J Mar Sci Technol DOI 10.1007/s00773-016-0382-1

ORIGINAL ARTICLE

Roll-induced bifurcation in ship maneuvering under model uncertainty Anil Kumar Dash1 • Praveen Perumpulissery Chandran2 • Mohammed Kareem Khan2 Vishwanath Nagarajan1 • Om Prakash Sha1

•

Received: 20 July 2015 / Accepted: 6 April 2016 Ó JASNAOE 2016

Abstract In this paper, a mathematical model is developed for the maneuvering motion of a naval ship and bifurcations of its equilibrium are identified in roll-coupled motion. The subject ship is a high-speed surface combatant with twin-propeller twin-rudder system. Captive model tests are conducted for the ship using planar motion mechanism. Maneuvering coefficients are calculated by polynomial curve fitting of the test data. Uncertainty distribution in the coefficients is assumed same as that of the curve fitting errors. Uncertainty in the model coefficients is propagated to full-scale simulation results by the stochastic response surface method (SRSM). This method is computationally efficient as compared to standard Monte Carlo simulation technique. The SRSM uses polynomial chaos expansion of orthogonal to fit any probability distribution. Bifurcation analysis of the mathematical model is performed by varying the vertical center of gravity as the bifurcation parameter. Hopf bifurcation is identified. It is found that the bifurcations occur due to the coupling of roll motion with sway, yaw motion and rudder angle. In the presence of wind, roll angle response in bifurcation diagram is discussed.

List of symbols a0 ; a1 ; a2 a; b

aH

AR 0 0 CRd ; CRr

CR ; hR DP fa Fn FRX , FRY g Ix , Iz IP

Keywords Ship maneuvering  Twin-propeller twinrudder  PMM  Uncertainty assessment and propagation  Roll motion coupling  Bifurcations  Wind

& Vishwanath Nagarajan [email protected] 1

Department of Ocean Engineering and Naval Architecture, IIT Kharagpur, Kharagpur, West Bengal 721302, India

2

Naval Science and Technological Laboratory (NSTL), Visakhapatnam, India

Jx , Jz Kv_ , Kr_, Nv_ and Nr_ KT , KQ LOA LPP l1 m mx ¼ Xu_

Coefficients representing KT Proportional and derivative control parameters, respectively, of autopilot equation Ratio of additional lateral force induced on ship hull by rudder action to the rudder force Lateral area of a rudder (=span 9 chord) Rudder flow straightening coefficients for rudder angle and yaw rate, respectively Rudder chord and span, respectively Propeller diameter Rudder normal force coefficient Froude number Surge and sway forces acting on rudder stock, in rudder-fixed coordinate system Acceleration due to gravity Moment of inertia of the ship about x and z axis, respectively Moment of inertia of the propeller and shaft Added mass moment of inertia of ship with respect to x and z axis, respectively Added mass moment of inertia Thrust and torque coefficient, respectively Overall ship length Ship length between perpendiculars First Lyapunov coefficient Mass of the ship Added mass in x direction of the ship

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my ¼ Yv_ n0 nP nPMM PE Q0 q0 ; q1 ; q2 QE ; QP ; QR Ry T TD TE tP0 ; tP

TP TPMM tR u; v; p; r UC UR UX ; UY ; UN wP0 ; wP

wR ðxG ; yG ; zG Þ xH xP ; xR X; Y; N; K

XH ; YH ; NH ; KH

XP ; NP

XR ; YR ; NR ; KR

XW ; YW ; NW ; KW

yP ; yR

123

Added mass in y direction of the ship Initial engine speed Propeller revolutions Revolution of PMM Engine power Initial engine torque Coefficients representing KQ Engine, propeller, and rudder torque (acting on rudder stock), respectively Rudder lift Ship draft Turning diameter Time constant of steering gear Thrust deduction factor in straight moving and maneuvering motion, respectively Propeller thrust Time period of PMM Rudder resistance deduction factor Surge, sway, roll, and yaw velocity at midship, respectively Towing carriage speed Inflow velocity to rudder Uncertainty in measured surge, sway force and yaw moment Wake coefficient at propeller location in straight moving and maneuvering motion, respectively Wake coefficient at rudder location in maneuvering motion Position of center of gravity of ship from the origin O Rudder-hull interaction coefficients Longitudinal position of propeller and rudder, respectively Total surge force, sway force, yaw moment, and roll moment at the midship, respectively Surge force, sway force, yaw moment, and roll moment due to the bare hull at the midship, respectively Surge force and yaw moment due to the twin-propeller at the midship, respectively Surge force, sway force, yaw moment, and roll moment due to the twin-rudder at the midship, respectively Surge force, sway force, yaw moment, and roll moment due to wind at the midship, respectively Lateral position of propeller and rudder, respectively

ZR aR

b bP bR cR d d0 e j gS gR q qA w wW wR wA / /_

Vertical center of rudder force Effective rudder inflow angle or effective angle of attack of rudder to flow Ship drift angle Geometrical drift angle induced at the propeller position due to ship motions Geometrical drift angle induced at the rudder position due to ship motions Rudder flow straightening coefficient for drift angle Rudder angle Neutral rudder angle for straight motion Ratio of effective wake fraction in way of propeller and rudder Propeller race amplification factor Shaft efficiency Relative rotative efficiency Water density Air density Heading angle of ship Heading angle of wind Relative heading angle between wind and ship Absolute heading angle of ship in wind Roll angle of ship The time derivative of a variable is denoted by a dot above the variable, e.g., /_ is the time derivative of /

Superscript 0 Non-dimensional value Subscript {S} Starboard {P} Portside

1 Introduction In ship maneuvering motions, the hydrodynamic forces acting on the hull, propeller and rudder can be mathematically expressed by a set of nonlinear and cross-coupled ordinary differential equations. After developing the mathematical model, the standard maneuvers of the ship could be simulated. Coefficients present in the mathematical model can be calculated from measured hydrodynamic forces by conducting captive model tests or computational fluid dynamics (CFD) simulations for different vessel dynamics. In captive model tests, planar motion mechanism (PMM), circular motion test (CMT) and rotating arm (RA) tests are popular. A scaled model of the ship is towed

J Mar Sci Technol

in the towing tank with definite motion parameters and hydrodynamic forces and moments acting on the model are measured using load cells. In CFD, hydrodynamic forces are usually calculated by solving Reynolds averaged Navier–Stokes equation. The fluid motion around a ship is usually unstable, due to turbulence, flow around a curved surface, viscosity, hull-propeller-rudder interaction, etc. In both experimental and CFD approach, some approximations are involved. In experiments, the measured forces are assumed steady and averaged over time. In CFD, the complex fluid dynamic motion is modeled using approximate turbulence equation. Therefore, there is some uncertainty in the developed mathematical model. Due to this reason, uncertainty analysis (UA) of the ship maneuvering motion is essential. This will help in designing model experiments better; understand the limitations of CFD outputs, and finally improve the mathematical model for wider applications. Uncertainty analysis can be carried out in two steps. First, by finding out the uncertainty in model coefficients. Second, by propagating these uncertainties to full-scale simulations. Strøm-Tejsen and Chislett [1] developed a maneuvering model where hull, propeller, and rudder forces are combined together. On the other hand, we have the modular type Japanese mathematical modeling group (MMG) model. For high-speed vessels, with low metacentric height, the coupling of roll motion with maneuvering motions becomes critical. This aspect has been investigated by several researchers [2, 3]. For a twin-propeller twinrudder (TPTR) system, the asymmetric behavior of twinpropeller [4] and twin-rudder [5, 6] during maneuvering motion has been well investigated. In the TPTR system, even though both the rudders are operated in a synchronized manner, the torque for port and starboard unit is expected to be different. Coraddu et al. [7] studied asymmetry in twin-propeller performance by conducting the model free running test. They recommended that the propeller wake and flow straightening effect for such system during maneuvering motion must be investigated in detail. Therefore, an accurate mathematical model is required, which can handle the hull, twin-propeller, and twin-rudder and their interactions separately. In real sea condition, it is difficult to predict accurately the maneuvering characteristics of a ship where the effect of the waves, wind and current need to be considered [8]. For a large class of engineering tests, the uncertainty quantification methodology and procedures are based on the 95 % confidence level [9]. Coleman and Steele [10] provided a detailed derivation and discussion of these procedures. The International Towing Tank Conference (ITTC) procedure [11] is based on this methodology. Simonsen [12] developed a procedure for carrying out UA of the bare hull PMM captive maneuvering tests. Yoon [13]

applied UA to the stereoscopic particle image velocimetry (SPIV) flow measurements during PMM model experiments. In the literature [13, 14], uncertainty is calculated in non-dimensional forces, moments, and motions in PMM model tests (static drift, pure sway, pure yaw and drift ? yaw) for bare hull. Uncertainties in other PMM tests such as static heel, drift ? heel, yaw ? heel, static rudder, drift ? rudder and yaw ? rudder, propeller wake variation and maneuvering model coefficients have not been calculated. MCS method is popular for propagating uncertainty in any analytical or numerical model. Output distribution is determined by a large number of simulation runs with a different combination of input parameters. Input variables are randomly sampled from their respective distributions at each run. This method has been successfully applied to a large number of nonlinear problems. However, for uncertainty propagation, it requires more computer time and shows slow convergence. The response surface method (RSM) is a good alternative to MCS method. However, the number of input variables handled by this method is typically limited to eight or nine and may not be a convergent series [15]. SRSM has been proposed to overcome these problems. SRSM takes less computer time and converges faster. Isukapalli [16] developed the SRSM for uncertainty propagation in environmental and biological systems with uncertain input variables. Li et al. [17] applied the SRSM for reliability analysis of rock slope stability. It is shown that 378 runs on the SRSM are equivalent to 106 runs on the MCS for the rock slope factor of safety determination. Detailed theory and implementation of SRSM can be found in the literature [6, 16, 17]. In SRSM, each uncertain input or output is expressed as a polynomial chaos (PC) expansion of any orthogonal polynomial of standard random variable (SRV). For different orthogonal polynomials used, the nature of SRV will be different. PC expansion is a mean square convergent series expansion and is orthogonal in L2 space. Therefore, any function in space can be represented by PC expansion. The PC concept originated from Wiener’s homogeneous chaos concept [18]. Further developments on this method could be found in Ghanem and Spanos [19]. Bulian et al. [20] propagated uncertainty of maneuvering coefficients to the outputs of turning and zigzag simulations. A linearized Gaussian approach is proposed for the study. This approach is compared with MCS method and found to be computationally efficient. Fine form ships, especially naval vessels are required to operate in heavy weather conditions at high-speed where rolling motion may be considerable. At high-speed, roll and its coupling with the maneuvering motions, become significant. Therefore, maneuvering prediction and stability of high-speed naval vessels with roll-coupled motion are more critical. The roll-coupling motion has significant

