Dynamical Simulation - Euler Model

Euler Model

The inertial model is much more complex than we typically need but it is the most simple to use. In this model, we do not need to change our forces or constrain our system. However, if we make a few intelligent changes to our system, simulation will become much easier, and our calculation time will decrease. The first constraint will be to put each torque in terms of the principal axes. This makes each torque much more difficult to program, but it simplifies our equations significantly. When we apply this constraint, we diagonalize the moment of inertia tensor, which simplifies our three equations into a special set of equations called Euler's equations. These equations describe all rotational momentum in terms of the principal axes:

$\begin{matrix} I_1\dot{\omega}_{1}+(I_3-I_2)\omega_2\omega_3 &=& N_{1}\\ I_2\dot{\omega}_{2}+(I_1-I_3)\omega_3\omega_1 &=& N_{2}\\ I_3\dot{\omega}_{3}+(I_2-I_1)\omega_1\omega_2 &=& N_{3} \end{matrix}$

The N terms are applied torques about the principal axes
The I terms are the principal moments of inertia
The terms are angular velocities about the principal axes

The drawback to this model is that all the computation is on the front end, so it is still slower than we would like. The real usefulness is not apparent because it still relies on a system of non-linear differential equations. To alleviate this problem, we have to find a method that can remove the second term from the equation. This will allow us to integrate much more easily. The easiest way to do this is to assume a certain amount of symmetry.

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