F-16 Quadratic LCO Identiﬂcation - Virginia Tech

Chapter 4

F-16 Quadratic LCO Identification

The store configuration of an F-16 influences the flight conditions at which limit cycle oscilla-

tions 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 mod-

els 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 com-

pared. 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 388th Tactical Fighter Wing at Hill Air

Force Base, Utah [56]. As a successor to the F-15, 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 [56]. 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 [57]. During supersonic flight, the center of gravity is in front of the center of pressure

101

Figure 4.1: F-16 with stores on the wing [42]

yielding a statically stable configuration [56].

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

102

between the ground crew and the pilot during LCO testing ([36] and [5]).

Variations of the peak magnitude of the LCO have been shown to vary with the Mach num-

ber 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 = 0.85, 5, 000 ft. and all conditions at M = 0.80 and

below. The other data points at M = 0.85 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 = 0.9.

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 = 0.95. Both Runs 2 and 5 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 10, 000

ft. and M = 0.95 and is referred to as Run 5. In this run, limit cycle oscillations occurred over a

period of about 20 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 10, 000 ft. and M = 0.8.

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 5 0.95 10,000 [40− 60] seconds Maneuver2 2 0.80 10,000 [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

103

0.7 0.75 0.8 0.85 0.9 0.950

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Mach Number

Max

|Acc

el.|

(g)

Max Magnitude; ID 1; Inst. 6

Run 2

Run 5

10000 ft.5000 ft.2000 ft.

(a)

0.7 0.75 0.8 0.85 0.9 0.950

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Mach Number

RM

S A

ccel

. (g)

RMS; ID 1; Inst. 6

Run 2

Run 5

10000 ft.5000 ft.2000 ft.

(b)

Figure 4.2: Wing-tip launcher, ID 1, vertical accelerations at three different altitudes as a functionof increasing Mach number; max magnitude (a) and RMS (b).

Table 4.2: Accelerometer locations and instrumentation numbers for vertical and lateral accelerom-eters.

Name ID x (in) Baseline (in) Vert Inst. # Lat Inst. # Plot LetterWing-tip launcher 1 318 183 6 5 a

Underwing Launcher 4 308 157 18 17 bPylon-wing Interface 6 368.3 156.3 49 48 cPylon-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.

104

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

105

Maneuver Induced LCO; Vertical Accelerations

Several parameters of the flight conditions for Run 5 are shown in Figure 4.4. Starting with

the top plot, the Mach number increases from 0.87 to 0.95, the altitude is constant around 10, 000

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

0.9 near t = 20 seconds and persist until t = 70 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 = 40 seconds.

0.8

0.9

1Run 5

Mac

h N

umbe

r

1

1.05x 10

4

Alt.

(ft)

0

5

AoA

(°)

0 20 40 60 80 100−4−2

024

Acc

el. (

g)

Time (s)

Figure 4.4: Mach number, altitude, angle of attack, and vertical wing-tip acceleration for Run 5[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. 157 in), the pylon-wing in-

terface ID 6 (B. L. 156.3), and the pylon-wing interface ID 8 (B. L. 117.6 in), are shown in Figure

4.5. 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. 156.3 and ±0.25g

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

106

−5

0

5A

ccel

. (g)

Run 5; Vertical Accelerations

ID 1

Inst. 6

−5

0

5

Acc

el. (

g)

ID 4

Inst. 18

−1

0

1

Acc

el. (

g)

ID 6

Inst. 49

10 20 30 40 50 60 70 80 90 100 110−1

0

1

Time (s)

Acc

el. (

g)

ID 8

Inst. 29

Figure 4.5: Vertical acceleration at ID 1, 4, 6, and 8. A large response is present from 20 − 70seconds at all four locations; Run 5

presented in Figure 4.6. Limit cycle oscillations are indicated by strong harmonic content at 8 Hz

starting at the time near 20 seconds and lasting for about 50 seconds with the largest amplitude

near t = 55 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 5.5 Hz, the wing’s symmetric bending mode [39] and

