Solving Equations of Motion by Direct Time Integration

Demonstrates the technique of converting a system of n 2nd order (ordinary) differential equations into a 2n system of 1st order ODEs that can be integrated by a computer by the implementation of a time-marching solution.

[@TheBehfarataei]

Thanks for amazing videos.

[@AJ-et3vf]

This is nice and interesting! I have actually implemented this kind of solution too in Python recently, but I used the state-space formulation in Inman's Mechanical Vibrations which inverts the n x n mass matrix to obtain the state vector, xdot(t, x). The A matrix is still a 2n x 2n matrix, but it is A = [zeros(n) eye(n): m^-1*k m^-1*c] . Since the mass matrix is a sparse, diagonal matrix, it wouldn't be computationally expensive to invert it compared to the 2n x 2n matrix, A, here.