7.1.4: Performances in a steady linear flight
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- 78150
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.