Blending Manoeuvring and Seakeeping: Heading-keeping Simulations for 5415M

16th International Symposium on Practical Design of Ships and Other Floating Structures PRADS 2025 Ann Arbor, MI, USA, October 19th-23rd 2025 Blending Manoeuvring and Seakeeping: Heading-keeping Simulations for 5415M Ruddy Kurnia*, Roberto Tonelli, Nicolas Carette, and Tim Bunnik Maritime Research Institute Netherlands (MARIN), Wageningen, The Netherlands Abstract. This paper presents two modelling frameworks for fast time-domain simulations of a ship manoeuvring in waves. The frameworks are developed based on the “two-time scale” and “unified” methods. The two-time scale method solves the low-frequency and wave-frequency ship motions separately. The mean and slowly varying drift forces combined with the manoeuvring and appendages ones are applied to compute the slowly varying path. The wave-frequency motions are then solved through the linear superposition of response amplitude operators (RAOs) for each motion along this path. The RAOs are computed using the boundary element potential flow solver SEACAL using Rankine sources. The unified method on the other hand solves the lowand wave-frequency ship motions in one integrated system using force superposition. Imposing first-order wave-frequency forces based on RAOs into a time-domain simulation can generate undesired (or ghost) drift forces. To overcome this, analytic ghost drift force are pre-computed in SEACAL for later correction during the time-domain simulations. Moreover, radiation forces in time domain need to be properly modelled in order to avoid additional spurious drift forces due to low frequency damping components arising from Rankine source solutions. Furthermore, the paper compares heading-keeping simulations, using both methods, with the experiments, showcasing the methods’ capabilities and shortcomings. For this comparison, the 5415M naval combatant ship has been used as a case study to illustrate how the methods perform in practice. The validation cases show that the simulations predict the ship behavior in regular and irregular waves with fairly good accuracy. Key words: Manoeuvring in waves, two-time scale, unified, 5415M 1. Introduction A reliable quantification of manoeuvring in waves is of paramount importance for ship design and performance assessment. This is mainly driven by the Energy Efficiency Design Index (EEDI) requirement to decrease installed power of merchant ships. The need has also increased due to requirements for heading and track-keeping, and to quantify the controllability in adverse weather, especially naval ships. The performance assessment of ship manoeuvring in waves can be studied using numerical simulations at earlier design stage and physical model tests at a later stage. The model has to be equipped with actuators (e.g. rudders and propellers) and autopilot to control the speed and heading. These tests are challenging not only for the model tests but also for the numerical simulations. Practically, heading-keeping or trackkeeping simulations have to be performed with a duration of at least 30 minutes up to 3 hours with a variety of wave directions and ship speeds. Those simulations can only be performed efficiently using a fast time-domain simulation tool. In performing time-domain simulations, two frameworks are available in literature. The frameworks are known as the unified and two-time scale methods. The unified methods have been introduced by W. R. McCreight [1986], Hooft and Pieffers [1988], P. A. Bailey, W.G. Price, P. Temarel [1997]. The method solves the equation of motions for the manoeuvring and seakeeping in one integrated system. Convolution integral [Cummins, 1962] is used to evaluate frequency-domain based transfer functions of hydrodynamic coefficients. The convolution integral requires an accurate estimation of the impulse response functions. First-order excitation forces are computed in *Correspondence to: r[email protected] 1 time-domain through the linear superposition of the force transfer functions and wave spectrum amplitudes. As also mentioned in [R. Skejic, 2013], the evaluation of first-order forces in time-domain simulations can gradually