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Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

Chapter 5 Seakeeping Theory

4.1 Hydrodynamic Concepts and Poten5al Theory 4.2 Seakeeping and Maneuvering Kinema5cs 4.3 The Classical Frequency-Domain Model 4.4 Time-Domain Models including Fluid Memory Eects

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Chapter 5 - Seakeeping Theory Equa5ons of Mo5on Seakeeping theory is formulated in equilibrium (SEAKEEPING) axes {s} but it can be

transformed to BODY axes {b} by including uid memory eects represented by impulse response func5ons.

The transforma5on is is done within a linear framework such that addi5onal nonlinear viscous damping must be added in the 5me-domain under the assump5on of linear superposi5on.

is an addi5onal term represen5ng the uid memory eects.

Inertia forces: !MRB !MA" "! ! CRB#!$! ! CA#!r$!r

Damping forces: !#Dp ! DV$!r ! Dn#!r$!r ! "

Restoring forces: !g##$ ! goWind and wave forces: # $wind ! $wave

Propulsion forces: !$

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Strip Theory (2-D Poten5al Theory) For slender bodies, the mo5on of the uid can be formulated as a 2-D problem. An accurate es5mate of the hydrodynamic forces can be obtained by applying strip theory (Newman, 1977; Fal/nsen, 1990; Journee and Massie, 2001). The 2-D theory takes into account that varia5on of the ow in the cross-direc5onal plane is much larger than the varia5on in the longitudinal direc5on of the ship. The principle of strip theory involves dividing the submerged part of the craZ into a nite number of strips. Hence, 2-D hydrodynamic coecients for added mass can be computed for each strip and then summed over the length of the body to yield the 3-D coecients. Commercial Codes: MARINTEK (ShipX-Veres) and Amarcon (Octopus Oce)

5.1 Hydrodynamic Concepts and Potential Theory

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ShipX (VERES) by MARINTEK

VERES - VEssel RESponse program is a Strip Theory Program which calculates wave-induced loads on and mo5ons of mono-hulls and barges in deep to very shallow water. The program is based on the famous paper by Salvesen, Tuck and Fal?nsen (1970). Ship Mo/ons and Sea Loads. Trans. SNAME.

MARINTEK - the Norwegian Marine Technology Research Ins5tute - does research and development in the mari5me sector for industry and the public sector. The Ins5tute develops and veries technological solu5ons for the shipping and mari5me equipment industries and for oshore petroleum produc5on.

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ShipX (Veres)

ShipX (VERES) by MARINTEK

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OCTOPUS SEAWAY by Amarcon

and AMARCON cooperate in further development of SEAWAY

SEAWAY is developed by Professor J. M. J. Journe at the DelZ Univiversity of Technology SEAWAY is a Strip Theory Program to calculate wave-induced loads on and mo5ons of mono-

hulls and barges in deep to very shallow water. When not accoun5ng for interac5on eects between the hulls, also catamarans can be analyzed. Work of very acknowledged hydromechanic scien5sts (such as Ursell, Tasai, Frank, Keil, Newman, Fal5nsen, Ikeda, etc.) has been used, when developing this code.

SEAWAY has extensively been veried and validated using other computer codes and experimental data.

The Mari5me Research Ins5tute Netherlands (MARIN) and AMARCON agree to cooperate in further development of SEAWAY. MARIN is an interna5onally recognized authority on hydrodynamics, involved in fron5er breaking research programs for the mari5me and oshore industries and navies.

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Copyright 2005 Marine Cybernetics AS

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5.1 Hydrodynamic Concepts and Potential Theory Panel Methods (3-D Poten5al Theory) For poten5al ows, the integrals over the uid domain can be transformed to integrals over the boundaries of the uid domain. This allows the applica5on of panel or boundary element methods to solve the 3-D poten5al theory problem. Panel methods divide the surface of the ship and the surrounding water into discrete elements (panels). On each of these elements, a distribu5on of sources and sinks is dened which fulll the Laplace equa5on. Commercial code: WAMIT (www.wamit.com)

-40

-30

-20

-10

0

10

20

30

40 -30-20

-10

010

2030

-12

-10

-8

-6

-4

-2

0

2

4

Y-axis (m)

3D Visualization of the Wamit file: supply.gdf

X-axis (m)

Z-axis

(m)

3D Panelization of a Supply Vessel

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WAMIT

WAMIT is the most advanced set of tools available for analyzing wave interactions with offshore platforms and other structures or vessels.

WAMIT was developed by Professor Newman and coworkers at MIT in 1987, and it has gained widespread recognition for its ability to analyze the complex structures with a high degree of accuracy and efficiency.

