## The hypocycloid

A hypocycloid is defined as the locus of a point on the circumference of a circle which rolls without slip around the inside of another circle.

The construction for the hypocycloid (Fig. 10.16) is very similar to that for the epicycloid, but note that the rolling circle rotates in the opposite direction for this construction.

It is often necessary to study the paths taken by parts of oscillating, reciprocating, or rotating mechanisms; from a knowledge of displacement and time,

Hypocycloid

Base circle

Final position of rolling circle

Direction of rotation of rolling circle

Fig. 10.16 Hypocycloid

Hypocycloid

Base circle

Direction of rotation of rolling circle

### Fig. 10.16 Hypocycloid

Final position of rolling circle information regarding velocity and acceleration can be obtained. It may also be required to study the extreme movements of linkages, so that safety guards can be designed to protect machine operators.

Figure 10.17 shows a crank OA, a connecting rod

AB, and a piston B which slides along the horizontal axis BO. P is any point along the connecting rod. To plot the locus of point P, a circle of radius OA has been divided into twelve equal parts. From each position of the crank, the connecting rod is drawn, distance AP measured, and the path taken for one revolution lined in as indicated.

The drawing also shows the piston-displacement diagram. A convenient vertical scale is drawn for the crank angle. and in this case clockwise rotation was assumed to start from the 9 o'clock position. From each position of the piston, a vertical line is drawn down to the corresponding crank-angle line, and the points of intersection are joined to give the piston-displacement diagram.

The locus of the point P can also be plotted by the trammel method indicated in Fig. 10.18. Point P1 can be marked for any position where B1 lies on the horizontal line, provided A1 also lies on the circumference of the circle radius OA. This method of solving some loci problems has the advantage that an infinite number of points can easily be obtained, and these are especially useful where a change in direction in the loci curve takes place.

Figure 10.19 shows a crank OA rotating anticlockwise about centre O. A rod BC is connected to the crank at point A, and this rod slides freely through a block which is allowed to pivot at point S. The loci of points B and C are indicated after reproducing the mechanism in 12 different positions. A trammel method could also be used here if required.

Part of a shaping-machine mechanism is given in Fig. 10.20. Crank OB rotates about centre O. A is a fixed pivot point, and CA slides through the pivoting block at B. Point C moves in a circular arc of radius

AC, and is connected by link CD, where point D slides horizontally. In the position shown, angle OBA is 90°, and if OB now rotates anti-clockwise at constant speed it will be seen that the forward motion of point D takes more time than the return motion. A displacement diagram for point D has been constructed as previously described.

Fig. 10.19

Forward stroke

Return stroke

Displacement diagram

Forward stroke

Return stroke

Displacement diagram

### Fig. 10.21

In Fig. 10.21 the radius OB has been increased, with the effect of increasing the stroke of point D. Note also that the return stroke in this condition is quicker than before.

The outlines of two gears are shown in Fig. 10.22, where the pitch circle of the larger gear is twice the pitch circle of the smaller gear. As a result, the smaller gear rotates twice while the larger gear rotates once. The mechanism has been drawn in twelve positions to plot the path of the pivot point C, where links BC and CA are connected. A trammel method cannot be applied successfully in this type of problem.

Figure 10.23 gives an example of Watt's straight-line motion. Two levers AX and BY are connected by a link AB, and the plotted curve is the locus of the

Fig. 10.23 Watt's straight line motion

mid-point P. The levers in this instance oscillate in circular arcs. This mechanism was used in engines designed by James Watt, the famous engineer.

A toggle action is illustrated in Fig. 10.24, where a crank rotates anticlockwise. Links AC, CD and CE are pivoted at C. D is a fixed pivot point, and E slides along the horizontal axis. The displacement diagram has been plotted as previously described, but note that, as the mechanism at E slides to the right, it is virtually stationary between points 9, 10 and 11.

The locus of any point B is also shown.

 ( Fig. 10.24

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### Responses

• folcard
How to make hypocycloide in geometry in engineering?
9 years ago
• April
How to draw a hypocycloid?
9 years ago
• alfrida
How do you plot a locus of a crank mechanism with a pivoted lever?
8 years ago
• william peters
How to make a hypocycloid in engg drawing?
8 years ago
• julia
How to construct hypocycloid construction in drawing?
8 years ago
How to draw hypocycloid drawing?
8 years ago
• asphodel burrows
How to draw crank angle diagram?
8 years ago
• florian
How to plot locus technical drawing?
8 years ago
• faramond
What is locus of a point in Technical drawing?
8 years ago
• tess
How to draw a hypo cycloid ?
7 years ago
• IOLE
How to draw a hypocycloid video?
7 years ago
• penelope
How to construct hypocycloid in engineering drawing?
7 years ago
• Dennis Beecher
How to draw a crank in engineering drawing that slides through a pivoted block?
4 years ago
• sam
How to dra hypocycloid detail explanation?
4 years ago
• Christina
How to draw hypocycliod?
4 years ago
• Ensio
How to draw hypocycloid in engineering drawing?
4 years ago
• lachlan
How to draw a loci on a pivot?
3 years ago
• sara
How to divide the base line into twelve equal parts in hypocycloid?
3 years ago
• Robel
How to construct epi hypo cycloid?
3 years ago
• Biniam
How to construct hypocloid?
3 years ago
• innes
How to plot locus on revolving cranks rotating oppositly joined by two rods?
3 years ago
• hugo hill
How is epicycloid in one revolution?
3 years ago
• Manu
How to draw an hypoclycloid loci?
2 years ago
• MARILYN
How to draw loci in technical drawing showing steps?
2 years ago
• Barry Jamieson
How to construct epicycloid and hypocycloid problems?
2 years ago
• melanie
How to consyrct epi and hypo cycloid in engineering dtawing?
2 years ago
• LACEY
How to draw an hypocycloid and epicycliod?
2 years ago
• Lodovico
How to draw hypocycloid in stright line base?
2 years ago
• Temshe
How to make two revolution hypocycloid?
2 years ago
• stephen
How to construct an epicycliod?
2 years ago
• ponto
How to construct the locus of watt's straight line motion?
2 years ago
• ermias
How to divide the angle for hypocycloid?
2 years ago
• tyler
How to make an epicycloid and a hypocycloid?
2 years ago
• Rosaria
How to draw an hypocycloid in engineering graphics?
2 years ago
• Liberato
How to construct epicycloid and hypocycloid in a circle?
1 year ago
• Bowman
How to draw epicycloid and hypocycloids?
1 year ago
• Enrico
How to draw a hypocloid?
1 year ago
• miika
How to draw hypercycloid?
12 months ago
• menegilda
How to draw epicycloid and hypocyloid?
5 days ago