- restoring forces due to the mooring system,

Transcription

1 OTC 2882 THE BEHAVIOUR OF MOORED SH l PS IN WAVES. by Gerard van Oortmerssen, Nether1 ands Ship Model Basin Copyright Offshore Technology Conference This paper was presented at the 9th Annual OTC in Houston, Tex., May 2-5,1977. The material is subject to correction by the author. Permisslon to copy Is restricted to an abstract of not more &an 300 words ABSTRACT A method is described for the mathematical simulation of the behaviour of a moored ship in waves. The method is based on the equations of motion in the time domain according to Cummins, while the hydrodynamic loads on the ship are obtained by means of the three,-from dimensional source technique. of computations for a ship moored to a jetty are discussed and compared with experimental results. INTRODUCTION - UP till a few decades ago, the mooring of ships has been mainly a matter of practical experience. Ships were moored in harbours or sheltered areas only, where the external force: are in general limited to the rather steady current and wind forces. With the development of the ocean industry and the advent of very large ships, which can only be accommodated in a few harbours with sufficient water depth, the need arose to moor ships in exposed areas. To this Purpose special mooring facilities were aesignea to absorb the loads exerted by the environment onvthe moored ship. Nowadays a variety of mooring arrangements is in operation. Because of the short history and fast development of mooring in exposed areas, the design of terminals can not be based on empirism. On the other hand the problem is too complicated for an analytical treatment. fore it is common practice to study the behaviour of a moored,ship by means of experiments with small scale models. Although model testing provides an effective tool to determine mooring forces and maximum motions of the moored ship for design purposes, this method inheres a few drawbacks. First, model tests are expensive and time consuming. The test set-up is complicated, it is essential that elasticity properties of mooring lines and fenders are simulaced very carefully, and sophisticated facilities are needed to simulate the relevant environmental conditions. For this reason test programs are usuazly restricted to final design configurations and selected weather conditions which are assumed to be the most critical' Further, the fundamental insight gained model tests on these complicated systems is limited. Only the resulting output is measured without learning much of the mechanism which causes this output. As an example the low frequency motion of a moored ship observed in tests in irregular waves may be mentioned. On a basis of model test results it is not possible to conclude whether this motion is caused by second order effects in the wave loads, or by the fact that the elasticity properties of the mooring system as- non- linear. It is, therefore, highly desirable to dispose ol a mathematical simulation method for the behaviour of a moored object. Such a method can help to increase the understanding of moored ship behaviour, and can be used as a tool in optimization studies in the early stages of design, prior to model testing. The basis of each mathematical model for the simulation of the behaviour of moored objects is the law of dynamics of ~ewton: a(&) -=p dt There-or,since the inertia m of the ship may be regarded as constant: m; = F The external force P is composed of - arbitrarily in time varying forces due to the waves; - hydrodynamic and hydrostatic restoring forces, which are a function of the motians of the ship; - restoring forces due to the mooring system,

