Simple Estimation of Wave Added Resistance from Experiments in Transient and Irregular Water Waves

Simple Estimation of Wave Added Resistance from Experiments in Transient and Irregular Water Waves by Tsugukiyo Hirayama*, Member Xuefeng Wang*, Member Summary Experiments in transient water waves are well known in getting the transfer functions of ship motions. Here, we introduce a new simple method to estimate added resistance coefficients from these experiments. Using this method, we can obtain the added resistance coefficients in single transient water wave experiment instead of doing many experiments in regular waves. This new estimation method is based on Hsu's assumption, and using the Fourier spectra to separate the added resistance from surge force. Based on NSM (new strip method) and Maruo's theory, we calculated the added resistance coefficients theoretically, and compared them with the estimation results. It is another important point that the data in encounter frequency must be changed into absolute frequency domain. In experiments, three models were used : SR208 (a tanker), SR108 (a container ship), and SHSS (a super high speed container ship). Three models have different shapes and examined in different advance velocities. Here, all of their results will be reported. Present method is not only limited in transient water wave experiments. It can also be used in general irregular waves directly. In irregular long crested wave experiments, we obtained relatively good results as in transient water wave experiments. Using present method, we can also analyse the surge force from experimental data and can simulate the total resistance time history numerically by using the results from experiments in transient water waves. 1. Introduction About the calculation of added resistance, many research papers have been published already. About theory, the Maruo's formula is famous 1, 2), and many other theories are also utilized in many different researches. We used the Maruo's formula to calculate the added resistance coefficients in this paper, and compared these results with the estimated values which were obtained from the experiments. About the details of theoretical calculations, please see Reference 3). On the other hand, J. F. Dalzell, Hosoda etc. discussed the experimental analysing method called Cross- Bispectral Method4,5,6) for variable resistance by waves including added resistance. They explained the structure of resistance successfully in that method. But some problems are difficult to resolve. First, it needs 10-12 longer time histories than general spectrum analysis in order to get a certain accuracy. Second, it is difficult to calculate the two-dimensional added resistance Department of Naval Architecture & Ocean Engineering of Yokohama National University Received 9th July 1993 Read at the Autumn meeting 9, 10th Nov. 1993 coefficients theoretically. How to select filter is another problem. The analysis is too complex, needs a lot of data and experiences. So there are little examples about that analysis method. We transform the total surge force time history including added resistance into Fourier spectrum. In frequency domain, we separate the added resistance from surge force. Then by inverse Fourier transformation we get the added resistance time history which is considered to be in proportion to square of the wave height. Finally, use the Hsu's assumption to get the added resistance coefficients. Irregular wave is considered to be composed of regular waves in linear theory. Using present method, we can get the added resistance coefficients with a comparative good accuracy from single transient water wave. And it also can be used to obtain the added resistance in general irregular long crested waves. Even in directional spectrum waves7,8,9), present method is available theoretically, but some techniques have not be resolved yet and will let it to next chance. We consider this method will also be used to estimate the added resistance coefficients indirectly from the ship in actual sea through speed reduction, if the sea waves could be obtained by any way. Inversely, the simulation of the resistance time history from the experimentally

