7.2.3: Longitudinal balancing

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The longitudinal balancing is the problem of determining the state of equilibrium of a longitudinal movement in which the lateral and directional variables are considered uncoupled. For the longitudinal analysis, one must consider forces on \(z\)-axis (\(F_z\)) and torques around \(y\)-axis (\(M_y\)). Generally, it is necessary to consider external actions coming from aerodynamics, propulsion, and gravity. However, it is common to consider only the gravity and the lift forces in wing and horizontal stabilizer. Additional hypotheses include: no wind; mass and velocity are constant.

The equations to be fulfilled are:

\[\sum F_z = 0,\]

\[\sum M_y = 0.\]

Which results in

\[mg - L - L_t = 0,\]

\[-M_{ca} + Lx_{cg} - L_t l = 0,\]

where \(L_t\) is the lift generated by the horizontal stabilizer, \(M_{ca}\) is the pitch torque with respect to the aerodynamic center, \(x_{cg}\) is the distance between the center of gravity and the aerodynamic center, and \(l\) is the distance between the center of gravity and the aerodynamic center of the horizontal stabilizer.