F-16 Quadratic LCO Identification

Chapter 4 F-16 Quadratic LCO Identification The store configuration of an F-16 influences the flight conditions at which limit cycle oscillations develop. Reduced-order modeling of the wing/store system with the objective of identifying unstable flight conditions is the subject of several research efforts. The need to validate these models and other computational procedures require validation of the physical aspects causing LCO. In this chapter, nonlinear dynamics leading to observed LCO in F-16 flight tests are identified using higher order spectral moments. Two cases of mechanically and maneuver-induced LCO are compared. The results show that nonlinear couplings present in the wing/store system resulting from maneuver induced LCO are different in both order, and location, from nonlinear couplings resulting from mechanically forced LCO. This new information about the couplings of the wing/store system can be used to validate and increase the accuracy and scope of LCO modeling efforts. 4.1 Historical Context of the F-16 The F-16A was introduced in January, 1979 with the 388 th Tactical Fighter Wing at Hill Air Force Base, Utah [6]. As a successor to the F-1, the F-16 was designed as a high performance fighter. More technologically advanced than its predecessor, the F-16 was designed with relaxed longitudinal static stability [6]. In subsonic flight, the center of gravity is aft of the center of pressure resulting in negative stability. This is a more efficient configuration since the tail and the wings both act to generate lift, although it requires fly-by-wire computer control to maintain stability [7]. During supersonic flight, the center of gravity is in front of the center of pressure 11

Figure 4.1: F-16 with stores on the wing [42] yielding a statically stable configuration [6]. Other design considerations include a bubble cockpit and a side stick controller, both of which aid in high performance flight. Structural limits allow up to 9 g s of acceleration during extreme maneuvers. This places severe demands on both the pilot and the airframe. As a result of the high performance capability, the flight environment can change rapidly. Aeroelastic effects can cause the wings to exhibit limit cycle oscillations which can take place over a variety of flight conditions. Structural vibrations resulting from LCO could have detrimental effects on the lifetime of the aircraft and the pilot. Particularly, the pilot may experience increased fatigue and blurred vision by severe lateral vibrations. The conditions under which limit cycle oscillations develop are important to the mission of the Air Force and many research efforts have focused on predicting flight conditions and store configurations that induce LCO. An F-16 with one particular store configuration is shown in Figure 4.1 [42]. 4.2 LCO Testing procedure Flight tests for LCO on different wing/store configurations of the F-16 are conducted on a specially outfitted aircraft, administered by the Air Force s SEEK EAGLE program. The vehicle is instrumented with accelerometers on the wings, stores, and pylon-wing interfaces. The telemetry data is recorded by ground operators for the specific purpose of LCO identification. A strict protocol for safe LCO clearance limits has been established and constant communication is maintained 12

between the ground crew and the pilot during LCO testing ([36] and []). Variations of the peak magnitude of the LCO have been shown to vary with the Mach number and flight altitude. Figure 4.2 shows the peak magnitude of the wing-tip launcher s vertical acceleration as a function of the Mach number for three different altitudes. LCO was induced by flaperon motion for flight conditions at M =.8,, ft. and all conditions at M =.8 and below. The other data points at M =.8 and above are the result of maneuver induced LCO. The peak magnitude of the wing-tip launcher s vertical acceleration is a maximum near M =.9. A similar analysis using the RMS of the wing-tip launcher s vertical acceleration is shown in Figure 4.2b. Again, an increase in the mean LCO level is observed with increasing Mach number with the maximum value occurring near M =.9. Both Runs 2 and are also indicated in Figures 4.2a and b. In this work, data from two different testing procedures usually performed to induce LCO are analyzed. In the first procedure, LCO is induced by a specific maneuver consisting of straight and level flight followed by a wind up turn. The case considered here is one at an altitude of 1, ft. and M =.9 and is referred to as Run. In this run, limit cycle oscillations occurred over a period of about 2 seconds. In the second testing procedure, limit cycle oscillations were induced by mechanical excitation of the flaperons at a frequency close to that of the first wing antisymmetric bending mode. The case considered, Run 2, took place at an altitude of 1, ft. and M =.8. These testing conditions are summarized in Table 4.1. Table 4.1: Nominal flight conditions and LCO description for the two runs analyzed. Example Run # Mach Alt (ft.) Interval of LCO Origin of LCO 1.9 1, [4 6] seconds Maneuver 2 2.8 1, [26 32] seconds Mechanical forcing 4.3 Nonlinear Aspects of Maneuver-Induced LCO In each run analyzed, both the vertical and lateral accelerations as measured at four different locations, ID 1, 4, 6, and 8. These locations are shown in Figure 4.3. The accelerometer locations begin furthest from the fuselage and progress toward the center. Locations 1 and 4 are located on the launchers, while locations 6 and 8 are located on the pylon-wing interface itself. The exact 13

Max Accel. (g) 4. 4 3. 3 2. 2 1. 1. 1 ft. ft. 2 ft. Max Magnitude; ID 1; Inst. 6 Run 2 Run RMS 1.6 1.4 1.2 1.8.6.4.2 1 ft. ft. 2 ft. Run 2 RMS; ID 1; Inst. 6 Run.7.7.8.8.9.9 Mach Number (a).7.7.8.8.9.9 Mach Number (b) Figure 4.2: Wing-tip launcher, ID 1, vertical accelerations at three different altitudes as a function of increasing Mach number; max magnitude (a) and RMS (b). Table 4.2: Accelerometer locations and instrumentation numbers for vertical and lateral accelerometers. Name ID x (in) Baseline (in) Vert Inst. # Lat Inst. # Plot Letter Wing-tip launcher 1 318 183 6 a Underwing Launcher 4 38 17 18 17 b Pylon-wing Interface 6 368.3 16.3 49 48 c Pylon-wing Interface 8 343.8 117.6 29 28 d locations of the accelerometers are presented in Table 4.2. The plot letter, used when all four instrumentation locations are analyzed and plotted in subsequent figures, is also given in the same table. 14

Figure 4.3: The accelerometers locations used in the following analysis, ID 1, 4, 6, and 8, are indicated [42]. 1

