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# 3DOF (Body Axes)

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

## Library

Equations of Motion/3DOF

## Description

The 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

$\begin{array}{l}\stackrel{˙}{u}=\frac{{F}_{x}}{m}-qw-g\mathrm{sin}\theta \\ \stackrel{˙}{w}=\frac{{F}_{z}}{m}+qu+g\mathrm{cos}\theta \\ \stackrel{˙}{q}=\frac{M}{{I}_{yy}}\\ \stackrel{˙}{\theta }=q\end{array}$

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

## Dialog Box

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 velocity

A scalar value for the initial velocity of the body, (V0 ).

Initial body attitude

A scalar value for the initial pitch attitude of the body, (θ0 ).

Initial incidence

A scalar value for the initial angle between the velocity vector and the body, (α0 ).

Initial body rotation rate

A scalar value for the initial body rotation rate, (q0 ).

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

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.

## Inputs and Outputs

InputDimension TypeDescription

First

Contains the force acting along the body x-axis, (Fx ).

Second

Contains the force acting along the body z-axis, (Fz ).

Third

Contains the applied pitch moment, (M).

Fourth (Optional)

Contains the block is gravity in the selected units.

OutputDimension TypeDescription

First

Contains the pitch attitude, in radians (θ).

Second

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

Third

Contains the pitch angular acceleration, in radians per second squared $\left(\stackrel{˙}{q}\right)$.

Fourth

Two-element vectorContains the location of the body, in the flat Earth reference frame, (Xe, Ze).

Fifth

Two-element vectorContains the velocity of the body resolved into the body-fixed coordinate frame, (u, w).

Sixth

Two-element vectorContains the acceleration of the body resolved into the body-fixed coordinate frame, (Ax, Az).

## Examples

See aero_guidanceaero_guidance for an example of this block.