12.4.3: Kinematic relations

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Hypothesis 12.8 Moving Atmosphere

The atmosphere is considered moving, i.e., wind is taken into consideration. Vertical component is neglected due its low influence. Only kinematic effects are considered, i.e., dynamic effects of wind are also neglected due its low influence. The wind velocity \(\vec{W}\) can be expressed in local horizon axes as:

\[\vec{W}_h = \begin{bmatrix} W_x \\ W_y \\ 0 \end{bmatrix}\]

Considering the earth axes reference system as a inertial system, and assuming that earth axes are parallel to local horizon axes, the absolute velocity \(\vec{V}^G = \vec{V}^A + \vec{W}\) can be expressed referred to a wind axes reference as follows:

\[\vec{V}_e^G = \begin{bmatrix} \dot{x}_e \\ \dot{y}_e \\ \dot{z}_e \end{bmatrix} = L_{hw} \vec{V}_w^A + \vec{W}_h = L_{hw} \begin{bmatrix} u \\ v \\ w \end{bmatrix} + \begin{bmatrix} W_x \\ W_y \\ 0 \end{bmatrix} = L_{wh}^T \begin{bmatrix} V \\ 0 \\ 0\end{bmatrix} + \begin{bmatrix} W_x \\ W_y \\ 0\end{bmatrix}.\]

Remark 12.4

Notice that the absolute aerodynamic velocity \(\vec{V}^A\) expressed in a wind axes reference frame is \((V, 0, 0)\).

Therefore, the kinematic relations are as follows:

\[\dot{x}_e = V \cos \gamma \cos \chi + W_x,\label{eq12.4.3.3}\]

\[\dot{y}_e = V \cos \gamma \sin \chi + W_y,\]

\[\dot{z}_e = -V \sin \gamma,\label{eq12.4.3.5}\]

Equations (\(\ref{eq12.4.3.3}\))-(\(\ref{eq12.4.3.5}\)) provide the movement law and the trajectory of the aircraft can be determined.

Notice that Equation (\(\ref{eq12.4.3.5}\)) is usually rewritten as

\[\dot{h}_e = V \sin \gamma,\]

Remark 12.5

If one wants to model a flight over a spherical earth, since the radius of earth is sufficiently big and the angular velocity of earth is sufficiently small, it holds that the rotation of earth has very low influence in the centripetal acceleration and it is thus neglectable. Therefore, the hypothesis of flat earth holds in the dynamics of an aircraft moving over an spherical earthwith the following kinematic relations:

\[\dot{\lambda}_e = \dfrac{V \cos \gamma \cos \chi + W_x}{(R + h) \cos \theta},\]

\[\dot{\theta}_e = \dfrac{V \cos \gamma \sin \chi + W_y}{R + h},\]

\[\dot{h}_e = V \sin \gamma,\]

where \(\lambda\) and \(\theta\) are respectively the longitude and latitude and \(R\) is the radius of earth.