identification of flapper fin oscillations for active flow...
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Identification of Flapper Fin Oscillations for Active Flow Control Applications in Improved Watercraft
Propulsion
Kostas A. Belibassakis
School of Naval Architecture and Marine Engineering, National Technical University of Athens
Zografos, Athens, Greece
Nikolaos I. Xiros School of Naval Architecture and Marine Engineering, University of New Orleans
New Orleans, Louisiana, United States
Gerassimos K. Politis School of Naval Architecture and Marine Engineering, National Technical University of Athens
Zografos, Athens, Greece
Evangelos Filippas School of Naval Architecture and Marine Engineering, National Technical University of Athens
Zografos, Athens, Greece
Erdem Aktosun School of Naval Architecture and Marine Engineering, University of New Orleans
New Orleans, Louisiana, United States
Vasileios Tsarsitalidis School of Naval Architecture and Marine Engineering, National Technical University of Athens
Zografos, Athens, Greece
ABSTRACT
In this study, the data analysis of an oscillating flapping wing is
conducted for the development of a describing function model
especially for heave force data series obtained using a Boundary
Element Method (BEM) for different geometrical kinds of flapping
wings. The wing experiences a combination of vertical and angular
oscillatory motion, while travelling at constant forward speed. The
vertical motion is induced by the random motion of the ship in waves,
essentially due to ship heave and pitch, while the wing pitching motion
is selected as a proper function of wing vertical motion and it is
imposed by an external mechanism. The data series obtained by
simulation of the unsteady lifting flow around the system was applied
to develop a closed-form lumped phenomenological model for fin
motion control synthesis. Using this model a state-space controller for
thrust augmentation flappers will be later developed. Our study
concerning post-processing data series of thrust-producing flapping
foils can in effect be a useful application for feedback control law
design.
KEY WORDS: flapper fin, active flow control, system identification,
describing function.
INTRODUCTION
Biomimetic propulsors is the subject of intensive investigation, since
they are ideally suited for converting environmental (sea wave) energy
to useful thrust. Recent research and development results concerning
flapping foils and wings, supported also by extensive experimental
evidence and theoretical analysis, have shown that such systems at
optimum conditions could achieve high thrust levels; see, e.g.,
(Triantafyllou et al., 2000; Triantafyllou et al., 2004). A main
difference between a biomimetic propulsor and a conventional
propeller is that the former absorbs its energy by two independent
motions: the heaving motion and the pitching (wing) motion, while for
the propeller there is only rotational power feeding. In real sea
conditions, the ship undergoes a moderate or higher-amplitude
oscillatory motion due to waves, and the vertical ship motion could be
exploited for providing one of the modes of combined/complex
oscillatory motion of a biomimetic propulsion system (Triantafyllou et
al., 2000; Triantafyllou et al., 2004). At the same time, due to waves,
wind and other reasons, ship propulsion energy demand in rough sea is
usually increased well above the corresponding value in calm water for
the same speed, especially in the case of bow/quartering seas.
729
Proceedings of the Twenty-sixth (2016) International Ocean and Polar Engineering ConferenceRhodes, Greece, June 26-July 1, 2016Copyright © 2016 by the International Society of Offshore and Polar Engineers (ISOPE)ISBN 978-1-880653-88-3; ISSN 1098-6189
www.isope.org
METHODOLOGY
Biomimetic Wing Thruster Kinematics
For the description of the kinematic characteristics of the oscillating
wing and of the induced flow dynamics various reference systems are
considered, as the motionless inertial system as shown in Fig. 1, the
ship-fixed coordinate system which is steadily translated with velocity
U with respect to the former and oscillating with respect to the
fundamental degrees of freedom (heave, pitch) of the floating ship due
to waves and the body-fixed coordinate system attached to the flapping
wing, that undergoes a complex translational and oscillatory motion.
In the case of simple periodic oscillations, two distinct frequencies
enter into play, the relative heave frequency due to the waves
1 12 fω π� (1)
and the wing pitching frequency,
2 22 fω π� (2) that in the simpler thrust producing case are assumed to be equal:
1 2ω ω ω� � & 1 2f f f� � (3)
The translational motion of the flapping wing is
( )x t Ut�� (4) where h0 is amplitude of vertical oscillation of the flapping foil.
