Non-linear coupled ODE system of equations

Greetings,

I want to solve the following set of coupled ODEs:

I went ahead and used the following assumptions: [ y(1)=θ(t), y(2)=d(θ(t))/dt ,y(3)=φ(t) and y(4)=d(φ(t))/dt ] in order to later on employ the ode45 command to find the solution. Eventhough the script I made runs with no errors or issues, the output matrix y is filled with NaN (except the first row of course, which corresponds to the initial conditions given).

Could you please help me pin point the error in my code, or alternatively propose another method perhaps more suitable to solve these equations?

Thank you in advance!

Here's the script:

% Definition of Parameters

g=9.81; % m/s2

l= input('Strings length: ');

omega=(g/l);

Ttot=input('Total Simulation Time (sec): ');

% Initial Conditions

theta_0 = input('Theta_0 (in form of n*pi): ');

theta_t0 = input('Theta_t0: ');

phi_0 = input('Phi_0 (in form of n*pi): ');

phi_t0 = input('Phi_t0: ');

y0 = [theta_0, theta_t0, phi_0, phi_t0];

% Solution

f=@(t,y) Pendulum3D(t,y,omega);

[t,y]=ode45(f,[0 Ttot],y0);

% Function

function dy=Pendulum3D(t,y,omega)

dy=[y(2);(sin(y(1))*cos(y(1))*(y(4))^2)-omega*sin(y(1)); y(4); (-2*y(4)*y(2)*cot(y(1)))];

end

[Bjorn Gustavsson]

Check what happens when cot(theta) goes to infinity, that is the first worrisome point in your code.

Answer 1

[Jon]

As Bjorn has suggested, you must guard against values of theta where cot(theta) goes to +/- infinity, e.g. 0,pi

Your code seems to run ok for the initial conditions I've edited in below (I commented out the prompts for user input and just supplied values so the results are reproducible)

% Definition of Parameters
g=9.81; % m/s2
l= 1; %input('Strings length: ');
omega=(g/l);
Ttot=10;% input('Total Simulation Time (sec): ');
% Initial Conditions
theta_0 = pi/2; %input('Theta_0 (in form of n*pi): ');
theta_t0 = 0; % input('Theta_t0: ');
phi_0 = pi/4; %input('Phi_0 (in form of n*pi): ');
phi_t0 = 1; %input('Phi_t0: ');
y0   = [theta_0, theta_t0, phi_0, phi_t0];
% Solution
f=@(t,y) Pendulum3D(t,y,omega);
[t,y]=ode45(f,[0 Ttot],y0);

% plot results
plot(t,y)

I would suggest making your equations more obvious as follows, but it seems that yours are equivalent, just hard to read

function ydot = Pendulum3D(~,y,omega)
    % use actual variable names to make code more understandable, help avoid
    % errors in formulas

    theta = y(1);
    thetaDot = y(2);
    phi = y(3);
    phiDot = y(4);

    ydot(1) = thetaDot;
    ydot(2) = sin(theta)*cos(theta)*phiDot^2 - omega*sin(theta);
    ydot(3) = y(4);
    ydot(4) = -2*phiDot*thetaDot*cot(theta);

    % make sure it returns as a column
    ydot = ydot(:);
end