7.1.3: Aircraft equations of motion
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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:
\[m \dot{V} = T - D - mg \sin \gamma;\]
\[m V \dot{\chi} \cos \gamma = L \sin \mu;\]
\[m V \dot{\gamma} = L \cos \mu - mg \cos \gamma;\]
\[\dot{x}_e = V \cos \gamma \cos \chi + W_x;\]
\[\dot{y}_e = V \cos \gamma \sin \chi + W_y;\]
\[\dot{h}_e = V \sin \gamma;\]
\[\dot{m} = -T \eta.\]
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:
A parabolic drag polar is assumed, \(C_D = C_{D_0} + C_{D_i} C_L^2\).
A standard atmosphere is defined with \(\Delta_{ISA} = 0\).
Vertical motion
Considerer the additional hypothesis for a symmetric flight in the vertical plane:
- \(\chi\) 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., \(W_y = 0\).
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:
We consider flight in the horizontal plane, i.e., \(\dot{h}_e = 0\) and \(\gamma = 0\).
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.