OTC Statistics of High- and Low-Frequency Motions of a Moored Tanker. sensitive to lateral loading such as the SAL5 and

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1 OTC Statistics of High- and Low-Frequency Motions of a Moored Tanker by J.A..Pinkster, Maritie Research Inst. Netherlands Copyright 1989, Offshore Technology Conference This paper was presented at the 21st Annual OTC in Houston, Texas, May 1-4, This paper was selected for presentation by the OTC Progra Coittee following review of inforation contained in an abstract subitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Offshore Technology Conference and are subject to correction by the author@). The aterial, as presented, does not necessarily reflect any position of the Offshore Technology Conference or its officers. Perission to copy is restricted to an abstract of not ore than 300 words. Illustrations ay not be copied. The abstract should contain conspicuous acknowledgent of where and by who the paper is presented. ABSTRACT The separation of the total otions of a tanker oored in head seas into wave frequency and low frequency coponents with the purpose of developing siplified siulation coputations is discussed. The relationship between these otion coponents is clarified using results of long duration odel tests in irregular head seas. Results of tie doain siulations using a siple odel are used to illustrate soe of the probles facing the designer of a ooring syste. INTRODUCTION Rational design of ooring systes for peranently oored floating production and/or storage systes requires accurate and statistically reliable data on the otions of the vessel and the forces in the ooring syste. Tanker-based FPS systes oored in high seas I carry out wave frequency otions and superiposed, low frequency horizontal otions induced by ean I and slowly varying environental forces such as the second order wave drift forces or wind gusts. Low frequency otions are generally resonant due to the fact that the excitation frequencies coincide with the natural frequency resulting fro the ass of the vessel and the stiffness of the ooring syste. Such otion coponents can be large partly due to the relatively low syste daping. The cobined high and low frequency otions of the vessel cobined with the hydrodynaic forces acting on the coponents of the syste excite forces in the ooring syste. The design of the ooring sys te and its coponents requires insight References and illustrations at end of paper. I in the frequency content, distribution and extrees of the ooring syste loads. Data on the syste loads can be obtained directly fro odel tests carried out with a scale odel of the vessel and the ooring syste in odel basins in which all relevant environental effects can be generated on scale. This generally requires odel basins in which it is possible to generate siultaneously current, irregular waves and wind. Model tests of this kind are generally carried out in a later stage of the design process and nowadays are often carried out to confir previous design decisions ade on the basis of existing data for siilar systes or based on coputations. In the earlier stages of the design process the otions of the vessel and the associated ooring loads can be estiated based on coputations. The phenoenon involved are however very coplex and atheatical odels which describe in detail the ajor effects of the environent, the response properties of the vessel and of the ooring syste can becoe very tie consuing to develop and too difficult to use for all but the ost experienced (ref. [l]). For a detailed insight in the syste behaviour for general cases such odels are however necessary. For preliinary design purposes sipler odels which require less experience in use and less coputational power are needed. One of the ost iportant siplifications is based on the knowledge that in any practical cases the ooring syste design is governed by the survival head sea condition in which all environental effects are parallel and fro ahead. This assuption has proved to be one which has reduced the probles associated with the atheatical odelling by an order of agni'tude. In ters of the applicability of this case for the design of the ooring syste this assuption has also been confired in any cases. Only for ooring systes sensitive to lateral loading such as the SAL5 and SALM systes involving rigid yokes is it necessary

