Nonlinear Dyn (2016) 83:2035–2053 DOI 10.1007/s11071-015-2463-9
ORIGINAL PAPER
Bifurcation analysis of a high-speed twin-propeller twin-rudder ship maneuvering model in roll-coupling motion Anil Kumar Dash · Vishwanath Nagarajan · Om Prakash Sha
Received: 9 August 2014 / Accepted: 17 October 2015 / Published online: 31 October 2015 © Springer Science+Business Media Dordrecht 2015
Abstract In this paper, bifurcation analysis of a highspeed twin-propeller twin-rudder ship maneuvering mathematical model has been carried out. Surge, sway, yaw, and roll are the degrees of freedom considered in the model. Coupling of roll with sway and yaw motion during zigzag and turning maneuvers is shown. Hopf, fold, and period-doubling-type bifurcations are identified by allowing one-parameter numerical continuation of equilibrium. The vertical center of gravity is considered as the bifurcation parameter. Physical behavior of roll motion and capsize during the bifurcations are discussed. The bifurcation analysis is carried out using MATCONT. MATCONT is an MATLAB-based program that computes curves of equilibrium and its bifurcation points for any dynamical system. Influence of the wind on roll angle during turning maneuver is shown. Keywords Ship maneuvering · Roll motion coupling · MATCONT · Local bifurcations · Capsize · Wind
List of symbols aH CRr CRδ
CRQa , CRQb CT DP Fn fα FxR , FyR Ix Iz IP
A. K. Dash · V. Nagarajan (B) · O. P. Sha Department of Ocean Engineering and Naval Architecture, IIT Kharagpur, Kharagpur, West Bengal 721302, India e-mail:
[email protected] A. K. Dash e-mail:
[email protected] O. P. Sha e-mail:
[email protected]
JP Jz Jx KH
Ratio of additional lateral force induced on ship hull by rudder action to the rudder force Rudder flow-straightening coefficients for yaw rate Rudder flow-straightening coefficients for rudder angle Rudder torque coefficients Coefficient of total resistance Propeller diameter Froude number Rudder normal force coefficient Surge and sway forces acting on rudders, in rudder-fixed coordinate system Moment of inertia of the ship about xaxis Moment of inertia of the ship about zaxis Moment of inertia of the propeller and shaft Propeller advance ratio Added mass moment of inertia of ship with respect to z-axis. Added mass moment of inertia of ship with respect to x-axis. Bare hull hydrodynamic moment in x direction at midship
123
2036
K v˙ , K r˙ , Nv˙ and Nr˙ KR
A. K. Dash et al.
are the added mass moment of inertia
Hydrodynamic moment acting about xaxis on ship due to twin rudders Hydrodynamic moment acting about xKP axis on ship due to twin propellers Hydrodynamic moment due to wind actKW ing on ship about x-axis K Rudder form factor Thrust coefficient KT Torque coefficient KQ L Ship length First Lyapunov coefficient l1 m Mass of the ship Bare hull hydrodynamic moment in z NH direction at midship Hydrodynamic moment due to twin rudNR ders acting on ship about z direction Hydrodynamic moment due to twin proNP pellers acting on ship about z direction Hydrodynamic moment due to wind actNW ing on ship about z direction Propeller revolutions nP p Roll rate of ship about x-axis Engine torque QE Propeller torque QP Rudder torque QR r Yaw rate of ship about z-axis Reynolds number Rn T Ship draft Natural period of roll Tφ Thrust deduction factor tP tR , and x H are the rudder–hull interaction coefficients u Surge velocity of ship in x direction Absolute surge velocity of ship in x uA direction in wind Absolute wind velocity UW v Sway velocity of ship in y direction Absolute sway velocity of ship in y vA direction in wind Sway inflow velocity twin rudders vR Propeller wake fraction wP (xG , yG , z G ) Position of center of gravity of ship from the origin O Resistance of ship in longitudinal direcX∗ tion Added mass X u˙ , Yv˙ , Yr˙ , Y p˙
123
XH xlpR xlsR xspR XP XR XW YH YR YW YP yP zH
zR αR β βP βR γR δ δ0 ρ ψ ψW ψR ψA φ
Bare hull hydrodynamic force in x direction at midship Distance between leading edge to rudder center of pressure Distance between leading edge to rudder stock Distance between rudder stock to center of pressure Hydrodynamic force due to twin propellers acting on ship in x direction Hydrodynamic force due to twin rudders acting on ship in x direction Hydrodynamic force due to wind acting on ship in x direction Bare hull hydrodynamic force in y direction at midship Hydrodynamic force due to twin rudders acting on ship in y direction Hydrodynamic force due to wind acting on ship in y direction Hydrodynamic force due to twin propellers acting on ship in y direction Offset distance of rudder stock from the ship center line Vertical distance between the acting point of sway hydrodynamic force on hull and the origin of the body-fixed frame Vertical distance between the acting point of lift force on rudder Y R and the origin of the body-fixed frame Effective rudder inflow angle 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 coefficients for drift angle Rudder angle Neutral rudder angle for straight motion Water 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 superscript ( ) represents the non-dimensional value.
Bifurcation analysis of a high-speed twin-propeller twin-rudder ship maneuvering model...
The subscript {S} and {P} denote starboard and port side, respectively. The time derivative of a variable is denoted by a dot above the variable, e.g., φ˙ is the time derivative of φ.