Journal of The Society of Naval Architects of TaDan. Vol. 174 obtained coefficients will be useful for ships in actual seas. 2. Estimation method 2. 1 Theoretical calculation At first, for comparison with estimated results, we tried conventional theoretical calculation. The added resistances in regular waves are expressed by famous Maruo's formula as follows where, In this article, we only consider the case in heading seas, the wave incident angle The results of calculations will be shown after. The H(Ki, a) function is based on the Hosoda's formula"). The transfer functions of ship motions are calculated by NSM (new strip method), shown at Fig. 1. For calculation of added resistance coefficient, in order to improve the accuracy, amplitude of those transfer functions are modified using experimental data from transient water waves. In short wave length region, we added the Fujii-Takahasi formula'2) to improve the calculations considering the reflection at the bow of ship. In Fig. 1, abscissa is absolute wave angular frequency and ordinate is transfer functions of heave or pitch. The left figures are heaves, and the right figures are pitches. Fom top, amplitude of SR208 (tanker), phase of SR208, amplitude of SR108 (container), phase of SR108, amplitude of SHSS (container) and phase of SHSS are shown. The amplitudes include the results by NSM, transient water wave and regular wave. But phases only include calculations and transient water wave results. Three models have different shapes and speeds, the experiments in regular waves were used to check the calculations and estimation results. The 'calculation' represent these theoretical calculations in all figures. In these calculations, we didn't consider the surge force. In fact, even in regular waves, the amplitudes of surge forces are bigger than the variation of added resistances in most cases. When we separate the added resistances from the total resistance, the surge forces can also be compared with the theoretical calculations. In order to obtain the surge force, we integrate the wave components along the ship length regarding to the wave slopes. These results of the calculations will be compared with the experimental data after. ( 3 ) where is wave elevation, yw is half breadth of water line, dx is depth of the station, is the length of ship fore body, and la is the length of ship after body. 2. 2 Hsu's assumption In order to avoid the complex Cross-Bispectral analysis, we apply the Hsu's assumption" to running ships as simple estimation method. This assumption is very closely related Newman's approximation"). The Hsu's basic assumption is : 1) An irregular wave (long crested wave) are the same with the wave linked together by many regular waves which have the same amplitudes and the same half periods on the time history. 2) This irregular wave produces the same drift force time history with that produced by regular waves mentioned at 1). This assumption was utilized to estimate the drift forces of offshore structures which have no speeds. If we consider ships, they have speeds and we must treat the phenomena in encounter frequency base. So, we developed the assumption as that : 1) An irregular wave (long crested wave) are the same with the wave linked together by many regular waves which have the same amplitudes and the same encounter half periods on the time history like Fig. 2. 2) This irregular wave produces the same added resistance time history with that produced by regular waves mentioned above at the same time. If we get the wave time history and the theoretical added resistance coefficients (or obtained from the regular wave experiments), we can estimate the corresponding time history of added resistance. Fig. 2 shows those assumption which can be described by equation where ( 4 ) in detail. : is the peak point value of wave, and tp is the time of peak point. The encounter frequency We and the absolute frequency co have the relationship as equation ( 5 ) because the incident angle is 180. or obtained from regular waves as follows. If represents the added resistance in regular wave with frequency co, we get : where H(w) is the double amplitude of regular wave. In this article, our main aim is not to get theoretical time history of added resistance, but we will conduct that simulation after. We are more interested in how to get the added resistance coefficients from transient water waves or irregular long crested waves. If we know and the wave information (amplitude and frequency), of course added resistance coefficients E(w) can be obtained according to equation ( 4 ).

Simple Estimation of Wave Added Resistance from Experiments in Transient and Irregular Water Waves 277 (1) HEAVE AMPLITUDE OF SR208 (2) PITCH AMPLITUDE OF SR208 (3) HEAVE PHASE OF SR208 (4) PITCH PHASE OF SR208 (5) HEAVE AMPLITUDE OF SR108 (6) PITCH AMPLITUDE OF SR108 (7) HEAVE PHASE OF SR108 (8) PITCH PHASE OF SR108 (9) HEAVE AMPLITUDE OF SHSS (10) PITCH AMPLITUDE OF SHSS (11) HEAVE PHASE OF SHSS (12) PITCH PHASE OF SHSS Fig. 1 Transfer functions of heave and pitch

