12.3.1: Dynamic relations
- Page ID
- 78411
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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:
\[\vec{G} = \dfrac{d\vec{h}}{dt}, \nonumber \]
\[\vec{h} = I \vec{\omega}, \nonumber \]
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:
\[I = \begin{bmatrix} I_x & -I_{xy} & -J_{xz} \\ -J_{xy} & I_y & -J_{yz} \\ -J_{xz} & -J_{yz} & I_z \end{bmatrix}. \nonumber \]
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:
\[\{ \dfrac{\partial \vec{A}}{\partial t} \}_1 = \{ \dfrac{\partial \vec{A}}{\partial t} \}_0 + \vec{\omega}_{01} \wedge \vec{A}_1 \nonumber \]
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:
\[\vec{F} = (F_x, F_y, F_z)^T, \nonumber \]
\[\vec{G} = (L, M, N)^T, \nonumber \]
\[\vec{V} = (u, v, w)^T, \nonumber \]
\[\vec{\omega} = (p, q, r)^T. \nonumber \]
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.