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# 7.1.7: Performances in turn maneuvers


Horizontal stationary turn

Consider the additional hypotheses:

• Consider a symmetric flight in the horizontal plane.
• There is no wind.
• The mass and the velocity of the aircraft are constant.

The 3DOF equations governing the motion of the airplane are:

$T = D,\label{eq7.1.7.1}$

$m V \dot{\chi} = L \sin \mu ,\label{eq7.1.7.2}$

$L \cos \mu = mg,\label{eq7.1.7.3}$

$\dot{x}_e = V \cos \chi, \label{eq7.1.7.4}$

$\dot{y}_e = V \sin \chi.\label{eq7.1.7.5}$

In a uniform (stationary) circular movement, it is well known that the tangential velocity is equal to the angular velocity ($$\dot{\chi}$$) multiplied by the radius of turn $$(R)$$:

$V = \dot{\chi} R.$

Therefore, System ($$ref{eq7.1.7.1}$$, $$ref{eq7.1.7.2}$$, $$ref{eq7.1.7.3}$$, $$ref{eq7.1.7.4}$$, $$ref{eq7.1.7.5}$$) can be rewritten as:

$T = \dfrac{1}{2} \rho SC_{D_0} + \dfrac{2kn^2 (mg)^2}{\rho V^2 S},$

$n \sin \mu = \dfrac{V^2}{gR},$

$n = \dfrac{1}{\cos \mu} \to n > 1,$

$\dot{x}_e = V \cos \chi,$

$\dot{y}_e = V \sin \chi.$

where $$n = \dfrac{L}{mg}$$ is the load factor. Notice that the load factor and the bank angle are $$mg$$ inversely proportional, that is, if one increases the other reduces and vice versa, until the bank angle reaches $$90^{\circ}$$, where the load factor is infinity.

The stall speed in horizontal turn is defined as:

$V_S = \sqrt{\dfrac{2mg}{\rho S C_{L_{\max}}} \dfrac{1}{\cos \mu}}$

Ideal looping

The ideal looping is a circumference of radius R into a vertical plane performed at constant velocity. Consider then the following additional hypotheses:

• Consider a symmetric flight in the vertical plane.
• $$\chi$$ can be considered constant.
• The aircraft performs a leveled wing flight, i.e., $$\mu = 0$$.
• There is no wind.
• The mass and the velocity of the aircraft are constant.

The 3DOF equations governing the motion of the airplane are:

$T = D + mg \sin \gamma,$

$L = mg \cos \gamma + mV \dot{\gamma},$

$\dot{x}_e = V \cos \gamma,$

$\dot{h}_e = V \sin \gamma,$

Figure 7.4: Aircraft forces in a vertical loop.

In a uniform (stationary) circular movement, it is well known that the tangential velocity is equal to the angular velocity ($$\dot{\gamma}$$ in this case) multiplied by the radius of turn ($$R$$):

$V = \dot{\gamma} R.$

The load factor and the coefficient of lift in this case are:

$n = \cos \gamma + \dfrac{V^2}{gR},$

$C_L = \dfrac{2mg}{\rho V^2 S} (\cos \gamma + \dfrac{V^2}{gR}).$

Notice that the load factor vaires in a sinusoidal way along the loop, reaching a maximum value at the superior point ($$n_{\max} = 1 + \tfrac{V^2}{gR})$$ and a minimum value at the inferior point ($$n_{\min} = \tfrac{V^2}{gR} - 1$$).

7.1.7: Performances in turn maneuvers 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.

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