# Practical maneuvering simulation method of ships considering the roll-coupling effect

## Abstract

In this study, a practical maneuvering simulation method is presented considering the roll-coupling effect by extending an ordinal simulation model (3D-MMG model) proposed by Yasukawa and Yoshimura (J Mar Sci Technol 20:37–52, 2015), and adding the motion equation of roll. The roll moment acting on the hull is estimated by multiplying the hull lateral force with the vertical acting point. With respect to the surge force, lateral force, and yaw moment, the derivative expression model is employed. Subsequently, hydrodynamic derivatives with the exception of the roll-related terms are obtained by a captive model test based on the 3D-MMG model. The roll-related derivatives and the vertical acting point of the hull lateral force are estimated by simple formulae constructed based on the experimental data of four ship models. To validate the proposed simulation method, turning simulations are conducted for a pure car carrier model with variations in the metacentric height \({\overline{\mathrm{GM}}}\) and are compared with free-running model test results. The simulation method exhibits sufficient accuracy with respect to its practical use and is useful to conventionally predict turning motions by considering the roll-coupling effect.

## Keywords

Maneuvering simulation Roll-coupling effect Pure car carrier## List of symbols

- \(A_\mathrm{D}\)
Advance

- \(A_\mathrm{R}\)
Rudder profile area

*a*,*b*Coefficients of the roll-extinction curve

- \(a_\mathrm{H}\)
Rudder force increase factor

- \(a_{24}, a_{26}, a_{46}\)
Added masses coupled among sway, roll, and yaw

*B*Ship breadth

- \(C_\mathrm{b}\)
Block coefficient

- \(c_0, c_1\)
Constants in the proposed formulae

- \(D_\mathrm{P}\)
Propeller diameter

- \(D_\mathrm{T}\)
Tactical diameter

*d*Ship draft

- \(F_\mathrm{N}\)
Rudder normal force

- \(Fn\)
Froude number based on ship length

- \(F_x, F_y\)
Surge force and lateral force acting on the ship, respectively

- \(f_{\alpha }\)
Rudder normal force gradient coefficient

- \({\overline{\mathrm{GM}}}\)
Metacentric height

*g*Gravity acceleration

- \(H_\mathrm{R}\)
Rudder span length

- \(I_{xx}, I_{zz}\)
Moment of inertia of the ship around

*x*- and*z*-axes, respectively- \(J_\mathrm{P}\)
Propeller advance ratio

- \(J_{xx}, J_{zz}\)
Added moment of inertia around

*x*- and*z*-axes, respectively- \({\overline{\mathrm{KM}}}\)
Metacenter height above baseline

- \(K_\mathrm{T}\)
Propeller thrust open water characteristic

- \(K_{{{\dot{\phi }}}}\), \(K_{{{\dot{\phi }}}{{\dot{\phi }}}}\)
Roll-damping coefficients

- \(k_{xx}\)
Radius of roll gyration including added moment of inertia with respect to the roll

- \(k_2, k_1, k_0\)
Coefficients that represent \(K_\mathrm{T}\)

*L*Ship length between perpendiculars

- \(l_\mathrm{P}\)
Longitudinal coordinate of the propeller position in the formula for \(\beta _\mathrm{P}\)

- \(l_\mathrm{R}\)
Effective longitudinal coordinate of the rudder position in the formula for \(\beta _\mathrm{R}\)

- \(M_x, M_z\)
Roll moment and yaw moment acting on ship around the center of gravity, respectively

*m*Ship’s mass

- \(m_x\), \(m_y\)
Added masses of the

*x*-axis direction and*y*-axis direction, respectively- \(n_\mathrm{P}\)
Propeller revolution

- \(o-xyz\)
Horizontal body-fixed coordinate system considering the origin at midship

- \(o_0-x_0y_0z_0\)
Space-fixed coordinate system

- \(R_0'\)
Ship resistance coefficient in straight movement

- \(R^2\)
Coefficient of determination

*r*Yaw rate

*T*Propeller thrust

*t*Time

- \(t_\mathrm{P}\)
Thrust deduction factor

- \(t_\mathrm{R}\)
Steering resistance deduction factor

*U*Resultant speed (\(=\sqrt{u^2+v_\mathrm{m}^2}\))

- \(U_0\)
Approach ship speed

- \(U_\mathrm{R}\)
Resultant inflow velocity to the rudder

*u*,*v*Surge velocity and lateral velocity at the center of gravity, respectively

- \(u_\mathrm{R}\), \(v_{\rm{R}}\)
Longitudinal and lateral inflow velocity components of the rudder, respectively

- \(v_\mathrm{m}\)
Lateral velocity at midship

- \(w_\mathrm{P}\)
Effective wake fraction at the propeller position in maneuvering motions

- \(w_{\mathrm{P}0}\)
Effective wake fraction at the propeller position in straight movement

*X*,*Y*,*N*,*K*Surge force, lateral force, yaw moment, and roll moment with the exception of added mass components, respectively

- \(X_\mathrm{H}\), \(Y_\mathrm{H}\), \(N_\mathrm{H}\), \(K_\mathrm{H}\)
Surge force, lateral force, yaw moment, and roll moment acting on the ship hull with the exception of added mass components, respectively

- \(X_\mathrm{P}\)
Surge force due to the propeller

- \(X_\mathrm{R}\), \(Y_\mathrm{R}\), \(N_\mathrm{R}\), \(K_\mathrm{R}\)
Surge force, lateral force, yaw moment, and roll moment by steering, respectively

- \(x_{G}\)
Longitudinal coordinate of the center of gravity of the ship

- \(x_\mathrm{H}\)
Longitudinal coordinate of the acting point of the additional lateral force component induced by steering

