Circular motion tests and uncertainty analysis for ship maneuverability

DOI 1.17/s773-9-65-2 ORIGINAL ARTICLE Circular motion tests and uncertainty analysis for ship maneuverability Michio Ueno Æ Yasuo Yoshimura Æ Yoshiaki Tsukada Æ Hideki Miyazaki Received: 13 October 2 / Accepted: 11 July 29 Ó JASNAOE 29 Abstract Circular motion test data and uncertainty analysis results of investigations of the hydrodynamic characteristics of ship maneuvering are presented. The model ships used were a container ship and two tankers, and the measured items were the surge and sway forces, yaw moment, propeller thrust, rudder normal and tangential forces, pitch and roll angles, and heave. The test parameters were the oblique angle and yaw rate for the conditions of a hull with a rudder and propeller in which the rudder angle was set to zero and the propeller speed was set to the model self-propulsion conditions. Carriage data showing the accuracy of the towing conditions in the circular motion test are also presented. It was confirmed that the uncertainties in the hydrodynamic forces such as the surge and sway forces, yaw moment, rudder tangential and normal forces, and propeller thrust were fairly small. The reported uncertainty analysis results of the circular motion test data may be beneficial in validating data quality and in discussing reliability for simulation of ship maneuvering performance. Keywords Uncertainty analysis Circular motion test Ship maneuverability Hydrodynamic coefficients Oblique motion test M. Ueno (&) Y. Tsukada H. Miyazaki Marine Dynamics Group, National Maritime Research Institute, 6-3-1 Shinkawa, Mitaka, Tokyo 11, Japan e-mail: ueno@nmri.go.jp Y. Yoshimura Graduate School of Fisheries Sciences, Hokkaido University, 3-1-1 Minato-cho, Hakodate, Hokkaido 41-611, Japan 1 Introduction From a viewpoint of maritime safety and global environmental protection against marine accidents, the maneuvering standards [1, 2] of the International Maritime Organization are widely applied. In order to confirm whether or not a ship would comply with the maneuvering standards in its design phase, the maneuverability of the ship must be estimated reliably. One of the most reliable methods for estimating ship maneuverability is to simulate the maneuvering motion of ships using a mathematical model with hydrodynamic derivatives determined from tank test data. The circular motion test (CMT) [3] and/or the planar motion mechanism (PMM) test [3] are widely employed in the field of ship maneuverability. The PMM test uses model ship sinusoidal motions and is used to determine hydrodynamic derivatives concerning acceleration terms related to the added mass and added moment of inertia together with the terms related to the sway velocity and yaw rate as functions of frequency. The CMT uses model ship steady circular and/ or oblique motions and is suitable for determining most of the hydrodynamic derivatives defined at zero frequency of motion since steady hydrodynamic forces and moments are obtained by the CMT. Although the mathematical models, especially the MMG (Mathematical Modeling Group) model [3], for maneuvering simulation are well established and their validity has been confirmed in previous research [3, 4], the reliability of the maneuvering simulation results depends on how accurately and adequately the hydrodynamic derivatives are determined. Although the sensitivity of hydrodynamic derivatives to maneuvering simulation results has been investigated [5], it has not yet been clarified how reliable are the determined hydrodynamic

Table 1 Model ship dimensions for a container ship (KCS) and two tankers ( and ) Item KCS Length, perpendiculars, L (m) 3.464 2.991 2.991 Breadth (m).4265.5273.5273 Draft, d (m).143.191.191 Longitudinal coordinate of center of gravity, x G (m) -.451.19.11 Nondimensional yaw gyration radius, j zz /L.2263.2.2 Metacentric height, GM (m).97.199.199 Displaced volume (m 3 ).129.235.2349 Propeller diameter, D P (m).146.96.96 Propeller pitch ratio, P P /D P.9967.695.695 Rudder aspect ratio 2.164 2.224 2.224 Rudder area ratio, A R /Ld 1/54.6 1/59.29 1/59.29 derivatives themselves. The reliability of determined hydrodynamic derivatives depends mainly on that of the experimental data. Uncertainty analysis [6 ] is an effective methodology to validate the quality of experimental data. Although the uncertainty of experimental data for forces and moments in oblique motion, wake flow fields, sinkage and trim conditions, and various wave profiles has been assessed and reported [9, 1], no CMT data have been discussed in terms of uncertainty to date. The uncertainty analysis of CMT data would be beneficial in validating data quality and in discussing the reliability of simulation methods for estimating ship maneuvering performance. In this report, CMT data for the hydrodynamic forces and moments and the positions of the three model ships, a container ship (KCS) and two tankers ( and ), are presented. The CMTs were carried out at the ocean engineering basin at the National Maritime Research Institute, Japan (NMRI). Uncertainty analysis following ANSI/ASME Performance Test Code (PTC19.1-195) [6] and AIAA Standard S-71-1995 [7] was applied to the CMT data. 1 The results of uncertainty analysis are presented and discussed. The test facility is introduced and data on the towing precision of the carriage are also presented. 2 Circular motion test 2.1 Model ships and circular motion The model ships [11] undergoing the CMT were a container ship, KCS, and two tankers, and. 1 The use of these standards does not reflect the current state of the art for calculating uncertainties in experimental hydrodynamics, and therefore this article is for the academic understanding of how old uncertainty analysis standards may be applied to tow tank maneuvering tests. 3 11/2 2 1 A.P. 1/2 These ships were designed by the Korea Research Institute of Ships and Ocean Engineering. The principal dimensions of the model ships are shown in Table 1. The scale ratios are 1/75.5 for KCS and 1/11. for the KVLCCs. The body plans for the model ships are shown in Fig. 1 for KCS and Fig. 2 for the KVLCCs. The two tanker models, and, have almost the same particulars but slightly different aft frame lines. The longitudinal coordinate of the center of gravity is represented by x G. The rudder area A R stands for that of the movable area of the rudder and its aspect ratio is defined by H 2 /A R where H stands for the rudder height. The coordinate system o xy was fixed for the model ships as shown in Fig. 3. The origin is at the midship. The circular motion of each model ship is described using ship speed U, oblique angle b, and yaw rate r. The nondimensional yaw rate r is defined as r(l/u), where L is the ship length between perpendiculars. F.P. 91/2 9 F.P. Fig. 2 and tanker models 3 A.P. 1/2 1 11/2 2 1/2 7 6 Fig. 1 KCS container ship model. AP aft perpendicular, FP forward perpendicular 91/2 9 7

