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

Equations of Motion/3DOF

The Custom Variable Mass 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

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

where the applied forces are assumed to act at the center of
gravity of the body. *Vre*_{w} is
the relative velocity in the wind axes at which the mass flow ($$\dot{m}$$)
is ejected or added to the body in wind axes.

**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.**Initial airspeed**A scalar value for the initial velocity of the body, (

*V*_{0}).**Initial flight path angle**A scalar value for the initial pitch attitude 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.

**Gravity Source**Specify source of gravity:

`External`

Variable gravity input to block

`Internal`

Constant gravity specified in

**Acceleration due to gravity****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`

. This parameter appears if you set**Gravity source**to`Internal`

.**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.

**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) | Vector | Contains one or more rates of change of mass, $$(\dot{m})$$ (positive if accreted, negative if ablated). |

Fifth | Contains the mass, (m). | |

Sixth | Contains the rate of change of inertia tensor matrix, $$({\dot{I}}_{yy})$$. | |

Seventh | Contains the inertia tensor matrix, (I)._{yy} | |

Eighth (Optional) | Contains the gravity in the selected units. | |

Ninth (Optional) | Two-element vector | Contains one or more relative velocities at which the mass is accreted to or ablated from the body in wind axes. |

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

First | Contains the flight path angle, 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). |

Eight (Optional) | 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. |

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)

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