A Simple VBA Tutorial: Newmark’s dynamic solutions of a 2 dof mass spring damper system.
This code solves a two dof mass spring damper system using a Newmark’s time stepping solution. This is a direct dynamics solution in the time domain and may be extended to a systam with more then 2 degrees of freedom.
Is quite involved so refer to “The Finite Element Method” by Thomas J.R. Hughes, but the implementation is fairly straight forward as follows.
1) For our example we take g=0.5 and b=0.25 which are integration parameters and a time step dt= 0.01.
2) we are going to solve (M+g*dt*C+b*dt^2*K)*an+1=-C*vbn+1-K*dbn+1 where M,C,K are the mass, damping and stiffness matrices.
3) from (M+g*dt*C+b*dt^2*K) we get our effective stiffness matrix and we invert this so we can calculate the new acceleration vector from the effective force.
4) -C*vbn+1-K*dbn+1 gives us the effective force where :-
we start with dn our displacement vector set to initial conditions [0.1] and vn and an set to [0,]0]
5) we calculate our effective force from 4 starting with our initial conditions (dn,vn,an) multiplying this by 3 the inverse of the effective stiffness matrix we get are new acceleration vector an+1.
6) We now calculate the new displacement and velocity vectors from an+1 as :-
dn+1=dbn+1 + b*dt^2*an+1
vn+1=vbn+1 + g*dt^2*an+1
7) now we goto 4 where dn and vn are now the newly calculated dn+1 vn+1