### 1. Introduction

### 2. Test facilities and test conditions

### 2.1 Test facilities

### 2.2 Test condition

*λ*= 223. The principal particulars of the real and scaled model are listed as in Table 1.

### 3. Mathematical model and sensitivity analysis

### 3.1 Coordinate system

### 3.2 Equation of motion

*m*is the mass of ship and

*I*is the moment of inertia of ship in yaw motion.

_{zz}*x*is the longitudinal position of ship’s gravity center.

_{G}*u*and

*υ*are the component of the velocity in the x-axis and y-axis direction, respectively.

*X*,

*Y*and

*N*are the hydrodynamic force and yaw moment around z-axis. These forces can be described separating into the three components by Eq. (2).

### 3.3 Hull forces

*m*,

_{x}*m*and

_{y}*J*are the added mass of

_{zz}*x*axis direction,

*y*axis direction, and added moment of inertia, respectively. Prime (′) inserted to the symbol refers to non-dimensional value. Non-dimensional of added mass and added moment of inertia can be estimated using Eq. (4). Non-dimensional sway velocity

*υ*′, non-dimensional yaw rate

*r*′ and resultant velocity

*U*can be estimated using Eq. (5).

### 3.4 Propeller forces

*X*can be estimated using Eq. (6).

_{P}*t*,

_{P}*ρ*and

*D*are the thrust deduction factor, water density, and the diameter of propeller, respectively. Thurst deduction factor

_{P}*t*is assumed to be constant at given propeller load for simplicity.

_{P}*K*is the thurst coefficient and can be expressed as a 3

_{T}^{rd}polynomial of propeller advance ratio

*J*using Eq. (7),

_{P}*k*

_{0},

*k*

_{1},

*k*

_{2}, and

*k*

_{3}are coefficients representing

*K*.

_{T}### 3.5 Rudder forces

*F*represents the rudder normal force,

_{N}*t*represents the steering resistance deduction factor, and

_{R}*a*is the rudder force increase factor, representing the additional lateral force acting on ship via steering.

_{H}*x*is the longitudinal position of rudder, and

_{R}*x*is the longitudinal acting point of the additional lateral force component.

_{H}*A*represent the rudder area,

_{R}*Λ*represents the rudder aspect ratio, while,

*υ*, and

_{R}*u*are the lateral and longitudinal inflow velocity induced by propeller rotation to the rudder, respectively. Resultant inflow velocity to rudder

_{R}*U*can be estimated using Eq. (10).

_{R}*f*represents the rudder lift gradient coefficient.

_{α}*η*denotes the ratio of the propeller diameter to rudder span.

*κ*represents the interaction between propeller and rudder.

*∈*represents a ration of wake fraction at rudder position to that of the propeller position.

*α*denotes the effective inflow angle to a rudder.

_{R}*γ*is flow straightening coefficient,

_{R}*β*represents the effective inflow angle to rudder in maneuvering motion.

_{R}*w*represents the wake coefficient at propeller position in maneuvering motion.

_{P}*l*′

_{R}is treated as an experimental constant for expressing

*υ*accurately.

_{R}### 3.6 Sensitivity analysis

*S*parameter is estimated using Eq. (11).

_{ijk}*H*

^{*}denotes the experimental value of hydrodynamic derivatives.

*H*represents the deviated value of hydrodynamic derivative from experimentally derived value.

*R*

^{*}denotes the value of the corresponding maneuvering parameters obtained from standard maneuvering tests such as turning circle and zig-zag.

*R*represents the corresponding values of maneuvering parameters obtained from the maneuvering tests performed using

*H*.

*S*is the sensitivity index for at the

_{ijk}*i*standard maneuvering parameter for

^{th}*k*% change in the

*j*hydrodynamic derivative. The influence of hydrodynamic derivatives on the standard maneuvering test parameters like tactical diameter, transfer in the case of turning circle and first overshoot angle and time to reach 1

^{th}^{st}over shoot angle in the case of zig-zag test are considered.

