Roll-induced bifurcation in ship maneuvering under model uncertainty

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.

Appendix

1.1 Maneuvering model for hull forces and moments

The hull forces and moments are expressed at the midship. The mathematical expression for $X_{\text{H}}$ is shown in Eq. 6.

$$ \left. \begin{array}{l} X_{\text{H}} = X_{0} + X_{{\dot{u}}} \dot{u} \hfill \\ X_{0} = X_{*} + X_{vv} v^{2} + X_{rr} r^{2} + X_{\phi \phi } \phi^{2} \hfill \\ \end{array} \right\} $$

(6)

where $X_{*}$ represents the resistance of the ship.

The mathematical expression for $Y_{\text{H}}$ and $N_{\text{H}}$ is shown in Eq. 7.

$$ \left. \begin{array}{l} \left( \begin{aligned} Y \hfill \\ N \hfill \\ \end{aligned} \right)_{\text{H}} = \left( \begin{aligned} Y \hfill \\ N \hfill \\ \end{aligned} \right)_{0} + \left( \begin{aligned} Y \hfill \\ N \hfill \\ \end{aligned} \right)_{{\dot{v}}} \dot{v} + \left( \begin{aligned} Y \hfill \\ N \hfill \\ \end{aligned} \right)_{{\dot{r}}} \dot{r} \hfill \\ \left( \begin{aligned} Y \hfill \\ N \hfill \\ \end{aligned} \right)_{0} = \left( \begin{aligned} Y \hfill \\ N \hfill \\ \end{aligned} \right)_{v} v + \left( \begin{aligned} Y \hfill \\ N \hfill \\ \end{aligned} \right)_{vvv} v^{3} + \left( \begin{aligned} Y \hfill \\ N \hfill \\ \end{aligned} \right)_{r} r + \left( \begin{aligned} Y \hfill \\ N \hfill \\ \end{aligned} \right)_{rrr} r^{3} + \left( \begin{aligned} Y \hfill \\ N \hfill \\ \end{aligned} \right)_{vvr} v^{2} r \hfill \\ \qquad \quad+ \left( \begin{aligned} Y \hfill \\ N \hfill \\ \end{aligned} \right)_{vrr} vr^{2} + \left( \begin{aligned} Y \hfill \\ N \hfill \\ \end{aligned} \right)_{\phi } \phi + \left( \begin{aligned} Y \hfill \\ N \hfill \\ \end{aligned} \right)_{\phi \phi \phi } \phi^{3} + \left( \begin{aligned} Y \hfill \\ N \hfill \\ \end{aligned} \right)_{vv\phi } v^{2} \phi \hfill \\ \qquad \quad + \left( \begin{aligned} Y \hfill \\ N \hfill \\ \end{aligned} \right)_{v\phi \phi } v\phi^{2} + \left( \begin{aligned} Y \hfill \\ N \hfill \\ \end{aligned} \right)_{rr\phi } r^{2} \phi + \left( \begin{aligned} Y \hfill \\ N \hfill \\ \end{aligned} \right)_{r\phi \phi } r\phi^{2} \hfill \\ \end{array} \right\} $$

(7)

The mathematical expression for $K_{\text{H}}$ is shown in Eq. 8.

$$ \left. \begin{array}{l} K_{\text{H}} = K_{0} + K_{{\dot{v}}} \dot{v} + K_{{\dot{r}}} \dot{r} + K_{{\dot{p}}} \dot{p} \hfill \\ K_{0} = - mg\overline{GZ} - z_{H} Y_{H} (v,r) + K_{\phi } \phi + K_{\phi \phi \phi } \phi^{3} + K_{vv\phi } v^{2} \phi \hfill \\ \qquad \quad + K_{v\phi \phi } v\phi^{2} + K_{rr\phi } r^{2} \phi + K_{r\phi \phi } r\phi^{2} + K_{p} p + K_{ppp} p^{3} \hfill \\ \end{array} \right\} $$

(8)

where $K_{p}$ is the roll damping coefficient, $mg\overline{GZ}$ is the roll restoring moment, $z_{\text{H}}$ is the $z$ coordinate of the point where the $Y_{\text{H}}$ acts. $KN$ depends on the geometric characteristics of 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 $\overline{GZ}$ can be computed for any vertical center of gravity $VCG$ corresponding to that displacement condition using the formula $\overline{GZ} = KN - VCG\sin \phi$.

