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Helicopter Flight Dynamics Using Linear and
Nonlinear Analyses
Taikang Ning1,2
, John F. Wei3, Rong-Huei Chen
2, Chia-Wei Huang
2, and Ting-Yu Ho
2
1Department of Engineering, Trinity College, Hartford, Connecticut, USA
2Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan
3Department of Mechanical Engineering, Central Connecticut State University, New Britain, USA
Abstract This paper compares measures derived from linear
power spectral analysis and nonlinear chaotic analysis of
helicopter vibratory data obtained during different flight tests. In
particular, the strange attractor behavior of the trajectory of
reconstructed state vectors in phase space was examined and
quantified using the correlation dimension, which provides a
fractal dimension measure in regards to the number of activedegrees of freedom of helicopter vibration. We have examined
the helicopter vibratory data collected from a well tuned four-
blade servo-flap rotor system conducting hover, low-speed, and
cruising-speed forward flights. In this paper, we examined servo-
flap mid-span bending and rotor blade flat-wise bending. The
results have shown that during helicopter hover, vibratory data
of servo-flap bending and rotor flat-wise bending are more
random than chaotic. On the other hand, the complexity (fractal
dimension) of vibratory data increase when the helicopter
switched from the minimum forward speed flight to cruising-
speed forward flight.
Keywords- Servo-flap; helicopter flight vibration; power spectrum;correlation dimension
I. INTRODUCTION
One important design approach for advanced rotor-based
aircrafts is to shift the existing rotor blade control from the
fixed system to the rotating system using the servo-flap [1]-[4]. The rotor system of a helicopter is the most critical one
that provides the helicopter necessary maneuvering abilities
but also adversely causes vibrations that impede helicopter
performance, reduce helicopter life span, and increasemaintenance costs. Therefore, recent helicopter research and
development activities have placed greater emphasis on rotor
blade design and flight control systems [3]-[4]. One uniqueand revolutionary approach adopted by Kaman Aerospace was
to move the rotor blade control design from a typical fixed
system to the rotating system through servo-flaps [1]-[4], [8],which are small blade airfoils located at the 75% radius on the
trailing edge of the blades. Using aerodynamic pitching
moments generated by the flap to fly the helicopter, the servo-flap is designed as the primary rotor control. As such, it
eliminates the need for a heavy hydraulic control system
required to control the blade attack angle.However, studies on system behavior of servo-flap based
rotor systems are still lacking in the literature [2]-[5], [8]. This
underlying study compares performance measures derived
from traditionally used linear measures of power spectral
analysis to nonlinear dynamics of servo-flap based rotorsystems. The power spectrum is a useful linear system
analysis tool. It has been widely used to analyze helicopter
vibratory data and reveal the energy distribution in the
frequency domain at harmonics of the 1/rev frequency; usefulsystem design parameters are identified through power
spectral analysis [6], [8]-[9]. However, the physicalmechanism which generates helicopter vibrations is highly
nonlinear, and the power spectrum provides only the linear
analysis aspect of helicopter vibrations.Limitations of linear analysis necessitate the development
of nonlinear dynamics to search for better tools that can
provide meaningful theoretic explanations and effectivequantifiers to measure complex phenomena. This underlying
study utilizes nonlinear dynamics measures to characterize the
strange attractor behavior of the state trajectory in phase spacefrom servo-flap helicopter vibratory data. Chaos has captured
the attention of scientists, mathematicians, and engineers in
recent years due to the fact that most physical systems arenonlinear in nature. For example, in complex systems, there
are usually many degrees of freedom that contribute to some
random/chaotic behavior. Chaos can furnish a conceptual
understanding and provides an alternative explanation of this
nonlinear random behavior. Examples of chaos and theirstudies are abundant in the literature [10]-[13].
In our study, we use the correlation dimension quantifier
to characterize the strange attractor behavior of helicoptervibratory data collected from both the rotating and fixed
systems during different flight conditions. Unlike an integergeometric dimension, the correlation dimension provides a
measure of the fractal dimension that reflects the dimension
complexity or the number of active degrees of freedom in thereconstructed state space of an underlying signal. The
correlation dimension was often calculated using the popularGrassberger and Procaccia (GP) method [14]. The author
proposed a modified GP method [7] that uses a Euclidean
distance normalized by the embedding dimension to estimatethe correlation dimension.