2 2 STATISTICS OF HIGH AND LOW FREQUENCY MOTIONS OF A MOORED TANKER OTC 6178 to consider ore coplex atheatical odels involving not only the surge otions of the vessel but also the sway and yaw otions. In this paper we will only consider the head sea case. This is always the governing case for, for instance, turret oored vessels. In this case the otions of interest are the surge, heave and pitch otions of the tanker. SIMPLIFIED MODEL OF VESSEL AND MOORING SYSTEM BEHAVIOUR A coplicating factor in setting up a siplified atheatical odel of the horizontal otions of the vessel is fored by the fact that the otions are carried out in two distinctly different frequency regies, i.e. at wave frequencies and at frequencies corresponding to the natural surge period. To all intents and purposes the vertical otions of tankers only consist of wave frequency otion coponents. In integrating the equations of otion the tie step is governed by the highest frequencies present in the otions. When considering wave frequency coponents this results in tie steps of the order of 0.5 to 1.0 S. If it is desired to carry out siulation coputations for durations which give results with sufficient statistical reliability, the total siulation tie is governed by the nuber of oscillations of the low frequency horizontal otions. Cobined with the sall tie step of the siulations as required by the wave frequency otion coponents, this leads to a very large nuber of tie steps in the siulation with correspondingly large coputational loads. This is an undesirable situation in the preliinary design stage. An obvious approach to this proble is to view the wave frequency otions and the.10~ frequency otions as separate processes which can be analyzed independently. Wave frequency otions are coputed based on standard linear hydrodynaic theory, often in the frequency doain, while the low frequency otions are coputed in the tie doain. The extree in the total otion is often deterined by adding the extree low frequency otion to the extree wave frequency otion. This is a practical approach by eans of which it is possible to assess the extree loads in the ooring syste. This ethod has, however, shortcoings fro the point of view of obtaining statistical data on the otions and ooring forces. Based on this siplification it is not possible to assign probability levels to the extree ooring forces or otions. The question also arises whether it is perissible to view the wave frequency otions and the low frequency otions as independent processes. Even,though these otions take place in different frequency regies, both are derived fro the wave elevations and as such ust be related. The separation into independent processes does however have its attractions so that it is of interest to investigate the validity of this assuption. MODEL TEST WITH A 300 kdwt TANKER In order to evaluate the independence of the wave frequency and low frequency otions of a tanker in head seas, long duration odel tests previously carried out at MARIN with a scale 1:80 od'el of a fully loaded 300 kdwt tanker were analyzed. The vessel was oored by a spring syste with linear and with non-linear restoring characteristics siulating a SALS and a Turret type ooring. The tests were carried out for a duration of 12 hours full scale in a North Sea survival condition with the following particulars: Significant wave height: 13.0 Mean wave period : 12.0 S The ain particulars of the vessel are given in Table 1. The test set-up is shown in Figure 1. The wave spectru is shown in Figure 2 and the restoring characteristics of the ooring syste are shown, in Figure 3. The surge otions of the vessel were easured by eans of an optical tracking device consisting of a point light source on the odel being tracked by a basin-fixed tracking syste. In Figure 4 results of the easureents in ters of the wave elevation record and the associated surge otion record for the linearly and non-linearly oored vessel are shown. This figure shows that the surge otions are doinated by the low frequency otion coponents. This does not ean however, that the wave frequency otions can be neglected. For instance, when considering such aspects as dynaic agnification effects in chain forces, the cobination of low frequency otions which give the static offset to the ooring syste and the wave frequency otion coponent which induce additional dynaic agnification effects, is of great iportance. In order to deterine whether, as a first approxiation, the high and low frequency otion coponents can be regarded as independent quantities the following analyses were carried out. The total surge otion record was filtered to yield separate tie records of the wave frequency and low frequency otion coponents. The cut-off frequency of the filter was chosen based on the spectru of the total otions. In Figure 5 the high and low frequency surge otion coponents are shown along with the wave elevation record. The dependence of the wave frequency otions on the low frequency.otions (or the reverse) can be viewed in different ways. One way is to deterine the coherence between the two coponents. Since the otion coponents cover non-overlapping frequency bands this will indicate that there is no coherence. A second way of analyzing the dependence is to deterine the coherence between the low frequency otions and the low frequency part of the square of the high frequency otions. The background to this ethod is the knowledge that