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influence on ship stability and capsizing behavior [21]. Yasukawa and Yoshimura [22] proposed a formula to check the course stability of ships by considering the rollcoupling effect. Besides the hydrodynamic hull coefficients, the proposed formula also considers the influence of forward speed and metacentric height of the ship on the roll-coupled course stability coefficient. Eda [3] investigated course stability of a high-speed TPTR naval hull. Oscillatory instability in yaw and roll motions in the presence of stepwise beam wind is demonstrated. Twinpropeller and twin-rudder are not modeled as separate entities. Additionally, sway-roll, yaw-roll coupling coefficients are not included in the maneuvering model. Son and Nomoto [2] investigated the roll motion instability of a high-speed single-propeller single-rudder ship (SR 108 container ship). The influence of metacentric height on a sway-yaw-roll-steering gear coupling motion and the choice of autopilot on self-excited coupled yaw/roll motions are investigated. It is shown that the ship may capsize due to self-excited sway-yaw-roll-rudder coupled motion during maneuvering. Spyrou [23] investigated yaw stability of a super tanker ship and a river boat in the steady wind. The wind direction is taken as a bifurcation parameter. Hopf bifurcation point is identified when ship enters from beam wind to head wind. At this point, static stability is lost and self-sustained oscillations emerged. Limit cycles for the vessel are shown at the unstable region. Falzarano et al. [24] treated the ship’s roll motion as a nonlinear bifurcation problem. They studied the roll behavior of a fishing vessel during turning in a wave at low speed. The heading angle of the ship is taken as a bifurcation parameter. Period doubling bifurcations for parametric excitation, external excitation and parametric ? external excitation are investigated. Francescutto and Contento [25] investigated the roll motion of a destroyer in beam sea condition through model experiments. Umeda et al. [26] identified the period doubling, chaos, and capsizing characteristics in parametric rolling for a container ship due to wave load. Virgin [27] investigated the nonlinear roll response of a low freeboard ship model in regular waves. Period doubling bifurcation leading to chaotic roll motions of the vessel prior to capsizing in regular waves is shown. Instability characteristics such as multiple solutions, jumps, period multiplying bifurcation, dynamic chaos, sub and super harmonic resonance often occur in highly nonlinear multiple-degree-of-freedom systems. These have not been investigated in detail for maneuvering motions of a ship. Dash et al. [28] carried out bifurcation analysis of DTMB 5415 surface combatant. Hopf, fold, and flip type local bifurcations are identified due to roll-coupled motion during turning maneuver. The uncertainty of model coefficients is considered during bifurcation analysis.

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The paper is organized as follows. In Sect. 2, ship maneuvering motion equations and PMM tests conducted are described. In Sect. 3, a nonlinear MMG model for estimating the hydrodynamic forces acting on the hull, twin-propeller, and twin-rudder is developed. The coefficients of the model are determined from PMM test data. The present model is validated by free running model tests. Uncertainty in the model coefficients is calculated. Section 4 covers propagation of model coefficients’ uncertainty to full-scale maneuvering simulations; the SRSM is employed for the propagation. Simulation results and the uncertainty are discussed. In Sect. 5, vertical center of gravity induced local bifurcations of the equilibria and periodic orbits are investigated numerically using MATCONT [29]. Local bifurcations of periodic orbits such as Hopf are found. The roll angle responses in different bifurcations are discussed. In Sect. 6, turning maneuver is simulated in a steady wind and roll behavior is discussed.

2 Mathematical model and test cases The Japanese MMG type maneuvering model for predicting the hydrodynamic forces acting on the hull, twin-propeller, and twin-rudder is developed by referring some of the earlier work on the subject, [2, 4, 6, 30–32]. Heave and pitch motions are ignored since their influence in maneuvering motion of large ships is less as compared to other motions. The maneuvering motion equations are given in Eq. 1. Coordinate system and sign convention followed in the paper is shown in Fig. 1. 9 ðm þ mx Þu_ ¼ ðm þ my Þrv þ mxG r 2  mzG pr þ X0 þ XP þ XR þ XW > > > ðm þ my Þv_  mzG p_ þ ðmxG  Yr_Þr_ ¼ ðm þ mx Þur þ Y0 þ YR þ YW = > ðIz þ Jz Þr_ þ ðmxG  Nv_ Þv_ ¼ mxG ur þ N0 þ NP þ NR þ NW > > ; ðIx þ Jx Þp_  ðmzG þ Kv_ Þv_  Kr_r_  Kp p ¼ mzG ur þ K0 þ KR þ KW

ð1Þ

Fig. 1 Coordinate system and sign convention

J Mar Sci Technol

The starboard and port rudders are controlled by proportional cum derivative type autopilot as shown in Eq. 2.

Table 1 Principal particulars of ship model (1:19.2) Items

Magnitude

_ d_ fSg TE ¼ ðdfSg  aw  bwÞ

LPP =B

9.469

T=LPP

0.033

LCB (from midship), fwd?

-0.021 LPP

CB

0.511

KG=LPP

44.75 9 10-3

KM=LPP

54.125 9 10-3

Fn(U)

0.24 (18 kn),

P

P

ð2Þ

The mathematical model for propeller revolution rate is expressed as shown in Eq. 3.   . 2 5 K S 2pIP n_ PfSg ¼ gS QEfSg  qnPfSg DP S fPg QfPg gRfSPg P P P ð3Þ

0.41 (30 kn)

where the relation QEfSg ¼ Q0fSg is used for diesel engine P P model (constant torque control strategy) and the relation . QEfSg ¼ Q0fSg ðnPfSg n0fSg Þn is used for gas turbine P

P

P

Propellers

P

model [33]. We have taken n ¼ 1 for analysis in this paper. The complete mathematical model is shown in ‘‘Appendix’’. The subject ship is a high speed hull form (HSHF) with TPTR system. Hull model is fabricated with appendages such as sonar dome, skeg, shafts, bossing, ‘A’ brackets, rudders, stabilizer fins, and bilge keels. Its main particulars are given in Table 1. A set of PMM tests is carried out separately on bare and appended hull in the high-speed towing tank of Naval Science and Technological Laboratory (NSTL) [34]. During the bare hull test, the hull is fitted with fins, bilge keels, skegs and ‘A’ brackets. Twin-propellers and twin-rudders are additionally fitted during appended hull test. Details of static and dynamic tests carried out are given in Table 2. All the tests were conducted at Fn = 0.24. Appended tests are conducted at model self-propulsion point nP = 11.85 rps. The total surge and sway forces and the yaw moment are found by combining the measured forces from each of the load cells as shown in Eq. 4. Inertial forces due to the mass of the model and the moving part of the measurement system are subtracted from measured forces and moment to get the pure hydrodynamic forces as shown in Eq. 5. 9 ( ) ( ) ( ) X X X > > = ¼ þ ð4Þ Y total Y measured;fore Y measured;aft > > ; Ntotal ¼ Ymeasured;fore  Lfore þ Ymeasured;aft  Laft 9 FX þ m ðu_  rv  xG r 2 Þ > > XH0 ¼ measured > > > 0:5qLPP TU 2 > > =  _ _ F þ m ð v þ ur þ x r Þ Ymeasured G 0 YH ¼ > 0:5qLPP TU 2 > > >   > _ _ M þ F x þ I x r þ m ð v þ ur Þ > Z Y PMM G measured measured 0 Z > ; NH ¼ 2 2 0:5qLPP TU ð5Þ where, Lfore and Laft is the distance of the forward and aft gauge from PMM frame center, respectively. FXmeasured and

Type

Fixed pitch

Number of blades D

5 0.025 LPP

P=D ð0:7RÞ

1.117

Ae =A0

0.59

Hub ratio

0.133

Rotation

Outward on top

Rudders Type

Spade

Span/chord

1.2

Turn rate (°/s)

5.5

Table 2 PMM bare and appended hull test program in NSTL, DRDO, India [34] Test

Range of test parameters (in deg)

TPMM (s)

H

/ = -10, 5, 8, 12

–

D?H

b = 0, 5, 7.5, 10, 12.5, / = 5, 8

–

S

bmax = 5, 10

15

Y

0 rmax = 0.1, 0.2, 0.3

15

S?Y

bmax = bmax =

Y?H

/ = 5,

0 5 and rmax = 0.2 0 10 and rmax = 0.1 0 rmax = 0.05, 0.1, 0.2

16 18

Appended hull tests R

d = -35 to ?35

–

D?R

b = 10, d = -35 to ?35

–

b = -10, d = -35 to ?35 Y?R

0 rmax = 0.3, d = -35 to 0

15

H heel, D drift, S sway, Y yaw, R rudder

FYmeasured are the measured total X (Xtotal ) and Y force (Ytotal ), respectively. MZmeasured is the measured yaw moment (Ntotal ). m is the mass and IZ is the yaw mass moment of inertia of the model undergoing dynamic maneuver. It may be noted that m and IZ are not equal to model’s mass and yaw mass moment of inertia in air. This is because the inertial forces of a portion of tow post and balance cannot be measured by load cells although their mass is supported by model ship’s

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XH' × 103

buoyancy. The coordinates of the PMM frame center from O is given by (xPMM , yPMM , zPMM ). The layout of PMM setup for dynamic motion is given in ‘‘Appendix’’.

