can be easily calculated), and each succeeding tangent would increase by one-twelfth of the circumference, if the circle is equally divided.

This construction is conveniently effected by drawing the tangent P, P12, equal in length to the circumference of the circle, and dividing it into the same number of equal parts as the circle. The length of each tangent can then be taken from it, as, for example, P1 1 P1, P2 2 = P 2, &c.

=

Normals and Tangents.-Normals to the involute are tangents to the evolute, as in the cycloidal curves. Therefore, to draw a normal at any point O, it is only necessary to draw from that point a tangent to the circle. This is done by the method of Fig. 23, the point O is joined to C, and a semicircle is drawn upon it cutting the circle in the point N. Then the line NO is a normal, and the line ST at right angles through O is a tangent.

If in Fig. 56 we regard P, P12 as a straight line having one end touching the circle at P, then the involute is evidently the path of the end P12, as the line rolls around the circle in an anticlockwise direction. But as a line may be regarded as a circle of infinite radius, an involute is evidently an epicycloid having a rolling or generating circle of infinite radius. The involute has also the properties of an archimedean spiral, and if used as a cam would impart linear motion to a point uniform with the circular movement of the cam.

EX. 13.

EXAMPLES.

Draw the involute of a circle 21" diameter, and draw a normal and tangent at any point in the curve not found when constructing it. Show that the radius at any point in the curve is proportional in length to the angle passed through by the radius from the starting point of the curve.

EX. 14.-Draw the curve traced out by the end of a straight line 3" long as it rolls round the circumference of a circle 4′′ diameter. (The curve is an involute.)

EX. 15. Draw two circles of 5" diameter in contact at a point P. From P draw part of an involute to each circle (about 2′′ long), the curves for the two circles to be in opposite directions.

EX. 16.-Draw the curve traced by a point on a straight line which rolls on a semicircle of 3" diameter. (Vict. Hon., 1892.)

SECTION

VIII.

CONSTRUCTION OF CURVES FOR TEETH
OF WHEELS.

THE most common and useful practical application of cycloidal and involute curves is to shape the teeth of geared wheels. The diameter and proportions of wheels for different speeds, and the number and sizes of the teeth, in order to transmit a required power is a question not of constructive geometry, but of machine design, and owing to its difficulty will not be dealt with in this book. The object of this section is merely to give the student a sufficient knowledge of the principle and method of shaping the teeth of wheels, as to fit him better for their complete design at a later stage. But in order to effect this the following general principles must be understood:

When two toothed wheels are in gear it is most important that their relative velocity shall not vary during the revolution-that is, one wheel must not at one instant be moving 3 times as fast .as the other, and at another instant only 27 times as fast. This fact is expressed in mechanics by saying that the velocity ratio of the wheels must be constant at every part of the revolution. When two simple circular discs transmit motion by the frictional contact of their rims, without slip, it is evident that their velocity ratio is constant and is equal to , where R and R' are the radii

R" of the two discs. Her PRO wo toothed wheels in gear, the distance from the cented pinion heel to the point of contact with the tooth of the other wheel measured along the line joining the wheel centres, must be the same for each pair of teeth, otherwise, the velocity ratio will not be constant. It follows from this, that all these points of contact must lie on the circumferences of circles described from the centres of the wheels, and that if R R and R' be the radii of these circles, the velocity ratio is and is R"

precisely the same as if the wheels were replaced by two friction discs of radii, R and R'. These circles are called the pitch circles.* The diameter of a toothed wheel is the diameter of its pitch circle, and not its diameter outside or inside the teeth. The teeth must evidently be equally spaced around the pitch circle, the distance between the centre of one

The pitch line or pitch circle in toothed gearing corresponds to the directing line or circle of the cycloids (pp. 69, 70).

tooth and the centre of the next tooth, measuring along the circumference of the pitch circle or the pitch line, being called the pitch of the teeth. It is easy to see that if any two of the three sizes, radius of pitch circle, pitch of teeth, and number of teeth are known, the third can be found. Also that the velocity ratio of the wheels equals

number of teeth in driver number of teeth in follower' revolutions of follower revolutions of driver

