Longitudinal balancing refers to the process of determining the state of equilibrium for a longitudinal movement of an aircraft, while considering lateral and directional variables as uncoupled. In this analysis, the focus is on forces acting along the z-axis (Fz) and torques around the y-axis (My). Typically, external influences from aerodynamics, propulsion, and gravity are considered, although for simplicity, only gravity and lift forces on the wing and horizontal stabilizer are often taken into account.

## What is Longitudinal Balancing & Stability?

Several assumptions are made during longitudinal balancing calculations:

- No wind: External wind factors are disregarded to simplify the analysis.
- Constant mass and velocity: It is assumed that the mass and velocity of the aircraft remain constant throughout the analysis.

In the given equation:

*M* represents the pitch torque with respect to the aerodynamic center._{ca}*L*_{t}denotes the lift generated by the horizontal stabilizer.*x*_{cg} is the distance between the center of gravity and the aerodynamic center.*l*represents the distance between the center of gravity and the aerodynamic center of the horizontal stabilizer.

### Longitudinal Stability of Aircraft

In the scenario, where an aircraft in horizontal, steady, linear flight experiences a perturbation due to a vertical wind gust, the angle of attack increases, resulting in a perturbation in the lift forces acting on both the main wing (L) and the horizontal stabilizer (Lt). Assuming that the behavior of the horizontal stabilizer is similar to that of the wing, both lift forces increase to *L*+Δ*L* and *L*_{t}+Δ*L _{t}*.

If the increase in lift force on the horizontal stabilizer Δ*L _{t}* multiplied by its lever arm distance l is greater than the increase in lift force on the main wing Δ

*L*, then the angle of attack tends to decrease, leading to static stability. Conversely, if Δ

*L*⋅

_{t}*l*<Δ

*L*indicating that the increase in lift on the horizontal stabilizer is not sufficient to counteract the increase in lift on the main wing, the aircraft is statically unstable.

In essence, the stability of the aircraft after a perturbation depends on the relationship between the changes in lift forces on the main wing and the horizontal stabilizer, as well as the relative positions of their aerodynamic centers with respect to the aircraft’s center of gravity. Therefore, aerodynamic design plays a crucial role in determining the static stability of the aircraft.

The external longitudinal moments acting on the center of gravity can be made dimensionless as follows:

*M*,*cg* represents the coefficient of moments of the aircraft with respect to its center of gravity. This coefficient captures the rotational tendencies of the aircraft caused by aerodynamic forces and moments around its center of gravity.

The *cM*0 denotes the coefficient of moments that remains constant regardless of the angle of attack or the deflection of the elevator. It represents the baseline contribution to the aircraft’s moments.

*cMα* stands for the derivative of the coefficient of moments of the aircraft with respect to the angle of attack. This derivative reflects how the aircraft’s rotational tendencies change in response to variations in angle of attack.

Similarly, *cMδe* represents the derivative of the coefficient of moments of the aircraft with respect to the deflection of the elevator. It quantifies how adjustments in elevator deflection influence the aircraft’s rotational behavior.

For an aircraft to be statically stable, the following condition must be satisfied:

The coefficient of moments with respect to the center of gravity (*cM*,*cg*) must be positive when the angle of attack is less than the angle of equilibrium (*αe*). This condition ensures that a perturbation causing a decrease in the angle of attack will result in a moment that tends to pitch up the aircraft, restoring it to its initial equilibrium state.

In Figure above, this condition is illustrated by comparing the coefficient of moments curves for two different aircraft. For aircraft (a), when the angle of attack (*α*1) is less than the equilibrium angle (*αe*), *cM*,*cg*,*a*>0, indicating a moment that pitches up the aircraft and stabilizes it. Conversely, for aircraft (b), *cM*,*cg*,*b*<0, indicating a moment that pitches down the aircraft and renders it statically unstable.

Therefore, statically stable aircraft exhibit positive values of *cM*,*cg* when the angle of attack is less than the equilibrium angle. This condition ensures that the aircraft tends to return to its equilibrium state following a perturbation, contributing to its overall stability.