Construction of Cycloid

CONSTRUCTION OF CYCLOID

Points to understand

1. Cycloid is a curve generated by a point on the circumference of a circle which rolls without slipping along a fixed straight line, called Base line.

2. If the circle rolls in clockwise direction the cycloid is drawn on the right hand side and if the circle rolls in counter clockwise direction the cycloid is drawn on the left hand side.

3. When the rolling circle makes one revolution on the base line the distance travelled along the base line is equal to the circumference of the rolling circle.

4. A normal drawn at any point on the cycloid will pass through the point of contact between the directing line and rolling circle at that position.

5. If the rolling circle rolls in clockwise direction, number the division points on its circumference in anticlockwise direction as shown above and vice-versa.

Example 1: Construct a cycloid having a generating circle of 50 mm diameter. Also draw tangent and normal at any point on the curve.

Construction:

1. Draw a horizontal line Ca - Cb of length equal to the circumference of the rolling circle.

ie., Ca - Cb = 2π × Radius of the rolling circle

= 2π × 25

= 157 mm

2. With Ca as centre and radius equal to 25 mm draw a circle which is the initial position of rolling circle.

3. Divide the circle into any number of equal parts (say 12 parts; more the divisions more will be the accuracy) and name the divisions 1, 2, etc., in anticlockwise direction (assuming that the rolling circle rolls in clockwise direction).

4. Divide the line Ca - Cb (which is the locus of the centre of rolling circle) also into same number of equal parts (say 12 parts) and mark the parts as C1, C2...etc.

5. Draw a line AB parallel to Ca - Cb and equal in length. AB is the tangent drawn to the circle at its initial position and represents the Fixed line (or Base line) along which the rolling circle rolls without slipping.

6. Now the required cycloid is the path of a point P on the periphery of the rolling circle. Let Po be the initial position of the point P, which is the point of contact of rolling circle and the Base line. (ie., the point A (or) 12).

7. After one complete revolution the point P coincides with the point B. Now the cycloid is to be drawn between the points Po and B (or between the points A and B)

To Locate intermediate positions of the point P.

8. Draw horizontal lines through the points 1, 2.... etc. marked on the circumference of circle.

9. Draw perpendicular lines through the points C1, C2 … etc., perpendicular to the base line AB to intersect at 1', 2' ... etc.,

10. With C1 as centre and radius equal to (C1 - 1') [ie., the radius of rolling circle 25 mm] draw an arc to cut the horizontal line drawn through the point 1. Name this point as P1.

11. Similarly with C2, C3 .... etc. as centres and radius equal to C2 - 2', C3 -3'... etc. draw arcs to cut the horizontal lines drawn through the points 2, 3... etc. at P2, P3 …. etc.

Note: As C1 - 1', C2 - 2' - 2′, C3 - 3'.... are equal and equal to the radius of rolling circle, taking the radius of circle and C1, C2 … as centres draw arcs to cut the horizontal lines drawn through the points 1, 2.... etc., at P1, P2 … etc.

Hence the step (9) can be avoided.

12. Draw a smooth curve joining the points Po, P1, P2, … P11, P12 and B to get the required cycloid.

To draw a Tangent and Normal at any point on the cycloid.

Let M be a point on the curve through which tangent and normal are to be drawn.

i) With M as centre and radius equal to the radius of rolling circle (= 25 mm, given in the problem) draw an arc to cut the line Ca - Cb at C'.

ii) From C', erect a perpendicular to the base line AB to intersect at N.

iii) Join the points N and M and extend which is the required normal, NN'.

iv) Draw a line perpendicular to the normal NN' and passing through the point M. Let this line be TT' which is the required tangent drawn through the point M.