The mathematical expressions for $X_{\text{P}}$, $N_{\text{P}}$ and $Q_{\text{P}}$ are shown in Eqs. 9 and 10.

$$ \left. \begin{aligned} {\text{X}}_{\text{P}} = \rho \left( {\left( {1 - t_{{{\text{P}}\left\{ {\text{S}} \right\}}} } \right)n_{{{\text{P}}\left\{ {\text{S}} \right\}}}^{2} D_{{{\text{P}}\left\{ {\text{S}} \right\}}}^{4} K_{{{\text{T}}\left\{ {\text{S}} \right\}}} + \left( {1 - t_{{{\text{P}}\left\{ {\text{P}} \right\}}} } \right)n_{{{\text{P}}\left\{ {\text{P}} \right\}}}^{2} D_{{{\text{P}}\left\{ {\text{P}} \right\}}}^{4} K_{{{\text{T}}\left\{ {\text{P}} \right\}}} } \right) \hfill \\ N_{\text{P}} = \,y_{{{\text{P}}\left\{ {\text{S}} \right\}}} \rho \left( {(1 - t_{{{\text{P}}\left\{ {\text{P}} \right\}}} )n_{{{\text{P}}\left\{ {\text{P}} \right\}}}^{2} D_{{{\text{P}}\left\{ {\text{P}} \right\}}}^{4} K_{{{\text{T}}\left\{ {\text{P}} \right\}}} - (1 - t_{{{\text{P}}\left\{ {\text{S}} \right\}}} )n_{{{\text{P}}\left\{ {\text{S}} \right\}}}^{2} D_{{{\text{P}}\left\{ {\text{S}} \right\}}}^{4} K_{{{\text{T}}\left\{ {\text{S}} \right\}}} } \right) \hfill \\ Q_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} = \pm \rho n_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}}^{2} D_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}}^{5} K_{{{\text{Q}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} \hfill \\ \end{aligned} \right\} $$

(9)

$$ \left. \begin{aligned} K_{{{\text{T}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} = a_{0} + a_{1} J_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} + a_{2} J_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}}^{2} \hfill \\ K_{{{\text{Q}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} = q_{0} + q_{1} J_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} + q_{2} J_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}}^{2} \hfill \\ J_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} = (u - y_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} r)(1 - w_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} )/(n_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} D_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} ) \hfill \\ 1 - t_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} = 1 - t_{{{\text{P}}0}} + \lambda_{{\beta_{\text{P}} \left\{ {\text{S}}\atop{\text{P}} \right\}}} \beta_{\text{P}} + \lambda_{{3\beta_{\text{P}} \left\{ {\text{S}}\atop{\text{P}} \right\}}} \beta_{\text{P}}^{3} \hfill \\ 1 - w_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} = 1 - w_{{{\text{P}}0}} + \tau_{{\beta_{\text{P}} \left\{ {\text{S}}\atop{\text{P}} \right\}}} \beta_{\text{P}} + \tau_{{3\beta_{\text{P}} \left\{ {\text{S}}\atop{\text{P}} \right\}}} \beta_{\text{P}}^{3} \hfill \\ \beta_{\text{P}} = \beta - x^{\prime}_{\text{P}} r^{\prime},\tau_{{\beta_{\text{P}} \left\{ {\text{P}} \right\}}} = - \tau_{{\beta_{\text{P}} \left\{ {\text{S}} \right\}}} , \, \tau_{{3\beta_{\text{P}} \left\{ {\text{P}} \right\}}} = - \tau_{{3\beta_{\text{P}} \left\{ {\text{S}} \right\}}} \hfill \\ \lambda_{{\beta_{\text{P}} \left\{ {\text{P}} \right\}}} = - \lambda_{{\beta_{\text{P}} \left\{ {\text{S}} \right\}}} , \, \lambda_{{3\beta_{\text{P}} \left\{ {\text{P}} \right\}}} = - \lambda_{{3\beta_{\text{P}} \left\{ {\text{S}} \right\}}} \hfill \\ \end{aligned} \right\} $$

