# 7.1.4: Performances in a steady linear flight


• Consider a symmetric flight in the horizontal 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:4

$T = D,\label{eq7.1.4.1}$

$L = mg, (which\ implies \ n = 1),\label{eq7.1.4.2}$

$\dot{x}_e = V,\label{eq7.1.4.3}$

Recall the following expressions already exposed in Chapter 3:

• $$L = \tfrac{1}{2} \rho SV^2 C_L (\alpha ); C_L = C_{L_0} + C_{L_{\alpha}} \alpha,$$,
• $$D = \tfrac{1}{2} \rho SV^2 C_D (\alpha ); C_D = C_{D_0} + k C_L^2$$,
• $$E = \tfrac{L}{D} = \tfrac{C_L}{C_D} = \tfrac{C_L}{C_{D_0} + k C_L^2}$$, with $$E_{\max} = \tfrac{1}{2\sqrt{C_{D_0} k}}$$.5

Considering these expressions, System of equations ($$\ref{eq7.1.4.1}$$), ($$\ref{eq7.1.4.2}$$) and ($$\ref{eq7.1.4.3}$$) can be expressed as:

$T = \dfrac{1}{2} \rho S V^2 C_{D_0} + \dfrac{2k(mg)^2}{\rho SV^2},\label{eq7.1.4.4}$

$mg = \dfrac{1}{2} \rho SV^2 (C_{L_0} + C_{L_{\alpha}} \alpha ),\label{eq7.1.4.5}$

$\dot{x}_e = V.$

Expression ($$\ref{eq7.1.4.5}$$) says that in order to increase velocity it is necessary to reduce the angle of attack and vice-versa. Expression ($$\ref{eq7.1.4.4}$$) gives the two velocities at which an aircraft can fly for a given thrust.

4. $$n = \tfrac{L}{mg}$$ is referred to as load factor

5. remember that $$E_{\max}$$ refers to the maximum efficiency.

7.1.4: Performances in a steady linear flight 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.