Simultaneously, the wing undergoes a pitch oscillatory motion and as
mentioned before in the simple harmonic thrust producing case where
the frequencies are equal, pitch oscillatory motion becomes
0( ) sin(2 )m
t ftθ θ θ π ψ� � � (5) where �m is mean angle of attack, �0 is amplitude of pitch oscillation of
the flapping foil and � is phase angle between the two oscillatory
motions.
Fig. 1: Description of motion of the flapping wing
Dynamics
The phase difference, � is very important as far as the efficiency of the
thrust development by the flapping system is concerned. As it is
discussed before in the simple harmonic thrust producing case where:
1 2ω ω ω� � , it usually takes value � = 900. With the pivot point for
the angular motion of the wing located around the 1/3 chord length
from the leading edge, a minimization of the required torque for
pitching is achieved (Belibassakis & Politis, 2013). For flapping
systems steadily advancing in unbounded liquid as shown in Fig. 2 the
main flow parameter controlling the unsteady lift production
mechanism is the Strouhal number,
02 /St fh U� (6) while the Reynolds number has a secondary role affecting viscous drag
corrections. As a result of the simultaneous heaving and pitching
motions of the biomimetic wing the instantaneous angle of attack is
given by: 1 1( ) ( ) ( ) tan ( / ) ( )Ht t t U dh dt tα θ θ θ� �� � � � (7)
For relatively low amplitudes of purely harmonic motion, �H(t) and
optimum phase difference � =90o, the angle of attack becomes; 1
0 0( ) ( )cos( )t U h tα ω θ ω�� � (8)
which is equivalently achieved by setting the pitch angle �(t)
proportional to �H(t) 1 1( ) tan ( / )t w U dh dtθ � �� (9)
and thus
0 0 /wh Uθ ω� (10)
where w is the ‘pitch control parameter’ after (9), usually taking values
in 0<w<1, which is amenable to optimization. Increasing the value of
w, the maximum angle of attack is reduced and the wing operates at
lighter loads. On the contrary, by decreasing the above parameter the
wing loading becomes higher and so is the danger of leading edge
separation that would lead to significant dynamic stall effects. We can
now use the above relations, as an active pitch control rule of the
flapping-wing thruster in the general multi-chromatic case, based on the
time history of vertical motion. In this case, the instantaneous angle of
attack is 1 1( ) (1 ) tan ( / )t w U dh dtα � �� � (11)
Fig. 2: BEM simulation model
Free Surface Effects
In the case of the biomimetic system under the calm or wavy free
surface, additional parameters enrich the above set, as the Froude
number, 1/2/ ( )F U gL� (12)
where L denotes the characteristic (ship) length and g is gravitational
acceleration, as well as various frequency parameter(s) associated with
the incoming wave, like and , � has distinguish subcritical (� < 1/ 4)
from supercritical (� > 1/4) condition.
Geometrical Parameters
As far as a standalone flapping wing is concerned, the selection of plan-
form area, in conjunction with horizontal/vertical sweep and twist
angles, and generating shapes ranging from simple orthogonal or
trapezoidal-like wings like in Fig. 3 to fish-tail like forms e.g. Fig. 4 &
5, constitutes the set of the most important geometrical parameters
730
(Belibassakis & Politis, 2013). Other important parameters are the wing
aspect ratio (s/c), skewback angle S, span-wise distribution of chord,
thickness and possibly camber of wing sections, as well as the specific
wing-sectional form(s).
Data Analysis
Data series have been created from the BEM simulation model applied
to model the unsteady lifting flow around the system based on selection
of geometry, aspect ratio AR, heave to chord ratio h/c and all files
correspond to Strouhal number St and �0 is amplitude of pitch
oscillation of the flapping foil.
The typical set has simulations for five Strouhal numbers from 0.1 to
0.7 and �0 values ranging from 50 to maximum that corresponds to zero
mean thrust. Each data file has time running surge (fx), heave (fy), sway
(fz) forces and roll (mx), yaw (my), and pitch (mz) moments.
For all cases, mean angle of attack �m =00, phase angle between the two
movements �=900 chord length of the wings, c=1m and the flow
velocity is U = 2.3m/s . By known heave to chord ratio h/c, h0 the
amplitude of vertical oscillation of the flapping foil can be calculated.