(10)

The mathematical expression for forces ($X_{\text{R}}$ and $Y_{\text{R}}$) and moment ($N_{\text{R}}$ and $K_{\text{R}}$) induced at midship due to rudder are shown in Eq. 11.

$$ \left. \begin{aligned} \hfill X_{\text{R}} = - \left( {1 - t_{{{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} } \right)\left[ {\left( {F_{{{\text{RY}}\left\{ {\text{S}} \right\}}} + F_{{{\text{RY}}\left\{ {\text{P}} \right\}}} } \right)s\delta + \left( {F_{{{\text{RX}}\left\{ {\text{S}} \right\}}} + F_{{{\text{RX}}\left\{ {\text{P}} \right\}}} } \right)c\delta } \right] \\ \hfill Y_{\text{R}} = - \left( {1 + a_{\text{H}} } \right)\left[ {\left( {F_{{{\text{RY}}\left\{ {\text{S}} \right\}}} + F_{{{\text{RY}}\left\{ {\text{P}} \right\}}} } \right)c\delta - \left( {F_{{{\text{RX}}\left\{ {\text{S}} \right\}}} + F_{{{\text{RX}}\left\{ {\text{P}} \right\}}} } \right)s\delta } \right] \\ \hfill N_{\text{R}} = - \left( {x_{\text{R}} + a_{\text{H}} x_{\text{H}} } \right)\left[ {\left( {F_{{{\text{RY}}\left\{ {\text{S}} \right\}}} + F_{{{\text{RY}}\left\{ {\text{P}} \right\}}} } \right)c\delta - \left( {F_{{{\text{RX}}\left\{ {\text{S}} \right\}}} + F_{{{\text{RX}}\left\{ {\text{P}} \right\}}} } \right)s\delta } \right] \\ \hfill \, + \left( {1 - t_{{{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} } \right)y_{{{\text{R}}\left\{ {\text{S}} \right\}}} \left[ {\left( {F_{{{\text{RX}}\left\{ {\text{S}} \right\}}} - F_{{{\text{RX}}\left\{ {\text{P}} \right\}}} } \right)c\delta + \left( {F_{{{\text{RY}}\left\{ {\text{S}} \right\}}} - F_{{{\text{RY}}\left\{ {\text{P}} \right\}}} } \right)s\delta } \right] \\ \hfill K_{\text{R}} = - z_{\text{R}} Y_{\text{R}} \\ \end{aligned} \right\} $$

(11)

where, $s$: sin, $c$: cos

The forces $F_{\text{RX}}$, $F_{\text{RY}}$ and torque $Q_{\text{R}}$ are expressed as shown in Eq. 12.

$$ \left. \begin{aligned} F_{{{\text{RY}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} = \frac{1}{2}\rho A_{\text{R}} U_{{{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}}^{2} f_{\alpha } \sin \alpha_{{{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} \hfill \\ F_{{{\text{RX}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} = \frac{1}{2}\rho C_{\text{T}} 2A_{\text{R}} U_{{{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}}^{2} \hfill \\ Q_{{{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} = F_{{{\text{RY}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} \left( {x_{{lp{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} - x_{{ls{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} } \right) \hfill \\ C_{\text{T}} = 0.075(1 + k)/(\log_{10} Rn - 2)^{2} \hfill \\ x_{{lp{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} = C_{\text{R}} \left( {C_{{{\text{RQ}}a\left\{ {\text{S}}\atop{\text{P}} \right\}}} + C_{{{\text{RQ}}b\left\{ {\text{S}}\atop{\text{P}} \right\}}} \sin \alpha_{{{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} } \right) \hfill \\ \end{aligned} \right\} $$

(12)

The $U_{\text{R}}$ and $\alpha_{\text{R}}$ are expressed as shown in Eq. 13.

$$ \left. \begin{aligned} U_{{{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} = \sqrt {u_{{{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}}^{2} + v_{{{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}}^{2} } \hfill \\ \alpha_{{{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} = \delta_{{\left\{ {\text{S}}\atop{\text{P}} \right\}}} - \delta_{{0\left\{ {\text{S}}\atop{\text{P}} \right\}}} - \tan^{ - 1} \left( {\frac{{v_{{{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} }}{{u_{{{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} }}} \right) \hfill \\ \end{aligned} \right\} $$