Maneuver Induced LCO; Vertical Accelerations Several parameters of the flight conditions for Run are shown in Figure 4.4. Starting with the top plot, the Mach number increases from.87 to.9, the altitude is constant around 1, feet, and the angle of attack is around 2. The bottom plot shows the wing-tip launcher s vertical acceleration, ID 1, instrument 6. Limit Cycle Oscillations develop as the Mach number approaches.9 near t = 2 seconds and persist until t = 7 seconds. At the onset of LCO, the wing-tip launcher s vertical acceleration is around ±2g s and increases to roughly ±3g s by t = 4 seconds. Mach Number Alt. (ft) 1.9.8 1. x 14 1 Run AoA ( ) 4 2 2 4 2 4 6 8 1 Figure 4.4: Mach number, altitude, angle of attack, and vertical wing-tip acceleration for Run [42] The vertical accelerations of the four instruments mentioned earlier; namely the wing-tip launcher ID 1 (B. L. 183 in), the underwing launcher ID 4 (B. L. 17 in), the pylon-wing interface ID 6 (B. L. 16.3), and the pylon-wing interface ID 8 (B. L. 117.6 in), are shown in Figure 4.. All instruments show similar behavior as LCO develops. The wing-tip launcher (B. L. 183 in) shows the largest vertical acceleration of about ±4g. The magnitude of the vertical acceleration decreases at the other instrumentation locations with values near ±1g at B. L. 16.3 and ±.2g at B. L. 117.6 in. Harmonic oscillations in the wing-tip launcher s vertical accelerations (ID 1) are confirmed by the magnitude of the wavelet transform of the measured accelerations at the wing-tip launcher, as 16

1 1 1 1 Run ; Vertical Accelerations 1 2 3 4 6 7 8 9 1 11 ID 1 Inst. 6 ID 4 Inst. 18 ID 6 Inst. 49 ID 8 Inst. 29 Figure 4.: Vertical acceleration at ID 1, 4, 6, and 8. A large response is present from 2 7 seconds at all four locations; Run presented in Figure 4.6. Limit cycle oscillations are indicated by strong harmonic content at 8 Hz starting at the time near 2 seconds and lasting for about seconds with the largest amplitude near t = seconds. Magnitudes of the wavelet transform at the other locations are presented below. Although LCO does not persist for the entire length of the signal, the power spectrum can provide valuable information about the frequency content of the accelerations. The power spectra of the vertical accelerations of all four instruments (ID 1, 4, 6, and 8) are presented in Figure 4.7a, b, c, and d respectively. All spectra show a strong response at 8.2 Hz, the wing s anti-symmetric bending mode, and a much smaller response at. Hz, the wing s symmetric bending mode [39] and [8]. In addition, spectra of the accelerations at ID 1, 4, and 6 show an increased response at 24. Hz, three times the frequency of the antisymmetric wing bending mode. The vertical accelerations of the pylon-wing interface closest to the fuselage (ID 8) contain no significant power at this higher harmonic. The vertical accelerations, and wavelet transform magnitudes, at all four locations are shown during the interval of strongest LCO, from 4 seconds to 7 seconds in Figure 4.8a, b, c, and d 17

3 2 1 Run ; ID 1; Inst. 6 2 4 6 8 1 Figure 4.6: Vertical acceleration of the wing-tip launcher (top) and its wavelet transform magnitude (bottom); LCO is strongest around seconds. 1 2 Run ; ID 1; Inst. 6 1 Run ; ID 4; Inst. 18 1 1 1 1 2 Power 1 2 Power 1 3 1 4 1 4 1 1 6 1 2 3 4 Frequency (Hz) (a) 1 6 1 2 3 4 Frequency (Hz) (b) 1 Run ; ID 6; Inst. 49 1 2 Run ; ID 8; Inst. 29 1 1 1 2 1 3 Power 1 3 Power 1 4 1 4 1 1 1 6 1 2 3 4 Frequency (Hz) (c) 1 6 1 2 3 4 Frequency (Hz) (d) Figure 4.7: Power spectra of vertical accelerations at ID 1 (a), 4 (b), 6 (c), and 8 (d). 18

Run ; ID 1; Inst 6 2 Run ; ID 4; Inst 18 4 6 6 3 2 4 6 6 3 Freq (Hz) 2 1 Freq (Hz) 2 1 4 6 6 (a) 4 6 6 (b) 1 Run ; ID 6; Inst 49. Run ; ID 8; Inst 29. 1 4 6 6 3 1 4 6 6 3 Freq (Hz) 2 1 Freq (Hz) 2 1 4 6 6 (c) 4 6 6 (d) Figure 4.8: Expanded view of the vertical accelerations and wavelet transform magnitudes of ID 1 (a), 4 (b), 6 (c), and 8 (d) during LCO. respectively. All instruments contain a strong harmonic component around 8 Hz. Additionally, the measured accelerations at ID 1, plot a, contain intermittent higher harmonics. The pylon-wing interface, ID 8, has the lowest magnitude of vertical acceleration. Quadratic coupling was evaluated using the wavelet-based auto-bicoherence. Due to the localized nature of wavelet-based higher order spectra, three 1. second long intervals were analyzed in order to cover the duration of the strongest limit cycle oscillations. As presented in Figure 4.9a, b, c, and d, no quadratic coupling is detected in the vertical components of the accelerations at ID 1, 4, 6, and 8 during [2. 3.] seconds. Contour levels set at [.3 :.1 :.9] were chosen to span a greater range than normal in order to detect even small coupling levels. The same observations 19

can be made over the interval [. 6.] seconds as presented in Figure 4.1a, b, c, and d and again over the interval [8. 9.] seconds, presented in Figure 4.11a, b, c, and d. 2 Run ; ID 1; Inst 6 2 Run ; ID 4; Inst 18 1 1 1 1 1 2 3 4 (a) 1 2 3 4 (b) 2 Run ; ID 6; Inst 49 2 Run ; ID 8; Inst 29 1 1 1 1 1 2 3 4 (c) 1 2 3 4 (d) Figure 4.9: Wavelet-based auto-bicoherence of the vertical accelerations at ID 1, 4, 6, and 8 during the interval t = [2. 3.] seconds. Contour levels are set at ([.3 :.1 :.9]). 11

2 Run ; ID 1; Inst 6 2 Run ; ID 4; Inst 18 1 1 1 1 1 2 3 4 (a) 1 2 3 4 (b) 2 Run ; ID 6; Inst 49 2 Run ; ID 8; Inst 29 1 1 1 1 1 2 3 4 (c) 1 2 3 4 (d) Figure 4.1: Wavelet-based auto-bicoherence of the vertical acceleration of ID 1, 4, 6, and 8 during the interval t = [. 6.] seconds. Contour levels are set at ([.3 :.1 :.9]). 2 Run ; ID 1; Inst 6 2 Run ; ID 4; Inst 18 1 1 1 1 1 2 3 4 (a) 1 2 3 4 (b) 2 Run ; ID 6; Inst 49 2 Run ; ID 8; Inst 29 1 1 1 1 1 2 3 4 (c) 1 2 3 4 (d) Figure 4.11: Wavelet-based auto-bicoherence of the vertical acceleration of of ID 1, 4, 6, and 8 during the interval t = [8. 9.] seconds. Contour levels are set at ([.3 :.1 :.9]). 111