Fig. 3 Coordinate system U β x, X model test and 14. C for the and model tests. 2.3 Measured items Table 2 Test conditions Item Towing speed; U 2.2 Test conditions Midship Fn r, N y, Y The test conditions are shown in Table 2. Towing speeds U were 1.1 m/s for KCS and.76 m/s for the two KVLCCs. Propeller revolutions n were 12.93 rps for KCS, 17.5 rps for, and 17.75 rps for. These propeller revolutions are the model self-propulsion conditions. The CMT data reported here are for the hull with rudder and propeller conditions in which the model ships have the constant ship speeds mentioned above and zero rudder angle. The test condition parameters are the oblique angle b and nondimensional yaw rate r. The range of oblique angle b was from -2 to 2 for all model ships, and the range of nondimensional yaw rate r was from -.6 to.6 for the KCS and from -. to. for the KVLCCs. The model ships were free in pitch, roll, and heave motions. The water depth was 1.2 m throughout the tank test. The average water temperature was 1.2 C for the KCS Ft Description T o 1.1 (m/s): KCS.76 (m/s): and Propeller revolutions; n 12.93 (rps): KCS 17.69 (rps): and Rudder angle Motion parameters Oblique angle, b -2 to 2 Nondimensional yaw rate, r -.6 to.6: KCS -. to.: and Captive mode Pitch free, roll free, heave free Water depth 1.2 (m) Water temperature 1.2 C (KCS), 14. C (KVLCC) The measured items were the towing speed, oblique angle, yaw rate, propeller revolutions, rudder angle, model ship position, and nine items for the uncertainty analysis. These nine items were the surge force X, sway force Y, yaw moment N, rudder tangential force F t, rudder normal force F n, propeller thrust T, pitch angle h, roll angle /, and heave z. The surge force X, sway force Y, and yaw moment N are the hydrodynamic forces and moment acting on the whole model ship system and the inertia forces originating from the model ship mass when the yaw rate is not zero. The positive direction of surge force X, sway force Y, and yaw moment N are forward, starboard, and clockwise, respectively. The rudder tangential and normal forces, F t and F n, have positive values when the tangential force is directed to the rudder trailing edge and the normal force is directed to the port side. These are as shown in Fig. 3. Pitch and roll angles are positive for the bow up and the port side up, respectively. Heave z is upward positive. The pitch and roll angles h and / and heave z were measured by potentiometers. These measured items were used in the uncertainty analysis. All forces and moments, including rudder forces and propeller thrust, were measured by load cells. Note that the hydrodynamic sway force and yaw moment were obtained by the outputs of two load cells, both for measuring sway force components, placed.3 m fore and aft from the midship. A part of the mass of a load cell may influence the measured forces as a part of inertia force. This part of the mass was estimated at the manufacturing stage of the CMT apparatus based on its material and dimensions. The additional masses originating from the load cells were.434 kg in the surge direction and.764 kg in the sway direction,.36 and.63% of the KCS model ship mass and.1 and.33% of the KVLCC model ship mass. There was no effect on the measured yaw moment since the additional masses in the sway direction were equally distributed at points.3 m fore and aft from the midship. These additional mass effects [12] must be taken into consideration when the centrifugal force is subtracted from the measured CMT data to obtain the pure hydrodynamic components. This report processes and presents the raw data, including these additional mass effects. 2.4 Towing precision Tank tests were carried out at the ocean engineering basin [13] at NMRI. Figure 4 shows the main facilities of the

Fig. 4 Test facility for the circular motion test Table 3 Specifications of the test facility Item Tank size Moving area Motion freedom Motion mode Maximum speed Related facilities Specifications Length (X C -direction): 4. m Width (Y C -direction): 27.1 m Depth:.3 2. m X C : 27.2 m, Y C : 16.4 m (for the subcarriage center) (height of subcarriage adjustable within 1.25 m range) Three degrees of freedom (X C, Y C, and w C directions) (X C by main carriage; Y C by subcarriage; w C, horizontal rotation by turntable on subcarriage) Circular motion, planar motion, oblique motion, forced oscillation, autotracking X C : 2. m/s (main carriage), Y C : 1.5 m/s (subcarriage), w C : 36. /s (turntable) Wave generator, current generator, large subcarriage for a fixed large model basin, the specifications of which are listed in Table 3. The size of the tank was 4. m 9 27.1 m and its depth was up to 2. m. The movement of main carriage and subcarriage were orthogonal and the subcarriage had a turntable that rotates in a horizontal plane. This carriage system was controlled by a computer and was able to provide a model ship with arbitrarily designed surge sway yaw combined motion such as circular motion, planar motion, oblique motion, or forced oscillation. The towing precision influences the uncertainty of tank test data. The precision of ship motion in CMT is evaluated by how precisely the designated ship speed U, oblique angle b, and yaw rate r are realized. The ship speed U, oblique angle b, and yaw rate r are based on the speed of the main carriage and subcarriage and also on the turning rate and direction of the turntable, as recorded at 5 Hz by the carriage control system. These values are relative to the ground. Figure 5 shows the residuals of the average model ship speed du, oblique angle db, and yaw rate dr in the CMT du (m/s).3.2 U =1.1(m/s).1. -.1 -.2 -.3 -.4-2 -1 1 2 r (degree/s) dβ (degree) dr (degree/s).2.1. -.1 -.2 U =1.1(m/s) -.3-2 -1 1 2 r (degree/s).4.2. -.2 U =1.1(m/s) -.4-2 -1 1 2 r (degree/s) Fig. 5 Towing precision of the carriage. du towing speed error, db oblique angle error, dr yaw rate error, U ship speed, r yaw rate for the KCS model. Since the designated ship speed is 1.1 m/s, the residual of the towing speed is within.3%. The residual of oblique angle correlates more with the yaw rate than with the designated oblique angle. The maximum residual of the oblique angle is around.2. The residual of the yaw rate is proportional to the magnitude of yaw rate and is about 2%. These towing residuals are among the factors of the precision index in the uncertainty analysis that is mentioned later.