generates nonlinearities that can affect the ship manoeuvers. The two-time scale method was introduced in [Skejic and Faltinsen, 2008, H. Yasukawa and Y. Nakayama, 2009]. The method solves equations of motion separately for different time scale. It relies on pre-computed response amplitude operators (RAOs) of the wave-frequency motions of the ship. These are usually obtained using frequency-domain calculations. Then, motions in time-domain are computed in two following stages. Firstly, the low-frequency surge, sway, yaw and roll are computed. This is done by considering the manoeuvring forces, including control surfaces such as propellers and rudders, and mean second order wave drift forces. Then, the wave-frequency motions, for the six degrees of freedom, are computed at each location based on the product of the RAO and the complex wave amplitude. This paper presents the MARIN blending of manoeuvring and seakeeping models based on the unified and two-time scale methods. Analytical formulations are derived to remove the un-physical non-linearities (due to ghost drift forces) that are generated through the evaluation of first-order excitation forces and radiation forces transfer functions in a time-domain simulation. Additionally, the pure second-order wave excitation forces are imposed both to the unified and two-time scale methods through the full quadratic transfer functions provided by a frequency-domain boundary element solver with Rankine source formulation. Both models use the following extra modelling aspects: the Unified Damping model as derived in [F. H. H. A. Quadvlieg, R. Tonelli, and A. Bedos, 2021] for the manoeuvring coefficients, the four quadrants propeller approach as in [Kuiper, 1992, J. Moulijn and R. Tonelli and U. Shipurkar, 2024] for the propellers’ thrust and torque modelling, and MARIN’s rudder model [Tonelli and Vogels, 2024]. The hydrodynamic coefficients and wave excitation forces are obtained by solving double-body steady potential, radiation and diffraction potential problems. The velocity potentials are obtained from the source distribution solutions of a boundary element solver with Rankine source over hull and free-surface domain. The drift forces quadratic transfer functions (QTFs) are computed with the quadratic part based on the direct pressure integration method and the potential part based on the Pinkster approximation [Pinkster, 1975]. The solver is called SEACAL, developed under Cooperative Research Ship framework (CRS 1) [MARIN, 2024a,b]. Validation of the models are performed against the model tests of the 5415M naval combatant ship. The model tests were performed under CRS project. The data have also been used in [F. H. H. A. Quadvlied, S. Rapuc, 2019] to validate their heading-keeping simulations. This paper is organised as follows. The modelling framework of manoeuvring in waves is described in Section 2.. Description of the validation material is presented in Section 3.. Verifications of second-order excitation forces and ghost drift force are discussed in Section 4., and the validations of heading-keeping simulations are discussed in Section 5.. Conclusions wrap-up this paper in Section 6.. 2. Modelling of manoeuvring in waves This section describes the modelling frameworks for simulating ship manoeuvres in waves. This section starts with the description of coordinate systems and transformation. Derivation of the equations of motion and the modelling of hydrodynamic forces, wave excitation forces, and the actuators forces are described. 2.1. Coordinate systems and transformation Ship motions in six degrees of freedom (DOF) are described in three orthogonal coordinate systems: a global inertial, body-fixed and seekeeping (equilibrium), following the notation of Fossen [2005]. The global inertial (fixed) coordinate system is denoted by {n}with (xn, yn, zn)axes and origin on;xn,ynare horizontal axes perpendicular to upward vertical axis zn. The body-fixed reference frame {b}is a moving coordinate frame with axis (xb, yb, zb)axis and origin ob. The seakeeping frame {s}= (xs, ys, zs)is fixed to the equilibrium state. The distance vectors for the reference frames are denoted as rs/n,rb/s, and