1 CAD/CANR HEADQUARTERS
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 C12-114-000/MW-000 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 cross-section 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: Cross-Section of the CL-41
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
high-g 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 counter-clockwise 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}.
P_{d} P_{i}
Figure 18:
Horizontal
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/A-18, 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 CF-188 (F/A-18). 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 CF-188 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.