identification of flapper fin oscillations for active flow...

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


Evangelos Filippas School of Naval Architecture and Marine Engineering, National Technical University of Athens


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


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


731


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

identification of flapper fin oscillations for active flow...

Documents