Cycloid And Trochoid | Problems on Cycloid And Trochoid
Cycloid :
1 ] Draw the locus of a point on the periphery of a circle which rolls on a straight-line path. Take a circle which rolls on a straight-line path. Take circle diameter as 50 mm.
Solution Steps:
- From center c draw a horizontal line equal to πD (2πr = periphery ) distance.
- Divide the circle into 8 number of equal parts and in a clockwise direction, after P names 1, 2, 3 up to 8.
- Divide πD distance into 8 number of equal parts and name them C1, C2, C3...C8.
- From all these points on circle draw horizontal lines parallel to the locus of C.
- With a fixed distance CP in compass, C1 as a center, mark a point on a horizontal line from 1. Name it P.
- Repeat this procedure from C2, C3, C4 up to C8 as centers. Mark point P2, P3, P4, P5 up to P8 on the horizontal lines draw from 2, 3, 4, 6, 7 respectively.
- Join all these points by curve. It is cycloid.
SUPERIOR TROCHOID
2 ] Draw the locus of a point, 5 mm away from the periphery of a circle that rolls on a straight-line path. Take circle diameter as 50 mm.
Solution Steps:
- Draw a circle of given diameter ( 50 mm) and draw a horizontal line from its center C of length πD and divide it in 8 number of equal parts and name them C1, C2, C3, up to C8.
- Draw circle by CP radius, as in this case CP is large then the radius of the circle.
- Now divide a new circle into 8 numbers of equal parts and in a clockwise direction, after P names 1, 2, 3 up to 8.
- From all these points on circle draw horizontal lines parallel to the locus of C.
- With a fixed distance CP in compass, C1 as a center, mark a point on a horizontal line from 1. Name it P.
- Repeat this procedure from C2, C3, C4 up to C8 as centers. Mark point P2, P3, P4, P5 up to P8 on the horizontal lines draw from 2, 3, 4, 6, 7 respectively.
- Join all these points by curve, this curve is called as Superior Trochoid.
INFERIOR TROCHOID
3 ] Draw the locus of a point, 5 mm inside from the periphery of a circle that rolls on a straight-line path. Take circle diameter as 50 mm.
Solution Steps:
- Draw a circle of given diameter ( 50 mm) and draw a horizontal line from its center C of length πD and divide it in 8 number of equal parts and name them C1, C2, C3, up to C8.
- Draw circle by CP radius, as in this case CP is shorter than the radius of the circle.
- Now divide a new circle into 8 numbers of equal parts and in a clockwise direction, after P names 1, 2, 3 up to 8.
- From all these points on circle draw horizontal lines parallel to the locus of C.
- With a fixed distance CP in compass, C1 as a center, mark a point on a horizontal line from 1. Name it P.
- Repeat this procedure from C2, C3, C4 up to C8 as centers. Mark point P2, P3, P4, P5 up to P8 on the horizontal lines draw from 2, 3, 4, 6, 7 respectively.
- Join all these points by curve, this curve is called as Inferior Trochoid.
EPI CYClOIDE
4 ] Draw the locus of a point on the periphery of a circle that rolls on a curved path. Take the diameter of rolling circle 50 mm and radius of directing circle ( curved path ) 75mm.
Solution Steps:
- When smaller circle will roll on a larger circle for one revolution it will cover πD distance on the arc and it will be decided by included arc angle θ.
- Calculate θ by formula θ = (r/R) × 3600.
- Construct angle θ with radius OC and draw an arc by taking O as center OC as radius and from a sector of angle θ.
- Divide this sector into 8 numbers of equal angular parts. And from C onward name them C1, C2, C3 up to C8.
- Divide smaller circle ( Generating circle ) also in 8 number of equal parts. And next to P in clockwise direction name those 1, 2, 3, up to 8.
- With O as center, O1 as radius draw an arc in the sector. Take O2, O3, O4, up to O8 distance with center O, Draw all concentric arcs in the sector. Take fixed distance CP in compass, C1 center, cut arc of 1 at P1. Repeat procedure and locate P2, P3, P4, P5 up to P8 and join them by smooth curve.this is EPI-CYCLOID.
HYPO CYCLOID
5 ] Draw the locus of a point on the periphery of a circle which rolls from the inside of a curved path. Take the diameter of rolling circle 50 mm and radius of directing circle (curved path) 75 mm.
Solution Steps:
- Smaller circle is rolling here, it has to rotate anticlockwise to move ahead.
- When smaller circle will roll on a larger circle for one revolution it will cover πD distance on the arc and it will be decided by included arc angle θ.
- Calculate θ by formula θ = (r/R) × 3600.
- Construct angle θ with radius OC and draw an arc by taking O as center OC as radius and from a sector of angle θ.
- Divide this sector into 8 numbers of equal angular parts. And from C onward name them C1, C2, C3 up to C8.
- Divide smaller circle ( Generating circle ) also in 8 number of equal parts. And next to P in anticlockwise direction name those 1, 2, 3, up to 8.
- With O as center, O1 as radius draw an arc in the sector. Take O2, O3, O4, up to O8 distance with center O, Draw all concentric arcs in the sector. Take fixed distance CP in compass, C1 center, cut arc of 1 at P1. Repeat procedure and locate P2, P3, P4, P5 up to P8 and join them by smooth curve.this is HYPO-CYCLOID.
