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 , the tail lift , the centre of gravity distance , the tail arm and the aerodynamic centre moment .
The moment equation around the center of gravity of the King Air 200 can be derived as follows:
Total moment = Lift moment + Tail moment equation
Lift moment = (Total lift) * (Center of gravity distance)
Tail moment = (Tail lift) * (Tail arm)
Therefore, the moment equation is:
Total moment = (Total lift) * (Center of gravity distance) + (Tail lift) * (Tail arm) + (Aerodynamic center moment)
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. Here's how we can proceed:
1. Draw a free-body diagram of the aircraft, showing the forces and moments acting on it.
2. Identify the forces and moments relevant to this analysis:
- Total lift (L): This is the lift force acting on the wings, which is assumed to act through the center of lift.
- Tail lift (Lt): This is the lift force acting on the tail, which is assumed to act through the tail's center of lift.
- Center of gravity distance (d): This is the distance between the center of gravity and the wing's center of lift.
- Tail arm (Lc): This is the distance between the center of gravity and the tail's center of lift.
- Aerodynamic center moment (Mc): This is the moment acting on the aircraft about the center of gravity due to the aerodynamic forces acting through the aircraft's aerodynamic center.
3. Write down the moment equation:
Sum of moments about the center of gravity = -L * d + Lt * Lc + Mc
4. Simplify the equation:
-L * d + Lt * Lc + Mc = 0
This is the moment equation around the center of gravity of the King Air 200 as a function of the total lift (L), tail lift (Lt), center of gravity distance (d), tail arm (Lc), and aerodynamic center moment (Mc).
To derive the moment equation around the center of gravity (CG) of the King Air 200, we need to consider the equilibrium of moments.
1. We start by defining the variables:
- Total lift: L
- Tail lift: L_tail
- Center of gravity distance from aerodynamic center (AC): x_cg
- Tail arm (distance between CG and tail AC): x_tail
- Aerodynamic center moment: M_ac
2. The total moment around the CG can be written as the sum of the individual moments:
Sum of moments = Moment due to total lift + Moment due to tail lift + Moment due to aerodynamic center
3. Moment due to total lift:
The moment due to total lift is the product of the total lift (L) and the distance from the CG to the AC.
Moment due to total lift = L * x_cg
4. Moment due to tail lift:
The moment due to tail lift is the product of the tail lift (L_tail) and the distance from the CG to the tail AC.
Moment due to tail lift = L_tail * x_tail
5. Moment due to aerodynamic center:
The moment due to the aerodynamic center is the product of the aerodynamic center moment (M_ac) and the distance from the CG to the AC.
Moment due to aerodynamic center = M_ac * x_cg
6. Summing up all the moments, we have the moment equation around the CG:
Sum of moments = L * x_cg + L_tail * x_tail + M_ac * x_cg
This is the derived moment equation around the center of gravity of the King Air 200 as a function of the total lift, tail lift, center of gravity distance, tail arm, and the aerodynamic center moment.