It

and that either of these is equal to

is shown in text-books on mechanics that the conditions of constant velocity ratio for toothed wheels, as specified above, is only obtained when the normal to the two teeth at the point of contact is common to both, and that this condition is met by shaping the teeth to cycloidal or involute curves. It is also necessary that the teeth should roll smoothly when in contact, and not rub or grind, a condition which is also satisfied by using these

curves, for suppose a pinion (which is the name given to a small toothed wheel) is gearing with a rack as in Fig. 58, then we may suppose the rack to be fixed and the pinion to roll along it, and we see at once that a point on the pinion will describe a cycloidal path, so that if we wish to make the pinion leave the

rack smoothly the teeth of the rack should be shaped to a cycloid which is exactly what is done in practice. A similar reasoning applies to the use of epi- and hypocycloids for the teeth of spur wheels.

In Fig. 57 part of two wheels are shown in gear, and sizes are marked on giving the usual proportions of the teeth, as taken from Professor Unwin's Machine Design, and which should be adhered to by students in working the examples of this section. They may be stated as follows:-Thickness of tooth along pitch circle 048 p, height outside pitch circle 0·3 p, depth inside pitch circle 04 p; where p pitch of teeth, this gives a clearance between the teeth of 0·52 p. The pitch circles are marked Q PR and S P T, P being the pitch point. The face of the tooth is the part marked BA outside the pitch circle, and the flank of the tooth is the part marked B C inside the pitch circle. It is impor

tant to remember this distinction, as in working the faces of one wheel make contact with the flanks of the other wheel, and the curves of the faces and flanks must be described with rolling circles of the same radius.

The size of rolling circles used in drawing the curves for wheel teeth do not bear any fixed ratio to the size of the wheels, and vary with different makers. The size adopted in any particular case does not change the conditions of velocity ratio or smooth rolling, but only affects the thickness of tooth above and below the pitch circles. The first of the examples at the end of this section is intended to show the effect on the shape of the teeth of rolling circles of different diameters.

Rack and Pinion.-PROBLEM XLI. (Fig. 58).—To draw the teeth of a rack and pinion in gear, knowing sizes of pitch and rolling circles and pitch of teeth.

Draw the straight line QPR to represent the pitch line of the rack, and from centre C draw the pitch circle S PT of the pinion, touching the pitch line of the rack in the point P, called the "pitch point."

The faces of the rack teeth gear with the flanks of the pinion teeth, and these had better be considered first. If we decide to have radial flanks for the pinion, a usual construction, we know that they will be obtained by using a rolling circle of a diameter equal to the radius of the pinion, as this gives a hypocycloid which is a straight line. Therefore, draw a circle with centre A, and diameter equal to C P, and this will be the rolling circle for the faces of the rack teeth, which we know are to be cycloids. Then draw part of a cycloid, starting from the pitch point P, taking the pitch line QPR of the rack for the directing line, and rolling the circle towards the right hand. The most con

venient way of doing this, geometrically, since the complete curve is not required, is as follows:-Draw the locus of the rolling circle A 5', and mark off four or five equal parts of short length, as 1', 2',.. 5', and with each of these points as centre draw arcs of the rolling circle as shown. Mark the contact points of the rolling circles with the pitch line of the rack, as at 1, 2, 3, 4, 5. Then take the distance of the equal parts A 1', 1' 2' (equal of course to P 1, 1 2, &c.) in the compasses, and mark off distances along each of the circles just drawn, from the points 1, 2, 3, 4, 5 to the left-that is, from 1 mark one distance, from

2 two distances, and so on, thus finding a sufficient number of points in the cycloid, through which the curve from P can be drawn. The radial line from P to A is the flank of the pinion tooth, in contact with the rack tooth at P.

Next consider the flanks of the rack teeth and the faces of the pinion teeth. Following a usual construction we will also make the rack teeth flanks radial-that is, they must be drawn perpendicular to the rack pitch line, as that may be regarded as the arc of a circle of infinite radius; the line PN, therefore, gives the flank of the rack tooth. But this line may be regarded as part of a cycloid traced by a generating or rolling circle of infinite radius, and we know, therefore, that we must take the rolling circle for the faces of the pinion teeth of infinite radius-that is, it must be a straight line. But the epicycloid traced by a line rolling round a circle is an involute of the circle, therefore the faces of the