Fig. 3: Wing outline for s/c = 2, 4, 6, respectively
Also, by using Strouhal number 02 /St fh U� , the frequency of the
heave and pitch oscillatory motion, f (1/s) can be calculated for each
data. Now, due to known all parameters, the motions heave oscillatory
motion h(t) and pitch oscillatory motion �(t) are created for each data.
Fig. 4: Wing outline s/c=4 and S = 150, 300, 450, respectively.
Fig. 5: Wing outline s/c=6 and S = 150, 300, 450 , respectively.
Fig. 6: FFT analysis for one data series
Fig. 7: FFT analysis for one data series
731
Fig. 8: FFT analysis for one data series
A DESCRIBING FUNCTION FOR THE HEAVE FORCE
SIGNAL
Outline of the method
Data series have been created from the BEM simulation model applied
to model the unsteady lifting flow around the system based on selection
of geometry, aspect ratio AR, heave to chord ratio h/c and all files
correspond to Strouhal number St and �0 is amplitude of pitch
oscillation of the flapping foil. Describing function theory and
techniques represent a powerful mathematical approach for
understanding (analyzing) and improving (designing) the behavior of
nonlinear systems. In order to present describing functions, certain
mathematical formalisms must be taken for granted, most especially
differential equations and concepts such as step response and sinusoidal
input response. In addition to a basic grasp of differential equations as a
way to describe the behavior of a system (circuit, electric drive, robot,
aircraft, chemical reactor, ecosystem, etc.) certain additional
mathematical concepts are essential for the useful application of
describing functions Laplace transforms, Fourier expansions and the
frequency domain being foremost on the list.
The main motivation for describing function techniques is the need to
understand the behavior of nonlinear systems, which in turn is based on
the simple fact that every system is nonlinear except in very limited
operating regimes. Nonlinear effects can be beneficial (many desirable
behaviors can only be achieved by a nonlinear system, e.g., the
generation of useful periodic signals or oscillations), or they can be
detrimental (e.g., loss of control and accident at a nuclear reactor).
Unfortunately, the mathematics required to understand nonlinear
behavior is considerably more advanced than that needed for the linear
case.
The elegant mathematical theory for linear systems provides a unified
framework for understanding all possible linear system behaviors. Such
results do not exist for nonlinear systems. In contrast, different types of
behavior generally require different mathematical tools, some of which
are exact, some approximate. As a generality, exact methods may be
available for relatively simple systems (ones that are of low order, or
that have just one nonlinearity, or where the nonlinearity is described
by simple relations), while more complicated systems may only be
amenable to approximate methods. Describing function approaches fit
in the latter category: approximate methods for complicated systems.
One way to deal with a nonlinear system is to linearize it. The strong
attraction of small-signal linearization is that the elegant theory for
linear systems may be brought to bear on the resulting linear model.
However, this approach can only explain the effects of small variations
about the linearization point, and, perhaps more importantly, it can only
reveal linear system behavior. This approach is thus ill-suited for
understanding phenomena such as nonlinear oscillation or for studying
the limiting or detrimental effects of nonlinearity.
The basic idea of the describing function approach for modeling and
studying nonlinear system behavior is to replace each nonlinear
element with a (quasi)linear descriptor or describing function whose
gain is a function of input amplitude. The functional form of such a
descriptor is governed by several factors: the type of input signal,
which is assumed in advance, and the approximation criterion, e.g.,
minimization of mean squared error. This technique is dealt with very
thoroughly in a number of texts for the case of nonlinear systems with a
single nonlinearity or for systems with multiple nonlinearities in
arbitrary configurations, as well as with random-input describing
functions or sinusoidal-input describing functions (SIDF). For an
excellent account see (Gelb & Vander Velde, 1968).
