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# Simple Variable Mass 6DOF Wind (Wind Angles)

Implement wind angle representation of six-degrees-of-freedom equations of motion of simple 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 Simple Variable Mass 6DOF (Quaternion) block.

The relationship between the wind angles, [$\mu \gamma \chi$]T, can be determined by resolving the wind rates into the wind-fixed coordinate frame.

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

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.

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

## Dialog Box

Units

Specifies the input and output units:

UnitsForcesMomentAccelerationVelocityPositionMassInertia
Metric (MKS)NewtonNewton meterMeters per second squaredMeters per secondMetersKilogramKilogram meter squared
English (Velocity in ft/s)PoundFoot poundFeet per second squaredFeet per secondFeetSlugSlug foot squared
English (Velocity in kts)PoundFoot poundFeet per second squaredKnotsFeetSlugSlug 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 Simple 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.

Initial mass

The initial mass of the rigid body.

Empty mass

A scalar value for the empty mass of the body.

Full mass

A scalar value for the full mass of the body.

Empty inertia matrix

A 3-by-3 inertia tensor matrix for the empty inertia of the body, in body-fixed axes.

Full inertia matrix

A 3-by-3 inertia tensor matrix for the full inertia of the body, in body-fixed axes.

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

Scalar or vectorContains one or more rates of change of mass. This value is positive if the mass is added (accreted) to the body in wind axes. It is negative if the mass is ejected (ablated) from the body in wind axes.

Fourth (Optional)

Three-element vectorContains one or more relative velocities at which the mass is accreted to or ablated from the body in wind axes.

OutputDimension TypeDescription

First

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

Second

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

Third

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

Fourth

3-by-3 matrixApplies 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 vectorContains the angular rates in body-fixed axes, in radians per second.

Ninth

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

Tenth

Three-element vectorContains the accelerations in body-fixed axes.

Eleventh

Scalar elementContains a flag for fuel tank status:
• 1 indicates that the tank is full.

• 0 indicates that the integral is neither full nor empty.

• -1 indicates that the tank is empty.

## Assumptions and Limitations

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

## References

Mangiacasale, L., Flight Mechanics of a μ-Airplane with a MATLAB Simulink Helper, Edizioni Libreria CLUP, Milan, 1998.

Stevens, B. L., and F. L. Lewis, Aircraft Control and Simulation, John Wiley & Sons, New York, 1992.