On the Teeth of Wheels

If two cylinders be placed in close contact, motion cannot be communicated to the one without that motion, by means of the irregularities of their surfaces, being communicated to the other, and the smaller cylinder shall perform exactly as many revolutions to one revolution of the larger cylinder, as the larger cylinder contains upon its circumference so many measured circumferences of the smaller cylinder.

Wheels, however, which act by their surfaces only, are ill- calculated to transmit motion to any considerable extent, as the motion which the follower has acquired is not of sufficient power to overcome the great resistance which would, in such case, be opposed to it ; consequently it becomes necessary to have projections or teeth, and that form of the teeth will be the best which causes the wheel to act as though the motion were communicated by contact of the pitch lines.

If the three circles 1, 2, 3, in contact at the point a, be made to revolve about their centres, so that they shall continually touch at the point a, their motions will be similar to what would have been generated by one communicating motion to the other two by contact ; and circle 3 will move as though rolling on the external surface of circle 1, and internal surface of circle 2, and consequently become the generating circle of the exterior epicycloid on circle 1, and the generating circle of the interior epicycloid on circle 2.

As the diameter of circle 3 is equal to the radius of circle 2, the interior epicycloid will be a straight line passing through ? the centre of circle 2 ; and, supposing the point a to have performed that portion of a revolution which places it at ?, ? portion of the exterior epicycloid will be represented by the line E, K, and a portion of the interior epicycloid by D K.

Therefore, as the epicycloids D K and E K are both generated by one motion of the same point on the same circle, they will continually touch at the generating point, and the total surface of E ? will pass over the total surface of D K ; and if the epicycloid E K be affixed to the external surface of circle 1, and act upon the portion of the epicycloid D K, it will transmit motion to circle 2, as though that motion were communicated by contact of the pitch lines ; which proves that E A presents us with the best form of tooth, and which tooth would, when acting upon the radii of the wheel to be driven, more it as though the motion were communicated by contact.

Trundle Wheel Teeth

Fig. 40 represents a mode of forming the teeth of wheels when they are to act upon a truiidle. Circle 1 represents the pitch line of the wheel ; and circle 2 the pitch line of the trundle ; which are supposed to act by contact at the point a. When point a arrives at a1, it will have traced that portion of an epicycloid represented by a1, a2 and as a is the generating point of the epicycloid, the distance from a to a1, and from a to a2, will be equal : and the epicycloid a1 a2, being generated by the proportional circle or pitch line of.the trundle, presents us with the properest form for the tooth of a wheel that is to drive a trundle with circular staves posited in its pitch line.

We shall now proceed to the practical mode of applying these rules. Let circle 2 be the proportional circle or pitch line of a trundle ; and circle 1 the pitch line of a wheel which is to drive that trundle ; and by the revolutions of these two circles let the portion of an epicycloid a1 a3 be generated, so that when a line is drawn from a3 to the centre of circle 1, it will intersect it at circle at b, whose distance from a1 is such, that when the semi-diameter of a staff of the trundle is subtracted from it, the remainder will be equal to half the intended thickness of the tooth of the wheel.

Set off perpendicularly to the epicycloid inwards, the semi-diameter of one of the staves at so many points that you will be able to trace through the points thus set off, a line parallel to the epicycloid a1 a3, which line will be the face of the tooth of the wheel, being less than the tooth formed by the epicycloid a1 a3 by the semi-diameter of a staff of the trundle, indeed the diminution must be rather more, as the width g g must be made sufficient for the staves to clear themselves, as the whole of the epicycloidal line must act upon their surface.