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7.1.3: Aircraft equations of motion

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    78149
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    3D motion3

    Under Hypotheses 7.1-7.10, the 3DOF equations governing the translational 3D motion of an airplane are the following:

    • 3 dynamic equations relating forces to translational acceleration.

    • 3 kinematic equations giving the translational position relative to an Earth reference frame.

    • 1 equation defining the variable-mass characteristics of the airplane versus time.

    The equation of motion is hence defined by the following Ordinary Differential Equations (ODE) system:

    Definition 7.3 (3DOF equations of 3D motion)

    \[m \dot{V} = T - D - mg \sin \gamma;\]

    \[m V \dot{\chi} \cos \gamma = L \sin \mu;\]

    \[m V \dot{\gamma} = L \cos \mu - mg \cos \gamma;\]

    \[\dot{x}_e = V \cos \gamma \cos \chi + W_x;\]

    \[\dot{y}_e = V \cos \gamma \sin \chi + W_y;\]

    \[\dot{h}_e = V \sin \gamma;\]

    \[\dot{m} = -T \eta.\]

    截屏2022-01-21 下午10.15.37.png
    Figure 7.2: Aircraft forces.

    Where in the above:

    • the three dynamics equaitions are expressed in an aircraft based reference frame, the wind axes system \(F_w (O, x_w, y_w, z_w)\), usually \(x_w\) coincident with the velocity vector.
    • the three kinematic equations are expressed in a ground based reference frame, the Earth reference frame \(F_e (O_e, x_e, y_e, z_e)\) and are usually referred to as down range (or longitude), cross range (or latitude), and altitude, respectively.
    • \(x_e, y_e\) and \(h_e\) denote the components of the center of gravity of the aircraft, the radio vector \(\vec{r}\), expressed in an Earth reference frame \(F_e (O_e, x_e, y_e, z_e)\).
    • \(W_x\), and \(W_y\) denote the components of the wind, \(\vec{W} = (W_x, W_y, 0)\), expressed in an Earth reference frame \(F_e (O_e, x_e, y_e, z_e)\).
    • \(\mu, \chi\), and \(\gamma\) are the bank angle, the heading angle, and the flight-path angle, respectively.
    • \(m\) is the mass of the aircraft and \(\eta\) is the specific fuel consumption.
    • \(g\) is the acceleration due to gravity.
    • \(V\) is the true air speed of the aircraft.
    • \(T\) is the engines' thrust, the force generated by the aircraft's engines. It depends on the altitude \(h\), Mach number \(M\), and throttle \(\pi\) by an assumedly known relationship \(T = T(h, M, \pi)\).
    • lift, \(L = C_L S \hat{q}\), and drag, \(D = C_D S \hat{q}\) are the components of the aerodynamic force, where \(C_L\) is the dimensionless coefficient of lift and \(C_D\) is the dimensionless coefficient of drag, \(\hat{q} = \tfrac{1}{2} \rho V^2\) is referred to as dynamic pressure, \(\rho\) is the air density, and \(S\) is the wet wing surface. \(C_L\) is, in general, a function of the angle of attack, Mach and Reynolds number: \(C_L = C_L (\alpha, M, \text{Re})\). \(C_D\) is, in general, a function of the coefficient of lift: \(C_D = C_D (C_L (\alpha, M, \text{Re}))\).

    Additional assumptions are:

    Hypothesis 7.11 Parabolic drag polar

    A parabolic drag polar is assumed, \(C_D = C_{D_0} + C_{D_i} C_L^2\).

    Hypothesis 7.12 Standard atmosphere model

    A standard atmosphere is defined with \(\Delta_{ISA} = 0\).

    Vertical motion

    Considerer the additional hypothesis for a symmetric flight in the vertical plane:

    Hypothesis 7.13 Vertical motion
    • \(\chi\) can be considered constant.
    • The aircraft performs a leveled wing flight, i.e., \(\mu = 0\).
    • There are no actions out of the vertical plane, i.e., \(W_y = 0\).
    Definition 7.4 (3DOF equations of vertical motion)

    The 3DOF equations governing the translational vertical motion of an airplane is given by the following ODE system:

    \[m \dot{V} = T - D - mg \sin \gamma,\]

    \[m V \dot{\gamma} = L - mg \cos \gamma,\]

    \[\dot{x}_e = V \cos \gamma \cos \chi + W_x,\]

    \[\dot{h}_e = V \sin \gamma,\]

    \[\dot{m} = -T \eta.\]

    Horizontal motion

    Considerer the additional hypothesis for a symmetric flight in the horizontal plane:

    Hypothesis 7.14 Horizontal motion

    We consider flight in the horizontal plane, i.e., \(\dot{h}_e = 0\) and \(\gamma = 0\).

    Definition 7.5 (3DOF equations of horizontal motion)

    The 3DOF equations governing the translational horizontal motion of an airplane is given by the following ODE system:

    \[m \dot{V} = T - D\]

    \[m V \dot{\chi} = L\sin \mu,\]

    \[0 = V \cos \mu - mg,\]

    \[\dot{x}_e = V \cos \chi + W_x,\]

    \[\dot{y}_e = V \sin \chi + W_y,\]

    \[\dot{m} = -T \eta.\]


    3. The reader is encouraged to read Appendix A for a better understanding.


    7.1.3: Aircraft equations of motion 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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