3 CMT data and its uncertainty analysis 3.1 Procedure of uncertainty analysis The measurement of uncertainty [6 ] is indicated by the 95% confidence uncertainty U RSS which is defined as follows: p >< U RSS ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pp 2 þ B 2 P ¼ ts= ffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M ð1þ >: P B ¼ ðsi B i Þ 2 where P is the precision index that is to be obtained by a number of repeated measurements with the same test conditions. S and M are the standard deviation of the measured values and the degrees of freedom, respectively, t is the Student number of the t distribution, and B is the bias limit consisting of several factors B i of which the sensitivity is s i. The intended test conditions for the uncertainty analysis are those in which the combination of oblique angle and nondimensional yaw rate (b, r ) is equal to (12.,.) and (.,.4). The former is a steady oblique towing condition and the latter is a pure yaw condition. For the KCS model, the measurements were repeated nine times for the (12.,.) condition and ten times for the (.,.4) condition. For the KVLCC models, the measurements were repeated ten times for both the (12.,.) and (.,.4) conditions. The student number t was equal to 2.22, 2.262, and 2.36 when the value of M was equal to 1, 9, and, respectively. Outlier analysis using the modified Thompson s technique [6] was applied to remove erroneous data before the uncertainty analysis was performed. The uncertainty analysis procedure was applied to nondimensional values other than the pitch and roll angles. The nondimensional forms are as follows, in which d and q stand for the model ship draft and the density of water, respectively: X ¼ X=ð:5qLdU 2 Þ Y ¼ Y=ð:5qLdU 2 Þ >< N ¼ N=ð:5qL 2 du 2 Þ Fn ¼ F n=ð:5qldu 2 Þ ð2þ Ft ¼ F t=ð:5qldu 2 Þ T ¼ T=ð:5qLdU 2 Þ >: z ¼ z=l The sensitivity s i was calculated using the following equation in which Q Bi is a quantity related to the factor of bias limit B i, such as the force or speed, and A represents one of the nine items to which the uncertainty analysis was applied, i.e., the nondimensional surge force X, sway force Y, yaw moment N, rudder tangential force F t, rudder normal force F n, propeller thrust T, heave z, pitch angle h, and roll angle /: s i ¼ oa ð3þ oq Bi Data acquisition for the above-mentioned nine items was done through the following steps: (1) analogue voltage signals from the sensors, the load cell, and potentiometer were transferred to the amplifiers; (2) the signal was amplified and filtered with a 1-Hz low-pass filter; (3) the PC received the signal and converted it to a digital signal of 16 bits at 2 Hz using 14 channels. The measurement duration was about 6 s. This procedure was same for both the CMTs and the calibration of the sensors. Measured raw data were averaged, the zero-reference data were subtracted from the averaged raw data, and physical values were obtained by multiplying the subtracted data by the calibration coefficient. The uncertainties in this system are considered to be included implicitly in the standard error of the estimate, SEE, of the sensors obtained by calibration. 3.2 Surge and sway forces and yaw moment The CMT data for the nondimensional surge force X, sway force Y, and yaw moment N for KCS,, and are shown in Figs. 6, 7, and, respectively. The uncertainty of CMT data is discussed based on the nondimensional values of the raw data. This means that the surge force X and sway force Y include centrifugal force components due to the model ship mass. This treatment is considered to have validity to a certain extent because the centrifugal force component of ship mass contributes to the ship maneuverability. In fact, the linear sway force hydrodynamic derivative concerning yaw rate Y r appears in a mathematical model [3] together with the ship mass and longitudinal added mass components m? m x in the form Y r -(m? m x ). Although a part of the load cell mass was also included in the measured value, the effect was quite small because the relevant part of the load cell mass was quite small, as stated in Sect. 2.3. For the surge force X in pure sway motion, the sway motion apparently increases the resistance for KCS; however, the KVLCC models do not show such a clear tendency. For the effect of yaw rate on surge force X, the yaw motion-induced resistance increase for KCS but decrease for the KVLCCs. For the sway force Y and yaw moment N, the qualitative tendency seems to be similar for KCS and the KVLCCs, although the hull shapes are quite different. Although the differences in hydrodynamic forces and moments are not major between and, since the hull forms are quite similar for these

X '.4.2. -.2 KCS -.4 r'=.4 -.6 r'= -.2 -. r'= -2-.4-1 1 2 r'= -.6 Y '.2.15 KCS.1.5. -.5 -.1 r'=.4 -.15 -.2 r'= -.2 -.25 r'= -.4-2 -1 1 2 r'= -.6 N '..6.4.2. -.2 -.4 -.6 -. -.1 KCS r'=.4-2 -1 1 r'= -.2 2 r'= -.4 r'= -.6 Fig. 6 Nondimensional surge force X, sway force Y, and yaw moment N for model KCS. r nondimensional yaw rate, b oblique angle X '.15.1.5. -.5 r'=. -.1 r'= -.2 -.15 r'= -.6 r'= -. (r'= -2.4) -1 1 2 Y '.3.2.1. -.1 -.2 -.3-2 -1 r'=. r'= -.2 r'= -.6 1 r'= -. 2 (r'=.4) N '.1..6.4.2. -.2 -.4 -.6 -. -.1 -.12 r'=. r'= -.2 r'= -.6-2 r'= -. (r'=.4) -1 1 2 Fig. 7 Nondimensional surge force X, sway force Y, and yaw moment N for model X '.15.1.5. -.5 r'=. -.1 r'= -.2 -.15 r'= -.6 r'= -2 -. -1 1 2 (r'=.4) Y '.3.2.1. -.1 r'=. -.2 -.3 r'= -.2-2 -1 1 r'= -.6 2 r'= -. (r'=.4) N '.1..6.4.2. -.2 -.4 -.6r'=. -. -.1 -.12r'= -.2-2 r'= -.6-1 1 2 r'= -. (r'=.4) Fig. Nondimensional surge force X, sway force Y, and yaw moment N for model model ships, differences can be seen in some cases in the maneuvering simulation results by authors [12, 14] who made use of hydrodynamic derivatives determined based on these CMT data. A comparison of the turning trajectories for and for a rudder angle of 5 is shown in Fig. 9 as an example. The ordinate and abscissa

