Implement wind angle representation of six-degrees-of-freedom equations of motion of custom variable mass

**Library:**Aerospace Blockset / Equations of Motion / 6DOF

The Custom Variable Mass 6DOF Wind (Wind Angles) block implements a wind angle representation of six-degrees-of-freedom equations of motion of custom variable mass. For a description of the coordinate system employed and the translational dynamics, see the block description for the Custom Variable Mass 6DOF Wind (Quaternion) block.

For more information of the relationship between the wind angles, see Algorithms

The block assumes that the applied forces act at the center of gravity of the body.

The relationship between the wind angles, [*μ γ
χ*]^{T}, can be determined by resolving the wind
rates into the wind-fixed coordinate frame.

$$\left[\begin{array}{l}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=\left[\begin{array}{l}\dot{\mu}\\ 0\\ 0\end{array}\right]+\left[\begin{array}{lll}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & \mathrm{cos}\mu \hfill & \mathrm{sin}\mu \hfill \\ 0\hfill & -\mathrm{sin}\mu \hfill & \mathrm{cos}\mu \hfill \end{array}\right]\left[\begin{array}{l}0\\ \dot{\gamma}\\ 0\end{array}\right]+\left[\begin{array}{lll}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & \mathrm{cos}\mu \hfill & \mathrm{sin}\mu \hfill \\ 0\hfill & -\mathrm{sin}\mu \hfill & \mathrm{cos}\mu \hfill \end{array}\right]\left[\begin{array}{lll}\mathrm{cos}\gamma \hfill & 0\hfill & -\mathrm{sin}\gamma \hfill \\ 0\hfill & 1\hfill & 0\hfill \\ \mathrm{sin}\gamma \hfill & 0\hfill & \mathrm{cos}\gamma \hfill \end{array}\right]\left[\begin{array}{l}0\\ 0\\ \dot{\chi}\end{array}\right]\equiv {J}^{-1}\left[\begin{array}{l}\dot{\mu}\\ \dot{\gamma}\\ \dot{\chi}\end{array}\right]$$

Inverting *J* then gives the required relationship to determine the
wind rate vector.

$$\left[\begin{array}{l}\dot{\mu}\\ \dot{\gamma}\\ \dot{\chi}\end{array}\right]=J\left[\begin{array}{l}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=\left[\begin{array}{lll}1\hfill & (\mathrm{sin}\mu \mathrm{tan}\gamma )\hfill & (\mathrm{cos}\mu \mathrm{tan}\gamma )\hfill \\ 0\hfill & \mathrm{cos}\mu \hfill & -\mathrm{sin}\mu \hfill \\ 0\hfill & \frac{\mathrm{sin}\mu}{\mathrm{cos}\gamma}\hfill & \frac{\mathrm{cos}\mu}{\mathrm{cos}\gamma}\hfill \end{array}\right]\left[\begin{array}{l}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]$$

The body-fixed angular rates are related to the wind-fixed angular rate by the following equation.

$$\left[\begin{array}{l}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=DM{C}_{wb}\left[\begin{array}{c}{p}_{b}-\dot{\beta}\mathrm{sin}\alpha \\ {q}_{b}-\dot{\alpha}\\ {r}_{b}+\dot{\beta}\mathrm{cos}\alpha \end{array}\right]$$

Using this relationship in the wind rate vector equations, gives the relationship between the wind rate vector and the body-fixed angular rates.

$$\left[\begin{array}{l}\dot{\mu}\\ \dot{\gamma}\\ \dot{\chi}\end{array}\right]=J\left[\begin{array}{l}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=\left[\begin{array}{lll}1\hfill & (\mathrm{sin}\mu \mathrm{tan}\gamma )\hfill & (\mathrm{cos}\mu \mathrm{tan}\gamma )\hfill \\ 0\hfill & \mathrm{cos}\mu \hfill & -\mathrm{sin}\mu \hfill \\ 0\hfill & \frac{\mathrm{sin}\mu}{\mathrm{cos}\gamma}\hfill & \frac{\mathrm{cos}\mu}{\mathrm{cos}\gamma}\hfill \end{array}\right]DM{C}_{wb}\left[\begin{array}{c}{p}_{b}-\dot{\beta}\mathrm{sin}\alpha \\ {q}_{b}-\dot{\alpha}\\ {r}_{b}+\dot{\beta}\mathrm{cos}\alpha \end{array}\right]$$

[1] Stevens, Brian, and Frank Lewis.
*Aircraft Control and Simulation*, 2nd ed. Hoboken, NJ: John
Wiley & Sons, 2003.

[2] Zipfel, Peter H. *Modeling
and Simulation of Aerospace Vehicle Dynamics*. 2nd ed: Reston, VA: AIAA
Education Series, 2007.

6DOF (Euler Angles) | 6DOF (Quaternion) | 6DOF ECEF (Quaternion) | 6DOF Wind (Quaternion) | 6DOF Wind (Wind Angles) | Custom Variable Mass 6DOF (Euler Angles) | Custom Variable Mass 6DOF (Quaternion) | Custom Variable Mass 6DOF ECEF (Quaternion) | Custom Variable Mass 6DOF Wind (Quaternion) | Simple Variable Mass 6DOF (Euler Angles) | Simple Variable Mass 6DOF (Quaternion) | Simple Variable Mass 6DOF ECEF (Quaternion) | Simple Variable Mass 6DOF Wind (Wind Angles)