2 which are a function of the instantaneous position of the ship. In the classical ship motion theory, it is common practice to formulate the equations as follows: a, b and c are coefficients which describe the hydrodynamic and hydrostatic restoring forces. In fact, (1) is not a real equation of motion, in the sense that it relates the instantaneous motion variables to the instantaneous value of the exciting forces. It can only be used as a description in the frequency domain of a steady oscillatory motion, since the hydrodynamic coefficients a and b depend on the frequency of motion. Analytical work on the moored ship uro- I blem published so far has been based on In this paper a mathematical model is described which.is based on the equations of motion in the time domain as they have first been formulated by Cummins [l31. These equations of motion may be considered as true differential equations; they give the instantaneous relationship between the motion variables and the external forces. In these equations the various factors governing the response of the ship are separated into clearly identifiable units. The only assumption involved is linearity of the hydrodynamic restoring forces. Nonlinear and asymmetric mooring characteristics can be dealt with, and the exciting force may be arbitrary, which means that besides first. order wave forces also slowly varying drift. forces and wind- and current forces can be included in the forcing function. equation (1 ), where three categories can be I DESCRIPT1O' OF THE -. discerned with regard to the simplifying assumptions made. Some investigators, as for instance Kaplan and Putz Cl?, Leendertse 121, Muga 131 and ~eidl L41 linmieed the elasticity characteristics of the mooring system. The restoring forces of the mooring aids can then be incorpor~ted in the hydrostatic term cx and the equations (1) of motion in the frequency domain can be solved easily, with the restsiction that only harmonic excitalions can be used. Others, as Abramson and Wilson C51, Yang [6] and ~ilner [TJ add non-linear terms to equation (1) to account for the restoring forces of the mooring system, ahd solve the equations by means of the method of equivalent linearization, assuming that the excitation is pure sinusoidal and that, as in the earlier mentioned method, the response of the ship is simple harmonic too, with a frequency equal to that of the excitation. This is not realistic, since observations both in model and full S-cale situations have revealed that also other modes of motion may occur. The work of Wilson and Awadalla C8, 91, Lean C101, Wilson [l 13 and Bomze C121 belongs to the third category, which is characterized by the assumption that the hydrodynamic coefficients a and h in equation (1) are independent of the frequency, so that this equation is regarded as a* actual differential equation. The solution, which is found either by approximate analytical methods (ref. C81, C101 ) or by finite difference integration in the time domain (ref. C91, C111 and t.123)~ Mkj may Contain components with frequencies lower (subharmonic) or higher (superharmonic) than that of the forcing function. Unfortunately, the assumption of constant hydrodynamic coefficients can not be justified: especially in shallow water these coefficients appear to be very sensitive to changes in frequency. Consequently, a timedomain description of the behaviour of the moored ship is needed which takes into account the frequency dependency of the fluid reaction forces. In this section, a general description is given of a mathematical technique to simulate the behaviour of a ship, moored to a jetty. For a more detailed description reference is made to [l41. Consider a ship, moored by means of a number of mooring lines with non-linear elastic characteristics, to a jetty equipped with fenders, as schematized in Figure 1. A space fixed, right handed system of coordinates GXIX2X is used, with its origin in the centre of g?avity of the ship. The equations of motion in the time domain can be written as: 6 t Z {(M.+m )f + Kkj(t-r);.(r)dr+Ckjxjl = j=1 k~ kj J' -m J n 1 n f = xk(t) + C ~ ~ ~ + ( X t ~ ) ~ ~ ( t ) i=l i=l k = 1' 2, a. e *** (2) is an inertia matrix. Since the origin of the system of axes coincides with the centre of gravity of the ship in its rest position, it is found that

3 m between the motions and the fluid reactive o m o o forces. For the computation of constant added o o m o mass coefficients, retardation functions and time histories of wave loads, the added mass 0 0 and damping coefficients and wave loads are needed over the complete range of frequencies of interest O - '6 For the computation of deep water ship motions, the so-called strip theory, has where m = mass of the ship, proven its usefulness. This theory takes adth vantage of the fact that for ships the longi- Ik = moment of inertia in the k tudinal dimension is large relative to the mode, lateral and vertical dimensions. For such a I = product of inertia. slender body the three-dimensional problem k j can be reduced successfully to a local twodimensional problem. After its presentation, is the matrix of hydrostatic restoring the method has been refined by many authors Ck. fogce coefficients. and the results are in general reliable. A he hydrodynamic reaction forces acting on the drawback is, however, that no informat'on ship are expressed by a constant "added mass" Can be the surge coefficient m. and a retardation function Unlike the deep water problem, very few K which ta%ds into account the memory studies have been presented on the motions 09 e$jict of the free water surface. a ship in shallow water. m and K describe the hydrodynamic force Kim C161 has adapted the strip theory ikjthe kthjmode, due to motion of the ship for a restricted water depth. For the vertiin the jth mode. cal modes of motion this approach yields use- Ogilvie [l?] has shown the relationship be- ful but it can be used tween these quantities and the frequency- lateral motions, especially in the lower fredependent added mass and damping coef'~~cr@nts:- quency since the is basically two-dimensional, requiring that the flow m 2 Kkj(t) = 7 1 -(W) cos cot du - m S...(3) k~ 0 m m = akj(w ) + 7 / Kkj(t) Sin t dt...(4) k j W o where a kj b kj = frequency-dependent added mass coefficient, = frequency-dependent damping coefficient. u1 is an arbitrarily chosen value of the with this method, Figure 2 shows a comparison result for m iven by (4) is independent of Of wave forces on a 200,000 taw the value ofkjw$. tanker in shallow water (keel clearance 20 ~ ~ ~ is ( the t component ) in the kth mode of the percent of the draft) with values obtained tension in the 5th mooring line, which depends from model tests. Figure 3 shows a similar on the instantaneous length of the line, the Of the added mass and damping of the line and the load-elonga- coefficient for motion in the sway mode. It tion characteristics. becomes obvious from this figure, that the ~ ~ ~ is ( the t component ) in the kth mode the added mass in sway, which is quite important force in the ith which is a function for the behaviour of a moored ship, is strongof the deflection of the fender and the elasti :ly dependent On the frequency Of properties. n is the total number of mooring lines, np tie number of fenders. xk(t) represents the time history of the total environmental load in the k-mode. the equations of in the time domain an arbitrary motion is described as a succession of small impulsive displacements. The basic assumption is, that at any time the total fluid reactive force is the sum of the reactions to the individual impulsive displacements, each reaction being calculated an appropriate time lag from the instant of the corresponding impulsive motion, or in there must,axis+ a linear words, of water passes entirely underneath the keel of the ship. In shallow water, however, threedimensional effects become important. The water flows partly underneath the ship and partly around the ends. In the extreme case, the ship sitting on bottom, water can move only around the ends of the ship. A method of solution, of which the validity is unrestricted as long as linearity is ascertained, is provided by the three-dimensional source technique. This technique has been applied successfully during the last few years for the computation of wave loads on large offshore structures, but it can be used just as well for ship shaped bodies (see ref. [l71 ). As an example of the results obtained As has been stated, the constant "added mass" coefficients and retardation functions which appear in the equations of motion in the time domain, can be from the added mass and damping coefficients in the frequency domain. As an example, Figure 4 shows the retardation functions for surge, sway and heave motions of a 200,000 tanker in water with a depth, 20 percent in excess of the ship's draft. With the three-dimensional source witk.techni ue it is a190 possible to take the influence Of a quay On the othe:6coefficient~ into account. Figure 5 shows computed and measured values of the added mass

