Implement Euler angle representation of six-degrees-of-freedom equations of motion of simple variable mass
Equations of Motion/6DOF
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. Vreb is the relative velocity in the body axes at which the mass flow () is ejected or added to the body in body axes.
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.
Specifies the input and output units:
|Newton||Newton meter||Meters per second squared||Meters per second||Meters||Kilogram||Kilogram meter squared|
|Pound||Foot pound||Feet per second squared||Feet per second||Feet||Slug||Slug foot squared|
|Pound||Foot pound||Feet per second squared||Knots||Feet||Slug||Slug foot squared|
Select the type of mass to use:
|Mass is constant throughout the simulation.|
|Mass and inertia vary linearly as a function of mass rate.|
|Mass and inertia variations are customizable.|
Simple Variable selection conforms to
the previously described equations of motion.
Select the representation to use:
|Use Euler angles within equations of motion.|
|Use quaternions within equations of motion.|
Euler Angles selection conforms to the
previously described equations of motion.
The three-element vector for the initial location of the body in the flat Earth reference frame.
The three-element vector for the initial velocity in the body-fixed coordinate frame.
The three-element vector for the initial Euler rotation angles [roll, pitch, yaw], in radians.
The three-element vector for the initial body-fixed angular rates, in radians per second.
The initial mass of the rigid body.
A scalar value for the empty mass of the body.
A scalar value for the full mass of the body.
A 3-by-3 inertia tensor matrix for the empty inertia of the body.
A 3-by-3 inertia tensor matrix for the full inertia of the body.
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.
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.
|Vector||Contains the three applied forces.|
|Vector||Contains the three applied moments.|
|Scalar||Contains one or more rates of change of mass.|
|Three-element vector||Contains one or more relative velocities at which the mass is accreted to or ablated from the body in body-fixed axes.|
|Three-element vector||Contains the velocity in the flat Earth reference frame.|
|Three-element vector||Contains the position in the flat Earth reference frame.|
|Three-element vector||Contains the Euler rotation angles [roll, pitch, yaw], in radians.|
|3-by-3 matrix||Applies to the coordinate transformation from flat Earth axes to body-fixed axes.|
|Three-element vector||Contains the velocity in the body-fixed frame.|
|Three-element vector||Contains the angular rates in body-fixed axes, in radians per second.|
|Three-element vector||Contains the angular accelerations in body-fixed axes, in radians per second squared.|
|Three-element vector||Contains the accelerations in body-fixed axes with respect to body frame.|
|Scalar element||Contains a flag for fuel tank status:|
|Three-element vector||Contains the accelerations in body-fixed axes with respect to inertial frame (flat Earth). You typically connect this signal to the accelerometer.|
The block assumes that the applied forces are acting at the center of gravity of the body.
Stevens, Brian, and Frank Lewis, Aircraft Control and Simulation, Second Edition, John Wiley & Sons, 2003.
Zipfel, Peter H., Modeling and Simulation of Aerospace Vehicle Dynamics. Second Edition, AIAA Education Series, 2007.