F-16 Quadratic LCO Identiflcation - Virginia Tech
Transcript of 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
Run 5; ID 1; Inst. 6
(a)
0 10 20 30 40 5010
−6
10−5
10−4
10−3
10−2
10−1
100
Frequency (Hz)
Pow
er
Run 5; ID 4; Inst. 18
(b)
0 10 20 30 40 5010
−6
10−5
10−4
10−3
10−2
10−1
100
Frequency (Hz)
Pow
er
Run 5; ID 6; Inst. 49
(c)
0 10 20 30 40 5010
−6
10−5
10−4
10−3
10−2
Frequency (Hz)
Pow
er
Run 5; ID 8; Inst. 29
(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)
Run 5; ID 6; Inst 49
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)
Run 5; ID 8; Inst 29
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)
Run 5; ID 4; Inst 18
0 10 20 30 400
5
10
15
20
(b)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 6; Inst 49
0 10 20 30 400
5
10
15
20
(c)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 8; Inst 29
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)
Run 5; ID 4; Inst 18
0 10 20 30 400
5
10
15
20
(b)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 6; Inst 49
0 10 20 30 400
5
10
15
20
(c)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 8; Inst 29
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)
Run 5; ID 4; Inst 18
0 10 20 30 400
5
10
15
20
(b)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 6; Inst 49
0 10 20 30 400
5
10
15
20
(c)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 8; Inst 29
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
Run 5; ID 1; Inst. 5
(a)
0 10 20 30 40 5010
−8
10−6
10−4
10−2
100
Frequency (Hz)
Pow
er
Run 5; ID 4; Inst. 17
(b)
0 10 20 30 40 5010
−7
10−6
10−5
10−4
10−3
Frequency (Hz)
Pow
er
Run 5; ID 6; Inst. 48
(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)
Run 5; ID 4; Inst 17
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)
Run 5; ID 6; Inst 48
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)
Run 5; ID 8; Inst 28
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)
Run 5; ID 4; Inst 17
0 10 20 30 400
5
10
15
20
(b)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 6; Inst 48
0 10 20 30 400
5
10
15
20
(c)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 8; Inst 28
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)
Run 5; ID 4; Inst 17
0 10 20 30 400
5
10
15
20
(b)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 6; Inst 48
0 10 20 30 400
5
10
15
20
(c)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 8; Inst 28
0 10 20 30 400
5
10
15
20
(d)
Figure 4.16: Wavelet-based auto-bicoherence of the lateral 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 5
0 10 20 30 400
5
10
15
20
(a)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 4; Inst 17
0 10 20 30 400
5
10
15
20
(b)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 6; Inst 48
0 10 20 30 400
5
10
15
20
(c)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 8; Inst 28
0 10 20 30 400
5
10
15
20
(d)
Figure 4.17: Wavelet-based auto-bicoherence of the lateral acceleration of ID 1, 4, 6, and 8 duringthe interval t = [58.0− 59.5] seconds. Contour levels are set at ([0.3 : 0.1 : 0.9]).
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)
Run 5; ID 4; Inst 18 & Inst 17
0 20 40−40
−30
−20
−10
0
10
20
(b)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 6; Inst 49 & Inst 48
0 20 40−40
−30
−20
−10
0
10
20
(c)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 8; Inst 29 & Inst 28
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)
Run 5; ID 1; Inst 6 & Inst 5
0 20 40−40
−30
−20
−10
0
10
20
(a)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 4; Inst 18 & Inst 17
0 20 40−40
−30
−20
−10
0
10
20
(b)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 6; Inst 49 & Inst 48
0 20 40−40
−30
−20
−10
0
10
20
(c)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 8; Inst 29 & Inst 28
0 20 40−40
−30
−20
−10
0
10
20
(d)
Figure 4.19: Cross-bicoherence between the lateral and vertical accelerations at ID 1, 4, 6, and 8during the interval [55.0− 56.5] seconds. Contour levels are set at ([0.3 : 0.1 : 0.9]).
119
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)
Run 5; ID 4; Inst 18 & Inst 17
0 20 40−40
−30
−20
−10
0
10
20
(b)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 6; Inst 49 & Inst 48
0 20 40−40
−30
−20
−10
0
10
20
(c)
Freq. (Hz)
Fre
q. (
Hz)
Run 5; ID 8; Inst 29 & Inst 28
0 20 40−40
−30
−20
−10
0
10
20
(d)
Figure 4.20: Cross-bicoherence between the lateral and vertical accelerations at ID 1, 4, 6, and 8during the interval [58.0− 59.5] seconds. Contour levels are set at ([0.3 : 0.1 : 0.9]).
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)
Run 5; ID 1; Inst. 5
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)
Run 2; ID 4; Inst 18
Time (s)
Fre
q (H
z)
20 25 30 35
10
20
30
(b)
20 25 30 35−2
0
2
Acc
el. (
g)
Run 2; ID 6; Inst 49
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)
Run 2; ID 8; Inst 29
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)
Run 2; ID 4; Inst 18
0 10 20 30 400
5
10
15
20
(b)
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 6; Inst 49
0 10 20 30 400
5
10
15
20
(c)
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 8; Inst 29
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)
Run 2; ID 4; Inst 18
0 10 20 30 400
5
10
15
20
(b)
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 6; Inst 49
0 10 20 30 400
5
10
15
20
(c)
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 8; Inst 29
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)
Run 2; ID 4; Inst 17
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)
Run 2; ID 6; Inst 48
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)
Run 2; ID 8; Inst 28
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)
Run 2; ID 4; Inst 17
0 10 20 30 400
5
10
15
20
(b)
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 6; Inst 48
0 10 20 30 400
5
10
15
20
(c)
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 8; Inst 28
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)
Run 2; ID 4; Inst 17
0 10 20 30 400
5
10
15
20
(b)
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 6; Inst 48
0 10 20 30 400
5
10
15
20
(c)
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 8; Inst 28
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)
Run 2; ID 1; Inst 6 & Inst 5
0 20 40−40
−30
−20
−10
0
10
20
(a)
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 4; Inst 18 & Inst 17
0 20 40−40
−30
−20
−10
0
10
20
(b)
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 6; Inst 49 & Inst 48
0 20 40−40
−30
−20
−10
0
10
20
(c)
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 8; Inst 29 & Inst 28
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)
Run 2; ID 1; Inst 6 & Inst 5
0 20 40−40
−30
−20
−10
0
10
20
(a)
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 4; Inst 18 & Inst 17
0 20 40−40
−30
−20
−10
0
10
20
(b)
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 6; Inst 49 & Inst 48
0 20 40−40
−30
−20
−10
0
10
20
(c)
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 8; Inst 29 & Inst 28
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.
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 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)
Run 2; ID 1; Inst. 6
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)
Run 2; ID 1; Inst 6; Interval 2
0 10 20 30 400
5
10
15
20
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 1; Inst 6; Interval 3
0 10 20 30 400
5
10
15
20
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 1; Inst 6; Interval 4
0 10 20 30 400
5
10
15
20
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 1; Inst 6; Interval 5
0 10 20 30 400
5
10
15
20
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 1; Inst 6; Interval 6
0 10 20 30 400
5
10
15
20
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 1; Inst 6; Interval 7
0 10 20 30 400
5
10
15
20
Freq. (Hz)
Fre
q. (
Hz)
Run 2; ID 1; Inst 6; Interval 8
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.
151