123

5

10

15 t (sec)

20

25

30

(c)

(b)

0.1

0.2

NH' × 103 0.15

0.075

0.1

0.05

0.1

0.05

0.025

0.05

0 -6

-4

0

-2

2

4

0 -18

0.15

-12

U %

-6

0 3

0 -15 -10 -5

U %

X

0

5

10

U %

Y

N

XH' × 103

Fig. 3 Probability distribution of modeling uncertainty in pure sway test data. Sample size is 3000. Distribution type and parameters: a generalized extreme value, l = -2.01231, r = 2.15186, m = -0.26847, b generalized extreme value, l = -8.35906, r = 5.09425, m = -0.38923 and c normal, l = -4.22084, r = 4.12883

YH' × 103

0 0 0 The coefficients Xvv , Yv0 , Yvvv , Yv0_ , Nv0 , Nvvv , and Nv0_ are 0 computed by Fourier series analysis of XH , YH0 , and NH0 measured during the pure sway tests. Comparison between the math-model fit and experimental data at bmax = 108 is shown in Fig. 2. Distribution of uncertainty in XH0 , YH0 , and NH0 is shown in Fig. 3. The probability distribution type is assumed generalized extreme value for surge and sway force and normal for the yaw moment. It is assumed that the distribution of uncertainty in model coefficients is same as that of the corresponding force or moment data. For example, it is assumed that the distribution of uncertainty 0 0 in Xvv is same as that of UX ; Yv0 , Yvvv , and Yv0_ is same as that 0 0 0 of UY ; Nv , Nvvv , and Nv_ is same as that of UN . This assumption is also valid for following tests. The coeffi0 0 cients Xrr0 , Yr0 , Yrrr , Yr0_, Nr0 , Nrrr , and Nr0_ are computed by 0 0 Fourier series analysis of XH , YH , and NH0 measured during the pure yaw tests. Comparison between the math-model fit 0 and experimental data at rmax = 0.3 is shown in Fig. 4.

0

(a)

0.2

NH' × 103

3.1 Hull model and uncertainty

Expt. Fit

Fig. 2 Curve fitting of pure sway test data at bmax ¼ 10

PDF

In this section, test data are curve fitted to the described mathematical model for calculating the maneuvering coefficients related to the hull, propeller, and rudder model. The difference between the polynomial fit and experimental data is considered as the uncertainty in the model. The predicted uncertainty is imposed on model coefficients. During dynamic tests, the ship model follows an oscillatory path of two complete cycles without transients. The resulting time-varying forces and moments are obtained by fairing or filtering with Fourier series consisting of eight harmonics [12]. The uncertainty introduced via the dynamic fairing is not considered in the present work. During static tests, in each run the ship model is towed at steady speed along the towing tank for 20 s. The hydrodynamic forces and moments are measured at time interval of 0.01 s. Therefore, 2000 number of measured data points is available for each run. The model coefficients are calculated by least square fitting of all data points simultaneously. Uncertainty (difference between the model fit and measured data) introduced via static averaging is considered in the present work. Forces and moments are non-dimensionalized by 0:5qLPP TU 2 and 0:5qL2PP TU 2 , respectively, while KG and KN are non-dimensionalized by LPP .

YH' × 103

3 Model coefficients and their uncertainty

-17.5 -18 -18.5 -19 -19.5 -20 -20.5 80 40 0 -40 -80 30 20 10 0 -10 -20 -30

1 0.5 0 -0.5 -1 -1.5 20 10 0 -10 -20 15 10 5 0 -5 -10 -15

Expt. Fit

0

5

10 15

20 25 30 35 40 45 t (sec)

0 Fig. 4 Curve fitting of pure yaw test data at rmax ¼ 0:3

Distribution of uncertainty in XH0 , YH0 , and NH0 is shown in Fig. 5. The probability distribution type is assumed t location-scale for surge force, logistic for sway force and t

J Mar Sci Technol

0.075

0.2

0.1

0.05 0.1

0.05

0.025

0 -6

-3

0

3

6

0 -20

U % X

-10

0 10 U %

20

0 -10

0

Y

10 UN %

20

NH' × 103

-17 -18 -19 -20 -21 -22 80 40 0 -40 -80 20 15 10 5 0 -5 -10 -15

φ (deg) -15-10 -5 0 5 10 15 0 -0.05 -0.1 -0.15 -0.2 -0.25

Expt. Fit

4

15 20 t (sec)

25

(b)

300

30

35

120

(c)

0.06

0.09

0.04

0.06

0.02

0.03

80

0.1 0.05 0 U % X

4

8

0 -40

-20 0 U % Y

20

0 -30

0 -300

-150

0 150 U %

300

0 -40

-20

Y

0 20 U %

40

N

Fig. 9 Probability distribution of modeling uncertainty in static heel test data. Sample size is 4000. Distribution type and parameters: a t Location-scale, l = 18.3741, r = 125.09800, m = 1.62527, b t Location-scale, l = -0.65567, r = 91.61580, m = 3.343450 and c t Location-scale, l = 0.34290, r = 10.47500, m = 2.48373

YH' × 103

0.15

PDF

150

X

10

0.02 0.01

0 U %

5

0.03

2

0 -300 -150

0

(c) 0.05 0.04

2 1

(a)

-4

0.4 0.3 Expt. 0.2 Fit 0.1 0 -0.1 -0.2 -0.3 -0.4 -15-10 -5 0 5 10 15 φ (deg)

(b)

-3

6 x 10

3

100

0 -8

(a)

-3

4 x 10

0 Fig. 6 Curve fitting of sway ? yaw test data at bmax 10 , rmax ¼ 0:1

0.2

1.5 1 0.5 0 -0.5 -1 -1.5 -15-10 -5 0 5 10 15 φ (deg)

Fig. 8 Curve fitting of static heel test data at / = -10°, 0°, 5°, 8°, 12°

PDF

YH' × 103

XH' × 103

Fig. 5 Probability distribution of modeling uncertainty in pure yaw test data. Sample size is 4500. Distribution type and parameters: a t Location-scale, l = 0.18605, r = 1.79095, m = 55.00971, b logistic, l = 2.22312, r = 2.91785 and c t Location-scale, l = 3.73797, r = 3.30238, m = 3.10072

NH' × 103

(c) 0.15

40

φ = 80

φ = 50

Expt. Fit

30

NH' × 103

PDF

(b) 0.1

YH' × 103

(a)

0.3

0 0 0 , Y/0 , Y/// , N/0 and N/// are comThe coefficients X// 0 0 0 puted by least square fit of XH , YH , and NH measured during the static heel tests. Comparison between the math-model fit and experimental data is shown in Fig. 8. Distribution of uncertainty in XH0 , YH0 and NH0 is shown in Fig. 9. The probability distribution type is assumed t location-scale for surge and sway force and yaw moment. The coefficients 0 0 0 0 Yvv/ , Yv// , Nvv/ and Nv// are computed by least square fit 0 0 0 of XH , YH , and NH measured during the drift ? heel tests. The uncoupled force and moment components due to v, vvv, / and /// are deducted from the measured forces and moment before analysis. Comparison between the mathmodel fit (contains v, / and v  / related coefficients) and experimental data is shown in Fig. 10. Distribution of uncertainty in YH0 and NH0 is shown in Fig. 11. The probability distribution type is assumed generalized extreme

XH' × 103

0 0 location-scale for yaw moment. The coefficients Xvr , Yvvr , 0 0 0 Yvrr , Nvvr and Nvrr are computed by Fourier series analysis of XH0 , YH0 , and NH0 measured during the sway ? yaw tests. Comparison between the math-model fit and experimental 0 data at bmax = 108, rmax = 0.1 is shown in Fig. 6. Distribution of uncertainty in XH0 , YH0 and NH0 is shown in Fig. 7. The probability distribution type is assumed logistic for surge force, normal for sway force and generalized extreme value for yaw moment.

60 40

Expt. Fit

20 10

20 -20 -10 U %

0

0

N

0 -20

Fig. 7 Probability distribution of modeling uncertainty in sway ? yaw test data. Sample size is 3200. Distribution type and parameters: a logistic, l = 0.80452, r = 1.90295, b normal, l = -11.60281, r = 13.31171 and c generalized extreme value, l = -12.38850, r = 6.51133, m = -0.57140

0 2.5 5 7.5 10 12.5 β (deg)

-10

0 2.5 5 7.5 10 12.5 β (deg)

Fig. 10 Curve fitting of drift ? heel test data. The math-model fit curve contains v, /, and v  / related coefficients

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value for sway force and yaw moment. The coefficients 0 0 0 0 Yrr/ , Yr// , Nrr/ , and Nr// are computed by Fourier series 0 0 analysis of XH , YH , and NH0 measured during the yaw ? heel tests. Comparison between the math-model fit and experimental data at r 0 = 0.2 and / = 58 is shown in Fig. 12. Distribution of uncertainty in YH0 and NH0 is shown in Fig. 13. The maneuvering coefficients in hull model are given in Table 3.

