# 12.1: Reference frames

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##### Definition 12.1: Inertial Reference Frame

According to classical mechanics, a inertial reference frame $$F_I (O_I , x_I , y_I , z_I)$$ is either a non accelerated frame with respect to a quasi-fixed reference star, or either a system which for a punctual mass is possible to apply the second Newton’s law:

$\sum \vec{F}_1 = \dfrac{d(m \cdot \vec{V}_I)}{dt}\nonumber$

##### Definition 12.2: 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 $$z_e$$ 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 it is sometimes referred to as navigational system since it is very useful to represent the trajectory of an aircraft from the departure airport.

##### Hypothesis 12.1: 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 $$F_1$$ can be written using the well-known formula of acceleration composition from the classical mechanics:

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

where the centripetal acceleration and the Coriolis acceleration 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 the aircraft in flight (200-300 [m/s]). This means $$\vec{a}_I^{CG} = \vec{a}_e^{CG}$$ and therefore $$F_e$$ can be considered inertial reference frame.

##### Definition 12.3: 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 $$(x_h, y_h, z_h)$$.

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

##### Definition 12.4: 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 xb 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 xb 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 the right wing side of the aircraft).

##### Definition 12.5: 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, $$y_w \equiv y_b$$.

12.1: Reference frames 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.