Maneuver Induced LCO; Lateral Accelerations Time series of the measured lateral accelerations of the wing-tip launcher ID 1, the underwing launcher ID 4, the pylon-wing interface ID 6, and the pylon-wing interface ID 8, are presented in Figure 4.12. The onset of LCO is indicated by the large accelerations, ±.7 g, of the underwing launcher. This instrument follows a similar growth envelope to the vertical component plotted in Figure 4.. The lateral accelerations at the other stations do not show a distinct growth envelope as limit cycle oscillations develop, nor do they exhibit a simple relationship between the horizontal distance from the fuselage and the magnitude of acceleration... 1 1...2.2 Run ; Lateral Accelerations 1 2 3 4 6 7 8 9 1 11 Figure 4.12: Lateral accelerations of ID 1, 4, 6, and 8. ID 1 Inst. ID 4 Inst. 17 ID 6 Inst. 48 ID 8 Inst. 28 The power spectra of the lateral accelerations of wing-tip launcher ID 1, underwing launcher ID 4, pylon-wing interface ID 6, and the pylon-wing interface ID 8, are shown in Figure 4.13. All instruments show a strong 8.2 Hz component; the same frequency observed in the vertical accelerations. A quadratic nonlinearity in the lateral acceleration of the wing-tip launcher is suggested by the strong harmonic power near 16. Hz which was not present in the vertical accelerations. Quadratic and cubic nonlinearities are also suggested by the strong response near 24. Hz on the spectra of the wing-tip launcher s accelerations, ID 1, which is roughly the same order of magnitude as the 8.2 Hz component. A cubic nonlinearity is also suggested in the under-wing launcher, ID 4, 112

by the strong 24. Hz component, although it is significantly attenuated compared to the harmonic in the wing-tip launcher. 1 2 Run ; ID 1; Inst. 1 Run ; ID 4; Inst. 17 1 3 1 4 1 2 Power 1 Power 1 4 1 6 1 7 1 6 1 8 1 2 3 4 Frequency (Hz) (a) 1 8 1 2 3 4 Frequency (Hz) (b) 1 3 Run ; ID 6; Inst. 48 1 4 Run ; ID 8; Inst. 28 1 4 Power 1 Power 1 1 6 1 7 1 2 3 4 Frequency (Hz) (c) 1 6 1 2 3 4 Frequency (Hz) (d) Figure 4.13: Power spectra of lateral accelerations at ID 1 (a), 4 (b), 6 (c), and 8 (d). An expanded view of the lateral accelerations of the wing-tip launcher ID 1, underwing launcher ID 4, pylon-wing interface ID 6, and pylon-wing interface ID 8, and their wavelet transform magnitudes are presented in Figures 4.14a, b, c, and d. Although the time series do not indicate the onset of LCO, the wavelet transform magnitudes of all instruments indicate harmonic oscillations near 8 Hz. The lateral accelerations of the wing-tip launcher, ID 1, plot a, shows a consistent 8 Hz and an intermittent 16 Hz component as well as higher harmonics. The underwing launcher, ID 4 (plot b), and both pylon wing interfaces, ID 6 and ID 8 (plots c and d), contain energy at the 8 Hz component and more intermittently at the higher frequency components as well. 113

Run ; ID 1; Inst Run ; ID 4; Inst 17 1.. 4 6 1 4 6 2 1 6 6 4 (b) Run ; ID 6; Inst 48 Run ; ID 8; Inst 28.2 6.2 6.2 4 6 6 6 6 6 3 Freq (Hz) 3 Freq (Hz) 6 1 (a) 2 1 4 6 2.2 4 6 3 Freq (Hz) Freq (Hz) 3 4 6 6 2 1 4 (c) (d) Figure 4.14: Expanded view of the lateral accelerations and wavelet transform magnitudes at ID 1 (a), 4 (b), 6 (c), and 8 (d) during LCO. 114

2 Run ; ID 1; Inst 2 Run ; ID 4; Inst 17 1 1 1 1 1 2 3 4 (a) 1 2 3 4 (b) 2 Run ; ID 6; Inst 48 2 Run ; ID 8; Inst 28 1 1 1 1 1 2 3 4 (c) 1 2 3 4 (d) Figure 4.1: Wavelet-based auto-bicoherence of the lateral acceleration of ID 1, 4, 6, and 8 during the interval t = [2. 3.] seconds. Contour levels are set at ([.3 :.1 :.9]). Wavelet auto-bicoherence plots of the lateral accelerations over three different time intervals are presented in Figures 4.1, 4.16, and 4.17. The results show a high and consistent quadratic coupling in the wing-tip launcher, ID 1, plot a, at (8Hz, 8Hz, 16Hz) over the three intervals chosen. A weaker nonlinearity also exists between (8Hz, 16Hz, 24Hz). Intermittent quadratic coupling is present in the lateral acceleration of the underwing launcher, ID 4, plot b, around (3Hz, 8Hz, 38Hz) and at (24Hz, 8Hz, 32Hz). Neither pylon-wing interface, ID 6, nor 8, plots c and d, show any significant quadratic coupling. 11

2 Run ; ID 1; Inst 2 Run ; ID 4; Inst 17 1 1 1 1 1 2 3 4 (a) 1 2 3 4 (b) 2 Run ; ID 6; Inst 48 2 Run ; ID 8; Inst 28 1 1 1 1 1 2 3 4 (c) 1 2 3 4 (d) Figure 4.16: Wavelet-based auto-bicoherence of the lateral acceleration of ID 1, 4, 6, and 8 during the interval t = [. 6.] seconds. Contour levels are set at ([.3 :.1 :.9]). 2 Run ; ID 1; Inst 2 Run ; ID 4; Inst 17 1 1 1 1 1 2 3 4 (a) 1 2 3 4 (b) 2 Run ; ID 6; Inst 48 2 Run ; ID 8; Inst 28 1 1 1 1 1 2 3 4 (c) 1 2 3 4 (d) Figure 4.17: Wavelet-based auto-bicoherence of the lateral acceleration of ID 1, 4, 6, and 8 during the interval t = [8. 9.] seconds. Contour levels are set at ([.3 :.1 :.9]). 116

