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# Simple Variable Mass 6DoF (Euler Angles)

Implement Euler angle representation of six-degrees-of-freedom equations of motion of simple variable mass

## Library

Equations of Motion/6DoF

## Description

The Simple Variable Mass 6DoF (Euler Angles) block considers the rotation of a body-fixed coordinate frame (Xb, Yb, Zb) about a flat Earth reference frame (Xe, Ye, Ze). The origin of the body-fixed coordinate frame is the center of gravity of the body, and the body is assumed to be rigid, an assumption that eliminates the need to consider the forces acting between individual elements of mass. The flat Earth reference frame is considered inertial, an excellent approximation that allows the forces due to the Earth's motion relative to the "fixed stars" to be neglected.

The translational motion of the body-fixed coordinate frame is given below, where the applied forces [Fx Fy Fz]T are in the body-fixed frame.

The rotational dynamics of the body-fixed frame are given below, where the applied moments are [L M N]T, and the inertia tensor I is with respect to the origin O.

The inertia tensor is determined using a table lookup which linearly interpolates between Ifull and Iempty based on mass (m). While the rate of change of the inertia tensor is estimated by the following equation.

The relationship between the body-fixed angular velocity vector, [p q r]T, and the rate of change of the Euler angles, [ ]T, can be determined by resolving the Euler rates into the body-fixed coordinate frame.

Inverting J then gives the required relationship to determine the Euler rate vector.

## Dialog Box

Units

Specifies the input and output units:

UnitsForcesMomentAccelerationVelocityPositionMassInertia
Metric (MKS)NewtonNewton meterMeters per second squaredMeters per secondMetersKilogramKilogram meter squared
English (Velocity in ft/s)PoundFoot poundFeet per second squaredFeet per secondFeetSlugSlug foot squared
English (Velocity in kts)PoundFoot poundFeet per second squaredKnotsFeetSlugSlug 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 Simple Variable selection conforms to the previously described equations of motion.

Representation

Select the representation to use:

 Euler Angles Use Euler angles within equations of motion. Quaternion Use quaternions within equations of motion.

The Euler Angles selection conforms to the previously described equations of motion.

Initial position in inertial axes

The three-element vector for the initial location of the body in the flat Earth reference frame.

Initial velocity in body axes

The three-element vector for the initial velocity in the body-fixed coordinate frame.

Initial Euler rotation

The three-element vector for the initial Euler rotation angles [roll, pitch, yaw], in radians.

Initial body rotation rates

The three-element vector for the initial body-fixed angular rates, in radians per second.

Initial mass

The initial mass of the rigid body.

Empty mass

A scalar value for the empty mass of the body.

Full mass

A scalar value for the full mass of the body.

Empty inertia matrix

A 3-by-3 inertia tensor matrix for the empty inertia of the body.

Full inertia matrix

A 3-by-3 inertia tensor matrix for the full inertia of the body.

## Inputs and Outputs

InputDimension TypeDescription

First

VectorContains the three applied forces.

Second

VectorContains the three applied moments.

Third

ScalarContains the rate of change of mass.

OutputDimension TypeDescription

First

Three-element vectorContains the velocity in the flat Earth reference frame.

Second

Three-element vectorContains the position in the flat Earth reference frame.

Third

Three-element vectorContains the Euler rotation angles [roll, pitch, yaw], in radians.

Fourth

3-by-3 matrixApplies to the coordinate transformation from flat Earth axes to body-fixed axes.

Fifth

Three-element vectorContains the velocity in the body-fixed frame.

Sixth

Three-element vectorContains the angular rates in body-fixed axes, in radians per second.

Seventh

Three-element vectorContains the angular accelerations in body-fixed axes, in radians per second squared.

Eight

Three-element vectorContains the accelerations in body-fixed axes.

Ninth

Scalar elementContains a flag for fuel tank status:
• 1 indicates that the tank is full.

• 0 indicates that the integral is neither full nor empty.

• -1 indicates that the tank is empty.

## Assumptions and Limitations

The block assumes that the applied forces are acting at the center of gravity of the body.

## Reference

Mangiacasale, L., Flight Mechanics of a μ-Airplane with a MATLAB Simulink Helper, Edizioni Libreria CLUP, Milan, 1998.

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