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

Equations of Motion/3DOF

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

$$\begin{array}{l}{A}_{xb}=\dot{u}=\frac{{F}_{x}}{m}-\frac{\dot{m}Ur{e}_{b}}{m}-qw-g\mathrm{sin}\theta \\ Vr{e}_{b}={\begin{array}{cc}[Ure& Wre]\end{array}}_{b}\\ {A}_{yb}=\dot{w}=\frac{{F}_{z}}{m}-\frac{\dot{m}Wr{e}_{b}}{m}+qu+g\mathrm{cos}\theta \\ \dot{q}=\frac{M-{\dot{I}}_{yy}q}{{I}_{yy}}\\ \dot{\theta}=q\\ {A}_{be}=\left[\begin{array}{c}{A}_{xe}\\ {A}_{ze}\end{array}\right]=\frac{{F}_{b}-\dot{m}Ur{e}_{b}}{m}-\overline{g}\\ {A}_{bb}=\left[\begin{array}{c}{A}_{xb}\\ {A}_{yb}\end{array}\right]\end{array}$$

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 ($$\dot{m}$$)
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.

**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 | Scalar | Contains the force acting along the body -axis,
(x ).F_{x} |

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

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

Fourth (Optional) | Vector | Contains the rate of change of mass, ($$\dot{m}$$) (positive if accreted, negative if ablated). |

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

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

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 $$(\dot{q})$$. |

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 |

Seventh (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. |

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