4 L and damping coefficients in the sway mode for a 200,000 tdw tanker beside a quay. The numerical solution of the 6 coupled second order differential equations (2) is carried through according to the following procedure. Suppose the simulation has arrived to the moment t, A t is the kime increment applied, so the equations of motion have to be solved for the moment t + At. First, the velocities for t + and t + A t are predicted 2 by extrapolating the obtained time histories. To this end, the velocity is expanded in a Taylor series: ~t~ ( t + At) = ( t ) r At%.(t) i 7 jij(t) J J J 2. A t = ( t + A t t. ( t ) + {fj(t) - J J - t.(t - at))...(5) J A t In a similar way x. (t + --)is found. Subse- J 2 quently, the new position and orientation are predicted by numerical integration of the velocities, applying Simpson's rule: x.(t + At) = r.(t) + -g A t {x.(t) A t -) + J J J J 2 + x.(t + At))...( 6) J The time history of the velocities is now known until the moment + At, so the convolution integrals can be computed. The numeri- cal integration of these convolution integrals is carried out by means of Simpson's rule, using a time increment equal to the time step At, applied for the solution of the equations the main findings w i l l be summarized* of motion. The upper bound of the integrals, in theory infinity, is fixed at 25 seconds, which apparently is sufficient. Subsequently, the values of the mooring line and fender forces as well as the hydrostatic restoring forces can be calculated for the new coordinates. After substitution of these forces in (2), 6 linear equations are obtained from which the accelerations fj(t + At) can be found. Finally, the predicted velocities are checked by integration of the accelerations. In case the difference is acceptable, the computatiun continues for the next time step; if not, the time increment has to be decreased. When starting the computation process at t = 0, no reliable prediction of i.(~t) can be made. Therefore, an iteration pgocedure is used in that case: the values of the velocities found after solving the equations of motion are used as a new prediction and the process is repeated until the predicted and computed velocities at t = At are in satisfactory agreement. The mathematical simulation process described before has been programmed in FORTRAN for use on a Control Data 6600 computer. inautb~~i~~s some para:eters9 the - inertia matrix of the ship, - matrix of hydrostatic restoring coefficients - added mass coefficients and retardation functions of the ship, - coordinates of fenders and bollards, - elasticity characteristics of fenders and lines, - pretension in mooring lines, - time history of the external loads. The retardation functions and elasticity functions of lines and fenders are read in as a number of discrete points, sufficient to fix the curves. Intermediate values are, when needed, found by means of interpolation subroutines. As output of the program, the computed time histories of motions and forces can be printed, plotted or dumped on digital tape for further analysis. The computing time is linearly proportional to the inverse of the time step. Systematic computations with varying time step have shown that a step of 0.2 seconds is sufficiently small for an accurate numerical solution of the equations of motion in case the ship is moored against stiff fenders. With this time step, the required computing time amounts to 1 second for 10 seconds real time. EXPERIMENTAL VERIFICATION To check the adequacy of the mathematical model for moored ships, an extensive experimental program has been carried out to analyse the motion behaviour of a moored ship in regular and irregular waves. Afterwards, typical test situations were selected for simulation on the computer to see whether the phenomena be mean'' of the mathematical simulation. For a detailed description and complete results reference is made to c141. Here, only The study was conducted for a loaded 200,000 tdw tanker, moored to an open jetty in water with a depth amounting to 7.2 times the draft af the vessel. The main particulars of the ship are given in Table I. The mooring lay-out is depicted in Figure 6. The vessel was moored by means of 4 lines, each representing two or three wires with nylon tails in reality. The load-elongation curves of the lines are shown in Figure 7. In each line a pretension was of 20 tons. The pair of stiff fenders had a linear elasticity amounting to 1575 ton/m. The model tests were carried out in the Shallow Water Laboratory of the Netherlands Ship Model Basin on a scale of 1:82.5, The loads in mooring lines and fenders were measured by means of strain gauge transducers. The motions were measured by means of potentiometers. All signals were recorded both on magnetic tape and paper chart. The model tests were based on Froude's law of similitude. First an extensive series of tests in regular beam paves (a = 90~) was carried out to investigate the behaviour of the moored ship over a wide range of periods, varying fron.

