Full Scale Measurements and Theoretical Predictions of 2nd Order Pitch and Roll Slow Motions of a...

Vinicius L. F. MatosPetrobras, E&P,

Av. Santa Monica 120,

Macae, RJ 27943-625, Brazil

e-mail: [email protected]

Eric O. RibeiroPetrobras, Cenpes/PDEP/TEO,

Av. Horacio Macedo 950,

Rio de Janeiro, RJ 21941-915, Brazil


Alexandre N. SimosUniversity of Sao Paulo,

Av. Prof. Mello Moraes 2231,

Sao Paulo, SP 05508-030, Brazil


Sergio H. SphaierCOPPE/Universidade Federal do Rio de Janeiro,

LabOceano—Parque Tecnologico do Rio,

Rua Paulo Emıdio Barbosa,

485, Quadra 07-A,

Cidade Universitaria,

Rio de Janeiro, RJ 21941-907, Brazil


Full Scale Measurementsand Theoretical Predictionsof 2nd Order Pitch and Roll SlowMotions of a SemisubmersiblePlatformIn Oct. 2007, the semisubmersible platform PETROBRAS 52 (P-52) was installed inCampos Basin (Roncador Field) offshore Brazil, in a depth around 1800 m through 16lines in taut-leg con. The maximum production capacity is 180.000 bpd with a displace-ment of 80,986t at the operational draft of 27.5 m. Slow drift motions in the vertical plane(heave, roll, and pitch) were observed in a model test performed in a wave basin duringthe design phase. As resonant responses vary considerably with the damping loads, slowmotion could be affected by scale effects. To observe the phenomena, by that time, it wasa doubt if this phenomenon would happen during the platform operation. Since June2008, PETROBRAS has been monitoring P-52 motions with the use of accelerometersand rate-gyros. Through spectral analysis of the measured signals, it was possible to ver-ify the presence of slow motions with frequencies around the natural frequencies of rolland pitch during almost the whole monitoring period. Sometimes, the 2nd order ampli-tudes were even greater than the 1st order ones. Furthermore, the environmental condi-tions have also been monitored through wave radars, ADCPS (current) andmeteorological stations (wind) in the vicinity of P-52 location, making the excitationloads identification possible. A comparative study confronting full-scale measurementsand theoretical predictions was performed. First and second-order forces and responseswere calculated using WAMIT

VR second order module. This study permitted the estimationof the full scale damping values of this offshore system (hull plus mooring and riser lines)for one of the environmental conditions measured. The results indicate the importance ofconsidering the resonant roll and pitch motions in the seakeeping analysis of large-volume semisubmersible platforms, contributing with an important feedback to futuredesigns. [DOI: 10.1115/1.4024206]

Introduction

In the P-52 model tests performed at the Norwegian MarineTechnology Research Institute in 2002, resonant heave, pitch, androll motions were observed (see Ref. [1]). Due to these motionsbeing, in some cases, even higher than the first order ones, itwas concluded that their contribution is significant for this type ofplatform—a large-volume semisubmersible.

Figure 1 shows a photo of the P-52 model in the tank. Figure 2shows a scheme of the platform in relation to the north. The x-axispoints from south to north and y-axis from East to West. Thez-axis points upward. Roll and Pitch are the motions around thex- and y-axis, respectively. The waves were multidirectional com-ing from 150 to 250 deg in relation to north.

Indeed, this seakeeping behavior, mentioned above, was pre-dicted in Ref. [2] for large volume structures with small waterplane areas and natural periods above the typical wave periods.As presented in Fig. 3, the trend of PETROBRAS’ semisubmersi-ble platforms is increasing the size of their hulls, raising the con-cern about possible resonant second order motions in the verticalplane.

Another identification obtained from the mentioned model testcampaign was that these resonant motions are strongly affected by

the level of the system damping, which has direct relation withthe presence of lines (risers and mooring) and current. To illus-trate this fact, Fig. 4 shows the pitch response spectrum in fourdifferent cases:

• With horizontal mooring lines and without current and risers(5130)

• With truncated mooring lines and without current and risers(2310)

Fig. 1 Model test with P-52 (see Ref. [1])

Contributed by the Ocean, Offshore, and Arctic Engineering Division of ASMEfor publication in the JOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING.Manuscript received October 6, 2010; final manuscript received June 30, 2012;published online June 6, 2013. Assoc. Editor: Moo-Hyun Kim.

