# Introduction of MMG standard method for ship maneuvering predictions

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## Abstract

A lot of simulation methods based on Maneuvering Modeling Group (MMG) model for ship maneuvering have been presented. Many simulation methods sometimes harm the adaptability of hydrodynamic force data for the maneuvering simulations since one method may be not applicable to other method in general. To avoid this, basic part of the method should be common. Under such a background, research committee on “standardization of mathematical model for ship maneuvering predictions” was organized by the Japan Society of Naval Architects and Ocean Engineers and proposed a prototype of maneuvering prediction method for ships, called “MMG standard method”. In this article, the MMG standard method is introduced. The MMG standard method is composed of 4 elements; maneuvering simulation model, procedure of the required captive model tests to capture the hydrodynamic force characteristics, analysis method for determining the hydrodynamic force coefficients for maneuvering simulations, and prediction method for maneuvering motions of a ship in fullscale. KVLCC2 tanker is selected as a sample ship and the captive mode test results are presented with a process of the data analysis. Using the hydrodynamic force coefficients presented, maneuvering simulations are carried out for KVLCC2 model and the fullscale ship for validation of the method. The present method can roughly capture the maneuvering motions and is useful for the maneuvering predictions in fullscale.

### Keywords

MMG standard method MMG model Maneuvering prediction KVLCC2 Captive model tests### List of symbols

- \(A_D\)
Advance

- \(A_{\rm R}\)
Profile area of movable part of mariner rudder

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

- \(B\)
Ship breadth

- \(B_{\rm R}\)
Averaged rudder chord length

- \(C_b\)
Block coefficient

- \(C_1, C_2\)
Experimental constants representing wake characteristic in maneuvering

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

- \(D_T\)
Tactical diameter

- \(d\)
Ship draft

- \(F_N\)
Rudder normal force

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

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

- \(f_{\alpha }\)
Rudder lift gradient coefficient

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

- \(I_{zG}\)
Moment of inertia of ship around center of gravity

- \(J_{\rm P}\)
Propeller advanced ratio

- \(J_z\)
Added moment of inertia

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

- \(k_2, k_1, k_0\)
Coefficients representing \(K_T\)

- \(L_{pp}\)
Ship length between perpendiculars

- \(\ell _{\rm R}\)
Effective longitudinal coordinate of rudder position in formula of \(\beta _{\rm R}\)

- \(M_z\)
Yaw moment acting on ship around center of gravity

- \(m\)
Ship’s mass

- \(m_x\), \(m_y\)
Added masses of \(x\) axis direction and \(y\) axis direction, respectively

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

- \(o-xyz\)
Ship fixed coordinate system taking the origin at midship

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

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

- \(r\)
Yaw rate

- \(T\)
Propeller thrust

- \(t\)
Time

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

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

- \(U\)
Resultant speed (\(=\sqrt{u^2+v_m^2}\))

- \(U_0\)
Approach ship speed (given speed)

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

- \(u\), \(v\)
Surge velocity, lateral velocity at center of gravity, respectively

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

- \(v_m\)
Lateral velocity at midship

- \(w_{\rm P}\)
Wake coefficient at propeller position in maneuvering motions

- \(w_{P0}\)
Wake coefficient at propeller position in straight moving

- \(w_{\rm R}\)
Wake coefficient at rudder position

- \(X\), \(Y\), \(N_m\)
Surge force, lateral force, yaw moment around midship except added mass components

- \(X_{H}\), \(Y_{H}\), \(N_{H}\)
Surge force, lateral force, yaw moment around midship acting on ship hull except added mass components \(X_{\rm P}\) Surge force due to propeller

- \(X_{R}\), \(Y_{R}\), \(N_{R}\)
Surge force, lateral force, yaw moment around midship by steering

- \(X_{\rm mes}\), \(Y_{\rm mes}\), \(N_{\rm mes}\)
Surge force, lateral force, yaw moment around midship measured in CMT

- \(x_G\)
Longitudinal coordinate of center of gravity of ship

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

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

- \(x_{\rm R}\)
Longitudinal coordinate of rudder position (=\(-0.5L_{pp}\))

- \(Y_v', N_v'\)
Linear hydrodynamic derivatives with respect to lateral velocity

