2.4: System references

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The atmospheric flight mechanics uses different coordinates references to express the positions, velocities, accelerations, forces, and torques. Therefore, before going into the fundamentals of flight mechanics, it is useful to define some of the most important ones:

Definition 2.1 (Inertial Reference Frame)

According to classical mechanics, an inertial reference from \(F_I (O_I, x_I, y_I, z_I)\) is either a non accelerated frame with respect to a quasifixed reference star, or either a system which for a punctual mass is possible to apply the second Newton's law:

\[\sum \vec{F}_I = \dfrac{d (m \cdot \vec{V}_I)}{dt} \nonumber\]

Definition 2.2 (Earth Reference Frame)

An earth reference frame \(F_e (O_e, x_e, y_e, z_e)\) is a rotating topocentric (measured from the surface of the earth) system. The origin Oe is any point on the surface of earth defined by its latitude \(\theta_e\) and longitude \(\lambda_e\). Axis ze points to the center of earth; \(x_e\) lays in the horizontal plane and points to a fixed direction (typically north); \(y_e\) forms a right-handed thrihedral (typically east).

Such system is sometimes referred to as navigational system since it is very useful to represent the trajectory of an aircraft from the departure airport.

Theorem 2.2 Flat earth

The earth can be considered flat, non rotating, and approximate inertial reference frame. Consider \(F_I\) and \(F_e\). Consider the center of mass of the aircraft denoted by \(CG\). The acceleration of \(CG\) with respect to \(F_I\) can be written using the well-known formula of acceleration composition from the classical mechanics:

\[\vec{a}_I^{CG} = \vec{a}_e^{CG} + \vec{\Omega} \wedge (\vec{\Omega} \wedge \vec{r}_{OICG}) + 2 \vec{\Omega} \wedge \vec{V}_e^{CG},\]

where the centripetal acceleration \((\vec{\Omega} \wedge (\vec{\Omega} \wedge \vec{r}_{OICG}))\) and the Coriolis acceleration \((2 \vec{\Omega} \wedge \vec{V}_e^{CG})\) are neglectable if we consider typical values: \(\vec{\Omega}\) (the earth angular velocity) is one revolution per day; \(\vec{r}\) is the radius of earth plus the altitude (around 6380 [km]); \(\vec{V}_e^{CG}\) is the velocity of aircraft in flight (200-300 [m/s]). This means \(\vec{a}_I^{CG} \approx \vec{a}_e^{CG}\) and therefore \(F_e\) can be considered inertial reference frame.

Definition 2.3 (Local Horizon Frame)

A local horizon frame \(F_h (O_h, x_h, y_h, z_h)\) is a system of axes centered in any point of the symmetry plane (assuming there is one) of the aircraft, typically the center of gravity. Axes \((x_h, y_h, z_h)\) are defined parallel to axes \((x_e, y_e, z_e)\).

In atmospheric flight, this system can be considered as quasi-inertial.

Definition 2.4 (Body Axes Frame)

A body axes frame \(F_b (O_b, x_b, y_b, z_b)\) represents the aircraft as a rigid solid model. It is a system of axes centered in any point of the symmetry plane (assuming there is one) of the aircraft, typically the center of gravity. Axis \(x_b\) lays in to the plane of symmetry and it is parallel to a reference line in the aircraft (for instance, the zero-lift line), pointing forwards according to the movement of the aircraft. Axis \(z_b\) also lays in to the plane of symmetry, perpendicular to \(x_b\) and pointing down according to regular aircraft performance. Axis \(y_b\) is perpendicular to the plane of symmetry forming a right-handed thrihedral (\(y_b\) points then to the right wing side of the aircraft).

Definition 2.5 (Wind Axes Frame)

A wind axes frame \(F_w (O_w, x_w, y_w, z_w)\) is linked to the instantaneous aerodynamic velocity of the aircraft. It is a system of axes centered in any point of the symmetry plane (assuming there is one) of the aircraft, typically the center of gravity. Axis \(x_w\) points at each instant to the direction of the aerodynamic velocity of the aircraft \(\vec{V}\). Axis \(z_w\) lays in to the plane of symmetry, perpendicular to \(x_w\) and pointing down according to regular aircraft performance. Axis \(y_b\) forms a right-handed thrihedral.

Notice that if the aerodynamic velocity lays in to the plane of symmetry, \(y_w \equiv y_b\).