Error: ode45 Must return a column vector?

I'm trying to solve this problem involving a 2DOF system involving springs and dampers, and i've coded all of this below, including the state space matrix, this is urgent as this is due in 2 days, any help would be appreciated :)

Many thanks in advance

function batch_timeresp
% the total time domain response

% System parameters
k3 = 15; k1 = 10.0; k2 = 10.0; m1 = 2.0; m2 = 6.0; L = 1.0; c = 1; ct = 0.5; g = 9.81;kequi=5;
%ball
zeta1 = (c/(2*sqrt(m2*k3)));
%block
zeta2 =  (ct/(2*sqrt(m1*kequi)));
%damped natural freq of block
om02=sqrt(kequi/m2);
omd2=om02*(sqrt(1-(zeta2)^2));
%damped natural freq of ball
om01=sqrt(k3/m1);
omd1=om01*(sqrt(1-(zeta1)^2));

T1 = 0.01*((2*pi)/om02);
T2 = 1.50*((2*pi)/om01);
Tpulse = T2-T1;

% Excitation parameters
Ft = 30;
Im = ((Ft*Tpulse)/2);

% State space system matrix and coupling
A=[0,0,1,0;0,0,0,1;-(k1+k2)/(m1+m2),0,-c/(m1+m2),0;0,-k3/(m1*(L^2)),0,-ct/(m1*(L^2))]; %state space matrix
S=[1,0,0,0;0,1,0,0;0,0,1,(m1*L)/(m1+m2);0,0,(m1*L)/(m1*(L^2)),1]; %coupling
Sinv=inv(S);
gmat = [0;0;-g;g/L]; %gravity terms
Fmat=[0;0;Im/(m1+m2);(2*Im*L)/(m1*(L^2))]; %impulse force

% Numerical integration
tspan=linspace(0,5,1e3);
X0=[0;0;0;0];
[T,X]=ode45(@ssmodel,tspan,X0);

% Results
figure
subplot(2,1,1), plot(T,X(:,1)), ylabel('x [m]')
subplot(2,1,2), plot(T,X(:,2)), ylabel('dx/dt [m/s]'), xlabel('time [s]')

function dx=ssmodel(t,x)
% State-space model of 1 DOF system
dx=(A*x+gmat+Fmat).*Sinv;
end
end

Answer 1

[Alan Stevens]

Like this?

% the total time domain response

% System parameters
k3 = 15; k1 = 10.0; k2 = 10.0; m1 = 2.0; m2 = 6.0; L = 1.0; c = 1; ct = 0.5; g = 9.81;kequi=5;
%ball
zeta1 = (c/(2*sqrt(m2*k3)));
%block
zeta2 =  (ct/(2*sqrt(m1*kequi)));
%damped natural freq of block
om02=sqrt(kequi/m2);
omd2=om02*(sqrt(1-(zeta2)^2));
%damped natural freq of ball
om01=sqrt(k3/m1);
omd1=om01*(sqrt(1-(zeta1)^2));

T1 = 0.01*((2*pi)/om02);
T2 = 1.50*((2*pi)/om01);
Tpulse = T2-T1;

% Excitation parameters
Ft = 30;
Im = ((Ft*Tpulse)/2);

% State space system matrix and coupling
A=[0,0,1,0;0,0,0,1;-(k1+k2)/(m1+m2),0,-c/(m1+m2),0;0,-k3/(m1*(L^2)),0,-ct/(m1*(L^2))]; %state space matrix
S=[1,0,0,0;0,1,0,0;0,0,1,(m1*L)/(m1+m2);0,0,(m1*L)/(m1*(L^2)),1]; %coupling
Sinv=inv(S);
gmat = [0;0;-g;g/L]; %gravity terms
Fmat=[0;0;Im/(m1+m2);(2*Im*L)/(m1*(L^2))]; %impulse force

% Numerical integration
tspan=linspace(0,5,1e3);
X0=[0;0;0;0];
[T,X]=ode45(@(t,x) ssmodel(t,x,A,gmat,Fmat,Sinv),tspan,X0);

% Results
figure
subplot(2,1,1), plot(T,X(:,1)), ylabel('x [m]')
subplot(2,1,2), plot(T,X(:,2)), ylabel('dx/dt [m/s]'), xlabel('time [s]')

function dx=ssmodel(~,x,A,gmat,Fmat,Sinv)
% State-space model of 1 DOF system
dx=Sinv*(A*x+gmat+Fmat);  %%%%%%%%%%%%%%%%%%%%%%
end

[King Hin Wong]

Yes! Thank you so much!!! If you wouldn't mind, could you please explain what you've done in the function?

dx=ssmodel(~,x,A,gmat,Fmat,Sinv)

What does the tilde do? And for this:

dx=Sinv*(A*x+gmat+Fmat)

Would the order of multiplication of the inverse matrix with the stuff the in brackets not matter?

[Alan Stevens]

The tilde is just a placeholder, since your function doesn't make explicit use of t. This isn't essential, you could keep the t there if you prefer.

The order in which matrices and vectors are multiplied is very important!

[King Hin Wong]

If I had to do

(A*x+gmat+Fmat)*Sinv

For the sake of the order of multiplication, why does it return an error?

[Alan Stevens]

You are solving for four parameters, which means the function must return a 4x1 vector. If you have

dx=(A*x+gmat+Fmat).*Sinv;
 

this returns a 4x4 matrix.

Your state-space equation is presumably

S*dx = A*x+gmat+Fmat;

which means you need to multiply from the left to get x' on its own.

Sinv*S*dx = Sinv*(A*x+gmat+Fmat);

or

dx = Sinv*(A*x+gmat+Fmat);

[King Hin Wong]

Solved! Thank you so much, have a nice day :)