Two categories of describing functions (DF) have been particularly
successful: sinusoidal-input describing functions and random-input
describing functions, depending, as indicated, upon the class of input
signals under consideration. A more detailed classification can also be
developed, e.g., SIDF for pure sinusoidal inputs, sine-plus-bias
describing if there is a constant nonzero offset ‘dc’ value, RIDF for
pure random inputs, random-plus-bias DF; however, this seems
unnecessary since sine-plus-bias and random-plus-bias can be treated
directly in a unified way, so we will use the terms SIDF and RIDF
accordingly. Other types of DF also have been developed and used in
studying more complicated phenomena, e.g., two-sinusoidal-input DF
may be used to study effects of limit cycle quenching via the injection
of a sinusoidal “dither” signal, but those developments are beyond the
needs of this application.
The SIDF approach generally can be used to study periodic
phenomena. It is applied for two primary purposes: limit cycle analysis
and characterizing the input/output behavior of a nonlinear plant in the
frequency domain. This latter application serves as the basis for a
variety of control system analysis and design methods. RIDF methods,
on the other hand, are used for stochastic nonlinear systems analysis
and design (analysis and design of systems with random signals), in
analogous ways with the corresponding SIDF approaches, although
SIDF may be said to be more general and versatile.
In conclusion, describing function approaches allow one to solve a
wide variety of problems in nonlinear system analysis and design via
the use of direct and simple extensions of linear systems analysis
methodology. In point of fact, the mathematical basis is generally
different (not based on linear systems theory); however, the application
often results in conditions of the same form which are easily solved.
Finally, we note that the types of nonlinearity that can be studied via
the DF approach are very general; nonlinearities that are discontinuous
and even multivalued can be considered. The order of the system is also
not a serious limitation. Given software such as Matlab for solving
problems that are couched in terms of linear system mathematics, e.g.,
plotting the polar or Nyquist plot of a linear system transfer function,
one can easily apply DF techniques to high-order nonlinear systems.
The real power of this technique lies in these factors.
732
Concept application
The fundamental basis for use of the SIDF approach can best be
introduced by overviewing the most common application, limit cycle
analysis for a system with a single nonlinearity. A limit cycle (LC) is a
periodic signal,
� � � �LC LCx t T x t� � (13)
for all t and for some T (the period), such that perturbed solutions either
approach xLC (a stable limit cycle) or diverge from it (an unstable one).
The study of LC conditions in nonlinear systems is a problem of
considerable interest in engineering. An approach to LC analysis that
has gained widespread acceptance is the frequency-domain SIDF
method. This technique, as it was first developed for systems with a
single nonlinearity, involved formulating the system in the form shown
in Fig. 9, where G(s) is defined in terms of a ratio of polynomials, as
follows
� �� �
� �� � � �,
p sY s E s e r f y
q s� �� (14)
where p(s) & q(s) represent polynomials in the Laplace complex
variable s, with order(p) < order(q) = n. the subsystem input is then
given to be the external input signal r(t) minus a nonlinear function of
y. There is thus one single-input/single-output (SISO) nonlinearity,
f(y), and linear dynamics of arbitrary order. Thus the system
description is a formulation of the conventional linear plant in the
forward path with a nonlinearity in the feedback path depicted in Fig. 9.
The single nonlinearity might be an actuator or sensor characteristic, or
a plant nonlinearity in any case, the following LC analysis can be
performed using this configuration.
Fig. 9: Describing Function for SISO input
In order to investigate LC conditions with no excitation, r(t) = 0, the
nonlinearity is treated as follows. First, we assume that the input y is
essentially sinusoidal, i.e., that a periodic input signal may exist,
� � � �cosy t a tω� (15)
Thus the output is also periodic.
Expanding in a Fourier series, we have
� �� � � �� �1
cos Re expk
k
f a t b a j tω ω�
�
� A (16)
By omitting the constant or DC term from the equation above we are
implicitly assuming that f(y) is an odd function, f(−y) = −f(y) for all y,
so that no ‘rectification’ occurs; cases when f(y) is not odd present no
difficulty, but are omitted to simplify this discussion. Then we make
the approximation
� �� � � �� �
� �� �1cos Re exp
Re exps
f a t b a j t
N a a j t
ω ω
ω
�
� (17)
This approximate representation for f(a cos(�t)) includes only the first
term of the Fourier expansion given previously. Therefore, the
approximation error, difference
� �� � � �� �� �cos Re expsf a t N a a j tω ω� (18)
is minimized in the Mean Squared Error (MSE) sense.
