MODEL TESTS OF THE MOTIONS OF A CATAMARAN HULL IN WAVES

MODEL TESTS OF THE MOTIONS OF A CATAMARAN HULL IN WAVES C. Guedes Soares 1 N. Fonseca P. Santos A. Marón 2 SUMMARY The paper describes the results of model tests of a catamaran in regular waves. The program concentrated on the heave, pitch and roll behavior of the catamaran and considered the effects of speed and heading on the motion behavior. Some derived responses were also investigated such as the vertical accelerations and relative motions and the mean added resistance in waves. The tests were carried out for several wave headings and Froude numbers and for a wide range of wave frequencies. INTRODUCTION The seakeeping behavior of catamarans in waves have been studied since the seventies when there was the successful implementation of strip theory to seakeeping problems. However the amount of work applied to this ship type is much smaller than for monohulls. This is especially true for the published experimental data, which is essential for the validation of numerical models. Some of the earlier work is discussed here so as to point out the main features that have been covered in the experimental work. A comprehensive set of sea motions and loads experiments have been conducted at the Naval Ship Research and Development Center (NSRDC) by Wahab et al (1971) for an ASR catamaran model advancing in waves. The ASR catamaran was intended to service the deep sea rescue vehicle and she has asymmetric forebodies and symmetric aftbodies. The heave, pitch, roll and the relative motions at the bow were measured for 5 hull separations, several forward speeds and 13 headings between head and following regular waves. The hull separations varied from H/B =.7 to H/B = 2.11, where H is the hull separation and B is the beam at he waterline of each hull. Froude numbers varied from. to.41. In addition to the motions, five components of the wave induced loads at the centre-line of the cross-deck structure were measured for the 5 hull separations and several forward speeds in beam, bow and head regular waves. Some tests were also carried out in irregular waves. Lee et al (1973) studied numerically and experimentally the motions of two catamarans in head seas and the loads in beam seas. The tests were conducted at the NSRDC in order to obtain the response amplitude operators (RAOs) of the induced motions. The most extensive experimental work was carried out with a model of the ASR catamaran. One hull spacing was used, corresponding to H/B = 1.58. Three groups of tests were carried out, namely: forced oscillation tests to obtain the added mass and damping coefficients, restrained model tests to obtain the wave-exciting force and moment and tests in regular waves to obtain the heave and pitch motions. For the motion tests the model was self-propelled and free to oscillate in all six modes of motion. Four Froude numbers were used between. and.414, and a range of head regular wavelength-to-shiplength between.5 and 3.5. The experimental heave and pitch results are presented only for the Froude number.14 and head waves. Additional heave and roll data are also presented for zero speed condition and beam regular waves. A model of the CVA catamaran was also tested at zero Froude number and beam regular waves. The CVA is a conventional catamaran of large beam, shallow draft and symmetric hulls. Measurements are presented of the heave and roll motions and also of the bending moment and vertical shear force acting on the crossbeam structure. Faltinsen et al (1992) conducted a series of self propelled tests at the Ocean Environment Laboratory of MARINTEK with a catamaran model, in order to measure the induced motions and loads on the cross deck structure induced by regular waves. The objective was to obtain data to validate a 2 1/2 - dimensional linear method. The model has symmetric hulls, with deep V-shape at the bow, a relation H/B of 2.44. Heave, roll and 1 Unit of Marine Engineering and Technology, Technical University of Lisbon,Av. Rovisco Pais, 149-1 Lisbon, Portugal 2 Canal de Experiencias Hidrodinámicas de El Pardo, calle Sierra, El Pardo, 2842 Madrid, Spain - 1 -

