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Question 13.4: A gearbox is needed to provide an exact 30:1 increase in spe...

A gearbox is needed to provide an exact 30:1 increase in speed, while minimizing the overall gearbox size. Specify appropriate teeth numbers.

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The previous example demonstrated the difficulty with finding integer numbers of teeth to provide an exact ratio. In order to obtain integers, factor the overall ratio into two integer stages.

 

e=30=(6)(5)

 

N_{2} / N_{3}=6 \quad \text { and } \quad N_{4} / N_{5}=5

 

With two equations and four unknown numbers of teeth, two free choices are available. Choose N_{3} \text { and } N_{5} to be as small as possible without interference. Assuming a 20° pressure angle, Eq. (13–11) gives the minimum as 16.

Then

 

N_{P}=\frac{2 k}{(1+2 m) \sin ^{2} \phi}\left(m+\sqrt{m^{2}+(1+2 m) \sin ^{2} \phi}\right) (13–11)

 

N_{2}=6 N_{3}=6(16)=96

 

N_{4}=5 N_{5}=5(16)=80

 

The overall train value is then exact.

 

e=(96 / 16)(80 / 16)=(6)(5)=30

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