(b)

(a) 0.2

PDF

0.06

0.15

0.04

0.1 0.02

0.05

0 -20

-10

0 10 U %

20

0 -4

0

4

8

U %

Y

N

NH' × 103

YH' × 103

Fig. 11 Probability distribution of modeling uncertainty in drift ? heel test data. Sample size is 4000. Distribution type and parameters: a generalized extreme value, l = -1.04406, r = 7.28488, m = -0.33106, and b generalized extreme value, l = -0.44450, r = 2.10089, m = -0.00418

P

3.3 Model validation 5

10

15

20 25 t (sec)

30

35

40

Fig. 12 Curve fitting of static heel ? yaw test data at / ¼ 5 , 0 = 0.2. The fit curve contains r, /, and r  / related coefficients rmax

(a)

(b) 0.045

0.04 0.03 PDF

P

sponding to the rudder, sway, and yaw motions, respectively. Distribution of uncertainty in FRY is shown in Fig. 16, probability distribution type is assumed normal. The other flow straightening coefficients are calculated from static drift and pure yaw test at d ¼ 0 . The hull, rudder and propeller interaction coefficients are given in Table 4.

Expt. Fit

0

0.03

0.02 0.015

0.01 0

20 UY %

40 50

0 -15

0

15 30 UN %

45

Fig. 13 Probability distribution of modeling uncertainty in yaw ? heel test data. Sample size is 4000. Distribution type and parameters: a generalized extreme value, l = 8.76099, r = 14.51850, m = -0.21183 and b generalized extreme value, l = 2.03496, r = 11.57830, m = 0.03614

123

The static rudder test is performed by varying d in the range -35° to ?35°. Hydrodynamic interaction coefficients between ship hull and rudder aH and x0H are calculated from measured forces. Hydrodynamic interaction coefficients between propeller and rudder d0 , e and j are calculated from the rudder normal force at d ¼ 0 . The estimated value of aH and x0H is shown in Fig. 14. Here, aH = 0.325. Comparison of the math-model fit and experimental data of rudder normal force is shown in Fig. 15. After d = 28°, the trend of the test data differs. Therefore, the twin-rudder model is made discontinuous at d ¼ 28 . During maneuvering of ships with twin-propeller and twin-rudder, the port and starboard rudders or propellers do not get same flow field. Because of this reason, both twin-rudder and twin-propeller may exhibit asymmetric characteristics during port or starboard maneuvering. Similar behavior is also observed in the present ship. This aspect has been investigated earlier [3–5], and is therefore not explained elaborately in this paper. The asymmetric behavior of twin-rudder is handled by the flow straightening coefficients CRdfSg , cRfSg and CRrfSg correP

12 8 4 0 -4 -8 8 6 4 2 0 -2 -4 -6 -8

0 -20

3.2 Twin-rudder and twin-propeller model and uncertainty

Simulation of 20° zigzag and 35° turning circle maneuvers in full-scale are carried out using the present mathematical model. The powering of the ship is done with the constant propeller revolution strategy. Full-scale simulation results of zigzag (see Fig. 17) and turning (see Fig. 18) maneuver are compared with the free running test data. Simulation results match well with the experimental data. During maneuvering simulations in full-scale, scale effects for skin friction correction for both ship and propeller are considered as per ITTC recommended procedure.

4 Uncertainty analysis For UA, SRSM is used in this paper. For comparison purposes, MCS UA results are also shown. In SRSM, Hermite polynomial of the third order and Gaussian SRV are used in the PC expansion. A linear sensitivity

J Mar Sci Technol Table 3 Ship maneuvering coefficients

Items

Value

Items

Xu0_

Value

Items

Value

0 Nv//

-5.008E-01

0 Nrr/

9.463E-02 -2.326E-01

z0H Y0 vv/  z0H Y0 rr/ 

-5.477E-03

0 Yvv/

4.716E?00

0 Xvv Xrr0 0 X// 0 Xvr Yv0_ Yv0

-2.274E-02

-6.661E-01

-2.524E-01

0 Yv// 0 Yrr/ 0 Yr// Nv0_ Nv0 0 Nvvv

-1.361E?00

0 Nr// K/0 0 K/// z0H 0  

0 Yvvv

-1.507E?00

Nr0_

-1.381E-02

0  K rr/

Yr0_

-2.538E-02

Nr0

-3.707E-02

Kv0_

-1.900E-03

Yr0

3.283E-02

0 Nrrr

-4.552E-02

Kr0_

3.255E-04

N/0 0 N/// 0 Nvvr 0 Nvrr 0 Nvv/

-1.253E-03

Kp0

-1.441E-04

-1.001E-02

0 Kppp

-1.355E-05

-1.677E?00

Kp0_

-2.408E-05

-2.512E-03 -6.194E-03 1.201E-01 -1.905E-01

1.155E-01 2.514E-01 -7.675E-03 -8.497E-02

K

0.3 Y/0 0.0 0.02

vv/ v//

v//

r//

0 Yrrr Y/0 0 Y/// 0 Yvvr 0 Yvrr

-2.479E-01 -2.400E-03 -7.797E-02 -2.268E?00 -2.909E?00

r//

-3.421E-01 -2.667E-01

The added mass in surge direction is computed from Motora [35]

8

Ry'× 103 12 16

16

-8 -12

14 δ=5

12

δ = 10

-20 aH = 0.325 -24

-32

N'× 103

Y'× 103

-16

δ = 20

0

10

Distribution: Normal

0.06

δ = 200

0.04 0.02

δ = 10 ' δ = 100 xH = -0.485 0

8

δ = 250 δ = 300 δ = 350

0.08

δ = 250

0 0

-28

δ = 350 δ = 300

20

PDF

4

6 4

8

12 16 Ry'× 103

0 -12

-8

UF

20

8

4

0

-4

12

% RY

Fig. 16 Uncertainty in rudder model

Fig. 14 Hull-rudder interaction coefficient aH and x0H

Table 4 Interaction coefficients of TPTR model 0.050

cRfSg

0.785

xP0

0.050

cRfPg

0

sbP fSg

0.108

0 CRrfSg

-0.35

s3bP fSg

0.500

0 CRrfPg

-0.35

0.267

0 CRdfSg

-0.267

k3bP fSg

1.625

0 CRdfPg

-0.267

-10

aH

0.325

0 CRdfPg

-0.750

-15

x0H

0 CRdfSg

-0.750

(a)

(b)

(c)

15 10

FRY{S}' × 103

For bR \0

tP0

20

rmax'= 0.3

5

kbP fSg

0 -5

β = 100

-20 -40 -20 0

20 40 -40 -20 0

δ (deg)

20 40 -40 -20 0

δ (deg)

20 40

δ (deg)

Fig. 15 Comparison of model prediction and experimental data, a static rudder, b drift ? rudder and c yaw ? rudder test

-0.485

For d\  28 

For d [ 28

z0R



0.006

For aRfPg \0

CRQbfPg

-0.075

e

1.047

For aRfPg [ 0

CRQbfPg

0.032



K

0.600

For aRfSg \0

CRQbfSg

-0.045

CRQafSg

0.3

For aRfSg [ 0

CRQbfSg

0.075

P

123

J Mar Sci Technol

(b)

(a)

(UL), and lower limit (LL) at 95 % confidence interval (CI), are obtained by finding the 2.5 and 97.5 % percentiles.

1

0.995 u' (-)

δ and ψ (deg)

35 30 25 20 15 10 5 0 -5 -10 -15 -20 -25 -30 (c) 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4

0.99 At constant prop. rev., Fn = 0.24 Simulation Free run expt.

0.985 0.98 8 6 4 2 0 -2 -4 -6 -8

r' (-)

φ (deg)

(d)

0

50

100 150 t (sec)

200

0

50

100 150 t (sec)

200

Fig. 17 Comparison of free running experiments and simulation of 20° port zigzag test. Both the model free run test and simulations results are extrapolated to full scale

4

(b) 6

3

4

φ (deg)

2

x' (-)

(a)

1

-4

-3

-2 -1 y' (-)

0

2 0 Simulation Free run expt.