Over the past 20 years WAMIT has been licensed to more than80 industrial and research organizations worldwide.

Panelization of semi-submersible using WAMIT user supplied tools

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5.1 Hydrodynamic Concepts and Potential Theory Poten5al theory programs typically compute:

Frequency-dependent added mass, A(w) Poten5al damping coecients, B(w) Restoring terms, C 1st- and 2nd-order wave-induced forces and mo5ons

(amplitudes and phases) for given wave direc5ons and frequencies and much more

One special feature of WAMIT is that the program solves a boundary value problem for zero and innite added mass. These boundary values are par5cular useful when compu5ng the retarda5on func5ons describing the uid memory eects. Processing of Hydrodynamic Data using MSS HYDRO www.marinecontrol.org The toolbox reads output data les generated by the hydrodynamic programs:

ShipX (Veres) by MARINTEK AS WAMIT by WAMIT Inc.

and processes the data for use in Matlab/Simulink.

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5.2 Seakeeping and Maneuvering Kinematics Seakeeping Theory (Perturba5on Coordinates) The SEAKEEPING reference frame {s} is not xed to the craZ; it is xed to the equilibrium state:

e1 ! !1,0,0,0,0,0"!

! ! !"

L :!

0 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 !1 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

!" ! !#" ! !$

! ! !!1,!2,!3,!4,!5,!6"T #

!sb ! !!4,!5,!6"! ! !"#,"#,"$"! #

!! ! ! !U!L!" " e1"!!# ! !# !UL!

# #

Transforma5on between {b} and {s}

!

"

#

!

00#"

#

$!

$"

$#

#

vnsn ! !Ucos!, Usin!, 0"!

!nsn ! !0,0,0"!

"ns ! !0,0,!" "!

# # #

-In the absence of wave excita5on, {s} coincides with {b}. - Under the ac5on of the waves, the hull is disturbed from its equilibrium and {s} oscillates, with respect to its equilibrium posi5on.

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Seakeeping Analysis The seakeeping equa5ons of mo5on are considered to be iner5al:

5.3 The Classical Frequency-Domain Model

! ! !" ! !!x,!y,!z,!",!#,!$"! #

Equa5ons of Mo5on

MRB!" ! #hyd " #hs " #exc #

Cummins (1962) showed that the radia5on-induced hydrodynamic forces in an ideal uid can be related to frequency-dependent added mass A() and poten5al damping B() according to:

!hyd ! !"# ! "0

tK$ !t ! !""%!!"d! #

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! A!!"

Frequency-dependent added mass A22() and poten5al damping B22() in sway

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Cummins Model


K!t" ! 2! !0"B!""cos!"t"d" #

!MRB ! A!!""!" ! "0

tK# !t # !"!$!!"d! ! C! " %exc #

If linear restoring forces hs = -C are included in the model, this results in the 5me-domain model:

Matrix of retarda5on func5ons given by

!hyd ! !"# ! "0

tK$ !t ! !""#!!"d! #

The uid memory eects can be replaced by a state-space model to avoid the integral

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Longitudinal added mass coecients as a func5on of frequency.

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Lateral added mass coecients as a func5on of frequency.

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Longitudinal poten5al damping coecients as a func5on of frequency. Exponen5al decaying viscous damping is included for B11.

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Lateral poten5al damping coecients as a func5on of frequency. Exponen5al decaying viscous damping is included for B22 and B66 while viscous IKEDA damping is included in B44

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5.3.1 Potential Coefficients and the Concept of Forced Oscillations

for each frequency .

The matrices A(),B() and C represents a "hydrodynamic mass-damper-spring system" which varies with the frequency of the forced oscilla5on. This model is rooted deeply in the literature of hydrodynamics and the abuse of nota5on of this false 5me-domain model has been discussed eloquently in the literature (incorrect mixture of 5me and frequency in an ODE). Consequently, we will use Cummins 5me-domain model and transform this model to the frequency domain no mixture of 5me and frequency!