pitch results are presented for beam and bow waves and a Froude number of.49. Loads were measured at a longitudinal cross section close to the centre-line include the vertical shear force, vertical bending moment and pitch connecting moment. Kashiwagi (1993) compared experimental data from a catamaran with a Wigley demi-hull with numerical predictions from unified slender body theory. The experiments were carry out in head regular waves and Froude numbers of.15 and.3 in order to measure the heave and pitch responses. Van't Veer and Siregar (1995) presented experimental results for a Wigley catamaran model at forward speed in head regular waves. The tests were carried out at the towing tank of the Delft Ship Hydrodynamic Laboratory with the objective of validating a strip theory program. In fact the results were obtained using a Wigley monohull towed in the vicinity of the tank wall, thus simulating a Wigley catamaran by using the tank wall as the symmetry plane of the vessel. The experiments consisted of forced heave oscillation tests with a segmented model, restrained model tests and motion tests. The tests were carried out with three different hull spacing, corresponding to H/B = 1.4, 2.1 and 3.14, where H is the hull separation and B is the beam at the waterline of each hull. For each hull spacing three Froude numbers were used,.15,.3 and.45. van't Veer and Siregar (1995) paper presents only part of the experimental data. Siregar (1995) publishes the complete set of experimental data in a report. Hermundstad et al (1995,1999) validated a linear method for hydroelastic analysis of high-speed vessels with test data from a high-speed catamaran model in regular waves. The experiments were carried out at MARINTEK, with a hinged model where each hull consists of three separate rigid segments that are connected by elastic hinges. The model with a relation H/B = 2.82, was self propelled and used an autopilot system. The relevant measured quantities were the heave, roll and pitch motions and the vertical shear force and vertical bending moment at each hinge. Two speeds were tested corresponding to Froude numbers of.47 and.63. In regular waves three headings were used, namely: head waves (º), bow waves (3º) and beam waves (9º). Tests were also performed in calm water, in impulsive waves and in two long-crested irregular seastates. Only the results in regular waves are present in Hermundstad et al (1995,1999). Complete information on the tests can be found in Hermundstad (1995). They compared numerical predictions from linear and quasinonlinear strip methods with experimental results from three catamarans in head regular waves. Two of the catamarans were tested by other authors, namely the ASR catamaran from Wahab et al (1971) and the MARINTEK catamaran from Faltinsen et al. (1992). A catamaran was tested by Fang et al. (1996,1997) at the Hydrodynamic Laboratory of the University of Glasgow. The catamaran had a high-speed hard-shine hull form with V-type cross sections on the forebody and a cut-off transom stern. Details of the catamaran and additional motion results can be found in Incecik et al (1991). A vertical post towed the model and it was free to heave and pitch around the centre of gravity. One of the objectives of these tests was to assess the nonlinearity of the heave and pitch RAOs with the wave height. For this reason the RAOs were obtained for three different wave amplitudes and three Froude numbers (Fn =.,.226 and.677). They compared numerical predictions from linear and quasinonlinear strip methods with experimental results from three catamarans in head regular waves. Two of the catamarans were tested by other authors, namely the ASR catamaran from Wahab et al (1971) and the MARINTEK catamaran from Faltinsen et al. (1992). More recently Centeno et al (1999) presented the results of an experimental program carried out at the Hydrodynamic Laboratory of the University of Glasgow, with a hard-chine catamaran model in head regular waves. The demi-hull is the same as used by Fang et al. (1996,1997) and Incecik et al (1991), but with two different hull spacings corresponding to H/B of 2.53 and 3.8, where H is the hull spacing and B is the beam at the waterline of the demi-hull. The heave and pitch RAOs were obtained for four different Froude numbers:.,.25,.625 and.75. From the review of published experimental data on wave induced motions and loads on catamarans, one may conclude that the amount of experimental data is not large and great part of the results are for head waves only. In fact only a few authors present results for bow and beam waves and there seems to be no results for quartering and following seas. In