How to give input function dz/dt=Z*omega*cos(omega) to state space system?

EOm=4mu''+30 cu'+3ku=12cz'

Let

m=1000

k=5000

c=0.02*2*sqrt(Ke*Me)

How to write above equation in state space veriables?

I am confused with the part at R.H.S. 12cz', input= z(t)=Zcos(omega)

Kindly gude.

Answer 1

[Alan Stevens]

Do you mean something along these lines:

tspan = [0 2];  % Or whatever you need
IC = [1 0];     % Set your own initial conditions IC = [u(0) dudt(0)]

[t, U] = ode45(@fn, tspan, IC);

u = U(:,1); dudt = U(:,2);
plot(t,u,'r',t,dudt,'b'),grid


function dUdt = fn(t,U)

        m=1000;
        k=5000;
        Ke = 1000;      % Set to your own value
        Me = 1000;      % Ditto
        Z = 1000;       % Ditto
        omega = 10;     % Ditto
        c=0.02*2*sqrt(Ke*Me);
        % I suspect you want z = Z*sin(omega*t), so that
        % dz/dt = Z*omega*cos(omega*t)
        dzdt = Z*omega*cos(omega*t);

        M = [1 0; 0 4*m];
        K = [0 -1; 3*k 30*c];
        V = [0; 12*c*dzdt];
    
        dUdt = M^-1*(V - K*U);

end

% Let v = u', then your 2nd order equation becomes two ist order equations
% u' - v = 0 
% 4*m*v' + 30*c*v + 3*k*u = 12*c*Z*omega*cos(omega*t)
% [1   0][u'] + [0       -1][u] = [0                        ]  
% [0 4*m][v']   [3*k  30*c][v]   [12*c*Z*omega*cos(omega*t)]

[Sadia]

Thank you very much for the response. Yes you are right it was cos(omega*t) sorry about it. I also have to find Frequency response function, i-e U/Z and have to plot against frequency omega. I am working on it.