| VCL 7 - Introduction to Numerical Computing III | Page 6 of 10 |
The Epitrochoid
The triangular rotor of the Wankel engine spins in a metal compartment called the bore whose shape is an epitrochoid.
| Despite the haughty name, an epitrochoid is just the curve traced by a point P on one circle as it rolls around another circle. For example, if you were to roll a penny around a nickel the curve traced by any point such as the president's nose would be an epitrochoid. If you were to bicycle around the world, any point on the front wheel would trace out a very large epitrochoid. Of course the equation for an epitrochoid must depend on the radii r and R of the two circles and on the distance h of the point P from the center of the circle. Choose this hyperlink for a java animation of an epitrochoid generation. |
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The equations for an epitrochoid are:
, 
where r is the radius of the small moving circle and R is the radius of the larger fixed circle; h is the distance from the center of the moving circle to the point in question. Don't panic! Once the constants are plugged in, each equation is just the sum of a sine and a cosine.
The best way to understand these epitrochoids is to see what they look like by plotting them. Enter the following equations for an epitrochoid into your Matlab worksheet. For each choice of the constants R, r and h we get a different epitrochoid. We can create a parametric plot of the epitrochoid if we choose r=1, R = 2 and h = 1/2.
Enter the values of the constants r, R and h.
r = 1; R = 2; h = 1/2;
Create the plot using the plot command. Here are creating arrays with indices of the form x(c) and y(c) where c is an integer from 1 to N.
c=1;
for th = 0: pi/20: 2*pi;
x(c) = (r+R)* cos(th) - h* cos(((r+R)/r)*th);
y(c) = (r+R)* sin(th) - h* sin(((r+R)/r)*th);
c=c+1;
end;
plot(x, y, 'r');
axis equal
Recall that h is the distance from the center of the moving circle to the point P. To investigate the effect of the parameter h, create an animation of the resulting epitrochoid as h changes from 0 to 1.
Now vary h from -2 to 2. How do the epitrochoids vary with negative h compared to those with positive h?