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

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

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

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

- \(\beta _{R0}\)
Geometrical inflow angle to rudder in maneuvering motions

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

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

- \(\delta\)
Rudder angle

- \(\delta _{FN0}\)
Rudder angle where rudder normal force becomes zero

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

- \(\Lambda\)
Rudder aspect ratio

- \(\kappa\)
An experimental constant for expressing \(u_{\rm R}\)

- \(\nabla\)
Displacement volume of ship

- \(\psi\)
Ship heading

- \(\rho\)
Water density

- \(\varepsilon\)
Ratio of wake fraction at propeller and rudder positions (\(=(1-w_{\rm R})/(1-w_{\rm P})\))

## 1 Introduction

MMG model is one of the solutions for ship maneuvering motion simulations developed in Japan. The model was proposed by a research group called Maneuvering Modeling Group (MMG) in Japanese Towing Tank Conference (JTTC), and the outline was reported in the Bulletin of Society of Naval Architects of Japan [1] in 1977. In the report, the concept for maneuvering simulations was mainly described, but concrete simulation model was not described in detail. According to MMG model concept, afterward, concrete methods including expression of hydrodynamic forces acting on ships were presented by Ogawa and Kasai [2], Matsumoto and Suemitsu [3], Inoue et al. [4] and so on. Nowadays, a lot of simulation methods based on MMG model are existing.

Many simulation methods sometimes harm the adaptability of hydrodynamic force data for the maneuvering simulations since one method may be not applicable to other method in general. To avoid this, basic part of the method should be common. The test procedure and the data analysis to determine the hydrodynamic force coefficients for the simulations should be also common since those often involve the quantitative value of the hydrodynamic coefficients.

Under such a background, the research committee on “standardization of mathematical model for ship maneuvering predictions” organized by the Japan Society of Naval Architects and Ocean Engineers has checked the details of existing MMG models such as the coordinate system, the motion equations, the hull and rudder hydrodynamic force models etc., in view of accuracy, simplicity, physical/theoretical background and adoptability to the captive model tests for capturing the hydrodynamic force characteristics. As the conclusion, a prototype of maneuvering simulation method for ships called “MMG standard method”, has been proposed [5].

maneuvering simulation model,

procedure of the required captive model tests to capture the hydrodynamic force characteristics,

analysis method for determining the hydrodynamic force coefficients for maneuvering simulations, and

prediction method for maneuvering motions of a ship in fullscale.

## 2 Maneuvering simulation model

First, the motion equations to express the maneuvering motions for a ship with single propeller and single rudder, and the simulation model of hydrodynamic forces acting on the ship are described.

In this article, prime \('\) putting to the symbol means non-dimensionalized value. Force and moment are non-dimensionalized by \((1/2)\rho L_{pp} dU^2\) and \((1/2)\rho L_{pp}^2 dU^2\), respectively. In addition, mass and moment of inertia are non-dimensionalized by \((1/2)\rho L_{pp}^2 d\) and \((1/2)\rho L_{pp}^4 d\), respectively. Velocity component is non-dimensionalized by \(U\) and length component is by \(L_{pp}\).

### 2.1 Assumptions and coordinate systems

Ship is a rigid body.

Hydrodynamic forces acting on the ship are treated quasi-steadily.

Lateral velocity component is small compared with longitudinal velocity component.

Ship speed is not fast that wave-making effect can be neglected.

Metacentric height \(\overline{GM}\) is sufficiently large, and the roll coupling effect on maneuvering is negligible.

One of the special feature of the present model is the use of the coordinate system fixed to the midship position. This may be convenient when considering the captive model tests with different load conditions like full and ballast loads. When employing the origin of the center of gravity, for instance, the coordinate of the rudder/propeller position changes in full and ballast load conditions since the longitudinal position of the center of gravity generally changes in different load conditions. Employing the midship-based coordinate system can avoid such the troublesome.

### 2.2 Motion equations

### 2.3 Hydrodynamic forces acting on ship hull

### 2.4 Hydrodynamic force due to propeller

In the expression of \(X_{\rm P}\), the steering effect on the propeller thrust \(T\) is excluded. Instead of this, the effect is taken into account at the rudder force component \(X_{\rm R}\) as shown in the next section.

### 2.5 Hydrodynamic forces by steering

The \(\gamma _{\rm R}\) characteristic considerably affects the maneuvering simulation, so we have to capture it correctly. Value of \(\gamma _{\rm R}\) generally takes different magnitude for port and starboard turning and this is one of the reasons for asymmetrical turning motions in port and starboard. The flow straightening effect was pointed out by Fujii and Tuda [13] first, and after that a form of Eq. 23 was proposed by Kose et al. [9].

