ratio

The axial contact ratio, \(m_a\), is given by \(m_a = (m_t \cdot \tan{\psi}) / \pi\). Substituting the given values, \(m_a = (8.994 \cdot \tan{30^{\circ}}) / \pi\). After calculating, we get \(m_a = 1.552\).

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Most popular questions from this chapter

A 3 -start wormset has \(d=1.75\) in, \(p_{x}=0.2 \mathrm{in}, m_{G}=17: 1\). Find the lead, lead angle, worm gear diameter, and center distance. Will it self- lock? The input speed is 1400 rpm.

A 2-start wormset has \(L=2.00 \mathrm{in}, C=9.00 \mathrm{in}, m_{G}=20\), and the angle between the shafts is \(90^{\circ}\). Find the pitch diameters of the worm and worm gear, the lead angle, and the axial pitch.

A \(90^{\circ}\) spiral-bevel gearset is needed to give a \(5: 1\) reduction. Determine the pitch cone angles, pitch diameters, and gear forces if the \(20^{\circ}\)-pressure-angle pinion has 16 teeth of \(p_{d}=7\), and the transmitted power is \(3 \mathrm{hp}\) at 600 pinion rpm.

A 3-start wormset has \(d=60 \mathrm{~mm}, p_{x}=12 \mathrm{~mm}, m_{G}=60\). Find the lead, lead angle, worm gear diameter, and center distance. Will it self-lock? The input speed is \(2500 \mathrm{rpm}\).

A 78-tooth, \(30^{\circ}\)-helix-angle, helical gear is in mesh with a 27-tooth pinion. The \(p_{d}=6\) and \(\phi=20^{\circ}\). Find the transverse and axial contact ratios.
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