rb/n 1Cooperative Research Ships (CRS) was started in 1969 and focuses on hydrodynamics, structural and related problems of all kind of ship types from a fundamental, design and operational perspective. Today, the CRS consists of 24 member organizations and companies carrying out a joint work program. More information on http://www.crships.org 2 for the distance vector of {s}with respect to {n}, of {b}with respect to {s}, and of {b}with respect to {n}, respectively. The vectors satisfy the following relation rb/n =rs/n +rb/s. The position vector of point obwith respect to {n}expressed in {n}is denoted as pn b/n = (pE, pN, pU) with the x-axis pointing toward east, y-axis towards north and z-axis upwards. A Euler angles vector is defined as Θnb = (ϕ, θ, ψ). The ship motion in 6 DOF with the coordinate origin obcan be expressed in the generalized ship position and orientation vector ζ= (pn b/n,Θnb)Tin which Tis the transpose operator. Correspondingly, the body-fixed velocity vector is denoted v= (vb b/n,ωb b/n)T= (v1,v2)T= (u, v, w, p, q, r)Tand the force vector is F= (fb b/n,mb b/n)T= (X, Y, Z, K, M, N)T. The seakeeping reference frame {s}is considered inertial fixed to an equilibrium state. The equilibrium state is defined as vn s/n = (Ucos( ¯ ψ), U sin( ¯ ψ))T,ωn s/n = (0,0,0)T,Θns = (0,0,¯ ψ)where U is the ship speed and ¯ ψis the mean heading. The perturbed motion is defined as ξ= (rs b/s,Θsb)T= (ξ1, ξ2, ξ3, ξ4, ξ5, ξ6)Tthat corresponds to surge, sway, heave, roll, pitch and yaw perturbations. The corresponding velocity and acceleration in {s}are then expressed as ˙ ξ,¨ ξ. The perturbed velocity expressed in {b}is denoted as δv= (vb b/s,ωb b/s) = (δu, δv, δw, δp, δq, δr). Following [Fossen, 2005], a transformation between body and seakeeping frames can be obtained as follows. The velocities in {s}and in {b}relates through the transformation matrix JΘ(ξ), forming kinematic equation, as ˙ ξ=JΘ(ξ)δv(1) where the transformation matrix JΘ(ξ)is expressed with the Euler angles rotation matrix Rs b(Θsb) = Rz,ξ6Ry,ξ5Rx,ξ4and the angular velocity transformation matrix TΘ(Θsb)for small angles as JΘ(ξ) = Rs b(Θsb)03×3 03×3TΘ(Θsb)where TΘ(Θsb) =   10ξ5 01−ξ4 0ξ41 . The equilibrium linear velocity in {b}is defined as ¯ v1=Rb n(Θbn)vn s/n =Rb s(Θbs)Ue1while the angular velocity ¯ v2= (0,0,0)Tsince {s}do not rotate with respect to {n},ωs/n = 0. In small angles approximation, ¯ v≈U(1,−ξ6, ξ5,0,0,0)T=U(e1−Lξ)and ˙ ¯v≈U(0,−δr, δq, 0,0,0)T=−ULδv with L= [06×4,−e3,e2], with ejis the six dimension unit vector. The relation of the rigid body velocity, acceleration and the Euler angles between the references of frames are then expressed as follows v=¯ v+δv=U(e1−Lξ) + δv,˙ v=˙ ¯v+δ˙ v=−ULδv+δ˙ v,Θnb =Θns +Θsb.(2) 2.2. Equation of motions Ship sailing at forward speed (U) encounters waves with frequency ωe=ω−kU cos(µ), where ω,k and µare respectively wave frequency, wave number and wave direction. In frequency domain, the ship motions vector is denoted by ˆ ξas a complex amplitude of ξ. The equation of motion is then expressed in terms of rigid body mass matrices MRB, hydrodynamic coefficients of added mass A(ωe)and damping B(ωe), restoring coefficient C, linear excitation forces ˆ F(1) exc(ωe)and other external forces ˆ Fext(ωe), as follows. −ω2 e[MRB +A(ωe)] −iωeB(ωe) + Cˆ ξ(ωe) = ˆ F(1) exc +ˆ Fext.(3) The hydrodynamic coefficients and wave excitation forces are obtained from the frequency-domain boundary solver, SEACAL. In time domain, the seakeeping equation follows Cummins [1962] as [MRB +A(∞)] ¨ ξ+Zt 0 K(t−τ)˙ ξ(τ)dτ +Cξ =Fwave +Fext,(4) where the impulse response (retardation) function is defined as K(t) = 2 πZ∞ 0 B(ωe) cos(ωet)dωe.(5) 3 Equation (4) (time-domain) relates to Equation (3) (frequency-domain), through the Ogilve transformations [T. F. Ogilve, 1964]. It can be shown that the frequency dependent added mass and damping coefficients can be obtained from the retardation function. Equation (4) can be expressed in body coordinates {b}as hMb RB +Ab(∞)i(˙ v+ULδv) + Zt 0 Kb(t−τ)δv(τ)dτ +Cbξ=Fb wave +Fb ext.