In this study, linear spectral analysis and nonlinear
dynamics measures were both used to examine helicoptervibratory data during three different helicopter flight
conditionshover, low-speed, and cruising-speed flights.
The research was supported in part by the National Science Council of
Taiwan under the contract NSC 98-2221-E002-138-MY3.
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II. SERVO-FLAP ROTORCONTROL
The servo-flap control is divided into primary and the
feedback inputs. Since the blade control feedback washes out
the pilot's input, the input and output of the servo-flap controlpositions are different for a servo-flap rotor. The angular
position of the servo-flap with respect to the trailing edge of
the main rotor is critical; the control system that describes the
behavior of rotors and servo-flap control is given by
vKKwKcsinput cc G]GTGE T\G\GGG cossin 110 (1)
where0G is the collective and ,1sG c1G are cyclic azimuth
coefficients andGEK , GTK , G]K are mechanical feedback
couplings [3]-[5]. It is the control input that varies to
manipulate different flights. A production servo-flap
helicopter rotor blade for Kaman Aerospace is shown in Fig.1[3]. Though the servo-flap controls the helicopter flight, its
flatwise bending, on the other hand, was increased when the
helicopter changed from hover to low-speed and to cruising-speed forward flights. For example, the bending magnitude
more than doubled when the flight condition was shifted fromhover to cruising-speed forward flight.
III. METHODS
A. Linear Spectral Analysis
The linear spectral analysis has been the most widely
employed tool to examine vibrations in various applications.Single channel power spectrum and multichannel coherence
analyses, for example, are two typically used methods [5]-[6],[8]-[9]. Linear power spectrum is computationally efficient
and it can effectively display the energy distribution of the
underlying signal in the frequency domain. However, thepower spectrum bears the fundamental assumption, as do other
linear analysis methods, that the target signal is stationary and
the energy sources that make up the distribution in the
frequency domain are linearly uncorrelated. The validity ofsuch an assumption invites suggestions of more complicated
and more realistic approaches as new theories rapidly emerge
in recent years [10]-[14] and computation concern is muchrelieved with the advancement of fast computing software and
hardware. In this study, vibratory data were analyzed using an
FFT-based periodogram to estimate the spectral peaks at
harmonics of the rotor blade rotating frequency (1/rev).
B. Correlation Dimension
Theoretic work has paved the path to use a single variable
time series to capture the behavior of multidimensional
nonlinear systems through time-delay embedding [10]-[14].Multidimensional state space vectors {v(i)} can be constructed
from data {x(n)} via time-delay embedding [13]
])1((),...(),([)( LmixLixixiv , (2)
where L is the delay time (lag) and m is the embeddingdimension. Lag (L) is usually chosen to produce a state vector
with independent or uncorrelated entries. The state space
trajectory graphically represents the evolution of a dynamicsystem over time; it is composed of a series of points that
show precise moments in time where each point is a graphical
representation of the independent variables of the system. Aninteresting property that may be estimated from the trajectory
is the fractal dimension of the attractor.
A computationally efficient algorithm to estimate the
correlation dimension (fractal dimension) was proposed by
Grassberger and Procaccia [14] and quickly became themethod of choice in a wide variety of applications. The
Grassberger and Procaccia (GP) method estimate the
correlation dimension by way of computing an intermediateindexcorrelation sum/integral C()
||))()(||()1(
2)(
1 1
jvivNN
CN
i
N
ij
4
WW . (3)
is the Heaviside function described below
-
t4
0,0
0,1)(
O
OO , (4)
and ||.|| denotes the Euclidean distance measure between v(i)
and v(j). The author proposed a modified GP (mGP) method
that uses a scaled Euclidean distance normalized by the
embedding dimension (m) [7]
m
kLjxkLixjviv
m
k
1
0
2)]()([
||)()( . (5)
The correlation sum C() was repeatedly computed for an
incrementally increasing scaling radius . A linear region in
the log(C()) versus log() plot reflects the region where thecorrelation dimension (D2) can be reliably estimated by the
slope of the identified linear scaling region as the following,
)log()log(
))(log())(log(lim
02
WWW
WWW
W '
'
o'
CCD . (6)
Figure 1. A composite rotor blade with an external servo-flap.
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D2 was estimated for different embedding dimensions (m) to
ensure estimation consistency.
