# Custom Variable Mass 6DOF Wind (Wind Angles)

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

## Library

Equations of Motion/6DOF

## Description

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.

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

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

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

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

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

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

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

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

## Parameters

Units

Specifies the input and output units:

Units

Forces

Moment

Acceleration

Velocity

Position

Mass

Inertia

`Metric (MKS)`

Newton

Newton meter

Meters per second squared

Meters per second

Meters

Kilogram

Kilogram meter squared

`English (Velocity in ft/s)`

Pound

Foot pound

Feet per second squared

Feet per second

Feet

Slug

Slug foot squared

`English (Velocity in kts)`

Pound

Foot pound

Feet per second squared

Knots

Feet

Slug

Slug foot squared

Mass Type

Select the type of mass to use:

 `Fixed` Mass is constant throughout the simulation. `Simple Variable` Mass and inertia vary linearly as a function of mass rate. `Custom Variable` Mass and inertia variations are customizable.

The `Custom Variable` selection conforms to the previously described equations of motion.

Representation

Select the representation to use:

 `Wind Angles` Use wind angles within equations of motion. `Quaternion` Use quaternions within equations of motion.

The `Wind Angles` selection conforms to the previously described equations of motion.

Initial position in inertial axes

The three-element vector for the initial location of the body in the flat Earth reference frame.

Initial airspeed, sideslip angle, and angle of attack

The three-element vector containing the initial airspeed, initial sideslip angle and initial angle of attack.

Initial wind orientation

The three-element vector containing the initial wind angles [bank, flight path, and heading], in radians.

Initial body rotation rates

The three-element vector for the initial body-fixed angular rates, in radians per second.

Include mass flow relative velocity

Select this check box to add a mass flow relative velocity port. This is the relative velocity at which the mass is accreted or ablated.

## Inputs and Outputs

InputDimension TypeDescription

First

VectorContains the three applied forces in wind-fixed axes.

Second

VectorContains the three applied moments in body-fixed axes (+/-).

Third (Optional)

VectorContains one or more rates of change of mass (positive if accreted, negative if ablated).

Fourth

ScalarContains the mass.

Fifth

3-by-3 matrix Applies to the rate of change of inertia tensor matrix in body-fixed axes.

Sixth

3-by-3 matrixApplies to the inertia tensor matrix in body-fixed axes.

Seventh (Optional)

1-by-1-by-m arrayContains one or more relative velocities at which the mass is accreted to or ablated from the body in wind axes. m is three times the size of the third input vector.

OutputDimension TypeDescription

First

Three-element vectorContains the velocity in the flat Earth reference frame.

Second

Three-element vector Contains the position in the flat Earth reference frame.

Third

Three-element vector Contains the wind rotation angles [bank, flight path, heading], in radians.

Fourth

3-by-3 matrix Applies to the coordinate transformation from flat Earth axes to wind-fixed axes.

Fifth

Three-element vectorContains the velocity in the wind-fixed frame.

Sixth

Two-element vectorContains the angle of attack and sideslip angle, in radians.

Seventh

Two-element vectorContains the rate of change of angle of attack and rate of change of sideslip angle, in radians per second.

Eighth

Three-element vector Contains the angular rates in body-fixed axes, in radians per second.

Ninth

Three-element vector Contains the angular accelerations in body-fixed axes, in radians per second squared.

Tenth

Three-element vector Contains the accelerations in body-fixed axes.

## Assumptions and Limitations

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

## References

Stevens, Brian, and Frank Lewis, Aircraft Control and Simulation, Second Edition, John Wiley & Sons, 2003.

Zipfel, Peter H., Modeling and Simulation of Aerospace Vehicle Dynamics. Second Edition, AIAA Education Series, 2007.