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 Eﬀects 1 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 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 ﬂuid memory eﬀects 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. Inertia forces: !MRB ! MA "!" ! C RB #!$! ! C A #! r $!r Damping forces: !#D p ! D V $! r ! D n #!r $!r ! " Restoring forces: !g##$ ! g o Wind and wave forces: # $ wind ! $ wave !$ Propulsion forces: μ is an addi5onal term represen5ng the ﬂuid memory eﬀects. 2 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.1 Hydrodynamic Concepts and Potential Theory Strip Theory (2-‐D Poten5al Theory) For slender bodies, the mo5on of the ﬂuid 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 ﬂow 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 ﬁnite number of strips. Hence, 2-‐D hydrodynamic coeﬃcients for added mass can be computed for each strip and then summed over the length of the body to yield the 3-‐D coeﬃcients. Commercial Codes: MARINTEK (ShipX-‐Veres) and Amarcon (Octopus Oﬃce) 3 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) ShipX (VERES) by MARINTEK 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 veriﬁes technological solu5ons for the shipping and mari5me equipment industries and for oﬀshore petroleum produc5on. 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. 4 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) ShipX (VERES) by MARINTEK ShipX (Veres) 5 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) OCTOPUS SEAWAY by Amarcon and AMARCON cooperate in further development of SEAWAY 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 oﬀshore industries and navies. SEAWAY is developed by Professor J. M. J. Journée 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 eﬀects 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 veriﬁed and validated using other computer codes and experimental data. 6 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 7 7 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.1 Hydrodynamic Concepts and Potential Theory Panel Methods (3-‐D Poten5al Theory) For poten5al ﬂows, the integrals over the ﬂuid domain can be transformed to integrals over the boundaries of the ﬂuid domain. This allows the applica5on of panel or boundary element methods to solve the 3-‐D poten5al theory problem. 3D Panelization of a Supply Vessel 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 deﬁned which fulﬁll the Laplace equa5on. Commercial code: WAMIT (www.wamit.com) 3D Visualization of the Wamit file: supply.gdf 4 2 Z-axis (m) 0 -2 -4 -6 -8 -10 -12 -40 -30 -20 -10 20 0 10 10 0 20 -10 -20 30 40 X-axis (m) -30 Y-axis (m) 8 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 3 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. Panelization of semi-submersible using WAMIT user supplied tools 9 Over the past 20 years WAMIT has been licensed to more than 80 industrial and research organizations worldwide. Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.1 Hydrodynamic Concepts and Potential Theory Poten5al theory programs typically compute: • Frequency-‐dependent added mass, A(w) • Poten5al damping coeﬃcients, 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 inﬁnite added mass. These boundary values are par5cular useful when compu5ng the retarda5on func5ons describing the ﬂuid memory eﬀects. Processing of Hydrodynamic Data using MSS HYDRO – www.marinecontrol.org The toolbox reads output data ﬁles generated by the hydrodynamic programs: • ShipX (Veres) by MARINTEK AS • WAMIT by WAMIT Inc. and processes the data for use in Matlab/Simulink. 10 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.2 Seakeeping and Maneuvering Kinematics Seakeeping Theory (Perturba5on Coordinates) The SEAKEEPING reference frame {s} is not ﬁxed to the craZ; it is ﬁxed to the equilibrium state: ! ! !! 1 , ! 2 , ! 3 , ! 4 , ! 5 , ! 6 " T # ! ! !" !" ! !#" ! !$ ! sb ! !! 4 , ! 5 , ! 6 " ! ! !"#, "#, "$" ! -‐ 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. v nns ! !U cos !, U sin!, 0" ! # Transforma5on between {b} and {s} !! ! ! !U!L!" " e 1 " # !!# ! !# !UL! # L :! e 1 ! !1, 0,0,0, 0, 0" ! 0 0 0 0 0 0 0 0 0 0 0 1 ! nns ! !0, 0, 0" ! # " ns ! !0, 0, ! " "! # 0 0 0 0 !1 0 0 0 0 0 0 0 ! 0 0 0 0 0 0 " 0 0 0 0 0 0 # $! 0 ! # 0 # " $# 11 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) $" # 5.3 The Classical Frequency-Domain Model Seakeeping Analysis The seakeeping equa5ons of mo5on are considered to be iner5al: Equa5ons of Mo5on ! ! !" ! !!x, !y, !z, !", !#, !