addition important results like the relative motions at the bow and vertical accelerations seem to be not published also. Concerning the wave induced loads at the cross deck structure, which are of primary importance for the design of medium to large size catamarans, such data is presented only for a few hull forms and speed / heading conditions. Finally, one of the seakeeping characteristics that naval architects are most interested is the added resistance in waves. Again it was found that there are no published experimental data of the added resistance in waves for catamarans. In the present experimental program data was obtained for a new catamaran model, covering some of the aspects where there seems to exist no data published. EXPERIMENTAL PROGRAM AND SETUP Facilities The experimental programme was carried out at the Laboratory of Ship Dynamics of El Pardo Model Basin in Madrid. The laboratory is made up of three basic facilities: the tank, the wave generator and associated power plant and the computerized planar motion carriage. The lay out and main characteristics of the testing facilities are shown in figure 1. The snake-type wave generator is formed by sixty elements. By adjusting the phase of the almost sinusoidal motion of each of the elements, the direction of the waves can be adjusted at will within - 2 -

certain limits. Each element is of the flap type, hinged at two meters from the tank bottom. Figure 1 Layout of testing facilities from below reducing the effect of meniscus as much as possible. Ship Model Scale 1 1:1 Length overall (m) 43.5 4.35 Length between pp (m) 43. 4.3 Beam overall (m) 11.4 1.14 Demi-hull beam at wl (m) 2.7.27 Ship depth (m) 4.55.45 Draft (m) 1.354.14 Trim (m).. Hull spacing (m) 8.6.86 LCG aft of midship (m) 3.38.3 VCG from baseline (m) 3.71.371 Displacement (Kg) 18411 184.1 Roll radius of gyration (m).267lpp Pitch radius of gyration (m).443boa Table 1 Main characteristics of the catamaran An inclination test was made with a lateral displacement of a 1 Kg with the model in the water in order to determine the vertical position of the center of gravity. Figure 2 Catamaran body lines The Computerized Planar Motion Carriage (CPMC) can follow any path in the horizontal plane either towing a captive model, as in the present case, or tracking a free-running, self-propelled model. It is used both for seakeeping and manoeuvring studies. The tank has a depth of five meters, which makes any bottom effects negligible for typical model sizes. A pit with a total depth of ten meters allows the testing of deep-water structures. The dimensions of the basin are adequate for testing models of about 4 to 6 meters in length. Model characteristics and instrumentation The tests were carried out with a model of a catamaran passenger ferry designed to operate in sea environment. The main characteristics of the catamaran and model are given in Table 1 and Figure 2 presents the bodylines. The model was constructed in FRP at a scale of 1:1. In the preparation of the model checks were made for the displacement, draft and trim. The model weight was obtained from the scale. The drafts were observed with the model on the water against draft marks previously painted on the model. Transparent windows allowed the observation of draft marks The longitudinal and transversal radius of inertia were adjusted to the value of.267 Lpp and.443 B respectively by measuring the natural period of oscillation of the model on an inertia table designed for such purpose. This table consists of a platform that can rotate about a horizontal axis. The table is maintained horizontally by means of calibrated springs on each axis. The oscillation period is related to the inertia of the table itself, which is known, the inertia of the model and the constant of the springs, which is also known. Therefore the measurement of this period allows the calculation of the inertia of the model itself. The period was measured by means of an inclinometer whose electric signal is fed to the data acquisition system of a PC that performs the necessary computations. Instrumentation Figure 3 is a sketch of the model instrumentation. The model was towed at the required speed by means of two vertical bars connected to the carriage. These bars were free to move vertically along a low friction linear guide allowing the model to heave. The forward bar was allowed to rotate in the longitudinal plane to compensate the small variations in the longitudinal distance when pitching. The bars were connected to the model through a double axis hinge allowing free pitch and roll. The list of measured quantities and related instrumentation is: The heave, pitch and roll motions were measured by a noncontact optical tracking system. - 3 -

