CYCLOIDS

What is a Cycloid?A cycloid is a curve generated by a point on the

circumference of a circle as the circle rolls along a straight line without slipping

The moving circle is called a generating circle and the straight line is called a directing line or base line.

The point on the generating circle which traces the curve is called the generating point.

Construction of a Cycloid

Step 1:Draw the generating circle and base line equal to circumference of generating circle.

Step 2: Divide the circle and base line into an equal number of parts. Also erect the perpendicular lines from the divisions of line.

Step 3: With your compass set to the radius of the circle and centers as C1,C2,C3, etc cut the arcs on the lines from circle through 1,2,3, etc.

Step 4: Locate the points which are produced by cutting arcs and join them by a smooth curve.

By joining these new points you will have created the locus of the point P for the circle as it rotates along the straight line without slipping.

And your final result is a Cycloid.

Construction of a Tangent and a Normal to a point on a Cycloid.

Mark any point P1 on the curve and with the radius of the circle mark on the centre line of the rotating circle. From that point draw horizontal line which meets the base line at some point. Now join both the points with a line which is the required normal and draw a perpendicular to normal, tangent is obtained.

EPICYCLOIDS

What is Epicycloid?

The cycloid is called the epicycloid when the generating circle rolls along another circle outside (directing circle)

The curve traced by a point on a circle which rolls on the outside of a circular base surface.

Construction of Epicycloid

Steps 1: Draw and divide rolling circle into 12 equal divisions.

Step 2: Transfer the 12 divisions on to the base surface.

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P

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Step 3: Mark the 12 positions of the circle – centre (C1,C2, …) as the circle rolls on the base surface.

Step 4: Project the positions of the point from the circle.

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C1

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CENTRE LINE

Step 5: Using the radius of the circle and from the marked centres C1,C2,C3 etc cut ff the arcs through 1,2,3

Step 6: Darken the curve.

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Draw an epicycloid of rolling circle diameter 40 mm which rolls outside another circle (base circle) of 150mm diameter for one revolution

Step 1: Draw an arc PQ with radius75 mm and centre O, subtending and angle 96º. P is the generating point. On OP produced mark PC = 20mm. Draw a circle with centre C and radius 20 mm.

Step 2 : Divide rolling diameter in to12 equal parts and name them as 1,2,3,4… 12 in Clock Wise direction.

Step 3 : With O as centre draw concentric arcs passing through1,2,3,…,12.

Step 4 : With O as centre and OC as radius, draw an arc to represent locus of centre.

Step 5 : Divide arc PQ in to 12 equal parts and name them as1’, 2’, …., 12’.

Step 6 : JoinO1’, O2’, …and produce them to cut the locus of centers atC1, C2, ….

Step 7 : Taking C1 as centre, and radius equal to 20 mm, draw an arc cutting the arc through1 at P1. Similarly obtain pointsP2, P3,…., P12.

HYPOCYCLOIDS

CONSTRUCTION OF A HYPOCYCLOIDCONSTRUCTION OF A HYPOCYCLOIDThe curve traced by a point on a circle which rolls on the inside of a circular base surface.

Step 1: Divide rolling circle into 12 equal divisions.

Step 2: Transfer the 12 divisions on to the base surface.

Step 3: Mark the 12 positions of the circle – centre (C1,C2, …) as the circle rolls on the base surface.

Step 4: Project the positions of the point from the circle.

Step 5: Using the radius of the circle and from the marked centres step off the position of the point.

Step 6: Darken the curve.

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Applications of cycloid curves:

Cycloid curves are used in the design of gear tooth profiles.

It is also used in the design of conveyor of mould boxes in foundry shops.

Cycloid curves are commonly used in kinematics (motion studies) and in mechanisms that work with rolling contact.