$" ! # MRB !" ! #hyd " #hs " #exc # Cummins (1962) showed that the radia5on-‐induced hydrodynamic forces in an ideal ﬂuid can be related to frequency-‐dependent added mass A(ω) and poten5al damping B(ω) according to: t $ !t ! !""%!!"d! # !hyd ! !Ā"# ! " K 0 12 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.3 The Classical Frequency-Domain Model Frequency-‐dependent added mass A22(ω) and poten5al damping B22(ω) in sway Ā ! A!!" 13 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.3 The Classical Frequency-Domain Model !hyd t $ !t ! !""#!!"d! # ! !Ā"# ! " K 0 Matrix of retarda5on func5ons given by 2 K!t" ! ! " ! 0 B!"" cos!"t"d" # Cummins Model If linear restoring forces τhs = -Cξ are included in the model, this results in the 5me-‐domain model: t " # !t # !"!$!!"d! ! C! " %exc # !MRB ! A!!""! ! " K 0 The ﬂuid memory eﬀects can be replaced by a state-‐space model to avoid the integral 14 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.3 The Classical Frequency-Domain Model 15 Longitudinal added mass coeﬃcients as a func5on of frequency. Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.3 The Classical Frequency-Domain Model 16 Lateral added mass coeﬃcients as a func5on of frequency. Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.3 The Classical Frequency-Domain Model 17 Longitudinal poten5al damping coeﬃcients as a func5on of frequency. Exponen5al decaying viscous damping is included for B11. Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.3 The Classical Frequency-Domain Model 18 Lateral poten5al damping coeﬃcients as a func5on of frequency. Exponen5al decaying viscous damping is included for B22 and B66 while viscous IKEDA damping is included in B44 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.3.1 Potential Coefficients and the Concept of Forced Oscillations MRB !" ! #hyd " #hs " f cos!!t" # harmonic excita5on In an experimental setup with a restrained scale model, it is possible to vary the wave excita5on frequency ω and the amplitudes fi of the excita5on force. Hence, by measuring the posi5on and aptude vector η, the response of the 2nd-‐order order system can be ﬁqed to a linear model: !MRB ! A"!#$!" ! B"!#!# ! C! " f cos"!t# # 19 for each frequency ω. The matrices A(ω),B(ω) 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! Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.3.2 Frequency-Domain Seakeeping Models !MRB t # !t # !"!$!!"d! ! C! " %exc # ! A!!""!" ! " K 0 Cummins equa5on can be transformed to the frequency domain (Newman, 1977; Fal5nsen 1990) according to: !!! 2 "MRB ! A!!#$ ! j!B!!# ! C#!!j!# " "exc !j!# # where the complex response and excita5on variables are wriqen as: ! i !t" ! !" i cos!"t # # i " $ ! i !j"" ! !" i exp!j# i " $ exc,i !t" ! $" i cos!"t # % i " $ $ exc,i !j"" ! $" exc,i exp!j% i " # # The poten5al coeﬃcients A(ω) and B(ω) are usually computed using a seakeeping program but the frequency response will not be accurate unless viscous damping is included. The op5onal viscous damping matrix BV(ω) can be used to model viscous damping such as skin fric5on, surge resistance and viscous roll damping (for instance IKEDA roll damping). 20 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.3.2 Frequency-Domain Seakeeping Models Viscous frequency-‐dependent damping: !! 2 !MRB ! A"!#$ ! j!B total"!# ! C !"j!# " "wave "j!# ! "wind "j!# ! """j!# # B total!!" ! B!!" " B V !!" # " 1 e !#! "N ITTC !A 1 " 0 0 B V !!" ! 0 0 0 0 " 2 e !#! 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 " 6 e !#! Viscous skin fric5on: ! i e !"# # Quadra5c ITTC drag: X ! !X |u|u |u|u " N ITTC !A1 "u # Quadra5c damping is approximated using describing func5ons (similar to the equivalent lineariza5on method): u ! Asin!!t" # y ! c 1 x " c 2 x|x|"c 3 x 33 # 2 y ! N!A"u # N!A" ! c 1 " 8A c 2 " 3A c 3 # 3! 4 21 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.4 Time-Domain Models including Fluid Memory Effects Cummins equa?on in SEAKEEPING coordinates (linear theory which includes ﬂuid memory eﬀects) Transform from SEAKEEPING to BODY coordinates (linearized kinema?c transforma?on) Linear seakeeping equa?ons in BODY coordinates (ﬂuid memory eﬀects are approximated as state-‐space models) Uniﬁed maneuvering and seakeeping model (nonlinear viscous damping/maneuvering coeﬃcients are added manually) 22 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.4.1 Cummins Equation in SEAKEEPING Coordinates Cummins (1962) Equa5on !MRB ! Ā"!" ! ! t "# # ! " %wind ! %wave ! "% # # !t " !"!$!!"d! ! C K The Ogilvie (1964) Transforma5on gives Ā ! A!!" # ! !t" ! 2 K ! " ! 