1 Introduction Due to commercial considerations, fine-form ships such as container carrier, pure car carrier, roll-on/rolloff ship, liquefied natural gas carrier, and naval ship are required to operate in heavy weather condition at high speed. Therefore, ship designers need to predict maneuvering characteristics of these vessels with giving additional emphasis on roll-coupled motion. Naval vessels such as cruisers, destroyers, and frigates need to operate in severe conditions. Therefore, maneuvering prediction and control of high-speed naval vessels with the roll-coupled maneuvering motion is critical. Ship’s roll motion becomes significant during maneuvers at high speed [1]. The roll coupling is compounded in the presence of wind and waves. High rolling motion is undesirable from the safety and stability point of view. The roll motion is influenced by ship’s speed, hull lines, propeller and rudder configuration and geometry, rudder sway flow asymmetry, etc. The roll coupling is usually nonlinear, and since it affects ship stability and capsizing behavior, developing a mathematical model for predicting and simulating the roll motion is important. Eda [2] investigated the course stability of a high-speed twin-propeller twin-rudder (TPTR) naval hull. This vessel has a sonar dome at the bow part. He found the existence of oscillatory instability in yaw and roll motions in the presence of stepwise beam wind. Change in underwater section along the length, when the ship is heeled, is shown to influence the sway– yaw–roll coupling. At the low metacentric height, roll motion is shown to be high. It is also reported that beam wind created greater difficulties, while a suitably designed autopilot can increase the region of stability. Twin propellers and twin rudders are not modeled as separate entities. Additionally, sway–roll, yaw–roll coupling coefficients are not included in the maneuvering model. Son and Nomoto [3] investigated the roll motion instability of a high-speed single-propeller single-rudder ship (SR 108 container ship). Influence of the metacentric height on a sway–yaw–roll–steering gear coupling motion and the choice of autopilot on
2037
self-excited coupled yaw–roll motions are investigated. It is shown that the ship might capsize due to selfexcited sway–yaw–roll–rudder coupled motion during maneuvering. Spyrou [4] investigated yaw stability of a tanker ship and a river boat in the steady wind. Both ships are single-propeller single-rudder system. For the river boat, it is shown that if wind speed is exceeded beyond a limit in following/quartering condition, then it could not be kept on track even by using the maximum rudder. Hopf bifurcation is identified by varying the wind direction as a control parameter. Limit cycles are shown in the unstable region. In case of the tanker in ballast condition, oscillations near Hopf bifurcation point are shown. All the ships showed dynamical instability in following beam wind. Oh and Nayfeh [5] investigated nonlinearly coupled pitch and roll response of a vessel in regular waves. The natural frequency in pitch is twice that of the roll. The wave encounter frequency similar to pitch and roll natural frequency is considered for investigation. Hopf bifurcations in motion response are shown in both the cases. It is reported that nonlinear roll damping is a critical parameter in the investigation. The roll motion has been extensively investigated as a single degree-of-freedom (1DoF) problem. Falzarano et al. [6] treated the ship’s roll motion as a nonlinear bifurcation problem. They studied the roll behavior of a fishing vessel during turning in the wave at low speed. Heading angle of the ship is taken as a bifurcation parameter. The analysis is carried out for a 1DoF and a 3DoF model. In case of 1DoF model, the advance speed is taken as zero. Period-doubling bifurcations for parametric excitation, external excitation, and parametric + external excitation are investigated. In 3DoF model, sway, yaw, and roll motion are considered. Only external excitation is applied. Effect of wave height on the period-doubling bifurcation is shown. Bifurcation is influenced by initial maneuvering conditions of the ship with respect to wave. Francescutto and Contento [7] investigated roll motion of a destroyer in beam sea condition through model experiments. Roll restoring moment curve of the ship had soft spring characteristics. Bifurcations in roll motion at two different wave frequencies are shown in experiments. A mathematical model is developed for simulating the nonlinear roll motion. Stability analysis of the model is carried out to predict the bifurcations (observed during the experiment) numerically. Umeda et al. [8] identified period doubling, chaos, and capsizing characteristics in parametric rolling due to wave load. They reported that
123
2038
anti-roll tanks are more effective as compared to sponsons in reducing parametric rolling of a container ship. They could also optimize the volume of anti-roll tank for suppressing parametric rolling. Virgin [9] 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 capsize in regular waves is shown. Instability characteristics such as multiple solutions, jumps, period multiplying bifurcation, dynamic chaos, sub- and superharmonic resonance often occur in highly nonlinear multiple degree-of-freedom systems [10,11]. These have not been investigated much for maneuvering motions of a ship. The paper is organized as follows. In Sect. 2, we develop a nonlinear MMG-type mathematical model for estimating the hydrodynamic forces and moments, twin-propeller and twin-rudder forces and moment acting on the hull. Coefficients of the model are determined from available captive maneuvering test data. Influences of sway–yaw–roll–rudder coupling in high-speed maneuvers are shown in Sect. 3. In Sect. 4, vertical center of gravity-induced local bifurcations of the equilibria and periodic orbits are investigated numerically using MATCONT. Local bifurcations of periodic orbits such as subcritical Hopf, supercritical Hopf, fold, and period doubling are found. The roll angle responses in different bifurcations are discussed. MATCONT is a bifurcation toolbox, which is compatible with the MATLAB ODE representation of differential equations [12]. Bifurcation of limit cycles with two control parameters can be solved in MATCONT [13]. Turning maneuver is simulated in a steady wind, and roll behavior is discussed.
2 Mathematical model Most of the ships are now required to comply with maneuverability standards formulated by the International Maritime Organization (IMO) [14]. Classification societies have a guide for assessment of parametric roll motion in the design of container ships [15]. For commercial displacement-type vessels, the regulations require rudder angle to be rotated up to the maximum limit to both port and starboard side at full speed. For high-speed vessels and naval surface craft, besides above, the regulations additionally
123
A. K. Dash et al.
specify the maximum heel angle of 10◦ at a maximum rudder angle and maximum ship speed [16]. One of the methods for ensuring compliance with these guidelines is numerical simulations. This requires a nonlinear mathematical model for simulating various maneuvering motions to check compliance with the prescribed standards. Coefficients in the mathematical model can be determined by captive model test, empirical formulas, etc. A ship maneuvering mathematical model for predicting hydrodynamic forces and moments acting on the hull, twin propellers, and twin rudders during maneuvering motion is adopted from Khanfir et al. [17], Dash and Nagarajan [18]. The mathematical model for estimating wind forces and moments is adopted from [19,20]. The subject ship considered for investigation is DTMB 5415. It is a TPTR naval vessel used as a benchmark design for CFD validation. Captive model experiment data for the vessel are made available by MARIN [21] and FORCE [22]. Main particulars of the ship are given in Table 1. For detailed information about the ship, we can refer the SIMMAN-2008 website [23]. The Earth- and ship-fixed coordinate system for the forces and moments are shown in Fig. 1. The origin of the ship-fixed coordinate system is fixed at the intersection of midship and central plane at the level of water surface. Mathematical expressions for forces (surge and sway) and moments (yaw and roll) acting on the hull can be written as shown in Eq. (1).
Table 1 Main hull particulars of US Navy surface combatant DTMB 5415 Particulars
Ship
Particulars
L O A (m)
153.3
CP
L P P (m)
142.2
S (m2 )
Ship 0.616 2972.6
BW L (m)
19.06
CM
0.821
B (m)
19.38
KM (m)
9.5
D at midship (m)
11.22
KG (m)
7.53
D (m)
12.47
GM (m)
1.97
T (m)
6.15
LCG (m)
−0.652
∇(m3 ) CB Tφ (s)
8424.4 0.507
k X X (m)
6.932
k Z Z (m)
36.802
11.5
* For particulars of full-scale propellers and rudders, please refer [23]
Bifurcation analysis of a high-speed twin-propeller twin-rudder ship maneuvering model...
2039
Fig. 1 Coordinate system
(m + m x )u˙ (m + m y )v˙ − mz G p˙ + (mx G − Yr˙ )˙r (Iz + Jz )˙r + (mx G − Nv˙ )v˙ (I x + Jx ) p˙ − (mz G + K v˙ )v˙ − K r˙ r˙ − K φ˙ φ˙ φ˙
⎫ = (m + m y )r v + mx G r 2 − mz G pr + X 0 + X P + X R + X W ⎪ ⎪ ⎪ ⎪ = −(m + m x )ur + Y0 + Y R + YW ⎬ = −mx G ur + N0 + N P + N R + N W (1) ⎪ ⎪ = mz G ur + K 0 + K R + K W ⎪ ⎪ ⎭ =p
The forces and moments are non-dimensionalized by 1 1 2 2 2 2 ρLTU and 2 ρ L TU , respectively. Detailed expressions of the forces and moments on right-hand side of the above equations are given in “Appendix.” The linear, nonlinear, cross-coupling velocity, and acceleration-related model coefficients are calculated from different captive maneuvering experimental data, which are listed in Table 2. Forces due to heave and pitch motion are ignored, as their influence in maneuvering motion of large ships is negligible. We computed the maximum permissible one side roll angle for this ship as 30◦ . Therefore, if the roll angle is more than 30◦ , then the ship is considered to have capsized. The present rudder model is discontinuous at δ = 20◦ , and the reason is explained in “Appendix.” Equation (1) can be represented in state space form as shown in Eq. (2). ⎡
m + mx 0 ⎢0 m + my ⎢ ⎢0 mxG − Nv˙ ⎢ ⎣0 −mz G − K v˙ 0 0 ⎧ ⎫ ⎧ ⎫ u ˙ ⎪ ⎪ ⎪ f1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ v˙ ⎪ ⎬ ⎬ ⎪ ⎨ f2 ⎪ r˙ = f 3 ⎪ ⎪ ⎪ ⎪ ⎪ p˙ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ f4 ⎪ ⎪ ⎩ ˙⎪ ⎭ ⎭ ⎪ ⎩ ⎪ f5 φ
0 mxG − Yr˙ Iz + Jz −K r˙ 0
0 −mz G 0 I x + Jx 0
⎤ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ −K φ˙ ⎦ 1
(2)
where f 1 to f 5 are the abbreviations for right-hand side part of the Eq. (1), respectively.