278 Journal of The Society of Naval Architects of Japan, Vol. 174 Fig. 2 Description of Hsu's assumption 2. 3 Estimation method About experiment on drift forces of offshore structure model, we can select suitable mooring system to separate the surge force from drift forces, and can get the drift force time history directly. But about towed ships experiments, it is not easy to get the added resistance time histories directly from experiments. We should find a way to separate the added resistances from the total surging force time histories. So we express variable resistances by next functional polynomia14'5-6) : ( 7 ) where Rw(t) is wave resistance, gi(ti) is impulse response of surge force, and g2(tl, t2) is that of added resistance. About the spectrum of Rw(t), we have : ( 8 ) Here, G1 corresponds to first order surge force transfer function and G2 corresponds to that of second order variable resistance. S means wave spectrum. g2 is obtained by Fourier transformation of G2, and G2 is obtained by cross-bispectral analysis of experimental data or caculated by potential theory5-6). The mean of the surge force is zero and the mean of the added resistance is not equal zero and have already discussed in Reference 3). In irregular waves or in directional spectrum waves, the mean of added resistance is important and usually can be described by following equation5''6) : purpose, we replace in equation ( 7 ) by simple and approximated expression ( 4 ). We consider the R0 (still water resistance) is a constant in any waves at a given ship speed. We obtain it from still water experiment, and eliminate them from the total resistance time histories in transient water or irregular waves. Next, in order to eliminate surge forces, we apply the Fourier transformation to the total resistance time history (here the still water resistance have been eliminated). We suppose that the Fourier spectrum in low frequency area where no wave energy exist correspond to the added resistance component. As zero frequency, the amplitude of Fourier spectrum corresponds to mean added resistance. Using inverse Fourier transformation at this low frequency part, we will get the time history of the added resistance. The added resistance coefficients can be obtained by using Hsu's assumption finally. The obtained added resistance coefficients have some scattering at the same frequency by the present method. So we cut away the too small wave amplitude data and get the mean of the rest. Some times, the estimation time histories of the added resistances have the minus values, especially when the advanced velocity is slow as SR208. In heading seas, the minus added resistance is not probable, so we consider this is an error of estimation. In order to decrease this error, we introduce modification like Fig. 3 when the minus data appeared. The minus values usually appeared when the wave height is small. So we altered the time history to the half cycle sinusoidal curves with the same area as that of integrated one. This method is applicable to transient water wave experiments. And it can also be applied in irregular waves. In the case of transient water wave experiments, the number of wave peaks are limited but the period changes gradually, so this wave is suitable to analyse. Inversely, in irregular long crested waves, the number of wave peaks is very large but the sequences of period are random. So it is difficult to obtain accurate results. Finally, the surge force transfer function can be analysed in frequency domain by usual method to compare with the calculations. Using the added resistance coefficients, the mean added resistance can be estimated like examples in References 3). So considering the transfer functions of surge force, we can estimate the ( 9 ) Here the E(w) is added resistance coefficient of regular wave which can be calculated by the method mentioned at section 2.1 or can be obtained from regular wave experiments. Now we try to get them from transient or irregular water wave experiment efficiently. For this Fig. 3 Method of elimination of minus data

Simple Estimation of Wave Added Resistance from Experiments in Transient and Irregular Water Waves 279 time history of total towing force in arbitrary irregular waves (long crested waves for this time). 3. Experiments 3. 1 Models We selected three models whose principal dimensions are shown in Table 1 : a tanker with a full body called SR208 (=SR196, because it was used in SR208 panel ; a container ship model SR108 which is well known ; and SHSS (a super high speed container ship). About the details, please see the Reference 3). ( 1 ) SR208 (=SR196 model : the part of the added resistance caused by reflection at the blunt bow of model are great because its block coefficient is very high (Cb =0.802). The service speed is slow (Fn =0.129, V=0.639 m/sec in model scale, it is 14 knot in ship scale). ( 2 ) SR108 model : A standard container model. Service speed is 22 knot in ship scale, (Froude number Fn=0.275, V=1.217 m/sec in model scale). Body shape is normal, so the calculations of motions by NSM method agree well with the experimental ones. ( 3 ) SHSS model : A slender and fine body ship. The service speed is high (V =2.656 m/sec in model scale, it is 46 knot in ship scale, Fn =- 0.537). Deck wetness and slamming are depressed as relatively small in spite of high speed. 3. 2 Facilities The experiments were done in the long tank of the Yokohama National University using the before mentioned three models. The schematic view of the measuring facility for SR108 model is shown at Fig. 4. The other models are examined in similar measuring facilities. The dimension of the towing tank is 100 m length, 8 m breadth and 3.5 m depth. Snake type wave generator composed of 24 segmented plunger type wave generators is equipped at the end of the towing tank in order Fig. 4 Arrangement for model experiment to generate many kinds of waves including the directional spectrum waves"). In this paper, we used the long crested transient water waves, regular waves and irregular waves. As irregular waves, we selected the ITTC type one dimensional spectra which has the mean period To2=1.0 sec and its significant wave height equals H1/3=5 cm. The potentiometer at the bottom of the model measure pitch, the one at the top of the guide rod measure heave. Sway, yaw, surge motions are constrained. The measured transfer functions of motions of heave and pitch were used in theoretical calculations for added resistance coefficients. The dynamometer at the middle of the guide rod measure the longitudinal force including surge force and added resistance. Through this guide rod the carriage tows the model at a given speed. Waves were measured by laser type wave surface probe which can analyse the directional spectrum waves"), and by servo-needle type wave probes. In the experiments, we set up two servo-needle type wave probes, one in front of the model, and another one at the side of the center of gravity of the model. So the waves measured by side wave probe correspond to that of theoretical calculation. At low speed, we must notice Table 1 Principal dimensions

