Is this equation numerically solvable?
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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?)
2 个评论
David Goodmanson
2018-9-7
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
2018-9-7
回答（1 个）
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
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2018-9-7
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