3 Simulation and model validation In this section, influences of the vertical center of gravity (hereafter, KG) and nonlinear model on zigzag (hereafter, ZZ) and turning (hereafter, TC) maneuvers are investigated. Righting lever arm (hereafter, GZ) is calculated from KN curve for different KG values with keeping the displacement constant using commercial software [24]. The GZ curve is fitted with a fifth-order odd polynomial equation. The KN curve of the present ship is given in Fig. 2. It seems that the maximum value of KG is 9.5 m for keeping GM positive at φ = 0◦ . All maneuvering coefficients except the roll-related coefficients are extracted from the captive model test data at Fn = 0.248. FORCE [22] has conducted static heel test on the appended hull at Fn = 0.248 and 0.413 which is made available through SIMMAN [23]. In this test, β and r are zero. It is found that the trend of the yaw moment at Fn = 0.248 is opposite to that of at Fn = 0.413 as shown in Fig. 3. Therefore, it is noted that the roll-related coefficients are speed dependent.
123
2040 Table 2 Ship maneuvering coefficients for DTMB 5415
A. K. Dash et al.
Items
Value
Items
Value
Items
Value
X 0
−1.6550E−02
Yvrr
−3.6761E−01
2.1091E+00
X vv
−5.8910E−02
Yvvφ
Nvφφ
1.3927E+00
(a) Hull
X rr X φφ X vr Yv Yvvv Yr Yrrr Yφ Yφφφ Yvvr
−1.4561E−02 −2.2055E−02 6.5426E−03 −3.6663E−01 −2.0415E+00 3.7452E−02 −1.0412E−02 −2.0169E−02 −1.2014E−01 −8.2264E−01
Yvφφ Nv Nvvv Nr Nrrr Nφ Nφφφ Nvvr Nvrr Nvvφ
−2.4552E+00 −1.1671E−01 −5.1760E−01 −8.2431E−02 −8.3312E−03 −1.0234E−04 −4.2441E−02 −1.5637E+00 −5.3745E−01
K φ
K φφφ K vvφ K vφφ Yv˙ Nv˙ Yr˙ Nr˙ K p z H
−5.8993E−04 2.3286E−02 −1.7666E−02 3.6785E−01 −1.1576E−01 −1.5494E−02 −9.8130E−03 −1.0008E−02 −2.6600E−04 2.2100E−02
−5.8824E−01
(b) Twin propellers and twin rudders
* The r and φ cross-coupled coefficients are adopted from Son and Nomoto [3]
1.00
0.050
γ R{P}
0.100
0.108
C Rr {S}
−0.410
0.150
ω P0 τβ P {S} τ3β P {S}
0.500
λβ P {S}
0.267
λ3β P {S}
1.625
C Rr {P}
0.063
For δ
xH
−0.391
For δ > 20◦
−0.410
C Rδ{S} C Rδ{P} C Rδ{P} C Rδ{S}
0.153
0◦
<-20◦
aH
0.023 −0.700 −0.700
z R
0.035
For α R{P} <
C R Qb{P}
−0.075
ε
0.925
For α R{P} > 0◦
C R Qb{P}
0.032
K
0.650
For α R{S} < 0◦
C R Qb{S}
−0.045
C R Qa{P}
0.3
For α R{S} > 0◦
C R Qb{S}
0.075
C R Qa{S}
0.3
Some noise in the experimental data of roll moment can be observed as it is not passing through the origin at φ = 0◦ . Therefore, the linear roll coefficients Yφ , Nφ and K φ are different at two different Fn. The v −φ and r − φ coupling coefficients are extracted from captive tests at Fn = 0.248 and 0.212, respectively. The roll damping coefficient K p is adopted from Wilson et al. [25]. The influence of these coefficients on bifurcation analysis will be shown later. Full-scale standard maneuvers are simulated using the present mathematical model at Fn = 0.413. Figure 4 compares the trajectories of 10◦ turning maneuver simulations with and without roll motion coupling. It is observed that if the roll coupling is considered in maneuvering model, then the turning diam-
123
For β R < 0◦
γ R{S}
t P0
eter decreases. The maneuvering characteristics of the ship during turning and zigzag maneuvers at different KG are shown in Fig. 5. It is seen that the tactical diameter and the steady-state heel angle increase marginally during turning maneuver with the increase in KG. It may be noted that the increment is higher in the maximum roll angle. The trend of turning characteristics at different KG for the subject hull is slightly different from that of Eda [2]. This is because, the total lateral 1 ) for the subject ship is larger area of rudders ( LATR = 28 1 ) [2] and Son as compared to that of Eda ( LATR = 40 AR 1 and Nomoto ( L T = 45 ) [3]. During the starboard turning maneuver, a large transient roll angle appears on the starboard side. The steady heel angle during starboard maneuver is on the port side. A similar pattern is
Bifurcation analysis of a high-speed twin-propeller twin-rudder ship maneuvering model... 5
2041
30 20
4
N'×104
KN (m)
10 3
2
0 -10 Fn = 0.248 Fn = 0.413
-20
1
-30 -12
0 0
5
10
15
20
25
-8
30
φ (deg)
-12
Fig. 2 KN curve of DTMB 5415 at the design draft
-8
-4
0 4 φ (deg) φ (deg) -4 0
8
4
12
8
0 -0.5 -1 K'×104
observed for port turning maneuver. This shows that the ship’s initial heel angle is dominated by the direction of the rudder-induced roll moment, since hydrodynamic and inertial forces are negligible. Additionally, the high rudder-turning rate, i.e., 9◦ /s, is also contributing to the large transient roll angle. This behavior of roll motion is not reported in earlier investigations of other ship types [2,3]. This is also one of the reasons for the ship’s roll instability, which will be shown later. The influence of KG during zigzag maneuver will be discussed. The maximum roll angle and the time of reach to overshoot angles become more with the increase in KG. The change in overshoot angle is marginal, but the increase in roll response is significant. Full-scale 10◦ zigzag maneuver is simulated at KG = 8 m and 8.5 m. The comparison of roll angle response predicted by linear and the nonlinear mathematical model is shown in Fig. 6. The linear model consists of coefficients related to v, r and φ. The roll angle increases with increase in ship speed and/or increase in KG. It happens both in the linear and nonlinear model, which is expected. The nonlinear model predicts larger roll angle as compared to a linear model. This is due to the presence of nonlinear and coupling coefficients. A linear model may give good results at lower rudder deflections (below 10◦ ). However, for higher rudder deflections (for 15◦ −35◦ ), the nonlinear model is required to properly explain the sway–yaw–roll–rudder coupling motion. It is noted that the effect of nonlinear-
-1.5 -2 -2.5 -3
Fig. 3 Comparison of yaw and roll moment from static heel test for appended hull [22] at Froude number 0.248 and 0.413
ity is significant at high-speed maneuvering motions. Therefore, these coefficients need to be estimated accurately for higher speed. Furthermore, if the roll-related captive tests are conducted in bare hull condition, the hull and rudder components can be separated.