5 9 to 41 seconds. The long period waves are not merely of theoretical interest, but may also be associated with seiches or harbour oscillations, which sometimes cause problems in -. practice. - In all conditions tested, the ship attained a periodic motion, after a period of transience. Characteristic of the sway motion was the occurrence of a mean displacement in addition to the oscillatory motion. In short waves this motion had the same frequency as the waves, with bounces against the fenders of equal strength at time intervals equal to the wave period. In certain long waves, however, a sub-harmonic motion was observed: the ship motion was composed of a motion with frequency w equal to that of the exciting waves, on which a motion was superimposed with frequency either w/3 or w/2. Impacts against the fenders occurred then at intervals Of 3 Or wave periods Subharmonic motion was observed only in waves with such a frequency, that the frequency of the subharmonic (w/3 or w/2) was close to the "natural" frequency of the moored which in the sway to 0.07 rad.sec.-l (due to the non-linear elasticity characteristic of the mooring system there is no well-defined-natural frequency; in fact the resonance Frequency depends on the amplitude of motion). When the amplitude of motion was decreases, the subharmonic motiolithe disappeared. In some cases another mode of regular motion was found, for instance with alternating light and heavy bounces against the fenders. The computations were carried out with pure harmonic forcing Tunctions, which were obtained from the three-dimensional source technique, taking into account a wave height equal to that, measured in the basin without the ship model being there. Time histories were computed for a period of 1000 seconds, although it appeared that after 500 seconds the results became stationary. It was found, that in all cases the typical motion behaviour observed in the was predicted by the mathematical model. Besides, the quantitative agreement between and apgeared to be satisfactory: the differences between measured and computed values of motion and mooring force amplitndes were in general less than 20 percent. As an example Figure 8 shows the results for a wave condition in which subharmonic response occurred. Also long crested irregular waves have been taken into consideratitin. One sea condition was used, of which the spectral density is shown in Figure 9, with three angles of wave attack, a = 90, 135 and 180 degrees- The measurements lasted a period corresponding to 2100 prototype and began looo seconds after starting the wave generator, avoiding that transient phenomena would infllrence the results. The computations were performed for a period corresponding to 2500 seconds. The firsl. 400 seconds represent a period OF transience, thus leaving '2100 seconds for analysis. Of all measured and computed time histories of motions and forces a spectral analysis was carried out. Besides spectra, this analysis yielded the following statistical quantities: - mean value, root mean square value, - significant double amplitude, -max~mum and minimum value, The wave spectrum was simulated in the computation by means of 15 sine waves, ranging in frequency from to rad. see.-l, To that end the measured wave spectrum fsee Figure 9) was subdivided into 15 bands of constant width. Each band was represented by a component, having the centre frequency of that band and a height, following from the band area. These wave romponents were summea with arbitrary phase angles, and with the aid of the transfer functions computed with the three-dimensional source technique the functions in the 6 modes were determined. Mathematical simulations showed that in beam seas the roll motion at the frequency is overestimated by the theory. Therefore it appeared desirable to take some extra roll as determined from model tests, into account. In waves from 90 and 135 degrees a good agreement was observed between the =esults of experiments and the mathematical simulation. As an example, Figure 10 shows the measured and computed spectra of motions and forces for 135 degrees waves, while the most interesting digital information for this case is tabulated in It appeared that the spectra of horizontal motions and mooring forces show peaks in the low frequency range. For the case of head waves (a = l80 degrees), in first instance a very bad torre- lation *as found: the theoretical forcing functions consist only of small oscillatory loads in surge, heave and pitch mode, resulting in small variations of the mooring forces around the pretension and a small surge oscillation around the zero position. In reality, however, the surge consists of a larger amplitude oscillation around a mean displacement, and the variations in the mooring loads are also larger. It was assumed that in this case the second order wave drift force plays an important role. To check this assumption, two calculations were made: one with an additional constant and one with a slowly varying drift force, which is much more realistic in irregular waves. As an approximation of the magnitude of the drift force, t18,, performed on measurements by Pinkster a different ship model in a similar sea tion were used. With Lhe additional slowly varying drift force a good agreement was found. Obviously, the influence of the low-frequency drift force is thur8essential in this condition, ANALYSIS OF RESULTS The results of the investigations describ in the previous section have revealed inte-

