4 Performance Calculation
4.2 Air Distance
The previous sections discussed the aircraft accelerating on the runway, covering the Liftoff Distance. After the Liftoff Distance has been covered and the aircraft has reached VLOF, the aircraft will have to cover a certain distance in the air until passing the obstacle or screen height. In the case of a BFL determination, the Air Distance is always part of the runway field length required and needs to be considered.
A simplified approach not taking into account the acceleration of the aircraft between VLOF
and V2 is outlined in this section. This simplification can be made as the screen height is sig-nificantly reduced on a wet runway, and the aircraft under this condition does not have to reach V2 when clearing the reduced screen height of 15 feet on a wet runway. This has been discussed in Section 3.3.
The method to determine the Air Distance is based on a simplified Roskam 1997 method from Scholz 1999. This approach divides the aircraft trajectory after liftoff into two separate trajectories, the bow-shaped rotation phase trajectory and the linear climb phase trajectory at a constant climb angle as shown in Fig. 4.19.
Fig. 4.19 Climb out to Obstacle Height method (Roskam, Lan 1997)
As the incidence angle of the aircraft changes and multiple effects influence the lift force act-ing on the aircraft, the lift coefficient in flight differs from the lift coefficient on ground. It is a sufficiently accurate method as proposed by Scholz 1999 to assume that the lift at the liftoff point needs to balance the weight of the aircraft. It bases on the assumption that the aircraft rotates to a new attitude where the aircraft lift equals its weight.
At small climb angles, it can be assumed that . This leads to the assump-tion that
or
(4.89)
With
Lift force on the aircraft at liftoff
Weight force of the aircraft due to its mass
Lift coefficient at liftoff
Earth gravity
By transformation of Eq. 4.89, the lift coefficient of the aircraft at liftoff yields:
(4.90)
As shown in equation 4.13 from Section 4.1.2, the liftoff speed can be estimated from the stall speed of the aircraft as provided by the AFM.
4.2.1 Rotation and Climb Trajectory
During the bow shaped flight phase, the aircraft experiences an augmented g-load along its z axis (yaw axis). The lift during the rotation phase must therefore balance the weight of the aircraft augmented by the centrifugal acceleration which is greater than earth gravity. In analogy to Eq. 4.89, the following assumption is made.
(4.91)
Substituting the acceleration and aircraft mass yields
(4.92)
With
Aircraft mass
Acceleration in z-axis of the aircraft during rotation
Radius of bow shaped rotation trajectory
Load Factor during Rotation Phase
In order to determine the radius of the bow shaped rotation trajectory, the load factor needs to be determined. It results from the acceleration experienced by the aircraft in the z-axis.
(4.93)
From Eq. 4.93 follows through substitution:
( )
(4.94)
It can be assumed that the ratio of lift coefficients according to Scholz 1999 is in the order of:
(4.95)
This yields an estimation for the load factor experienced by the aircraft during the bow shaped rotation trajectory.
Rotation Trajectory Bow Radius
Introducing the relation for L from Eq. 4.93 into equation 4.92 yields:
( ) (4.96)
This permits to determine the radius R by transformation of Eq. 4.96.
( )
(4.97)
The bow radius R can now be calculated by using the estimation for the load factor .
Transition Height
With R being known, the transition height between the bow shaped rotation phase and the linear climb phase can be determined. The transition height takes place at the point where the aircraft trajectory reaches the maximum constant climb angle possible for the aircraft configu-ration. This means that the aircraft cannot further increase its climb angle and will follow a linear climb trajectory from the transition height on.
From Fig. 4.19, the transition height can be determined geometrically to be:
( ) (4.98) With
Transition Height
The climb angle the aircraft is able to assume during the climb phase is a function of the aircraft thrust, weight, lift and drag.
For small angles of , it can be assumed that W L. This yields:
(4.99)
Thus
( ) (4.100)
4.2.2 Rotation and Climb Distances over Ground
For the rotation phase and the linear climb phase, two cases need to be distinguished:
A) The screen height has already been cleared when the transition height is reached
B) The screen height has not been cleared when the transition height is reached
Case A – Screen Height Cleared when Transition Height Reached
In this case, the Air Distance is determined from the rotation trajectory radius R intersection with the screen height :
√ ( ) (4.101)
With
Screen Height
This can be further simplified if , the single summand allowing to simplify with
to:
√ (4.102)
Case B – Screen height not Cleared when Transition Height Reached
If the screen height is not yet cleared when the bow shaped rotation phase trajectory blends into the climb phase trajectory, the distance over ground covered at constant climb angle has to be considered until the screen height is passed.
In this case, the climb phase distance needs to be added to the rotation phase distance.
For the distance covered over ground until reaching the transition height:
(4.103)
For the distance covered over ground in a linear climb trajectory at :
(4.104)
The total Air Distance is then calculated from the sum of the rotation phase distance and the climb distance.
(4.105)