Bifurcation analysis of a high-speed twin-propeller twin-rudder ship maneuvering model in roll-coupling motion

Abstract

In this paper, bifurcation analysis of a high-speed 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.

Appendix

Ship maneuvering mathematical model for TPTR system.

Fig. 14

Wind force coefficients versus effective wind heading angle

1.1 Hull forces and moments

All hull forces and moments are expressed at midship of hull. The mathematical expression for $X_H$ is given in Eq. 3, where $X_*$ represents the resistance of the ship.

$$\begin{aligned}&\left. {\begin{array}{l} X_H =X_0 +X_{\dot{u}} \dot{u} \\ X_0 =X_*+X_{vv} v^{2}+X_{rr} r^{2}+X_{\phi \phi } \phi ^{2} \\ \end{array}} \right\} \end{aligned}$$

(3)

The mathematical expression for $Y_H$ is given in Eq. 4.

$$\begin{aligned}&\left. {\begin{array}{l} Y_H =Y_0 +Y_{\dot{v}} \dot{v}+Y_{\dot{r}} \dot{r} \\ Y_0 =Y_v v+Y_{vvv} v^{3}+Y_r r+Y_{rrr} r^{3}+Y_{vvr} v^{2}r \\ \quad \quad \quad +Y_{vrr} vr^{2}+Y_\phi \phi +Y_{\phi \phi \phi } \phi ^{3}+Y_{vv\phi } v^{2}\phi \\ \quad \quad \quad +Y_{v\phi \phi } v\phi ^{2}+Y_{rr\phi } r^{2}\phi +Y_{r\phi \phi } r\phi ^{2} \\ \end{array}} \right\} \nonumber \\ \end{aligned}$$

(4)

The mathematical expression for $N_H$ is given in Eq. 5.

$$\begin{aligned}&\left. {\begin{array}{l} N_H =N_0 +N_{\dot{v}} \dot{v}+N_{\dot{r}} \dot{r} \\ N_0 =N_v v+N_{vvv} v^{3}+N_r r+N_{rrr} r^{3}+N_{vvr} v^{2}r \\ \quad \quad \quad +N_{vrr} vr^{2}+N_\phi \phi +N_{\phi \phi \phi } \phi ^{3}+N_{vv\phi } v^{2}\phi \\ \quad \quad \quad +N_{v\phi \phi } v\phi ^{2}+N_{rr\phi } r^{2}\phi +N_{r\phi \phi } r\phi ^{2} \\ \end{array}} \right\} \nonumber \\ \end{aligned}$$

(5)

The mathematical expression for $K_H$ is given in Eq. 6, where $K_{\dot{\phi }}$is the roll damping coefficient, $mg\overline{GZ}$ 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).

$$\begin{aligned}&\left. {\begin{array}{l} K_H =K_0 +K_{\dot{v}} \dot{v}+K_{\dot{r}} \dot{r}+K_p p+K_{\dot{p}} \dot{p} \\ K_0 =-mg\overline{GZ} -z_H Y_H (v,r)+K_\phi \phi \\ \quad \quad \quad +K_{\phi \phi \phi } \phi ^{3}+K_{vv\phi } v^{2}\phi \\ \quad \quad \quad +K_{v\phi \phi } v\phi ^{2}+K_{rr\phi } r^{2}\phi +K_{r\phi \phi } r\phi ^{2} \\ \end{array}} \right\} \end{aligned}$$

(6)

1.2 Twin-propeller forces and torque

The mathematical expressions for $\hbox {X}_P , \hbox {N}_P$ and $\hbox {Q}_P$ are given in Eq. 7.

$$\begin{aligned}&\left. {\begin{array}{l} \hbox {X}_P =\rho \left( {\begin{array}{l} \left( {1-t_{P\{ S \}} } \right) n_{P\{ S \}}^2 D_{P\{ S \}}^4 K_{T\{ S \}} \\ +\left( {1-t_{P\{ P \}} } \right) n_{P\{ P \}}^2 D_{P\{ P \}}^4 K_{T\{ P \}} \\ \end{array}} \right) \\ N_P = y_{P\{ S \}} \rho \left( {\begin{array}{l} (1-t_{P\{ P \}} )n_{P\{ P \}}^2 D_{P\{ P \}}^4 K_{T\{ P \}} \\ -(1-t_{P\{ S \}} )n_{P\{ S \}}^2 D_{P\{ S \}}^4 K_{T\{ S \}} \\ \end{array}} \right) \\ Q_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } =\pm \rho n_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} }^2 D_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} }^5 K_{Q\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } \\ \end{array}} \right\} \nonumber \\ \end{aligned}$$