Maneuver Induced LCO; Cross-Coupling Between Vertical and Lateral Accelerations In order to complete the characterization of maneuver induced quadratic coupling, the waveletbased cross-bicoherence between the lateral and vertical accelerations at each location were determined over the same three time periods. The results are presented in Figures 4.18, 4.19, and 4.2. The wing-tip launcher, ID 1, contains persistent quadratic coupling at (8Hz, 8Hz, 16Hz). Both the wing-tip and under-wing launchers, ID 1 and ID 4, show weak and inconsistent coupling at (8Hz,.Hz, 2.Hz). No persistent quadratic coupling is indicated at either of the pylon-wing interfaces ID 6 and ID 8. 117

2 Run ; ID 1; Inst 6 & Inst Run ; ID 4; Inst 18 & Inst 17 2 1 1 1 1 2 2 3 3 4 2 4 (a) Run ; ID 6; Inst 49 & Inst 48 2 4 2 4 (b) Run ; ID 8; Inst 29 & Inst 28 2 1 1 1 1 2 2 3 3 4 2 4 (c) 4 2 4 (d) Figure 4.18: Cross-bicoherence between the lateral and vertical accelerations at ID 1, 4, 6, and 8 during the interval [2. 3.] seconds. Contour levels are set at ([.3 :.1 :.9]). 118

2 Run ; ID 1; Inst 6 & Inst Run ; ID 4; Inst 18 & Inst 17 2 1 1 1 1 2 2 3 3 4 2 4 (a) Run ; ID 6; Inst 49 & Inst 48 2 4 2 4 (b) Run ; ID 8; Inst 29 & Inst 28 2 1 1 1 1 2 2 3 3 4 2 4 (c) 4 2 4 (d) Figure 4.19: Cross-bicoherence between the lateral and vertical accelerations at ID 1, 4, 6, and 8 during the interval [. 6.] seconds. Contour levels are set at ([.3 :.1 :.9]). 119

2 Run ; ID 1; Inst 6 & Inst Run ; ID 4; Inst 18 & Inst 17 2 1 1 1 1 2 2 3 3 4 2 4 (a) Run ; ID 6; Inst 49 & Inst 48 2 4 2 4 (b) Run ; ID 8; Inst 29 & Inst 28 2 1 1 1 1 2 2 3 3 4 2 4 (c) 4 2 4 (d) Figure 4.2: Cross-bicoherence between the lateral and vertical accelerations at ID 1, 4, 6, and 8 during the interval [8. 9.] seconds. Contour levels are set at ([.3 :.1 :.9]). 12

Maneuver Induced LCO; Cubic Nonlinearity The power spectrum of lateral motion of ID 1, the wing-tip launcher, suggested a cubic nonlinearity between the antisymmetric wing bending mode, 8.2 Hz, and its frequency triple at 24. Hz. An expanded view of the time series of the lateral and vertical accelerations during the interval of LCO is presented in Figure 4.21. The lateral motion is shown in blue and has a more complicated structure than the simple harmonic motion displayed by the vertical component (green). The components are in phase, however, the lateral component contains higher harmonics in addition to the 8.2 Hz component. Lat..4.2.2.4 Run ; ID 1; Inst. and 6 1 1. 2 2. 3 3. 4 2 2 4 Vert. Figure 4.21: Lateral (blue) and vertical (green) accelerations of the the wing-tip launcher ID 1 during an interval of LCO. The wing-tip launcher s lateral acceleration contains a cubic nonlinearity at (8.2Hz, 8.2Hz, 8.2Hz, and 24.6Hz) as calculated by the auto-tricoherence, shown in Figure 4.22. Only the highest levels of auto-tricoherence are shown. The results are repeated in Figure 4.23, presented in a two dimensional plot, as introduced in Chapter 2. Frequency summation is indicated clearly at (8.2Hz, 8.2Hz, 8.2Hz, and 24.6Hz) in the figure. A cubic nonlinearity was not found in the underwing launcher ID 4, however Hajj and Beran [8], found evidence of a cubic nonlinearity in this instrument by calculating the tricoherence over a different window. 121

Run ; ID 1; Inst..6.8.6.4.2 f 3 (Hz) 1 4..48 2 1 1 f 2 (Hz) 1 2 3 f 1 (Hz).46.44.42 Figure 4.22: A cubic nonlinearity is identified in the wing-tip launcher s lateral acceleration ID 1, as indicated by the high auto-tricoherence value at (8.2Hz, 8.2Hz, 8.2Hz, 24.6Hz). 1 Run ; ID 1; Inst..9.8 Tricoherence Value.7.6..4.3.2 f 1 f 2.1 f 3 Σ(f i 1 1 2 2 3 3 4 Figure 4.23: The auto-tricoherence is repeated using a novel plotting technique. A high value of auto-tricoherence is indicated at (8.2Hz, 8.2Hz, 8.2Hz, 24.6Hz). 122

4.4 Nonlinear Aspects of Mechanically-Induced LCO Details of the flight conditions for Run 2 are presented in Figure 4.24. Starting with the top plot, the Mach number is roughly constant with a value near.8. The altitude is also constant around 1, ft., and the angle of attack is constant around 2. The vertical acceleration of the wing-tip launcher ID 1 is shown in the final plot of Figure 4.24. The sudden increase in acceleration to ±3 g s around 27 seconds was caused by deliberate excitations of the flaperons. Mach Number Alt. (ft).9.8.7 1.2 x 14 1.1 1 Run 2 AoA ( ) 4 2 2 4 1 2 3 4 6 Figure 4.24: Mach number, Altitude, angle of attack, and vertical acceleration of the wing-tip launcher ID 1 for Run 2 [42] The amplitude of the right flaperon and its wavelet transform magnitude are presented in Figure 4.2. Distinct harmonic oscillations are present at 8 Hz between [24. 27.] seconds. obviously, the flaperon s motion, 3/4 in amplitude, forces the wing into limit cycle oscillations. The flaperons on both wings were actuated in an antisymmetric motion, as shown in Figure 4.26a. The amplitude of the right flaperon, shown in blue, is always 18 out of phase with the amplitude of the left flaperon, shown in black, over the chosen interval. The vertical accelerations of the wing-tip launcher (green), appear closely related to the actuation of the flaperon (blue), in both frequency and duration as shown in Figure 4.26b. 123