280 Journal of The Society of Naval Architects of Japan. Vol. 174 the wall effect and the influence of radiated wave on side wave probe. 3. 3 Experimental data The transfer functions of heave and pitch in transient water waves or in regular waves are shown at Fig. 1. We can see the tendency of them are similar with the theoretical calculations by NSM (new strip method). There are some differences especially about SHSS model. The details are discussed in Reference 3)8)9). By considering experimental results, we used the modified transfer functions when we calculate the added resistance coefficients by method mentioned at section 2.1. The time history of transient water waves and the total resistance by that waves are shown at ( 1 ) and ( 2 ) of Fig. 5. They are the examples of the experimental data. The wave data were measured by the side wave probe. In experiments of SR208 (=SRI.% and SR108, we all used 20 Hz sampling time. But for SHSS, because it had a super high speed, we used 40 Hz sampling time in order to realize the enough resolution. SHSS is high speed, so the effective measuring time is short in one run, so many runs are needed. The time histories of total towing force excluding steady towing force in still water is composed of surge forces, low frequency parts and the high frequency parts of the added resistances. They even include the elastic vibration of the dynamometer and other electrical noise. Usually, the suitable filter is used to cut off those complex factors. But we did not use the filter to avoid losing the useful parts. 4. Comparison 4. 1 Transient water waves Taking one example of SR208 (tanker) in a transient water wave, the estimation flow is shown at Fig. 5. The time history of the transient water wave is shown at ( 1 ), its data number is 1024, and its duration is about 50 sec. Of course, its mean is 0. The corresponding towing force time history is shown at ( 2 ), and the mean is about 11.5 gram. That is the mean added resistance. By Fourier transformation, we get their Fourier spectra as ( 3 ). The solid line is the spectrum of towing force and the broken line represent the wave spectrum. We separate the parts of the resistance : ( a ) About the low frequency part where there is no corresponding wave energy, we can consider that this is the low frequency part showing the added resistance. Considering the Hsu's assumption, this part is essential part to estimate the added resistance coefficients, because the variation of added resistance are concentrated at low frequency area discussed at section 2.1. ( b ) The middle part corresponds to wave energy part, and this part is essential for surge force. ( c ) The high frequency part where there is no wave energy. This part is the high frequency part of the added resistance and other noise like vibrations. Then we cut off the middle part of the surge force. Cut off criteria is that the value of amplitude of wave is greater than the 1/20 of maximum amplitude. As ( 4 ) of Fig. 5, the left spectrum has two parts : the low frequency part 0-2.2 rad/sec and the high frequency part that is greater than 8.7 rad/sec. We think this high frequency part only causes the fast vibrations and has less contribution to the added resistance. Finally, we selected the rest low frequency part to estimate the time history of the added resistance. For getting the time history of the added resistance, phases of the wave and the resistance are also needed as shown ( 5 ), ( 6 ) of Fig. 5. The phase of the towing force is also separated into three parts and we use the low frequency part for the added resistance. By inverse transformation of Fourier spectrum, using the low frequency parts of the amplitude and its phase, we got the time history of the variable added resistance shown at ( 7 ) of Fig. 5. We see that some parts show minus values. For this phenomena, there will be two reasons. One is that, for this part the wave amplitude is too small as can be seen in the time history of wave at ( 1 ) of Fig. 5. So there will be some error. Another reason will be that there exist repeat phenomena about Fourier inverse transformation and a ghost time history appears where there is no waves. So we cut off the part where the amplitude of wave is smaller than the 1/10 of the wave maximum amplitude, and we get the time history of the added resistance shown at ( 8 ) of Fig. 5. Some times, the minus data still exist after that treatment. For those cases we make further modification discussed at section 2.3 or neglect them. Using Hsu's assumption, we get the table about periods, heights and the added resistances from individual waves. Transform the encounter periods into the absolute wave angular frequency and normalize the added resistances into the added resistance coefficients, and we can get the curve shown at ( 9 ) and (10). The abscissa of ( 9 ) is absolute wave angular frequency and that of (10) is the wave length/model length. The ordinate of ( 9 ) and (10) are the added resistance coefficients normalized by equation ( 6 ). There are many values at a given frequency, but mean of those values is shown in this figure. The theoretical calculations and the results of regular waves are both shown at ( 9 ) and (10). They show relatively good correspondence each other except the short wave length part. At short wave length region, the increasing tendency of the added resistance coefficients can be seen, and from the results of regular waves, the same tendency exists at that area. We used the same estimation method to analyse the data of SR108 (container) and SHSS (a super high speed ship). Fig. 6 show those results. The left side figure correspond to SR108 and the right side figure