Fig. 9 Turning trajectories for a 5 rudder angle for and (left port turn, right starboard turn). Xe, Ye earth-fixed coordinate system, Lpp length between perpendiculars Xe /Lpp 6 4 2 Xe /Lpp 6 4 2 (5deg. Turn) (-5deg. Turn) -2-1 - -6-4 -2 2 Ye /Lpp -2-2 2 4 6 1 Ye /Lpp represent nondimensional length using the ship length between perpendiculars. The ships approach the origin from the negative part of the ordinate at a speed of.76 m/s in model scale and they commence steering at the origin. For KCS, outlier analysis removed one data point each for the surge force X and yaw moment N for (b, r ) equal to (12.,.). For, outlier analysis removed one data point each for the sway force Y and yaw moment N for the (.,.4) condition. For, outlier analysis removed one data point for the yaw moment N for both the (12.,.) and (.,.4) conditions. The bias error components taken into consideration were the standard error of estimate, SEE, of the load cell obtained by its calibration (B 1 ), the towing speed error originating from the error related to the touch roller diameter (.2 mm, B 2 ), the force component originating from the alignment error of model ship with respect to the carriage (.25, B 3 ), and the force component originating from the model ship mass error (.1 kg, B 4 ). The standard error of estimate, SEE (B 1 ) was calculated from the scatter of the linear regression of the calibration curve of the load cells. The touch roller diameter is 95.5 mm and its circumference is 3 mm. The values for the error of touch roller diameter and the alignment error were chosen based on the examples given by Toda []. The model ship mass error is assumed as a possible maximum value in the CMT. The alignment error is assumed to be the directional error of measured forces. The error of the model ship mass is assumed to be the error of the centrifugal component of the measured forces. The bias limits of the water density q, model ship length L, and draft d are assumed to be small and were neglected [1] although they are considered to be included partially in the model ship mass error. The bias error components and corresponding sensitivities are given by Eqs. 4 6 for surge force X, sway force Y, and yaw moment N, respectively: >< >: >< >: >< >: B 1 ¼ SEE of X;s 1 ¼ ox ox ¼ 1 :5qLdU 2 B 2 ¼ :2ðmmÞ 3ðmmÞ=2p U;s 2 ¼ ox ou ¼ 2X U B 3 ¼f1 cosð:25 ÞgX;s 3 ¼ s 1 B 4 B 1 ¼ :1ðkgÞ U2 r sinb L ;s 4 ¼ s 1 ¼ SEE of Y;s 1 ¼ oy oy ¼ 1 :5qLdU 2 B 2 ¼ :2ðmmÞ 3ðmmÞ=2p U;s 2 ¼ oy ou ¼ 2Y U B 3 ¼f1 cosð:25 ÞgY;s 3 ¼ s 1 B 4 B 1 ¼ :1ðkgÞ U2 r cosb L ;s 4 ¼ s 1 ¼ SEE of N;s 1 ¼ on on ¼ 1 :5qL 2 du 2 B 2 ¼ :2ðmmÞ 3ðmmÞ=2p U;s 2 ¼ on ou ¼ 2N U B 3 ¼f1 cosð:25 ÞgN;s 3 ¼ s 1 B 4 ¼ :1ðkgÞ U2 r cosb L x G ;s 4 ¼ s 1 ðfor surge force X Þ ðfor sway force Y Þ ð4þ ð5þ ðfor yaw moment N Þ ð6þ In Eqs. 4 6, B 3 represents the difference between the true value and the measured value in a system with.25 alignment error. The results of uncertainty analysis for the nondimensional surge force X, sway force Y, and yaw moment N for KCS,, and are shown in Tables 4, 5, 6, respectively. Only the bias limits for the error of load cell calibration B 1 and the error of touch roller diameter B 2 are shown in the tables because other components have values of less than.1% of the total bias limit B. The precision index P makes a larger contribution to U RSS than the bias limit B does for the nondimensional surge force X, whereas the bias limit B makes a much larger contribution to U RSS than the precision limit P does for the nondimensional sway force Y and yaw moment N. The reason for this is considered to be mainly that the

Table 4 Uncertainty analysis for the nondimensional surge force X, sway force Y, and yaw moment N for model KCS Item X Y N % of Bias factor Unit % % % (b = 12., r =.) Average -6.164E-3 7.36E-2 2.276E-2 U RSS 5.2E-5. 2.72E-3 3.9 2.37E-4 1.2 Ave. P 3.33E-5 54.3 5.5E-4 3.1 5.127E-5 3.3 U RSS B 3.514E-5 45.7 2.27E-3 96.9 2.79E-4 96.7 U RSS s 1 3.72E-2 3.72E-2 1.221E-2 Load cell cal. s 2 1.121E-2-1.34E-1-4.139E-2 Touch roller dia. s 1 B 1 3.476E-5 97. 2.27E-3 1. 2.74E-4 99.5 B Load cell cal. s 2 B 2 5.164E-6 2.2-6.173E-5. -1.97E-5.5 B Touch roller dia. (b =., r =.4) Average -4.33E-3-4.765E-2-2.549E-2 U RSS 1.4E-4 4.1 2.9E-3 6.1 2.93E-4 1.1 Ave. P 1.77E-4 96.3 6.59E-4 5.2.65E-5 7.7 U RSS B 3.491E-5 3.7 2.24E-3 94. 2.7E-4 92.3 U RSS s 1 3.715E-2 3.715E-2 1.219E-2 Load cell cal s 2 7.964E-3.659E-2 4.632E-2 Touch roller dia. s 1 B 1 3.471E-5 9.9 2.23E-3 99.9 2.7E-4 99.4 B Load cell cal. s 2 B 2 3.672E-6 1.1 3.992E-5. 2.136E-5.6 B Touch roller dia. b oblique angle, r nondimensional yaw rate, U RSS 95% confidence uncertainty, P precision index, B total bias limit, B 1 bias limit for the error in load cell calibration, B 2 the error in the touch roller diameter, s 1, s 2 sensitivities of B 1, B 2, Ave. average, cal. calibration, dia. diameter Table 5 Uncertainty analysis for X, Y, and N for model Item X Y N % of Bias factor Unit % % % (b = 12., r =.) Average -2.E-3.967E-2 2.472E-2 U RSS 2.31E-4 9.7 4.169E-3 4.6 4.324E-4 1.7 Ave. P 2.3E-4 97.3 1.527E-3 13.4 1.627E-4 14.2 U RSS B 3.321E-5 2.7 3.E-3 6.6 4.6E-4 5. U RSS s 1 6.173E-2 6.173E-2 2.122E-2 Load cell cal. s 2 5.496E-3-2.36E-1-6.56E-2 Touch roller dia. s 1 B 1 3.316E-5 99.7 3.79E-3 1. 4.E-4 99.7 B Load cell cal. s 2 B 2 1.75E-6.3-7.512E-5. -2.71E-5.3 B Touch roller dia. (b =., r =.4) Average 1.37E-3 -.27E-2-2.65E-2 U RSS 2.77E-4 2.3 3.965E-3 4. 4.119E-4 1.4 Ave. P 2.75E-4 9.6.152E-4 4.2 9.467E-5 5.3 U RSS B 3.319E-5 1.4 3.1E-3 95. 4.E-4 94.7 U RSS s 1 6.174E-2 6.174E-2 2.122E-2 Load cell cal. s 2-3.65E-3 2.11E-1 7.54E-2 Touch roller dia. s 1 B 1 3.317E-5 99.9 3.E-3 1. 4.1E-4 99.6 B Load cell cal. s 2 B 2-1.14E-6.1 6.943E-5. 2.4E-5.4 B Touch roller dia.

