CENTRE FOR OPERATIONAL RESEARCH AND ANALYSIS
BRIEFING NOTE
An Analysis of Debris Trajectories
Associated with Aerobatic Manoeuvres
by
Norman C. Corbett
Ph. D. (Applied Mathematics)
Looping Manoeuvre
For
this analysis we make the following assumptions:
1.
The
aircraft is executing a looping manoeuvre when the incident occurs. The loop
begins at 300 feet AGL and extends as high as 3,500 feet AGL (see Figure
2
). For this analysis, we restrict our attention to the part of the loop
where the aircraft has a positive angle of attack and is flying towards the
audience. Here, we assume that the audience is to the right of the pull
up point and that the aircraft is travelling from left to right. As a
result, we can restrict our attention to points along the bottom, right quarter
of the loop.
2.
We
ignore possible changes in orientation of the aircraft (e.g. yaw and roll). As
well, we have ignored possible lift forces. Consequently, after the
incident occurs, the aircraft is treated as a ballistic object.
3.
We
have assumed that there is no significant wind during the manoeuvre.
The
physical parameters which govern how far the debris will travel are:
4.
The
mass of the debris, M:
A mass of 7,800 pounds was indicated initially. However, depending on when
(during the air show) the looping manoeuvre is executed, a significant amount of
fuel may have been expended. CFTO C12114000/MW000 indicates that if all but
150 pounds of reserve fuel are expended, then the weight of the total weight of
the aircraft will be about 5,731 pounds. Accordingly we consider two scenarios,
corresponding to the choices M=5,731 and 7,800.
5.
The
drag coefficient, C_{D}:
This parameter depends on the size and shape of the debris. Documents obtained
from Bombardier give the estimate C_{D}=0.75.
This estimate is based on a supposition of “normal” orientation (i.e. angle
of attack). An out of control aircraft would have a much higher drag
coefficient. As well, this coefficient is for a “clean” aircraft
configurations (i.e. no smoke tank attached). Our estimates here are on the
conservative side.
6.
The
frontal surface area, A:
Using detailed dimensional data (again obtained from Bombardier), the Tutor
crosssection depicted in Figure
1 was constructed. This same
information allowed for an accurate estimate of the frontal surface. We found A=61
square feet. Oddly enough this estimate agrees with our previous (crudely
constructed) estimate. Again we point out that this estimate is for a
“clean” configuration and that the addition of an object, like a smoke tank,
would tend to increase the frontal surface area.
Figure 1: CrossSection of the CL41
7.
The
density of air, In
our previous work, we assumed that the density of air was the constant value =1.25 kg/m^{3}. It is well known that the density of
air varies with both temperature and altitude. The trajectory calculation
program has since been updated to account for an air density which varies with
altitude. In particular,
where z is the altitude in feet. Unfortunately, this refinement tends to result in longer trajectories (all parameters/factors held constant).
8.
The
initial[1] speed of the aircraft,
V_{0}: We have assumed that this quantity is fixed at 290 knots for
all scenarios involving the Tutor.
9.
The
initial position of the aircraft,
X_{0}, Z_{0}: Here,
the number X_{0 }denotes the
distance beyond the pull up point, while Z_{0}
denotes the altitude AGL. Once again, the initial position is taken to be
various points along the bottom half of the loop depicted in Figure
2.
It
is important to know how the parameters above can influence the quantities of
interest (e.g. scattering distance). For future reference, we list some simple
qualitative relationships. It is important to note that when the influence of
one parameter is specified, it is assumed that all other parameters are held
constant.
Figure 3: Aircraft Trajectories (Scenario 1)
Figure 4: Speed vs. Angle (Scenario 1)
Figure 5: Aircraft Trajectories (Scenario 2)
Figure 6: Speed vs. Angle (Scenario 2)
·
For
this scenario, we have once again assumed that the mass of the aircraft is 7,800
pounds. However, in this case, we have set the gravitational force to twice
normal. This is intended to simulate the negative authority of the aircraft due
to trim configuration.
Figure 7: Aircraft Trajectories (Scenario 3)
Figure 8: Stabilizer Trajectory
· From Figure 8 we see that lighter debris tends to exhibit smaller maximum scattering distances and lower maximum impact velocities (around 125 knots in this case).
· We now consider the case in which half the airfoil comes away from the aircraft. Once again, we use the estimate C_{D}=0.5. The results of this run are given in Figures 11 and 12. We point out that, in this instance, we have taken the frontal surface area to be 14 square feet and the mass to be 850 pounds.
Figure 9: Speed vs. Angle (Stabilizer)
Figure 10: Airfoil Trajectory
Figure 11: Speed vs. Angle (Airfoil)
where is the frequency of A(t) in hertz (cycles per second) and . An example of the function A(t), in the case =0.5[3] is depicted in Figure 12 .
Figure 12: Time Varying Frontal Surface Area of an Airfoil
· We assume that the mass, drag coefficient, initial positions, and velocities are the same as in our previous analysis. However, in this case we replace the constant frontal surface area with the function A(t) described above (with w=0.5).
