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hw8.sol.Fall.2010x

# hw8.sol.Fall.2010x - ME 452 Machine Design II Solution to...

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1 ME 452: Machine Design II Solution to Homework Set 8. Problem 1 (Problem 13-10, page 721). For a spur-gear set with φ = 20 degrees, while avoiding interference, find: (a) The smallest pinion tooth count that will run with itself, (b) The smallest pinion tooth count at a ratio of m G = 2.5, and the largest gear tooth count possible with this pinion, and (c) The smallest pinion that will run with a rack. SOLUTION: (a) The problem statement does not specify whether the teeth are full-depth (k=1) or stub teeth (k=0.8). Unless specified otherwise, generally assume full-depth teeth. So we assume here k=1. The smallest number of teeth on the pinion that will allow the pinion to run in mesh with a similar gear without interference is given by Equation (13-10), page 686 as 2 2 2 (1 1 3sin ) 3sin P k N φ φ = + + For full depth teeth, see page 686, the factor k is given as 1 k = Substituting the values into Equation (13-10), the smallest number of teeth on the pinion is 2 2 2 1 (1 1 3 sin 20 ) 12.3 3 sin 20 P N × = + + × ° = × ° Since it is not possible to have 12.3 teeth then 13 teeth represents the least number of teeth on the pinion. Therefore, a spur-gear set with full depth teeth, pressure angle of 20 φ = ° and 13 teeth on both the pinion and the gear is free of interference. (b) The smallest number of teeth on the pinion for meshing with a gear to obtain a gear ratio of 2.5 G m = without interference, see Equation (13-11), page 686, is ( 2 2 2 2 (1 2 )sin (1 2 )sin P k N m m m m φ φ = + + + + For full depth teeth, see page 686, the factor k is 1 k = Substituting the values into Equation (13-11), the smallest number of teeth on the pinion is

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2 ( 2 2 2 2 1 2.5 (2.5) (1 2 2.5) sin 20 14.63708 (1 2 2.5) sin 20 P N × = + + + × × ° = + × × ° Since it is not possible to have 14.6 teeth then 15 teeth represents the smallest number of teeth on the pinion. Therefore, a spur-gear set with full depth teeth, pressure angle of 20 φ = ° with 15 teeth on the pinion and a gear ratio of 2.5 is free of interference. Note also that the required N G
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hw8.sol.Fall.2010x - ME 452 Machine Design II Solution to...

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