4 Bifurcation analysis Bifurcation analysis deals with the identification of equilibrium points and their bifurcations, bifurcations of limit cycles, the coexistence of solutions, etc. In bifurcation analysis, the change in solutions of a
123
2042
A. K. Dash et al. 12
dynamical system due to a change in the parameter in the system is investigated. In this paper, bifurcation analysis of equilibria and periodic orbits in ship maneuvering motion is carried out using the GUI version of MATCONT. The procedure is as follows. At each variation of bifurcation parameter, equilibrium of the system is computed and its stability is examined by monitoring the eigenvalues of the Jacobian. By seeing the nature of eigenvalues and corresponding stability coefficients (i.e., Lyapunov coefficient, Floquet multipliers), the type of bifurcation is decided. If bifurcation is detected in a small neighborhood of the equilibrium, then it is called local bifurcation. According to Kuznetsov [26], local bifurcations 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: suband supercritical ones. Type of Hopf bifurcation point
8
100 TC Fn = 0.413 KG = 8.5 m
0
-4
-8
With roll motion -12 -4
Without roll motion 0
4
8
12
16
20
y/LPP (-) Fig. 4 Comparison of trajectory of 10◦ turning circle maneuvers with and without roll motion
5
16
Turning maneuver Fn = 0.413
14
TD/LPP (-)
Fig. 5 Influence of vertical center of gravity KG on the full-scale maneuvers at different rudder angle δ. LHS: turning circle maneuver, RHS: zigzag maneuver
12
1st Overshoot of heading (deg)
x/LPP (-)
4
KG = 7.5 m KG = 8 m KG = 8.5 m
10 8 6
Zigzag maneuver Fn = 0.413 4
3
2
1
4 10
15
20
25
30
5
35
20
1st Overshoot of roll (deg)
0
Maximum roll (deg)
16
12
8
4
0 20
25
δ (deg)
123
-5
-10
-15
-20
-25 15
15
δ (deg)
δ (deg)
10
10
30
35
20
Bifurcation analysis of a high-speed twin-propeller twin-rudder ship maneuvering model... 25 20
Hopf bifurcation point indicates where the Hopf point changes from sub- to supercritical, i.e., at this point the first Lyapunov coefficient will be zero. The following abbreviations will be used hereafter in this paper: H: Hopf or flutter bifurcation point, LPC: limit point cycle or fold bifurcation point, PD: period doubling or flip bifurcation point, GH: generalized Hopf bifurcation point.
100 zigzag (at constant torque) Fn = 0.413
KG = 8 m
15 10
φ (deg)
2043
5 0 -5 -10 -15
4.1 Continuation of equilibrium point
Mathematical model Linear Non linear
-20 -25 25
KG = 8.5 m
20 15
φ (deg)
10 5 0 -5 -10 -15 -20 -25 0
50
100
150
t (sec)
Fig. 6 Comparison of the roll angle response predicted by linear and nonlinear mathematical model during 10◦ zigzag simulation at Froude number 0.413 and vertical center of gravity 8 and 8.5 m
can be identified by calculating the first Lyapunov coefficient. It is greater than zero for the subcritical and less than zero for the supercritical- type Hopf. 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 [27]. 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. The generalized
In each continuation, the value of δ is being set, and the equilibrium solutions are located by starting from arbitrary initial points using the time integration scheme. The computed equilibrium solutions are used as initial conditions for the next step. KG is the bifurcation parameter, initialized at 7.5 m and incremented with a minimum step size of 10−5 m to maximum step size of 10−2 m. The continuation is stopped when KG reaches 9.5 m. If stability loss occurs in continuation, the type of bifurcation, state space, and associated coefficient are recorded. Initial ship speed is kept constant throughout the analysis and is 30 knots. The powering of the ship is done with constant torque strategy. No external force (e.g., wind) is applied in bifurcation analysis. In our model, the instability of the equilibrium occurs through Hopf bifurcation. Hopf points are detected at different rudder angles as shown in Fig. 7, and the corresponding equilibrium solutions at each Hopf point are tabulated in Table 3. It is observed that the trend of Hopf points is discontinuous as the mathematical model is discontinuous at δ = 20◦ . At lower δ, the bifurcations occur at a relatively higher value of KG. As δ increases, the bifurcation boundary occurs for lower values of KG. Both subcritical and supercritical Hopf points are found. Subcritical Hopf points exist in the range of rudder angle 10◦ to 32◦ and KG from 9.5 m to 8.647213 m. In each case, the first Lyapunov coefficient is found greater than zero. Supercritical Hopf points exist in the range of rudder angle 32.0857◦ to 35◦ and KG from 8.608239 m to 8.401784 m. In each case, the first Lyapunov coefficient is found less than zero. There is no bifurcation for δ < 10◦ . Generalized Hopf point is identified at δ = 32.028o , and the corresponding KG = 8.612232 m is determined by performing two-parameter bifurcation analysis. Equilibrium of this point is given in Table 4.
