ratio, and a shifting train with the velocity-ratio n, the required velocity-ratio may be obtained with precision by means of wheels of moderate size.* Another use of epicyclic trains is to make the train-arm move, for purposes of regulation (as in certain governors), with a velocity proportional to the difference between the velocities of the primary wheels B and C. This is best effected by causing the primary wheels B and C to rotate in contrary directions, and to connect them by means of a shifting train such that, when the train-arm is at rest, the angular velocities of those wheels are equal and opposite. This amounts to making n = - 1 in equation 3, and c = a negative quantity, say - k; and then the expression for the angular velocity of the train-arm becomes A For example, in Fig. 176, O is a vertical spindle, about which the equal and similar bevel wheels B and C turn in opposite directions. A is the train-arm, being a horizontal spindle carried by a collar which turns about the vertical spindle. The shifting train consists of a bevel wheel turning about the spindle A, and gearing with the wheels B and C. In order to produce a balance of forces, two, and sometimes three or four, equal and similar horizontal spindles like A project from the collar, and carry equal and similar bevel wheels. In the figure two are shown. The result is, that when the wheels B and C turn in opposite directions with equal speed, the train-arm stands still; but when the velocities of those wheels become unequal, the train-arm turns in the direction Fig. 176. • The solution of this problem is to be obtained in any particular case by a series of trials conducted generally in the following manner :-Let n be an approximation to the ratio not containing factors exceeding what is considered a convenient limit (values of n may be found by the method of continued fractions, Article 117, page 107). Then make a = a C 1 b n - n ; and try whether the ratio contains inconveniently large factors. The trial is b to be repeated with the various different values of n, until a satisfactory result is arrived at. This method cannot fail, provided it is c only, and not b, which contains inconveniently large factors. of the greater of the two velocities, with a speed equal to half their difference. Other applications of epicyclic trains, where the last follower is a secondary piece, will be mentioned under the head of aggregate paths. SECTION III.-Production of varying Aggregate Velocity-Ratios. 235. The Reciprocating Endless Screw may be used where it is desired that there shall be periodic fluctuations in the ratio of the speed of the follower to that of the driver. In this combination a wheel is driven by a rotating screw, as in fig. 112, page 164, which screw has at the same time a reciprocating motion along its axis. 236. Epicyclic Trains with Periodic Action are used for the same purpose. This is effected by communicating, by means of suitable mechanism, such as a cam, or a crank and link, the required reciprocating motion to the train-arm A, fig. 175, page 243. The angular velocity of the follower, C, is expressed, as in Article 234, by in which n b is a constant term, and (1 n) a a periodically varying term; the factor 1 n being constant, and the factor a periodic. 236 A.—The Sun-and-Planet Motion is a sort of epicyclic train with periodic action. In fig. 177, C is a shaft which overhangs B Fig. 177. its bearing, and carries on its overhanging end a toothed wheel, CE, called the sun-wheel. This gears with another toothed wheel, D E, called the planet-wheel, which is made fast to the connecting-rod D B, which hangs from one end of the lever or walking-beam, A B. At the centre, D, of the wheel D E is a pin which is connected with the shaft C by a link or bridle, C D (shown by dotted lines); so that it revolves round the axis of C like a crank-pin, making one revolution for each double-stroke of the beam A B. In the first place, to determine the mean ratio of the linear velocity of the pin D to that of the pitch-circle of the sun-wheel, CE, it is to be observed that the latter velocity is at every instant equal to that of the pitch-point E in the planet-wheel. Now, the motion of the planet-wheel is one of translation in a circle along with the pin D, compounded with an angular oscillation to and fro along with the rod D B. Hence the mean linear velocity of Dis equal to that of the pitch-circle of C E. Secondly, as to the mean ratio of the angular velocity of the bridle CD to that of the sun-wheel C E, it is obvious that as the mean linear velocities of D, and of the pitch-circle of C E, are equal, their mean angular velocities are inversely as the radii C E and C D; or in symbols In the sun-and-planet motion, as originally contrived and constructed by Watt, the sun-wheel and planet-wheel were made of equal radii; so that CD was 2 CE; and the sun-wheel made two turns for each revolution of the planet-wheel round it. = Thirdly, as to the ratio of the linear velocities of the points D and E at any instant; this is to be found by producing D ̊C till it cuts A B in I, which will be the instantaneous axis of the planetwheel; and then taking the proportion, The mean value of this ratio is unity, as already stated. It attains its greatest and least values in the two positions of the combination when B D and CD are in one straight line, so that I coincides with B; and then its values are respectively 237. Eccentric Gearing. This is a combination for producing a periodically varying velocity-ratio by means of a train of circular wheels, one of which turns eccentrically about an axis. It is nearly, but not exactly, equivalent in its action to a pair of elliptic toothed wheels (Article 100, page 95). In fig. 178, A is the axis of a shaft, which carries an eccentric circular toothed wheel. This A Fig. 178. gears with a second toothed wheel, centred on a moveable axis, C, which gears with a third toothed wheel, centred on a fixed axis, D. The centres of the three wheels are linked together by the two train-arms B C, C D; so that the wheels are kept always in gearing, while the centre pin B revolves round the axis A. Suppose the wheels on B and D to be of equal size. Then, if the trainarms were fixed, the rotation of the first wheel about B would produce a rotation of the third wheel about D, with equal speed and in the same direction. The effect of the revolving of B about A is to combine that rotation of D with an alternate increase and diminution of speed, corresponding to the alternate diminution and increase of the angle B C D. The greatest and least values of the velocity-ratio take place when the line of connection, C B, touches the two sides of the circle described by B about A; that is to say, when that line is in the two positions marked C B I and C J B' respectively. Let I and J be the points where C B cuts the line of centres, D A, when in those positions; then the two corresponding values of the velocity-ratio of D to A are respectively 238. Aggregate Linkwork in General.-A combination in aggregate linkwork is usually of the following kind :-A bar, or other rigid body, capable of moving parallel to a given plane, has two of its points connected by means of rods with two drivers:-A third point is connected by means of a rod with a follower. The motions 4 of the first two points, as compared with those of their drivers, are determined by the principles of elementary combinations in linkwork, and so also is the motion of the follower, as compared with that of the third point; but the determination of the motion of the third point from that of the first two is a problem to be solved by the principles of the motion of secondary pieces, Article 69, pages 45 to 48; that is, by the process of finding the instantaneous axis, or by some equivalent process. In most of the particular cases of aggregate velocities obtained by linkwork which occur in practice, the three points in the bar are either situated in one straight line to which their motions are perpendicular, or are so nearly in that position that their comparative motions, as determined on the supposition of their being in it exactly, are sufficiently near to the truth for practical purposes. In such cases let A and B, figs. 179 and 180, be the two points whose velocities at a given instant are given, and C the third point. Draw A a and B b perpendicular to A B, and of lengths proportional to the given velocities, and in the proper direction; join a b; draw C c also perpendicular to A B, cutting a b in c; Cc will represent the velocity of C. The following formula is the symbolical expression of the same rule, in which a, b, and c denote the velocities of A, B, and C respectively :- The formula, as it stands, is applicable to the case in which C lies between A and B, the velocities a, b, and c being treated as positive or negative according to their directions. When Clies beyond B, BC is to be treated as negative, and CA as positive; when beyond A, C A is to be treated as negative, and B C as positive. The velocity of C may be regarded as the resultant of two coma BC which would be its velocity if B were fixed; and ponents, b.CA AB AB which would be its velocity if A were fixed. Trains of aggregate linkwork may be used to combine any |