0 B total!"" cos!"t"d" # From a numerical point of view is it beqer to integrate the diﬀerence: K!t" ! 2 ! "#B !"" # B !""$ cos!"t"d" # total ! 0 total This can be don by rewri5ng Cummins equa5on as: !MRB t ! A"!#$!" ! B total"!#!# ! " K"t # !#!#"!#d! ! C! " $wind ! $wave ! "$ # 0 23 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.4.2 Linear Time-Domain Seakeeping Equations in BODY Coordinates !MRB t ! A"!#$!" ! B total"!#!# ! " K"t # !#!#"!#d! ! C! " $wind ! $wave ! "$ # 0 It is possible to transform the 5me-‐domain representa5on of Cummins equa5on from {s} to {b} using the kinema5c rela5onships: !! ! ! !U!L!" " e 1 " # ! ! !" !!# ! !# !UL! # !" ! !# This gives: t !MRB ! A"!#$!!" !UL!$ ! B total"!#!! !U"L!# " e 1 #$ ! # K"t " "#!!""#d" ! C!# " $wind ! $wave ! "$ " $%# # 0 The steady-‐state control force τ needed to obtain the forward speed U when τwind = τwave= 0 and δη = 0 is: !" ! B total!!"Ue 1 # Hence, t !MRB ! A"!#$!!" !UL!$ ! B total"!#!! ! UL!#$ ! " K"t # "#!!""#d" ! C!# " $wind ! $wave ! $ # 0 24 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.4.2 Linear Time-Domain Seakeeping Equations in BODY Coordinates t !MRB ! A"!#$!!" !UL!$ ! B total"!#!! ! UL!#$ ! " K"t # "#!!""#d" ! C!# " $wind ! $wave ! $ # 0 When compu5ng the damping and retarda5on func5ons, it is common to neglect the inﬂuence of δη on the forward speed such that: !! ! v !U!L!" " e 1 " ! v "Ue 1 # Finally, let use replace ν by the rela5ve velocity νr to include ocean currents and deﬁne: M = MRB + MA such that : t M!" ! C !RB ! ! C !A !r ! D!r ! " K!t # !"#!!!"#Ue 1 $d! ! G# " $wind ! $wave ! $ # 0 where MA ! A!!" " C A ! UA!!"L C "RB ! UMRB L Linear Coriolis and D ! B total!!" 25 G!C centripetal forces due to a rota5on of {b} about {s} Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.4.2 Linear Time-Domain Seakeeping Equations in BODY Coordinates Fluid Memory Eﬀects The integral in the following equa5on represents the ﬂuid memory eﬀects: M!" ! ! :! C !RB ! ! t ! D!r ! " K!t # !"#!!!"#Ue 1 $d! ! G# " $wind ! $wave ! $ # 0 t ! 0 K!t " !"#"!!""Ue 1 $ d! # "" x 10 Approximated by a state-‐ space model K22(t) 7 2.5 C !A !r 2 ! ! H!s"#" !Ue 1 $ # x! " A rx # B r!" ! " C rx 1.5 1 0.5 Impulse response func5on 0 2 K!t" ! ! -0.5 -1 0 5 10 15 20 " ! 0 #B!"" # B!""$ cos!"t"d" 25 time (s) 26 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) # # 5.4.3 Nonlinear Unified Seakeeping and Maneuvering Model with Fluid Memory Effects Linear Seakeeping Equa5ons (BODY coordinates) M!" ! C !RB ! ! C !A !r ! D!r ! # ! G$ " %wind ! %wave ! % # Copyright © Bjarne Stenberg/NTNU Uniﬁed Nonlinear Seakeeping and Maneuvering Model Copyright © The US Navy • Use nonlinear kinema5cs • Replace linear Coriolis and centripetal forces with their nonlinear counterparts • Include maneuvering coeﬃcients in a nonlinear damping matrix (linear superposi5on) !" ! J " !!"# # M#"r # C RB !#"# # C A !#r "#r # D!#r"#r # $ # G! ! % wind # %wave # % # 27 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.5 Case Study: Identification of Fluid Memory Effects The ﬂuid memory eﬀects can be approximated using frequency-‐domain iden5ﬁca5on. 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 ﬂuid memory eﬀects μ can be approximated by a matrix H(s) containing rela5ve degree one transfer func5ons: ! ! H!s"!" # h ij !s" ! p r sr "p r!1 sr!1 "..."p 0 sn "q n!1 sn!1 "..."q 0 r ! n ! 1, n " 2 State-‐space model: H!s" ! C r!sI ! A r" !1 B r # x! " A r x # B r !" ! " C rx # 28 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 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 inﬁnite added mass matrix is unknown. Hence, we can es5mate the ﬂuid 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; 29 [KradNum,KradDen,Ainf] = FDIRadMod(W,A,0,B,FDIopt,Dof) Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.5.1 Frequency-Domain Identification using the MSS FDI Toolbox 30 FPSO iden5ﬁca5on results for h₃₃(s) without using the inﬁnite added mass A₃₃(∞). The leZ-‐hand-‐side plots show the complex coeﬃcient and its es5mate while added mass and damping are ploqed on the right-‐hand-‐side. Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) 5.5.1 Frequency-Domain Identification using the MSS FDI Toolbox 1. 672e007 s 3 " 2. 286e007 s 2 " 2. 06e006 s h 33 !s" ! 4 s " 1. 233 s 3 " 0. 7295 s 2 " 0. 1955 s " 0. 01639 ! ! H!s"#" !Ue 1 $ # x! " A rx # B r!" !1. 2335 !0. 7295 !0. 1955 !0. 0164 Ar ! 1 00 00 0 1 0 1 0 0 0 ! " C rx 1 Br ! 0 0 0 Cr ! 1. 672e007 2. 286e007 2. 06e006 0 Dr ! 0 31 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) #

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