2 Run 2; Inst. Amp. ( ) 2 3 2 1 1 2 3 4 6 Figure 4.2: Right flaperon motion and wavelet transform magnitude, notice the harmonic content around 27 seconds. Amp. ( ) Amp. ( ) 1. 1. Run 2; Right and Left Flaperons Right Left. 24 24. 2 2. 26 26. 27 27. 28 28. 29 (a) Run 2; Flaperon and ID 1; Inst 6 3 1 2.8 1.6 1.4 2.2 3 24 24. 2 2. 26 26. 27 27. 28 28. 29 4 (b) Figure 4.26: Right (blue) and left (black) flaperon motion during mechanically forced event (top), and right flaperon (blue) and vertical wing-tip acceleration, ID 1 instrument 6, (green) during the mechanically forced event (bottom). 124

Mechanically Forced LCO; Vertical Accelerations Time series of the vertical accelerations of the wing-tip launcher ID 1 (B. L. 183 in), the underwing launcher ID 4 (B. L. 17 in), the pylon-wing interface ID 6 (B. L. 16.3), and the pylonwing interface ID 8 (B. L. 117.6 in), are presented in Figure 4.27. All four accelerations display a sudden increase in magnitude near 24. seconds, the beginning of flaperon excitation, and start to decay after 27. seconds, as the flaperon excitation ceases. The peak magnitude of the accelerations varies between ±.2 g at B. L. 117.6 and ±2 g at B. L. 183. 2 2 1 1 Run 2; Vertical Accelerations 1 2 3 4 6 ID 1 Inst. 6 ID 4 Inst. 18 ID 6 Inst. 49 ID 8 Inst. 29 Figure 4.27: Vertical accelerations of the wing-tip launcher ID 1 (B. L. 183 in), the underwing launcher ID 4 (B. L. 17 in), the pylon-wing interface ID 6 (B. L. 16.3), and the pylon-wing interface ID 8 (B. L. 117.6 in) during Run 2. Detailed plots of the measured accelerations at the four locations and the corresponding magnitudes of their wavelet transforms are presented in Figure 4.28a, b, c, and d. The results show that all accelerations contain a strong 8 Hz component coincident with the actuation of the flaperon. The vertical acceleration of the wing-tip launcher, ID 1, and the pylon-wing interface, ID 8, contain a strong 16 Hz component. The vertical accelerations of the underwing launcher shows an attenuation of response at 8 Hz around 27 seconds where a low energy 16 Hz component is indicated. The vertical acceleration of the pylon-wing interface ID 6 contains a weak 16 Hz component as well. Wavelet-based auto-bicoherence estimates of the above signals were calculated over the two 12

Run 2; ID 1; Inst 6 Run 2; ID 4; Inst 18 2 2 3 3 3 2 2 3 3 3 Freq (Hz) 2 1 Freq (Hz) 2 1 2 2 3 3 (a) 2 2 3 3 (b) 2 Run 2; ID 6; Inst 49. Run 2; ID 8; Inst 29. 2 2 2 3 3 3 1 2 2 3 3 3 Freq (Hz) 2 1 Freq (Hz) 2 1 2 2 3 3 (c) 2 2 3 3 (d) Figure 4.28: Expanded view of the vertical accelerations of the wing-tip launcher ID 1, the underwing launcher ID 4, the pylon-wing interface ID 6, and the pylon-wing interface 8 and their wavelet transform magnitudes during the forcing event. 126

2 Run 2; ID 1; Inst 6 2 Run 2; ID 4; Inst 18 1 1 1 1 1 2 3 4 (a) 1 2 3 4 (b) 2 Run 2; ID 6; Inst 49 2 Run 2; ID 8; Inst 29 1 1 1 1 1 2 3 4 (c) 1 2 3 4 (d) Figure 4.29: Wavelet-based auto-bicoherence of the vertical acceleration of ID 1, 4, 6, and 8 over the interval t = [26. 27.] second. Contour levels are set at ([. :.1 :.9]). intervals t = [26. 27.] seconds and t = [27. 28.] seconds. The results are plotted in Figures 4.29 and 4.3 respectively, with contour levels of (. :.1 :.9). The results show strong quadratic coupling in the wing-tip launcher ID 1, and both pylon-wing interfaces ID 6 and ID 8 at (8Hz, 8Hz, 16Hz) over both intervals. Quadratic coupling is not present in the underwing launcher ID 4, shown in plots b of the different figures. For this particular instrument, the analysis interval of t = [26. 27.] seconds, presented in Figure 4.29b, corresponds to the largest response. The interval t = [27. 28.] seconds, presented in Figure 4.3b, corresponds to significantly attenuated accelerations. 127

2 Run 2; ID 1; Inst 6 2 Run 2; ID 4; Inst 18 1 1 1 1 1 2 3 4 (a) 1 2 3 4 (b) 2 Run 2; ID 6; Inst 49 2 Run 2; ID 8; Inst 29 1 1 1 1 1 2 3 4 (c) 1 2 3 4 (d) Figure 4.3: Wavelet-based auto-bicoherence of the vertical acceleration of ID 1, 4, 6, and 8 over the interval t = [27. 28.] second. Contour levels are set at ([. :.1 :.9]). 128

Mechanically Forced LCO; Cross Coupling Between Vertical and Lateral Accelerations Lateral accelerations of the wing-tip launcher ID 1 (B. L. 183 in), the underwing launcher ID 4 (B. L. 17 in), the pylon-wing interface ID 6 (B. L. 16.3), and the pylon-wing interface ID 8 (B. L. 117.6 in), all experience an increase in magnitude coincident with the flaperon excitation as shown in Figure 4.31. The accelerations at ID 4 experience a distinct growth and decay envelope similar to the vertical acceleration at this location, while the other three instruments experience a nominal increase during the period of excitation... 1. 1..2.2.2.2 Run 2; Lateral Accelerations 1 2 3 4 6 ID 1 Inst. ID4 Inst. 17 ID 6 Inst. 48 ID 8 Inst. 28 Figure 4.31: Lateral accelerations of the wing-tip launcher ID 1 (B. L. 183 in), the underwing launcher ID 4 (B. L. 17 in), the pylon-wing interface ID 6 (B. L. 16.3), and the pylon-wing interface ID 8 (B. L. 117.6 in) during Run 2. Detailed plots of the lateral accelerations at the four locations and the corresponding magnitudes of their wavelet transforms are presented in Figure 4.32a, b, c, and d. These results show that all four lateral accelerations contain a strong 8 Hz component from 24. 27 seconds. The wing-tip launcher ID 1, and both pylon-wing interfaces ID 6 and ID 8 contain strong 16 Hz harmonics and intermittent higher frequencies. The 16 Hz harmonic is curiously absent in the underwing launcher ID 4. Wavelet-based auto-bicoherence estimates of the above signals over the two intervals t = [26. 27.] seconds and t = [27. 28.] seconds are plotted in Figures 4.33 and 4.34 respectively with 129

