A cycloid is a curve traced by a point on the circumference of a circle which rolls without slipping along a line.
The circle which rolls along the line is called rolling circle, and the line on which the circle rolls is called director.
PRACTICAL APPLICATION
The cycloid was formerly used most extensively in the design of profile of gear teeth, but modern production methods tend to limit its application to small gear such as those used in instruments.
TO DRAW CYCLOID
The following example shows the procedure for drawing a cycloid.
Q.A circle of 50 mm diametre rolls along a line. A point on the circumference of the circle is in contact with the line in the begining and after one complete revolution, draw the cycloidal path of the point.
SOLUTION:
Draw a line PQ equal to the circumference of the rolling circle. Draw the rolling circle with O as centre and radius r = 25 mm and touching the line PQ at P. P is the common point lying on the circle and also on the line PQ. Divide the circle into twelve equal parts at the division points 1, 2, 3 etc., Draw horizontal lines through 1, 2, 3 etc.
Divide PQ also into the same number of equal parts at the division points 1′, 2, 3 etc., At 1′, 2′, 3′, draw lines perpendicular to PQ to intersect the horizontal line through C, called the locus of centres respectively at C_{1} , C_{2} Сз.
When the circle rolls forward in the clockwise direction by 1/12 of a revolution the division point 1 on the rolling circle will come in the contact with 1′ and the centre of the rolling circle moves the new poisition O_{1} which is vertically above 1′. Since the point “1” is in contact with the line at 1′, the generating point P will have moved in the clockwise direction to the level of the point 1 in the original position before rotation, and also it will be at distance r = OP from the new centre C with the radius of the rolling circle, cut the horizontal line through the point 1 at P₁ as centre C_{1} , similarly with C_{2} as center and with the same radius, cut the horizontal through point 2 at P_{2} Similarly, obtain P3, P4 etc., Passing through the point P, P1, P_{2} P3, etc., draw a smooth curve.