Five accelerometers were fitted to measure vertical accelerations and one for transversal acceleration. Two load cells at the towing bar allowed the measurement of added resistance. A capacitative wave probe was installed forward of the model to measure the incoming wave elevation and four others close to the bow (two on each hull). Two pressure transducers at the central hull. Figure 3 Instrumentation of the model All signals were digitized at 2 Hz, corresponding to 6.3 Hz at full scale. The time series were recorded on a data acquisition PC while another PC connected through a local area network allows to observe graphically the time traces of all the signals in real time. This allows a quick check and quality control of the instrumentation. Experimental program The testing program in regular waves includes four Froude numbers between. and.6, and several headings for each speed of advance. For each of these conditions a range of wave frequencies was used corresponding to a ratio wave-length to ship-length from.5 to 4.. The objective of the tests was to determine experimentally the linear responses of the model, in order to validate linear procedures. For this reason the waves selected had to be in one hand of small amplitude and on the other hand they had to generate responses suitable for accurate measurements. Having this in mind, the wave height (at full scale) varied from.43m for short waves, to.5m for medium length waves and.7m for long waves. These correspond to wave slopes kζ a (k is the wave number and ζ a the wave amplitude) of.6 for short waves and.1 for long waves. Around the vertical motions resonance frequency, the wave slope was about.4. The majority of the tests were performed with the third bow installed in the model. Those conditions in which impact pressures were measured at the third bow, were repeated without the third bow. Part of the experimental results are presented here, namely the responses for Froude number.4, for head and bow waves and the responses for head waves and all four Froude numbers. EXPERIMENTAL RESULTS Figures 4 to 8 show several responses of the catamaran model in regular waves and advancing with a Froude number of.4. Three different headings are considered, namely: head waves (18º) and two bow wave conditions ( and ). The results are presented in an x-axis scale of nondimensional wave frequencies, where? represents the wave frequency, L is the length between perpendiculars and g is the acceleration of gravity. The symbols represent experimental points. In order to better understand the tendencies of the responses, a "smooth" line connects the experimental points. However it should be noted that the line does not necessarily represents the real experimental curve, since a limited number of discrete points have been tested. Each response is represented by its response amplitude operator (RAO) and phase angle. The RAO is defined by the first harmonic of the response divided by the first harmonic of the incident wave elevation. Other terms beside the wave elevation may be used to nondimensionalise the responses. The phase angles represent the delay of the response (first harmonic) with respect to the maximum wave elevation at the center of gravity of the model. Figure 4 represents the heave amplitudes (ξ 3 ) divided by the wave amplitudes (ζ a ) and respective phase angles. One resonance peak can be observed approximately at the nondimensional frequency of 2.5, which corresponds to a relation wave length - ship length of 1. The peak is higher for head waves than for bow waves. It is possible that the true resonance peak is not accurately captured by the experiments because of the frequency separation between experimental points. The ideal would be to test more points around the nondimensional frequency 2.5. As expected, the phase angles for the three headings are similar for long waves and start to deviate for shorter waves. The pitch linear response is shown in figure 5. The amplitudes are nondimensionalised by the wave slope ( kζ a ). A resonance peak is now identified for the nondimensional frequency around 2.1, which corresponds to a wave length to ship length ratio of 1.4. The amplitudes tend to decrease from head to bow waves, following the decrease of effective wave slope. - 4 -