6 resting features of the behaviour of a ship, moored to a jetty. In regular beam seas, the sway motion consists 09 an oscillation around a mean displacement. In certain long waves a subharmonic sway motion was found with frequency w/2 or w/3. In irregular seas, the spectra of horizontal motions and mooring forces show lowfrequency peaks, clearly distinct from the range of wave frequencies. From the comgutations it followed that in head waves the second order wave force plays an important role in exciting this low frequency behaviour, but in the other wave directions considered, low frequency peaks were found due to first order wave excitation only. This low frequency behaviour in irregular waves must also be distinguished from the subharmonic motion in regular waves: the longest wave component in the spectrum had a frequency of rad. sec.-l whereas subharmonic motions only occurred in waves with frequencies lower than approximately 0.21 rad.sec.-l. As was shown, this typical behaviour of a moored ship can be predicted by the complicated mathematical model described here, but to understand why these modes of motion occur, it is helpful to use a simplified analytical approach. The above mentioned special features of moored ship motions occur mainly in the horizontal modes, surge, sway and yaw. When these motions are considered as being uncoupled, it is observed that the restoring force and momen'; in surge and yaw are non-linear, but symmetric, - with F = Fa cos wt, subharmonic solutions exist with frequency w/3, - with excitations consisting of two harmonic components, F = F, cos wlt + F2 cos w t, 2 the motions contaln comwonents with frequencies 2b1 2 w2 and 2w2 + U,, the so-callec. combination t-ones, besides the basic frequencies w, and w2. The elasticity of the mooring system in the sway mode is essentially different in that sense, that the restoring force is asymmetrical: when pushing against the fenders, the force is different from the case that the ship pulls at the lines. The most simple way to schematize such a restoring force is: 2 3 f(x) = ax + Bx + yx...(9) When again ignoring frequency-dependency of added mass and damping, the equation of motion becomes: ux + fix + yx3 = ~ ( t... ) (10) Solutions of this equation are discussed in ref. C141. It appears that: - with an excitation F = F cos wt, the first order approximation of tae resulting motion shows an oscillation around a mean value: X = A cos at + B - subharmonic motion may occur with frequencies w/2 as well as w/3, - with bi-frequency excitation (F = F l w,t + F2 cos w 2 t) combination tones can be found wlth frequencies 2wl 2 w2, 2w2 2 wl and w, 2 w2. Thus this simplified analytical approach and hence they can be schematized as: shows how a mean displacement and low frequency motions originate from the non-linear f(x) = ax + 3 particulars of the mooring It will be clear, that the equations of motion in the frequency domain can only be used Lo explain certain particulars of the where f(x) is the restoring force, X is the behaviour of the moored ship, brought about displacement and a and B are constants. When by the peculiarities of the mooring system, moreover the frequency-dependency of the the coefficients of the inertia and damping added mass is ignored and the damping is neg- terms being strongly dependent on frequency. lected, the simplified equation of motion be- In the present case, the inertia coefficient comes at the subharmonic frequency is around four times larger than at the wave frequency, which clearly illustrates the necessaty to take 2 + ux + 8x3 = ~ ( t ) (8) the frequency-dependency of the coefficients into account. The low frequency response in irregular This is the well-known Duffing equation, which seas from 90 and 135 degrees is obviously a has been treated extensively in the literature result of the phenomenon of combination tones, of non-linear vibrations (see for instance since subharmonic response was not found in Stoker, 1193 ). The solutions of the Duffing the separate components of the spectrum. Comequation show the folloving particulars: putations with low frequency drift forces - the first order of the motion added to the force input did not yield signidue to a harmonic excitation F = F cos ficantly different results for these wave cona wt is: X = A cos wt, ditions. For an other mooring configuration - the response shows a I I ~ shape, with soft fenders, however, the drift force ~ ~ ~ ~ ~ ~ ~ ~ I did have FUTURE DEVELOPMENTS The program package available at N.S.M.B. is operational for the analysis of the behaviour of ships, moored to open or solid jetties, and can also be applied for the determination of impact loads during berthing J

