7.1.3: Aircraft equations of motion
- Page ID
- 78149
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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:
\[m \dot{V} = T - D - mg \sin \gamma; \nonumber \]
\[m V \dot{\chi} \cos \gamma = L \sin \mu; \nonumber \]
\[m V \dot{\gamma} = L \cos \mu - mg \cos \gamma; \nonumber \]
\[\dot{x}_e = V \cos \gamma \cos \chi + W_x; \nonumber \]
\[\dot{y}_e = V \cos \gamma \sin \chi + W_y; \nonumber \]
\[\dot{h}_e = V \sin \gamma; \nonumber \]
\[\dot{m} = -T \eta. \nonumber \]
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:
\[m \dot{V} = T - D - mg \sin \gamma, \nonumber \]
\[m V \dot{\gamma} = L - mg \cos \gamma, \nonumber \]
\[\dot{x}_e = V \cos \gamma \cos \chi + W_x, \nonumber \]
\[\dot{h}_e = V \sin \gamma, \nonumber \]
\[\dot{m} = -T \eta. \nonumber \]
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:
\[m \dot{V} = T - D \nonumber \]
\[m V \dot{\chi} = L\sin \mu, \nonumber \]
\[0 = V \cos \mu - mg, \nonumber \]
\[\dot{x}_e = V \cos \chi + W_x, \nonumber \]
\[\dot{y}_e = V \sin \chi + W_y, \nonumber \]
\[\dot{m} = -T \eta. \nonumber \]
3. The reader is encouraged to read Appendix A for a better understanding.