Motion of a Cycloid
With today's modern computers, there are many ways to model the motion of
a cycloid. As a cycloid is the locus of a point on a circle that is rolling
along a line, a graphing program would be the easiest way to generate one.
To simplify the graphing procedure , we will examine the motion of the point
seperately in the horizontal and vertical plane. By breaking up the motion
into two components, it will be easier to visualize and the results will
be able to be graphed parametrically.
The first motion that we should consider is the motion in the horizontal
plane. The center of the circle will be moving forward at a constant rate
t. The rotation of the circle will also affect the motion of the point.
Before the circle begins rotating, the point is directly in line with the
center, below it. As the circle begins to rotate, the point falls behind
the center until it has rotated one fourth the way around, it then begins
catching upto to the center, until it is even with it after one half of
a rotation. As sine is a function that begins at zero, increases and then
decreases, the second part of the motion in the horizontal plane is sin[t].
As both of these motions will depend upon the radius, the function that
will define horizontal motion is r(t-sin[t]).
The point will also be moving in the vertical direction as the circle rotates.
The point will start below the center, as the rotation begins it will get
closer to the center, until it reaches the height of the center one fourth
of the way through a revolution. A function that begins at its maximum and
decreases is cosine. To start at floor level, the function needs more terms
that just the cosine term. The function that defines the motion in the vertical
direction is r(1-cos[t]).
To achieve one complete revolution, t needs to begin at 0 and stop at two
Pi. The parametric equations that define the motion of a cycloid are
x=r(t-sin[t])
y=r(1-cos[t])
where t measures the angle of rotation of the cycloid. Many different graphing
programs can be used to graph the results of this parametirc equation, and
Mathematica 3.0 was chosen for this investigation. The following lines,
when entered into Mathematica, will produce a cycloid where the circle has
undergone two complete rotations.
r = 1;
g1=ParametricPlot[{(r t - r Sin[t]),r(1 - Cos[t])},{t,0,4Pi}]
The following graphs show the cycloid generated by the equations above.
The green point on the circle is the point whose locus is being used to
generate the cycloid.
Each time the circle and point have been drawn, t has increased by a Pi/3.
It is interesting to note that the majority of the horizontal motion occurs
from 2Pi/3 to 4Pi/3. This is easier to see in the graph below that has multiple
circles on the same graph.
As the radius is increased, two results occur. The first result is that
the highest point on the cycloid increases, it is the diameter of the circle.
The other result is that the distance in the horizontal direction that is
required to complete a revolution increases. The distance is the circumference
of the circle. The following graphs illustrate the point. The red cycloid
has a radius of 0.5, the purple cycloid has a radius of 1, the aqua cycloid
has a radius of 2 and the green cycloid has a radius of 4.
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