[58]. In addition, spectra of the accelerations at ID 1, 4, and 6 show an increased response at 24.5

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 45 seconds to 70 seconds in Figure 4.8a, b, c, and d

107

−5

0

5A

ccel

. (g)

Run 5; ID 1; Inst. 6

Time (s)

Fre

q. (

Hz)

0 20 40 60 80 100

10

20

30

Figure 4.6: Vertical acceleration of the wing-tip launcher (top) and its wavelet transform magnitude(bottom); LCO is strongest around 55 seconds.

0 10 20 30 40 5010

−6

10−4

10−2

100

102

Frequency (Hz)

Pow

er


(a)

0 10 20 30 40 5010

−6

10−5

10−4

10−3

10−2

10−1

100

Frequency (Hz)

Pow

er


(b)

0 10 20 30 40 5010

−6

10−5

10−4

10−3

10−2

10−1

100

Frequency (Hz)

Pow

er


(c)

0 10 20 30 40 5010

−6

10−5

10−4

10−3

10−2

Frequency (Hz)

Pow

er


(d)

Figure 4.7: Power spectra of vertical accelerations at ID 1 (a), 4 (b), 6 (c), and 8 (d).

108

45 50 55 60 65−5

0

5

Acc

el. (

g)

Run 5; ID 1; Inst 6

Time (s)

Fre

q (H

z)

45 50 55 60 65

10

20

30

(a)

45 50 55 60 65−2

0

2

Acc

el. (

g)

Run 5; ID 4; Inst 18

Time (s)

Fre

q (H

z)

45 50 55 60 65

10

20

30

(b)

45 50 55 60 65−1

0

1

Acc

el. (

g)


Time (s)

Fre

q (H

z)

45 50 55 60 65

10

20

30

(c)

45 50 55 60 65−1

−0.5

0

0.5

Acc

el. (

g)


Time (s)

Fre

q (H

z)

45 50 55 60 65

10

20

30

(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 local-

ized nature of wavelet-based higher order spectra, three 1.5 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 [52.0 − 53.5] seconds. Contour levels set at [0.3 : 0.1 : 0.9] were chosen to span

a greater range than normal in order to detect even small coupling levels. The same observations

109

can be made over the interval [55.0 − 56.5] seconds as presented in Figure 4.10a, b, c, and d and

again over the interval [58.0− 59.5] seconds, presented in Figure 4.11a, b, c, and d.

Freq. (Hz)

Fre

q. (

Hz)

Run 5; ID 1; Inst 6

0 10 20 30 400

5

10

15

20

(a)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(b)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(c)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(d)

Figure 4.9: Wavelet-based auto-bicoherence of the vertical accelerations at ID 1, 4, 6, and 8 duringthe interval t = [52.0− 53.5] seconds. Contour levels are set at ([0.3 : 0.1 : 0.9]).

110

Freq. (Hz)

Fre

q. (

Hz)

Run 5; ID 1; Inst 6

0 10 20 30 400

5

10

15

20

(a)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(b)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(c)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(d)

Figure 4.10: Wavelet-based auto-bicoherence of the vertical acceleration of ID 1, 4, 6, and 8 duringthe interval t = [55.0− 56.5] seconds. Contour levels are set at ([0.3 : 0.1 : 0.9]).

Freq. (Hz)

Fre

q. (

Hz)

Run 5; ID 1; Inst 6

0 10 20 30 400

5

10

15

20

(a)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(b)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(c)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(d)

Figure 4.11: Wavelet-based auto-bicoherence of the vertical acceleration of of ID 1, 4, 6, and 8during the interval t = [58.0− 59.5] seconds. Contour levels are set at ([0.3 : 0.1 : 0.9]).

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, ±0.75 g, of the underwing

launcher. This instrument follows a similar growth envelope to the vertical component plotted in

Figure 4.5. 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.