-2

0

-4

-1 1

(d) 1

At constant prop. rev., Fn = 0.24

r' (-)

(c) 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7

0

u' (-)

0.995

t (sec) 50 100 150 200

0.99 0.985 0.98 0.975 0

50

100 150 200 t (sec)

Fig. 18 Comparison of full-scale simulation of 35° turning with free running model test

study has been carried out to select the sensitive inputs for each output (see ‘‘Appendix’’). Uncertainty propagation in simulation results of full-scale zigzag and turning maneuver will be discussed. When the ship turns to the port side, the port is the leeward side and starboard is the windward side. For turns to the starboard side, it is vice versa. The ship is powered by two gas turbines, one on each shaft. Uncertainty in model inputs is propagated to simulation results by SRSM. Instead of showing complete uncertainty distribution, outputs are presented by their deterministic values, upper limit

123

4.1 UA of zigzag maneuver Full-scale simulation of starboard 20° zigzag maneuver at Fn = 0.4 is carried out using the present mathematical model. Asymmetric behavior and uncertainty limit of propeller and rudder performances during full-scale simulation of zigzag maneuver are shown in Fig. 19. The SRSM result of starboard engine torque is compared with MCS by finding the 50 % percentile, shown in Fig. 19a. It is observed that the SRSM result agrees well with MCS. Uncertainty in w, r, u, v, / is given in Table 5. The simulation results of twin-engine, twin-propeller, and twin-rudder performances and their uncertainties will be discussed. Simulation result of propeller revolution is shown in Fig. 19b. At a particular time level, the leeward propeller revolution is about 2–4 % higher than the corresponding windward value. Uncertainty in either propeller revolution is 5–7 % of deterministic value. The overall variation is about 3–3.9 rps. Simulation result of propeller thrust is shown in Fig. 19c. At a particular time level, the leeward propeller thrust is about 1–2 % higher than the corresponding windward value. Uncertainty in either propeller thrust is 4–5 % of deterministic value. The overall variation is between 590 and 720 kN. Simulation result of engine torque is shown in Fig. 19d. At a particular time level, the leeward engine torque is about 2–4 % higher than the corresponding windward value. Uncertainty in either engine torque is 5–6 % of deterministic value. The overall variation is between 400 and 520 kN-m. Engine torque fluctuates between its maximum to minimum values in 12–15 s. Propeller and shaft experience different loading during starboard and port turn of the ship. This information may be useful in designing the control algorithm for starboard and port engine. Simulation result of engine power is shown in Fig. 19e. At a particular time level, the leeward engine power is about 3–7 % higher than the corresponding windward value. Uncertainty in either engine power is 10–12 % of deterministic value. The overall variation is about 7.5–13 MW. Simulation result of rudder normal force is shown in Fig. 19f. At a particular time level, the leeward rudder normal force is about 50–100 % higher than the corresponding windward value. Uncertainty in either rudder normal force is 10–12 % of deterministic value. The overall force variation is about -650 to 550 kN. Simulation result of rudder torque is shown in Fig. 19g. The trend of rudder torque is similar to the rudder normal force. The overall variation is about -45 to 35 kNm. The torque for the leeward and windward rudder is not same during

J Mar Sci Technol Table 5 Total uncertainty in full-scale maneuvering simulation motion parameters

Deterministic

Q E{S} × 10-5 (Nm)

6 5.8 5.6 5.4 5.2 5 4.8 4.6 4.4

MCS mean (at 50% CL) (105 samples)

(a)

Port side 3rd order SRSM (UL at 95% C.I)

Starboard side 3rd order SRSM (UL at 95% C.I)

Deterministic

Deterministic

3rd order SRSM (LL at 95% C.I)

(c)

TP × 10-5 (N)

20° zigzag

35° turning

u

8–10

10–12

v

8–9

10–14

r w

8–12 8–10

8–10 –

/

7–11

7–8

TD

–

10–13

torque on bearings. There is significant variation in the pressure center of rudder normal force. Additionally, the torque is not same during displacement and restoring motion of the rudder [21]. In TPTR system, besides above, the port and starboard rudder torque is asymmetric. This information will be useful for designing steering gear mechanism.

(b)

QE × 10-5 (Nm) P E × 10-5 (Watt)

4.2 UA of turning maneuver

(d)

(e)

20

δ (deg)

10 0 -10 -20

FRY × 10-5 (N) QR × 10-5 (Nm)

Items

3rd order SRSM (LL at 95% C.I)

nP (rps)

4.8 4.4 4 3.6 3.2 2.8 8.4 8 7.6 7.2 6.8 6.4 6 5.6 6.4 6 5.6 5.2 4.8 4.4 4 180 160 140 120 100 80 60 8 6 4 2 0 -2 -4 -6 -8 0.45 0.3 0.15 0 -0.15 -0.3 -0.45

Uncertainty (%)

3rd Order SRSM mean (1632 samples)

(f)

(g)

0

50

100

150

t (sec) Fig. 19 Asymmetric behavior and uncertainty limit of propeller and rudder performances during full-scale simulation of zigzag maneuver

maneuvering. It may be noted that the rudder torque shown here is pure hydrodynamic torque. During full-scale ship trials, the rudder torque is hydrodynamic and the frictional

Full-scale simulation of starboard 35° turning maneuver at Fn = 0.4 is carried out using the present mathematical model. Asymmetric behavior and uncertainty limit of propeller and rudder performances during full-scale simulation of turning maneuver are shown in Fig. 20. The SRSM result of starboard engine torque is compared with MCS by finding the 50 % percentile, shown in Fig. 20a. It is observed that the SRSM result agrees well with MCS. Uncertainty in u, v, r, /, TD is given in Table 5. The simulation results of twin-propeller, twin-engine and twin-rudder performances and their uncertainties will be discussed. Simulation result of propeller revolution is shown in Fig. 20b. At a particular time level, the leeward propeller revolution is about 4–8 % higher than the corresponding windward value. Uncertainty in either propeller revolution is 5–6 % of deterministic value. The overall variation is about 2.3–3.1 rps. Simulation result of propeller thrust is shown in Fig. 20c. At a particular time level, the leeward propeller thrust is about 2–3 % higher than the corresponding windward value. Uncertainty in propeller thrust is 2.5–5 % of deterministic value. The overall variation is about 500–605 kN. Simulation result of engine torque is shown in Fig. 20d. At a particular time level, the leeward engine torque is about 3.5–9 % higher than the corresponding windward value. Uncertainty in engine torque is 5–6.6 % of deterministic value. The overall variation is about 320–410 kNm. Simulation result of engine power is shown in Fig. 20e. At a particular time level, the leeward engine power is about 9–11 % higher than the corresponding windward value. Uncertainty in engine power is

123

J Mar Sci Technol

QE{S} × 10-5 (Nm)

0

overall rudder force variation is about 160–370 kN. Simulation result of rudder torque is shown in Fig. 20g. The trend of rudder torque is similar to the rudder normal force. The overall variation is about 3–21 kNm. In both zigzag and turning maneuvers, it is observed that the uncertainty in rudder normal force and rudder torque is higher as compared to propeller revolution, propeller thrust and engine torque. The reason is that the inflow velocity to rudders gets more disturbed as compared to propellers. Therefore, it is necessary to focus on accurate modeling of inflow to rudder. Asymmetry and uncertainty in twin-propeller and twin-rudder forces and moments may create hull vibration, load imbalance in the prime mover, etc. The characteristics presented here will help in the further investigation of these aspects.

Deterministic MCS mean (at 50% CL) (105 samples) 3rd order SRSM mean (1632 samples)

(a)

6 5.5 5 4.5 4 3.5

50

100

150

t (sec) 5

8.5

(b)

(c)

8

4.5

7.5

TP × 10-5 (N)

nP (rps)

4 3.5 3

7 6.5 6 5.5

2.5

5 Bifurcation analysis

5 4.5 180

2 6.5

(e)

6

160

5.5

140

PE × 10-5 (Watt)

QE × 10-5 (Nm)

(d)

5 4.5 4

120 100 80

3.5

60

3 8

40 0.4

(f) 6

0.3

QR × 10-5 (Nm)

FRY × 10-5 (N)

(g)

0.35

4 2

0.25 0.2 0.15 0.1 0.05

0

0 -2

-0.05 0

50 100 t (sec)

150

0

50 100 t (sec)

150

Fig. 20 Asymmetric behavior and uncertainty limit of propeller and rudder performances during full-scale simulation of turning maneuver. The legend for b–g is same as that for Fig. 19b–g

9–12 % of deterministic value. The overall variation is about 4.5–8.05 MW. Simulation result of rudder normal force is shown in Fig. 20f. At a particular time level, the leeward rudder normal force is about 25–35 % higher than the corresponding windward value. Uncertainty in either rudder normal force is 12–25 % of deterministic value. The

123

Bifurcation defines the changes in the equilibrium of a dynamical system by varying a parameter in the system. In this section, bifurcation analysis of equilibrium points and periodic orbits of roll motion in ship maneuvering motion are carried out. In this context, GUI version of MATCONT is used for bifurcation analysis. At each variation of bifurcation parameter, equilibrium of the system is computed; its stability is tested by monitoring the eigenvalues of the Jacobian. Eigenvalues and corresponding stability coefficients (i.e., Lyapunov coefficients, Floquet multipliers) decide the type of bifurcation. According to Kuznetsov [36], local bifurcations (in a small neighborhood) of the equilibrium occur via the Hopf, fold, and/or flip type bifurcations. A fold bifurcation occurs when one real Eigenvalue becomes zero or positive. A Hopf bifurcation occurs when the real parts of a complex Eigenvalue pair become zero or positive. There are two types of Hopf bifurcations, sub- and supercritical ones. Type of Hopf bifurcation point can be identified by calculating the first Lyapunov coefficient. It is greater than zero for subcritical and less than zero for supercritical type Hopf bifurcation. At a subcritical Hopf bifurcation, a branch of unstable limit cycles bifurcates from the steady state. This branch may fold back at a limit point cycle and become stable at higher amplitudes [37]. The branches of stable and unstable limit cycles overlap with the branch of stable steady states. At a supercritical Hopf bifurcation, a branch of stable limit cycles bifurcates from the steady state. This branch may grow in amplitude with the parameter increment. The branch of stable limit cycles overlaps with the unstable steady states. We carried out bifurcation analysis for the turning circle maneuver. The initial speed considered corresponds to Fn = 0.4. The powering of the ship is done by twin-gas turbine. No external force (e.g., wind) is applied in bifurcation analysis.