3/ a 1.4 1.2 1..8.6.4.2. 3 25 2 15 1 5-5 Heave motion - RAO 18º Heave motion - phase angle 18º Figure 4 Heave motion response for Froude number.4 5/k a 1.4 1.2 1..8.6.4.2. 36 3 24 18 12 6 Pitch motion - RAO 18º Pitch motion - phase angle 18º Figure 5 Pitch motion response for Froude number.4 Figure 6 represents the roll response in bow waves. The amplitudes are also dimensionalised by the wave slope. One can say that the roll amplitudes of the catamaran model are very small in bow waves, reaching to a maximum value of only 6% of the wave slope for heading. More interesting are the vertical accelerations measured on the model. Figure 7 presents the vertical acceleration responses measured at the bow of the weather hull. The longitudinal position is 15.3m forward of midship. The amplitudes are shown in two vertical axis. One represents nondimensional values, zl/ζag, being z the vertical acceleration and g the acceleration of gravity. The other represents acceleration amplitudes per unit wave amplitude, divided by the acceleration of gravity. One can observe the presence of a large peak of around 2g, for wave lengths similar to the ship length. This is mainly related to the high encounter frequency, which occurs simultaneously with relatively large amplitude of the vertical motions. Also the "adverse" effects of heave and pitch on the vertical accelerations tend to sum up for this condition. The relative vertical motions between the ship and the waves was measured at the two bows, and at each bow inside and outside. The longitudinal position is 16.1m forward of midship for the inside sensors and 17.3m for the outside sensors. The response curves showed that for head and bow waves there are small differences in the relative motions inside and outside of the hulls. For this reason results are presented only for the outside of the weather hull. Figure 8 shows the relative motion amplitudes divided by the wave amplitudes and the related phase angles. The peak of the transfer function is about 3.5 for waves of the same length as the ship. 4/k a.7.6.5.4.3.2.1. 5 4 3 2 1 Roll motion - RAO Roll motion - phase angle Figure 6 Roll motion response for Froude number.4-5 -

(zl)/( ag) 1 8 6 4 2 25 2 15 1 5 Vertical acceleration at the bow - RAO 18º 2. 1.5 1..5. Vertical acceleration at the bow - phase angle 18º z/ ag [m -1 ] Figure 7 Vertical acceleration at the bow for Froude number.4 r/ a 4 3 2 1 2 15 1 Relative Motion at the Bow - RAO 18º ω sqrt(l/g) 5 Relative Motion at the Bow - phase angle 18º 1. 1.5 2. 2.5 3. 3.5 Figure 8 Relative motion at the bow for Froude number.4 Figures 9 and 1 represent the magnitudes of several responses of the catamaran in head waves and for four Froude numbers, respectively.,.2,.4 and.6. Figure 9 shows the heave and pitch nondimensional amplitudes. As expected, one can observe that the resonance peak increases very much with the the Froude number. Although the entire justification is not straight forward due to the complexity of the phenomena envolved, this increase of the resonance peak with the Froude number is mainly due to the shift of the resonance frequency to the longer wave lengths where the exciting forces are of higher amplitude. The increase of the vertical motions with the Froude number is followed by the other related vertical responses, namelly the vertical acceleration at the bow and relative motion at the bow (figure 1). 3/ a 5/k a 2.5 2. 1.5 1..5. 1.6 1.2.8.4. Heave motion - RAO Fn =. Fn =.2 Fn =.4 Fn =.6 Pitch motion - RAO Fn =. Fn =.2 Fn =.4 Fn =.6 Figure 9 Heave and pitch nondimensional amplitudes in head waves Finally the mean added resistance in waves is presented in figure 11. The left side graph shows the mean added resistance in head and bow waves for Froude number.4. The right side graph shows the same response in head waves for four Froude numbers between. and.6. The amplitudes are 2 nondimensionalised by ρgζa B 2 / L, where the term ρ represents the mass density of the water and B is the beam overall at the waterline. - 6 -

