Implement wind angle representation of sixdegreesoffreedom equations of motion
Aerospace Blockset / Equations of Motion / 6DOF
The 6DOF Wind (Wind Angles) block implements a wind angle representation of sixdegreesoffreedom equations of motion. For a description of the coordinate system employed and the translational dynamics, see the block description for the 6DOF Wind (Quaternion) block.
For more information on the relationship between the wind angles, see Algorithms
The block assumes that the applied forces act at the center of gravity of the body, and that the mass and inertia are constant.
The relationship between the wind angles $${[\mu \gamma \chi ]}^{\text{T}}$$ can be determined by resolving the wind rates into the windfixed coordinate frame.
$$\left[\begin{array}{c}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=\left[\begin{array}{c}\dot{\mu}\\ 0\\ 0\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\mu & \mathrm{sin}\mu \\ 0& \mathrm{sin}\mu & \mathrm{cos}\mu \end{array}\right]\left[\begin{array}{c}0\\ \dot{\gamma}\\ 0\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\mu & \mathrm{sin}\mu \\ 0& \mathrm{sin}\mu & \mathrm{cos}\mu \end{array}\right]\left[\begin{array}{ccc}\mathrm{cos}\gamma & 0& \mathrm{sin}\gamma \\ 0& 1& 0\\ \mathrm{sin}\gamma & 0& \mathrm{cos}\gamma \end{array}\right]\left[\begin{array}{c}0\\ 0\\ \dot{\chi}\end{array}\right]\equiv {J}^{1}\left[\begin{array}{c}\dot{\mu}\\ \dot{\gamma}\\ \dot{\chi}\end{array}\right]$$

Inverting J then gives the required relationship to determine the wind rate vector.
$$\left[\begin{array}{c}\dot{\mu}\\ \dot{\gamma}\\ \dot{\chi}\end{array}\right]=J\left[\begin{array}{c}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=\left[\begin{array}{ccc}1& (\mathrm{sin}\mu \mathrm{tan}\gamma )& (\mathrm{cos}\mu \mathrm{tan}\gamma )\\ 0& \mathrm{cos}\mu & \mathrm{sin}\mu \\ 0& \frac{\mathrm{sin}\mu}{\mathrm{cos}\gamma}& \frac{\mathrm{cos}\mu}{\mathrm{cos}\gamma}\end{array}\right]\left[\begin{array}{c}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]$$
The bodyfixed angular rates are related to the windfixed angular rate by the following equation.
$$\left[\begin{array}{c}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=DM{C}_{wb}\left[\begin{array}{c}{p}_{b}\dot{\beta}\mathrm{sin}\alpha \\ {q}_{b}\dot{\alpha}\\ {r}_{b}+\dot{\beta}\mathrm{cos}\alpha \end{array}\right]$$
Using this relationship in the wind rate vector equations, gives the relationship between the wind rate vector and the bodyfixed angular rates.
$$\left[\begin{array}{c}\dot{\mu}\\ \dot{\gamma}\\ \dot{\chi}\end{array}\right]=J\left[\begin{array}{c}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=\left[\begin{array}{ccc}1& (\mathrm{sin}\mu \mathrm{tan}\gamma )& (\mathrm{cos}\mu \mathrm{tan}\gamma )\\ 0& \mathrm{cos}\mu & \mathrm{sin}\mu \\ 0& \frac{\mathrm{sin}\mu}{\mathrm{cos}\gamma}& \frac{\mathrm{cos}\mu}{\mathrm{cos}\gamma}\end{array}\right]DM{C}_{wb}\left[\begin{array}{c}{p}_{b}\dot{\beta}\mathrm{sin}\alpha \\ {q}_{b}\dot{\alpha}\\ {r}_{b}+\dot{\beta}\mathrm{cos}\alpha \end{array}\right]$$
[1] Stevens, Brian, and Frank Lewis. Aircraft Control and Simulation. New York: John Wiley & Sons, 1992.
6DOF (Euler Angles)  6DOF (Quaternion)  6DOF ECEF (Quaternion)  6DOF Wind (Quaternion)  Custom Variable Mass 6DOF (Euler Angles)  Custom Variable Mass 6DOF (Quaternion)  Custom Variable Mass 6DOF ECEF (Quaternion)  Custom Variable Mass 6DOF Wind (Quaternion)  Custom Variable Mass 6DOF Wind (Wind Angles)  Simple Variable Mass 6DOF (Euler Angles)  Simple Variable Mass 6DOF (Quaternion)  Simple Variable Mass 6DOF ECEF (Quaternion)  Simple Variable Mass 6DOF Wind (Quaternion)  Simple Variable Mass 6DOF Wind (Wind Angles)