In the picture below you see a King Air 200, an aircraft with a so-called T-tail. One of the primary reasons of placing the tail higher up is to keep the tail out of the wake and downwash of the wing. Here, you may assume that . Image courtesy of Mark Jones Jr., CC - BY For this aircraft, we will in this exercise investigate the required position of the centre of gravity to guarantee static, longitudinal stability. The first step in this analysis is to set up the moment equation around the centre of gravity of this aircraft. 1. Derive the moment equation around the centre of gravity of this King Air 200, as a function of the total lift L , the tail lift L_H, the centre of gravity distance l_cg, the tail arm l_H and the aerodynamic centre moment M_ac .
The moment equation around the centre of gravity of the King Air 200 can be derived as follows:
M_cg = (L × l_cg) - (L_H × l_H) - M_ac
Where:
M_cg = Moment around the centre of gravity
L = Total lift
l_cg = Centre of gravity distance
L_H = Tail lift
l_H = Tail arm
M_ac = Aerodynamic centre moment
To derive the moment equation around the center of gravity (CG) of the King Air 200 aircraft, we need to consider the forces and moments acting on the airplane.
Let's denote the following variables:
- L: Total lift
- L_H: Tail lift
- l_cg: Distance from the CG to the aerodynamic center (AC)
- l_H: Tail arm
- M_ac: Aerodynamic center moment
The moment equation can be derived as follows:
1. First, consider the wing lift and the tail lift moments:
- The moment generated by the wing lift is L_cg = L * l_cg. This is because the wing lift acts at the CG.
- The moment generated by the tail lift is L_H * l_H. This is because the tail lift acts at a distance l_H from the CG.
2. Next, consider the moment generated by the aerodynamic center:
- The moment generated by the aerodynamic center is M_ac.
3. Finally, sum up all the moments and set them equal to zero (for static, longitudinal stability):
L_cg + L_H * l_H + M_ac = 0
This is the moment equation around the center of gravity of the King Air 200 aircraft, as a function of the total lift L, the tail lift L_H, the center of gravity distance l_cg, the tail arm l_H, and the aerodynamic center moment M_ac.
To derive the moment equation around the center of gravity of the King Air 200, we need to consider the forces and moments acting on the aircraft.
First, let's define the variables:
- L: Total lift generated by the wings
- L_H: Lift generated by the tail
- l_cg: Distance from the center of gravity to the reference point (usually the wing root or nose)
- l_H: Distance from the center of gravity to the tail
- M_ac: Aerodynamic center moment
Now, let's analyze the forces and moments acting on the aircraft:
1. Lift forces:
The total lift generated by the wings can be divided into two components: the main wing lift (L) and the tail lift (L_H).
2. Moment arms:
The distance from the center of gravity to the main wing (l_cg) and the distance from the center of gravity to the tail (l_H) determine the moment arms for the lift forces.
3. Aerodynamic center moment:
The aerodynamic center moment (M_ac) represents the moment generated by the distribution of lift along the wing and horizontal tail.
Now, let's derive the moment equation:
The total moment acting around the center of gravity (M_cg) can be expressed as the sum of moments from the lift forces and the aerodynamic center moment:
M_cg = (L * l_cg) + (L_H * l_H) + M_ac
This equation states that the total moment is the sum of the moments generated by the lift forces and the aerodynamic center moment.
By analyzing this equation, we can determine the required position of the center of gravity for static, longitudinal stability. This analysis involves ensuring that the moment generated by the tail (L_H * l_H) is sufficient to counteract the moment generated by the wings (L * l_cg) and the aerodynamic center moment (M_ac).
Note that specific values for L, L_H, l_cg, l_H, and M_ac would need to be provided to solve this equation accurately for a given King Air 200 aircraft.