(6) The influence of ξon the forward speed is often neglected, so that δv≈v−Ue1. Equation (6) can be rewritten with the mass matrix Mb=Mb RB +Ab(∞), Coriolis and centripetal forces coefficients due to rotation of {b}about {s}C∗ b/s =MbUL, the hydrostatic force Cbξ=Gζ and the steady control force Fb sc =C∗ b/sUe1to obtain u=Uas Mb˙ v+C∗ b/sv+Zt 0 Kb(t−τ) (v(τ)−Ue1)dτ +Gζ =Fb wave +Fb ext +C∗ b/sUe1−Fb sc.(7) Equation (7) is a linear equation with the linearized Coriolis and centripetal matrices. Nonlinearity can be introduced in Equation (7) by replacing the Coriolis and centripetal matrices to the one in {b}with respect to {n}and adding non-linear damping D(v)vor manoeuvring coefficients. Furthermore, the wave drift forces should also be taken into account, so that Fb wave =F(1) exc +F(2) exc. A unified seakeeping and manoeuvering model is then expressed with the kinematic and dynamic equations as ˙ ζ=JΘ(ζ)v,(8) Mb˙ v+Cb/n(v)v+D(v)v+Zt 0 Kb(t−τ) (v(τ)−Ue1)dτ +Gζ =Fb wave +Fb ext. On the other hand, two-time scale model solves low-frequency and wave-frequency motions separately. The equation of motion for the low-frequency motion is expressed as ˙ ζLF =JΘLF (ζLF )vLF ,(9) Mb˙ vLF +Cb/n(vLF )vLF +D(1) +D(vLF )vLF +GζLF =F(2) exc +Fb ext. in which the added mass contribution in the mass matrix Mband the Coriolis and centripetal matrices Cb/n, and the linear damping coefficient D(1) are computed with zero-encounter frequency. The perturbed position and orientation vector δζ=ξis then obtained through a linear superposition of the motion RAOs (obtained in Equation (3)) and wave spectrum components ˆηj, with the wave frequency that correspond with wave number  k= (kx, ky)at ship position p = (pE, pN), as follows ξ(t) = Re   Nω X j=1 ˆ ξ(U, ωj, µ)ˆηjei( kj·p−ωet) .(10) A transformation is needed to express ξ= (rs b/s,Θsb)in {n}, so that ξn= (Rz, ¯ ψrs b/s,Θsb). The final ship position and orientation vector ζis then given as, ζ=ζLF +ξn.(11) 2.3. Blended model Two variants of unified model can be derived from Eq. 8, further called Blended and Fully blended models. On the one hand, the Blended model computes a linear hydrostatic force based on the wetted hull surface at rest in calm water. On the other hand, the Fully blended model computes non-linear hydrostatic force based on the instantaneous wetted hull surface due to the unperturbed wave elevation and ship motions. This paper focuses only to the Blended model, the Fully blended will be published in a future publication. The following subsections describe the corrections to be applied to the Blended model. 4 2.3.1. Hydrodynamic coefficients correction The accuracy of the retardation functions depends on the limiting behavior of the hydrodynamic coefficients (added mass and damping) at zero and infinite frequencies. The hydrodynamic coefficients are obtained from SEACAL that considers the forward speed effect in the double-body steady potentials. In that case, a pre-processing before constructing the retardation functions needs to be applied to the hydrodynamic database. A smoothing function is applied to the damping coefficients so that the values go to zero for ω→0. This avoids an interference with manoeuvring damping. That correction is not necessary when the hydrodynamic database are obtained from the boundary element solver with zero-speed Green-function method. Furthermore, maximum frequency and frequency step in the hydrodynamic database determine also the accuracy of the retardation function. Time step in the retardation function could be too coarse if the maximum frequency was too small. While, the maximum time of the retardation function could be too short if the frequency step was too coarse. 2.3.2. Ghost drift force removal In the blended models, the evaluation of the first-order forces in non-equilibrium state generates undesired non-linear forces that could significantly influence the manoeuvring path. The undesired forces, that are later called ghost drift forces, are produced in the following process: • The first-order excitation force is evaluated with the phase of wave at the actual position. • The first-order excitation force is evaluated at the actual wave-frequency varying heading. • The hydrodynamic reaction forces due to added mass, damping and hydrostatic are computed with ship-fixed motions, velocity