. Run 2; ID 1; Inst 1 Run 2; ID 4; Inst 17. 2 2 3 3 3 1 2 2 3 3 3 Freq (Hz) 2 1 Freq (Hz) 2 1 2 2 3 3 (a) 2 2 3 3 (b).1 Run 2; ID 6; Inst 48.2 Run 2; ID 8; Inst 28.1.2 2 2 3 3 3.2 2 2 3 3 3 Freq (Hz) 2 1 Freq (Hz) 2 1 2 2 3 3 (c) 2 2 3 3 (d) Figure 4.32: Expanded view of the lateral accelerations of the wing-tip launcher ID 1, the underwing launcher ID 4, the pylon-wing interface ID 6, and the pylon-wing interface 8 and their wavelet transform magnitudes during the forcing event. 13

2 Run 2; ID 1; Inst 2 Run 2; ID 4; Inst 17 1 1 1 1 1 2 3 4 (a) 1 2 3 4 (b) 2 Run 2; ID 6; Inst 48 2 Run 2; ID 8; Inst 28 1 1 1 1 1 2 3 4 (c) 1 2 3 4 (d) Figure 4.33: Wavelet-based auto-bicoherence of the lateral acceleration of ID 1, 4, 6, and 8 over the interval of [26. 27.] seconds. Contour levels are set at ([. :.1 :.9]). contour levels of (. :.1 :.9). Results for both intervals are similar. Strong quadratic coupling is present in the lateral accelerations at ID 1, 6, and 8 at (8Hz, 8Hz, 16Hz). The pylon-wing interface, ID 6, contains strong coupling at (16Hz, 16Hz, 32Hz) as well. The underwing launcher, ID 4, plot b, contains almost insignificant quadratic coupling near (8Hz, 8Hz, 16Hz). 131

2 Run 2; ID 1; Inst 2 Run 2; ID 4; Inst 17 1 1 1 1 1 2 3 4 (a) 1 2 3 4 (b) 2 Run 2; ID 6; Inst 48 2 Run 2; ID 8; Inst 28 1 1 1 1 1 2 3 4 (c) 1 2 3 4 (d) Figure 4.34: Wavelet-based auto-bicoherence of the lateral acceleration of ID 1, 4, 6, and 8 over the interval of [27. 28.] seconds. Contour levels are set at ([. :.1 :.9]). 132

Mechanically Forced LCO; Combined Accelerations The cross-bicoherence between the vertical and lateral accelerations of each instrument were calculated over both intervals in order to gain additional understanding into the physical interactions of the flaperon induced limit cycle oscillations. Quadratic coupling is present between the vertical and lateral acceleration of the wing-tip launcher, ID 1, and both pylon-wing interfaces, ID 6 and ID 8 at (8Hz, 8Hz, 16Hz) as well as (16Hz, 8Hz, 8Hz), as shown in Figures 4.3 and 4.36. The frequency triplet (16Hz, 8Hz, 8Hz) is a point of symmetry since both the vertical and lateral accelerations contain phase coupled 8 Hz and 16 Hz components. Quadratic coupling is also present in the pylon-wing interface, ID 8, plot d, at (16Hz, 16Hz, 32Hz). Quadratic coupling is absent between the vertical and lateral components of acceleration of the underwing launcher, ID 4, plot b. This result is consistent with the lack of coupling indicated by the auto-bicoherence calculations, and is quite remarkable considering that all other stations analyzed exhibit strong quadratic coupling. 133

2 Run 2; ID 1; Inst 6 & Inst Run 2; ID 4; Inst 18 & Inst 17 2 1 1 1 1 2 2 3 3 4 2 4 (a) Run 2; ID 6; Inst 49 & Inst 48 2 4 2 4 (b) Run 2; ID 8; Inst 29 & Inst 28 2 1 1 1 1 2 2 3 3 4 2 4 (c) 4 2 4 (d) Figure 4.3: Cross-bicoherence between the lateral and vertical acceleration at ID 1, 4, 6, and 8 over the interval of [26. 27.] seconds. Contour levels are set at ([. :.1 :.9]). 134

2 Run 2; ID 1; Inst 6 & Inst Run 2; ID 4; Inst 18 & Inst 17 2 1 1 1 1 2 2 3 3 4 2 4 (a) Run 2; ID 6; Inst 49 & Inst 48 2 4 2 4 (b) Run 2; ID 8; Inst 29 & Inst 28 2 1 1 1 1 2 2 3 3 4 2 4 (c) 4 2 4 (d) Figure 4.36: Cross-bicoherence between the lateral and vertical acceleration at ID 1, 4, 6, and 8 over the interval of [27. 28.] seconds. Contour levels are set at ([. :.1 :.9]). 13

Summary of Nonlinear Couplings in Lateral and Vertical Accelerations A summary of the strongest nonlinear couplings in the lateral and vertical accelerations of all analyzed locations during maneuver induced (Run ) and mechanically forced LCO (Run 2) is presented in Table 4.3. During maneuver induced LCO, the wing-tip launcher, ID 1, exhibits significant and persistent quadratic coupling between the anti-symmetric wing bending mode and its second harmonic. The underwing launcher, ID 4, exhibits weak intermittent coupling at frequencies related to the symmetric and anti-symmetric wing bending modes. The wing-tip launcher, ID 1, also exhibits cubic coupling between anti-symmetric wing bending mode and its third harmonic. During mechanically forced LCO, quadratic coupling is both stronger and more prevalent. The vertical and lateral accelerations of the wing-tip launcher, ID 1, and both pylon wing interfaces, ID 6 and ID 8, exhibit coupling between the anti-symmetric wing bending mode and its second harmonic. The under-wing launcher, ID 4, does not exhibit quadratic coupling, however, a noteworthy observation is that its growth and decay envelope of its lateral acceleration is similar to that of its vertical acceleration. Accelerations measured at the other three locations exhibit lateral acceleration growth and decay envelopes that are quite different from their vertical counterparts. Table 4.3: Summary of the primary coupling at all eight vertical and lateral instrumentation locations for maneuver induced LCO (Run ) and mechanically forced LCO (Run 2). Description ID & Direction Maneuver Induced (Run ) Flaperon Forced (Run 2) Wing-tip launcher ID 1 Vertical (8Hz, 8Hz, 16Hz) Wing-tip launcher ID 1 Lateral (8Hz, 8Hz, 16Hz) (8Hz, 8Hz, 16Hz) Wing-tip launcher ID 1 Cross (8Hz, 8Hz, 16Hz) (8Hz, 8Hz, 16Hz) Underwing Launcher ID 4 Vertical Underwing Launcher ID 4 Lateral Inconsistent Underwing Launcher ID 4 Cross Inconsistent Pylon-wing interface ID 6 Vertical (8Hz, 8Hz, 16Hz) Pylon-wing interface ID 6 Lateral (8Hz, 8Hz, 16Hz) Pylon-wing interface ID 6 Cross (8Hz, 8Hz, 16Hz) Pylon-wing interface ID 8 Vertical (8Hz, 8Hz, 16Hz) Pylon-wing interface ID 8 Lateral (8Hz, 8Hz, 16Hz) Pylon-wing interface ID 8 Cross (8Hz, 8Hz, 16Hz) (8Hz, 8Hz, 16Hz) Persistent Cubic nonlinearity Acceleration growth and decay envelope similar to vertical instruments Indistinct high frequency weak (16Hz, 16Hz, 32Hz) as well weak 136

