number of axes requisite for producing a given synchronal ratio of rotation between the first and last axes, when the number of teeth in the drivers cannot exceed N, and the number in the followers cannot be less than n1.

Find m, in eq. (1), equal to the highest whole number, which does not make the right member greater than the left; then the least number of axes will be m + 2. But if m, a whole number, can be found so as to make the right-hand member exactly equal to the left, then, in this case, the least number of axes will be m + 1.

Example.-Required the least number of axes in a train of wheels which shall cause the last axis to revolve 180 times as fast as the first axis, allowing that none of the drivers can contain more than 54 teeth, and none of the followers less than 9. Here we must find the greatest whole number for m, so that

(54)TM

or (6) shall not exceed 180. This value of m is obviously 2; and the least number of axes will be 4.

Idle Wheels.

62. The wheel c, placed between two other wheels, A and B,

does not affect the velocity ratio of these wheels; and hence the wheel c is called an idle wheel. This intermediate wheel, however, causes the wheels A and B to revolve in the same direction, whereas, if A and B were in contact, they would revolve in opposite directions.

Annular Wheels.

А

63. Fig. 44 represents an annular wheel A, having its teeth cut on the internal edge of the annulus or rim. The toothed wheel B, revolving within the annular wheel A, causes it to revolve in the same direction; whereas two ordinary spur wheels revolve in opposite directions.

Concentric Wheels.

64. When two separate wheels revolve about the same centre

of motion they are called concentric wheels. The pinion D is fixed to the axis FE, whilst the concentric wheel c is fixed to a tube, or cannon, N, which revolves freely upon the axis FE. The driving wheels, a and B, fixed to the parallel axis H G, communicate the relative velocities to the axis FE and to the cannon N.

זי

N

Fig. 45.

B

C

Wheel-work when the axes are not parallel to each other.

65. When the axes of two wheels are not parallel to each other, motion is generally communicated from the one to the other by bevel wheels or bevel gear. When the axes are perpendicular to each other, the face wheel and lantern and the crown wheel are frequently employed.

L.

Face Wheel and Lantern.

66. In fig. 46, F represents a face wheel, with its lantern Here motion is transmitted

from the vertical axis A B to the horizontal axis a C. The teeth F on the face of the face wheel are called cogs, which are usually made of iron, whilst the round staves forming the teeth of the lantern, L, are made of hard wood. The axes A B and C D

should, when produced, intersect in a point.

Fig. 46.

B

L

F

C

A

Fig. 47.

B

Crown Wheels.

67. Fig. 47 represents a crown wheel B, with its pinion A, having their axes at right angles to each other. The teeth of the crown wheel are cut on the edge of a hoop, the plane of which is at right angles to its axis, and the pinion is thicker than wheels are commonly made.

CASE I. To construct Bevel Wheels or Bevel Gear when the axes are in the same plane.

68. Let A c and A B be two axes of rotation in the same

E

K

Fig. 48.

B

F

plane, and cutting each other in the point A. On these axes two right cones, a D F and A DE, may be formed, touching each other in the line AHD; and also two right frusta, DFGH and DHK E, of these cones may be formed.

Now, if the frustum D F G H revolve on its axis B A, it will communicate, by rolling contact, a rotatory motion to the frustum DH KE upon its axis C A.

These frusta of cones will obviously perform their rotations in the same time as the ordinary spur wheels previously described.

On the surfaces of these frusta a series of equidistant teeth are cut, directed to the apex a of the cones, so that a straight line passing through the apex to the outline of the teeth upon the bases D F and D E of the frusta shall touch the teeth in every part, as shown in the diagram.

Wheels cut in this manner are called bevel gear.

Two wheels of this construction will always transfer motion, with a constant velocity ratio, from one axis to the other, provided these axes meet each other in a point, which point being always made the apex of the frusta forming the bevel of the wheels.

69. General Problem.-Given the radii of two bevel wheels, and the position of their axes, to construct the frusta forming the wheels, the two axes being in the same plane.

Fig. 49.

с

L

H

K

Let A B and A c be the position of the axes cutting each other in A. Draw IJ parallel to a в at a distance equal to the radius of the wheel on the axis A B; and draw ML parallel to AC, at a distance equal to the radius of the wheel on the axis a C, cutting the line I J in the point D. From the point D, draw D B F perpendicular to a B, and DCE perpendicular to A c. Take BF equal to BD, and CE equal to C D. Join A E, A D, and A F. distance equal to the thickness of the wheel, draw HG parallel to D F, cutting AD in H; and through H, draw HK parallel to Then D F G H and D H K E will be the frusta required.

DE.

At a

M

CASE II. To construct Bevel Gear when the axes are not in the same plane.

70. This is usually done by introducing an intermediate wheel with two frusta formed upon it, one frustum rolling in contact with the driving wheel, and the other frustum in contact with the driven wheel.

Fig. 50.

71. Let A B and C D be the direction of the given axes; take A D as a third axis, meeting the axes A B and C D at any convenient points, A and D; then A will be the vertex of two rolling frusta of cones & and H, and D will be the vertex of two other rolling frusta of cones I and K. Whilst the intermediate axis, with its two frusta of cones, revolves, the teeth of the frustum н will have a rolling contact

E

I H

B

A

with the teeth of the frustum &, and at the same time the teeth of frustum I will have a rolling contact with the teeth of the frustum K; and thus motion will be transmitted from the axis A B to the axis CD with a constant velocity ratio.

Let Q1 and Q3 be the number of rotations performed by the axes A B and C D respectively in the same time; N1 =the number of teeth in the bevel wheel &; n=the number in the edge н; N2 the number in the edge 1; and n, the number in the bevel wheel K; then,

=

=

which is similar to the expression given in eq. (1), Art. 57. When n1 = N2, then this equality becomes,

In this case the intermediate bevel wheel, I н, may be regarded as an idle wheel.

VARIABLE MOTIONS PRODUCED BY WHEEL-WORK HAVING ROLLING

Fig. 51.

E

D

P

B

CONTACT.

72. Two curved wheels, EP and FP, having rolling contact, revolve on the axes A and B. In order that these wheels may roll on each other without slipping, or without producing any strain upon the axes A and B, these axes must always be in the line of contact a P B, and if the curve PE on the one wheel be equal to the curve P F on the other wheel, the sum of the lines A E and B F must always be equal to A B, the distance between the centres of motion. Various curves may be constructed having this property. For example, two equal ellipses, E P and FP, revolving on their foci, A and B, and having A E and BF in the line of their major axes, will have a perfect rolling contact. Two equal logarithmic spirals have also the same property.

Let DPC be the common tangent to the point of contact P; from A and B let fall AC and BD perpendicular to DPC; then, angular velocity A P BD

BP

=

or

...

A P

(1).

This result may be expressed in language as follows:-The