PLANETARY WHEEL TRAINS.
By Prof. C.W. MacCORD, Sc. D.
IV.
The arrangement of planetary wheels which has been applied in practice to the greatest extent and to the most purposes, is probably that in which the axial motions of the train are derived from a fixed sun wheel. Numerous examples of such trains are met with in the differential gearing of hoisting machines, in portable horse-powers, etc. The action of these mechanisms has already been fully discussed; it may be remarked in addition that unless the speed be very moderate, it is found advantageous to balance the weights and divide the pressures by extending the train arm and placing the planet-wheels in equal pairs diametrically opposite each other, as, for instance, in Bogardus' horse power, Fig. 31.
PLANETARY WHEEL TRAINS.
In trains of this description, the velocity ratio is invariable; which for the above-mentioned objects it should be. But the use of a planetary combination enables us to cause the motions of two independent trains to converge, and unite in producing a single resultant rotation. This may be done in two ways; each of the two independent trains may drive one sun-wheel, thus determining the motion of the train-arm; or, the train-arm may be driven by one of them, and the first sun-wheel by the other; then the motion of the second sun-wheel is the resultant. Under these circumstances the ratio of the resultant velocity to that of either independent train is not invariable, since it may be affected by a change in the velocity of the other one. To illustrate our meaning, we give two examples of arrangements of this nature. The first is Robinson's rope-making machine, Fig. 32. The bobbins upon which the strands composing the rope are wound turn freely in bearings in the frames, G, G, and these frames turn in bearings in the disk, H, and the three-armed frame or spider, K, both of which are secured to the central shaft, S. Each bobbin-frame is provided with a pinion, a, and these three pinions engage with the annular wheel, A. This wheel has no shaft, but is carried and kept in position by three pairs of rollers, as shown, so that its axis of rotation is the same as that of the shaft, S; and it is toothed externally as well as internally. The strands pass through the hollow axes of the pinions, and thence each to its own opening through the laying-top, T, fixed upon S, which completes the operation of twisting them into a rope. The annular wheel, A, it will be perceived, may be driven by a pinion, E, engaging with its external teeth, at a rate of speed different from that of the central shaft; and by varying the speed of that pinion, the velocity of the wheel, A, may be changed without affecting the velocity of S.
It is true that in making a certain kind of rope, the velocity ratio of A and S must remain constant, in order that the strands may be equally twisted throughout; but if for another kind of rope a different degree of twist is wanted, the velocity of the pinion, E, may be altered by means of change-wheels, and thus the same machine may be used for manufacturing many different sorts.
The second combination of this kind was devised by the writer as a "tell-tale" for showing whether the engines driving a pair of twin screw-propellers were going at the same rate. In Fig. 33, an index, P, is carried by the wheel, F: the wheel, A, is loose upon the shaft of the train-arm, which latter is driven by the wheel, E. The wheels, F and f, are of the same size, but a is twice as large as A; if then A be driven by one engine, and E by the other, at the same rate but in the opposite direction, the index will remain stationary, whatever the absolute velocities. But if either engine go faster than the other, the index will turn to the right or the left accordingly. The same object may also be accomplished as shown in Fig. 34, the index being carried by the train-arm. It makes no difference what the actual value of the ratio A/a may be, but it must be equal to F/f: under which condition it is evident that if A and F be driven contrary ways at equal speeds, small or great, the train-arm will remain at rest; but any inequality will cause the index to turn.
In some cases, particularly when annular wheels are used, the train-arm may become very short, so that it may be impossible to mount the planet-wheel in the manner thus far represented, upon a pin carried by a crank. This difficulty may be surmounted as shown in Fig. 35, which illustrates an arrangement originally forming a part of Nelson's steam steering gear. The Internal pinions, a, f, are but little smaller than the annular wheels, A, F, and are hung upon an eccentric E formed in one solid piece with the driving shaft, D.
The action of a complete epicyclic train involves virtually and always the action of two suns and two planets; but it has already been shown that the two planets may merge into one piece, as in Fig. 10, where the planet-wheel gears externally with one sun-wheel, and internally with the other.
But the train may be reduced still further, and yet retain the essential character of completeness in the same sense, though composed actually of but two toothed wheels. An instance of this is shown in Fig. 36, the annular planet being hung upon and carried by the pins of three cranks, c, c, c, which are all equal and parallel to the virtual train-arm, T. These cranks turning about fixed axes, communicate to f a motion of circular translation, which is the resultant of a revolution, v', about the axis of F in one direction, and a rotation, v, at the same rate in the opposite direction about its own axis, as has been already explained. The cranks then supply the place of a fixed sun-wheel and a planet of equal size, with an intermediate idler for reversing the, direction of the rotation of the planet; and the velocity of F is
V'= v'(1 - f/F).