J Mar Sci Technol

5.1 Continuation of equilibrium point

Table 6 Equilibrium at different Hopf bifurcation points

In each continuation, the value of d is being set, equilibrium solutions are located by starting from arbitrary initial points using the time integration scheme. The computed equilibrium solution is used as initial conditions for the next step. KG0 is the bifurcation parameter, initialized at 44.75 9 10-3 and incremented with a minimum step size of 10-5 to maximum step size of 10-4. The continuation is stopped when KG0 reaches 54.12 9 10-3. In each step, the nonlinear GZ ¼ C1 / þ C2 /3 curve is calculated from the KN curve. The KN value at different heel angle at design draft is shown in Fig. 21. If stability loss occurs in continuation, the type of bifurcation, state space, and associated coefficient are recorded. In our model, instability of the equilibrium occurs through Hopf bifurcation. The Hopf points are detected at different rudder angle as shown in Fig. 22. The corresponding equilibrium solutions at each Hopf point are given in Table 6. It is observed that the trend of Hopf points is discontinuous as the mathematical model is discontinuous at d ¼ 28 . At lower d, the bifurcations occur at a relatively higher value of KG0 . As d increases, bifurcation boundary occurs for lower values of KG0 . These Hopf points are identified as supercritical type since the first Lyapunov coefficient is less than zero. Sensitivity analysis is conducted to know the influencing coefficients in bifurcation. The percentage change in the Hopf point is determined by incrementing 1 % of each maneuvering coefficient separately. Sensitivity of Hopf point to different maneuvering coefficients is plotted in

KN' (-)

0.06 0.04 0.02 0 0

10

20 30 φ (deg)

40

50

Fig. 21 KN curve at design draft

KG' × 103

53 52

Supercritical Hopf points

51

Unstable region

50 49

Stable region

Stable region

48 0

10

20 δ (deg)

30

40

Fig. 22 Hopf bifurcation boundary at different rudder angle

d (°)

u0

v0

r0

p0

/ (rad)

KG0  103

l1

6

0.979

-0.031

0.108

0

-0.266

50.926

-1.956

10

0.946

-0.053

0.169

0

-0.286

49.985

-0.881

20

0.822

-0.093

0.274

0

-0.323

49.365

-0.031

25

0.747

-0.105

0.308

0

-0.318

49.195

-0.076

28

0.704

-0.108

0.324

0

-0.304

48.926

-0.162

28.1

0.762

-0.101

0.295

0

-0.325

49.432

-0.032

30 35

0.734 0.665

-0.104 -0.107

0.305 0.329

0 0

-0.320 -0.296

49.350 48.921

-0.093 -0.219

Fig. 23. It can be observed that the influence of coupling coefficients on bifurcation is significant. We also observed that the bifurcation of equilibrium does not occur if the v  / and r  / coupling coefficients in the mathematical model are ignored. Uncertainty of Hopf bifurcation point at d ¼ 25 is done by third order SRSM. The probability distribution of Hopf point is shown in Fig. 24. It is observed that the detected Hopf point has an uncertainty of 2.5–3 %. 5.2 Continuation of limit cycle Post processing of one sample of supercritical Hopf bifurcation point will be discussed in this section. Continuation of the limit cycle is started from the supercritical Hopf point when the ship turns to starboard side at d ¼ 25 . As a result, stable limit cycle manifold appears as shown in Fig. 25. The Hopf point (hereafter H) occurs at KG0 = 49.195 9 10-3. Time responses of roll angle in the bifurcation diagram (see Fig. 25) are discussed in Fig. 26a–c. In Fig. 26a, at KG0 = 49.190 9 10-3 (KG\KGH ), ship attains a stable steady state. In Fig. 26b, at KG0 = 49.195 9 10-3 (at H), the roll angle shows stable limit cycle of small amplitude, which oscillates between -17.18° to -19.29° with a period of about 11 s. In Fig. 26c at KG0 = 49.200 9 10-3 (KG [ KGH ), the roll angle shows stable limit cycle, which oscillates between -16.25° to -20.3° with a period of about 11 s. All detected H shown in Fig. 22 are analyzed, but not shown in the paper due to lack of space. It is found that the size of overlap of stable limit cycles and unstable steady states increases with the increase of rudder angle. It is observed that during starboard turning the ship oscillates about a port side heel angle. It is noted that the ship oscillates at higher period than its natural roll period, i.e., 9.2 s. It may be noted that the increment in KG0 for observing the bifurcation phenomena is low. For actual ships, the uncertainty in the measurement of KG0 is usually of a much higher order. For high-speed passenger ships,

123

J Mar Sci Technol Fig. 23 Sensitivity of Hopf bifurcation point to different maneuvering coefficients

1.2

× 0.05

Change %

0.8 0.4 0 -0.4 -0.8 -1.2

t (sec)

1 0 -10

0.6 0.4 0.2 0

48.5

49

49.5

50

50.5

t (sec) 80 100 0

KG' = 49.190 × 10-3 KG < KGH

100 200 300 400 500

(d)

KG' = 49.190 × 10-3 KG < KGH

-17.5

-25 -10

(b)

-12.5

φ (deg)

KG' × 103 49.188 49.191 49.194 49.197 -16

-15

KG' = 49.195 × 10-3 KG = KGH

(e)

KG' = 49.195 × 10-3 KG = KGH

-17.5 -20

49.2

49.203

-22.5 -25 -10

-17 -18

-12.5

-19

Bifurcation point Hopf

-21 Fig. 25 Branch of limit cycles bifurcating from the supercritical Hopf point during starboard turning at d = 25°. Solid line branch of stable steady states, dash dot line branch of unstable steady states, dash line branch of stable limit cycles

classification rules limit the maximum permissible heel angle as 10° when the ship is proceeding at maximum speed and has maximum rudder angle [38]. For commercial aircrafts, maximum permissible rudder angle for different speed ranges are specified [39]. The influence of speed in maneuvering behavior of fine form ships with low metacentric height has been shown earlier [40, 41]. The dependence of maneuvering behavior on speed was also observed for the subject hull form although not reported in this paper. To capture the bifurcation phenomena accurately, selection of appropriate mathematical model is important. Using the nonlinear-coupled

φ (deg)

φ (deg)

60

-22.5

Fig. 24 Distribution of Hopf bifurcation point at 25° turning. Distribution type: normal with mean 49.195 and standard deviation 0.4

123

-15

40

-20

48

KG' X 103

-20

(a)

-12.5

φ (deg)

PDF

0.8

20

-15

(c)

KG' = 49.200 × 10-3 KG > KGH

(f) KG' = 49.200 × 10-3 KG > KGH

-17.5 -20 -22.5 -25

Fig. 26 LHS: time response of roll angle in different regions of the bifurcation diagram during starboard turning at d = 25° (see Fig. 24), a stable steady state, b stable limit cycle of small amplitude, and c stable limit cycle. RHS: redraw of LHS in the presence of steady wind of speed 30 knots and direction 90° in Earth coordinate

maneuvering model and a suitable numerical method, a chart for the safe working limit can be developed for high-speed fine form ships with low metacentric height. For preparing such a design chart, information regarding ship’s rolling behavior using the above bifurcation analysis will be useful. This will help in the design and control of the ship in high-speed operations. Further work in this direction is required.

J Mar Sci Technol

of all results is presented at 95 % confidence interval. Uncertainty is more on rudder performances as compared to the propeller and engine. Hopf bifurcation is identified in the roll-coupled maneuvering model. It happens due to the coupling of roll with sway, yaw, and rudder during maneuvering. The safe limit of the vertical center of gravity for different rudder angles is determined. Roll-coupling motion is more pronounced at a higher rudder angle. To capture the Hopf bifurcation point accurately, the coupling coefficients in hull model needs to be estimated accurately. As the magnitude of roll oscillations become considerable, the coupled captive maneuvers at higher heel angle need to be carried out. Additionally, to accurately capture the influence of Froude number on maneuvering behavior of fine form ships with low metacentric height, selection of an appropriate maneuvering mathematical model is necessary.

6 Roll response in steady wind We consider the wind speed and direction fixed in the Earth-fixed coordinate axis. Wind model is adopted from literature [42, 43] given in ‘‘Appendix’’. To compute wind forces and moment, we need to estimate the effective wind speed and direction in the ship-fixed coordinate system. During maneuvering, the ship’s speed and heading angle in the Earth-fixed coordinate system keep varying. Consequently, the effective wind speed and direction in ship fixed coordinate axis keeps varying. Full-scale simulation of turning circle maneuver at d  25 is carried out in the presence of steady wind of speed 30 knots and direction 90° in Earth coordinate. The roll angle responses in the presence of the steady wind at different bifurcation points are shown in Fig. 26d–f. It is interesting to note that the secondary oscillations of large frequency are observed in the roll in unstable zone (at and after H), which is absent in stable zone (before H). Analysis of such secondary oscillations and determination of their stability boundaries will be helpful for defining the safe working limits for such ship types.

3.

Appendix Maneuvering model for hull forces and moments

7 Conclusions In this paper, uncertainty, stability and bifurcation analysis of a twin-propeller twin-rudder ship in maneuvering motion has been investigated. The following are the main conclusions of this work. 1.

2.