−0.5

0

0.5

Acc

el. (

g)

Run 5; Lateral Accelerations

ID 1

Inst. 5

−1

0

1

Acc

el. (

g)

ID 4

Inst. 17

−0.5

0

0.5

Acc

el. (

g)

ID 6

Inst. 48

10 20 30 40 50 60 70 80 90 100 110−0.2

0

0.2

Time (s)

Acc

el. (

g)

ID 8

Inst. 28

Figure 4.12: Lateral accelerations of ID 1, 4, 6, and 8.

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 accel-

erations. A quadratic nonlinearity in the lateral acceleration of the wing-tip launcher is suggested

by the strong harmonic power near 16.5 Hz which was not present in the vertical accelerations.

Quadratic and cubic nonlinearities are also suggested by the strong response near 24.5 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 24.5 Hz component, although it is significantly attenuated compared to the harmonic

in the wing-tip launcher.

0 10 20 30 40 5010

−8

10−7

10−6

10−5

10−4

10−3

10−2

Frequency (Hz)

Pow

er


(a)

0 10 20 30 40 5010

−8

10−6

10−4

10−2

100

Frequency (Hz)

Pow

er


(b)

0 10 20 30 40 5010

−7

10−6

10−5

10−4

10−3

Frequency (Hz)

Pow

er


(c)

0 10 20 30 40 5010

−6

10−5

10−4

Frequency (Hz)

Pow

erRun 5; ID 8; Inst. 28

(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 mag-

nitudes 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

45 50 55 60 65−0.5

0

0.5

Acc

el. (

g)

Run 5; ID 1; Inst 5

Time (s)

Fre

q (H

z)

45 50 55 60 65

10

20

30

(a)

45 50 55 60 65−1

0

1

Acc

el. (

g)


Time (s)

Fre

q (H

z)

45 50 55 60 65

10

20

30

(b)

45 50 55 60 65−0.2

0

0.2

Acc

el. (

g)


Time (s)

Fre

q (H

z)

45 50 55 60 65

10

20

30

(c)

45 50 55 60 65−0.2

0

0.2

Acc

el. (

g)


Time (s)

Fre

q (H

z)

45 50 55 60 65

10

20

30

(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

Freq. (Hz)

Fre

q. (

Hz)

Run 5; ID 1; Inst 5

0 10 20 30 400

5

10

15

20

(a)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(b)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(c)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(d)

Figure 4.15: Wavelet-based auto-bicoherence of the lateral acceleration of ID 1, 4, 6, and 8 duringthe interval t = [52.0− 53.5] seconds. Contour levels are set at ([0.3 : 0.1 : 0.9]).

Wavelet auto-bicoherence plots of the lateral accelerations over three different time intervals are

presented in Figures 4.15, 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 (30Hz, 8Hz, 38Hz) and at

(24Hz, 8Hz, 32Hz). Neither pylon-wing interface, ID 6, nor 8, plots c and d, show any significant

quadratic coupling.

115

Freq. (Hz)

Fre

q. (

Hz)

Run 5; ID 1; Inst 5

0 10 20 30 400

5

10

15

20

(a)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(b)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(c)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(d)


Freq. (Hz)

Fre

q. (

Hz)

Run 5; ID 1; Inst 5

0 10 20 30 400

5

10

15

20

(a)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(b)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(c)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(d)


116

Maneuver Induced LCO; Cross-Coupling Between Vertical and Lateral Accelerations

In order to complete the characterization of maneuver induced quadratic coupling, the wavelet-

based cross-bicoherence between the lateral and vertical accelerations at each location were deter-

mined over the same three time periods. The results are presented in Figures 4.18, 4.19, and 4.20.

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,−5.5Hz, 2.5Hz). No persistent quadratic coupling is indicated at either of the pylon-wing

interfaces ID 6 and ID 8.