A modification of this train better suited for practical use is shown in Fig. 37, in which the sun-wheel, instead of the planet, is annular, and the latter is carried by the two eccentrics, E, E, whose throw is equal to the difference between the diameters of the two pitch circles; these eccentrics must, of course, be driven in the same direction and at equal speeds, like the cranks in Fig. 36.
PLANETARY WHEEL TRAINS.
A curious arrangement of pin-gearing is shown in Fig. 38: in this case the diameter of the pinion is half that of the annular wheel, and the latter being the driver, the elementary hypocycloidal faces of its teeth are diameters of its pitch circle; the derived working tooth-outlines for pins of sensible diameter are parallels to these diameters, of which fact advantage is taken to make the pins turn in blocks which slide in straight slots as shown. The formula is the same as that for Fig. 36, viz.:
V' = v'(1 - f/F),
which, since f = 2F, reduces to V' = -v'.
Of the same general nature is the combination known as the "Epicycloidal Multiplying Gear" of Elihu Galloway, represented in Fig. 39. Upon examination it will be seen, although we are not aware that attention has previously been called to the fact, that this differs from the ordinary forms of "pin gearing" only in this particular, viz., that the elementary tooth of the driver consists of a complete branch, instead of a comparatively small part of the hypocycloid traced by rolling the smaller pitch-circle within the larger. It is self-evident that the hypocycloid must return into itself at the point of beginning, without crossing: each branch, then, must subtend an aliquot part of the circumference, and can be traced also by another and a smaller describing circle, whose diameter therefore must be an aliquot part of the diameter of the outer pitch-circle; and since this last must be equal to the sum of the diameters of the two describing circles, it follows that the radii of the pitch circles must be to each other in the ratio of two successive integers; and this is also the ratio of the number of pins to that of the epicycloidal branches.
Thus in Fig. 39, the diameters of the two pitch circles are to each other as 4 to 5; the hypocycloid has 5 branches, and 4 pins are used. These pins must in practice have a sensible diameter, and in order to reduce the friction this diameter is made large, and the pins themselves are in the form of rollers. The original hypocycloid is shown in dotted line, the working curve being at a constant normal distance from it equal to the radius of the roller; this forms a sort of frame or yoke, which is hung upon cranks as in Figs. 36 and 38. The expression for the velocity ratio is the same as in the preceding case:
V¹ = v'(1 - f/F); which in Fig. 39 gives
V¹ = v'(1 - 5/4)= -¼v':
the planet wheel, or epicycloidal yoke, then, has the higher speed, so that if it be desired to "gear up," and drive the propeller faster than the engine goes (and this, we believe, was the purpose of the inventor), the pin-wheel must be made the driver; which is the reverse of advantageous in respect to the relative amounts of approaching and receding action.
In Figs. 40 and 41 are given the skeletons of Galloway's device for ratios of 3:4 and 2:3 respectively, the former having four branches and three pins, the latter three branches and two pins. Following the analogy, it would seem that the next step should be to employ two branches with only one pin; but the rectilinear hypocycloid of Fig. 38 is a complete diameter, and the second branch is identical with the first; the straight tooth, then, could theoretically drive the pin half way round, but upon its reaching the center of the outer wheel, the driving action would cease: this renders it necessary to employ two pins and two slots, but it is not essential that the latter should be perpendicular to each other.
In these last arrangements, the forms of the parts are so different from those of ordinary wheels, that the true nature of the combinations is at least partially disguised. But it may be still more completely hidden, as for instance in the common elliptic trammel, Fig. 42. The slotted cross is here fixed, and the pins, R and P, sliding respectively in the vertical and horizontal lines, control the motion of the bar which carries the pencil, S. At first glance there would seem to be nothing here resembling wheel works. But if we describe a circle upon R P as a diameter, its circumference will always pass through C, because R C P is a right angle, and the instantaneous axis of the bar being at the intersection O of a vertical line through P, with a horizontal line through R, will also lie upon this circumference. Again, since O is diametrically opposite to C, we have C O = R P, whence a circle about center C with radius R P will also pass through O, which therefore is the point of contact of these two circles. It will now be seen that the motion of the bar is the same as though carried by the inner circle while rolling within the outer one, the latter being fixed; the points P and R describing the diameters L M and K N, the point D a circle, and S an ellipse; C D being the train-arm. The distance R P being always the diameter of one circle and the radius of the other, the sizes of the wheels can be in effect varied by altering that distance.
Thus we see that this combination is virtually the same in its action as the one shown in Fig. 43, known as Suardi's Geometrical Pen. In this particular case the diameter of a is half of that of A; these wheels are connected by the idler, E, which merely reverses the direction without affecting the velocity of a's rotation. The working train arm is jointed so as to pivot about the axis of E, and may be clamped at any angle within its range, thus changing the length of the virtual train arm, C D. The bar being fixed to a, then, moves as though carried by the wheel, a¹, rolling within A¹; the radius of a¹ being C D, and that of A¹ twice as great.
