Skip to main content
Engineering LibreTexts

7.1.4: Performances in a steady linear flight

  • Page ID
    78150
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    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.


    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.