The Fourier coefficient b1 (and thus the gain Ns(a)) is generally
complex unless f(y) is single-valued; the real and imaginary parts of b1
represent the in-phase (cosine) and quadrature (sine) fundamental
components of f(a cos(�t)), respectively. The so-called describing
function Ns(a) introduced previously is, as noted, amplitude dependent,
thus retaining a basic property of a nonlinear operation.
By the principle of harmonic balance, the assumed oscillation if it is to
exist must result in a loop gain of unity (including the summing
junction), i.e., substituting
� � � �sf y N a yB (19)
yields the requirement
� � � � � � � �1 1s sN a G j G j N aω ω�� C �� (20)
The condition in the equation above is easy to verify using the polar or
Nyquist plot of G(j�); in addition the LC amplitude aLC and frequency
�LC are determined in the process.
It is generally well understood that the conventional analysis as
outlined above is only approximate, so caution is always recommended
in its use. The standard caveat that G(j�) should be low-pass to
attenuate higher harmonics (so that the first harmonic is dominant)
indicates that the analysis has to proceed with caution. Nonetheless, this
approach is simple to apply, very informative, and in general quite
accurate. The main circumstances in which SIDF limit cycle analysis
may yield poor results is in a borderline case, i.e., one where the DF
just barely cuts the Nyquist plot, or just barely misses it.
Describing functions as neural nets
Artificial Neural Networks (ANNs) represent an engineering discipline
concerned with non-programmed adaptive information processing
systems that develop associations (transforms or mappings) between
objects in response to their environment. That is, they learn from
examples. ANNs are a type of massively parallel computing
architecture based on brain-like information encoding and processing
models and as such they can exhibit brain-like behaviors such as
learning, association, categorization, generalization, feature extraction
and optimization.
Given noisy inputs, ANNs build up their internal computational
structures through experience rather than preprogramming according to
a known algorithm as shown in the diagram of Fig. 10. Usually the
neurons or Processing Elements (PEs) that make up the ANN are all
similar and may be interconnected in various ways. The ANN achieves
its ability to learn and then recall that learning through the weighted
interconnections of those PEs. The interconnection architecture can be
very different for different networks. Architectures can vary from
feedforward, and recurrent structures to lattice and many other more
complex and novel structures.
733
Fig. 10: Training process for an ANN
From an engineering perspective many ANNs can often be thought of
as “black box” devices for information processing that accept inputs
and produce outputs. Fig. 11 shows the ANN as a black box, which
accepts a set of N input vectors paired with a corresponding set of N
output vectors. The input vector dimension is p and the output vector
dimension is K where p, K � 1. The output vector set may represent the
actual network outputs given the corresponding input vector set or it
may represent the desired outputs.
Fig. 11: ANN as a black box without feedback internally or externally.
Major features of the neural torque approximators are:
The three-layer feed-forward Multi-layer Perceptron has a parallel
input, one parallel hidden layer and a parallel output layer. The input
layer is only a “fan-out” layer, where the input vector is distributed to
all hidden layer PEs. There is no real processing done in this layer. The
hidden layer is the key to the operation of the MLP. Each of the hidden
nodes is a single PE, which implements its own decision surface. The
output layer is a set of decision surfaces in which each of its PEs has
decided what part of the decision space the input vector lays. The role
of the output layer is essentially to combine all of the “votes” of the
hidden layer PEs and decide upon the overall classification of the
vector. The nonlinearity provided is by the nonlinear activation
functions of the hidden and output PEs and this allows this network to
solve complex classification problems that are not linearly separable.
This is done by forming complex decision surfaces by a nonlinear
combination of the hidden layer’s decision surfaces.
Fig. 12 represents a three-layer feed-forward MLP model. After
training the feed-forward equations relating the inputs to the outputs are
described by a general equation presented in the next section.
Fig. 12: Perceptron architecture.