Simple Estimation of Wave Added Resistance from Experiments in Transient and Irregular Water Waves 281 (1) Time History of Wove (2) Time History of Resistance (3) Fourier Spectrum of SR208 (4), Added Resistance Fourier Spectrum (5) Phase of Transient Wave (6) Phase of Resistance (7) Time History of Added Resistance (8) Time History of Added Resistance (9) Added Resistance Coefficient (10) Added Resistance Coefficient Fig. 5 Estimation of added resistance coefficiemts of SR208 (tanker) in transient water waves correspond to SHSS. From the top, time histories of transient waves, time histories of total towing force, time histories of added resistances and the added resistance coefficients are shown. The added resistance time history transformed from the Fourier spectrum of SR108 has less minus values than SR208 because its mean added resistance is greater than SR208. So the modification mentioned at section 2.3 was not introduced here. The data of SHSS is similar and the ( 6 ) of Fig. 6 shows the final one which is the result of cutted off the small wave amplitude data. Both of them show good coincidence comparing with

282 Journal of The Society of Naval Architects of Japan, Vol. 174 (1) Wove of SR108 (2) Wove of SHSS (3) Resistance of SR 108 (4) Resistance of SHSS (5) Added Resistonce of SR108 (6) Added Resistonce of SHSS (7) Results of SR108 (8) Results of SHSS Fig. 6 Estimation of added resistance coefficients of SR108 and SHSS in transient water waves the data of the regular waves. Theoretical calculation is also similar to the results from regular waves. Considering the both results of SR108 and SR208, we can say that present estimation method is available for different ship models with different speeds. Of course, the analysis are done for other transient water waves, and the same results had been obtained. It must be noticed that the number of result points are decided by sampling times and the number of wave peaks. Sampling time decides the frequency resolution, and wave peak number decides the data number. As ( 7 ) of Fig. 6, we could not get the point near the peak because the sampling time is 20 Hz. We should get data by higher sampling frequency like SHSS shown at ( 8 ) of Fig. 6. If we use transient water wave with longer duration, the accuracy will be improved. 4. 2 Irregular waves Usually, irregular waves have longer duration than transient water waves, and we can use the same method to estimate the added resistance coefficients. In order to compare with the results by transient water waves, we selected the same data number (1024, and this correspond to 50 sec for SR208, SR108 and 25 sec for SHSS) as a transient water wave example. For example, the time histories of long crested irregular wave and its towing force excluding still water resistance of SR208 are shown at ( 1 ) and ( 2 ) of Fig. 7. From the time history of the irregular wave, we find the number of wave peak is large but there are some small wave height parts near the zero cross line. From the time history of towing force, the mean added resistance is 189.4 gram and this is smaller than the amplitude of total towing force. Their Fourier spectra of ( 1 ) and ( 2 ) are shown at ( 3 ) of Fig. 7. Using the same analysis as before, we cut off the wave parts of the resistance shown at ( 4 ) of Fig. 7. And, also eliminate the high frequency part, we get the Fourier spectrum amplitude of the added resistance. ( 5 ) and ( 6 ) of Fig. 7 show the phases of wave and towing force. Same as the transient waves, we divided the phase of the resistance into three parts and use the low frequency part. The time history of the added resistance obtained by inverse Fourier transformation, is shown at ( 7 ) of Fig. 7. There are also minus value parts and modified by the method mentioned at section 2.3. The final time history of the added resistance is shown at ( 8 ) of Fig. 7. The added resistance coefficients are shown at ( 9 )