117

Freq. (Hz)

Fre

q. (

Hz)

Run 5; ID 1; Inst 6 & Inst 5

0 20 40−40

−30

−20

−10

0

10

20

(a)

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(b)

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(c)

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(d)

Figure 4.18: Cross-bicoherence between the lateral and vertical accelerations at ID 1, 4, 6, and 8during the interval [52.0− 53.5] seconds. Contour levels are set at ([0.3 : 0.1 : 0.9]).

118

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(a)

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(b)

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(c)

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(d)


119

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(a)

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(b)

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(c)

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(d)


120

Maneuver Induced LCO; Cubic Nonlinearity

The power spectrum of lateral motion of ID 1, the wing-tip launcher, suggested a cubic nonlin-

earity between the antisymmetric wing bending mode, 8.2 Hz, and its frequency triple at 24.5 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 com-

ponents are in phase, however, the lateral component contains higher harmonics in addition to the

8.2 Hz component.

51 51.5 52 52.5 53 53.5

−0.4

−0.2

0

0.2

0.4

Lat.

Acc

el. (

g)

Time (s)

Run 5; ID 1; Inst. 5 and 6

51 51.5 52 52.5 53 53.5−4

−2

0

2

4

Ver

t. A

ccel

. (g)

Figure 4.21: Lateral (blue) and vertical (green) accelerations of the the wing-tip launcher ID 1during 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 under-

wing launcher ID 4, however Hajj and Beran [58], found evidence of a cubic nonlinearity in this

instrument by calculating the tricoherence over a different window.

121

0

10

20

30

40

05

1015

200

5

10

f1 (Hz)


f2 (Hz)

f 3 (H

z)

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

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).

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Run 5; ID 1; Inst. 5

Tric

oher

ence

Val

ue

Freq. (Hz)

f1

f2

f3

Σ(fi)

Figure 4.23: The auto-tricoherence is repeated using a novel plotting technique. A high value ofauto-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 0.8. The altitude is also constant

around 10, 000 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.

0.70.80.9

Run 2

Mac

h N

umbe

r

11.011.02

x 104

Alt.

(ft)

0

5

AoA

(°)

0 10 20 30 40 50 60−4−2

024

Acc

el. (

g)

Time (s)

Figure 4.24: Mach number, Altitude, angle of attack, and vertical acceleration of the wing-tiplauncher ID 1 for Run 2 [42]

The amplitude of the right flaperon and its wavelet transform magnitude are presented in Figure

4.25. Distinct harmonic oscillations are present at 8 Hz between [24.5−27.5] 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 180◦ 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

0

2

Am

p. (°

)

Run 2; Inst. 55

Time (s)

Fre

q. (

Hz)

0 10 20 30 40 50 60

10

20

30

Figure 4.25: Right flaperon motion and wavelet transform magnitude, notice the harmonic contentaround 27 seconds.

24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29−0.5

0

0.5

1

1.5

Time (s)

Am

p. (°

)

Run 2; Right and Left Flaperons

RightLeft

(a)

24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29

0.20.40.60.8

1

Am

p. (°

)

Time (s)

Run 2; Flaperon and ID 1; Inst 6

24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29−4−3−2−10123

Acc

el. (

g)

(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 themechanically 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. 157 in), the pylon-wing interface ID 6 (B. L. 156.3), and the pylon-

wing interface ID 8 (B. L. 117.6 in), are presented in Figure 4.27. All four accelerations display a

sudden increase in magnitude near 24.5 seconds, the beginning of flaperon excitation, and start to

decay after 27.5 seconds, as the flaperon excitation ceases. The peak magnitude of the accelerations

varies between ±0.25 g at B. L. 117.6 and ±2 g at B. L. 183.