7 manoeuvres (see ref. L201). The analysis is theoretical, except for the wave drift forces, which still should be determined experimentally. In view of the importance of these 'forces to the behaviour of moored ships, much effort w i l l be needed for the development of reliable prediction methods. It is expected, that within the next two years means will become available to calculate the time history of the low frequency drift forces. On this subject, Progress is made by among others Dalzell c211 and Pinkster L181. It is intended to extend the applicability of the program in such a way, that single point moorings can be simulated. This means, that the influence of significant changes 09 heading on the wave loads and the dynamics of the buoy must be included in the mathematical model. CONCLUSIONS -- Xk a kj b kj g wave force or moment in the k-th mode, added mass coefficient in the k-th due to motian in the j-th mode, damping coefficient in the k-th mode due to motion in the j-th mode, acceleration of gravity, j, k subscripts ranging from 1 to 6 used for a direction or a degree of freedom, m mass of the ship, m frequency-independent added mass coeffikj cient in the k-th mode due to motion in the j-th mode, X j a displacement in the j-th mode, angle of wave incidence, The mathematical model described is suitable for the simulation of moored ship motions in six degrees of freedom in waves. The typical motion behaviour as observed in full scale and in model tests is reproduced by the mathematical model. A ship, moored in random seas, can experience three types of low frequency behaviour: - subharmonic response to certain harmonic components, with a frequency amounting to 112 or 113 of that of the exciting wave component. This subharmonic response is a result of the non-linearity of the mooring system, - "combination tones", a low frequency motion induced by the simultaneous action of more than one harmonic wave component. ~l~~ this phenomenon originates from the non-linear elasticity characteristics, - low frequency motions excited by the low frequency second order wave drift force. NOMENCLATURE C k j F G Ik I k j K kj L Lik M k j Nik matrix of restoring force coefficients, general force or moment, centre of gravity, moment of inertia in the k-th mode, product of inertia, retardation function in the k-th mode due to motion in the j-th mode, length of the ship, force or moment in the k-th mode due to the i-th mooring line, inertia matrix, force or moment in the k-th mode due to the i-th fender, w circular frequency. REFERENCES 1. Kaplan, P, and Putz, R.R.: "The motions of a moored construction-type barge in irregular waves and their influence on construction operation": N BY-32206, Marine Advisors, Inc. La Jolla, Leendertse, J.J.: "Analysis of the response of moored surface and subsurface vessels tc ocean waves": Rand Corporation Memorandum RM-3368 PR, Muga, B.J.: "Experimental and theoretical study of motion of a barge as moored in ocean waves": University of Illinois, Hydraulic Engineering Series No. 13, Seidl, L.H.: "Prediction of motions of ships moored in irregular seas": Proc. N.A.T.O. Advanced Study Institute on Analytical Treatment of Problems in the Berthing and Mooring of Ships, Wallingford, 1973, pp Abramson, H.N. and Wilson, Basil W.: "A further analysis of the longitudinal response of moored vessels to sea oscillations": Proc. A.S.C.E. 85, 1959, W W ~, p Yang, I-Min: "~otions of moored ships in six degrees of freedom": 9th Symposium on Naval Hydrodynamics, Paris, 1972, 7. Kilner, F.A.: "Model tests on the motion of moored ships placed on long wavest1: Proc. 7th Conf. on Coastal Engineering, The Hague, 1960, Volume 2 pp Wilson, J.F. and Awadalla, N.G.: "Subharmonic response in the non-linear oscillations of moored ships": Offshore Technology Conference, Houston, 1971, paper OTC 1420, Volume 11 pp S S spectral density oflthe waves,