zl/ ag 1 8 6 4 2 Vertical Acceleration at the Bow - RAO Fn =. 2. Fn =.2 Fn =.4 Fn =.6 1.5 1..5 z/ ag Rwa/( g a 2 B 2 /L) 8 6 4 2 Mean added resistance in waves 18º. r/ a 4. 3. 2. 1. Relative Motion at the Bow - RAO Fn =. Fn =.2 Fn =.4 Rwa g a L 1 8 6 4 2 Mean added resistance in waves Fn =. Fn =.2 Fn =.4 Fn =.6. Figure 1 Vertical acceleration and relative motions amplitudes in head waves Figure 11 Mean added resistance in head waves. Head and bow waves at Fn =.4 (left side). Head waves at several Froude numbers (right side). The procedure to obtain the mean added resistance forces is the following. Firstly, the total towing force is measured using two load cells with the model at speed in waves. This is an oscillatory force. Secondly, the mean value of the towing force is calculated. Finally, the mean added resistance in waves is the difference between the mean towing force in waves and the resistance to the advance in still water. The later was measured at the initial stage of the tests. The mean added resistance in waves is a second order quantity proportional to the square of the wave amplitude, and with a magnitude much smaller than the first order forces. For these reason a high degree of accuracy is needed in the experiments. The small degree of dispersion of the experimental points in figure 11 suggest that these forces were measured accurately. The results in figure 11 show a large peak for the nondimensional frequency of 2.5, which is mainly related with the energy spent with the large vertical relative motions at the bow. COMPARISON WITH OTHER EXPERIMENTAL DATA The hull form considered in this work is different from the ones considered in other experimental studies and thus a direct comparison is not possible. However some qualitative checks will allow conclusions about the consistency of data and general trends of the results. The heave and pitch results from Hermundstad et al. (1999) in head waves at Fn =.47, show resonance peaks for both responses about 1% higher than the results presented here for Fn =.4. However Hermundstad results of vertical motions in head waves for Fn =.47 and.63 do not differ much in terms of amplitudes, while the results present here show a large increase of resonance peaks between Fn =.4 and.6. His heave peaks for the two Froude numbers are approximately the same and the pitch peak is even around 2% lower for the higher speed. In the case of the present results between Fn =.4 and Fn =.6, the heave and pitch resonance increase respectively 6% and 2%. - 7 -

It is possible that the different behavior between the two hulls is partly related with the different stern geometry. Both catamarans have transom sterns, however Hermundstad's catamaran has a tunnel stern, while the other has a conventional transom stern. While the former two sets of results agree relatively well, the results from Wahab (1971) show much larger resonance peaks for heave and pitch motions, for head and bow waves at similar Froude numbers (Fn around.4). Wahab's heave peaks are about two times larger than those presented here and the pitch peaks around 4% higher. It is believed that the difference is mainly due to the lower relation Bdh/T, which is 2. for the present catamaran and 1.33 for the ASR catamaran (Bdh is the demi-hull beam at the waterline and T is the draft). In general, this lower relation results in smaller damping as percentage of critical damping, thus larger resonance peaks. Another consequence is the shift of natural frequencies to longer waves where excitation forces and moments are higher. This can also be observed in the graphs: while the present catamaran has the resonance peak for L / L = 1., the ASR has for L w / L = 12.. pp w Another characteristic which is believed to have some effect on the vertical motion resonance amplitudes is the stern geometry. For higher Froude numbers the transom stern generates lifting effects which may have an influence on the heave and pitch responses. The ASR has a conventional stern while the present catamaran has a transom stern. When the heave response from Fang et al (1996) in head waves at Fn =.667 is compared with the same response of the present catamaran at Fn =.6, one find a resonance peak about 3% lower. The pitch peak is around 1% lower. The relation Bdh/T for the V-1 is 2.3, thus a little higher than for the model studied here. In addition the hulls have a hard-chine below the water-line along a large part of the length, thus it is probable that viscous damping effects occur during the vertical motion due to flow separation. These two characteristics of the hulls result in relatively low