(13)

The longitudinal component of rudder inflow velocity $u_{\text{R}}$ is expressed as shown in Eq. 14.

$$ \left. \begin{array}{l} u_{{{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} = \varepsilon_{{\left\{ {\text{S}}\atop{\text{P}} \right\}}} u_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} \sqrt {\eta_{{{\text{P}}\left\{{\text{S}} \atop {\text{P}} \right\}}} \left\{ {1 + \kappa{\xi}} \right\}^{2} + 1 - \eta_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} } \hfill \\ \varepsilon_{{\left\{ {\text{S}}\atop{\text{P}} \right\}}} = \frac{{1 - w_{{{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} }}{{1 - w_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} }}, \, \kappa { = }\frac{K^{*}}{{\varepsilon_{{\left\{ {\text{S}}\atop{\text{P}} \right\}}} }}, \, \eta_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} = \frac{{D_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} }}{{h_{{{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} }}, \, u_{{{\text{P}}\left\{ \text{S} \atop {\text{P}} \right\}}} = \left( {1 - w_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} } \right)\left( {u - y_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} r} \right) \hfill \\ {{\xi} = {\sqrt {1 + \frac{{8K_{{{\text{T}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} }}{{\pi J_{{{\text{P}}\left\{ {\text{S}}\atop{\text{P}} \right\}}}^{2} }}} - 1}}\hfill \\ \end{array} \right\} $$

(14)

The lateral component of rudder inflow velocity $v_{\text{R}}$ is expressed as shown in Eq. 15.

$$ \left. \begin{aligned} v_{{{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} = v_{{{\text{RP}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} - \left( {\gamma_{{{\text{R}}\left\{ {\text{S}}\atop{\text{P}} \right\}}} v + C_{{{\text{R}}r\left\{ {\text{S}}\atop{\text{P}} \right\}}} r + C_{{{\text{R}}\delta \left\{ {\text{S}}\atop{\text{P}} \right\}}} \delta_{{\left\{ {\text{P}} \atop {\text{S}} \right\}}} } \right) \hfill \\ \beta_{\text{R}} = \beta - x^{\prime}_{\text{R}} r^{\prime} \hfill \\ \gamma_{{{\text{R}}\left\{ {\text{S}} \right\}}} {\text{ at }} \pm \beta_{\text{R}} = \gamma_{{{\text{R}}\left\{ {\text{P}} \right\}}} {\text{ at }} \mp \beta_{\text{R}} \hfill \\ C^{\prime}_{{{\text{R}}\left\{ {\delta}\atop {r} \right\}\left\{ {\text{S}} \right\}}} {\text{ at }} \pm \beta_{\text{R}} = C^{\prime}_{{{\text{R}}\left\{ {\delta} \atop {r} \right\}\left\{ {\text{P}} \right\}}} {\text{ at }} \mp \beta_{\text{R}} \hfill \\ \delta_{{0\left\{ {\text{S}} \right\}}} = - \delta_{{0\left\{ {\text{P}} \right\}}} \hfill \\ \end{aligned} \right\} $$

(15)

The wind forces and moment acting on the ship are estimated as shown in Eqs. 16 and 17.

$$ \left. \begin{array}{l} X_{\text{W}} = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}\rho_{\text{A}} A_{\text{T}} U_{\text{A}}^{2} C_{\text{X}} (\psi_{\text{A}} ) \hfill \\ Y_{\text{W}} = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}\rho_{\text{A}} A_{\text{L}} U_{\text{A}}^{2} C_{\text{Y}} (\psi_{\text{A}} ) \hfill \\ N_{\text{W}} = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}\rho_{\text{A}} A_{\text{L}} U_{\text{A}}^{2} L_{\text{OA}} C_{\text{N}} (\psi_{\text{A}} ) \hfill \\ K_{\text{W}} = \left( {\frac{{\rho_{\text{A}} \left( {A_{\text{L}} } \right)^{2} U_{\text{A}}^{2} }}{{2L_{\text{OA}} }}} \right)C_{\text{K}} (\psi_{A} ) \hfill \\ \end{array} \right\} $$