## 3 Captive model test and the results

In this section, outline of captive model tests is described to capture the hydrodynamic force characteristics. As an example, the experimental data opened in SIMMAN2008 workshop [6] for KVLCC2 model is introduced.

### 3.1 A sample ship: KVLCC2

Principal particulars of a KVLCC2 tanker

L3-model | L7-model | Fullscale | |
---|---|---|---|

Scale | 1/110 | 1/45.7 | 1.00 |

\(L_{pp}\) (m) | 2.902 | 7.00 | 320.0 |

\(B\) (m) | 0.527 | 1.27 | 58.0 |

\(d\) (m) | 0.189 | 0.46 | 20.8 |

\(\nabla\) (m\(^3\)) | 0.235 | 3.27 | 312,600 |

\(x_G\) (m) | 0.102 | 0.25 | 11.2 |

\(C_b\) | 0.810 | 0.810 | 0.810 |

\(D_{\rm P}\) (m) | 0.090 | 0.216 | 9.86 |

\(H_{\rm R}\) (m) | 0.144 | 0.345 | 15.80 |

\(A_{\rm R}\) (m\(^2\)) | 0.00928 | 0.0539 | 112.5 |

### 3.2 Outline of captive model tests

#### 3.2.1 Kind of tests

The captive tests were carried out at propelled condition of a ship model with a rudder model. Ship speed \(U_0\) was set at 0.76 m/s (equivalent to 15.5 kn in fullscale). As the propeller loading point the model point was selected in principle.

- 1.
Rudder force test in straight moving under various propeller loads.

- 2.
Oblique towing test (OTT) and circular motion test (CMT).

- 3.
Rudder force test in oblique towing and steady turning conditions (flow straightening coefficient test).

OTT and CMT are the test to measure the hydrodynamic forces acting on the ship model in oblique moving and/or steady turning. Then, the rudder angle should be zero. From the tests, the hydrodynamic forces acting on the ship and the wake fractions at propeller position in maneuvering motions can be obtained. Planar motion technique (PMM) test is widely used as a method to capture the hydrodynamic derivatives on turning. The hydrodynamic derivatives obtained by PMM test remarkably change due to influence of the motion frequency and the motion amplitude given in the test and it is difficult to select the proper values for the maneuvering simulations. To avoid the uncertainty, CMT was employed here instead of PMM test.

The flow straightening coefficient test is the test to capture the rudder angle where the normal force becomes zero (\(\delta _{FN0}\)) and the inclination of the normal force coefficient versus rudder angle at \(\delta _{FN0}\) in oblique moving and/or steady turning (\(dF_N'/d\delta\)). The flow straightening coefficient (\(\gamma _{\rm R}\)) is determined from the results of \(\delta _{FN0}\) and \(dF_N'/d\delta\).

All the tests were carried out in the free condition for trim and sinkage of the model.

#### 3.2.2 Measurement items

Surge force, lateral force and yaw moment around midship acting on the ship model (\(X\), \(Y\), \(N_m\)),

rudder normal force (\(F_{N}\)),

propeller thrust (\(T\)).

### 3.3 Test results

#### 3.3.1 Rudder force test results in straight moving

#### 3.3.2 OTT and CMT results

#### 3.3.3 Flow straightening coefficient test results

- 1.
Rudder normal forces are measured with changing 3 rudder angles. These 3 rudder angles have to be selected appropriately so as the rudder angle at zero normal force can be determined.

- 2.
\(\delta _{FN0}\) is determined by an interpolation based on 3 measured rudder normal force results versus \(\delta\).

- 3.
\(dF_{N}'/d\delta\) is numerically calculated by taking an inclination of the rudder normal force coefficient versus \(\delta\).

## 4 Determination of hydrodynamic force coefficients

Next, analysis methods are described to determine the hydrodynamic force coefficients defined in the simulation model referring to Ref. [5].

### 4.1 \(t_{\rm R}\), \(a_{\rm H}\) and \(x_{\rm H}'\)

(\(1-t_{\rm R}\)) is determined as an inclination of \(X'\) versus \(-F_{N}'\sin \delta\). Note that \(R_0'\) and \((1-t_{\rm P})T'\) are not related to the rudder angle \(\delta\) in the simulation model.