−5

0

5

Acc

el. (

g)

Run 2; Vertical Accelerations

ID 1

Inst. 6

−5

0

5

Acc

el. (

g)

ID 4

Inst. 18

−2

0

2

Acc

el. (

g)

ID 6

Inst. 49

10 20 30 40 50 60−1

0

1

Time (s)

Acc

el. (

g)

ID 8

Inst. 29

Figure 4.27: Vertical accelerations of the wing-tip launcher ID 1 (B. L. 183 in), the underwinglauncher ID 4 (B. L. 157 in), the pylon-wing interface ID 6 (B. L. 156.3), and the pylon-winginterface ID 8 (B. L. 117.6 in) during Run 2.

Detailed plots of the measured accelerations at the four locations and the corresponding magni-

tudes 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 attenua-

tion 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

125

20 25 30 35−5

0

5

Acc

el. (

g)

Run 2; ID 1; Inst 6

Time (s)

Fre

q (H

z)

20 25 30 35

10

20

30

(a)

20 25 30 35−5

0

5

Acc

el. (

g)


Time (s)

Fre

q (H

z)

20 25 30 35

10

20

30

(b)

20 25 30 35−2

0

2

Acc

el. (

g)


Time (s)

Fre

q (H

z)

20 25 30 35

10

20

30

(c)

20 25 30 35−1

−0.5

0

0.5

Acc

el. (

g)


Time (s)

Fre

q (H

z)

20 25 30 35

10

20

30

(d)

Figure 4.28: Expanded view of the vertical accelerations of the wing-tip launcher ID 1, the under-wing launcher ID 4, the pylon-wing interface ID 6, and the pylon-wing interface 8 and their wavelettransform magnitudes during the forcing event.

126

Freq. (Hz)

Fre

q. (

Hz)

Run 2; ID 1; Inst 6

0 10 20 30 400

5

10

15

20

(a)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(b)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(c)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(d)

Figure 4.29: Wavelet-based auto-bicoherence of the vertical acceleration of ID 1, 4, 6, and 8 overthe interval t = [26.0− 27.5] second. Contour levels are set at ([0.5 : 0.1 : 0.9]).

intervals t = [26.0 − 27.5] seconds and t = [27.0 − 28.5] seconds. The results are plotted in

Figures 4.29 and 4.30 respectively, with contour levels of (0.5 : 0.1 : 0.9). 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 = [26.0 − 27.5] seconds, presented in Figure 4.29b, corresponds to the largest response. The

interval t = [27.0−28.5] seconds, presented in Figure 4.30b, corresponds to significantly attenuated

accelerations.

127

Freq. (Hz)

Fre

q. (

Hz)

Run 2; ID 1; Inst 6

0 10 20 30 400

5

10

15

20

(a)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(b)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(c)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(d)

Figure 4.30: Wavelet-based auto-bicoherence of the vertical acceleration of ID 1, 4, 6, and 8 overthe interval t = [27.0− 28.5] second. Contour levels are set at ([0.5 : 0.1 : 0.9]).

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. 157 in), the pylon-wing interface ID 6 (B. L. 156.3), 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.

−0.5

0

0.5

Acc

el. (

g)

Run 2; Lateral Accelerations

ID 1

Inst. 5

−1.0

0

1.0

Acc

el. (

g)

ID4

Inst. 17

−0.2

0

0.2

Acc

el. (

g)

ID 6

Inst. 48

10 20 30 40 50 60−0.2

0

0.2

Time (s)

Acc

el. (

g)

ID 8

Inst. 28

Figure 4.31: Lateral accelerations of the wing-tip launcher ID 1 (B. L. 183 in), the underwinglauncher ID 4 (B. L. 157 in), the pylon-wing interface ID 6 (B. L. 156.3), and the pylon-winginterface 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 24.5 − 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 = [26.0−27.5] seconds and t = [27.0 − 28.5] seconds are plotted in Figures 4.33 and 4.34 respectively with

129

20 25 30 35−0.5

0

0.5

Acc

el. (

g)

Run 2; ID 1; Inst 5

Time (s)

Fre

q (H

z)

20 25 30 35

10

20

30

(a)

20 25 30 35−1

0

1

Acc

el. (

g)