3 OTC 6178 J.A. PINKSTER 3 the low frequency otions are caused by second order low frequency wave drift forces. These forces are related to the square of the wave elevation record and hence one ay - expect - that the low frequency otions will show soe coherence in relation to the low frequency part of the square of the wave frequency surge otions. The results of the coherence co'putations are shown in Figure 6 to a base of the otion frequency. It is seen that the coherence is close to zero over the coplete frequency range of interest. Based on these results it, can be concluded that, for a first approxiation we ay assue the wave frequency surge otion coponents to be independent of the low frequency surge otions coponents. It should be noted however, that it is probable that a greater degree of coherence will be detected using bi-spectral analysis techniques which take into account in a ore correct way the quadratic dependence of low frequency otions and wave frequency quantities. Having established that high and low frequency otion coponents can, at first approxiation, be viewed as separate independent processes, the question now is how to odel the separate processes and how to obtain the total result. When viewing such aspects as the otional behaviour of a oored tanker in head seas it will be clear that the low frequency otion coponents are influenced by the ooring syste characteristics. Deterining these otion coponents for the general case involving non-linear ooring syste characteristics requires tie doain siulation coputations. For the case of ooring systes with linear restoring characteristics, the low frequency otions can be deterined by analytical eans, assuing that the low frequency surge otion can be described by eans of a second order equation of otion with constant coefficients and the low frequency excitation by eans of a white noise spectral shape (see ref. [Z]): F c Mean offset: X = -... (1) RMS of the slowly varying part of the offset: in which: F = ean part of the surge force b = total syste daping c = ooring syste stiffness S = spectral density of the wave drift forces where: S<(@) -(a) I.F2 1 % = wave spectru = ean drift force coefficient. For the case that the ooring syste restoring characteristics are non-linear, tie doain siulation coputations can be carried out based on the sae assuptions with respect to the low frequency part of the surge excitation. We have chosen for a Monte Carlo process to describe the surge excitation force. In order to take into account the basic quadratic relationship between the wave elevation process, which is norally distributed, and the slowly varying wave drift force excitation, we have selected a odel which results in an exponentially distributed surge force. The following expression for the tie doain representation of the force coplies with these requireents: in which: Fd = ean surge drift force At = tie step of the siulation. In Figure 7 and Figure 8 the white noise odel and the exponential distribution are copared with 'exactt representations which take into account in a correct way the quadratic relationship between the wave and the drift force (ref. [3]). The coparisons show that with respect to the spectral shape differences occur at frequencies higher than about 0.5 rad/s. The influences of these differences on the resultant otions is generally sall due to the fact that the low frequency otions are resonant in nature and hence highly tuned around the natural surge frequency. Depending on the actual ooring syste stiffness this will lie in the frequency band fro rad/s. Results of siulation coputations copared to odel test results given in ref.(2) confir this effect. In Figure 9 an exaple is shown of a typical surge drift force record as obtained using the siple Monte Carlo ethod. The siplified ethod reflects the skewness in the 'truet force but differences occur around the zero force level. The selection of a siplified odel for the wave frequency part of the otions hinges on the effect that the ooring syste properties have on these otion coponents. It is generally assued that the effect of the ooring syste on the wave frequency otions is sall. In Figure 10 and in Figure 11 the distributions and aplitude response functions of the wave frequency surge otion co-