(16)

$$ \left. \begin{array}{l} \psi_{\text{R}} = \psi_{\text{W}} - \psi \hfill \\ u_{\text{A}} = U_{\text{W}} \cos \psi_{\text{R}} - u \hfill \\ v_{\text{A}} = U_{\text{W}} \sin \psi_{\text{R}} - v \hfill \\ U_{\text{A}} = \sqrt {u_{\text{A}}^{2} + v_{\text{A}}^{2} } \hfill \\ \theta_{\text{A}} = a{\text{tan 2}}\left( {v_{\text{A}} ,u_{\text{A}} } \right) \hfill \\ \end{array} \right\} $$

(17)

To convert effective wind angle from our coordinate system to Fujiwara’s coordinate system we use Eq. 18.

$$ \psi_{\text{A}} = \theta_{\text{A}} + \pi $$

(18)

where, $C_{\text{X}} (\psi_{\text{A}} )$, $C_{\text{Y}} (\psi_{\text{A}} )$, $C_{\text{N}} (\psi_{\text{A}} )$, $C_{\text{K}} (\psi_{\text{A}} )$ 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.

Fig. 27

Wind force coefficients versus effective wind heading angle

1.2 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.

$$ \begin{aligned} \left( \begin{aligned} y \hfill \\ \psi \hfill \\ \end{aligned} \right) &= \left( \begin{aligned} y \hfill \\ \psi \hfill \\ \end{aligned} \right)_{\hbox{max} } \sin \omega t \hfill \\ v &= \dot{y}, \, r = \dot{\psi } \hfill \\ \dot{v} &= \ddot{y }, { }\dot{r} = \ddot{\psi } \hfill \\ \end{aligned} $$

(19)

where $y$ is the lateral displacement and $\psi$ is the angular displacement of the model ship.

Fig. 28

Layout of PMM setup for dynamic motion

The model dynamics for sway + yaw motion are shown in Eq. 20.

$$ \left. \begin{array}{l} \left\{ \begin{aligned} u \hfill \\ v \hfill \\ r \hfill \\ \end{aligned} \right\} = \left[ {\begin{array}{*{20}c} {c\psi } & {s\psi } & 0 \\ { - s\psi } & {c\psi } & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]\left\{ \begin{aligned} u \hfill \\ v \hfill \\ r \hfill \\ \end{aligned} \right\}_{\text{PMM}} \hfill \\ \dot{u} = \dot{u}_{\text{PMM}} c\psi + \dot{v}_{\text{PMM}} s\psi + \hfill \\ \, r_{\text{PMM}} ( - u_{\text{PMM}} s\psi + v_{\text{PMM}} c\psi ) \hfill \\ \dot{v} = - \dot{u}_{\text{PMM}} s\psi + \dot{v}_{\text{PMM}} c\psi - \hfill \\ \, r_{\text{PMM}} (u_{\text{PMM}} c\psi + v_{\text{PMM}} s\psi ) \hfill \\ \dot{r} = \dot{r}_{\text{PMM}} \hfill \\ \end{array} \right\} $$

(20)

where $s$: sin and $c$: cos

The mathematical expressions for $X_{\text{H}}$, $Y_{\text{H}}$, $N_{\text{H}}$ and $K_{\text{H}}$ during pure sway motion are obtained by substituting $r$ = 0, $\dot{r}$ = 0, $\phi$ = 0, $p$ = 0 and $\dot{p}$ = 0 in Eqs. 6–8, respectively. The mathematical expressions for $X_{\text{H}}$, $Y_{\text{H}}$, $N_{\text{H}}$ and $K_{\text{H}}$ during pure yaw motion are obtained by substituting $v$ = 0, $\dot{v}$ = 0, $\phi$ = 0, $p$ = 0 and $\dot{p}$ = 0 in Eqs. 6–8, respectively. The mathematical expressions for $X_{\text{H}}$, $Y_{\text{H}}$, $N_{\text{H}}$ and $K_{\text{H}}$ during heel + yaw motion are obtained by substituting $v$ = 0, $\dot{v}$ = 0, $p$ = 0 and $\dot{p}$ = 0 in Eqs. 6–8, respectively. The mathematical expressions for $X_{\text{H}}$, $Y_{\text{H}}$, $N_{\text{H}}$ and $K_{\text{H}}$ during sway + yaw motion are obtained by substituting $r$ = 0, $\dot{r}$ = 0, $p$ = 0 and $\dot{p}$ = 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.

