A spiral reduction gear of ratio 3:2 is to be used on a machine with the angle between the shafts 80°. The approximate centre distance between the shafts is 125 mm. The normal pitch of the teeth is 10 mm and the gear diameters are equal. Find the number of teeth on each gear, pitch circle diameters

Question

A spiral reduction gear of ratio 3:2 is to be used on a machine with the angle between the shafts 80°. The approximate centre distance between the shafts is 125 mm. The normal pitch of the teeth is 10 mm and the gear diameters are equal. Find the number of teeth on each gear, pitch circle diameters and spiral angles. Find the efficiency of the drive if the friction angle is 5°.

Accepted Answer

[latex] ext { Given: } \quad i=1.5, \Sigma=80^{\circ}, C=125 mm , p_{n}=10 mm , d_{1}=d_{2}, \phi=5^{\circ} [/latex].

[latex] \cos \beta_{1} / \cos \beta_{2}=z_{1} / z_{2}=2 / 3 [/latex].

[latex] 3 \cos \beta_{1}=2 \cos \left(80^{\circ}-\beta_{1} ight) [/latex].

[latex] =2\left(\cos 80^{\circ} \cos \beta_{1}+\sin 80^{\circ} \sin \beta_{1} ight) [/latex].

[latex] 1.5 \cos \beta_{1}=0.17365 \cos \beta_{1}+0.9898 \sin \beta_{1} [/latex].

[latex] an \beta_{1}=1.34682 [/latex].

[latex] \beta_{1}=53.4^{\circ}, \beta_{2}=80^{\circ}-53.4^{\circ}=26.6^{\circ} [/latex].

[latex] \eta=\left[\cos (\Sigma+\phi)+\cos \left(\beta_{1}-\beta_{2}-\phi ight) ight] /\left[\cos (\Sigma-\phi)+\cos \left(\beta_{1}-\beta_{2}-\phi ight) ight] [/latex].

[latex] =\left[\cos 85^{\circ}+\cos 20.8^{\circ} ight] /\left[\cos 75^{\circ}+\cos 20.8^{\circ} ight] [/latex].

[latex] =85.62 \% [/latex].

[latex] C=\left[z_{1} p_{n} /(2 \pi) ight]\left[1 / \cos \beta_{1}+i / \cos \beta_{2} ight] [/latex].

[latex] 125=\left[\left(z_{1} imes 10 ight) /(2 \pi) ight] imes\left(1 / \cos 53.4^{\circ}+1.5 / \cos 26.6^{\circ} ight] [/latex].

[latex] z_{1}=23.41 \cong 24 [/latex].

[latex] z_{2}=24 imes 1.5=36 [/latex].

[latex] d_{1}=d_{2}=z_{1} p_{n} /\left(\pi \cos \beta_{1} ight)=(24 imes 10) /\left(\pi imes \cos 53.4^{\circ} ight)=128.13 mm [/latex].

Question 14.41: A spiral reduction gear of ratio 3:2 is to be used on a mach...

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