Is this equation numerically solvable?

r'' - r θ'^2 = g sinθ - (g-y'')

r'+θ'+y'=0

rθ''+2r'θ'+g cosθ = 0

Sorry may I know how I can solve this with ode45. I am attempting to convert all three equations into 1st Order Coupled equations, but I can't seem to get the second equation into the correct substitution. Is it possible? Or do I have to use another ODE solver (which?)

[David Goodmanson]

Hi tch,

The second equation is unlikely, on dimensional grounds to begin with, since theta' has units of 1/sec and the other two terms have units of meters/sec.

[tan chien hao]

Sorry, the original differential equation had theta' multiplied with radius. I just removed the constants to simplify the differential equation. With units aside, is it possible to rewrite these into a system of 1st Order ODEs? Thanks for your help.

Answer 1

[Stephan]

Hi,

the following code will transform the equations into a form that can be solved with ode45. It creates a file ode_system, that contains the first order system.

Please check the equations for errors i possibly made:

syms r(t) theta(t) y(t) g
eqn1 = diff(r,t,2) - (r*(diff(theta,t)))^2 == g*sin(theta) - (g-diff(y,t,2));
eqn2 = diff(r,t) + diff(theta,t) + diff(y,t) == 0;
eqn3 = r*diff(theta,t,2) + 2*diff(r,t)*diff(theta,t) + g*cos(theta) == 0;

eqn1 = subs(eqn1,g,9.81);
eqn3 = subs(eqn3,g,9.81);

eqn = [eqn1, eqn2, eqn3];

[eqn, vars] = reduceDifferentialOrder(eqn,[r(t) theta(t) y(t)]);
[eqn, vars] = reduceRedundancies(eqn,vars);
[eqn, vars] = reduceDAEToODE(eqn,vars);
[eqn, vars] = odeToVectorField(eqn);
disp(vars)
matlabFunction(eqn,'File','ode_system','Vars',{'t','Y'})

The order of the variables is displayed when running the script. If you have 6 initial conditions for y,r and theta and their derivatives it should be possible to solve the system.

Best regards

Stephan