(\(1+a_{\rm H}\)) is determined as an inclination of \(Y'\) versus \(-F_{N}'\cos \delta\).

(\(x_{\rm R}'+a_{\rm H}x_{\rm H}'\)) is determined as an inclination of \(N_m'\) versus \(-F_{N}'\cos \delta\). Then, \(x_{\rm H}'\) can be calculated since \(x_{\rm R}'\) is \(-0.5\) and \(a_{\rm H}\) is known.

### 4.2 Hydrodynamic derivatives on maneuvering

Resistance coefficient and hydrodynamic derivatives on maneuvering

\(R_0'\) | 0.022 | \(Y_v'\) | −0.315 | \(N_v'\) | −0.137 |

\(X_{vv}'\) | −0.040 | \(Y_{\rm R}'-m'-m_x'\) | −0.233 | \(N_{\rm R}'-x_G'm'\) | −0.059 |

\(X_{vr}'+m'+m_y'\) | 0.518 | \(Y_{vvv}'\) | −1.607 | \(N_{vvv}'\) | −0.030 |

\(X_{rr}'+x_G'm'\) | 0.021 | \(Y_{vvr}'\) | 0.379 | \(N_{vvr}'\) | −0.294 |

\(X_{vvvv}'\) | 0.771 | \(Y_{vrr}'\) | −0.391 | \(N_{vrr}'\) | 0.055 |

\(Y_{rrr}'\) | 0.008 | \(N_{rrr}'\) | −0.013 |

### 4.3 \(w_{\rm P}\)

### 4.4 \(\gamma _{\rm R}\) and \(\ell _{\rm R}'\)

### 4.5 \(\kappa\) and \(\varepsilon\)

## 5 Maneuvering simulations

### 5.1 Details of simulations

Hull resistance was calculated by a 3-dimensional extrapolation method based on Schoenherr’s frictional resistance coefficient formula.

Parameters of propeller thrust open water characteristic were as follows: \((k_{0}, k_{1}, k_{2})=(0.2931, -0.2753, -0.1385)\).

Effective wake in straight moving \(w_{\rm P0}\) was assumed to be 0.40 for L7-model and 0.35 for fullscale.

Added mass coefficients (\(m_x'\), \(m_y'\), \(J_z'\)) listed in Table 3 were estimated by Motora’s empirical charts [16, 17, 18].

- Rudder lift gradient coefficient \(f_\alpha\) was estimated using Fujii’s formula expressed as [13]:This formula can be regarded as a modified version of Prandtl’s formula based on the lifting line theory. Here, \(\Lambda\) is aspect ratio of a rudder including the horn part. Hirano et al. [15] proposed a practical treatment when applying Eq. 38 to Mariner rudder: a whole rudder with the horn part is used for determining \(f_\alpha\) and a movable part area is used as a representative rudder area. Values of \(f_\alpha\) and \(A_{\rm R}\) were determined by this treatment.$$\begin{aligned} f_{\alpha }&= \frac{6.13\Lambda }{\Lambda +2.25} \end{aligned}$$(38)
In the simulations, we set that an initial approach speed \(U_0\) is 15.5 kn in fullscale, the rudder steering rate is \(1.76\,^{\circ }/s\) in fullscale, and the radius of yaw gyration is 0.25\(L_{pp}\). Propeller revolution is assumed to be kept the revolution at \(U_0\) constant without torque rich.

