# Application of the MMG method for the prediction of steady sailing condition and course stability of a ship under external disturbances

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## Abstract

For the navigation safety of ships, it is essential to monitor the maneuvering characteristics under external disturbances due to wind and waves. For this purpose, evaluating the average steady sailing conditions such as check helm, speed drop, hull drift angle, etc. of a ship moving straight in steady wind and waves is beneficial. In addition, the dynamic stability (course stability) of the ship should be evaluated under its steady sailing condition. This study proposes a method for predicting the steady sailing conditions and course stability under external disturbances, based on the MMG standard method presented by Yasukawa and Yoshimura (J Mar Sci Technol 20(1):37–52, 2015). The calculation accuracy of the MMG method has been validated through experiments. The steady sailing conditions and the course stability of a pure car carrier are calculated using the proposed method under external disturbances in deep and shallow waters. In addition, the environmental conditions that limit safe navigation (maneuvering limit) are also discussed, while investigating the effect of the main engine output in particular. In both deep and shallow waters, a significant effect on the maneuvering limit is observed due to a reduction in engine output. Thus, the presented method is useful in capturing the maneuvering limit of the ship under external disturbances.

## Keywords

Steady sailing condition Course stability Maneuvering limit MMG method Pure car carrier Adverse weather condition Main engine output## List of symbols

- \(A_X\), \(A_Y\)
Front and side profile areas of the ship in air, respectively

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

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

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

*B*Ship breadth

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

- \(C_{{\text {XA}}}\), \(C_{{\text {YA}}}\), \(C_{\text {NA}}\), \(C_{\text {KA}}\)
Aerodynamic force coefficients with respect to surge force, lateral force, yaw moment and roll moment, respectively

- \(C_{{\text {XW}}}\), \(C_{{\text {YW}}}\), \(C_{{\text {NW}}}\), \(C_{{\text {KW}}}\)
Wave-induced steady force coefficients with respect to surge force, lateral force, yaw moment and roll moment in regular waves, respectively

- \(\overline{C_{{\text {XW}}}}\), \(\overline{C_{{\text {YW}}}}\), \(\overline{C_{{\text {NW}}}}\), \(\overline{C_{{\text {KW}}}}\)
Average wave-induced steady force coefficients with respect to surge force, lateral force, yaw moment and roll moment in irregular waves, respectively

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

*d*Ship draft

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

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

- \(f_{\text {A}}\)
Correction coefficient when ship heels

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

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

- \(G(\theta )\)
Wave direction distribution function

- \(G_1, G_2\)
Control gains for autopilot

*g*Gravity acceleration

- \(H_{1/3}\)
Significant wave height

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

*h*Water depth

- \(I_{xx}, I_{zz}\)
Moment of inertia of the ship around \(x\)- and \(z\)-axes, respectively

- \(J_{{\text {P}}}\)
Propeller advance ratio

- \(J_{{\text {P}}0}\)
Propeller advance ratio during forward motion

- \(J_{xx}, J_{zz}\)
Added moment of inertia around \(x\)- and \(z\)-axes, respectively

- \(\overline{KM}\)
Metacenter height above baseline

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

- \(K_{\text{Q}}\)
Propeller torque 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 \(K_{\text {T}}\)

*L*Ship length between perpendiculars

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

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

*m*Ship’s mass

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

*x*-axis direction and*y*-axis direction, respectively- \(N_{{\text {MCR}}}\)
Propeller revolution at Maximum Continuous Rating (MCR)

- \(N_{{\text {NOR}}}\)
Propeller revolution at Normal Rating (NOR)

- \(N_v'\), \(N_{\text {r}}'\), \(N_{\phi }'\), \(N_{vvv}'\) etc.
Hydrodynamic derivatives with respect to yaw moment

- \(n_{{\text {P}}}\)
Propeller revolution

*O*-\(x_0y_0z_0\)Space fixed coordinate system

*o*-*xyz*Horizontal body fixed coordinate system considering the origin at midship

- \(P_{{\text {E}}}\)
Effective power

- \(P_{{\text {MCR}}}\)
Engine power at MCR

- \(P_{{\text {NOR}}}\)
Engine power at NOR

*Q*Propeller torque

- \(Q_{{\text {EMAX}}}\)
Maximum propeller torque of the main engines

- \(q_2, q_1, q_0\)
Coefficients that represent \(K_{\text {Q}}\)

- \(R_0\)
Ship resistance in straight moving

*r*Yaw rate

- \(S_{\zeta \zeta }\)
Wave spectrum

*T*Propeller thrust

- \(T_{{\text {P}}}\)
Average wave period

*t*Time

- \(t_{{\text {P}}}\)
Thrust deduction fraction

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

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

- \(U_0\)
Approach ship speed

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

- \(U_{\text {W}}\)
Absolute wind velocity

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

- \(u_0\), \(v_0\), \(\psi _0\), \(\phi _0\), \(\delta _0\)
Steady components of surge velocity, lateral velocity, heading angle, heel angle and rudder angle, respectively

- \(u_{\text {A}}\), \(v_{\text {A}}\)
Relative surge velocity and lateral velocity component due to wind, respectively

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

- \(V_{\text {A}}\)
Relative wind velocity

- \(V_{\text {S}}\)
Design speed of the ship

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

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

- \(w_{{\text {P}}0}\)
Effective wake fraction at the propeller position during forward motion

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

- \(X_{\text {A}}\), \(Y_{\text {A}}\), \(N_{\text {A}}\), \(K_{\text {A}}\)
Surge force, lateral force, yaw moment, and roll moment due to wind, respectively

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

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

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

- \(X_{\text {W}}\), \(Y_{\text {W}}\), \(N_{\text {W}}\), \(K_{\text {W}}\)
Wave-induced steady surge force, lateral force, yaw moment, and roll moment, respectively

- \(X_{vv}'\), \(X_{rr}'\), \(X_{\phi \phi }'\), \(X_{vr}'\) etc.
Hydrodynamic derivatives with respect to surge force

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

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

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

- \(Y_v'\), \(Y_{\text {r}}'\), \(Y_{\phi }'\), \(Y_{vvv}'\), etc.
Hydrodynamic derivatives with respect to lateral force

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

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

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

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

- \(z_{\text {W}}\)
Vertical coordinate of the acting point of wave-induced lateral force

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

- \(\alpha _z\)
Vertical acting point of the lateral added mass component \(m_y\)

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

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

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

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

- \(\varDelta u\), \(\varDelta v\), \(\varDelta \psi\), \(\varDelta \phi\), \(\varDelta \delta\)
Unsteady components of surge velocity, lateral velocity, heading angle, heel angle and rudder angle, respectively

- \(\delta\)
Rudder angle

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

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

- \(\eta _{\text {R}}\)
Relative rotative efficiency

- \(\theta _{\text {A}}\)
Relative wind direction

- \(\theta _{\text {W}}\)
Absolute wind direction

- \(\kappa\)
Experimental constant for expressing

- \(\rho\)
Water density

- \(\rho _{\text {a}}\)
Air density

- \(\phi\)
Roll angle

- \(\chi\)
Absolute wave direction

- \(\chi _0\)
Relative wave direction

- \(\psi\)
Ship heading

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

## Notes

### Acknowledgements

This study was supported by JSPS KAKENHI Grant number JP26249135. The authors express their sincere gratitude to Ms. A. Nishiyama for her assistance with the calculations of the SSC and the CS under external disturbances.

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