Table 6 Uncertainty analysis for X, Y, and N for model Item X Y N % of Bias factor Unit % % % (b = 12., r =.) Average -1.539E-3.76E-2 2.6E-2 U RSS 2.467E-4 16. 3.999E-3 4.6 1.173E-3 4.5 Ave. P 2.444E-4 9.2 9.77E-4 5.9 1.41E-4 1.5 U RSS B 3.319E-5 1. 3.E-3 94.1 1.164E-3 9.5 U RSS s 1 6.173E-2 6.173E-2 6.173E-2 Load cell cal. s 2 4.51E-3-2.291E-1-6.62E-2 Touch roller dia. s 1 B 1 3.316E-5 99. 3.79E-3 1. 1.164E-3 1. B Load cell cal. s 2 B 2 1.29E-6.2-7.294E-5. -2.14E-5. B Touch roller dia. (b =., r =.4) Average 1.141E-3 -.147E-2-2.6E-2 U RSS 3.129E-4 27.4 4.114E-3 5.1 1.172E-3 4.1 Ave. P 3.111E-4 9.9 1.37E-3 11.1 1.39E-4 1.4 U RSS B 3.317E-5 1.1 3.E-3.9 1.164E-3 9.6 U RSS s 1 6.173E-2 6.173E-2 6.173E-2 Load cell cal. s 2-3.2E-3 2.144E-1 7.526E-2 Touch roller dia. s 1 B 1 3.316E-5 99.9 3.79E-3 1. 1.164E-3 1. B Load cell cal. s 2 B 2-9.55E-7.1 6.25E-5. 2.396E-5. B Touch roller dia. measured values are much smaller for the surge force than for the sway force. The 95% confidence uncertainty U RSS is also shown in Figs. 6, 7, by error bars for measured points of which the combination of oblique angle and nondimensional yaw rate (b, r ) is equal to (12.,.) and (.,.4). The maximum percentage of U RSS to the average value is 27.4% for the surge force X of in the condition in which oblique angle b is. and nondimensional yaw rate r is.4. However, all the error bars for these forces and moments are within the marks and are difficult to recognize since these values are suitably small. 3.3 Rudder tangential force, rudder normal force, and propeller thrust The CMT data of the nondimensional rudder tangential force F t, rudder normal force F n, and propeller thrust T for KCS,, and are shown in Figs. 1, 11, 12, respectively. Although the rudder tangential force F t is much smaller than the rudder normal force F n, F t shows a certain recognizable pattern in which large unsymmetrical and complicated sway and yaw motion effects can be seen for all these model ships data. For the rudder normal force F n, the sway motion or oblique angle effect seems to have similar tendencies for KCS and the KVLCCs, whereas the yaw motion effect seems to be larger for the KVLCCs than for KCS. On the other hand, the propeller thrust T for KCS shows larger unsymmetrical yaw motion effects than those of the KVLCCs, especially in the negative oblique angle region. All these clear unsymmetrical characteristics for the rudder tangential force F t, rudder normal force F n, and propeller thrust T are considered to result from the asymmetry of right-hand propeller revolution. For KCS, outlier analysis removed one data point for the rudder tangential force F t for (b, r ) equal to (12.,.) and one data point for thrust T for the (.,.4) condition. For, outlier analysis removed one data point for thrust T for the (12.,.) condition. For, outlier analysis removed one data point each for rudder tangential force F t and thrust T for the (12.,.) condition, and one data point for rudder normal force F n for the (.,.4) condition. The bias error components taken into consideration were the SEE of the load cell obtained by its calibration (B 1 ), the towing speed error originating from the error in the touch roller diameter (.2 mm, B 2 ), the force component originating from the alignment error of rudder and propeller with respect to the model ship (.25, B 3 ), and the force component originating from a rudder mass error of.1 kg and a propeller mass error of.1 kg (B 4 ). The SEE (B 1 ) was calculated from the scatter of the linear regression of the calibration curve of the load cell for

