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.
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Fig. 10.24 | |||
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