# Simple Variable Mass 6DOF Wind (Wind Angles)

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

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

## Description

The Simple Variable Mass 6DOF Wind (Wind Angles) block implements a wind angle representation of six-degrees-of-freedom equations of motion of simple variable mass. For more information of the relationship between the wind angles, see Algorithms. For a description of the coordinate system employed and the translational dynamics, see the block description for the Simple Variable Mass 6DOF (Quaternion) block.

## Limitations

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

## Ports

### Input

### Output

## Parameters

## Algorithms

The relationship between the wind angles, [$$\mu \gamma \chi $$]^{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.

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

## References

[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.

## Extended Capabilities

## See Also

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

**Introduced in R2006a**