Construction of Cycloids & Involutes

What is a Cycloid?

o   A cycloid is a curve generated by a point on the circumference of the circle as the circle rolls along a straight line with out slipping..

o   The moving circle is called the "Generating circle" and the straight line is called the "Directing line" or the "Base line".

o   The point on the Generating circle which generates the curve is called the "Generating point"

Construction of a Cycloid

·         Step1: Draw the generating circle and the base line equal to the circumference of the generating circle. By using the formula (2πr).

·         Step 2 : Divide the circle and the base line in to equal number of parts. also erect the perpendicular lines from the division of the 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 joining by a smooth curve.

·         Step 5: 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 with out slipping

·         As our final result is a cycloid

Types of cycloids

      1.      EPICYCLOIDS

       2.       HYPOCYCLOIDS

    1)   EPICYCLOIDS:

• 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 out side of a circular base surface

CONSTRUCTION OF EPICYCLOID

·         Step1: Draw and divide the circle in to 12 equal divisions.

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

·         step3: Mark the 12 positions of the circle- centers (C1,C2,C3,C4..) 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 centers C1,C2,C3,C4.. etc cut off the arcs through 1,2,3.. etc.

·         Step 6: Darken the curve

     2)   HYPOCYCLOIDS:

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

CONSTRURCTION OF HYPOCYCLOIDS

·                     Step 1: Divide the rolling circle in to 12 equal divisions.

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

·                     Step 3: Mark the 12 positions of the circle - center (C1,C2,C3..) 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 centers step off the position of the point.

·                     Step 6: Darken the curve.

Applications of the cycloids

·        cycloid curves are used in the design of the gear tooth profiles.

(Worm gears have cycloidal profile. The head of the tooth of such worm gear is an epicycloid and the tooth foot is hypocycloid).

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

·        cycloidal curves are mainly used in Kinematics.

·        Cycloid is used by engineers and designers for designing roller coasters.

What is a Involute?

An Involute is a curve traced by the free end of a thread unwound from a circle or a polygon in such a way that the thread is always tight and tangential to the circle or side of the polygon.

CONSTRURCTION OF HYPOCYCLOIDS

·         Draw a circle & divide it into 8 equal parts and label them 1, 2, 3, etc. Take the bottom most point as P and ensure that labelling is in the anticlockwise direction. P will coincide with 8.

·         On P, draw a horizontal line PA (tangent) of length (circumference = 2πR or πD) and divide this line also into 8 equal parts.

·          At points 1, 2, 3, 4, etc on the circle, draw tangents by keeping the drafter perpendicular to C1, C2, C3, etc.

·         To get the points of the involute, use the principle, P-1’ = 1-P1 ; P-2’ = 2-P2,etc. Keep 1 as center, radius = P-1’, cut arc on line 1 to get P1. Similarly with 2 as center, radius = P-2’, cut arc on line 2 to get P2. Thus using this principle, cut the other arcs on the tangents to get points P3, P4, etc. Last point is PA and need not mark.

·         Join all the points P1, P2, P3,...A to get the required involute.

Applications of the cycloids

·         The involute has some properties that makes it extremely important to the gear industry: If two intermeshed gears have teeth with the profile-shape of involutes (rather than, for example, a "classic" triangular shape), they form an involute gear system. Their relative rates of rotation are constant while the teeth are engaged, and also, the gears always make contact along a single steady line of force. With teeth of other shapes, the relative speeds and forces rise and fall as successive teeth engage, resulting in vibration, noise, and excessive wear. For this reason, nearly all modern gear teeth bear the involute shape

    ·         The involute of a circle is also an important shape in gas compressing, as a scroll compressor can be built based on this shape. Scroll compressors make less sound than conventional compressors, and have proven to be quite efficient.