IV. FLIGHT DATA ANALYSIS
Vibratory test data examined in this paper were measuredusing strain gauges from a well-tuned four-blade helicopter
during three different conditions at 6000 ft: hover, low-speed,
and cruising-speed forward flight. These data were sampledusing a rate of 500 Hz. The rotor of the test helicopter has a
rotating speed of 4.967revolutions per second (approximately
5 Hz). Vibratory measurements and power spectra of servo-
flap mid-span bending during three test flights are shown inFig.2. One can notice in Fig.2(a) that servo-flap bending
during hover shows more waveform complexities than those
during forward flights, while vibratory waveform duringcruising-speed flight in Fig.2(c) is less complex. However, this
signal complexity difference did not reflect in power spectra
analysis results. Fig.3 shows the vibratory data of three flightconditions and power spectra via the same approach of the
main rotor blade flat-wise bending at 24% of the radius. TableI summarized vibratory energy levels of servo-flap mid-span
bending and rotor blade flat-wise bending at harmonics of
1/rev frequency during the three flight conditions.
The correlation sum C() in (3) was computed using the
mGP method [7] described in (5). Fig.4 shows log(C()) vs.
log() plots and D2 estimates of the servo-flap mid-spanbending during hover, low-speed, and cruising-speed forward
flights. These estimates were obtained with using 2650 data
samples and three embedding dimensions (m=4, 6, 8) andL=24. In order to calculate the correlation dimension, a linear
scaling region must be identified first. In Fig.4, the linear
scaling region was visually identified between log()=1.6and
log()=1.8; the correlation dimensions (D2) for three tested
TABLE I. ENERGY LEVELS AT HARMONICS OF 1/REV
FREQUENCY (UNITS IN DB)
Flight
Cond.
Servo-flap bending Blade bending
hover low cruising hover low cruising
1/rev 27.51 40.52 42.98 36.75 49.57 55.67
2/rev 12.53 21.91 35.12 42.76 42.67 49.99
3/rev 12.18 16.9 24.91 41.97 59.20 58.374/rev 9.04 25.11 19.39 43.76 54.16 38.375/rev 22.69 20.61 22.07 52.74 49.97 51.56
Figure 2. Servo-flap mid-span bending and power spectra in three
different flights.
Figure 3. Rotor blade flat-wise bending and power spectra in threedifferent flights.
Figure 4. Correlation dimension (D2) of servo-flap mid-span bending
during flight (embedding dimension: (m=4); (m=6;); (m=8).
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embedding dimensions (m=4, 6, 8) were estimated by
estimating the slope in the linear scaling region.
Fig.5 shows correlation sums and estimated correlation
dimensions for rotor blade flat-wise bending at 24% of therotor radius. The linear scaling region was visually
determined in the region between log()=2.6 and log()=2.8.
We observed that servo-flap bending during hover did notexhibit apparent strange attractor behavior. However, during
low-speed forward flight, the state trajectory shows a clear
strange attractor behavior, more uniformly spanned with a lagvalue ofL=24 in (2). Similar results were observed during
cruising-speed flight. The correlation dimension (D2)
associated with Fig.4(b) is roughly 1.2 while D2 is about 2.0during cruising-speed forward flight. Though the estimated D2
in hover flight is higher, the lack of apparent strange attractor
behavior suggests the servo-flap mid-span bending duringhover is more randomly behaved rather chaotic.
In regards to rotor blade flat-wise bending shown in Fig.3,
we observed similar findings, i.e., correlation dimension (D2)is higher during cruising-speed forward flight than low-speed
flight. However, during hover, the state trajectory did show
patterns of strange attractor with a lag value of L=4, anindication of chaotic behavior.
V. CONCLUSION
In summary, the linear system based measure such as the
power spectrum can quantify helicopter vibratory data interms of dynamic range in the frequency domain. The
correlation dimension, on the other hand, can quantify the
change of nonlinear dynamics in regards to the active degreeof freedom via the appearance of strange attractor behavior
during different flight conditions. The nonlinear dynamic
measure provides a quantifier in regards to the fractal
dimension of helicopter vibratory data that captures the
chaotic behavior of helicopter vibratory data and allows a
different perspective to correlate the mechanism that causeshelicopter vibrations. However, one caveat in estimating the
correlation dimension lies in the process being sensitive to an
individuals choice of the linearly scaling region. To ensure aconsistent estimation, repeated simulation runs for different
linear scaling regions and embedding dimensions should be
conducted.
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Figure 5. Correlation dimension (D2) of rotor blade flat-wise bending
during flight (embedding dimension: (m=4); (m=6;); (m=8).
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