- \(x_\mathrm{P}\)
Longitudinal coordinate of the propeller position

- \(x_\mathrm{R}\)
Longitudinal coordinate of the rudder position (= \(-\,0.5\)

*L*)- \(Y_{v}', N_{v}'\)
Linear hydrodynamic derivatives with respect to the lateral velocity

- \(Y_{r}', N_{r}'\)
Linear hydrodynamic derivatives with respect to the yaw rate

- \(Y_{\phi }', N_{\phi }'\)
Linear hydrodynamic derivatives with respect to the roll

- \(z_{G}\)
Vertical coordinate of the center of gravity of the ship

- \(z_\mathrm{H}\)
Vertical coordinate of the acting point of the hull lateral force

- \(z_\mathrm{P}\)
Vertical coordinate of the propeller position

- \(z_\mathrm{R}\)
Vertical coordinate of the acting point of the rudder force

- \(z_\mathrm{v}\)
Vertical coordinate of the acting point of the hull lateral force in the oblique towing condition

- \(\alpha _\mathrm{R}\)
Effective inflow angle to the rudder

- \(\alpha _\mathrm{z}\)
Vertical acting point of the lateral added mass component \(m_y\)

- \(\beta \)
Hull drift angle at midship

- \(\beta _\mathrm{P}\)
Geometrical inflow angle to the propeller in maneuvering motions

- \(\beta _\mathrm{R}\)
Effective inflow angle to the rudder in maneuvering motions

- \(\gamma _\mathrm{R}\)
Flow-straightening coefficient

- \(\delta \)
Rudder angle

- \(\eta \)
Ratio of the propeller diameter to the rudder span (\(=D_\mathrm{P}/H_\mathrm{R}\))

- \(\phi \)
Roll angle

- \(\phi _\mathrm{S}\)
Steady heel in turning

- \(\kappa \)
Experimental constant for expressing \(u_\mathrm{R}\)

- \(\nabla \)
Displacement volume of the ship

- \(\psi \)
Ship heading

- \(\rho \)
Water density

- \(\varepsilon \)
Ratio of the wake fraction at the propeller and rudder positions

## Notes

### Acknowledgements

This study was supported by JSPS KAKENHI Grant number JP26249135.

## References

- 1.Eda H (1980) Rolling and steering performance of high speed ships. In: Proceedings of the 13th symposium on naval hydrodynamics, Tokyo, pp 427–439Google Scholar
- 2.Hirano M, Takashina J (1980) A calculation of ship turning motion taking coupling effect due to heel into consideration. Trans West Jpn Soc Nav Architects 59:71–81Google Scholar
- 3.Son K, Nomoto K (1982) On the coupled motion of steering and rolling of a high speed container ship. Nav Archit Ocean Eng Soc Nav Architects Jpn 20:73–83Google Scholar
- 4.Fossen TI (1994) Guidance and control of ocean vehicles. Wiley, New YorkGoogle Scholar
- 5.Yasukawa H, Yoshimura Y (2014) Roll-coupling effect on ship maneuverability. Ship Technol Res 61:16–32CrossRefGoogle Scholar
- 6.Kim YG, Kim SY, Kim HT, Lee SW, Yu BS (2007) Prediction of the maneuverability of a large container ship with twin propellers and twin rudders. J Mar Sci Technol 12:130–138CrossRefGoogle Scholar
- 7.Yasukawa H, Yoshimura Y (2015) Introduction of MMG standard method for ship maneuvering prediction. J Mar Sci Technol 20:37–52CrossRefGoogle Scholar
- 8.Wang XG, Zou ZJ, Xu F, Ren RY (2014) Sensitivity analysis and parametric identification for ship manoeuvring in 4 degrees of freedom. J Mar Sci Technol 19:394–405CrossRefGoogle Scholar
- 9.Blanke M, Jensen AG (1997) Dynamic properties of a container vessel with low metacentric height. Trans Inst Meas Control 19(2):78–93CrossRefGoogle Scholar
- 10.Dash AK, Chandran PP, Khan MK, Nagarajan V, Sha OP (2016) Roll-induced bifurcation in ship maneuvering under model uncertainty. J Mar Sci Technol 21:689–708CrossRefGoogle Scholar
- 11.Fukui Y, Yokota H, Yano H, Kondo M, Nakano T, Yoshimura Y (2016) 4-DOF mathematical model for manoeuvring simulation including roll motion. J Jpn Soc Nav Architects Ocean Eng 24:167–179 (in Japanese)CrossRefGoogle Scholar
- 12.Hamamoto M, Kim Y (1993) A new coordinate system and the equations describing manoeuvring motion of a ship in waves. J Soc Nav Architects Jpn 173:209–220 (in Japanese)CrossRefGoogle Scholar
- 13.Masuyama Y, Nakamura I, Tatano H, Takagi K, Miyakawa T (1992) Sailing performance of ocean cruising yacht by full-scale sea test (part 1: steady sailing performance and dynamic performance in waves). J Soc Nav Architects Jpn 172:349–364 (in Japanese)CrossRefGoogle Scholar
- 14.Masuyama Y, Nakamura I, Tatano H, Sakaguchi K, Kanekiyo T (1993) Sailing performance of ocean cruising yacht by full-scale sea test (part 2: maneuverability and tacking performance). J Soc Nav Architects Jpn 174:377–388 (in Japanese)CrossRefGoogle Scholar
- 15.Yoshimura Y (1984) Mathematical model for the manoeuvring ship motion in shallow water. J Kansai Soc Nav Architects Jpn 200:41–51
**(in Japanese)**Google Scholar - 16.Yasukawa H, Hirata N, Yamazaki Y (2018) Effect of bilge keels on maneuverability of a fine ship. J Mar Sci Technol 23:302–318CrossRefGoogle Scholar