123
2044
A. K. Dash et al. Bifurcation points Subcritical Hopf Supercritical Hopf
9.5
Generalized Hopf 9.25
Unstable region
KG (m)
9
8.75
Stable region
Stable region
8.5
8.25 10
15
20
25
30
35
δ (deg)
Fig. 7 Hopf bifurcation boundary using numerical continuation with the vertical center of gravity KG as the continuation parameter at different rudder angles δ
4.2 Continuation of subcritical Hopf point Post-processing of one sample of subcritical Hopf bifurcation point will be discussed in this section. ConTable 3 Equilibrium values of state space variables at different sub- and supercritical Hopf bifurcation points
123
tinuation of limit cycle is started from the subcritical Hopf point when the ship turns to starboard side at δ = 25◦ . As a result, limit cycle manifold with a cycle limit point labeled by LPC appears as shown in Fig. 8. The H occurs at KG = 9.045360 m, LPC at KG = 9.038060 m, PD at KG = 9.038447 m and capsize at KG = 9.039170 m. Floquet multipliers at LPC are listed in Table 5. A LPC bifurcation occurs when a Floquet multiplier crosses the unit cycle at (+1, 0). Branch of unstable limit cycles (i.e., periodic solutions converge to stable steady states) appears in between the H and LPC, specified by red dotted lines. With the decrease in KG, amplitude of the unstable limit cycle grows from H to LPC. LPC is the fold bifurcation of the limit cycle where a branch of unstable limit cycles approaches to a branch of stable limit cycles. The continuation process stops when LPC is encountered. By resuming the continuation, another branch of stable limit cycles starts from LPC. With the increase in KG, amplitude of the stable limit cycle grows from LPC to capsize state. Limit cycles are one-periodic between the LPC and PD and converge to one cycle before closure. Period doubling of limit cycles starts at PD. Time response of the roll angle in the bifurcation diagram (see Fig. 8) is shown in Fig. 9. In Fig. 9a, at KG = 9.041870 m (KG L PC < KG < KG H ), the roll angle converges to stable steady state. In Fig. 9b, at KG = 9.038170 m (KG L PC < KG < KG P D ),
δ (deg)
u
v
r
p
φ (rad)
KG (m)
l1
10
0.972661
−0.060123
0.137490
0
−0.017458
9.476968
81.489190
12
0.960398
−0.071332
0.164471
0
−0.024925
9.382071
83.434320
14
0.945663
−0.082232
0.191302
0
−0.032250
9.268034
73.898860
16
0.928254
−0.092813
0.218091
0
−0.039420
9.139415
59.871000
18
0.908070
−0.102973
0.244755
0
−0.046475
9.000270
45.508020
20
0.884788
−0.112705
0.271492
0
−0.053560
8.853072
32.624570
22
0.925300
−0.089534
0.209136
0
−0.036461
9.200653
72.771870
24
0.909206
−0.097371
0.229344
0
−0.042007
9.099055
56.999390
25
0.900353
−0.101258
0.239604
0
−0.044817
9.045360
49.134970
26
0.890840
−0.105152
0.250075
0
−0.047687
8.989379
41.381750
28
0.869846
−0.112871
0.271518
0
−0.053599
8.871940
26.643260
30
0.845728
−0.120525
0.293951
0
−0.059889
8.746917
12.991240
32
0.817747
−0.128113
0.317828
0
−0.066779
8.614130
0.174882
32.1
0.816180
−0.128502
0.319111
0
−0.067157
8.607059
−0.474921
34
0.784674
−0.135636
0.343953
0
−0.074658
8.472711
−12.618320
35
0.765497
−0.139387
0.358360
0
−0.079200
8.397860
−19.583370
Bifurcation analysis of a high-speed twin-propeller twin-rudder ship maneuvering model...
2045
Table 4 Equilibrium value of state space variables at Generalized Hopf bifurcation point δ (deg)
u
v
r
p
φ (rad)
KG (m)
l2
32.028
0.817327
−0.128217
0.318172
0
−0.066881
8.612232
−5467.87102
KG (m) 9.035 1
9.04
9.045
9.05
0
φ (deg)
-1
-2
-3
-4
-5
H LPC PD Capsize
-6
Fig. 8 Branch of limit cycles bifurcating from the subcritical Hopf point at the starboard rudder angle δ = 25◦ . Black solid line branch of stable steady states, black dash line branch of unstable steady states, red dash line branch of unstable limit cycles, and blue solid line: branch of stable limit cycles. (Color figure online) Table 5 The Floquet multipliers at LPC At LPC Modulus 1
5.98706E−11
Modulus 2
1.44758E−6
Modulus 3
0.02841
Modulus 4
0.99908
Modulus 5
1.00086
Argument 1
0
Argument 2
0
Argument 3
0
Argument 4
0
Argument 5
0
the roll angle shows a single peak in the time response, i.e., limit cycle converges to one-period cycle before closure. The ship oscillates between 0.27◦ and −5.25◦ with a period of 5.5 seconds. With further increase in KG, at KG = 9.038567 m (KG P D < KG <
KG Capsi ze ), the roll angle shows two peaks in the time response, i.e., limit cycle is stable as it converges to two-period cycle as shown in Fig. 9c. The ship oscillates from 0.6◦ to −5.5◦ and 0.25◦ to −5◦ with doubled period, i.e., 11 s. Figure 9d depicts capsize of the ship at KG = 9.039170 m. Capsizing occurs with the roll angle exceeding the limits on outboard side. Analysis of the Hopf points for all rudder angles is carried out, but is not shown in the paper due to lack of space. It is found that the size of overlap between the unstable limit cycles and stable steady states decreases with the increase in rudder angle as shown in Fig. 10.
4.3 Continuation of supercritical Hopf point Post-processing of one sample of supercritical Hopf bifurcation point will be discussed in this section. Continuation of limit cycle is started from the supercritical Hopf point when the ship turns to starboard side at δ = 32.1◦ . As a result, stable limit cycle manifold appears as shown in Fig. 11. The H occurs at KG = 8.607059 m, PD at KG = 8.613644 m and capsize at KG = 8.614017 m. Time response of the roll angle in the bifurcation diagram (see Fig. 11) is discussed in Fig. 12. In Fig. 12a, at KG = 8.607059 m (at H), a limit cycle of small amplitude is observed; the ship oscillates between −3.8◦ and −3.9◦ with a period of 3.8 s. In Fig. 12b, at KG = 8.607441 m (KG H < KG < KG P D ), the roll angle shows a single peak in the time response, while the ship oscillates between −2.5◦ and −5.15◦ with a period of 3.8 s. In Fig. 12c at KG = 8.614017 m (KG P D < KG < KG Capsi ze ), the roll angle shows two peaks in the time response, while the ship oscillates from −0.35◦ to −6.9◦ and −1◦ to −6.3◦ with approximately doubled period, i.e., 8.8 seconds. Further increase in the KG results in capsizing of the ship as depicted in Fig. 12d. Capsize happens due to increase in the roll angle toward outboard side. All detected supercritical Hopf points are analyzed, but not shown in the paper due to lack of space. It is found that the size of overlap between the stable limit cycles and unsta-