and accelerations in the yawed system of axis including wave frequency yaw motions. Those three contributions of ghost drift forces are later referred as first, second and third contributions, respectively. As described in [T. Bunnik, 2022], the ghost drift force can be computed analytically from frequencydomain boundary element solutions. The analytical ghost drift forces QTFs for the first contribution is derived as follows. Surge and sway motions response to a bichromatic wave (ω1and ω2) are expressed with the corresponding motion complex transfer functions ( ˆ ξjj= 1 for surge and j= 2 for sway) as ξj=Re hˆ ξj1e−iω1t+ˆ ξj2e−iω2ti.(12) The surge and sway motions are in {s}and they account for the phase change of the wave in {n}. The first-order excitation force at the actual wave phase (F(1∗) exc ) is then obtained as, with the wave number vector  kl= (klcos µ, klsin µ), F(1∗) exc =Re hˆ F(1) exc1e−iω1t+i k1·(ξ1,ξ2)+ˆ F(1) exc2e−iω2t+i k2·(ξ1,ξ2)i.(13) Linearizing the Equation (13) with an assumption of small motion gives F(1∗) exc ≈Re hˆ F(1) exc1e−iω1t1 + i k1·(ξ1, ξ2)+ˆ F(1) exc2e−iω2t1 + i k2·(ξ1, ξ2)i.(14) The first term with the multiplier one is the first-order excitation force evaluated at the equilibrium state of the body. The remainder is the second-order force contribution. The QTF of the first contribution ghost drift force is then obtained by substituting the surge and sway motions Equation (12) into Equation (14) and considering only the mean and difference-frequency contributions. The second contribution of ghost drift force is derived as follows. The first-order force at the actual heading (ψ)with the assumption of a small heading change around the mean heading can be approximated as F(1∗) exc (ψ) = F(1) exc(¯ ψ) + δψ dF(1) exc(¯ ψ) dψ (¯ ψ) + O(δψ2).(15) 5 In a complex form for a bichromatic wave, the force is then rewritten, with δψ =Re ξ61e−iω1t+ξ62e−iω2t,(16) as F(1∗) exc (ψ) = Re " ˆ F(1) exc1+δψ dˆ F(1) exc1 dψ !e−iω1t+ ˆ F(1) exc2+δψ dˆ F(1) exc2 dψ !e−iω2t#.(17) The first terms, ˆ F(1) exc1and ˆ F(1) exc2, are the correct first-order forces. The QTFs of the ghost drift forces are then obtained by substituting Equation (16) into the remainder terms of Equation (17) and considering only the mean and difference-frequency contributions. The third contribution of drift forces are derived as follows. In our time-domain simulation framework, the radiation forces and restoring forces are evaluated with the body fixed motions, velocities, and acceleration in the yawed reference frame. The velocity and acceleration vectors are denoted by vyawed = ( ˙x(¯ ψ),˙y(¯ ψ),˙z, ˙ ϕ(¯ ψ),˙ θ(¯ ψ),˙ ψ)and ˙ vyawed = (¨x(¯ ψ),¨y(¯ ψ),¨z, ¨ ϕ(¯ ψ),¨ θ(¯ ψ),¨ ψ), respectively. The velocity and acceleration can be transform to {b}as follows vb=R−1 z,ψvyawed,˙ vb=R−1 z,ψ ˙ vyawed,where R−1 z,ψ =Rz,ψ 03×3 03×3Rz,ψ −1 . The reaction forces are then expressed as Fb rad =AR−1 z,ψ ˙ vyawed +BR−1 z,ψvyawed (18) By taking linearization with a small yaw angle assumption, the radiation force Equation (18) contains firstand second-order contribution. The QTFs of the ghost drift force can then be obtained by applying the second-order force contribution of Equation (18) in a bichromatic wave with a complex form formulation. Following similar approach as above, the second-order contribution for the restoring force and the corresponding QTFs can be derived. The total ghost drift forces QTFs are then used in the time-domain simulation for the force correction. The wave excitation force for the blended model becomes FbB wave =F(1) exc +F(2) exc −F(2) ghost.(19) 2.4. Manoeuvring reaction forces and calm water resistance The manoeuvring reaction forces are modelled via coefficients to fit pre-determined forces and moments, via the so-called Unified Damping Model (UDM). The UDM models the manoeuvring reaction forces (damping and added inertia forces due to a motion) via coefficients multiplied by the motion derivatives (ship velocities and accelerations). These coefficients fit captive manoeuvring model experiment or CFD results, following the procedure described in [V. Ferrari, R. Tonelli, A.S. Kisjes and R. Hallman, 2022]. The Unified Damping Model considers a starboard-portside symmetry of the ship or floating structure. Calm water (sailing straight) resistance data from model tests is imposed to the simulations. Interpolation of the data over the ship speed is applied in the simulations. 