resonance heave peaks. Comparing different Froude numbers in Fang's heave and pitch data (Fn=.23 and.68), it is seen that only the higher speed shows peaks of the transfer functions above 1. Experimental results of the vertical motions are also presented by Kashiwagi (1993) for a Wigley twin-hull in head waves. This hull configuration does not have a transom stern, and it is interesting to see that the vertical motions are quite resonant even for a relatively small Froude number of.3. Similar results were obtained by van't Veer and Sireger (1995) with another Wigley twin-hull model. CONCLUSIONS The paper describes the results of model tests of a catamaran in regular waves. The measured quantities are the heave, pitch and roll motions, the relative motions at the bow, the vertical accelerations and the mean added resistance in waves. The influence of the heading and of the Froude number on the catamaran behavior was investigated. Concerning the influence pp of the heading, it was found that the vertical responses tend to be slightly higher for head waves than for bow waves but the differences are small. On the other hand, the vertical responses are very sensitive to the Froude number, increasing with the Froude number. The comparison of experimental data presented here with other published by other authors, shows that the results agree with each other and the differences may be attributed to the different hull configurations. ACKNOWLEDGEMENTS The experimental work was developed in the project Experimental Study of Motions on Catamarans which has been funded by the Commission of the European Communities, through the Spanwave Project under contract ERB FMGE CT95 74. REFERENCES Centeno, R., Varyani, K.S. and Guedes Soares, C., (1999), Experimental Study on the Influence of Hard Chine Hull Spacing on Catamaran Motions, to be published. Couser, P., Hudson, D.A., Price, W.G. and Temarel, P., (1995), "Prediction of Hydrodynamic Loads and Motions of a High Speed Catamaran in regular Waves", Proceedings of the 3 rd Symposium on High Speed Marine Vehicles, Naples. Faltinsen, O., Hoff, J. R., Kvalsvold, J. and Zhao, R., (1992), Global Loads on High-Speed Catamarans, Proceedings of PRADS 92, ed. J.B. Caldwell and G. Ward, Elsevier Applied Science, London and New York, Vol. 1, pp.136-1375. Fang, C. C., Chan, H. S. and Incecik, A., (1996), Investigation of Motions of Catamarans in Regular Waves-I, Ocean Engineering, 23, No. 1, pp. 89-15. Fang, C. C., Chan, H. S. and Incecik, A., (1997), Investigation of Motions of Catamarans in Regular Waves-II, Ocean Engineering, 24, No. 1, pp. 949-966. Hermundstad, O.A., (1995), Theoretical and Experimental Hydroelastic Analysis of High Speed Vessels, Dr.Ing. thesis, Dept. Marine Structures, Norwegian University of Science and Technology, 153 pp. Hermundstad, O.A., Aarsnes, J.V. and Moan, T., (1995), Hydroelastic Analysis of a Flexible Catamaran and Comparison with Experiments, Proceedings of the 3 rd International Conference on Fast Ship Transportation (FAST 95), Schiffbautechnische Gesellschaft, Vol. 1, pp. 487-5. Hermundstad, O.A., Aarsnes, J.V. and Moan, T., (1999), Linear Hydroelastic Analysis of Catamarans and Monohulls, Journal of Ship Research, Vol. 43, No. 1, March 1999, pp. 48-63. - 8 -

Incecik, A., Morrison, B.F. and Rodgers, A.J., (1991), "Experimental Investigation of Resistance and Seakeeping Characteristics of a Catamaran Design", Proc. 1st Int. Conf. on Fast Sea Transportation, (FAST'91), Norway, pp. 239-258. Kashiwagi, M., (1993), "Heave and Pitch Motions of a Catamaran Advancing in Waves", Proceedings of the 2 nd International Conference on Fast Ship Transportation (FAST 93), pp. 643-655. Lee, C. M., Jones, H. D. and Curphey, R. M., (1973), Prediction of Motions and Hydrodynamic Loads of Catamarans, Marine Technology, 1, pp.392-45, October 1973. Siregar, F.R.T. (1995), "Experimental results of a wigley hull form with advancing forward speed in head waves", Technical Report 124, Delft University of Technology, Ships Hydrodynamic Laboratory. van't Veer, A. P. and Siregar, F. R. T., (1995), The Interaction Effects on a Catamaran Travelling with Forward Speed in Waves,, Proceedings of the 3 rd International Conference on Fast Ship Transportation (FAST 95), Vol. 1, pp. 87-98, Lübeck, Germany. Wahab, R., Pritchett, C. and Ruth, L.C., (1971), "On the Behaviour of the ASR Catamaran in Waves", Marine Technology, 8, No. 3. - 9 -