Let O (fig. 106) be the common apex of a pair of bevel-wheels; OB1I, OB2I their pitch cones; OC1, OC2 their axes; OI their line of contact. Perpendicular to OI draw A1IA2, cutting the axes in A1, A2; make the outer rims of the patterns and of the wheels portions of the cones A1B1I, A2B2I, of which the narrow zones occupied by the teeth will be sufficiently near to a spherical surface described about O for practical purposes. To find the figures of the teeth, draw on a flat surface circular arcs ID1, ID2, with the radii A1I, A2I; those arcs will be the developments of arcs of the pitch-circles B1I, B2I, when the conical surfaces A1B1I, A2B2I are spread out flat. Describe the figures of teeth for the developed arcs as for a pair of spur-wheels; then wrap the developed arcs on the cones, so as to make them coincide with the pitch-circles, and trace the teeth on the conical surfaces.
§ 55. Teeth of Skew-Bevel Wheels.—The crests of the teeth of a skew-bevel wheel are parallel to the generating straight line of the hyperboloidal pitch-surface; and the transverse sections of the teeth at a given pitch-circle are similar to those of the teeth of a bevel-wheel whose pitch surface is a cone touching the hyperboloidal surface at the given circle.
§ 56. Cams.—A cam is a single tooth, either rotating continuously or oscillating, and driving a sliding or turning piece either constantly or at intervals. All the principles which have been stated in § 45 as being applicable to teeth are applicable to cams; but in designing cams it is not usual to determine or take into consideration the form of the ideal pitch-surface, which would give the same comparative motion by rolling contact that the cam gives by sliding contact.
§ 57. Screws.—The figure of a screw is that of a convex or concave cylinder, with one or more helical projections, called threads, winding round it. Convex and concave screws are distinguished technically by the respective names of male and female; a short concave screw is called a nut; and when a screw is spoken of without qualification a convex screw is usually understood.
The relation between the advance and the rotation, which compose the motion of a screw working in contact with a fixed screw or helical guide, has already been demonstrated in § 32; and the same relation exists between the magnitudes of the rotation of a screw about a fixed axis and the advance of a shifting nut in which it rotates. The advance of the nut takes place in the opposite direction to that of the advance of the screw in the case in which the nut is fixed. The pitch or axial pitch of a screw has the meaning assigned to it in that section, viz. the distance, measured parallel to the axis, between the corresponding points in two successive turns of the same thread. If, therefore, the screw has several equidistant threads, the true pitch is equal to the divided axial pitch, as measured between two adjacent threads, multiplied by the number of threads.
If a helix be described round the screw, crossing each turn of the thread at right angles, the distance between two corresponding points on two successive turns of the same thread, measured along this normal helix, may be called the normal pitch; and when the screw has more than one thread the normal pitch from thread to thread may be called the normal divided pitch.
The distance from thread to thread, measured on a circle described about the axis of the screw, called the pitch-circle, may be called the circumferential pitch; for a screw of one thread it is one circumference; for a screw of n threads, (one circumference)/n.
Let r denote the radius of the pitch circle;
n the number of threads;
θ the obliquity of the threads to the pitch circle, and of the normal helix to the axis;