Describe the cycloid described by a wheel rolling inside a circle

• JanClaesen
In summary, the conversation discusses the equation for the relative position vector of a point on a wheel rolling inside a circle, as well as the position vector of the origin in the absolute coordinate system. It is mentioned that the wheel does not slip and that the smaller wheel is also moving in a circle, and a correction is suggested to accurately plot the curve.
JanClaesen
I'm trying to describe the cycloid described by a wheel rolling inside a circle:
The (rotor)vector attached to the center of the rolling wheel is described by this equation:
r = r cos(w t)e(x') + r sin(w t)e(y')

w is the angular velocity of the wheel, e(x') and e(y') are the unit vectors and t is the time

This is the relative position vector of the point on the wheel that touches the big circle as the wheel rolls.
The origin of the absolute coordinate system x y is the center of the big circle, the origin (the center of the wheel) of the other coordinate system x' y' (where x' always stays parallel to x and y' to y) moves in a smaller circle with radius R - r, R is the radius of the big circle, r is the radius of the wheel. So the position vector of the origin in the absolute coordinate system is given by:

p = (R - r) cos( (r/R) w t)e(x) - (R - r) sin( (r/R) w t)e(y)

The angular velocity W of the smaller circle is given by (r/R)*w, since r*w = R*W (the wheel doesn't slip). The minus sign is because the wheel rolls clockwise.

r + p should describe the curve, now, when R = 4r I should plot an astroid, which is not the case, where have I gone wrong?

You have taken into account that it doesn't slip, but you haven't taken into account that the little wheel is additionally moving in a circle itself. So the amount the little wheel turns if the big wheel turns wt is (R/r)wt + wt. Try that.

1. What is a cycloid?

A cycloid is a curve that is traced by a point on the circumference of a circle as it rolls along a straight line. It is also known as a trochoid.

2. How is a cycloid described by a wheel rolling inside a circle?

A cycloid is formed by a point on the circumference of a smaller circle as it rolls around the inside of a larger circle. The path traced by this point is a cycloid.

3. What is the equation for a cycloid?

The equation for a cycloid is given by x = r(θ - sinθ), y = r(1 - cosθ), where r is the radius of the rolling circle and θ is the angle of rotation.

4. What are the properties of a cycloid?

A cycloid is a non-linear curve with infinite length, tangent to the base circle at the point of contact, and has a cusp at the lowest point. It also has constant curvature, meaning that the radius of curvature is constant at any point on the curve.

5. How is the cycloid used in real life?

The cycloid has many real-life applications, including in the design of gears and gear teeth, as well as in the construction of arches and suspension bridges. It is also used in physics to model the motion of a pendulum and in mathematics to study the area under the curve.

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