In either instrument, the semi-major axis C X is equal to S R, and the semi-minor axis to S P.
The ellipse, then, is described by these arrangements because it is a special form of the epitrochoid; and various other epitrochoids may be traced with Suardi's pen by substituting other wheels, with different numbers of teeth, for a in Fig. 43.
Another disguised planetary arrangement is found in Oldham's coupling, Fig. 44. The two sections of shafting, A and B, have each a flange or collar forged or keyed upon them; and in each flange is planed a transverse groove. A third piece, C, equal in diameter to the flanges, is provided on each side with a tongue, fitted to slide in one of the grooves, and these tongues are at right angles to each other. The axes of A and B must be parallel, but need not coincide; and the result of this connection is that the two shafts will turn in the same direction at the same rate.
The fact that C in this arrangement is in reality a planetary wheel, will be perceived by the aid of the diagram, Fig. 45. Let C D be two pieces rotating about fixed parallel axes, each having a groove in which slides freely one of the arms, A C, A D, which are rigidly secured to each other at right angles.
The point C of the upper arm can at the instant move only in the direction C A; and the point D of the lower arm only in the direction A D, at the same instant; the instantaneous axis is therefore at the intersection, K, of perpendiculars to A C and A D, at the points C and D. C A D K being then a rectangle, A K and C D will be two diameters of a circle whose center, O, bisects C D; and K will also be the point of contact between this circle and another whose center is A, and radius A K = C D. If then we extend the arms so as to form the cross, P K, M N, and suppose this to be carried by the outer circle, f, rolling upon the inner one, F, its motion will be the same as that determined by the pieces, C D; and such a cross is identical with that formed by the tongues on the coupling-piece, C, of Fig. 44.
A O is the virtual train-arm; let the center, A, of the cross move to the position B, then since the angles A O B at the center, and A C B in the circumference, stand on the same arc, A B, the former is double the latter, showing that the cross revolves twice round the center O during each rotation of C; and since A C B = A D B, C and D rotate with equal velocities, and these rotations and the revolution about O have the same direction. While revolving, the cross rotates about its traveling center, A, in the opposite direction, the contact between the two circles being internal, and at a rate equal to that of the rotations of C and D, because the velocities of the axial and the orbital motion are to each other as f is to F, that is to say, as 1 is to 2. Since in the course of the revolution the points P and K must each coincide with C, and the points M and N with D, it follows that each tongue in Fig. 44 must slide in its groove a distance equal to twice that between the axes of the shafts.
Another example of a disguised planetary train is shown in Fig. 46. Let C be the center about which the train arm, T, revolves, and suppose it required that the distant shaft, B, carried by T, shall turn once backward for each forward revolution of the arm. E is a fixed eccentric of any convenient diameter, in the upper side of which is a pin, D. On the shaft, B, is keyed a crank, B G, equal in length to C D; and at any convenient point, H, on B C, or its prolongation, another crank, H F, equal also to C D, is provided with a bearing in the train-arm. The three crank pins, F, D, G, are connected by a rod, like the parallel rod of a locomotive; F D, D G, being respectively equal to H C, C B. Then, as the train-arm revolves, the three cranks must remain parallel to each other; but C D being fixed, the cranks, H F and B G, will remain always parallel to their original positions, thus receiving the required motion of circular translation.
The result then is the same as though the periphery of E were formed into a fixed spurwheel, A, and another, a, of the same size, secured on a shaft, B, the two being connected by the three equal wheels, L, M, N. It need hardly be stated that instead of the eccentric, E, a stationary crank similar and equal to B G may be used, should it be found better suited to the circumstances of the case.
It is possible also to apply the planetary principle to mechanism composed partially of racks; in fact, a rack is merely a wheel of prodigious size—the limiting case, just as a right line is a circle of infinite radius. A very neat application of this principle is found in Villa's Pantograph, of which a full description and illustration was given in SCIENTIFIC AMERICAN SUPPLEMENT, No. 424; the racks, moving side by side, are the sun-wheels, and the planet-wheels are the pinions, carried by the traveling socket, by which the motion of one rack is transmitted to the other.
Thus far attention has been called only to combinations of circular wheels. In these the velocity ratios are constant, if we except the cases in which two independent trains converge, the two sun-wheels, or one of them and the train-arm, being driven separately—and even in those, a variable motion of the ultimate follower is obtained only by varying the speed of one or both drivers. It is not, however, necessary to employ circular wheels exclusively or even at all; wheels of other forms are capable of acting together in the relation of sun and planet, and in this way a varying velocity ratio may be produced even with a fixed sun-wheel and a single driver. We have not found, in the works of any previous writer, any intimation that noncircular wheels have ever been thus combined; and we propose in the following article to illustrate some curious results which may be thus obtained.