# 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:

##### Theorem 12.1 Quantity of movement

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.

##### Remark 12.3

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.

##### Theorem 12.2 Kinematic momentum

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.

##### Definition 12.8 Tensor of Inertia

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.

##### Theorem 12.3 Field of velocities

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.

12.3.1: Dynamic relations is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by Manuel Soler Arnedo via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.