Mathematical Model


    
Our mathematical model deals with solving six nonlinear ordinary differential equations:

a = w3 sinb + w1 cosb
b = (-w3 cosb + w1 sinb) tan a + w
g = (w3 cosb - w1 sinb) sec a
w1 = E1 / G
w2 = E2 / G
w3 = E3 / G

     It is divided into two sections, the kinematic and the kinetic analyses. Because of the length and number of equations involved, a complete reference to them can be found in Appendix A. These equations were derived from Ref [3].

     The kinematic analysis is a series of complex equations that relate various positions, velocities, and spin rates to each other through geometric and/or other constraints. Here the contact point as measured from the ellipsoid's geometric center is related to the position of the ellipsoid through the direction cosines.

     The contact point has zero translational velocity as a result of assuming there is sufficient friction to keep it from sliding. The kinetic analysis is the second set of complex equations. It relates various translational and rotational accelerations to forces and the movements caused by these forces using Newton's Law.