· We note that it is not unreasonable to expect that the drag coefficient C_{D} will vary periodically with time as well. Moreover, this time dependence would probably contribute to shorter scattering distances. For this example however, we simply use the conservative estimate C_{D}=0.5.
· We add that caution is warranted in the interpretation of the results in this section. As we have not fully justified the form of the function A(t), the results should be used for illustrative purposes only and are intended to suggest what might happen when debris from an incident tumbles.
· The results of the current analysis are summarised in Figure 13 and Figure 14 . The reader should compare Figure 13 with Figure 10 and Figure 14 with Figure 11 . We note that the maximum scattering distance is about 1,000 feet shorter and that the impact speeds are lower.
Figure 13: Trajectory of Tumbling Airfoil
Figure 14: Speed vs. Angle (Tumbling Airfoil)
Figure 15: Crossing Manoeuvre
· We assume that, as the result of a collision, one aircraft applies the impulsive force F(t) to the another. After the collision we assume that the aircraft (debris) is treated as a ballistic object. Our objective is to see if it is possible for the force F(t) redirect debris towards the audience. Since we are not currently concerned with scattering distances and in order to simplify the analysis, we neglect the effects of wind resistance.
· We take the mass of the debris to be 36,710 pounds. The initial speed of the debris is assumed to be 350 knots. We assume that the angle is and hence the aircraft approach one another at a angle. We also assume that, for the time period of interest, we can treat the motion as planar (the altitude of the debris is not changing). Accordingly, we do not need to specify the initial altitude.
· It is important to note that, it is not generally possible know the specific forces (impulsive or not) acting during collision. Hence, the results in this section reflect only one possible outcome resulting from a collision during a crossing manoeuvre. Care should be exercised in the interpretation of these results.
Figure 16: Examples of Impulsive Forces
Figure 17: Path of Debris After Collision
· Herein we consider the effect of two impulsive forces on debris. The results of our analysis are summarised in Figure 17 . Although the particular results will be highly dependent on magnitude, duration and orientation of F(t), the plot above does indicate that it is possible for F(t) to redirect debris towards the audience.
For
this analysis we make the following assumptions:
10.
The
aircraft is executing a highg horizontal loop when the incident occurs. The
diameter of the loop is assumed to be 6,000 feet with a “nearest approach”
, of 1,500 feet from the audience (see Figure
18
). We have assumed that the direction of travel around the loop is
counterclockwise when the manoeuvre is viewed from above and that the loop is
executed at 300 feet AGL.
11. We assume that the aircraft leaves the loop at the point P_{d} and travels a distance D_{T} before impacting the ground. Our objective in this case is to estimate the distance between the point of impact P_{i} and the audience D_{A}.


Figure 18: Horizontal Loop
12.
As
before, the debris is the whole aircraft which is treated as a ballistic object.
As well, the effect of wind is assumed to be negligible.
The
physical parameters which govern how far the debris will travel are:
13.
The
mass of the debris, M:
Janes indicates that the weight of an F/A18, configured for a fighter mission,
is 36,710 pounds. Hence, we M=36,710.
14.
The
drag coefficient, C_{D}:
We have no information regarding the drag coefficient of a CF188 (F/A18). For this analysis we have assumed that C_{D}=1
15.
The
frontal surface area, A:
Again, using information from Janes, we estimate that the frontal surface
area of the CF188 is about 120 square feet.
16.
The
initial speed of the aircraft,
V_{0}: We have assumed that this quantity is fixed at 350 knots.
17.
The
initial position of the aircraft,
X_{0}, Y_{0}: Here,
the number X_{0 }denotes the
initial distance to the right of the centre of the loop, while Y_{0}
denotes the initial distance forward (i.e. towards the audience) from the centre
of the loop. These numbers are simply the components of the point of departure P_{d}.
In particular P_{d}=(X_{0},Y_{0}).
The initial positions are taken to be various points along the loop depicted in Figure
18
. We note that both of these quantities can be written in terms of the
loop diameter and the angle of departure
. In particular,
,
where we have assume that
is measured in degrees.
,
And, once D_{T} is known, we can compute the distance to the audience for any given angle of departure.
· In this case, we found that the debris will travel a total distance of 2,307 feet, will impact with a speed of 269 knots at an angle of 16.31 degrees below the horizon. A plot of the corresponding function is given in Figure 19 . We see that the debris in this instance does not actually cross the show line. In fact, the debris comes within 716 feet of the show line. This distance corresponds to a departure angle of 52 degrees.
Figure 19: Distance to Audience (300 ft AGL, 350 kts)
· We point out that, for the debris to cross the show line, we must have
,
And,
if we substitute the values D_{N}=1,500 and R=3,000, then
we obtain the inequality
.
[1] In this case, initial denotes the point in time at which the incident occurs (i.e. loss of power, control, etc.)
[2] There is some evidence to suggest that C_{D}=0.5 is a typical value for foils on modern aircraft.
[3] Note that w=0.5 simply means that one complete revolution of the airfoil will take two seconds.
[4] This follows from the fact that factors like altitude, speed, angle of attack, etc. are being held (approximately) constant throughout the manoeuvre.