Maneuvering model of a twin-propeller twin-rudder ship is developed from PMM test results. The asymmetric behavior of twin-propeller, twin-engine, and twin-rudder is modeled by varying flow straightening coefficients with respect to sway, yaw motions, and rudder angle. Uncertainty and its distribution in the model coefficients are calculated for different tests. Metacentric height and wind have a significant influence on instability. Therefore, the safe working limit boundary of rudder angle, metacentric height, and wind speed needs to be defined from the aspect of safety. This information will be useful to ship designers. Stochastic response surface method (SRSM) for uncertainty propagation has been developed for the ship maneuvering application. The simulation results of different maneuvers have shown that the proposed method SRSM is superior to conventional Monte Carlo simulation (MCS) method in terms of taking less computational time and decreasing number of simulation runs. The simulation results of propeller revolution and thrust, engine torque, and power, rudder normal force and torque are described. The uncertainty

The hull forces and moments are expressed at the midship. The mathematical expression for XH is shown in Eq. 6.  XH ¼ X0 þ Xu_ u_ ð6Þ X0 ¼ X þ Xvv v2 þ Xrr r 2 þ X// /2 where X represents the resistance of the ship. The mathematical expression for YH and NH is shown in Eq. 7. Y N Y N

! ¼ !H ¼ 0

þ þ

Y N Y

! !0

N ! Y N Y N

Y þ N

!

Y

vþ

N

v 2

vr þ

!vrr 2

v/ þ v//

Y N

v_ þ v_ !

! r_ r_

Y

3

v þ

N

vvv !

Y

/þ

N Y N

!

/ ! 2

Y N

r /þ rr/

Y

rþ !r

!

N

r3 þ ! Y

rrr

3

/ þ

N

///!

Y

N

2

r/ r//

vv/

9 > > > > > > > > ! > > > Y 2 > > v r> > = N vvr

2

v/

> > > > > > > > > > > > > > > ;

ð7Þ The mathematical expression for KH is shown in Eq. 8. 9 KH ¼ K0 þ Kv_ v_ þ Kr_r_ þ Kp_ p_ = K0 ¼ mgGZ  zH YH ðv; rÞ þ K/ / þ K/// /3 þ Kvv/ v2 / ; þKv// v/2 þ Krr/ r 2 / þ Kr// r/2 þ Kp p þ Kppp p3 ð8Þ where Kp is the roll damping coefficient, mgGZ is the roll restoring moment, zH is the z coordinate of the point where the YH acts. KN depends on the geometric characteristics of

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J Mar Sci Technol

the hull. It is computed for different heel angles at the constant displacement condition. The vessel is free to sink and trim. Once the KN is determined, the righting arm GZ can be computed for any vertical center of gravity VCG corresponding to that displacement condition using the formula GZ ¼ KN  VCG sin /. The mathematical expressions for XP , NP and QP are shown in Eqs. 9 and 10.

9    > 1  tPfSg n2PfSg D4PfSg KTfSg þ 1  tPfPg n2PfPg D4PfPg KTfPg > > =

> 2 4 2 4 NP ¼ yPfSg q ð1  tPfPg ÞnPfPg DPfPg KTfPg  ð1  tPfSg ÞnPfSg DPfSg KTfSg > > > > ; QPfSg ¼ qn2PfSg D5PfSg KQfSg P P P P XP ¼ q



ð9Þ KTfSg ¼ a0 þ a1 JPfSg þ P

P

a2 JP2 S fP g

KQfSg ¼ q0 þ q1 JPfSg þ q2 JP2fSg P

P

P

JPfSg ¼ ðu  yPfSg rÞð1  wPfSg Þ=ðnPfSg DPfSg Þ P

P

P

P

P

1  tPfSg ¼ 1  tP0 þ kbP fSg bP þ k3bP fSg b3P P

P

P

1  wPfSg ¼ 1  wP0 þ sbP fSg bP þ P

bP ¼ b 

P

x0P r 0 ; sbP fPg

9 > > > > > > > > > > > > > > > =

> > > > > > > > > > > > ¼ s3bP fSg > > > ;

s3bP fSg b3P P

¼ sbP fSg ; s3bP fPg

kbP fPg ¼ kbP fSg ; k3bP fPg ¼ k3bP fSg

ð10Þ The mathematical expression for forces (XR and YR ) and moment (NR and KR ) induced at midship due to rudder are shown in Eq. 11. XR YR NR

KR

9    > > FRYfSg þ FRYfPg sd þ FRXfSg þ FRXfPg cd ¼  1  tRfSg > > P >     > > > ¼ ð1 þ aH Þ FRYfSg þ FRYfPg cd  FRXfSg þ FRXfPg sd > =     ¼ ðxR þ aH xH Þ FRYfSg þ FRYfPg cd  FRXfSg þ FRXfPg sd

>     > > > þ 1  tRfSg yRfSg FRXfSg  FRXfPg cd þ FRYfSg  FRYfPg sd > > > P > > ; ¼ z Y



R R

ð11Þ where, s: sin, c: cos The forces FRX , FRY and torque QR are expressed as shown in Eq. 12. 9 1 > > FRYfSg ¼ qAR UR2 fSg fa sin aRfSg > > P P P 2 > > > > 1 > 2 > FRXfSg ¼ qCT 2AR URfSg > > P = P 2

ð12Þ QRfSg ¼ FRYfSg xlpRfSg  xlsRfSg > > P P P P > > > > > CT ¼ 0:075ð1 þ kÞ=ðlog10 Rn  2Þ2 > >

> > > xlpRfSg ¼ CR CRQafSg þ CRQbfSg sin aRfSg ; P P P P The UR and aR are expressed as shown in Eq. 13.

123

U Rf S g ¼ P

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2R S þ v2R S fP g fP g

aRfSg ¼ dfSg  d0fSg  tan P P P

1

vRfSg P

u Rf S g P

!

9 > > > > = > > > > ;

ð13Þ

The longitudinal component of rudder inflow velocity uR is expressed as shown in Eq. 14. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 uRfSg ¼ efSg uPfSg gPfSg f1 þ jng2 þ1  gPfSg > > P P P P P > > > >  1  wRfSg D > S > P K f g P P > efSg ¼ ; j¼ ; g Pf S g ¼ ; > > P P 1  wPfSg h Rf S g > efSg = P P

P

ð14Þ > uPfSg ¼ 1  wPfSg u  yPfSg r > > P vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP P > > u > 8KTfSg > u > P > > 1 n ¼ t1 þ 2 > > pJP S ; fP g The lateral component of rudder inflow velocity vR is expressed as shown in Eq. 15.

9 vRfSg ¼ vRPfSg  cRfSg v þ CRrfSg r þ CRdfSg dfPg > > > P P P P P S > > > 0 0 > bR ¼ b  xR r > = cRfSg at  bR ¼ cRfPg at  bR > > > > CR0 d fSg at  bR ¼ CR0 d fPg at  bR > > fr g fr g > > ; d0fSg ¼ d0fPg ð15Þ The wind forces and moment acting on the ship are estimated as shown in Eqs. 16 and 17. 9 > XW ¼ 1=2qA AT UA2 CX ðwA Þ > > > 2 1 > YW ¼ =2qA AL UA CY ðwA Þ > = 2 1 NW ¼ =2qA AL UA LOA C ðw Þ ð16Þ N A ! > 2 2 > > qA ðAL Þ UA > KW ¼ CK ðwA Þ > > ; 2LOA 9 wR ¼ wW  w > > > uA ¼ UW cos wR  u > = ð17Þ vA ¼ Up sin wR  v Wffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > UA ¼ u2A þ v2A > > ; hA ¼ atan 2ðvA ; uA Þ To convert effective wind angle from our coordinate system to Fujiwara’s coordinate system we use Eq. 18. wA ¼ hA þ p

ð18Þ

where, CX ðwA Þ, CY ðwA Þ, CN ðwA Þ, CK ðwA Þ are the wind force coefficients. These coefficients are calculated for the present ship as per Fujiwara et al. [42] and Nagarajan et al. [43] and are shown in Fig. 27.

J Mar Sci Technol 0.8

0.8 0.6

0.6

0.2

CY

CX

0.4 0.4

0 0.2

-0.2 -0.4

0

-0.6 0.2

0.12

CK

CN

0.16

0.08 0.04 0 0

30 60 90 120 150 180

ψA (deg)

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

Fig. 28 Layout of PMM setup for dynamic motion 0

30 60 90 120 150 180

ψA (deg)

Fig. 27 Wind force coefficients versus effective wind heading angle

PMM experiment analysis In the Strøm–Tejsen PMM experiment set up [1], the aft and forward part of the model is connected to two rotating discs through connecting rods and yoke. The PMM experiment analysis procedure for this set up is well known. Recently, some of the marine dynamics laboratories have carriage capable of generating X, Y and rotation motion using computer controlled carriage motion. In this type of facility, circular motion test, rotating arm test and PMM test can be done with the same set up. The layout of this type of PMM set up is shown in Fig. 28 [34]. The 1:19.2 scale model is tested in this type of set up. The kinematics of the set up permits static drift, pure sway, pure yaw and yaw ? sway motions to be induced on the model. In Strøm–Tejsen’s set up, it was static drift, pure sway, pure yaw and drift ? sway motions. The extraction of coefficients from this type of set up is described below. The model dynamics for pure sway and yaw motion is shown in Eq. 19. ! ! y y ¼ sin xt w w max ð19Þ _ r ¼ w_ v ¼ y; € v_ ¼ y€; r_ ¼ w where y is the lateral displacement and w is the angular displacement of the model ship. The model dynamics for sway ? yaw motion are shown in Eq. 20.

