# 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:

##### Definition 7.3 (3DOF equations of 3D motion)

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

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

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

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

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

$\dot{h}_e = V \sin \gamma;$

$\dot{m} = -T \eta.$ 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: $$C_D = C_D (C_L (\alpha, M, \text{Re}))$$.

##### 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.