APPLIED MECHAMCS. 



ITOUTUBO w HELLS. 



of an intermediate wheel b (Fig. 233). Some- 

 tn. Un iwL u in B HI 



internally (Fig. 234) 

 where a is a portion o 

 the one, and o 6 part o 

 the circumference o 

 ->\\v_^yi the other. In this case, 



* Xj~ u c X^-^S* the direction of rota- 



--> tion u not reversed. 



By means of toothed gearing, the angular speed ol 

 fig. Ut. rotation may be altered at pleasure, 



as will be evident from the follow-in; 

 considerations. The larger whee 

 (Fig. 232) has 45 teeth, and therefore, 

 during each revolution, presents 45 

 successive recesses for the teeth of the 

 smaller pinion, which number 22. 

 For every revolution of the wheel, 

 therefore, the pinion must make 

 revolutions and advance the space ol 

 1 tooth, because 2 X 22 + 1 = 45 

 during 2 revolutions of the wheel, the 

 pinion makes 4 revolutions and 2 teeth; 

 and so on, until the wheel has made 

 22 revolutions, or caused 22 x 45 = 

 990 of its teeth to pass the point of gear, in which time 

 the pinion must have made 45 revolutions, or caused 

 45 X 22=990, the same number, to pass the point of 

 gear. And so it would be found with any other number 

 of cogs, that the angular velocities of the geared wheels, 

 or the number of revolutions they make in a given time, 

 are inversely as the numbers of their teeth. When an 

 intermediate wheel is employed, as in Fig. 233, it only 

 affects direction of rotation, not the speed of the extreme 

 wheels. Thus if a (Fig. 233) have 34 teeth, and 6, the 

 intermediate, have 20 teeth, the angular velocity of o is 

 to that of 6 as 20 to 34 ; or if we take the speed of o as 



34 

 1 , the speed of 6 is ^. Again, if c have 27 teeth, its speed 



i to that of 6 as 20 to 27, or it is ^ths of the speed of 



6, that '* 



20 34 34 



*** e pced of o. But, did we 



leave the intermediate wheel out of consideration, we 

 find that were a with 34 teeth to drive c with 27, the 



4JA 



peed of e would berths of that of o, the same result 



as before, though in the opposite direction. The same 

 principle applies, whatever be the number of intermediate 

 wheels ; for the speed of the first and last will always be to 

 one another inversely as their respective numbers of teeth. 

 f from the centres of two geared wheels A B (Fig. 

 235) circles be drawn, touching each other at a point 

 midway between the extreme projections of their teeth, 



Fig. 23*. 



mean that the distance measured from the centre of one 

 tooth to that of the next (these nits being 



taken on the pitch circle) is 1 inch or 2 inches, as the 

 case may be. In Fig. 235, the distance between any 

 two adjoining points, where the dotted radii from A anil 

 B cut the pitch circles, is the pitch of either wheel, and 

 those pitchit are necessarily equal in length, in order that 

 the wheels may gear. But as this distance is the interval 

 between the central points measured in a straight line, 

 or along the chord of the small arc intercepted between 

 them, it does not accurately correspond to the length of 

 the arc. The larger the circle of which the arc is a por- 

 tion, and the smaller the pitch or portions into which 

 the circumference is divided, the more nearly do the 

 circular arc and its chord approach to equality. Ac- 

 cordingly, in cases of nearly equal wheels, or whenever 

 the pitch is small in proportion to the dimensions of 

 either wheel, the more nearly do the numbers of teeth 

 express the proportions of the circumferences. But the 

 circumferences of circles being exactly proportional to 

 their diameters, the numbers of teeth are very nearly in 

 the same proportion ; and the angular velocities of 

 geared wheels are therefore inversely as the diameters 

 of their pitch circles. 



When there is a train of geared wheels, so arranged 

 that the first shall drive the second that the third, 

 fixed on the shaft of the second, shall drive the fourth 

 the fifth, fixed on the shaft of the fourth, shall drive the 

 sixth, and so on we readily find the angular velocity 

 of any wheel in the train thus. 



Ride. Multiply the angular velocity of the first driver 

 iy its number of teeth or diameter, and by the numbers 

 of teeth or diameters of all the drivers, and divide the 

 result by the product of the numbers of teeth or the 

 diameters of all the driven wheels. 



Example. Wheel I., 3ii inches diameter with 72 teeth, 

 making 120 revolutions per minute, drives wheel II., 

 12 ius. diain. with 24 teeth ; on the shaft of wheel II. is 

 ixed wheel III., 10 ins. dia. with 30 teeth, driving v. 

 ;V.. C ius. diaiu. with 13 teeth: required the speed of 

 wheel IV. 



120 rev. X 30 ins. x 10 ins. 



12 ins. X <i in*. 



120 rev. X 72 teeth x 30 teeth 

 teeth 



= COO rev. per min. 



24 teeth X 



From the general principle of mechanics, that no 

 lower can be gained or lost in any train of mechanism, 

 jut that only the two elements of power, pressure and 

 elocity, can be interchanged, irrespective of friction or 

 ther useless resistances, it follows that, at the end of 

 any train of wheels, the power is the same as at the be- 

 inning ; but the velocity being different, the pressure 

 or strain on the teeth must be different also. In the 

 example wo have given, if we suppose that 1 horse-power 

 baa given motion to wheel I., we should expect to get 

 1 horse-power from wheel IV. ; but as the speed of 

 wheel IV. is five times that of wheel I., so the pres- 

 sure exerted by it at any point is J th of that exerted 

 by wheel I. at a point equally distant from its axis. 

 To show that this is true, not only in general terms, 

 but in the particular case, wo shall suppose 1 horse- 



circles are called the pitch lines or circles of the 

 eeth, and their circumferences being equally divi.1,-,1, 

 give the intervals from tooth to tooth in each, or what 

 to technically called the pitch of the tooth. When we 

 peak of wheel of 1 inch pitch, or 2 inches pitch, we 



power passing through the teetli of wheel I., 18 ins. 

 from its axis. The teeth of wheel II. nturt 



with those of wheel I., receive its full power at li ISM. 

 from its axis, and convey it through ita sluift to 

 the teeth of wheel III., distant 5 ins. from its axis, 

 ainl, therefore, sustaining a pressure of Jths of 

 the original pressure. Again, this strain being 

 to the teeth of wheel IV. at 3 ins. from its axis, is 

 equivalent to Aths of = 1th of the original i !..- 

 snro estimated at 18 ins. from the axis of the last 

 wheel. In any train of wheelwork, then, wo may 

 safely diminish the sizes and strengths of tin 

 as their velocity increases; and, conversely, we should 

 increase their strengths as the velocity diminishes. 



FORMS OF TEETH. One of the most important 

 matters connected with toothed wheels refers to the 

 forms or outlines of the teeth. It U to be desired that 



