7.1.4: Performances in a steady linear flight

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Consider the additional hypotheses:

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.