DMD:Background:The DMD Model Equations

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The relationship between time and position gives:



where is the vector position of the particle and is the vector velocity.
Note: html to make bold typeface for vectors in math mode is unknown to this author. Sorry.
At the time of contact, , we have ,



Where

The relation leads to a quadratic equation for the ij collision time, .



Where :


discriminant. Note that in m/s is a large number while is very small.

DMD Model Eq 1.jpg

DMD Model Eq 2.jpg

DMD Model Eq 3.jpg

(forget about it)

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(schedule it)



At the time of the collision, the velocities of the particles change according to,


We can derive this formula by assuming that particle j is stationary (reference frame) and particle i is moving on the x-axis with equal mass. The j-direction after collision is given by the line of action , since that is all j feels about momentum change. Conservation of momentum means that with the geometric interpretation of a sum of vectors in the form of a triangle. Conservation of energy gives . The Pythagorean theorem applied to the conservation of energy means that this triangle must be a right triangle. Therefore, we can rotate the coordinate system such that

DMD Model Eq 4.jpg

Note that and must be updated to the point of collision before computing the velocity changes.