In an experimental setup with a restrained scale model, it is possible to vary the wave excita5on frequency and the amplitudes fi of the excita5on force. Hence, by measuring the posi5on and aptude vector , the response of the 2nd-order order system can be qed to a linear model:

MRB!" ! #hyd " #hs " fcos!!t" #

!MRB ! A"!#$!" ! B"!#!# ! C! " fcos"!t# #

harmonic excita5on

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!MRB ! A!!""!" ! "0

tK# !t # !"!$!!"d! ! C! " %exc #

5.3.2 Frequency-Domain Seakeeping Models

Cummins equa5on can be transformed to the frequency domain (Newman, 1977; Fal5nsen 1990) according to: where the complex response and excita5on variables are wriqen as:

!!!2"MRB ! A!!#$ ! j!B!!# ! C#!!j!# " "exc!j!# #

!i!t" ! !" i cos!"t # # i" $ !i!j"" ! !" i exp!j# i"$exc,i!t" ! $" i cos!"t # % i" $ $exc,i!j"" ! $" exc,i exp!j% i"

# #

The poten5al coecients A() and B() are usually computed using a seakeeping program but the frequency response will not be accurate unless viscous damping is included. The op5onal viscous damping matrix BV() can be used to model viscous damping such as skin fric5on, surge resistance and viscous roll damping (for instance IKEDA roll damping).

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5.3.2 Frequency-Domain Seakeeping Models

B total!!" ! B!!" " BV!!" #

!!2!MRB ! A"!#$ ! j!B total"!# ! C !"j!# " "wave"j!# ! "wind"j!# ! """j!# #

BV!!" !

"1e!#!"NITTC!A1" 0 0 0 0 0

0 "2e!#! 0 0 0 0

0 0 0 0 0 0

0 0 0 "IKEDA!!" 0 0

0 0 0 0 0 0

0 0 0 0 0 "6e!#!

#

u ! Asin!!t" #

y ! c1x " c2x|x|"c3x33 # N!A" ! c1 " 8A3! c2 "

3A24 c3 # y ! N!A"u #

X ! !X |u|u|u|u" NITTC!A1"u #

Viscous frequency-dependent damping:

Quadra5c damping is approximated using describing func5ons (similar to the equivalent lineariza5on method):

!ie!"#

Quadra5c ITTC drag:

Viscous skin fric5on:

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5.4 Time-Domain Models including Fluid Memory Effects

Unied maneuvering and seakeeping model (nonlinear viscous damping/maneuvering coecients

are added manually)

Linear seakeeping equa?ons in BODY coordinates (uid memory eects are approximated as state-space models)

Transform from SEAKEEPING to BODY coordinates (linearized kinema?c transforma?on)

Cummins equa?on in SEAKEEPING coordinates (linear theory which includes uid memory eects)

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!MRB ! A"!#$!" ! B total"!#!# ! "0

tK"t # !#!#"!#d! ! C! " $wind ! $wave ! "$ #

From a numerical point of view is it beqer to integrate the dierence: This can be don by rewri5ng Cummins equa5on as:

5.4.1 Cummins Equation in SEAKEEPING Coordinates

!MRB ! "!" ! !"#

tK# !t " !"!$!!"d! ! C# ! " %wind ! %wave ! "% #

! A!!" #

K! !t" ! 2! !0"B total!""cos!"t"d" #

Cummins (1962) Equa5on

The Ogilvie (1964) Transforma5on gives

K!t" ! 2! !0"#B total!"" # B total!""$ cos!"t"d" #

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It is possible to transform the 5me-domain representa5on of Cummins equa5on from {s} to {b} using the kinema5c rela5onships: This gives: The steady-state control force needed to obtain the forward speed U when wind = wave= 0 and = 0 is: Hence,

5.4.2 Linear Time-Domain Seakeeping Equations in BODY Coordinates !MRB ! A"!#$!" ! B total"!#!# ! "

0

tK"t # !#!#"!#d! ! C! " $wind ! $wave ! "$ #

!MRB ! A"!#$!!" !UL!$ ! B total"!#!! !U"L!# " e1#$ ! #0tK"t " "#!!""#d" ! C!# " $wind ! $wave ! "$ " $%# #

!! ! ! !U!L!" " e1"!!# ! !# !UL!

# #

!" ! B total!!"Ue1 #

! ! !"

!" ! !#

!MRB ! A"!#$!!" !UL!$ ! B total"!#!! ! UL!#$ ! "0

tK"t # "#!!""#d" ! C!# " $wind ! $wave ! $ #

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When compu5ng the damping and retarda5on func5ons, it is common to neglect the inuence of on the forward speed such that: Finally, let use replace by the rela5ve velocity r to include ocean currents and dene: M = MRB + MA such that : where

5.4.2 Linear Time-Domain Seakeeping Equations in BODY Coordinates

!MRB ! A"!#$!!" !UL!$ ! B total"!#!! ! UL!#$ ! "0

tK"t # "#!!""#d" ! C!# " $wind ! $wave ! $ #

!! ! v !U!L!" " e1" ! v "Ue1 #

M!" ! CRB! ! ! CA!!r ! D!r ! "0

tK!t # !"#!!!"#Ue1$d! ! G# " $wind ! $wave ! $ #

MA ! A!!"CA" ! UA!!"LCRB" ! UMRBLD ! B total!!"

