motions of the pieces, and is, for that reason, defective, as Willis pointed out in his admirable treatise On the Principles of Mechanism.
Willis’s classification is founded, in the first place, on comparative motion, as expressed by velocity ratio and directional relation, and in the second place, on the mode of connexion of the driver and follower. He divides the elementary combinations in mechanism into three classes, of which the characters are as follows:—
- Class A: Directional relation constant; velocity ratio constant.
- Class B: Directional relation constant; velocity ratio varying.
- Class C: Directional relation changing periodically; velocity ratio constant or varying.
Each of those classes is subdivided by Willis into five divisions, of which the characters are as follows:—
| Division | A: | Connexion | by | rolling contact. |
| ” | B: | ” | ” | sliding contact. |
| ” | C: | ” | ” | wrapping connectors. |
| ” | D: | ” | ” | link-work. |
| ” | E: | ” | ” | reduplication. |
In the Reuleaux system of analysis of mechanisms the principle of comparative motion is generalized, and mechanisms apparently very diverse in character are shown to be founded on the same sequence of elementary combinations forming a kinematic chain. A short description of this system is given in § 80, but in the present article the principle of Willis’s classification is followed mainly. The arrangement is, however, modified by taking the mode of connexion as the basis of the primary classification, and by removing the subject of connexion by reduplication to the section of aggregate combinations. This modified arrangement is adopted as being better suited than the original arrangement to the limits of an article in an encyclopaedia; but it is not disputed that the original arrangement may be the best for a separate treatise.
§ 39. Rolling Contact: Smooth Wheels and Racks.—In order that two pieces may move in rolling contact, it is necessary that each pair of points in the two pieces which touch each other should at the instant of contact be moving in the same direction with the same velocity. In the case of two shifting pieces this would involve equal and parallel velocities for all the points of each piece, so that there could be no rolling, and, in fact, the two pieces would move like one; hence, in the case of rolling contact, either one or both of the pieces must rotate.
The direction of motion of a point in a turning piece being perpendicular to a plane passing through its axis, the condition that each pair of points in contact with each other must move in the same direction leads to the following consequences:—
I. That, when both pieces rotate, their axes, and all their points of contact, lie in the same plane.
II. That, when one piece rotates, and the other shifts, the axis of the rotating piece, and all the points of contact, lie in a plane perpendicular to the direction of motion of the shifting piece.
The condition that the velocity of each pair of points of contact must be equal leads to the following consequences:—
III. That the angular velocities of a pair of turning pieces in rolling contact must be inversely as the perpendicular distances of any pair of points of contact from the respective axes.
IV. That the linear velocity of a shifting piece in rolling contact with a turning piece is equal to the product of the angular velocity of the turning piece by the perpendicular distance from its axis to a pair of points of contact.
The line of contact is that line in which the points of contact are all situated. Respecting this line, the above Principles III. and IV. lead to the following conclusions:—
V. That for a pair of turning pieces with parallel axes, and for a turning piece and a shifting piece, the line of contact is straight, and parallel to the axes or axis; and hence that the rolling surfaces are either plane or cylindrical (the term “cylindrical” including all surfaces generated by the motion of a straight line parallel to itself).
VI. That for a pair of turning pieces with intersecting axes the line of contact is also straight, and traverses the point of intersection of the axes; and hence that the rolling surfaces are conical, with a common apex (the term “conical” including all surfaces generated by the motion of a straight line which traverses a fixed point).
Turning pieces in rolling contact are called smooth or toothless wheels. Shifting pieces in rolling contact with turning pieces may be called smooth or toothless racks.
VII. In a pair of pieces in rolling contact every straight line traversing the line of contact is a line of connexion.
§ 40. Cylindrical Wheels and Smooth Racks.—In designing cylindrical wheels and smooth racks, and determining their comparative motion, it is sufficient to consider a section of the pair of pieces made by a plane perpendicular to the axis or axes.