123
2046
A. K. Dash et al.
Fig. 9 Time response of roll angle in different regions of the bifurcation diagram at starboard rudder angle δ = 25◦ (see Fig. 8), a unstable limit cycle, b stable limit cycle, c period doubling of limit cycle, and d chaotic behavior during capsize
(a)
t (sec) 100 200 300 400 500 600
(b) 1
0
0
-1
-1
-2
-2
-3 -4 -5 KG = 9.041870 m KGLPC < KG < KGH
-7
50
-4
KG = 9.038170 m KGLPC < KG < KGPD
-7
10
t (sec) 20 30
40
50
(d)
0 1
0
0
-1
-1
-2
-2 φ (deg)
1
-3 -4 -5 -7
40
-8 0
-6
t (sec) 20 30
-3
-6
-8
(c)
10
-5
-6
φ (deg)
0
1
φ (deg)
φ (deg)
0
10
t (sec) 20 30
40
50
-3 -4 -5
KG = 9.038567 m KGPD < KG < KGCapsize
-8
-6 -7
KG = 9.039170 m KG = KGCapsize Capsize
-8
50
KG (m) 8.595
8.6
8.605
8.61
8.615
8.62
0
-2
30
φ (deg)
(KGH - KGLPC) (m) × 10
3
40
20
-4
10 -6
H PD Capsize
0 10
15
20
25
30
35
δ (deg)
Fig. 10 Size of overlap of unstable limit cycle and stable steadystate branches at different rudder angles δ
123
-8
Fig. 11 Branch of limit cycles bifurcating from the supercritical Hopf point at the starboard rudder angle = 32.1◦ . Solid black line branch of stable steady states, dotted black line branch of unstable steady states, blue solid line branch of stable limit cycles. (Color figure online)
Bifurcation analysis of a high-speed twin-propeller twin-rudder ship maneuvering model... t (sec)
(a) 2
0
10
20
30
50
2
0
0
-2
-2
-4
KG = 8.607059 m KG = KGH
-8
20
30
40
50
40
50
-4
KG = 8.607441 m KGH < KG < KGPD
-10
t (sec)
(c) 0
10
20
30
t (sec)
(d) 40
50
2 0
-2
-2
φ (deg)
0
-4 -6 -8
10
-8
-10
2
0
-6
-6
φ (deg)
t (sec)
(b) 40
φ (deg)
φ (deg)
Fig. 12 Time response of roll angle in different regions of the bifurcation diagram at starboard rudder angle δ = 32.1◦ (see Fig. 11), a small amplitude limit cycle, b stable limit cycle, c period doubling of limit cycle, and d chaotic behavior during capsize
2047
0
10
20
30
-4 -6
KG = 8.614017 m KGPD < KG < KGCapsize
-10
ble steady states increases with the increase in rudder angle. It is observed that during starboard turning the ship oscillates in port side with higher amplitude as compared to starboard side. Period of oscillation is lower than the ship’s natural roll period (i.e., 11.5 s). Capsizing happens due to increasing the roll angle on outboard side. Behavior of the roll motion in the bifurcations and capsize will help in design and control of ships in high-speed operations. For high-speed naval ships, classification rules limit the maximum permissible heel angle as 10◦ when the ship is proceeding at maximum speed and has maximum rudder angle [16]. For commercial aircrafts, maximum permissible rudder angle for different speed ranges is specified [28]. By using the nonlinear coupled maneuvering model and a suitable numerical method, a chart for the safe working limit can be developed for high-speed naval ships. For preparing such a design chart, information regarding maximum heel angle and ship’s oscillatory behavior, as shown
-8
KG = 8.614750 m KG = KGCapsize
Capsize
-10
above, will be useful. It may be noted that the increment in KG for observing the above phenomena is low. For actual ships, the uncertainty in the measurement of KG due to various reasons is usually of a much higher order.
5 Roll response in steady wind We consider the wind speed and direction fixed in the Earth-fixed coordinate axis. To compute the wind forces and moments, we need to estimate the relative wind speed and direction in ship-fixed coordinate system. During maneuvering, the ship’s speed and heading angle keep varying. Consequently, the effective wind speed and effective wind direction in ship-fixed coordinate axis keep varying. Full-scale simulation of turning circle maneuver at δ = 25◦ is carried out in the presence of the steady wind of speed 30 knots. In this condition, the ship capsizes at KG = 8.44 m. This is lower than the value of the KG at which the ship capsizes when
123
2048
A. K. Dash et al. 10
identify the bifurcations of equilibrium and periodic orbits. The important conclusions are as follows:
KG = 7.5 m KG = 8.43 m
7.5
5
φ (deg)
2.5
0 -2.5 -5 -7.5 -10 -12.5
0
100
200
300
400
500
600
t (sec)
Fig. 13 Roll angle response just before the ship capsize at starboard rudder angle δ = 25◦ , KG = 8.43 in the presence of steady wind of speed 30 knots
the wind is absent (see Fig. 9). Drop in KG is nearly 0.50 m, which is a significant value. In the steady wind, the roll angle response just before of capsize is compared to that of a lower KG = 7.5 m in Fig. 13. It is interesting to note that secondary oscillations of large frequency are observed in the roll which is absent at lower KG. Analysis of such secondary oscillations and determination of their stability boundaries will be helpful for defining the safe working limits for such ship types. For bifurcation analysis, we need a steady-state value of the dynamical parameters. During turning simulations without wind, u, v, r, p, and φ reach a steady-state value for a constant δ. Therefore, in the absence of wind, bifurcation analysis is possible as shown in the previous section. In the presence of wind, the dynamical parameter ψ R does not attain a steady-state value since the ship is gradually turning. Therefore, the bifurcation analysis of the model investigated in this paper in the presence of wind needs some modified procedure. This will be the focus of our future investigations.
(i) Hopf, fold, and period-doubling bifurcations are identified in the model using MATCONT. Roll responses at different bifurcations validate the MATCONT predictions. The safe limit of the vertical center of gravity for different rudder angles is determined. This information will be useful to ship designers. (ii) Twin-propeller twin-rudder ship shows instability and bifurcations due to sway–yaw–roll–rudder coupled motion during maneuvering. Roll-coupling motion is more pronounced at higher rudder angle. 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. (iii) Roll-related captive model experiments are sensitive to speed variation. The hydrodynamic coefficient Nφ creates a tendency to turn the ship to starboard when the heel angle is to port. This behavior is observed to be opposite in low and high speed. Additionally, nonlinear terms in the model increase roll-coupling motion. Hence, captive maneuvering model tests for such ship types need to be done with more caution. Roll-related captive model tests should be conducted in bare hull condition for a wide range of motion parameters. This will help in predicting dynamic stability and bifurcations accurately from model tests. (iv) In the presence of external forces (e.g., wind), roll-coupling-induced secondary oscillations are observed in the model. The stability and bifurcations analyses of the model in the presence of external wind need to be further investigated. This will be helpful for ship safety.
7 Appendix Ship maneuvering mathematical model for TPTR system.
6 Conclusions
7.1 Hull forces and moments
The purpose of this study is to explore the roll–sway– yaw coupling motion of a high-speed TPTR model and
All hull forces and moments are expressed at midship of hull. The mathematical expression for X H is given
123
Bifurcation analysis of a high-speed twin-propeller twin-rudder ship maneuvering model...
2049
0.8 0.8
0.6 0.6 0.4
CX
CY
0.2
0.4
0 0.2
-0.2
-0.4 0 -0.6
0
30
60
90
120
150
0
180
30
60
ψΑ (deg)
90
120
150
180
150
180
ψΑ (deg)
0.2
1.8 1.6
0.16
1.4 1.2
CN
CK
0.12 1 0.8 0.08 0.6 0.4
0.04
0.2 0
0 0
30
60
90
120
150
180
0
30
60
90
120
ψΑ (deg)
ψΑ (deg) Fig. 14 Wind force coefficients versus effective wind heading angle
in Eq. 3, where X ∗ represents the resistance of the ship.
X H = X 0 + X u˙ u˙ X 0 = X ∗ + X vv v 2 + X rr r 2 + X φφ φ 2
(3)
The mathematical expression for Y H is given in Eq. 4.
⎫ Y H = Y0 + Yv˙ v˙ + Yr˙ r˙ ⎪ ⎪ ⎬ Y0 = Yv v + Yvvv v 3 + Yr r + Yrrr r 3 + Yvvr v 2 r 2 3 2 +Yvrr vr + Yφ φ + Yφφφ φ + Yvvφ v φ ⎪ ⎪ ⎭ +Yvφφ vφ 2 + Yrr φ r 2 φ + Yr φφ r φ 2 (4)
The mathematical expression for N H is given in Eq. 5.
