I am trying to implement a system of differential equations (equations of motion of a roll axis vehicle, which are part of a vehicle model) in Matlab but it does not work for me. I have figured out that what constrains me is the lack of understanding of these equations. That's what brings me here.
There are four equations with four dependent variables: velocities $u$ and $v$, angular velocity $r$, and the roll angle $\phi$. After introducing $\phi$ in the system the equations become confusing to me due to the interdependencies of the variables.
Parameters $\sum F_x, \sum F_y, \sum M_z $ come from the simulation and will be different at every iteration. The rest of the parameters are constant. The initial conditions are zero.
$\dot u=(\sum F_x + mrv + \dot r hm \phi + 2 \dot \phi hmr)/m$
$\dot v = (\sum F_y - \dot \phi hm - mru + \dot \phi hmr)/m $
$ \dot r = (\sum M_z + \ddot \phi(I_{xz} - I_z\theta_r) + hm\phi(\dot u - rv))/I_z $
$ \ddot \phi = -(\dot \phi k + \phi(c - ghm) - \dot r(I_{xz} - I_z \theta_r) + hm(\dot v + ru) - \phi r^2 (mh^2 + I_y - I_z))/(mh^2 + I_z) $
My question is: Is it possible to obtain a solution at any given time step? Please help me understand why it is or it is not.
My initial belief was that, given null initial conditions, the program should be able to resolve the algebraic loop, since there are independent variables in the system. The solver does not give any results so I lost confidence that it is possible to find a solution. I am sorry that I can't provide more input but I am just plain confused right now.