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

The Custom Variable Mass 3DOF (Body Axes) block considers the rotation in the vertical plane of a body-fixed coordinate frame about a flat Earth reference frame.

The equations of motion are

where the applied forces are assumed to act at the center of
gravity of the body. *Ure*_{b} and *Wre*_{b} are
the relative velocities of the mass flow (
)
being added to or ejected from the body in body-fixed 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 velocity**A scalar value for the initial velocity of the body, (

*V*_{0}).**Initial body attitude**A scalar value for the initial pitch attitude of the body, (

*θ*_{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.

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

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

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

Third | Scalar | Contains the applied pitch moment, (M). |

Fourth (Optional) | Vector | Contains the rate of change of mass, ( ) (positive if accreted, negative if ablated). |

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

Sixth | Scalar | Contains the rate of change of inertia tensor matrix, . |

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

Eighth (Optional) | Scalar | 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 body axes. |

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

First | Scalar | Contains the pitch attitude, in radians (θ). |

Second | Scalar | Contains the pitch angular rate, in radians per second (q). |

Third | Scalar | Contains the pitch angular acceleration, in radians per second squared . |

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 body-fixed
coordinate frame, (u, w). |

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

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