Neural approximators for heave force
A pilot application of neural networks in the case at hand is to derive
approximating functions (approximators) to the magnitude and phase
(principal argument) of the sinusoidal-input describing function of the
heave force (fy) data series. Some interpolation scheme could have been
used instead for an analytical formulation. However, under the
assumption that functions magnitude FyMAG(f,h0,Q0) and phase
�y(f,h0,Q0) are continuous and monotonous, appropriately sized neural
nets, referred to as neural torque approximators, are used for the
approximation of the generated torque maps, according to the theorems
of Kolmogorov (1957) and Hecht-Nielsen (1987) (Tsoukalas & Uhrig,
1997). The three independent variables are: the frequency f of the flap’s
heave and pitch sinusoidal oscillations; the heave sinusoidal motion
amplitude h0, and; the pitch sinusoidal motion amplitude Q0 (i.e. �0). In
effect, after the nets are trained and their weights determined the
describing function for the heave force is given by:
� �0 0( , , ) exp ,
0,
y yMAG y
yMAG y
F f h Q F j
F
ϕ
π ϕ π
�
D � E F �(21)
In the above, the neural approximators are given by:
0 0
,
210, 0
logsig
1y0 0, 0
,
00
0 0
,
0, 0
tansig
y0 0, 0
( , , )
� F
( , , )
�
yMAG
f i
h i
i
i Q i
b i
y
f i
h i
i
Q i
F f h Q
F f
F hF
F Q
F
F
f h Q
f
h
Q
ϕ
ϕ
ϕϕ
φ ϕ
�
�
� �� �� � � �� �� �� � ���� ���� �� ���� ��� � �� ���� ���� � �� ��� �� �� �� �� �� �� ��� �� �� �� �� ��� �� �
�
�
� �
� �
A
21
1
,
00
i
b iϕ
ϕ
�
� �� �� ��� �� �� � ���� ���� ���� ��� ��� ���� ��� �� ��� �� �� �� �� �� �� ��� �� �� �� �� ��� �� �
A
�(22)
734
a) The inputs and outputs of the nets are normalized in interval [0,1].
b) The nodes of the input layer do not have input weights and activation
functions; the number of nodes for the hidden layer is chosen using
trial-and-error to be 21; the output layer consists of one adder (with
weights) of the outputs of the hidden layer neurons.
c) The activation function for the hidden layer nodes is the
monotonically increasing logistic sigmoid function for the magnitude
approximator and the hyperbolic tangent (tangent sigmoid) for the
phase approximator:
� �
logsig
tansig 2
1� ( )
1 1
2� ( ) tanh 1
1
x
x x
x
ex
e e
x xe
�
�
� �� �
� � ��
(23)
d) Biases are added to the output of each hidden and input layer node.
e) The training sets are the torque maps; the training algorithm used is
Levenberg-Marquardt backpropagation.
The SIDF was obtained in a form of two neural approximators the
parameters of which are presented in the tables of the Annex.
Comparison between the CFD results and the neural approximators is
given in Fig. 13 for the Heave Force SIDF amplitude and in Fig. 14 for
the SIDF phase.
Fig. 13: ANN for heave force SIDF magnitude
Fig. 14: ANN for heave force SIDF phase
RESULTS AND CONCLUSIONS A sinusoidal input describing was derived for the heave force signal
developed on a flap undergoing synchronized sinusoidal pitch and
heave oscillations. Satisfactory matching has been obtained. However,
improvements will be made by retraining the nets, testing other
architectures etc.
In conclusion, a pilot application to determine feasibility of neural nets
as sinusoidal-input describing functions of a flap’s dynamic response
was performed. The use of neural nets permits the semi-automated post
processing of data obtained by CFD simulations. On the other hand,
describing functions will be useful to gain further insight in the
crucially nonlinear process but also to design appropriate control
algorithms ensuring stability and tuning to changing operating
conditions.
ACKNOWLEDGEMENTS This research has been co-financed by the European Union (European
Social Fund – ESF) and Greek national funds through the Operational
Program "Education and Lifelong Learning" of the National Strategic
Reference Framework (NSRF) 2007-2013: Research Funding Program
ARISTEIA - project BIO-PROPSHIP: «Augmenting ship propulsion in
rough sea by biomimetic-wing system».
REFERENCES
Belibassakis, K.A., Politis, G.K. (2013) “Hydrodynamic performance of
flapping wings for augmenting ship propulsion in waves”. Ocean
Engineering 72: 227-240.
Gelb, A., Vander Velde, W. E. (1968) Multiple-Input Describing
Functions and Nonlinear System Design. McGraw-Hill Book Co.,
New York, NY.