(7)

$$\begin{aligned}&\left. {\begin{array}{l} K_{T\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } =C_0 +C_1 J_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } +C_2 J_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} }^2 \\ K_{Q\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } =C_3 +C_4 J_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } +C_5 J_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} }^2 \\ J_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } =\left( u-y_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } r\right) \left( 1-w_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } \right) /\left( n_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } D_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} }\right) \\ 1-t_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } =\left( {1-t_{P0} } \right) +\lambda _{\beta _P \left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } \beta _P +\lambda _{3\beta _P \left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } \beta _P^3 \\ 1-w_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } =1-w_{P0} +\tau _{\beta _P \left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } \beta _P +\tau _{3\beta _P \left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} }\beta _P^3 \\ \beta _P =\beta -{x}'_P {r}',\tau _{\beta _P \{ P \}} =-\tau _{\beta _P \{ S \}} , \tau _{3\beta _P \{ P \}} =-\tau _{3\beta _P \{ S \}} \\ \lambda _{\beta _P \{ P \}} =-\lambda _{\beta _P \{ S \}} , \lambda _{3\beta _P \{ P \}} =-\lambda _{3\beta _P \{ S \}} \\ \end{array}} \right\} \nonumber \\ \end{aligned}$$

(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.

1.3 Twin-rudder forces and moment

The rudder forces $X_R ,Y_R ,N_R$ and $K_R$ may be written as Eq. 9

$$\begin{aligned}&\left. {\begin{array}{l} X_R =-\left( {1-t_{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } } \right) \left( {\begin{array}{l} F_{yR\{ S \}} \sin \delta +F_{xR\{ S \}} \cos \delta \\ +F_{yR\{ P \}} \sin \delta +F_{xR\{ P \}} \cos \delta \\ \end{array}} \right) \\ Y_R =-\left( {1+a_H } \right) \left( {\begin{array}{l} F_{yR\{ S \}} \cos \delta -F_{xR\{ S \}} \sin \delta \\ +F_{yR\{ P \}} \cos \delta -F_{xR\{ P \}} \sin \delta \\ \end{array}} \right) \\ N_R =-\left( {x_R +a_H x_H } \right) \left( {\begin{array}{l} F_{yR\{ S \}} \cos \delta -F_{xR\{ S \}} \sin \delta \\ +F_{yR\{ P \}} \cos \delta -F_{xR\{ P \}} \sin \delta \\ \end{array}} \right) \\ \quad +\left( {1-t_{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } } \right) y_{R\{ S \}} \left( {\begin{array}{l} F_{yR\{ S \}} \sin \delta +F_{xR\{ S \}} \cos \delta \\ -F_{yR\{ P \}} \sin \delta -F_{xR\{ P \}} \cos \delta \\ \end{array}} \right) \\ K_R =-z_R Y_R \\ \end{array}} \right\} \nonumber \\ \end{aligned}$$

(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 $F_{RY}$, drag force developed due to rudder $F_{RX}$ and torque developed due to rudder normal force $Q_R$ can be expressed as shown in Eq. 10.

$$\begin{aligned}&\left. {\begin{array}{l} F_{RY\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } =\frac{1}{2}\rho A_R U_{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } ^2 f_\alpha \sin \alpha _{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } \\ F_{RX\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } =\frac{1}{2}\rho C_T 2A_R U_{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} }^2 \\ Q_{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } =F_{RY\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } \left( x_{lpR\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } -x_{lsR\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } \right) \\ C_T =0.075(1+k)/(\log _{10} Rn-2)^{2} \\ x_{lpR\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } =C_R \left( {C_{RQa\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } +C_{RQb\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } \sin \alpha _{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } } \right) \\ \end{array}} \right\} \nonumber \\ \end{aligned}$$

(10)

The effective inflow velocity $U_R$ and angle $\alpha _R$ to the rudder are expressed as shown in Eq. 11.