4 4 STATISTICS OF HIGH AND MW FREQUEXCY MOTIONS OF A MOORED TANKER OTC 6178 l I l l ponents as obtained fro the previously described ade1 tests with the linearly, and non-linearly oored tanker in high irregular head seas are shown. The coparison confirs that the effect of the difference in restoring force characteristics of 'the ooring is negligible. This suggests that it is not necessary to solve the equation of otion for these coponents for a different ooring syste. All that is required is a representation of the wave frequency otion coponent which confors with the particular sea condition and which is given in a for consistent with the representation of the ooring syste response to the wave frequency otion coponent. For exaple, for a linear ooring syste with static restoring coefficient c and with negligible additional dynaic effects in the response to the vessel otions, it is sufficient to describe the wave frequency otions in the frequency doain in ters of its spectral density. The sae quantities are directly derivable for the corresponding ooring force coponent. The frequency doain description can easily be transferred into a representative tie record which can be added to the low frequency result obtained fro tie doain siulations. For ooring systes with non-linear restoring characteristics and significant additional dynaic effects such as turret oorings, which require tie doain solutions of the equations describing the oorings syste response, presentation of the wave frequency vessel otions as tie records can be ore appropriate. This approach to deterining the forces in the ooring syste requires considerable coputational effort. For preliinary design purposes analytical solutions to the ooring line dynaics proble ay be ore suitable (see ref. [4]). Selection of such ethods will require an appropriate representation of the wave frequency otion coponents. The total otions and ooring forces are found by addition of tie records of the wave frequency and low frequency coponents. The statistics of the total otions and forces can then be deterined and preliinary design data such as the ost probable axiu ooring force can be derived.. 1 APPLICATION TO THE LOW FREQUENCY SURGE MOTIONS AND 1 MOORING FORCES I In the following two exaples of the application of the above described approach for the low frequency coponents of the ooring force will be given. The first applies to the case of a sensitivity analysis with respect to the stiffness of a linearly oored tanker. The case concerns the 300 kdwt tanker oored in head seas in the aforeentioned survival condition. It is required to investigate the effect of a 20% increase in the stiffness of the linear ooring syste on the axiu ooring load. This will be carried out aking use of tie doain siulations of the low frequency otions based on the cpnstant coefficient equation of otion for surge. The following values were used for the siulation coputations: 2 Virtual ass = tfs / Daping coefficient b = 80.6 tfsf Mean drift force ' tf Drift force spectral density : tf S Such tie doain siulations are carried out for a liited duration. One of the questions arising fro this approach concerns the validity of the answers in view of the statistical reliability of the results. Due to the saple variance effect there is the possibility that coparison of tesults of two siulation calculations carried out for, identical environental conditions for two different values of the ooring stiffness ay yield incorrect conclusions. We will investigate the probability of incorrect conclusions being drawn by carrying out 10 siulation coputations for the original ooring stiffness and 10 siulations for the increased ooring stiffness. The environental force records will be the sae thus siulating the ease of repeated odel tests in the sae wave train. Coparison of all ten cases with respect to the axiu ooring force will reveal the probability level that an arbitrary realization can lead to ineorrect conclusions. This process will be repeated for three siulation durations of 0.5 hour, 1.0 hour and 1.5 hours. The ooring stiffnesses correspond to 15.5 tf/ and 18.6 tf/ respectively. The results of the ten siulations in ters of the axiu ooring force are given in Table 2. The results in the table show that for a test duration of 0.5 hour in six out of ten cases the axiu ooring force will be reduced when the ooring syste stiffness is increased. However, for a duration of 1.0 hour and of 1.5 hours, seven out of ten cases indicate that the axiu ooring force wi,ll increase due to the increase in stiffness. The results of this exercise show that care has to be exercised in drawing conclusions fro siulations or odel tests which have a relatively short duration fro the point of view of the phenoenon being investigated. The second exaple concerns the deterination of the design load of a ooring syste. The design load of a ooring syste is often based on the ost probable axiu value of the force which will occur in a selected sea condition for soe assued duration of the particular condition. The ost probable axiu value of a quantity is the value of the force for which the distribution function of the extrees of the force is at its axiu. For a quantity of which the distribution of the extrees confors with the Rayleigh distribution, the ost probable axiu value is found by intersecting the distribution of the extrees

