12.3.1: Dynamic relations
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The dynamic model governing the movement of the aircraft is based on two fundamental theorems of the classical mechanics: the theorem of the quantity of movement and the theorem of the kinetic momentum:
The theorem of quantity of movement establishes that:
\[\vec{F} = \dfrac{d(m \vec{V})}{dt},\label{eq12.3.1.1}\]
where \(\vec{F}\) is the resulting of the external forces, \(\vec{V}\) is the absolute velocity of the aircraft (respect to a inertial reference frame), \(m\) is the mass of the aircraft, and \(t\) is the time.
For a conventional aircraft holds that the variation of its mass with respect to time is sufficiently slow so that the term \(\dot{m} \vec{V}\) Equation (\(\ref{eq12.3.1.1}\)) could be neglected.
The theorem of the kinematic momentum establishes that:
\[\vec{G} = \dfrac{d\vec{h}}{dt},\]
\[\vec{h} = I \vec{\omega},\]
where \(\vec{G}\) is the resulting of the external momentum around the center of gravity of the aircraft, \(\vec{h}\) is the absolute kinematic momentum of the aircraft, \(I\) is the tensor of inertia, and \(\vec{\omega}\) is the absolute angular velocity of the aircraft.
The tensor of inertia is defined as:
\[I = \begin{bmatrix} I_x & -I_{xy} & -J_{xz} \\ -J_{xy} & I_y & -J_{yz} \\ -J_{xz} & -J_{yz} & I_z \end{bmatrix}.\]
where \(I_x , I_y, I_z\) are the inertial momentums around the three axes of the reference system, and \(J_{xy}, J_{xz}, J_{yz}\) are the corresponding inertia products.
The resulting equations from both theorems can be projected in any reference system. In particular, projecting them into a body-axes reference frame (also to a wind-axes reference frame) have important advantages.
Given a inertial reference frame denoted by \(F_0\) and a non-inertial reference frame \(F_1\) whose related angular velocity is given by \(\vec{\omega}_{01}\), and given also a generic vector \(\vec{A}\), it holds:
\[\{ \dfrac{\partial \vec{A}}{\partial t} \}_1 = \{ \dfrac{\partial \vec{A}}{\partial t} \}_0 + \vec{\omega}_{01} \wedge \vec{A}_1\]
The three components expressed in a body-axes reference frame of the total force, the total momentum, the absolute velocity, and the absolute angular velocity are denoted by:
\[\vec{F} = (F_x, F_y, F_z)^T,\]
\[\vec{G} = (L, M, N)^T,\]
\[\vec{V} = (u, v, w)^T,\]
\[\vec{\omega} = (p, q, r)^T.\]
Therefore, the equations governing the motion of the aircraft are:
\[F_x = m(\dot{u} - rv + qw),\label{eq12.3.1.10}\]
\[F_y = m(\dot{v} + ru - pw),\label{eq12.3.1.11}\]
\[F_z = m(\dot{w} - qu + pv),\label{eq12.3.1.12}\]
\[L = I_x \dot{p} - J_{xz} \dot{r} + (I_z - I_y) qr - J_{xz} pq,\label{eq12.3.1.13}\]
\[M = I_y \dot{q} - (I_z - I_x) pr - J_{xz} (p^2 - r^2),\label{eq12.3.1.14}\]
\[N = I_z \dot{r} - J_{xz} \dot{p} + (I_x - I_y) pq - J_{xz} qr,\label{eq12.3.1.15}\]
System (\(\ref{eq12.3.1.10}\) - \(\ref{eq12.3.1.15}\)) is referred to as Euler equations of the movement of an aircraft.