Simple Estimation of Wave Added Resistance from Experiments in Transient and Irregular Water Waves 283 (1) Time History of Wove (2) Time History of Resistonce (3) Fourier Spectrum of SR208 (4) Added Resistance Fourier Spectrum (5) Phose of Irregular Wove (6) Phase of Resistance (7) Time History of Added Resistonce (8) Time History of Added Resistonce (9) Added Resistonce Coefficient (10) Added Resistance Coefficient Fig. 7 Estimation of added resistance coefficients of SR208 (tanker) in irregular waves and (10) of Fig. 7. We can get the result from irregular wave, but not so good as the one in transient water waves. The number of wave peak is increased, but the wave itself is more complex, and we must modify the minus values in irregular waves. Transient water wave or irregular long crested wave, both of them can be used to estimate the added resistance coefficients. But considering the ship motions and surge force which will discussed after, expriment in transient water waves is the best one. The estimated results of SR108 (container) and SHSS (super high speed ship) are shown at Fig. 8. The

284 Journal of The Society of Naval Architects of Japan, Vol. 174 left correspond to SR108 and the right to SHSS. From the top, time histories of irregular water waves, time histories of towing force, time histories of added resistances and the added resistance coefficients are shown. The time history of the added resistance of SHSS has no minus values and need no modification. But the time history of the added resistance of SR108 is the result after modification by the method described before. 4. 3 Surge force Back to the transient water waves, when we separate the added resistance from the total resistance, the rest is surge force. The theoretical calculation of surge force had discussed in section 2.1. We can compare these theoretical results with the data analysed from the transient water wave experiments. Fig. 9 show the surge force transfer functions of SR208, SR108 and SHSS. ( 1 ) ( 2) ( 3 ) and ( 4 ) of Fig. 9 show the results of SR208 ; ( 5 ) and ( 6 ) of Fig. 9 are results of SR108 ; ( 7 ) and ( 8 ) of Fig. 9 are results of SHSS. For each model, the amplitude of transfer function and the phase of transfer function are shown. About amplitude, both the theoretical calculations and the regular wave experimental data are shown, but about phase, the results of regular waves are not shown. ( 1 ) of Fig. 9 show the added resistance time history, solid line is the one shown at ( 8 ) of Fig. 5 and the broken line is estimated from the obtained added resistance coefficients and the wave data of transient water wave. We approximate the obtained added resistance coefficients estimated from the transient water waves into a continuous curve, then apply the Hsu's assumption to get the broken line of the ( 1 ) of Fig. 9. There are some differences between two lines. It is because when we estimate the coefficients we cut off the small wave height region. We used the broken line data to analyse the surge force here. The theoretical curve or regular wave data also can be used to estimate the time history of the added resistance, and all of them have the similar results. Time history of surge force which is obtained by cutting off the added resistance and also cutting off the higher frequency than 2 Hz is shown at ( 2 ) of Fig. 9. Transfer function (amplitude and phase) is shown at ( 3 ) and ( 4 ) of Fig. 9. Looking at the results of the three ships, we can see the estimated results from transient water waves is very (1) Wove of SR108 (2) Wove of SHSS (3) Resistance of SR108 (4) Resistonce of SHSS (5) Added Resistance of SR108 (6) Added Resistonce of SHSS (7) Results of SR108 (8) Results of SHSS Fig. 8 Estimation of added resistance coefficients of SR108 and SHSS in irregular waves

