MATLAB: How to solve an equation of motion with the ode solver when the spring constant is dependent on the position of the solution of the equation of motion

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I have the following equation of motion of a mass m and a position dependable springconstant k(x):

m*x''(t)+k(x)*x(t)=0

Where k(x) is the spring constant that is dependable on the position of x at a the time step that x is solved.

I can solve this with the Forward Euler Method by substituting the position x(n) at time step n into k(x(n)) and calculating the next position x(n+1) ect.

But with the ode solver it seems I can't do this, because the solver continues from the first time step to the last without letting me do anything in between.

Now I am new to matlab and still a student, so i don't have a lot of experience with Matlab. I hope this is a simple problem that a more experienced Matlab user can help me to solve this problem.

Thanks,

Maarten

Best Answer

There is absolutely no problem in using MATLAB's ODE solvers to solve a problem like this. Say for example

k(x) = 1+x^2

Then you can solve your ODE

m*x''+(1+x^2)*x=0

just like this:

ode45(@(t,x) [x(2); -x(1)*(1 + x(1)^2)  ] ,[ 0 10 ] ,[ 1 0 ])

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MATLAB: How to solve this coupled ODE problem using ode solvers

If you want to use the odeNN functions you need to rewrite your 2 2nd-order ODEs into 4 1st-order ODEs, something like this:

function dxdydvxdvydt = your_eqs_of_motion(t,xyvxvy,other,parameters)
x = xyvxvy(1);  % If my ode-equations are complicated 
y = xyvxvy(2);  % I prefer to explicitly extract the
vx = xyvxvy(3); % variables into something that's
vy = xyvxvy(4); % easily human-readable
m = other;
Fx = force_functionx(t,x,y,vx,vy,other,parameters); % Adjust inputs as necessary
Fy = force_functiony(t,x,y,vx,vy,other,parameters);
% or possibly preferable:
% [Fx,Fy] = force_function_xny(t,x,y,vx,vy,other,parameters);
dxdydvxdvydt = [vx;vy;Fx/m;Fy/m];
end

Then you can integrate the equations of motion something like this:

m = 12; % you know what to set for this one
x0y0vx0vy0 = [1e-10,5.08e-5,0,0]; % Your initial conditions
t_span = [0,2*pi];                % a time-span of interest
[t_out,xyvxvy] = ode45(@(t,xyvxvy) your_eqs_of_motion(t,xyvxvy,m,[]),t_span,x0y0vx0vy0);

When integrating equations of motion you typically also have to carefully check that the constants of motion we know should be conserved (total energy, angular momentum, etc.) are reasonably conserved - the general ODE-integrating functions are not guaranteed to do this.

HTH

MATLAB: Motion Detection

Yes.

Best Answer

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