2.5. Propeller and rudder forces 2.5.1. Propeller The propeller thrust and torque modelling requires input of a propeller open water curve, thrust deduction factor and wake fraction. First, the instantaneous advance velocity (VA) at the propeller which includes the wake fraction (w) and the ship velocity (V) is computed as VA=V(1 −w). The advance ratio (J) is then computed for a given propeller RPM (n) and a diameter of propeller (D) as J=VA nD . In the regular open water characteristic method, the thrust (T) and torque (Q) are computed with the thrust (KT) and torque (KQ) coefficients as a function of the advance ratio J. However, this approach is limited to positive rotation rate and positive advance speed. Differently, a four quadrants representation 6 of the propeller open water characteristic can model propeller thrust and torque for any combination of propeller rotation rate and advance speed. This method has been implemented and described in [J. Moulijn and R. Tonelli and U. Shipurkar, 2024, Kuiper, 1992]. In the four quadrants approach, the thrust and torque are computed with the thrust (CT) and torque (CQ) coefficients as function of the propeller pitch angle (β∗). The propeller pitch angle is then defined as an angle between the advance and rotational velocities. The rotational velocity (VT) of the blade is computed for instance at section 0.7.R, where the propeller radius R=D/2, as VT= 0.7πnD. The pitch angle is then expressed as β∗= arctan VA VT. The pitch angle ranges from -180 deg to 180 deg and is divided into four quadrants. The thrust, transversal force, and torque are then computed with the inflow velocity VR=pV2 A+V2 Tas follows T=1 8CT(β∗)ρV 2 RπD2, Y =1 8CY(β∗)ρV 2 RπD2, Q =1 8CQ(β∗)ρV 2 RπD3.(20) 2.5.2. Rudder In order to maintain the ship on the track or on heading, the rudder is steered by an autopilot. The rudder mathematical model describes the flow around the rudder, averaging velocity and force components in a single point of application. The model is defined by its dimensions, location, and stock inclination angle in the y-z plane. The model includes the effect of an accelerated inflow when positioned in a propellers slip stream, and the effect of a straightened inflow due to the presence of the hull. The mathematical model is based on lifting line and axial momentum (actuator disc) theory, with the implementation of empirical coefficients based on wind tunnel and towing tank model tests. The model theory and validation is fully described in [Tonelli and Vogels, 2024]. 2.5.3. Controller Heading-keeping controller is applied in simulations, presented in Section 5., with the autopilot gains of 2 [-], 0.018 [1/s] and 0.549 [s] for the proportional, integrator and damping gains, respectively. Additionally a speed controller is applied with 0.2 [-], 0.01 [1/s] and 0 [s] of the proportional, integrator and damping gains, respectively. This speed controller is also restricted by a propeller controller when the maximum power is exceeded. The maximum power is predefined from an acceleration test to reach a target speed. 3. Validation material This section describes the model test data used for the validation of the numerical models. In the CRS MORE working group, manoeuvring and seakeeping experiments with a model of the 5415M were performed at MARIN. The 5415M is a published naval combatant that is widely used in manoeuvring studies. The CRS MORE model test campaign includes the following tests: i) Captive manoeuvring experiments, reported in [R. Tonelli, 2020a] ii) Free sailing manoeuvring experiments, reported in [R. Tonelli, 2020b], iii) Seakeeping experiments, reported in [S. Rapuc, 2020]. Additionally, the captive tests data obtained in [R. Hallman, 2007] were used to make the UDM. An overview of the model is shown in Figure 1.. The model is appended with propellers, rudders, bilge keels and fin stabilizers. The ship particulars are presented in Table 1.. The seakeeping model tests were performed with a headingand track-keeping autopilot. In this paper, only the data of heading-keeping tests with active fin stabilizers will be used for the validation test. An overview of the seakeeping tests is presented in Table 2. for regular waves cases and Table 3. for irregular waves with Pierson-Moskowitz spectra. 