$$ \left. \begin{array}{l} X_{\text{H}} = X_{*} + X_{vv} v^{2} \hfill \\ = X_{*} + \frac{1}{2}X_{vv} v_{\hbox{max} }^{2} +{\frac{c(2{\omega}t)}{2}X_{vv} v_{\hbox{max} }^{2} } \hfill \\ Y_{\text{H}} = Y_{v} v + Y_{vvv} v^{3} + Y_{{\dot{v}}} \dot{v} \hfill \\ { = }\; Y_{{\dot{v}}} \dot{v}_{\hbox{max} } s (\omega t) - \left( { Y_{v} v_{\hbox{max} } + \frac{3}{4}Y_{vvv} v_{\hbox{max} }^{3} } \right) c( \omega t) - { Y_{vvv} v_{\hbox{max} }^{3} \frac{c(3 \omega t)}{4}} \hfill \\ \end{array} \right\} $$

(21)

where, $s$: sin and $c$: cos

$$ \left. \begin{aligned} F_{vvv} = - \frac{4}{{v_{\hbox{max} }^{3} }} \times \frac{2}{T}\int\limits_{0}^{T} {F_{H} \left( t \right) \cdot \cos 3\omega t \cdot {\text{d}}t} \hfill \\ F_{v} = - \frac{1}{{v_{\hbox{max} } }}\left( {\frac{2}{T}\int\limits_{0}^{T} {F_{H} \left( t \right) \cdot \cos \omega t \cdot {\text{d}}t} + \frac{3}{4}F_{vvv} v_{\hbox{max} }^{3} } \right) \hfill \\ F_{{\dot{v}}} = \frac{1}{{\dot{v}_{\hbox{max} } }} \times \frac{2}{T}\int\limits_{0}^{T} {F_{H} \left( t \right) \cdot \sin \omega t \cdot {\text{d}}t} \hfill \\ \end{aligned} \right\} $$

(22)

where the variable $F$ corresponds to $Y$ or $N$ or $K$.

$$ \left. \begin{array}{l} X_{\text{H}} = X_{*} + X_{vv} v^{2} + X_{rr} r^{2} + X_{vr} r^{2} \hfill \\ \, = \frac{1}{2} \left( 2{X_{*} + X_{vv} v_{\hbox{max} }^{2} + X_{rr} r_{\hbox{max} }^{2} - X_{vr} v_{\hbox{max} } r_{\hbox{max} } } \right) \hfill \\ \, + \frac{c (2{\omega}t)}{2}\left( {X_{vv} v_{\hbox{max} }^{2} + X_{rr} r_{\hbox{max} }^{2} - X_{vr} v_{\hbox{max} } r_{\hbox{max} } } \right) \hfill \\ Y_{\text{H}} = Y_{v} v + Y_{vvv} v^{3} + Y_{{\dot{v}}} \dot{v} + Y_{r} r + Y_{rrr} r^{3} + Y_{{\dot{r}}} \dot{r} + Y_{vvr} v^{2} r + Y_{vrr} vr^{2} \hfill \\ \, = s (\omega t) \left( {Y_{{\dot{v}}} \dot{v}_{\hbox{max} } - Y_{{\dot{r}}} \dot{r}_{\hbox{max} } } \right) \hfill \\ \, + \frac{3c(\omega t)}{4}\left( { - \frac{4}{3} Y_{v} v_{\hbox{max} } - Y_{vvv} v_{\hbox{max} }^{3} + \frac{4}{3} Y_{r} r_{\hbox{max} } + Y_{rrr} r_{\hbox{max} }^{3} + Y_{vvr} v_{\hbox{max} }^{2} r_{\hbox{max} } - Y_{vrr} v_{\hbox{max} } r_{\hbox{max} }^{2} } \right) \hfill \\ \, + \frac{c(3{\omega}t)}{4}\left( { - Y_{vvv} v_{\hbox{max} }^{3} + Y_{rrr} r_{\hbox{max} }^{3} + Y_{vvr} v_{\hbox{max} }^{2} r_{\hbox{max} } - Y_{vrr} v_{\hbox{max} } r_{\hbox{max} }^{2} } \right) \hfill \\ \end{array} \right\} $$

