1 CAD/CANR HEADQUARTERS

CENTRE FOR OPERATIONAL RESEARCH AND ANALYSIS

BRIEFING NOTE

An Analysis of Debris Trajectories

Associated with Aerobatic Manoeuvres

Norman C. Corbett

Ph. D. (Applied Mathematics)

Looping Manoeuvre

Assumptions & Background

We want to predict the extent of a debris field (scattering distance) resulting from an aircraft accident or incident that occurs during an aerobatic manoeuvre. In other words, we want to know how far debris, resulting from such an incident, might travel.
Debris formation is a complex process. The sizes and shapes of objects in an aircraft debris field are difficult to predict and will vary from incident to incident. As a result, we restrict our attention to scenarios in which the debris is the whole aircraft or some major component like an airfoil or a horizontal stabilizer.
We point out that in order to increase accuracy, we have made several refinements to the equations governing the trajectory program. Unfortunately, these refinements have tended to result in larger debris scatter distances.

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/m³. 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, V₀: We have assumed that this quantity is fixed at 290 knots for all scenarios involving the Tutor.

9. The initial position of the aircraft, X₀, Z₀: Here, the number X₀denotes the distance beyond the pull up point, while Z₀ 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.

Figure 2: Looping Manoeuvre

Qualitative Relationships

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.

An increase in the mass M of the debris leads to larger scattering distances.
Larger drag coefficients C_D lead to smaller scattering distances.
Larger frontal surface areas A lead to smaller scattering distances.
Larger initial speeds V₀lead to larger scattering distances.
Larger initial altitudes Z₀lead to larger scattering distances.
The scattering distance depends on the initial angle of attack. Although we have not proved this conclusively, there is some evidence to suggest that the largest scattering distances will be associated with initial angles which are (approximately) 45 degrees above the horizontal.

Scenario 1

For this scenario, we consider the whole aircraft and assume that the mass is 7,800 pounds.
A word about the “Speed vs. Angle” plots is in order. There is a lot of extra information in these graphs. Of particular interest however is the impact information. This information is always given by the “left most” point on each of the curves. For example, from Figure 4, we see that see that four of the nine trajectories had impact angles “steeper” that 60 degrees below horizontal. In addition, the trajectory with the highest terminal velocity (over 200 knots) had the “shallowest” impact angle (about 20 degrees below horizontal).
We point out that shallow impact angles are of some concern as, in these circumstances, the debris may bounce and travel further than the distance indicated by trajectory calculations. However, in the case of the vertical loop, we consistently observe low impact angles along with the shortest scattering distances.

Figure 3: Aircraft Trajectories (Scenario 1)

Figure 4: Speed vs. Angle (Scenario 1)

Scenario 2

For this scenario, we assume that the mass of the aircraft is 5,731 pounds. All other parameters are assumed to be the same as in Scenario 1. In this case, we see a reduction of about 600 feet in the maximum scattering distance. The impact speed is also reduced (approximately 20 knots in this case).

Figure 5: Aircraft Trajectories (Scenario 2)

Figure 6: Speed vs. Angle (Scenario 2)

Scenario 3

· 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)

Here we observe a reduction of about 1,000 feet in maximum scattering distance as compared with that of Scenario 1. Although encouraging, caution is warranted as a full justification of these results may not be possible without a more detailed analysis of the debris trajectory.

Other Debris

An extraction from the flight safety information database seems to indicate that, of the approximately 20 Snowbird accidents to date, many involve a “touching” of either stabilizer or wings. Accordingly, we now consider two scenarios in which the debris (resulting from the incident) is part of either a horizontal stabilizer or an airfoil.
From our Bombardier documents, we estimate the weight of the horizontal stabilizer to be about 114 pounds. We consider a scenario in which half of the stabilizer (57 pounds) breaks away from the aircraft. We estimate the frontal surface area of the half stabilizer to be about 2 square feet. We have no direct information about the drag coefficient for the stabilizer of a Tutor. We used a coefficient of 0.5 for this run[2]. The results of this scenario are given in Figure 8 and Figure 9.

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

In this case, the results are comparable to those seen in Scenarios 1 and 2. Again we see rather large scattering distances (nearly a mile from the pull up point). We should point out that it is quite possible that partial debris would have an unstable and variable orientation (say tumbling) in flight. This would lead to larger effective drag coefficients and greater effective surface areas. It is however impossible to predict the exact motion (orientation and trajectory) of the debris without an extremely detailed analysis.

Figure 11: Speed vs. Angle (Airfoil)

Tumbling Debris

Here we revisit the scenario in which half an airfoil has come away from the aircraft during a looping manoeuvre. In this case, however, we assume that the debris is rotating (tumbling) about its long axis as it moves along its trajectory.
As a result of this rotation, the frontal surface area changes with time. Based on the Bombardier data, we assume that this area varies regularly between 14 square feet and 41 square feet. To approximate this effect, we suppose that the frontal surface area A is a periodic function of time. In particular, we take

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)

Collision Analysis

A collision analysis for a “crossing manoeuvre” has been undertaken. Although the results are not conclusive, indications are that it is not impossible for debris to be redirected towards the audience if a collision were to occur at origin of the coordinate axes depicted in Figure 15 .
This analysis is rather complicated as it involves an inelastic collision and additional work is needed. The outcome of this analysis will probably be of general interest in the study of collisions.
In lieu of a completed collision analysis, we provide an illustrative example based on impulsive type forces. Such forces tend to act over very short time durations and, as such, are often appropriate for ming collisions.

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.

Horizontal Loop

Assumptions & Background

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 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.

It is not too difficult to prove that, for this manoeuvre, the distance D_T will be (approximately) constant[4]. It follows that the required distance D_A depends only on P_d which is specified by the angle of departure (see below)

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, V₀: We have assumed that this quantity is fixed at 350 knots.

17. The initial position of the aircraft, X₀, Y₀: Here, the number X₀denotes the initial distance to the right of the centre of the loop, while Y₀ 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=(X₀,Y₀). 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.

Let R be the radius of the loop (in this case, 3,000 feet). It can be shown that

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)

For comparison, we consider the case where the aircraft is at 400 feet AGL with a speed of 400 knots when it departs the loop. In this case the debris travels a total of 2,939 feet and impacts at a angle of 17.37 degrees below the horizon at a speed of 284 knots. Here, the debris comes within 300 feet of the audience (corresponding to ).

· 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.