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Implement three-degrees-of-freedom equations of motion with respect to wind axes

Equations of Motion/3DOF

The 3DOF (Wind Axes) block considers the rotation in the vertical plane of a wind-fixed coordinate frame about a flat Earth reference frame.

The equations of motion are

$$\begin{array}{l}\dot{V}=\frac{{F}_{{x}_{wind}}}{m}-g\mathrm{sin}\gamma \\ \dot{\alpha}=\frac{{F}_{{z}_{wind}}}{mV\mathrm{cos}\beta}+q+\frac{g}{V\mathrm{cos}\beta}\mathrm{cos}\gamma \\ \dot{q}=\dot{\theta}=\frac{{M}_{{y}_{body}}}{{I}_{yy}}\\ \dot{\gamma}=q-\dot{\alpha}\\ {A}_{be}=\left[\begin{array}{c}{A}_{xe}\\ {A}_{ze}\end{array}\right]=DC{M}_{wb}\left[\frac{{F}_{w}}{m}-g\mathrm{sin}\gamma \right]\\ {A}_{bb}=\left[\begin{array}{c}{\dot{u}}_{b}\\ {\dot{w}}_{b}\end{array}\right]=DC{M}_{wb}\left[\frac{{F}_{w}}{m}-g\mathrm{sin}\gamma -{\overline{\omega}}_{w}\times {\overline{V}}_{w}\right]\\ {\overline{F}}_{w}=\left[\begin{array}{c}{F}_{{x}_{wind}}\\ {F}_{{z}_{wind}}\end{array}\right],{\overline{V}}_{w}=\left[\begin{array}{c}{V}_{{x}_{wind}}\\ {V}_{z}{}_{{}_{wind}}\end{array}\right],{\overline{\omega}}_{w}=q\end{array}$$

where the applied forces are assumed to act at the center of gravity of the body.

**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

`Fixed`

selection conforms to the previously described equations of motion.**Initial airspeed**A scalar value for the initial velocity of the body, (

*V*_{0}).**Initial flight path angle**A scalar value for the initial flight path angle of the body, (

*γ*_{0}).**Initial incidence**A scalar value for the initial angle between the velocity vector and the body, (

*α*_{0}).**Initial body rotation rate**A scalar value for the initial body rotation rate, (

*q*_{0}).**Initial position (x,z)**A two-element vector containing the initial location of the body in the flat Earth reference frame.

**Initial Mass**A scalar value for the mass of the body.

**Inertia body axes**A scalar value for the inertia of the body.

**Gravity Source**Specify source of gravity:

`External`

Variable gravity input to block `Internal`

Constant gravity specified in mask **Acceleration due to gravity**A scalar value for the acceleration due to gravity used if internal gravity source is selected. If gravity is to be neglected in the simulation, this value can be set to 0.

**Include inertial acceleration**Select this check box to enable an additional output port for the accelerations in body-fixed axes with respect to the inertial frame. You typically connect this signal to the accelerometer.

Input | Dimension Type | Description |
---|---|---|

First | Contains the force acting along the wind x-axis,
(F)._{x} | |

Second | Contains the force acting along the wind z-axis,
(F)._{z} | |

Third | Contains the applied pitch moment in body axes, (M). | |

Fourth (Optional) | Contains the block is gravity in the selected units. |

Output | Dimension Type | Description |
---|---|---|

First | Contains the flight path angle, within ±pi, in radians (γ). | |

Second | Contains the pitch angular rate, in radians per second (ω)._{y} | |

Third | Contains the pitch angular acceleration, in radians per second
squared (dω/_{y}dt). | |

Fourth | Two-element vector | Contains the location of the body, in the flat Earth reference
frame, (Xe, Ze). |

Fifth | Two-element vector | Contains the velocity of the body resolved into the wind-fixed
coordinate frame, (V, 0). |

Sixth | Two-element vector | Contains the acceleration of the body resolved into the body-fixed
coordinate frame, (Ax, Az). |

Seventh | Scalar | Contains the angle of attack, (α). |

Eighth | Two-element vector | Contains the accelerations in body-fixed axes with respect to inertial frame (flat Earth). You typically connect this signal to the accelerometer. |

The block assumes that the applied forces are acting at the center of gravity of the body, and that the mass and inertia are constant.

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

4th Order Point Mass (Longitudinal)

Custom Variable Mass 3DOF (Body Axes)

Custom Variable Mass 3DOF (Wind Axes)

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