4. Quadratic Coupling in Flaperon/Wing-Store System As shown in the previous section, flaperon excitations cause quadratic coupling between the vertical and lateral accelerations of ID 1, 6, and 8. Treating the flaperon motion as an input and the acceleration of the wing/store system as an output, the wavelet-based cross-bicoherence is used to characterize the quadratic coupling between the flaperon excitation and the different components of the wing/store system. Quadratic Flaperon Coupling; Vertical Motion Time series of the vertical acceleration of the wing-tip launcher, ID 1, indicated by the upper righthand illustration of Figure 4.37 over the interval of 27. 28.2 seconds is presented in the top plot of the same figure. The harmonic acceleration of the wing-tip launcher and the right flaperon motion are similar as seen in the top two time series. The wing-tip launcher s vertical acceleration contains an 8 Hz component and a faint 16 Hz component as shown in the wavelet transform magnitude, presented in the third plot. The wavelet transform magnitude of the right flaperon contains only an 8 Hz component as shown in the fourth plot. Strong linear coherence between the flaperon motion and the vertical wing-tip acceleration is indicated by the high value of wavelet-based linear coherence at 8 Hz, shown in the bottom plot. Finally, quadratic coupling between the right flaperon motion and the vertical acceleration of the wing-tip launcher is confirmed by wavelet-based cross-bicoherence, plotted with contours at (. :.1 :.9). A large response is evident at (8Hz, 8Hz, 16Hz) as expected. The vertical acceleration of the underwing launcher, ID 4 instrument 18, is shown in the top plot of Figure 4.38, over the interval from 26. 27.2 seconds. Harmonic motion is present, and a secondary high frequency motion develops around 26.4 seconds. The amplitude of the flaperon is shown in the next plot. The wavelet transform of the vertical acceleration of the underwing launcher indicates a strong harmonic component at 8 Hz, as shown in the third plot. The large value of linear coherence between the flaperon and the vertical acceleration of the underwing launcher confirms linear coupling is present. No quadratic coupling exists between the flaperon and the vertical acceleration of the underwing launcher as confirmed by a low value of the cross-bicoherence, shown in the final plot. 137

The vertical acceleration of the pylon-wing interface, ID 6 instrument 49, has a similar motion to the right flaperon as shown in the top two plots of Figure 4.39. The wavelet transform magnitude of the pylon-wing interface indicates harmonic content at 8 Hz and much weaker content at 16 Hz. Flaperon motion is linearly coupled to the pylon-wing interface s vertical acceleration as indicated by the large value of linear coherence. Quadratic coupling also exists between the flaperon motion and the pylon-wing interface s vertical acceleration at (8Hz, 8Hz, 16Hz) as indicated by the large value of bicoherence. The magnitude of the vertical acceleration of the pylon-wing interface, ID 8 instrument 29, is roughly four times smaller than the vertical acceleration at ID 6. The vertical acceleration of the pylon-wing interface at ID 8 shows linear coupling with the flaperon motion at 8 Hz and quadratic coupling at (8Hz, 8Hz, 16Hz) as indicated in Figure 4.4. 138

Figure 4.37: The instrument location is indicated in the top right illustration. The other plots are: the vertical acceleration the wing-tip launcher ID 1, (top), flaperon motion (second), wavelet transform magnitude of the vertical acceleration of the wing-tip launcher (third), wavelet transform magnitude of the flaperon motion (fourth), wavelet-based linear coherence between the flaperon and the vertical acceleration of the wing-tip launcher ID 1 (bottom), and wavelet-based crossbicoherence treating the flaperon as an input and the wing-tip launcher s vertical acceleration as an output. 139

Figure 4.38: The instrument location is indicated in the top right illustration. The other plots are: vertical acceleration of the underwing launcher ID 4, (top), flaperon motion (second), wavelet transform magnitude of the vertical acceleration of the underwing launcher (third), wavelet transform magnitude of the flaperon motion (fourth), wavelet-based linear coherence between the flaperon and the vertical acceleration of the underwing launcher ID 4 (bottom), and wavelet-based crossbicoherence between the flaperon and the underwing launcher s vertical acceleration. 14

Figure 4.39: The instrument location is indicated in the top right illustration. The other plots are: vertical acceleration of the pylon-wing interface ID 6, (top), flaperon motion (second), wavelet transform magnitude of the vertical acceleration of the pylon-wing interface (third), wavelet transform magnitude of the flaperon motion (fourth), wavelet-based linear coherence between the flaperon and the vertical acceleration of the pylon-wing interface (bottom), and wavelet-based cross-bicoherence between the flaperon and the pylon-wing interface s vertical acceleration. 141

Figure 4.4: The instrument location is indicated in the top right illustration. The other plots are: vertical acceleration of the pylon-wing interface ID 8, (top), flaperon motion (second), wavelet transform magnitude of the vertical acceleration of the pylon-wing interface (third), wavelet transform magnitude of the flaperon motion (fourth), wavelet-based linear coherence between the flaperon and the vertical acceleration of the pylon-wing interface (bottom), and wavelet-based cross-bicoherence between the flaperon and the pylon-wing interface s vertical acceleration. 142

