and as the arc PQ equals the circumference of the generating circle, this becomes

angle PAQ

=

360 x rad. of generating circle

rad. of directing circle

Set off this angle. Draw the arc BC which is the locus of the centre of the generating circle, and, as before, divide it and the circle into the same number of equal parts, and then proceed as with the cycloid. The construction is clearly shown in the figure.

Normals and Tangents are drawn exactly as to a cycloid. Thus in the figure, N is the centre of the generating circle corresponding to the position O in the curve of the generating point, and M is the contact point of the rolling and directing circles. Then MO is a normal, and TR at right angles a tangent.

The evolutes are the curves PR, RQ, drawn tangent to the normals of the curve as before. They are similar curves to the original curve, PS Q, and are, therefore, epicycloids, but are not equal to the original curve. The hypocycloid and its evolutes are drawn in precisely the same way as the epicycloid, and do not, therefore, need separate explanation. In Fig. 55 the curve P'S'Q' is the hypocycloid, the evolutes not being shown for want of space. They are, however, drawn touching the normals to the curve as before. EF is a tangent, and HK a normal. The generating circle rolls in the direction from P' towards Q'.

Notice that both the epi- and hypocycloids are traced by the end of a thread unwound from the evolutes, as with a cycloid. No difficulty should be experienced in drawing the epi- and hypotrochoids, as the construction is exactly similar.

EXAMPLES.

EX. 9.-Draw an epicycloid and its evolutes when the diameters of the directing and generating circles are 10" and 4" respectively, and draw a tangent and normal at any point in the curve not found in the construction. Show that the evolute is an epicycloid traced by a point on a circle of diameter equal to R G (Fig. 55) rolling on a circle of radius, A R.

EX. 10.-Draw a hypocycloid and its evolutes when the diameters of the directing and generating circles are 10" and 3′′ respectively, and draw a tangent and normal at any point in the curve not found in the construction.

EX. 11.-Show that when the diameter of the directing circle is twice the diameter of the generating circle, the hypocycloid is a straight line.

EX. 12.-Draw a hypotrochoid when the diameter of the directing circle is twice the diameter of the generating circle, and show that half the curve is a quadrant of an ellipse.

Involute of a Circle.-The involute of a circle is the curve traced out by the end of a piece of thread unwound from the circle, the thread being kept tight. The circle is then the evolute to this curve.

PROBLEM XL. (Fig. 56).-To draw the involute of a

circle.

Let the circle have the centre C, and let P be the starting point of the curve or end of the supposed thread.

Let the thread be partly unwound, so that it assumes the position P3 3.

It is evident P3 3 must be a tangent to the circle, and be, therefore, at right angles to the radius C 3. Also P33 must equal the length of the arc P 3. Then P is a point in the involute.

If the arc P3 be divided into a number of small parts, and the

same number of parts be marked off from 3 to P3, then the length P3 3 may be assumed equal to the chord P 3 and P be a point in the curve. But it is better to divide the circumference of the circle into, say, twelve equal parts, in which case the length of the tangent P3 3 would be one-quarter of the circumference (which

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 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, P1 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.

EXAMPLES.

EX. 13.-Draw the involute of a circle 23" 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 :—

R

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 2 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 of the two discs. Hence, with two toothed wheels in gear, the distance from the centre of each wheel 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 Ꭱ and R' be the radii of these circles, the velocity ratio is and is R"

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

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