how to numerically solve a coupled ode?

Hi,

Please help to solve the following ODE or numerically solve it (get theta(t) and phi(t)):

ODE 1 :

thetaDoubleDot = ((36297*cos(theta))/2 + (80*sin(phi)*(- CA*cos(phi - theta)*phiDot^2 + 4*cos(theta)*thetaDot^2 + CA*phiDoubleDot*sin(phi - theta)))/cos(phi - theta))/((320*sin(phi)*sin(theta))/cos(phi - theta) - 1300/3)

ODE 2:

phiDoulbeDot = (sin(phi - theta)*((1300*thetaDoubleDot)/3 + (36297*cos(theta))/2 + (80*sin(phi)*(- CA*sin(phi - theta)*phiDot^2 + 4*sin(theta)*thetaDot^2 + 4*thetaDoubleDot*cos(theta) + 981/100))/sin(phi - theta)))/(80*CA*cos(phi - theta)*sin(phi))

The thing is that they are coupled. As in the first ODE contain phiDoubleDot and the second ODE contain thetaDoubleDot.

Regards,

Answer 1

[Star Strider]

Use the Symbolic Math Toolbox for this.

First, declare all the appropriate variables as functions of time:

syms theta(t) phi(t) CA T Y 

Then define the derivatives as:

Dtheta = diff(theta);
D2theta = diff(theta,2);
Dphi = diff(phi);
D2phi = diff(phi,2);

and make the appropriate substitutions in these equations:

thetaDoubleDot = ((36297*cos(theta))/2 + (80*sin(phi)*(- CA*cos(phi - theta)*phiDot^2 + 4*cos(theta)*thetaDot^2 + CA*phiDoubleDot*sin(phi - theta)))/cos(phi - theta))/((320*sin(phi)*sin(theta))/cos(phi - theta) - 1300/3)
phiDoulbeDot = (sin(phi - theta)*((1300*thetaDoubleDot)/3 + (36297*cos(theta))/2 + (80*sin(phi)*(- CA*sin(phi - theta)*phiDot^2 + 4*sin(theta)*thetaDot^2 + 4*thetaDoubleDot*cos(theta) + 981/100))/sin(phi - theta)))/(80*CA*cos(phi - theta)*sin(phi))

so that they are in the form of:

Eqn1 = D2theta == ((36297*cos(theta))/2 + ...
Eqn2 = D2phi == (sin(phi - theta)* ...

Since the objective is to solve them numerically, then use odeToVectorField and matlabFunction:

[VF,Subs] = odeToVectorField(Eqn1,Eqn2)
odefcn = matlabFucntion(VF, 'Vars',{T,Y,CA})

Then it should be possible to use ‘odefcn’ with ode45 and the others.

I leave the details to you.

.