(23)

where, $s$: sin, $c$: cos

The explanation for sway + yaw motion is given in Eqs. 23 and 24.

$$ \left. \begin{aligned} X_{vr} = - \frac{1}{{v_{\hbox{max} } r_{\hbox{max} } }}\left( { \frac{4}{T}\int\limits_{0}^{T} {X_{\text{H}} \left( t \right) \cdot c (2 {\omega} t) \cdot {\text{d}}t - \left( {X_{vv} v_{\hbox{max} }^{2} + X_{rr} r_{\hbox{max} }^{2} } \right)} } \right) \hfill \\ F_{vvr} v_{\hbox{max} } - F_{vrr} r_{\hbox{max} } = \frac{4}{{v_{\hbox{max} } r_{\hbox{max} } }}\left( {\frac{2}{T}\int\limits_{0}^{T} {F_{\text{H}} \left( t \right) \cdot c (3 {\omega} t) \cdot {\text{d}}t - \frac{1}{4} \left( { - F_{vvv} v_{\hbox{max} }^{3} + F_{rrr} r_{\hbox{max} }^{3} } \right)} } \right) \hfill \\ \end{aligned} \right\} $$

(24)

where $c$: cos, the variable $F$ corresponds to $Y$ or $N$ or $K$.

1.3 Development of the roll moment model

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 $z^{\prime}_{\text{H}}$ and $z^{\prime}_{\text{R}}$ 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 $K^{\prime}_{{\dot{v}}}$ and $K^{\prime}_{{\dot{r}}}$ are calculated by Fourier series analysis of pure sway and pure yaw test data, respectively. The coefficients $K^{\prime}_{{\dot{p}}}$, $K^{\prime}_{p}$, $K^{\prime}_{ppp}$ are calculated from roll decay test. The coefficients $K^{\prime}_{\phi }$ and $K^{\prime}_{\phi \phi \phi }$ are from 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 $v - \phi$ and $r - \phi$ coupling motion is assumed to be about $-z^{\prime}_{\text{H}}$ times of the nondimensional sway force, this makes the coefficients $K^{\prime}_{vv\phi }$, $K^{\prime}_{v\phi \phi }$, $K^{\prime}_{rr\phi }$, and $K^{\prime}_{r\phi \phi }$ as $- z^{\prime}_{\text{H}}$ times of the $Y^{\prime}_{vv\phi }$, $Y^{\prime}_{v\phi \phi }$, $Y^{\prime}_{rr\phi }$, and $Y^{\prime}_{r\phi \phi }$, respectively.

Table 7 PMM test program conducted in SSPA [44]

Fig. 29

Variation of $z^{\prime}_{\text{H}}$ and $z^{\prime}_{\text{R}}$ coefficients. a static drift test at $\beta$ = −10° to +10°, b static rudder test at $\delta$ = −35° to +35°

Fig. 30

Estimation of coefficients $K^{\prime}_{\phi }$ and $K^{\prime}_{\phi \phi \phi }$ from static heel test

1.4 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.

$$ \theta_{yx} \% = \frac{{\left( {\frac{{\sum {y^{2} } }}{{N_{t} }}} \right)_{x = a + \varDelta a} - \left( {\frac{{\sum {y^{2} } }}{{N_{t} }}} \right)_{x = a} }}{{\left( {\frac{{\sum {y^{2} } }}{{N_{t} }}} \right)_{x = a} }} \, \% \, 100 $$

(25)

where $\theta_{yx}$ is the sensitivity of output $y$ to input $x$, $\Delta a$ the perturbed value of each input, $N_{t}$ is the total number of samples in time series. In Eq. 25, the model coefficients are individually multiplied by a factor of 1.1 ($\Delta a = 0.10$) 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 $\delta$ change command is given. The root mean square (rms) of various parameters is taken for the 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 uncertainty propagation [6]. Coefficient $Y^{\prime}_{{\dot{v}}}$ and $C^{\prime}_{Rr}$ 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.

Fig. 31

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