8 , 9. Wilson, J.F. and Awadalla, N.G.: "Computer simulation of non-linear motion of moored shipst1: Proc. N.A.T.O. Advanced Study Institute on Analytical Treatment of problems in the ~ ~ ~ and t~~~~i~~ h i of ~ ~ Ships, Wallingford, 1973, pp Lean, G.H.: "Subharmonic motions of moored ships subjected to wave action": Transactions of the Royal Institution of Naval Architects, 1971, Vol. 113, pp Wilson, B.W.: "Progress in the study of ships moored in waves": Proc. N.A.T.O. Advanced Study Institute on Ana-lytical Treatment of Problems in the Berthing and Mooring of Ships, Wallingford, 1973, pp B ~ H,: ~ " ~ ~ ~ detereination l, ~ t i ~ of ~ l ship motion and mooring forces": Offshore Technology Conference, Houston, 1974, - paper OTC Cummins, W.E.: "The impulse response function and ship motions", D.T.M.B. Report 1661, Washington, D.C., Oortmerssen, G. van: "The motions of a moored ship in waves", N.S.M.B. publication No. 510, Ogilvie, T.F.: "~ecent progress toward the understanding and prediction of shlp motions": 5th Symposium on Naval Bydrodynamics, Bergen, Kim, C.H.: h he influence of water depth on the heaving and pitching motions of a ship moving in longitudinal regular head waves": Schiffstechnik, Vol. 15, No. 79, 1968, pp Oortmerssen, G. van: "The motions of a shi: in shallow water": Ocean Engineering, Vol. 3, pp , Pergamon Press, Pinkster, J.A.: "Low frequency second order wave forces on vessels moored at sea": 11th Symposium on Naval Bydrodynamics, London, March, Stoker, J.J.: "Non-linear vibrations in mechanical and electrical systems": Interscience Publishers, Inc., New York, Oortmerssen, G. van: "The berthing of a large tanker to a jetty": Offshore Technology Conference, Houston, 1974, paper OTC Dalzell, J.F.: "Cross-bispectral analysis: application to ship resistance in waves": Journal of ship research, Vol. 18, No. 1, March, TABLE I.. Main dimensions 200,000 TDW Tanker Length between perpendiculars Breadth Draft Volume of displacement Block coefficient Midship section coefficient Prismatic coefficient Distance of centre of gravity to midship section Height of centre of gravity Metacentric height Longitudinal radius of gyration Transverse radius of gyration m m m 235,000 m m m 5.78 m m m

9 1 FENDER / Pi1 tpiptpi3 WAVE DIRECTION Fig. 1 - Definition sketch.

10 THEORETICAL o EXPERIMENTAL Fig. 2 - Transfer functions of wave excited forces and moments; a = 225 degrees; water depthldraft = ~L - THEORETLCAL EXPERIMENTAL o o o Sz : m L2 : m S2 : m 0 + * & A * OO " 6 :P: - Non-dimensional added mass and damping coefficients in sway; water depthldraft = 1.2; sway amp1 i tude.

11 P. ~ ~ p~ seconds 10 0 FIG, 4 - RETARDATION FUNCTIONS FOR SURGE, SWAY AND HEAVE, THEORETICAL -- 0 EXPERIMENTAL " ,l -P - - RI.RI m RI. EU a " u q 2 wd$ OO '0 FIG, 5 - NON-DIMENSIONAL ADDED MASS AND DAMPING COEFFICIENCTS IN SWAY; DISTANCE BETWEEN SHIP AND QUAY 16,5 M; WATER DEPTH/ DRAFT 1.2 B TDW TANKER : : m 1 13S.45 m m l 1m4Orn CHOCK FENDER DOLPHIN

12 " C m KILL LIK 1 LME 1 LINE 3 L M 4 FIG. 8 - COMPUTED AND MEASURED SHIP MOTIONS AND MOORING FORCES; REGULAR WAVES FROM 90 DEGREES; = RAD. SEC-1; WAVE HEIGHT 0.9 M, to SURGE mm FENDER I MOORIN5 LlNE I MOORlNO LlNE 2 A PITCH MOORING LINE 4 ' \ L' l