### 4. Experiment

### 4.1 Experimental setup

### 4.2 Hydrodynamic forces acting on a ship hull

### 4.3 Hydrodynamic coefficients from model test

### 5. Result and Discussion

### 5.1 Comparison with FRMT

^{o}/10

^{o}zig-zag and 20

^{o}/5

^{o}zig-zag are compared with the result of Free Running Model Test(FRMT), which were conducted on the false bottom of KRISO’s towing tank. The result of time histories of heading angle and rudder angle for 20

^{o}/10

^{o}zig-zag maneuvers are compared with the result of Free Running Model Test(FRMT) by Yeo et al.(2016) as shown in Figs. 12～15. There is a good agreement between Free Running Model Test(FRMT) and simulation obtained by hydrodynamic coefficients from model test conducted in a precise bottom tank in CWNU. As seen in Figs. 12～15, the predicted heading angle and yaw rate of the ship show a similar trend and have the same value as those measured using FRMT by Yeo et al.(2016). The time to 1st Overshoot Angle of FRMT of port and starboard has a little difference, it may be as a result of the starting time of FRMT. The result of 1st Overshoot Angle (OSA) and time to 1st OSA are listed in Table 5.

### 5.2 Turning test

*Y*′

_{υ}and

*Y*′

_{r}. Hydrodynamic coefficient

*N*′

_{υ}and

*N*′

_{r}were found to be sensitive at all water depths and

*N*′

_{r}has the highest impact on tactical diameter and advance.

### 5.3 Zig-Zag test

*N*′

_{r},

*N*′

_{υ}and sway derivatives,

*Y*′

_{υ}and

*Y*′

_{r}. Sensitivity of 1st OSA is the highest in very shallow water, hydrodynamic coefficient

*N*′

_{υ}and

*N*′

_{r}were identified to be sensitive at all water depths,

*N*′

_{υ}has highest impact on 1st OSA. In contrast, the hydrodynamic derivatives that commonly affect time to 1st OSA are the yaw derivatives,

*N*′

_{r},

*N*′

_{υ}and sway derivatives,

*Y*′

_{υ}and

*Y*′

_{r}. Sensitivity of time to 1st OSA is highest in shallow water (h/T=1.5), hydrodynamic coefficient

*N*′

_{υ}and

*N*′

_{r}were identified to be sensitive at all water depths, it has the highest impact on time to 1st OSA.

*N*′

_{υ}and

*N*′

_{r}were identified to be sensitive at all water depths, it has highest impact on 2nd OSA. Conversely, the hydrodynamic derivatives that commonly affect time to 2nd OSA are the yaw derivatives,

*N*′

_{r},

*N*′

_{υ}and sway derivatives,

*Y*′

_{υ}and

*Y*′

_{r}. Sensitivity of time to 2nd OSA is highest in shallow water (h/T = 1.5), hydrodynamic coefficient

*N*′

_{υ}and

*N*′

_{r}were identified to be sensitive at all water depths, and

*N*′

_{υ}has the highest impact on time to 2nd OSA.

*N*′

_{r},

*N*′

_{υ}and sway derivatives,

*Y*′

_{υ}and

*Y*′

_{r}. Sensitivity of 1st OSA is highest in very shallow water (h/T=1.2), hydrodynamic coefficients

*N*′

_{υ}and

*N*′

_{r}were identified to be sensitive at all water depths, it has the highest impact on time to 1st OSA. Conversely, the hydrodynamic derivatives that commonly affect time to 1st OSA are the yaw derivatives,

*N*′

_{r},

*N*′

_{υ}and sway derivatives,

*Y*′

_{υ}and

*Y*′

_{r}. Sensitivity of time to 1st OSA is highest in shallow water (h/T=1.5), hydrodynamic coefficient

*N*′

_{υ}and

*N*′

_{r}were identified to be sensitive at all water depths.

### 6. Conclusion

*Y*′

_{υ}and

*Y*′

_{r}. Conversely, hydrodynamic coefficient

*N*′

_{υ}and

*N*′

_{r}were identified sensitive at all water depths, while

*N*′

_{r}has biggest impact on tactical diameter and advance. However, hydrodynamic coefficients

*N*′

_{υ}and

*N*′

_{r}were identified sensitive at all water depths, while

*N*′

_{υ}has the highest impact on the overshoot angle and time to overshoot angle in zig-zag maneuvers.