8 9 2 9 38 u 9 > > cw sw 0 > = < > = > > > > v ¼ 4 sw cw 0 5 v > > > > > > > ; : ; : > 0 0 1 r PMM > r > = u_ ¼ u_ PMM cw þ v_PMM swþ > > rPMM ðuPMM sw þ vPMM cwÞ > > > > v_ ¼ u_ PMM sw þ v_PMM cw > > > > rPMM ðuPMM cw þ vPMM swÞ > ; r_ ¼ r_PMM

ð20Þ

where s: sin and c: cos The mathematical expressions for XH , YH , NH and KH during pure sway motion are obtained by substituting r = 0, r_ = 0, / = 0, p = 0 and p_ = 0 in Eqs. 6–8, respectively. The mathematical expressions for XH , YH , NH and KH during pure yaw motion are obtained by substituting v = 0, v_ = 0, / = 0, p = 0 and p_ = 0 in Eqs. 6–8, respectively. The mathematical expressions for XH , YH , NH and KH during heel ? yaw motion are obtained by substituting v = 0, v_ = 0, p = 0 and p_ = 0 in Eqs. 6–8, respectively. The mathematical expressions for XH , YH , NH and KH during sway ? yaw motion are obtained by substituting r = 0, r_ = 0, p = 0 and p_ = 0 in Eqs. 6–8, respectively. Maneuvering coefficients during static tests are calculated by the least-square fit method. For this, the number of test cases should be more than twice of the number of coefficients present in the mathematical model. In dynamic tests, the mathematical model is expressed in harmonic form by substituting the PMM motions and the coefficients are calculated by Fourier series analysis. This is explained for pure sway motion in Eqs. 21 and 22.

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J Mar Sci Technol

XH YH

9 ¼ X þ Xvv v2 > > > 1 cð2xtÞ > 2 2 > > Xvv vmax ¼ X þ Xvv vmax þ > > 2 2 > = 3 ¼ Yv v þ Yvvv v þ  Yv_ v_  3 > ¼ Yv_ v_max sðxtÞ  Yv vmax þ Yvvv v3max cðxtÞ > > > 4 > > > > cð3xtÞ > 3 ; Yvvv vmax 4

where c: cos, the variable F corresponds to Y or N or K. Development of the roll moment model

ð21Þ where, s: sin and c: cos 9 ZT > > 2 > > Fvvv ¼  3  FH ðtÞ  cos 3xt  dt > > > vmax T > > 0 > > 0 T 1> > > Z = 1 @2 3 3 A FH ðtÞ  cos xt  dt þ Fvvv vmax Fv ¼  > vmax T 4 > > 0 > > > > T > Z > > 1 2 > > Fv_ ¼  FH ðtÞ  sin xt  dt > > ; v_max T 4

0

ð22Þ where the variable F corresponds to Y or N or K. XH

YH

9 > ¼ X þ Xvv v2 þ Xrr r 2 þ Xvr r 2 > >  > 1 > 2 > ¼ 2X þ Xvv v2max þ Xrr rmax  Xvr vmax rmax > > 2 > > >  cð2xtÞ  > 2 > > Xvv v2max þ Xrr rmax þ  Xvr vmax rmax > > 2 > > 3 3 > ¼ Yv v þ Yvvv v þ Yv_ v_ þ Yr r þ Yrrr r > > > 2 2 > þYr_r_ þ Yvvr v r þ Yvrr vr > = _ r ¼ sðxtÞðYv_ v_  Y Þ r_ max max > 3cðxtÞ 4 4 > 3 >  Yv vmax  Yvvv v3max þ Yr rmax þ Yrrr rmax þ > > > 4 3 3 >  > > > > 2 2 > þYvvr vmax rmax  Yvrr vmax rmax > > > > > > cð3xtÞ  > 3 3 2 > > Yvvv vmax þ Yrrr rmax þ Yvvr vmax rmax þ > > 4  > ; 2 Y v r vrr max max

ð23Þ where, s: sin, c: cos The explanation for sway ? yaw motion is given in Eqs. 23 and 24. 1 Xvr ¼  vmax rmax 0 T 1 Z   4 2 2 @ XH ðtÞ  cð2xtÞ  dt  Xvv vmax þ Xrr rmax A T 0

9 > > > > > > > > > > > > > > =

> > > Fvvr vmax  Fvrr rmax ¼ > > 0 T 1> > > Z > >   4 2 1 > 3 3 > @ A > FH ðtÞ  cð3xtÞ  dt  Fvvv vmax þ Frrr rmax > ; vmax rmax T 4 0

ð24Þ

123

During bare and appended hull PMM tests with 1:19.2 scale geosym model, roll moment was not measured. Additional PMM tests are carried out at Fn = 0.24 for 1: 30 scale geosym model for measuring roll moment. For 1:30 scale geosym model, the PMM tests are conducted only for appended hull condition. The PMM test 1:30 scale model is tested in the Maritime Dynamics Laboratory (MDL) of SSPA, Sweden [44]. We analyzed the experiment data for determining the roll related coefficients. The tests cover static drift, static heel, static rudder, pure sway and pure yaw. The summary of tests conducted is shown in Table 7. The forces (surge and sway) and moments (yaw and roll) are recorded during the tests. The forces and torque acting on starboard rudder and total thrust for the propeller are recorded. In addition, roll decay and free running model tests are conducted. These test data are used to develop the roll moment model. The coefficients present in roll moment model have been described in the ‘‘Maneuvering model for hull forces and moments’’ in Appendix. The value of z0H and z0R are obtained from the static drift and static rudder test, respectively [45]. These are calculated by comparing the measured sway force and roll moment as shown in Fig. 29. A linear relationship is seen in both cases. The coefficients Kv0_ and Kr0_ are calculated by Fourier series analysis of pure sway and pure yaw test data, respec0 tively. The coefficients Kp0_ , Kp0 , Kppp are calculated from 0 are from roll decay test. The coefficients K/0 and K/// static heel test data as shown in Fig. 30. The drift ? heel test were not conducted in SSPA [44]. Although drift ? heel test were carried out at NSTL [34], the roll moment could not be measured. Therefore, the exact relation between measured sway force and roll moment is not known. Therefore, the non-dimensional roll moment due to v  / and r  / coupling motion is assumed to be about z0H times of the nondimensional sway force, this 0 0 0 0 , Kv// , Krr/ , and Kr// as z0H makes the coefficients Kvv/ 0 0 0 0 times of the Yvv/ , Yv// , Yrr/ , and Yr// , respectively. Sensitivity analysis A sensitivity analysis for the propeller revolution, propeller thrust, engine torque and rudder normal force in zigzag and turning simulations is carried out. In addition, sensitivity analysis for the overshoot angle and advance for zigzag and turning simulations, respectively, is also carried out. The sensitivity is calculated using Eq. 25.

J Mar Sci Technol 12

Table 7 PMM test program conducted in SSPA [44]

9

Appended hull tests

200 ZZ

350 TC

TPMM (s)

Range of test parameters (°)

Change %

6

Test

3 0

D

b =  0, 2, 4, 6, 8, 10

H

/ =  0, 2, 5, 10

–

R

d = -35 to ?35

–

-6

S

bmax = 5

16

-9

Y

0 rmax = 0.1, 0.15

16

-12

D drift, H heel, R rudder, S sway, Y yaw

(a)

(b)

1.2

0.15

zH' = 0.02

0.4 0 -0.4 -0.8

zR'=0.006233

0.05 0 -0.05 -0.1

-1.2 -60 -40 -20 0 20 40 60

Engine Torque Starboard side Port side

Fig. 31 Sensitivity of engine torque in zigzag and turning maneuver to different maneuvering coefficients

0.1

K' × 103

KH' × 103

0.8

-3

-0.15 -30 -20 -10 0 10 20 30

Y' × 103

YH' × 103

Fig. 29 Variation of z0H and z0R coefficients. a static drift test at b = -10° to ?10°, b static rudder test at d = -35° to ?35°

entire time duration. All coefficients are sorted in decreasing order of their sensitivity value (positive or negative) for each output, sorting is performed exclusively for starboard propeller and rudder outputs. The first 15 sensitive coefficients shown in Fig. 31 are considered for 0 uncertainty propagation [6]. Coefficient Yv0_ and CRr have the largest impact on the prediction of engine torque during zigzag and turning maneuvers. Besides the hull model coefficients, the hull, propeller, and rudder interaction coefficients are also significantly sensitive. The influence of the coefficients present in windward rudder model is more as compared to that of leeward. In addition, it is higher in turning as compared to zigzag maneuver.

KH' × 103

0.2

Kφφφ' = 0.0

K φ' = 0.00072

0.1 0 -0.1 -0.2 -12

References -8

-4

0

4

8

12

φ (deg) 0 Fig. 30 Estimation of coefficients K/0 and K/// from static heel test

P  y2

Nt

hyx % ¼

 x¼aþDa P 

P  2

y2

Nt

y

Nt

x¼a

% 100

ð25Þ

x¼a

where hyx is the sensitivity of output y to input x, Da the perturbed value of each input, Nt is the total number of samples in time series. In Eq. 25, the model coefficients are individually multiplied by a factor of 1.1 (Da ¼ 0:10) and the change in respective output is recorded as a percentage of the original value. This equation is valid for a constant time step simulation. For steady state analysis, turning maneuver is considered. The total simulation time is taken as 300 s. Only the steady state part (after 100 s of simulation start) is considered in sensitivity analysis. For transient analysis, the zigzag maneuver is considered. Three complete cycles of d change command is given. The root mean square (rms) of various parameters is taken for the

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