G ! C

Linear Coriolis and centripetal forces due to a rota5on of {b} about {s}

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5.4.2 Linear Time-Domain Seakeeping Equations in BODY Coordinates

! :! !0

tK!t " !"#"!!""Ue1$

""d! #

Fluid Memory Eects The integral in the following equa5on represents the uid memory eects:

! ! H!s"#" !Ue1$ #

!x " Arx # B r!"! " Crx

#

Approximated by a state-space model

K!t" ! 2! !0"#B!"" # B!""$ cos!"t"d" #

Impulse response func5on

0 5 10 15 20 25 -1 -0.5

0 0.5

1 1.5

2 2.5 x 10 7

time (s)

K 22 (t)

M!" ! CRB! ! ! CA!!r ! D!r ! "0

tK!t # !"#!!!"#Ue1$d! ! G# " $wind ! $wave ! $ #

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5.4.3 Nonlinear Unified Seakeeping and Maneuvering Model with Fluid Memory Effects Linear Seakeeping Equa5ons (BODY coordinates)

Unied Nonlinear Seakeeping and Maneuvering Model

Use nonlinear kinema5cs Replace linear Coriolis and centripetal forces with their nonlinear counterparts Include maneuvering coecients in a nonlinear damping matrix (linear superposi5on)

!" ! J"!!"#M#" r # CRB!#"# # CA!#r"#r # D!#r"#r # $ # G! ! %wind # %wave # %

# #

M!" ! CRB! ! ! CA!!r ! D!r ! # ! G$ " %wind ! %wave ! % # Copyright Bjarne Stenberg/NTNU

Copyright The US Navy

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5.5 Case Study: Identification of Fluid Memory Effects The uid memory eects can be approximated using frequency-domain iden5ca5on. The main tool for this is the MSS FDI toolbox (Perez and Fossen 2009) - www.marinecontrol.org When using the frequency-domain approach, the property that the mapping: has rela5ve degree one is exploited. Hence, the uid memory eects can be approximated by a matrix H(s) containing rela5ve degree one transfer func5ons:

!! ! " #

! ! H!s"!" #

H!s" ! Cr!sI ! Ar"!1B r #

!x " Arx # B r!"! " Crx

#

hij!s" ! prsr"pr!1sr!1"..."p0

sn"qn!1sn!1"..."q0

r ! n ! 1, n " 2

State-space model:

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5.5.1 Frequency-Domain Identification using the MSS FDI Toolbox Consider the FPSO data set in the MSS toolbox (FDI tool) and assumes that the innite added mass matrix is unknown. Hence, we can es5mate the uid transfer func5on h33(s) by using the following Matlab code: load fpso Dof = [3,3]; %Use coupling 3-3 heave-heave Nf = length(vessel.freqs); W = vessel.freqs(1:Nf-1)'; Ainf = vessel.A(Dof(1),Dof(2),Nf); % Ainf computed by WAMIT A = reshape(vessel.A(Dof(1),Dof(2),1:Nf-1),1,length(W))'; B = reshape(vessel.B(Dof(1),Dof(2),1:Nf-1),1,length(W))'; FDIopt.OrdMax = 20; FDIopt.AinfFlag = 0; FDIopt.Method = 2; FDIopt.Iterations = 20; FDIopt.PlotFlag = 0; FDIopt.LogLin = 1; FDIopt.wsFactor = 0.1; FDIopt.wminFactor = 0.1; FDIopt.wmaxFactor = 5; [KradNum,KradDen,Ainf] = FDIRadMod(W,A,0,B,FDIopt,Dof)

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5.5.1 Frequency-Domain Identification using the MSS FDI Toolbox

FPSO iden5ca5on results for h(s) without using the innite added mass A(). The leZ-hand-side plots show the complex coecient and its es5mate while added mass and damping are ploqed on the right-hand-side.

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5.5.1 Frequency-Domain Identification using the MSS FDI Toolbox

h33!s" !1.672e007 s3 " 2.286e007 s2 " 2.06e006 s

s4 " 1.233 s3 " 0.7295 s2 " 0.1955 s " 0.01639

Ar !

!1.2335 !0.7295 !0.1955 !0.01641 0 0 00 1 0 00 0 1 0

B r !

1000

Cr ! 1.672e007 2.286e007 2.06e006 0

Dr ! 0

! ! H!s"#" !Ue1$ #

!x " Arx # B r!"! " Crx

#

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