Simple Estimation of Wave Added Resistance from Experiments in Transient and Irregular Water Waves 285 (1) Time History of Added Resistance (2) Time History of Surge Force (3) Surge Force of SR208 (4) Phase of SR208 (5) Surge Force of SR108 (6) Phose of SR208 (7) Surge Force of SHSS (8) Phose of SHSS Fig. 9 Transfer functions of surge force similar to regular wave results, even in the case of SHSS. The calculated values of SR208 and SR108 are similar to the experimental data but the SHSS has large differences. The reason is the SHSS has very high speed and the present calculation method is not enough for it. 4. 4 Time history simulation It is shown that the added resistance coefficients and the transfer functions of surge force can be estimated from the experiments in transient water waves using method discussed above. Using those results, we can make numerical time simulation in long crested irregular waves. First, analyse the irregular wave into Fourier spectrum. Then calculate the Fourier spectrum of surge force from the transfer functions. Use the inverse Four-

286 Journal of The Society of Naval Architects of Japan, Vol. 174 ier transformation to get the surge force time history. Of course this estimation can be made by convolution integral as shown by equation ( 7 ). On the other hand, use the Hsu's assumption to get the added resistance time history from the coefficients (it is discussed at above section). Finally, get the sum of them as the simulation results. ( 1 ) and ( 2 ) of Fig. 10 show the simulated data of SR208, ( 2 ) is the expanded data of ( 1 ) from 5 sec to 10 sec. Solid line is the experimental data and the broken line represent the simulated data. ( 3 ) and ( 4 ) show the results of SR108, ( 5 ) and ( 6 ) of Fig. 10 show the results of SHSS. The second figure is the expanded (1) Time History Simulation of SR208 (2) Time History Simulation of SR208 (3) Time History Simulation of SR108 (4) Time History Simulation of SR108 (5) Time History Simulation of SHSS one about 5 sec time. The advancing speed of SR208 is slowest, the simulation is very close to the experimental data. For SR108 ship with speed of Fn =0.275, the result is a little worse than SR208, and the simulation of SHSS (its speed is Fn =0.537) is worst but tendency is similar to experiment. 5. Conclusions Using different type of three models with different speed : Tanker ship SR208 ( =SR196 C) in Fn =0.129, Container ship SR108 in Fn =0.275 and Super high speed container ship SHSS in Fn =0.537, we measured towing force in long crested transient water waves and irregular waves (the data of the regular waves are also used). Analysing the time history of wave and the towing force, we estimated the wave added resistance coefficients by newly proposed method from single experiment and also obtained the transfer functions of surge force. We also made theoretical calculations and simulation of the towing force time history and we obtained the following conclusions. ( 1 ) A simple estimation method of wave added resistance coefficients from single experiment in long crested transient water waves or in irregular waves could be proposed by separating the variable added resistance. ( 2 ) Though there is need to investigate about the effect of wave height and other factors on estimated results, the present method is useful because the results showed good coincidence with the results by regular waves. The estimated results are also compared with the conventional theoretical calculations. ( 3 ) Considering the necessary data length and the estimation accuracy, it is better to use the transient water waves than to use the irregular waves. ( 4 ) We could separate the surge force from the total towing force and could obtain the transfer functions of them. Obtained results were compared with the theoretical calculations and pointed out the necessity of theoretical improvement for high speed ship. Finally it will be said that the proposed method here will be applicable to the estimation of transfer function of slow drift force of moored semisubmersible offshore structures. Acknowledgement (6) Time History Simulation of SHSS Fig. 10 Simulation of time history of resistance The authors would like to acknowledge the great co-operation received from Mr. K. Miyakaya and Mr. T. Takayama in experiments. The authors appreciate the contributions of ship team's students who carried out the experiments in the towing tank of Yokohama National University.

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