The points where axes intersect the plane of section are called centres; the point where the line of contact intersects it, the point of contact, or pitch-point; and the wheels are described as circular, elliptical, &c., according to the forms of their sections made by that plane.
When the point of contact of two wheels lies between their centres, they are said to be in outside gearing; when beyond their centres, in inside gearing, because the rolling surface of the larger wheel must in this case be turned inward or towards its centre.
From Principle III. of § 39 it appears that the angular velocity-ratio of a pair of wheels is the inverse ratio of the distances of the point of contact from the centres respectively.
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| Fig. 96. |
For outside gearing that ratio is negative, because the wheels turn contrary ways; for inside gearing it is positive, because they turn the same way.
If the velocity ratio is to be constant, as in Willis’s Class A, the wheels must be circular; and this is the most common form for wheels.
If the velocity ratio is to be variable, as in Willis’s Class B, the figures of the wheels are a pair of rolling curves, subject to the condition that the distance between their poles (which are the centres of rotation) shall be constant.
The following is the geometrical relation which must exist between such a pair of curves:—
Let C1, C2 (fig. 96) be the poles of a pair of rolling curves; T1, T2 any pair of points of contact; U1, U2 any other pair of points of contact. Then, for every possible pair of points of contact, the two following equations must be simultaneously fulfilled:—
arc, T2U2 = T1U1.
A condition equivalent to the above, and necessarily connected with it, is, that at each pair of points of contact the inclinations of the curves to their radii-vectores shall be equal and contrary; or, denoting by r1, r2 the radii-vectores at any given pair of points of contact, and s the length of the equal arcs measured from a certain fixed pair of points of contact—
which is the differential equation of a pair of rolling curves whose poles are at a constant distance apart.
For full details as to rolling curves, see Willis’s work, already mentioned, and Clerk Maxwell’s paper on Rolling Curves, Trans. Roy. Soc. Edin., 1849.
A rack, to work with a circular wheel, must be straight. To work with a wheel of any other figure, its section must be a rolling curve, subject to the condition that the perpendicular distance from the pole or centre of the wheel to a straight line parallel to the direction of the motion of the rack shall be constant. Let r1 be the radius-vector of a point of contact on the wheel, x2 the ordinate from the straight line before mentioned to the corresponding point of contact on the rack. Then
is the differential equation of the pair of rolling curves.
To illustrate this subject, it may be mentioned that an ellipse rotating about one focus rolls completely round in outside gearing with an equal and similar ellipse also rotating about one focus, the distance between the axes of rotation being equal to the major axis of the ellipses, and the velocity ratio varying from (1 + eccentricity)/(1 − eccentricity) to (1 − eccentricity)/(1 + eccentricity); an hyperbola rotating about its further focus rolls in inside gearing, through a limited arc, with an equal and similar hyperbola rotating about its nearer focus, the distance between the axes of rotation being equal to the axis of the hyperbolas, and the velocity ratio varying between (eccentricity + 1)/(eccentricity − 1) and unity; and a parabola rotating about its focus rolls with an equal and similar parabola, shifting parallel to its directrix.
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| Fig. 97. |
§ 41. Conical or Bevel and Disk Wheels.—From Principles III. and VI. of § 39 it appears that the angular velocities of a pair of wheels whose axes meet in a point are to each other inversely as the sines of the angles which the axes of the wheels make with the line of contact. Hence we have the following construction (figs. 97 and 98).—Let O be the apex or point of intersection of the two axes OC1, OC2. The angular velocity ratio being given, it is required to find the line of contact. On OC1, OC2 take lengths OA1, OA2, respectively proportional to the angular velocities of the pieces on whose axes they are taken. Complete the parallelogram OA1EA2; the diagonal OET will be the line of contact required.
When the velocity ratio is variable, the line of contact will shift its position in the plane C1OC2, and the wheels will be cones, with eccentric or irregular bases. In every case which occurs in practice, however, the velocity ratio is constant; the line of contact is constant in position, and the rolling surfaces of the wheels are regular circular cones (when they are called bevel wheels); or one of a pair of wheels may have a flat disk
