Related Course: Aircraft Stability and ControlTextbook used: Performance, Stability, Dynamics, and Control of Airplanes, Second Edition (AIAA Education)
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Assumption: Aircraft as a rigid body
Consider the following two equations:
\[ \Sigma \vec F = \frac{d}{dt}(m \vec V) \tag 1 \]The summation of all external forces acting on a body is equal to the time rate of change of the momentum of the body.
\[ \Sigma \vec M = \frac{d}{dt} (\vec H) \tag 2 \]The summation of the external moments acting on the body is equal to the time rate of change of the moment of the momentum which is known as the Angular Momentum.
Equations 1 and 2 are defined using inertial reference frame. Now, write them in components forms as shown below:
Equation 1 in Components
\[ \vec F = (F_x, F_y, F_z) \] \[ \vec V = (u, v, w) \] \[ F_x = \frac{d}{dt}(mu) \tag {3.1} \] \[ F_y = \frac{d}{dt}(mv) \tag {3.2} \] \[ F_x = \frac{d}{dt}(mw) \tag {3.3} \] These forces are due to the Aerodynamic, propulsive and gravitational forces.
Equation 2 in Components
\[ \vec M = (L, M, N) \] \[ \vec H = (H_x, H_y, H_z) \] \[ L = \frac{d}{dt}(H_x) \tag {4.1} \] \[ M = \frac{d}{dt}(H_y) \tag {4.2} \] \[ N = \frac{d}{dt}(H_z) \tag {4.3} \]
Equations 4.1 - 4.3 explains the rotation about x, y, z axis that is attached to the aircraft because off the moments. Attached axis is called the Body-Axis of the Aircraft.
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