communities.bentley.com/.../7610.CycloidCurveTypes.avi

 

Hi all,

I have created three type of  Cycloid Curve  ,CurtateCycloid ,and ProtateCycloid and Cycloid  using GCSCript programming  and model based approach.

 Enjoy watching Cycloid Curve Avi

 

Here is some history about cycloid curve

 

Definition : The shape defined by a fixed point on a wheel as it rolls. More precisely, it is the locus of a point on the rim of a circle rolling along a perfectly straight line. The curve resembles a succession of arches, with cusps separated by distances equal to the circumference of the circle.

The cycloid was named by Galileo  in 1599. It is the solution to both the tautochrone problem and the brachistochrone problem. In 1634, the French mathematician Gilles de Roberval (1610–1675) showed that the area under a cycloid is three times the area of its generating circle. In 1658, the English architect Christopher Wren showed that the length of a cycloid is four times the diameter of its generating circle.

 The above two statement satisfies by the cycloid curve Created GC.the area under the cycloid curve is 84.37 and area of circle with radius 3 is 28.27 which is exactly 4 times of the area of the circle.and Length of the Cycloid Curve is 24 which is exactly four times of the diameter of the circle of radius 3.

 

As well as the ordinary cycloid there is the Curtate cycloid, which is the path traced out by a point on the inside of a rolling circle, and the Prolate cycloid, which is followed by a point on the outside of the circle. A Prolate cycloid is traced out, for example, by points on the flange of the wheels of a locomotive, which extends below the top of the tracks. This leads to the surprising conclusion that even as the locomotive is moving forward there are always parts of its wheels that are going backward for a moment before moving forward again.

Cycloids play a major role in gears. They are perfectly meshing (gives perfect rolling action), and earlier was used in manufacturing gears. Mechanical clocks still use cycloidal gears. Because the contact force direction (pressure angle) changes rapidly for cycloidal gears, however, they are subjected to  fatigue stresses and not used in modern gears now a days. Instead involute curve is used for these gears.  However, at the extremities of addenda and dedenda of the gears are still manufactured with cycloidal profiles, as involute rapidly changes its shape near these extremities.

thanks

mridul