Quadratic Flaperon Coupling; Lateral Motion A similar analysis is repeated for the lateral accelerations at locations ID 1, 4, 6, and 8. The wing-tip launcher s lateral acceleration contains an 8 Hz component, similar to the right flaperon motion, and higher frequencies, as indicated in the first four plots in Figure 4.41 over the interval 27. 28.2 seconds. Both linear and quadratic coupling exist between the flaperon motion and the lateral acceleration of the wing-tip launcher as indicated by the final two plots in the figure. The lateral acceleration of the underwing launcher, ID 4 instrument 17, contains an 8 Hz harmonic component and weak energy at higher frequencies as indicated by the first four plots in Figure 4.42. While the flaperon motion is linearly coupled to the lateral acceleration of the underwing launcher, quadratic coupling is almost absent as indicated by the single contour (.) at (8Hz, 8Hz, 16Hz). The lateral acceleration of the pylon-wing interface, ID 6 contains an 8 Hz component and many higher frequencies as indicated in Figure 4.43. Both linear, at 8 Hz, and quadratic coupling, at (8Hz, 8Hz, 16Hz), exist between the lateral acceleration of the pylon-wing interface ID 6 and the right flaperon as indicated in the figure. Further toward the fuselage, the lateral acceleration of the underwing launcher ID 8, exhibits similar behavior as shown in Figure 4.44. The lateral acceleration contains a strong 8 Hz component and higher frequencies as well. Both linear coupling, at 8 Hz, and quadratic coupling, at (8Hz, 8Hz, 16Hz) exist between the flaperon and the lateral acceleration of the underwing launcher ID 8. 143

Figure 4.41: The instrument location is indicated in the top right illustration. The other plots are: lateral acceleration of the wing-tip launcher ID 1, (top), flaperon motion (second), wavelet transform magnitude of the lateral acceleration of the wing-tip launcher (third), wavelet transform magnitude of the flaperon motion (fourth), wavelet-based linear coherence between the flaperon and the lateral acceleration of the wing-tip launcher (bottom), and wavelet-based cross-bicoherence between the flaperon and the wing-tip launcher s lateral acceleration. 144

Figure 4.42: The instrument location is indicated in the top right illustration. The other plots are: lateral acceleration of the underwing launcher ID 4, (top), flaperon motion (second), wavelet transform magnitude of the lateral acceleration of the underwing launcher (third), wavelet transform magnitude of the flaperon motion (fourth), wavelet-based linear coherence between the flaperon and the lateral acceleration of the underwing launcher (bottom), and wavelet-based cross-bicoherence between the flaperon and the underwing launcher s lateral acceleration. 14

Figure 4.43: The instrument location is indicated in the top right illustration. The other plots are: lateral acceleration of the pylon-wing interface ID 6, (top), flaperon motion (second), wavelet transform magnitude of the lateral acceleration of the pylon-wing interface (third), wavelet transform magnitude of the flaperon motion (fourth), wavelet-based linear coherence between the flaperon and the lateral acceleration of the pylon-wing interface (bottom), and wavelet-based cross-bicoherence between the flaperon and the pylon-wing interface s lateral acceleration. 146

Figure 4.44: The instrument location is indicated in the top right illustration. The other plots are: lateral acceleration of the pylon-wing interface ID 8, (top), flaperon motion (second), wavelet transform magnitude of the lateral acceleration of the pylon-wing interface (third), wavelet transform magnitude of the flaperon motion (fourth), wavelet-based linear coherence between the flaperon and the lateral acceleration of the pylon-wing interface (bottom), and wavelet-based cross-bicoherence between the flaperon and the pylon-wing interface s lateral acceleration. 147

4.6 Growth and Decay of Quadratically Coupled LCO Wavelet-based bicoherence can be used as a metric to determine the extent of quadratic nonlinearity leading to Limit Cycle Oscillations. The onset of quadratic coupling may indicate a change in mechanism of LCO and may be used as an indicator to avoid certain maneuvers. In addition, a subsequent low value of bicoherence, indicating the decay of coupling, may be used to identify the mitigation of LCO while the magnitude of LCO indicates otherwise. An example of this application is presented below, based on the data from Run 2. The vertical acceleration of the wing-tip launcher, ID 1, and its wavelet transform magnitude are shown in Figure 4.4. Strong 8 Hz harmonic motion develops due to the mechanical excitation of the flaperon. Quadratic coupling is suggested by the presence of a 16 Hz harmonic between 26. and 28. seconds. 3 2 1 Run 2; ID 1; Inst. 6 23 24 2 26 27 28 29 3 Figure 4.4: Vertical acceleration of the wing-tip launcher, ID 6 (top), and its wavelet transform magnitude (bottom), during an interval of flaperon induced LCO. The wavelet-based auto-bicoherence was calculated over eight 1. second long intervals starting at t = 23. seconds and advancing by 3/4 of a second to 28.2 seconds. The time span of each interval is indicated in Table 4.4. The resulting bicoherence levels, shown in Figure 4.46 with contours at (. :.1 :.9), track the level of quadratic couplings. The first three intervals show an absence of quadratic coupling. A response develops at (8Hz, 8Hz, 16Hz) during the fourth interval and grows to a maximum at the sixth interval. A sharp decrease in the coupling occurs in the seventh interval and an absence of coupling is shown in the eighth interval. The growth and decay of the quadratic coupling can also be discerned from Figure 4.47. The level of bicoherence is plotted 148

Table 4.4: Starting and ending times of the eight intervals used to track the strength of quadratic coupling in LCO. Interval Number Start End Time(s) 1 23. 24. 2 23.7 2.2 3 24. 26. 4 2.2 26.7 26. 27. 6 26.7 28.2 7 27. 29. 8 28.2 29.7 as a function of time using the end of the interval as a reference. The strength of the quadratic coupling increases with the onset of LCO and reaches a peak, around 28. seconds, while the envelope of limit cycle oscillations remains roughly constant. The coupling level decreases sharply, around 29. seconds, where the magnitude of the limit cycle oscillations is slightly decreased but otherwise, gives no indication of a distinct decoupling event. 149

2 Run 2; ID 1; Inst 6; Interval 1 2 Run 2; ID 1; Inst 6; Interval 2 1 1 1 1 1 2 3 4 1 2 3 4 2 Run 2; ID 1; Inst 6; Interval 3 2 Run 2; ID 1; Inst 6; Interval 4 1 1 1 1 1 2 3 4 1 2 3 4 2 Run 2; ID 1; Inst 6; Interval 2 Run 2; ID 1; Inst 6; Interval 6 1 1 1 1 1 2 3 4 1 2 3 4 2 Run 2; ID 1; Inst 6; Interval 7 2 Run 2; ID 1; Inst 6; Interval 8 1 1 1 1 1 2 3 4 1 2 3 4 Figure 4.46: Auto-bicoherence levels over eight intervals track the quadratic coupling of the wing-tip launcher s vertical acceleration in Run 2. 1

Bicoherence value 1. 24 2 26 27 28 29 Figure 4.47: Level of quadratic coupling as a function of time, using the end of the calculation interval as a reference. 11