Ci 75(5 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 us 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 sect. 49 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 respec tively. 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 iu 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 winch must exist between such a pair of curves. See fig. 10. Let Cj, C 2 be the poles of a pair of rolling curves; Tj, T 2 any pair of points of contact ; U 3 , TJ 2 any other pair of points of contact. Then, for every possible pair of points of contact, the two following equations must be simultaneously fulfilled: Sum of radii, + C. 2 U 2 = arc, TU- j + C 2 T 2 = constant ; (17). 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 r lt r 2 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 ds (18); 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 iu the Transactions of the Royal Society of Edinburgh, 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 r^ be the radius- vector of a point of contact on the wheel, x. 2 the ordinate from the straight line before mentioned to the corresponding point of contact on the rack. Then dx. dr- ds ds (19) 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 ani similar ellipse also rotating about one focus, the distance between the axes of rotation being equal to the major axis 1 + eccentricity of the ellipses, and the velocity ratio varying from ^ - , . . to ^^ an hyperbola rotating about its further focus 1 + eccentricity 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 hyper bolas, and the velocity ratio varying between - ccentnc * ty + * an d eccentricity 1 unity ; and a parabola rotating about its focus rolls with an equal and similar parabola, shifting parallel to its directrix. 51. Conical or Bevel and Disk Wheels. From Principles III. and VI. of sect. 49 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 follows the following construction (figs. 11 and 12). Let be the apex or point of meeting of the two axes OC,, OC 2 . The angular velocity ratio being given, it is required to line of contact. On OCj, OC 2 take lengths CM.-, , OA 2 , re- AVz find the Bpectively proportional to the ai jular velocities of the pieces on [APPLIED MECHANICS. whose axes they are taken. Complete the parallelogram OAjEA. ; 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 C]PC 2 , and the wheels will be cones, with eccentric or irregular bases. In every case which occurs in practice, how ever, 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 for its rolling surface, as W 2 in fig. 12, in which case it is a disk wheel. The rolling surfaces of actual wheels consist of frusta or i,ones of the complete cones or disks, as shown by W 1} W 2 in figs. 11 and 12. F . ., 52. Sliding Contact (lateral^: Skew- Bevel Wheels. An hyperboloid of revolution is a surface resembling a sheaf or a dice box, generated by the rotation of a straight line round an axis from which it is at a constant distance, and to which it is inclined at a constant angle. If two such hyperboloids, equal or unequal, be placed in the closest possible contact, as in fig. 13, they will touch each other along one of the generating straight lines of each, which will form their line of contact, and will be inclined to the axes AO, BH in opposite directions. The axes will not be parallel, nor will they intersect each other. The motion of two such hyper boloids, turning in contact with each other, has hitherto been classed amongst cases of rolling contact ; but that classification is not strictly correct, for, although the component velocities of a pair of points of con tact in a direction at right angles to the line of contact are equal, still, as the axes are neither parallel to each other nor to the line of contact, the velocities of a pair of points of contact have components along the line of contact which are unequal, and their difference constitutes a lateral sliding. The directions and positions of the axes being given, and the re quired angular velocity ratio, the following construction serves to determine the line of contact, by whose rotation round the two axes respectively the hyperboloids are generated : In fig. 14, let BjCj, B 2 C 2 be the two axes ; B^ their common Sjrpendicular. Through any point in this common perpendicular draw OAj parallel to BjCj and OA 2 parallel to B 2 C 2 ; make those lines proportional to the angular veloci ties about the axes to which they are respectively parallel ; complete the parallelogram OA^Ag, and draw the diagonal OE ; divide BjB 2 in D into two parts, inversely proportional to the angular veloci ties about the axes which they re spectively adjoin ; through D paral lel to OE draw DT. This will be the line of contact. A pair of thin frusta of a pair of hyperboloids are used in practice to communicate motion between a pair of axes neither parallel nor intersecting, and are called skew-bevel wheels. In skew-bevel wheels the properties of a line of connexion are not possessed by every line traversing the line of contact, but only by every line traversing the line of contact at right angles. If the velocity ratio to be communicated were variable, the point D would alter its position, and the line DT its direction, at different periods of the motion, and the wheels would be hyperboloids of an eccentric or irregular cross-section ; but forms of this kind are not used in practice. 53. Sliding Contact (circular) : Grooved Wheels. As the ad hesion or friction between a pair of smooth wheels is seldom sufficient to prevent their slipping on each other, contrivances are used to increase their mutual hold. One of those consists in forming the rim of each wheel into a series of alternate ridges and grooves
Fig. 13.