.2.1 KCS.3.2.1 KCS.3.2 KCS Ft '. Fn '. T ' -.1 r'=.4 -.2 r'= -2 -.2-1 1 2 r'= -.4 r'= -.6 b(degree) -.1 -.2 r'=.4 -.3 r'= -.2-2 -1 1 r'= -.4 r'= -.6 b(degree) 2.1 r'=.4 r'= -.2 r'= -.4 r'= -.6. -2-1 1 2 b(degree) Fig. 1 Nondimensional rudder tangential force F t, rudder normal force F n, and propeller thrust T for model KCS.2.1.3.2.1.4.3 Ft '. Fn '. T '.2 r'=. -.1 r'= -.2 -.2r'= -.6 r'= -2-. -1 1 2 (r'=.4) -.1r'=. -.2 r'= -.2 -.3-2 r'= -.6-1 1 2 r'= -. (r'=.4) r'=..1 r'= -.2 r'= -.6 r'= -. (r'=.4). -2-1 1 2 Fig. 11 Nondimensional rudder tangential force F t, rudder normal force F n, and propeller thrust T for model.2.1.3.2.1.4.3 Ft '. Fn '. T '.2 r'=. -.1 r'= -.2 -.2 r'= -2 -.6-1 1 2 r'= -. (r'=.4) -.1r'=. -.2 r'= -.2 -.3r'= -2-.6-1 1 2 r'= -. (r'=.4).1 r'=. r'= -.2 r'= -.6 r'= -. (r'=.4). -2-1 1 2 Fig. 12 Nondimensional rudder tangential force F t, rudder normal force F n, and propeller thrust T for model rudder forces and thrust. The alignment error, the rudder mass error, and the propeller mass error were assumed as possible maximum values, the same as in the previous section. The alignment error was assumed to be the directional error of the measured forces. The error in the rudder and propeller mass were assumed to be the error of the centrifugal component of measured forces. The bias error components and corresponding sensitivities are given by Eqs. 7 9 for the rudder tangential force F t, rudder normal force F n, and propeller thrust T, respectively:

B 1 ¼ SEE of Ft ; s 1 ¼ of t of t ¼ 1 :5qLdU >< 2 :2 ðmmþ B 2 ¼ 3 ðmmþ=2p U; s 2 ¼ of t ou ¼ 2F t ðfor rudder U B 3 ¼f1 cos ð:25 ÞgF t ; s 3 ¼ s 1 tangential force Ft >: Þ B 4 ¼ :1 ðkgþ U2 r sin b L ; s 4 ¼ s 1 B 1 ¼ SEE of F n ; s 1 ¼ of n of n ¼ 1 :5qLdU >< 2 :2 ðmmþ B 2 ¼ 3 ðmmþ=2p U; s 2 ¼ of n ou ¼ 2F n U B 3 ¼f1 cos ð:25 ÞgF n ; s 3 ¼ s >: 1 B 4 ¼ :1 ðkgþ U2 r cos b L ; s 4 ¼ s 1 B 1 ¼ SEE of T; s 1 ¼ ot ot ¼ 1 :5qL 2 du >< 2 :2 ðmmþ B 2 ¼ 3 ðmmþ=2p U; s 2 ¼ ot ou ¼ 2T U B 3 ¼f1 cos ð:25 ÞgT; s 3 ¼ s 1 >: B 4 ¼ :1 ðkgþ U2 r sin b L ; s 4 ¼ s 1 ð7þ ðfor rudder normal force Fn Þ ðfor thrust T Þ ðþ ð9þ The results of the uncertainty analysis for the nondimensional rudder tangential force F t, rudder normal force F n, and propeller thrust T for KCS,, and are shown in Tables 7,, 9, respectively. Since B 3 and B 4 are less than.1% of B, the tables do not show these values. For the nondimensional rudder tangential force F t and rudder normal force F n the bias limit for the error of load cell calibration B 1 has a dominant effect. The uncertainty U RSS is less than 3 and 4% of the average values for the rudder tangential force F t and rudder normal force F n, respectively. For the propeller thrust T, a slight effect of the error in the touch roller diameter can be seen for KVLCCs. However, the ratio of uncertainty U RSS to the average value is around 1% for the propeller thrust T. Since the ratio of the uncertainty U RSS to the average value is so small, the error bars shown in Figs. 1, 11, 12 for points where the oblique angle and nondimensional yaw rate (b, r ) are (12.,.) and (.,.4) are within the marks. 3.4 Pitch angle, roll angle, and heave The CMT data for the pitch angle h, roll angle /, and nondimensional heave z for KCS,, and are shown in Figs. 13, 14, 15, respectively. A positive sway motion or oblique angle induced a bow up trim moment and vice versa for both KCS and the KVLCCs, except for the no yaw rate conditions in which both positive and negative sway motions induced a slight bow down trim moment. The roll angle showed negative values for positive sway motion and vice versa for both KCS and the KVLCCs in no yaw rate conditions. The yaw rate effects were much larger for KCS than for the KVLCCs. Both pitch and roll angles were, in general, larger for KCS than for the KVLCCs. The data of nondimensional heave z indicate that the maneuvering motion makes ships sink for both KCS and the KVLCCs. For KCS, outlier analysis removed one data point for pitch angle h for (b, r ) equal to (12.,.) and one data point for roll angle / for the (.,.4) condition. For, outlier analysis removed one data point each for pitch angle h, roll angle /, and heave z for the (12.,.) condition and one data point for heave z for the (.,.4) condition. For, outlier analysis removed one data point for pitch angle h for the (12.,.) condition and one data point for heave z for the (.,.4) condition. The bias error components taken into consideration were the SEE of the potentiometer obtained by its calibration (B 1 ) for pitch angle h, roll angle /, and nondimensional heave z. For heave z, the component originating from the alignment error (.25, B 2 ) was also taken into consideration. The SEE (B 1 ) was calculated from the scatter of the linear regression of the calibration curve of the potentiometers. The alignment error was assumed to be the directional error of the measured forces. The bias error components and corresponding sensitivities are given by Eqs. 1 12 for pitch angle h, roll angle /, and nondimensional heave z, respectively: B 1 ¼ SEE of h; s 1 ¼ 1 ðfor pitch angle hþ ð1þ B 1 ¼ SEE of /; s 1 ¼ 1 ðfor roll angle /Þ ð11þ B 1 ¼ SEE of z ; s 1 ¼ oz oz ¼ 1 L B 2 ¼f1 cos ð:25 ðfor heave z Þ Þgz; s 2 ¼ s 1 ð12þ The results of the uncertainty analysis for pitch angle h, roll angle /, and nondimensional heave z are shown in Tables 1, 11, 12 for KCS,, and, respectively. The effect of B 2 for heave z is not shown in the tables because it was less than.1% of the total bias limit B. The contribution of the bias limit B to the uncertainty U RSS is dominant for the pitch angle of KCS; however, in other cases, the contributions of the precision index P and the bias limit B to the uncertainty U RSS vary test by test. The uncertainty U RSS does not exceeds.5 for both pitch and roll angles for KCS and the KVLCCs. The heave uncertainty does not exceed.7 mm at the model scale. The uncertainties U RSS for these values seem to be small, but their ratios to the average values are not so small. For example, the ratio of uncertainty U RSS to the average value for the roll angle is 237.% for because the measured values are also small. The uncertainty U RSS is relatively large compared to these measured items, so some error bars in Figs. 13, 14, 15 can be distinguished.

