Question 14.16: A 4-module, 20° pinion with 30 teeth drives a rack. Calculat...

Given:  m=4 mm , \alpha=20^{\circ}, z_{1}=30 .

For a rack,    L_{p}=\left(r_{a 1}^{2}-r_{b 1}^{2}\right)^{0.5}-r_{1} \sin \alpha+\frac{h_{a}}{\sin \alpha} .

h_{a}=m=4 mm , d_{1}=m z=4 \times 30=120 mm .

r_{a 1}=r_{1}+h_{a}=60+4=64 mm .

r_{b 1}=r_{1} \cos a=60 \cos 20^{\circ}=56.38 mm .

L_{p}=\left[(64)^{2}-(56.38)^{2}\right]^{0.5}-60 \sin 20^{\circ}+\frac{4}{\sin 20^{\circ}} .

=21.461 mm .

\text { Base pitch } p_{b}=\frac{2 \pi r_{b 1}}{z_{1}}=2 \pi \times \frac{56.38}{30}=11.8082 mm .

\text { Contact ratio }=\frac{L_{p}}{p_{b}}=\frac{21.461}{11.8082}=1.8174 \simeq 2 .