3.3.2 Six DOF (6DOF) Solver Theory

The 6DOF solver in ANSYS FLUENT uses the object's forces and moments in order to compute the translational and angular motion of the center of gravity of an object. The governing equation for the translational motion of the center of gravity is solved for in the inertial coordinate system (Equation  3.3-17).

where $\dot{\overrightarrow{\nu_G}}$ is the translational motion of the center of gravity, $m$ is the mass, and $\overrightarrow{f}_G$ is the force vector due to gravity.

The angular motion of the object, $\dot{\overrightarrow{\omega_B}}$, is more easily computed using body coordinates (Equation  3.3-18).

where $\bf {L}$ is the inertia tensor, $\overrightarrow{M_{\rm B}}$ is the moment vector of the body, and $\overrightarrow{\omega_{\rm B}}$ is the rigid body angular velocity vector.

The moments are transformed from inertial to body coordinates using

where $\bf {R}$ is the following transformation matrix:

$C_\theta C_\psi$	$C_\theta S_\psi$	- $S_\theta$
$S_\phi S_\theta C_\psi - C_\phi S_\psi$	$S_\phi S_\theta S_\psi + C_\phi C_\psi$	$S_\phi C_\theta$
$C_\phi S_\theta C_\psi + S_\phi S_\psi$	$C_\phi S_\theta S_\psi - S_\phi C_\psi$	$C_\phi C_\theta$

where, in generic terms, $C_\chi = cos(\chi)$ and $S_\chi = sin(\chi)$. The angles $\phi$, $\theta$, and $\psi$ are Euler angles that represent the following sequence of rotations:

• rotation about the x-axis (e.g., roll for airplanes)

• rotation about the y-axis (e.g., pitch for airplanes)

• rotation about the z-axis (e.g., yaw for airplanes)

Once the angular and the translational accelerations are computed from Equation  3.3-17 and Equation  3.3-18, the rates are derived by numerical integration [ 328]. The angular and translational velocities are used in the dynamic mesh calculations to update the rigid body position.