Table 7 Uncertainty analysis for the nondimensional rudder tangential force F t, rudder normal force F n, and propeller thrust T for model KCS Item F t F n T % of Bias factor Unit % % % (b = 12., r =.) Average 6.916E-4-3.314E-3 1.91E-2 U RSS 1.2E-5 1.4 3.79E-5 1.2 2.659E-4 1.4 Ave. P 3.93E-6 15. 3.449E-5 79.1 1.139E-4 1.4 U RSS B 9.194E-6 4.2 1.775E-5 2.9 2.42E-4 1.6 U RSS s 1 3.72E-2 3.72E-2 3.72E-2 Load cell cal. s 2-1.257E-3 6.25E-3-3.456E-2 Touch roller dia. s 1 B 1 9.175E-6 99.6 1.753E-5 97.6 2.397E-4 99.6 B Load cell cal. s 2 B 2-5.794E-7.4 2.776E-6 2.4-1.592E-5.4 B Touch roller dia. (b =., r =.4) Average 6.24E-4-5.5E-3 2.44E-2 U RSS 1.59E-5 1.7 4.41E-5.9 2.412E-4 1.2 Ave. P 5.275E-6 24. 4.49E-5 6. 2.3E-5 1. U RSS B 9.17E-6 75.2 1.13E-5 14. 2.4E-4 99. U RSS s 1 3.715E-2 3.715E-2 3.715E-2 Load cell cal. s 2-1.134E-3 1.14E-2-3.714E-2 Touch roller dia. s 1 B 1 9.163E-6 99.7 1.75E-5 93.2 2.394E-4 99.5 B Load cell cal. s 2 B 2-5.227E-7.3 4.675E-6 6.6-1.713E-5.5 B Touch roller dia. Table Uncertainty analysis for F t, F n, and T for model Item F t F n T % of Bias factor Unit % % % (b = 12., r =.) Average 1.112E-3-6.53E-3 2.41E-2 U RSS 2.159E-5 1.9 1.3E-4 2.1 1.147E-4.5 Ave. P 1.211E-5 31.5 5.757E-5 17.4.77E-5 5.7 U RSS B 1.77E-5 6.5 1.254E-4 2.6 7.366E-5 41.3 U RSS s 1 6.173E-2 6.173E-2 6.173E-2 Load cell cal. s 2-2.927E-3 1.732E-2-6.364E-2 Touch roller dia. s 1 B 1 1.75E-5 99.7 1.253E-4 99. 7.2E-5 92.4 B Load cell cal. s 2 B 2-9.317E-7.3 5.515E-6.2-2.26E-5 7.6 B Touch roller dia. (b =., r =.4) Average 1.441E-3-7.46E-3 2.636E-2 U RSS 2.2E-5 1.5 1.925E-4 2.6 1.359E-4.5 Ave. P 1.21E-5 33.9 1.459E-4 57.5 1.139E-4 7.2 U RSS B 1.79E-5 66.1 1.255E-4 42.5 7.419E-5 29. U RSS s 1 6.174E-2 6.174E-2 6.174E-2 Load cell cal. s 2-3.793E-3 1.97E-2-6.937E-2 Touch roller dia. s 1 B 1 1.75E-5 99.5 1.253E-4 99. 7.3E-5 91.1 B Load cell cal. s 2 B 2-1.27E-6.5 6.272E-6.2-2.2E-5.9 B Touch roller dia.

Table 9 Uncertainty analysis for F t, F n, and T for model Item F t F n T % of Bias factor Unit % % % (b = 12., r =.) Average 1.165E-3-4.49E-3 2.5E-2 U RSS 2.319E-5 2. 1.4E-4 3.4 2.43E-4.9 Ave. P 1.477E-5 4.6 7.56E-5 2.2 2.315E-4 9.7 U RSS B 1.7E-5 59.4 1.254E-4 71. 7.47E-5 9.3 U RSS s 1 6.173E-2 6.173E-2 6.173E-2 Load cell cal. s 2-3.66E-3 1.16E-2-6.12E-2 Touch roller dia. s 1 B 1 1.75E-5 99.7 1.253E-4 99.9 7.2E-5 91.4 B Load cell cal. s 2 B 2-9.76E-7.3 3.694E-6.1-2.16E-5.6 B Touch roller dia. (b =., r =.4) Average 1.24E-3-6.57E-3 2.742E-2 U RSS 2.166E-5 1.7 1.576E-4 2.3 1.E-4.7 Ave. P 1.223E-5 31.9 9.544E-5 36.7 1.647E-4 3. U RSS B 1.7E-5 6.1 1.254E-4 63.3 7.445E-5 17. U RSS s 1 6.173E-2 6.173E-2 6.173E-2 Load cell cal. s 2-3.264E-3 1.4E-2-7.216E-2 Touch roller dia. s 1 B 1 1.75E-5 99.7 1.253E-4 99. 7.1E-5 9.5 B Load cell cal. s 2 B 2-1.39E-6.3 5.745E-6.2-2.297E-5 9.5 B Touch roller dia. θ(degree).4 r'=.4.3r'= -.2 r'= -.4.2r'= -.6.1. -.1 -.2 -.3 -.4 -.5 KCS -.6-2 -1 1 2 φ (degree).6.4.2. -.2 KCS -.4 -.6 r'=.4 -. -2-1 1 r'= -.2 2 r'= -.4 r'= -.6. -.1 -.2 -.3 KCS -.4 r'=.4 -.5 r'= -.2-2 -1 r'= 1-.4 2 r'= -.6 Fig. 13 Pitch angle h, roll angle /, and nondimensional heave z for model KCS 4 Conclusions The CMT for three model ships, KCS,, and, were carried out at the ocean engineering basin at NMRI. The data for the towing precision of the carriage system are presented and it was confirmed that the system has a high accuracy of towing suitable for carrying out the CMT. The CMT data for the nondimensional surge force, sway force, yaw moment, rudder tangential force, rudder normal force, propeller thrust, pitch angle, roll angle, and nondimensional heave are presented. These data were discussed and were confirmed to reflect the hull forms and configurations. The uncertainty assessment procedure was applied to the CMT data for one of the steady pure sway conditions and for one of the steady pure yaw conditions. The uncertainties were relatively large for the pitch angle, roll angle, and heave. However, these items do not relate directly to the precision of hydrodynamic derivatives and hence are not crucial in the prediction of maneuvering. The most important knowledge obtained here is that the uncertainties for hydrodynamic forces such as the surge

