I am attempting to use some of the functions in MATLAB to numerically solve a pair of coupled second order ODEs of the form
\ddot{x} = f(x,y,\dot{x},\dot{y})
\ddot{y} = f(x,y,\dot{x},\dot{y}).
I am able to get it to work with just one second-order ODE, but the code I am trying to does not work for a pair of ODEs.
The function odeToVectorField effectively takes a second order ODE and writes it as a vector for a pair of coupled first order ODEs. ode45 is the usual Runge-Kutta solution method. xInit and yInit correspond to the initial conditions for x and y and the aim is then to plot both x and y against time over a certain interval of time.
gamma1=0.1;
gamma2=0.1;
a=1;
m=1;
g=9.8;
d=1;
syms x(t) y(t)
eqn1=diff(x,2)== (gamma1*diff(x))/(a m*d^2 (m/2)*d^2*cos(y-x)) (gamma2*diff(y))/(a (m/2)*cos(y-x)) - ( (m/2)*d^2*sin(y-x)*(diff(x)^2 - diff(y)^2))/(a m*d^2 (m/2)*d^2*cos(y-x)) - ((m/2)*d^2*diff(x)^2*(y-x))/(a (m/2)*cos(y-x)) - ((m/2)*d*(3*g*sin(x) g*sin(y)))/(a m*d^2 (m/2)*d^2*cos(y-x)) - ((m/2)*d*g*sin(y))/(a (m/2)*cos(y-x))
eqn2=diff(y,2)== (gamma1*diff(x))/((m/2)*d^2*cos(y-x)) (gamma2*diff(y))/a - ( (m/2)*d^2*sin(y-x)*(diff(x)^2 - diff(y)^2))/((m/2)*d^2*cos(y-x)) - ((m/2)*d^2*diff(x)^2*(y-x))/a - ((m/2)*d*(3*g*sin(x) g*sin(y)))/((m/2)*d^2*cos(y-x)) - ((m/2)*d*g*sin(y))/a
V = odeToVectorField(eqn1,eqn2)
M = matlabFunction(V,'vars',{'t','Y'})
interval = [0 20];
xInit = [2 0];
yInit = [2 0];
ySol = ode45(M,interval,xInit, yInit);
tValues = linspace(0,20,100);
yValues = deval(ySol,tValues,1);
plot(tValues,yValues)
CodePudding user response:
Just to compare, without using symbolic expressions one would implement this equation as
function dV = M(t,V)
gamma1=0.1;
gamma2=0.1;
a=1;
m=1;
g=9.8;
d=1;
x = V(1); dx = V(2); y = V(3); dy = V(4);
ddx = (gamma1*dx)/(a m*d^2 (m/2)*d^2*cos(y-x)) (gamma2*dy)/(a (m/2)*cos(y-x)) - ( (m/2)*d^2*sin(y-x)*(dx^2 - dy^2))/(a m*d^2 (m/2)*d^2*cos(y-x)) - ((m/2)*d^2*dx^2*(y-x))/(a (m/2)*cos(y-x)) - ((m/2)*d*(3*g*sin(x) g*sin(y)))/(a m*d^2 (m/2)*d^2*cos(y-x)) - ((m/2)*d*g*sin(y))/(a (m/2)*cos(y-x));
ddy = (gamma1*dx)/((m/2)*d^2*cos(y-x)) (gamma2*dy)/a - ( (m/2)*d^2*sin(y-x)*(dx^2 - dy^2))/((m/2)*d^2*cos(y-x)) - ((m/2)*d^2*dx^2*(y-x))/a - ((m/2)*d*(3*g*sin(x) g*sin(y)))/((m/2)*d^2*cos(y-x)) - ((m/2)*d*g*sin(y))/a;
dV = [dx ddx dy ddy];
end%function
interval = [0 20];
xInit = [2 0];
yInit = [2 0];
vSol = ode45(M,interval,[ xInit yInit]);
tValues = linspace(0,20,100);
xValues = deval(vSol,tValues,1);
plot(tValues,xValues)
This works, but reports a singularity between t=0.244
and t=0.245
.