$$\begin{aligned}&\left. {\begin{array}{l} U_{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } =\sqrt{u_{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } ^2 +v_{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} }^2 } \\ \alpha _{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } =\delta _{\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } -\delta _{0\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } -\tan ^{-1}\left( {\frac{v_{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } }{u_{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} }}} \right) \\ x_{lpR\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } =C_R \left( {C_{RQa\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } +C_{RQb\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } \sin \alpha _{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } } \right) \\ \end{array}} \right\} \nonumber \\ \end{aligned}$$

(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.

$$\begin{aligned}&\left. {\begin{array}{l} u_{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } =\varepsilon _{\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } u_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } \sqrt{\eta _{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } \left\{ {1+\kappa \left( {\sqrt{1+\frac{8K_{T\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } }{\pi J_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } ^2 }}-1} \right) } \right\} ^{2}+1-\eta _{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} }} \\ \varepsilon _{\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } =\frac{1-w_{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } }{1-w_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } }, \kappa \hbox {=}\frac{K}{\varepsilon _{\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } }, \eta _{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } =\frac{D_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } }{h_{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } } \\ u_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } =\left( 1-w_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } \right) \left( u-y_{P\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } r\right) \\ \end{array}} \right\} \end{aligned}$$

(12)

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 $u_R$.

$$\begin{aligned}&\left. {\begin{array}{l} v_{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } =v_{RP\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } -\left( {\gamma _{R\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } v+C_{Rr\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } r+C_{R\delta \left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} } \delta _{\left\{ \begin{array}{c} S \\ P \\ \end{array} \right\} }} \right) \\ \beta _\mathrm{R} =\beta -{x}'_R {r}' \\ \gamma _{R\{ S \}} \hbox { at }\pm \beta _\mathrm{R} =\gamma _{R\{ P \}} \hbox { at }\mp \beta _R \\ {C}'_{R\delta \{ S \}} \hbox { at }\pm \beta _\mathrm{R} ={C}'_{R\delta \{ P \}} \hbox { at }\mp \beta _R \\ {C}'_{Rr\{ S \}} \hbox { at }\pm \beta _\mathrm{R} ={C}'_{Rr\{ P \}} \hbox { at }\mp \beta _R \\ \delta _{0\{ S \}} =-\delta _{0\{ P \}} \\ \end{array}} \right\} \nonumber \\ \end{aligned}$$

(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 $\delta = 20^{\circ }$ for leeward rudder. Because after $\delta = 20^{\circ }$, magnitude of the rudder normal force does not follow the conventional pattern.

1.4 Forces and moments due to wind

$$\begin{aligned}&\left. {\begin{array}{l} \psi _R =\psi _W -\psi \\ u_A =U_W \cos \psi _R -u \\ v_A =U_W \sin \psi _R -v \\ U_A =\sqrt{u_A^2 +v_A^2 } \\ \theta _A =\hbox {atan2}\left( {v_A ,u_A } \right) \\ \end{array}} \right\} \end{aligned}$$

(14)

The wind forces and moment acting on the vessel are estimated as shown in Eq. 15 [19].

$$\begin{aligned}&\left. {\begin{array}{l} X_W =1/2\rho _A A_T U_A^2 C_X \\ Y_W =1/2\rho _A A_L U_A^2 C_Y \\ N_W =1/2\rho _A A_L U_A^2 LC_N \\ K_W =\left( {\frac{\rho _A \left( {A_L } \right) ^{2}U_A^2 }{2L}} \right) C_K \\ \end{array}} \right\} \end{aligned}$$

(15)

To convert the effective wind angle from our coordinate system to Fujiwara’s coordinate system, we use Eq. 16.

$$\begin{aligned} \psi _A =\theta _A +\pi \end{aligned}$$

(16)

Fig. 15

Validation of $20^{\circ }$ zigzag simulation in full scale with free run model test from Carrica et al. [30]

Fig. 16

Validation of $35^{\circ }$ port turning simulation in full scale with free run model test from Carrica et al. [30]

The values of $C_X ,C_Y ,C_N$ and $C_K$ are given in Fig. 14. The simulation results of $20^{\circ }$ zigzag and $35^{\circ }$ 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].