Fig. 310. 444. Toothed wheels are sometimes employed in parallel motions; their action is necessarily not so smooth as that of the link-work we have been considering, but on the other hand the rectilinear motion is strictly true, instead of being an approximation, as will appear by the two examples which follow. B 445. Ex. 1. In fig. 310 a fixed annular wheel D has an axis of motion A at the center of its pitch-line. An arm or crank AB revolves round this center of motion, and carries the center of a wheel B, whose pitch line is exactly of half the diameter of the annular wheel with whose teeth it geers. By the well-known property of the hypocycloid any point C in the circumference of the pitch-line of B will describe a right line coinciding with a diameter of the annular pitch-circle. If then the extremity C of a rod Cc, be jointed to this wheel B by a pin exactly coinciding with the circumference of its pitch-circle, the rotation of the arm AB will cause C to describe an exact right line Cf, passing through the center A. This is termed White's parallel motion, from the name of its inventor; and the law of its motion is exactly the same as that described above (p. 215, fig. 204), which is known as Booth's motion (patented in 1843). 446. Ex. 2. Two equal toothed wheels, A and B, fig. 311, carry pins c and d at equal radial distances; and symmetrically Fig. 311. E placed with respect to the common center of motion of the crank. * Vide White's Century of Inventions. If however r be the radius of the crank ac or bd, R the radius of the pitch-circles of the wheels, the length of the link ce or ed, π and the angle cab=+0, then it can be easily shown that the 2 distance of e from the line of centers ab is equal to √ l2 —(R±r sin @)2±r. cos 0. PART THE THIRD. ON ADJUSTMENTS. CHAPTER I. GENERAL PRINCIPLES. 447. In the elementary combinations which have occupied the two previous Parts of this subject, the angular velocity ratio and directional relation in any given combination are determined by the proportion and arrangement of the parts, and will either always remain the same, or their changes will recur in similar periods. But it is necessary in many machines that we should have the power of altering or adjusting these relations. These adjustments may be distributed under three heads. (1.) To break off or resume at pleasure the communication of motion in any combination. (2) To reverse the direction of motion of the follower with respect to that of the driver; that is, to change their directional relation. (3.) To alter the velocity ratio either by determinate or by gradual steps. These changes may be either made by hand at any moment, or they may be effected by the machine itself, by means of a class of organs especially destined for that purpose; and which are in fact a kind of secondary moving powers to the machine. 448. The communication of motion may be broken off by detaching pieces that remain united during the action of the combination, and therefore move as one. Thus wheels and pullies are connected with their shafts for this purpose, by means of catches or bolts; and shafts are connected endlong with each other by couplings, or other contrivances which admit of being released or put in action at pleasure. Otherwise the communication may be broken off by disengaging the driver from the follower, which in the two kinds of contact action is effected by withdrawing the pieces from each other; in wrapping connections, by either slackening the belt or by slipping it off the pully; and in link-work, by disengaging the joints of the links. 449. But the whole of these contrivances as well as those by which the directional relation is changed, belong to constructive mechanism, and as they involve no calculations relating to the velocity ratio, which is the principal object of the present work, I shall not enter into any details respecting them, referring in the mean time to the Encyclopædias and other treatises on machinery, in which they are fully explained.* The case is different with respect to the third kind of adjustments, in which the velocity ratio is the subject of alteration, and I shall therefore give examples of the principal methods of effecting this purpose. The adjustments of the velocity ratio may consist either of (1) Determinate changes, which for the most part require the machine to be stopped, or of (2) Gradual changes, which do not require the machine to be stopped. * Vide especially Buchanan's Essays on Mill-work by Rennie, in which these combinations are very fully treated of. CHAPTER II. TO ALTER THE VELOCITY RATIO BY DETERMINATE CHANGES. 450. LET there be two axes A, B, whose position in the machine is fixed; and let it be required to connect these by toothed wheels in such a manner that the velocity ratio may assume any one of a given set of values. The simplest method is to provide as many pairs of wheels as there are to be values, and let the sum of the pitch-radii of each pair equal the distance AB of the centers. Then to obtain any one of the required ratios, we have only to screw the proper pair of wheels to the ends of the axes. Sets of wheels for this purpose are commonly termed Change-wheels. It is generally convenient that all the change-wheels should be of the same pitch, and the numbers may be calculated as in the following example. Let the given set of values for the velocity ratio or the change-ratios be 1 2 3 4 3 5 I'I'I'I'2'4 Then, since the pitch and distance of the centers are the same in every pair, the sum of their numbers of teeth must be the same; and this sum must also be divisible by the sum of the numerator and denominator of each of the above fractions, or by 2, 3, 4, 5, 9. The number required is therefore a multiple of 22.32.5=180, and if 180 be taken as the least possible number, we have the following pairs of wheels, which manifestly fulfil the conditions: |