123
2050
A. K. Dash et al.
⎫ N H = N0 + Nv˙ v˙ + Nr˙ r˙ ⎪ ⎪ ⎬ N0 = Nv v + Nvvv v 3 + Nr r + Nrrr r 3 + Nvvr v 2 r 2 3 2 +Nvrr vr + Nφ φ + Nφφφ φ + Nvvφ v φ ⎪ ⎪ ⎭ +Nvφφ vφ 2 + Nrr φ r 2 φ + Nr φφ r φ 2 (5) The mathematical expression for K H is given in Eq. 6, where K φ˙ is the roll damping coefficient, mgG Z is the roll restoring moment, z H is the z coordinate of the point (from O) where the lateral hydrodynamic force Y H acts and Y H (v, r ) is the hydrodynamic lateral force acting on the hull and function of v and r (the first six terms in Eq. 4). ⎫ K H = K 0 + K v˙ v˙ + K r˙ r˙ + K p p + K p˙ p˙ ⎪ ⎪ ⎬ K 0 = −mgG Z − z H Y H (v, r ) + K φ φ (6) +K φφφ φ 3 + K vvφ v 2 φ ⎪ ⎪ ⎭ +K vφφ vφ 2 + K rr φ r 2 φ + K r φφ r φ 2 7.2 Twin-propeller forces and torque The mathematical expressions for X P , N P and Q P are given in Eq. 7. ⎫ 1 − t P{S} n 2P{S} D 4P{S} K T {S} ⎪ ⎪ ⎪ XP = ρ ⎪ ⎪ + 1 − t P{P} n 2P{P} D 4P{P} K T {P} ⎪ ⎪ ⎬ 2 4 (1 − t P{P} )n P{P} D P{P} K T {P} N P = y P{S} ρ ⎪ −(1 − t P{S} )n 2P{S} D 4P{S} K T {S} ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Q S = ±ρn 2 D 5 K S ⎭ S P P
P P
S P P
Q P
(7) S T P K S Q P
K
J
S P P
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ = u−y S r 1−w S / n S D S ⎪ + C J 2 2 S S P P P P 2 C3 + C4 J S + C5 J S P P P P
= C0 + C1 J
P P P P P P P P β3 = (1 − t ) + λ β + λ P0 P S S S P P P βP P 3β P P 1 − w S = 1 − w P0 + τ S β P + τ S β P3 P P βP P 3β P P β P = β − x P r , τβ P {P} = −τβ P {S} , τ3β P {P} = −τ3β P {S} λβ P {P} = −λβ P {S} , λ3β P {P} = −λ3β P {S}
1−t
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(8) The propeller open-water characteristics K T and K Q are often estimated by propeller open-water test. The thrust deduction 1 − t P and effective wake 1 − w P coefficients are used to express the hull–propeller interactions. These coefficients are influenced by v and r motion parameters of ship.
123
7.3 Twin-rudder forces and moment The rudder forces X R , Y R , N R and K R may be written as Eq. 9 XR = − 1 − t
⎫ Fy R{S} sin δ + Fx R{S} cos δ ⎪ ⎪ ⎪ ⎪ +Fy R{P} sin δ + Fx R{P} cos ⎪ δ ⎪ ⎪ ⎪ Fy R{S} cos δ − Fx R{S} sin δ ⎪ ⎪ Y R = − (1 + a H ) ⎪ ⎪ +Fy R{P} cos δ − Fx R{P} sin δ ⎬ Fy R{S} cos δ − Fx R{S} sin δ N R = − (x R + a H x H ) ⎪ ⎪ ⎪ +Fy R{P} cos δ − Fx R{P} sin δ ⎪ ⎪ ⎪ Fy R{S} sin δ + Fx R{S} cos δ ⎪ ⎪ ⎪ + 1 − t S y R{S} R P ⎪ −Fy R{P} sin δ − Fx R{P} cos δ ⎪ ⎪ ⎭ K R = −z R Y R S R P
(9) The coefficients a H and x H are used to express the hydrodynamic force induced on the hull due to rudder deflection. The normal force acting on the rudder FRY , drag force developed due to rudder FR X and torque developed due to rudder normal force Q R can be expressed as shown in Eq. 10. ⎫ F S = 21 ρ A R U 2 S f α sin α S ⎪ ⎪ RY P R ⎪ R P P ⎪ ⎪ ⎪ 1 2 ⎪ F = ρC 2 A U ⎪ T R S 2 ⎪ S RX P ⎪ R P ⎪ ⎬ x Q S =F S S − xls R S ⎪ R P RY P lp R P ⎪ P ⎪ ⎪ 2 ⎪ k)/(log Rn − 2) C T = 0.075(1 + 10 ⎪ ⎪ ⎪ ⎪ ⎪ x = CR C +C sin α S ⎪ ⎭ S S S lp R R Qa R Qb R P
P
P
P
(10) The effective inflow velocity U R and angle α R to the rudder are expressed as shown in Eq. 11. ⎫ ⎪ U S = u 2 S + v2 S ⎪ ⎪ R P R P R P ⎪ ⎪ ⎪ ⎪ v S ⎬ R −1 P α S = δ S − δ S − tan u R P 0 P ⎪ R S P P ⎪ ⎪ ⎪ ⎪ +C sin α ⎪ ⎪ x S = CR C ⎭ lp R R Qa S R Qb S R S P
P
P
P
(11) The longitudinal component of rudder effective flow u R can be expressed as Eq. 12. It is greatly influenced by the propeller slipstream. Effective flow at rudder location increases due to the presence of propeller disk ahead of it.
Bifurcation analysis of a high-speed twin-propeller twin-rudder ship maneuvering model...
u
S R P
= ε S u S η P P
P
⎧ ⎨
S P P ⎩
⎛ 1 + κ ⎝1 +
D S S P K P = 1−w , κ= ε , η S = h P P S S R S P P P P P u − y S r u S = 1 − w S P P P P P P
ε S P
1−w
8K
T S P π J 2 S P P
⎫ ⎞⎫2 ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ − 1⎠ + 1 − η S ⎪ ⎪ ⎪ P ⎭ P ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
R
The lateral component of effective flow velocity to rudder v R can be expressed as Eq. 13. The value of v R is smaller than u R . =v S S R P RP P βR = β − x R r
v
− γ
v S R P
+C
r S Rr P
γ R{S} at ± βR = γ R{P} at ∓ β R C Rδ{S} at ± βR = C Rδ{P} at ∓ β R C Rr {S} at ± βR = C Rr {P} at ∓ β R δ0{S} = −δ0{P}
+C
2051
δ S S Rδ P P
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(13) During the maneuvering motion, the windward (outward) side rudder blocks the flow to the leeward (inward) side rudder. Therefore, both rudders get different flow patterns. Thus, the rudder normal forces of both rudders are different in magnitude. This type of asymmetry is also seen in the propeller wake and thrust deduction factor. Validation of the twin-propeller wake and twin-rudder normal force model is shown in [29]. The rudder sway flow model is made discontinuous at δ = 20◦ for leeward rudder. Because after δ = 20◦ , magnitude of the rudder normal force does not follow the conventional pattern.
(12)
The wind forces and moment acting on the vessel are estimated as shown in Eq. 15 [19]. ⎫ X W = 1/2ρ A A T U A2 C X ⎪ ⎪ ⎪ ⎪ YW = 1/2ρ A A L U A2 CY ⎪ ⎬ 2 (15) N W = 1/2ρ A A L U A LC N ⎪ ⎪ ⎪ ⎪ ρ A (A L )2 U A2 ⎭ KW = CK ⎪ 2L
To convert the effective wind angle from our coordinate system to Fujiwara’s coordinate system, we use Eq. 16. ψA = θA + π
(16)
The values of C X , CY , C N and C K are given in Fig. 14. The simulation results of 20◦ zigzag and 35◦ turning maneuvers at Fn = 0.413, GM = 1.966 m are shown in Figs. 15 and 16. The simulation results match well with Carrica et al. [30].
