THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

374 14. Machinery Noise ControlGear TrainsConsider a pair of nonplanetary gears aand b in mesh. The tangential velocity V tat the mesh is the same for both teeth. But the V t of each tooth of gear i of radiusr i rotating at angular velocity ω i is given byV t = r i × ω (14.18)The number of teeth N i on a gear i is directly proportional to the radius r i ofthe gear, and ω i = 2πn i /60. Since r a × ω a = r b × ω b for gears a and b at theirmesh point, we obtainn a= N b(14.19)n b N aThus, the ratio of rotational speeds is inversely proportional to the ratio of numbersof teeth: If both gears are external spur gears, the speed ratio is negative, i.e., onegear will turn clockwise and the other counterclockwise. If one gear is an internalgear (i.e., a ring gear) the speed ratio is positive, meaning that both gears willrotate in the same angular direction. For a gear train comprised of several gears,the output to input speed ratio is given byn outputn input= ∏ i(NdrivingN driven)i(14.20)An idler gear serves as both a driving gear and a driven gear. Idler gears must beincluded in determining the direction of rotation of shafts in a gear train.Example Problem 4In Figure 14.8 showing a gear train, gear 1 on the input shaft has 40 teeth and rotatesat 3600 rpm. Gears 2, 3, and 4 have 90, 44, and 86 teeth, respectively. Narrow bandspectrum analysis of noise and vibration shows discrete tones and vibration energypeaks of 14, 27, 1227, 2400, 2455, and 4800. Establish the possible contributionsto the noise and vibration at these frequencies.SolutionGear speeds are determined fromfrom which we obtainn 2n 1= N 1N 2n 2 = n 3 = 3600 × 40 = 1600 rpm90