θ(degree).3.2.1. -.1 -.2 -.3 -.4 -.5 r'=. -2-1 r'=. 1 2 r'= -.2 r'= -.6 r'= -. (r'=.4) φ(degree).4.2. -.2 r'=. -.4 r'= -.2 r'= -.6 -.6 r'= -. (r'=.4) -2-1 1 2.1. -.1 -.2 r'=. -.3 -.4 r'= -.2-2 -1 1 r'= -.6 2 r'= -. (r'=.4) Fig. 14 Pitch angle h, roll angle /, and nondimensional heave z for model θ(degree).3.2.1. -.1 -.2 -.3 -.4 -.5r'=. -.6-2 r'= -.2-1 r'= -.6 1 2 r'= -. (r'=.4) φ(degree).4.2. -.2 -.4 -.6 r'=. r'= -.2 r'= -.6 r'= -. (r'=.4) -2-1 1 2. -.1 -.2 -.3 r'=. -.4 r'= -.2 r'= -.6 r'= -. (r'=.4) -.5-2 -1 1 2 Fig. 15 Pitch angle h, roll angle /, and nondimensional heave z for model Table 1 Uncertainty analysis for pitch angle h, roll angle /, and nondimensional heave z for model KCS Item h / z % of Bias factor Unit Degrees % Degrees % % (b = 12., r =.) Average -2.36E-1-3.47E-1-2.592E-3 U RSS 4.4E-2 19. 2.64E-2.2 1.214E-4 4.7 Ave. P 6.29E-3 2. 2.92E-2 53.3 5.246E-5 1.7 U RSS B 4.435E-2 9. 1.956E-2 46.7 1.95E-4 1.3 U RSS s 1 1.E? 1.E? 3.23E-1 Sensor cal. s 1 B 1 4.435E-2 1. 1.956E-2 1. 1.95E-4 1. B Sensor cal. (b =., r =.4) Average -2.69E-2-3.465E-1-1.665E-3 U RSS 4.619E-2 171.2 2.1E-2 6.3 1.47E-4.9 Ave. P 1.2E-2 7. 9.66E-3 19.4 9.921E-5 45.1 U RSS B 4.435E-2 92.2 1.956E-2.6 1.95E-4 54.9 U RSS s 1 1.E? 1.E? 3.23E-1 Sensor cal. s 1 B 1 4.435E-2 1. 1.956E-2 1. 1.95E-4 1. B Sensor cal.

Table 11 Uncertainty analysis for h, /, and z for model Item h / z % of Bias factor Unit Degrees % Degrees % % (b = 12., r =.) Average -1.11E-1-1.349E-1-1.545E-3 U RSS 1.297E-2 7.2 1.492E-2 11.1 1.339E-4.7 Ave. P 7.9E-3 29.2 1.17E-2 46.5.326E-5 3.7 U RSS B 1.92E-2 7. 1.92E-2 53.5 1.49E-4 61.3 U RSS s 1 1.E? 1.E? 3.43E-1 Sensor cal. s 1 B 1 1.92E-2 1. 1.92E-2 1. 1.49E-4 1. B Sensor cal. (b =., r =.4) Average -7.226E-2-7.751E-2-1.291E-3 U RSS 1.66E-2 25. 1.41E-2 23. 1.212E-4 9.4 Ave. P 1.514E-2 65. 1.42E-2 64. 6.7E-5 25.1 U RSS B 1.92E-2 34.2 1.92E-2 35.2 1.49E-4 74.9 U RSS s 1 1.E? 1.E? 3.43E-1 Sensor cal. s 1 B 1 1.92E-2 1. 1.92E-2 1. 1.49E-4 1. B Sensor cal. Table 12 Uncertainty analysis for h, /, and z for model Item h / z % of Bias factor Unit Degrees % Degrees % % (b = 12., r =.) Average -1.622E-1-4.413E-2-1.46E-3 U RSS 1.376E-2.5 2.366E-2 53.6 2.217E-4 15.1 Ave. P.31E-3 37.1 2.99E-2 7.7 1.953E-4 77.6 U RSS B 1.92E-2 62.9 1.92E-2 21.3 1.49E-4 22.4 U RSS s 1 1.E? 1.E? 3.43E-1 Sensor cal. s 1 B 1 1.92E-2 1. 1.92E-2 1. 1.49E-4 1. B Sensor cal. (b =., r =.4) Average -7.136E-2-1.97E-2-1.377E-3 U RSS 2.9E-2 4.7 2.6E-2 237. 1.665E-4 12.1 Ave. P 2.695E-2 5.9 2.36E-2 2.4 1.293E-4 6.3 U RSS B 1.92E-2 14.1 1.92E-2 17.6 1.49E-4 39.7 U RSS s 1 1.E? 1.E? 3.43E-1 Sensor cal. s 1 B 1 1.92E-2 1. 1.92E-2 1. 1.49E-4 1. B Sensor cal. and sway forces, yaw moment, rudder tangential and normal forces, and propeller thrust in the CMT are fairly small. It has thus been confirmed that the CMT data reported here are sufficiently reliable to determine the hydrodynamic derivatives with which maneuvering simulation calculations can be carried out to predict the maneuverability of these ships. These uncertainty analysis results are also believed to help further the discussion about validation of developing theoretical and computational prediction methods for hydrodynamics of ships in maneuvering motion such as the steady and unsteady computational fluid dynamics technique. Acknowledgments A part of this work was supported by KA- KENHI (136414). References 1. International Maritime Organization (22) Standards for ship manoeuvrability. Annex 6, Resolution MSC. 137(76), MSC/76/ 23/Add.1

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