# How can I model second order ODE with matrices and external forcing?

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Hello. I am new to MATLAB-based modeling of ODEs, and I was wondering if someone here can help me with simulating the dynamics of the following ODE. Links to resources, code, and/or vocabulary would be appreciated. M is the mass matrix, E is the damping matrix, K is the stiffness matrix, B describes inputs, and C describes outputs.
My goal is the eventually model an impulse response or simply the evolution of the system with a collection of initial values.
For background, I don't really have sufficient background in FEA or numerical analysis. I am an intern learning the math and the MATLAB on the fly. I read into the documentation of ODE solvers, but I was unable to find a simple way to incorporate the equation above.
• Thank you

Alan Stevens on 15 Jul 2021
Like this perhaps (I've made up arbitrary data; you will obviously have to replace it with your own)
x0 = [1; -1];
v0 = [0; 0];
X0 = [x0; v0];
tspan = [0 1];
[t,X] = ode45(@fn, tspan, X0);
x = X(:,1:2);
v = X(:,3:4);
C = [2, 0; 0, 2];
y = C*x';
plot(t,x,'k',t,y,'r'),grid
xlabel('t'), ylabel('x (black) & y (red)') function dXdt = fn(t,X)
M = [1, 0.1; 0.5, 0.5];
E = [2, 0; 0, 1];
K = [1, 1; 0.1, 0.2];
B = [1, 0; 0, 1];
u = [1; 0];
x = X(1:2);
v = X(3:4);
dxdt = v;
dvdt = M\(B*u - K*x - E*v);
dXdt = [dxdt; dvdt];
end
Justin Burzachiello on 19 Jul 2021
Thank you, Alan. Also, thanks for the help with inputting the data matrices as arguments.

R2020a

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