Hydrodynamic force coefficients used in the simulations

\(X_{vv}'\) | −0.040 | \(m_x'\) | 0.022 |

\(X_{vr}'\) | 0.002 | \(m_y'\) | 0.223 |

\(X_{rr}'\) | 0.011 | \(J_z'\) | 0.011 |

\(X_{vvvv}'\) | 0.771 | \(t_{\rm P}\) | 0.220 |

\(Y_v'\) | −0.315 | \(t_{\rm R}\) | 0.387 |

\(Y_{\rm R}'\) | 0.083 | \(a_{\rm H}\) | 0.312 |

\(Y_{vvv}'\) | −1.607 | \(x_{\rm H}'\) | −0.464 |

\(Y_{vvr}'\) | 0.379 | \(C_1\) | 2.0 |

\(Y_{vrr}'\) | −0.391 | \(C_2\) (\(\beta _{\rm P}>0\)) | 1.6 |

\(Y_{rrr}'\) | 0.008 | \(C_2\) (\(\beta _{\rm P}<0\)) | 1.1 |

\(N_v'\) | −0.137 | \(\gamma _{\rm R}\) (\(\beta _{\rm R}<0\)) | 0.395 |

\(N_{\rm R}'\) | −0.049 | \(\gamma _{\rm R}\) (\(\beta _{\rm R}>0\)) | 0.640 |

\(N_{vvv}'\) | −0.030 | \(\ell _{\rm R}'\) | −0.710 |

\(N_{vvr}'\) | −0.294 | \(\varepsilon\) | 1.09 |

\(N_{vrr}'\) | 0.055 | \(\kappa\) | 0.50 |

\(N_{rrr}'\) | −0.013 | \(f_\alpha\) | 2.747 |

### 5.2 Comparison with free-running model test results

Comparison of turning indices

Cal. | Exp. | |
---|---|---|

\(A_D'\) (\(\delta =35^{\circ }\)) | 3.31 | 3.25 |

\(D_T'\) (\(\delta =35^{\circ }\)) | 3.36 | 3.34 |

\(A_D'\) (\(\delta =-35^{\circ }\)) | 3.26 | 3.11 |

\(D_T'\) (\(\delta =-35^{\circ }\)) | 3.26 | 3.08 |

Comparison of overshoot angles of zig-zag maneuvers (L7-model)

Cal. (\(^{\circ }\)) | Exp. (\(^{\circ }\)) | |
---|---|---|

1st OSA (10/10Z) | 5.2 | 8.2 |

2nd OSA (10/10Z) | 15.8 | 21.9 |

1st OSA (20/20Z) | 10.9 | 13.7 |

1st OSA (−10/−10Z) | 7.6 | 9.5 |

2nd OSA (−10/−10Z) | 10.2 | 15.0 |

1st OSA (−20/−20Z) | 14.5 | 15.1 |

### 5.3 Simulation results in fullscale

Simulation results of turning indices

L7-model | fullscale | |
---|---|---|

\(A_D' \, (\delta =35^{\circ })\) | 3.31 | 3.62 |

\(D_T'\,(\delta =35^{\circ })\) | 3.36 | 3.71 |

\(A_D'\,(\delta =-35^{\circ })\) | 3.26 | 3.56 |

\(D_T'\,(\delta =-35^{\circ })\) | 3.26 | 3.59 |

Simulation results of overshoot angles of zig-zag maneuvers

L7-mode (\(^{\circ }\)) | Fullscale (\(^{\circ }\)) | |
---|---|---|

1st OSA (10/10Z) | 5.2 | 5.8 |

2nd OSA (10/10Z) | 15.8 | 20.5 |

1st OSA (20/20Z) | 10.9 | 11.8 |

1st OSA (\(-10\)/\(-10\)Z) | 7.6 | 8.8 |

2nd OSA (\(-10\)/\(-10\)Z) | 10.2 | 12.6 |

1st OSA (\(-20\)/\(-20\)Z) | 14.5 | 16.1 |

## 6 Concluding remarks

In this article, a prototype of maneuvering prediction method for ships, called ”MMG standard method”, was introduced. The MMG standard method was composed of 4 elements: the maneuvering simulation model, the procedure of the required captive model tests to capture the hydrodynamic force characteristics, the analysis method for determining the hydrodynamic force coefficients for maneuvering simulations, and the prediction method for maneuvering motions in fullscale. KVLCC2 tanker was selected as a sample ship and the captive mode test results were presented with a process of the data analysis. Using the hydrodynamic force coefficients obtained, maneuvering simulations were carried out for KVLCC2 model [8] and the fullscale ship for validation of the method. It was confirmed that the present method can roughly capture the maneuvering motions and is useful for the maneuvering predictions in fullscale.

Collecting the hydrodynamic force coefficients determined by the MMG standard method in various ship kinds is the next work to make a useful data base of the force coefficients for ship maneuvering predictions.

## Notes

### Acknowledgments

We would like to express our thanks to committee members of ”Research committee on standardization of mathematical model for ship maneuvering predictions” organized by the Japan Society of Naval Architects and Ocean Engineers. The experimental data analysis presented in this article was carried out by Mr. S. Ito as a part of his master course study. We would like to extend our thanks to him.

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