## Problem 3.3 Engineering Curves – A circle of 50 mm diameter rolls on and in another fixed circle of radius 80 mm. Draw the epicycloid and hypocycloid for the point P on the rolling circle, which is at the contact point of the rolling and fixed circles.

*Procedure:*

* Step-1* Draw a horizontal axis of some suitable length. And mark a point O on it.

* Step-2* From the equation R .θ = 2πr, find out the value of θ in degree.

Where,

R = Radius of directing or fixed circle.

θ = Angle covered by the rolling circle in rolling one revolution over the fixed circle.

r = Radius of rolling circle.

* Step-3* In this problem Radius of fixed circle (R) = 80 mm. And Radius of rolling circle (r) = 25 mm. So, the value of θ =112.5°

* Step-4* Draw another axis at an angle θ =112.5° with the previously drawn horizontal axis.

* Step-5* With point O as center and radius equal to R=80 mm draw an arc between these two axis. This represents the fixed circle.

* Step-6* On the vertical axis shown into the figure draw a rolling circle outside the fixed circle with the radius r = 25 mm such that the one point of the circumference of the circle should touch the fixed circle. And give the name C

_{0}of the center of the circle.

Note: It is assumed that the rolling circle is rolling in clockwise direction.

* Step-7* Divide this circle into 12 equal divisions and give the notations as 1,2,3, etc. up to 12.

* Step-8* With O as center and radii equal to O1, O2, O3 etc. draw arcs from all the points of the circle up to the second axis as shown into the figure. Also draw an arc with the point O as center and C

_{0 }as radius as a center line up to the second axis.

* Step-9* Divide the angle θ =112.5° in to 12 equal divisions. And give the notations C

_{1}, C

_{2}, C

_{3}etc. at the points of intersections of these lines with the arc passing through the center C

_{0 }as shown into the figure.

Note: The number of divisions of the rolling circle and the angle between the two axis that is θ =112.5° should be in same number.

* Step-10* Now with C

_{1}, C

_{2}, C

_{3}etc. as centers and radius equal to the rolling circle, which is 25 mm cut the arcs passing through the points 1,2,3, etc. of the rolling circle in clock wise direction and give the notations as p

_{1}, p

_{2}, p

_{3 }etc. up to p

_{12 }as shown into the figure.

* Step-11* Draw a smooth medium dark free hand curve passing through the points p

_{1}, p

_{2}, p

_{3 }etc. up to p

_{12 }in sequence to get Epycycloid.

* Step-12* As per the procedure explained above do the same inside the fixed circle to get Hypo cycloid.

* Step-13* Give the dimensions by any one method of dimensions and give the name of the components by leader lines wherever necessary.

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