3d trajectory of system of wheels

[snay]

Hello!

I tried to develop equations of motion of system of wheels, where the first one with radius of r is rotated with some angular velocity v, the center of the second wheel is connected with the first on its perimeter and its plane of rotation is perpendicular to the plane of rotation of the first wheel. The second wheel has a radius of r/2 and angular velocity of 2v. The third wheel is connected with the second in the same way and its plane of rotation is perpendicular to the plane of rotation of the second wheel. The radius of the third wheel is r/4 and the velocity - 4v. I tried to develop general equations, which allow to determine trajectory for the system of arbitrary number of wheels. I have developed equations for the system of two and three wheels, but stuck in developing the equations for the system of four wheels. Here I post my solutions:

- for the system of two wheels
x1=(cos(phi)+1/q*cos(phi)*cos(q*phi))*r;
y1=(sin(phi)+1/q*sin(phi)*cos(q*phi))*r;
z1=(1/q*sin(q*phi))*r;

- for the system of three wheels
x2=(cos(phi)+1/q*cos(phi)*cos(q*phi)+1/q^2*(cos(q^2*phi)*cos(phi-pi/2)*cos(phi+pi/2)+cos(q*phi)*cos(phi)*cos(phi)))*r;
y2=(sin(phi)+1/q*sin(phi)*cos(q*phi)+1/q^2*(sin(q^2*phi)*cos(phi-pi/2)*cos(phi+pi/2)-sin(q*phi)*cos(phi)*cos(phi)))*r;
z2=(-1/q*sin(q*phi)-1/q^2*sin(q^2*phi)*cos(phi)+1/q*sin(phi)*cos(phi)*cos(phi))*r;

I will be very thankful for any comments.

 

[dodo]

Just as a curiosity, I wonder if the third wheel will scratch the first.