5 OTC 6178 J.A. PINKSTER 5 at the probability level found fro the following equation: in which: N = nuber of oscillations of the quantity in the assued duration of the stor condition. In general the distribution of the extrees of a quahtity such as the ooring force will not be in accordance with the Rayleigh forulation. In such cases the probability level can be deduced directly fro the distribution of the extrees by deterining the force value at the peak of the distribution. One of the probles associated with this approach fs the aount of data, in ters of the duration of the record on which the distribution is based. In ost cases only a liited aount of data is available which eans that the results will always be influenced by finite saple effects. For the case in hand we will assue that we ay use equation (6) to deterine the probability level at which the distribution of the extrees ust be intersected in order to obtain the ost probable value. For the 300 kdwt tanker oored by eans of the non-linear ooring syste, siulation coputations were carried out to deterine the statistical variance of the ost probable axiu ooring force for the sae sea condition as before. It was assued that the ost probable axiu was to be deterined for a stor duration of 3 hours. In order to deterine the distribution of the extrees fro which the ost probable value was to be deterined, siulations were carried out for durations of 6 hours, 12 hours and for 18 hours. The effect of saple variance was deterined by carrying out each siulation ten ties. For each siulation the ost probable axiu ooring force was calculated according to the proc.edure outlined above. Finally, the ean, axiu, iniu and RMS of the ost probable axiu ooring force values were deterined fro the ten siulations carried out for each duration value. The results of the coputations are shown in Figure 12. This figure shows that as the test duration is increased, so the RMS value of the ost probable axiu ooring force decreases thus aking it ore probable that the results obtained fro a single siulation (test) will yield data which is close to the 'true' value. CONCLUSIONS In this paper an approach to developing siplified atheatical odels of the wave frequency and low frequency otions and ooring forces of a tanker oored in head seas has been discussed. It has been shown on the basis of odel test results that it is perissible fro the point of view of obtaining results in the early design stage, to view low frequency otion coponents as being statistically independent of wave frequency otions coponents. This is an iportant and practical siplification which can lead to a signiffcant reduction in the coputational effort required to s'iulate the otions and ooring forces. Results of siplified low frequency siulations were used to highlight soe of the pitfalls awaiting the designer when deterining the sensitivity of the ooring loads for sall changes in a ooring syste characteristics and when deterining the design loads. REFERENCES 1. Wichers, J.E.W. : "A Siulation Model for a Single Point Moored Tanker," Doctoral Dissertation, University of Delft, June Pinkster, J.A. and Wichers, J.E.W.: "The Statistical Properties of Low Frequency Motions of Non-Linearly Moored Tankers," Offshore Technology Conference, Paper No. 5457, Houston, Pinkster, J.A.: "Low Frequency Second Order Wave Exciting Forces on Floating structure^,^ MARIN Publication No. 650, Polderdijk, S.H.: "Response of Anchor Lines to Excitation at the Top," BOSS185, Delft, 1985.

6 Table 1 - Particulars of tanker Designation Sybol Unit 300 ktdw YLCC Length between perpendi cul ars Breadth Draft fore Draft ean' Draft aft Di spl aceent Centre of gravity above base Metacentric height L B T~ T~ T~ v K G , Centre of buoyancy forward of station 10 Longitudinal radius of gyration Natural roll period LCB k~~ T4 S Table 2 - Influence of ooring stiffness on axiu ooring forces Run duration in hour a5 Run No. Mooring stiffness in tf/ Mooring stiffness in tfl Mooring stiffness in tf/

7 - 9 +X - i G C 1 l A.P. F.P. Fig. l-model test setup. WRVE SPECTRUM Meaaured ;4% ; Tl e --a--*---- TheoreticuL l P.tl. 1 ;4T ; I! 12.0 S WRVE FREQUENCY IN RAO/S Fig. 2-Spectru of irregular waves. 627

8 Non-l inear ooring Linear ooring (15.5 tf/) Fig. 3-Restoring characteristlcs of ooring systes. Wave 5 () 0 Surge 15 () 0 Fig. 4-Total surge otion in head seas. Wave 5 () Wave frequency part Surge 1 () 0 ' U Y Low frequency part Fig. 5-Wave frequency and low-frequency otion coponents

9 1.o ---- Non-l inear ooring (U U c a, c 0 U w in rad/s Fig. 6-Coherence of squared wave frequency otions and low-frequency otions Spectral density, coplete expression Approxiation Fig. 7-Spectral density of low-frequency surge drift forces on a tanker. 629

10 F in t Fig. 8-Distribution of drift forces. t in in. Fig. 9-Drift force tie record. 630

11 99.9 c, S 99 L n S 90.P 80 S 60 a, Q) U 40 X aj h ' r n 6 1 n 0 L i near oor1 ng ---- Non-l i near ooring a X a in Fig. 10-Distribution of wave frequency surge otions. Fig. 1 l-aplitude response of wave frequency surge otions.

12 Fig. 12-Influence of siulation duration on variability of ost probable axiu ooring force in 3 hours.