Time (s)

Fre

q (H

z)

20 25 30 35

10

20

30

(b)

20 25 30 35−0.2

−0.1

0

0.1

Acc

el. (

g)


Time (s)

Fre

q (H

z)

20 25 30 35

10

20

30

(c)

20 25 30 35−0.2

0

0.2

Acc

el. (

g)


Time (s)

Fre

q (H

z)

20 25 30 35

10

20

30

(d)

Figure 4.32: Expanded view of the lateral accelerations of the wing-tip launcher ID 1, the underwinglauncher ID 4, the pylon-wing interface ID 6, and the pylon-wing interface 8 and their wavelettransform magnitudes during the forcing event.

130

Freq. (Hz)

Fre

q. (

Hz)

Run 2; ID 1; Inst 5

0 10 20 30 400

5

10

15

20

(a)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(b)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(c)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(d)

Figure 4.33: Wavelet-based auto-bicoherence of the lateral acceleration of ID 1, 4, 6, and 8 overthe interval of [26.0− 27.5] seconds. Contour levels are set at ([0.5 : 0.1 : 0.9]).

contour levels of (0.5 : 0.1 : 0.9). 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

Freq. (Hz)

Fre

q. (

Hz)

Run 2; ID 1; Inst 5

0 10 20 30 400

5

10

15

20

(a)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(b)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(c)

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

(d)

Figure 4.34: Wavelet-based auto-bicoherence of the lateral acceleration of ID 1, 4, 6, and 8 overthe interval of [27.0− 28.5] seconds. Contour levels are set at ([0.5 : 0.1 : 0.9]).

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 interac-

tions 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 4.35 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

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(a)

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(b)

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(c)

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(d)

Figure 4.35: Cross-bicoherence between the lateral and vertical acceleration at ID 1, 4, 6, and 8over the interval of [26.0− 27.5] seconds. Contour levels are set at ([0.5 : 0.1 : 0.9]).

134

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(a)

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(b)

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(c)

Freq. (Hz)

Fre

q. (

Hz)


0 20 40−40

−30

−20

−10

0

10

20

(d)

Figure 4.36: Cross-bicoherence between the lateral and vertical acceleration at ID 1, 4, 6, and 8over the interval of [27.0− 28.5] seconds. Contour levels are set at ([0.5 : 0.1 : 0.9]).

135

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 5) 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 note-

worthy 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 instrumentationlocations for maneuver induced LCO (Run 5) and mechanically forced LCO (Run 2).

Description ID & Direction Maneuver Induced (Run 5) 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

4.5 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 27.0 − 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 (0.5 : 0.15 : 0.9). 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 26.0− 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 4.40.

138

Figure 4.37: The instrument location is indicated in the top right illustration. The other plotsare: the vertical acceleration the wing-tip launcher ID 1, (top), flaperon motion (second), wavelettransform magnitude of the vertical acceleration of the wing-tip launcher (third), wavelet transformmagnitude of the flaperon motion (fourth), wavelet-based linear coherence between the flaperonand the vertical acceleration of the wing-tip launcher ID 1 (bottom), and wavelet-based cross-bicoherence treating the flaperon as an input and the wing-tip launcher’s vertical acceleration asan 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 trans-form magnitude of the vertical acceleration of the underwing launcher (third), wavelet transformmagnitude of the flaperon motion (fourth), wavelet-based linear coherence between the flaperonand the vertical acceleration of the underwing launcher ID 4 (bottom), and wavelet-based cross-bicoherence between the flaperon and the underwing launcher’s vertical acceleration.

140

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 trans-form magnitude of the vertical acceleration of the pylon-wing interface (third), wavelet transformmagnitude of the flaperon motion (fourth), wavelet-based linear coherence between the flaperon andthe vertical acceleration of the pylon-wing interface (bottom), and wavelet-based cross-bicoherencebetween the flaperon and the pylon-wing interface’s vertical acceleration.