7.4 Forces and moments due to wind ⎫ ψ R = ψW − ψ ⎪ ⎪ ⎪ u A = UW cos ψ R − u ⎪ ⎪ ⎬ v A = UW sin ψ R − v ⎪ ⎪ ⎪ U A = u 2A + v 2A ⎪ ⎪ ⎭ θ A = atan2 (v A , u A )
(14)
123
2052
A. K. Dash et al. 4
30
δ 3
20
δ, ψ (deg)
Expt., Present
1
0
x' (-)
2
10
0
-10 -1
-20 -2
Present Expt.,MARIN
-30 20
-5
200 ZZ, Fn = 0.413 KG = 7.5 m at constant rps
-4
-3
-2
-1
0
1
y' (-) 6 4
15
2
10 0
φ (deg)
φ (deg)
5 0
-2 0
35 port TC, Fn = 0.413 KG = 7.5 m at constant rps
-4 -6
-5
Expt., Present
-8
-10 -10
-15
-12
0
50
100
150
200
250
t (sec)
-20 2
0
1.5 -0.5
0.5
r (deg/s)
r (deg/s)
1
0
-1
-1.5
-0.5 -2
-1 -1.5
-2.5
-2 0
50
100
150
t (sec) Fig. 15 Validation of 20◦ zigzag simulation in full scale with free run model test from Carrica et al. [30]
123
Fig. 16 Validation of 35◦ port turning simulation in full scale with free run model test from Carrica et al. [30]
Bifurcation analysis of a high-speed twin-propeller twin-rudder ship maneuvering model...
References 1. Hirano, M., Takashina, J.: A calculation of ship turning motion taking coupling effect due to heel into consideration. Trans. West-Japan Soc. Nav. Archit. 59, 71–81 (1980) 2. Eda, H.: Maneuvering performance of high-speed ships with effect of roll motion. Ocean Eng. 7(3), 379–397 (1980) 3. Son, K.H., Nomoto, K.: On the coupled motion of steering and rolling of a high-speed container ship. Nav. Archit. Ocean Eng. 20, 73–83 (1982) 4. Spyrou, K.: Yaw stability of ships in steady wind. Ship. Technol. Res. 42, 21–30 (1995) 5. Oh, I.G., Nayfeh, A.: Nonlinearly interacting responses of the two rotational modes of motion—roll and pitch motions. In: 21st Symposium on Naval Hydrodynamics, Trondheim, Norway, June 24–28 (1996) 6. Falzarano, J., Steindl, A., Troesch, A., Troger, H.: Rolling motion of ships treated as bifurcation problem. In: Bifurcation and Chaos: Analysis, Algorithms, Applications, pp. 117–121. Birkhäuser, Basel (1991) 7. Francescutto, A., Contento, G.: Bifurcations in ship rolling: experimental results and parameter identification technique. Ocean Eng. 26(11), 1095–1123 (1999) 8. Umeda, N., Hashimoto, H., Minegaki, S., Matsuda, A.: An investigation of different methods for the prevention of parametric rolling. J. Mar. Sci. Technol. 13(1), 16–23 (2008) 9. Virgin, L.N.: The nonlinear rolling response of a vessel including chaotic motions leading to capsize in regular seas. Appl. Ocean Res. 9(2), 89–95 (1987) 10. El-Nagar, A., El-Bassiouny, A.: Harmonic, subharmonic, superharmonic, simultaneous sub/super harmonic and combination resonances of self-excited two coupled second order system to multi-frequency excitations. J. Acta Mech. 8, 61–71 (1993) 11. Nayfeh, A.H., Balachandran, B.: Applied Non-Linear Mechanics. Wiley/Interscience, New York (1994) 12. Dhooge, A., Govaerts, W., Kuznetsov, Yu A.: MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw. 29(2), 141–164 (2003) 13. Dhooge, A., et al.: New features of the software MatCont for bifurcation analysis of dynamical systems. Math. Comput. Model. Dyn. Syst. 14(2), 147–175 (2008) 14. International Maritime Organization: Explanatory notes to the standards for ship manoeuvrability, MSC/ Circ. 1053 (2002) 15. American Bureau of Shipping: Guide for the assessment of parametric roll resonance in the design on container carriers (2008)
2053
16. Det Norske Veritas: Rules for high speed, light craft and naval surface craft, Part 1, Chapter 2, Section 2 (1999) 17. Khanfir, S., Hasegawa, K., Nagarajan, V., Shouji, K., Lee, S.K.: Manoeuvring characteristics of twin-rudder systems: rudder–hull interaction effect on the manoeuvrability of twin-rudder ships. J. Mar. Sci. Technol. 16(4), 472–490 (2011) 18. Dash, A.K., Nagarajan, V.: A stochastic response surface approach for uncertainty propagation in ship maneuvering. Int. Shipbuild. Progr. 61(3), 29–161 (2014). doi:10.3233/ ISP140110 19. Fujiwara, T., Ueno, M., Nimura, T.: Estimation of wind forces and moments acting on ships. JASNAOE 183, 77– 90 (1998) 20. Nagarajan, V., Kang, D.H., Hasegawa, K., Nabeshima, K.: Comparison of the mariner Schilling rudder and the mariner rudder for VLCCs in strong winds. J. Mar. Sci. Technol. 13(1), 24–39 (2008) 21. Hallmann, R., Toxopeus, S.L.: Validation and Extension of the Rotating Arm Functionality in the Seakeeping and Manoeuvring Basin. MARIN., Report No. 21569-1DT/SMB (2007) 22. Simonsen, C.D.: PMM model test with DDG51 including uncertainty analysis. Project No. ONRII187.01, FORCE Technology, Denmark (2004) 23. Simman (2008). http://www.simman2008.dk/ 24. Maxsurf software. http://www.bentley.com/en-US/ Products/Maxsurf/ 25. Wilson, R.V., Carrica, P.M., Stern, F.: Unsteady RANS method for ship motions with application to roll for a surface combatant. Comput. Fluids 35, 501–526 (2006) 26. Kuznetsov, I.A.: Elements of Applied Bifurcation Theory, vol. 112. Springer, Berlin (1998) 27. Larger, L., Goedgebuer, J.P., Erneux, T.: Subcritical Hopf bifurcation in dynamical systems described by a scalar nonlinear delay differential equation. Phys. Rev. E 69(3), 036210 (2004) 28. National Transportation Safety Board, USA: Aircraft Accident Report, NTSB/AAR-04/04, Page 21 (2001) 29. Dash, A.K., Nagarajan, V., Sha, O.P.: Uncertainty assessment for ship maneuvering mathematical model. Int. Shipbuild. Progr. (2015). doi:10.3233/ISP-150117 30. Carrica, P.M., Ismail, F., Hyman, M., Bhushan, S., Stern, F.: Turn and zigzag maneuvers of a surface combatant using a URANS approach with dynamic overset grids. J. Mar. Sci. Technol. 18(2), 166–181 (2013)
123