7 Figure 1. Overview of 5415M model Table 1. Main particulars of the 5415M hull DESIGNATION SYMBOL MAGNITUDE UNIT Length between perpendiculars LP P 142 m Breadth moulded on waterline BW L 19.06 m Draught moulded on FPTF6.15 m Draught moulded on APTA6.15 m Displacement volume moulded ∇8428.6 m3 Transverse metacentre height above baseline KM 9.45 m Vertical center of gravity (from keel line) VCG 7.51 m Transverse metacentric height GMt1.94 m Longitudinal centre of buoyancy (fore of AP) LCB 70.05 m Approximate Roll radius of gyration kxx 7.62 m Approximate Pitch radius of gyration kyy 35.5 m Approximate Yaw radius of gyration kzz 35.5 m Table 2. Regular wave tests with heading-keeping autopilot and active fins Wave Amplitude [m] wave frequencies [rad/s] Wave directions [deg] Ship speed [kn] 1 0.45, 0.55, 0.65, 0.9,1.2 1.75 0.45, 0.55, 0.65, 0.9,1.2 180, 135, 60, 30 4, 12, 20 3.0 0.55, 0.9 Table 3. Irregular wave tests with heading-keeping autopilot and active fins Sea state Hs [m] Tp [s] γWave directions [deg] Ship speed [kn] SS-3 1.25 7.5 180, 135, 60, 30, 0 4 SS-4 2.5 8.8 1 180, 135, 60, 30 12, 20 SS-6 6.0 12.4 180, 135, 60, 30 4, 12, 20 8 4. Verification of ghost drift force A soft-mooring system was set up using linear spring coefficients applied for surge, sway and yaw moment. The restoring coefficients are set up so that the corresponding natural periods are 140 s, 120 s and 100 s for surge, sway and yaw motions, respectively. A linear spring damping of 7.5% of the critical damping in each mode was used for the stability. The simulations were performed in regular waves over a range of wave frequency in (0.2,1.6) rad/s with amplitude 1 m. For each simulation, a stable offset is measured and then drift forces are computed by multiplying the restoring coefficient with the corresponding offset of each mode. Time-domain simulation was set up with the following forces acting on the ship: linear hydrostatic force, radiation force, first-order excitation force and the mooring force. The computed QTFs obtained from the time-domain simulations (XMF) are compared against the database obtained using SEACAL. The results are shown in Figure 2. for zero-speed case in head and beam seas and in Figure 3. for the ship sailing at 4 knots in beam seas. The simulations showed that the second-order drift forces are generated by only imposing first-order excitation, linear hydrostatics and radiation forces. This confirms the existence of ghost drift forces in the time-domain simulations that agree with the analytic forces computed in SEACAL. Comparing the ghost and pure drift forces, while the ghost drift forces are much smaller with respect to the drift forces, the yaw ghost drift moment can be equal or larger than the pure drift moment. This suggest that the ghost drift moment cannot be ignored. Discrepancies of yaw drift moment between XMF and SEACAL at higher frequencies are observed. This is probably due to the coupled motions contributions which were not considered in the drift force calculation. Figure 2. Comparison of the drift forces between SEACAL and XMF at zero-speed in head seas (180 deg) and beam seas (90 deg). The top plot shows the surge drift forces in head seas. The middle and bottom plots show the sway drift forces and yaw drift moments in beam seas. 9 P. A. Bailey, W.G. Price, P. Temarel. A unified mathematical model describing the maneuvering of a ship traveling in a seaway. In Transactions of RINA, 1997. J.A. Pinkster. Low-frequency phenomena associated with vessels moored at sea. Society of Petroleum Engineers Journal, 15(06):487–494, 12 1975. R. Hallman. Applicability rotating arm: Validation and extension of the rotating arm functionality in the seakeeping and manoeuvring basin. Technical Report 21569-1-DT/SMB, MARIN, 2007. R. Skejic. 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