Triantafyllou, M. S., Techet, A. H., Hover, F. S. (2004) “Review of
experimental work in biomimetic foils”, IEEE J. Ocean Eng. Vol. 29,
585–594
Triantafyllou, M.S., Triantafyllou, G.S., Yue, D. (2000) “Hydro-
dynamics of fishlike swimming”, An. Rev. Fluid Mech. Vol. 32
Tsoukalas L.H., Uhrig R.E. (1997) Fuzzy And Neural Approaches In
Engineering. John Wiley & Sons, Inc., USA.
0 20 40 60 80 100 120 1400
20
40
60
80
100
120
140
Heave Force Magnitude (N)
CFD
Neural
-180 -120 -60 0 60 120 180-180
-120
-60
0
60
120
180
Heave Force Phase (deg)
CFD
Neural
735
ANNEX
HEAVE FORCE MAGNITUDE & ARGUMENT NEURAL NET APPROXIMATORS
Approximator of FyMAG
Heave Force Magnitude
i F fF 0h
F 0QF b
F
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
0.2461
-9.3318
-209.9266
125.2268
119.8562
-91.2324
498.9288
-17.4399
-313.6610
-14.6372
-27.0715
-57.4954
-5.6674
4.3260
-266.9741
120.9737
74.8973
1.9094
293.0120
-289.8887
148.4026
-167.0623
-29.8675
6.2182
21.8881
-3.3258
11.8964
7.9375
-21.8596
11.9797
-18.6421
22.379
-29.6207
-30.5515
-33.9913
5.3154
3.3946
-25.2791
158.3283
-2.439
-4.7411
-10.8034
57.6367
-5.2646
-26.6704
5.9979
3.1864
-11.2032
3.9488
-3.6874
4.5028
-1.7428
-26.6016
14.7488
-11.2809
7.9456
3.6822
-3.1728
-7.3364
-34.7898
22.1898
6.6731
-0.885
2.3286
0.0288
-0.0069
-0.0204
-0.0842
-0.0097
-0.0187
-0.0054
-0.0178
0.0226
-0.0038
0.0365
0.0019
-0.0074
-0.0205
0.0838
0.0173
-0.0469
-0.0162
0.0052
0.0460
44.8600
20.3253
24.9529
-20.3338
5.0803
12.0887
-11.6937
17.1093
-13.9008
8.6543
14.1720
7.4072
27.0123
-10.2655
-10.3712
-5.1106
24.0131
-5.0081
-48.0876
-6.1410
-4.6374
y0F 135.064 N�
00192.420F �
Approximator of �y
Heave Force Argument
i ϕ fϕ 0h
ϕ 0Qϕ b
ϕ
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
0.0082
-0.0083
-0.1227
-0.2274
-0.0791
-0.2714
-0.0159
-0.0782
-0.0075
-0.0006
1.0320
-0.0674
-0.1362
0.2126
0.0001
-1.0319
-0.0001
-0.2788
0.0147
-0.0176
0.4399
26.4873
-413.2522
-10.6132
-18.4833
7.2279
-16.9
4.2626
-7.2601
-12.3908
52.0368
23.6255
-72.3897
11.399
19.8647
-35.0347
26.067
780.1962
17.2593
-0.9435
-2.7022
-19.1293
-1.4495
166.7224
0.1421
-2.9826
2.5336
0.3893
1.8319
-2.5317
3.5773
-90.1387
4.3166
-0.0729
-0.092
3.1478
-25.6961
4.508
-30.5389
-0.4121
-2.0889
-2.5993
-3.0573
-3.2680
-0.8368
-0.0365
0.2164
-0.1514
2.1451
2.5880
0.1512
-0.0995
2.1334
-1.7860
0.0548
0.0390
-0.2285
-3.4918
-1.8314
-5.3822
-0.6666
-3.2140
5.2803
0.2220
4.6982
175.0547
9.6920
3.0479
-1.8738
-1.7143
-3.5754
1.8935
10.4347
-49.6384
14.9989
0.1424
-10.4098
-3.4194
51.6185
15.0724
65.0656
-5.8026
2.0514
3.3883
3.2251
y0 radφ π�
00-16.5970ϕ �
736