141

Figure 4.40: 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 trans-form magnitude of the vertical acceleration of the pylon-wing interface (third), wavelet transformmagnitude of the flaperon motion (fourth), wavelet-based linear coherence between the flaperon andthe vertical acceleration of the pylon-wing interface (bottom), and wavelet-based cross-bicoherencebetween 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

27.0− 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 (.5) 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 transformmagnitude of the lateral acceleration of the wing-tip launcher (third), wavelet transform magnitudeof the flaperon motion (fourth), wavelet-based linear coherence between the flaperon and the lateralacceleration of the wing-tip launcher (bottom), and wavelet-based cross-bicoherence between theflaperon and the wing-tip launcher’s lateral acceleration.

144

Figure 4.42: The instrument location is indicated in the top right illustration. The other plotsare: lateral acceleration of the underwing launcher ID 4, (top), flaperon motion (second), wavelettransform magnitude of the lateral acceleration of the underwing launcher (third), wavelet transformmagnitude of the flaperon motion (fourth), wavelet-based linear coherence between the flaperon andthe lateral acceleration of the underwing launcher (bottom), and wavelet-based cross-bicoherencebetween the flaperon and the underwing launcher’s lateral acceleration.

145

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 trans-form magnitude of the lateral acceleration of the pylon-wing interface (third), wavelet transformmagnitude of the flaperon motion (fourth), wavelet-based linear coherence between the flaperon andthe lateral acceleration of the pylon-wing interface (bottom), and wavelet-based cross-bicoherencebetween 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 trans-form magnitude of the lateral acceleration of the pylon-wing interface (third), wavelet transformmagnitude of the flaperon motion (fourth), wavelet-based linear coherence between the flaperon andthe lateral acceleration of the pylon-wing interface (bottom), and wavelet-based cross-bicoherencebetween the flaperon and the pylon-wing interface’s lateral acceleration.

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4.6 Growth and Decay of Quadratically Coupled LCO

Wavelet-based bicoherence can be used as a metric to determine the extent of quadratic nonlin-

earity 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 4.45. 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 26.5

and 28.5 seconds.

−5

0

5

Acc

el. (

g)


Time (s)

Fre

q. (

Hz)

23 24 25 26 27 28 29 30

10

20

30

Figure 4.45: Vertical acceleration of the wing-tip launcher, ID 6 (top), and its wavelet transformmagnitude (bottom), during an interval of flaperon induced LCO.

The wavelet-based auto-bicoherence was calculated over eight 1.5 second long intervals starting

at t = 23.00 seconds and advancing by 3/4 of a second to 28.25 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 (0.5 : 0.1 : 0.9), 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 quadraticcoupling in LCO.

Interval Number Start Time (s) End Time(s)1 23.00 24.502 23.75 25.253 24.50 26.004 25.25 26.755 26.00 27.506 26.75 28.257 27.50 29.008 28.25 29.75

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 28.0 seconds, while the

envelope of limit cycle oscillations remains roughly constant. The coupling level decreases sharply,

around 29.0 seconds, where the magnitude of the limit cycle oscillations is slightly decreased but

otherwise, gives no indication of a distinct decoupling event.

149

Freq. (Hz)

Fre

q. (

Hz)

Run 2; ID 1; Inst 6; Interval 1

0 10 20 30 400

5

10

15

20

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

Freq. (Hz)

Fre

q. (

Hz)


0 10 20 30 400

5

10

15

20

Figure 4.46: Auto-bicoherence levels over eight intervals track the quadratic coupling of the wing-tiplauncher’s vertical acceleration in Run 2.

150

−5

0

5

Acc

el. (

g)

24 25 26 27 28 290

0.5

1

Time (s)

Bic

oher

ence

val

ue

Figure 4.47: Level of quadratic coupling as a function of time, using the end of the calculationinterval as a reference.

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F-16 Quadratic LCO Identiﬂcation - Virginia Tech