Journal of Offshore Mechanics and Arctic Engineering AUGUST 2013, Vol. 135 / 031106-1Copyright VC 2013 by ASME

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• With truncated mooring lines and riser, without current(2110)

• With truncated mooring lines, current (U1¼ 1.96 m/s in fullscale) and risers (2220)

The four configurations described above were tested with thesame calibrated wave with significant height of HS¼ 6.3 m andpeak period of Tp¼ 10 s.

It is clear that by increasing the damping of the system; in otherwords, with the presence of lines and current, the second orderresonant pitch motion decreases considerably. Note that the pitchnatural period is around 31 s. The same type of behavior was alsoverified for heave and roll motions. Furthermore, it is possible to

observe the pitch response due to the coupling with surge motion,and that excited by second order surge forces.

Hence, including the effects of the lines damping in the compu-tation of second order resonant motions in the vertical planeseems to be essential to a realistic seakeeping analysis. However,one point can be raised on that discussion: how close are themotions in full scale to the ones observed in model tests as the vis-cous effects and, consequently, the level of damping are affectedby scale effects. A full scale measurement campaign appears as afundamental verification of the presence and importance ofthe second order resonant motions, as well as for a dampingestimative.

Thus, a case study was proposed with the P-52 platform, thistime with full scale measurements. Time series for roll and pitchwere obtained during June 2008. This data, collected simultane-ously with the environmental parameters (wave, wind, and cur-rent), allowed theoretical computation of the environmental loadsand the system response. The main goal of this case study was toprove that significant second order resonant motions in the verticalplane can be induced by the waves in full scale.

System Description

The P-52 platform was installed in Roncador field (Campos Ba-sin) in Oct. 2007. It is moored in a water depth around 1.800 mthrough a mooring system with 16 lines–taut leg configuration.The platform heads north. Main dimensions are shown in Table 1.Figure 5 shows a picture of P-52 after the topside installation.

Fig. 3 Displacement (a) and draft (b) of PETROBRASsemisubmersibles

Fig. 4 Spectral density functions of pitch movement(DOF 5 16) for different mooring configurations and same wavesea condition (a); same results, giving more emphasis to theresonant range (b)

Fig. 2 Scheme of the positioning of the platform P52 showingwave spectrum radar plot

031106-2 / Vol. 135, AUGUST 2013 Transactions of the ASME


Instrumentation

Since the beginning of P-52 operation, a motion reference unit(MRU) was installed for measuring its motions in the six degreesof freedom. The model of MRU chosen is the OCTANS

VR

, whichis composed by three accelerometers and three optic-fiber gyro-compasses (see Fig. 6). The acquisition rate for the motion mea-surement was 1 Hz.

Besides the motion measurement, it is very important for thiscase study to obtain the simultaneous environmental parameters atthe location of the P-52. However, the P-52 platform had no cur-rent measurement system installed during the period of this study.For that reason, it was necessary to obtain current measurementsfrom sites as close as possible to the P-52. The following equip-ment and platforms were then considered:

• FSI-2D-P-18 (50 km apart from P-52, depth of measurement:10 m)1

• ADCP-FPSO Brasil (10 km apart from P-52, depth of mea-surement: 57 m)2

• FSI 3D-PPG-1 (72 km apart from P-52, depth of measure-ment: 13 m)3

The next environmental agent to be considered is the wind. Itwas not possible to obtain a continuous wind time series as the ac-quisition standard for Campos Basin is set to measure only 10 minfor each hour. Nevertheless, in the specific case of P-52, the windacquisition system was set for 10 min each 20 min. Therefore,here it will be evaluated the wind acquisitions executed from 23 h40 of June 16 to 1 h 20 of June 17. This period shall have enoughdata for the identification of the excitation provided by the windspectra. The wind velocities and directions were measured at P-52in a height of 78 m above the water surface through a meteorologi-cal station YOUNG4. The acquisition rate was 1 Hz.

Finally, the wave incidence at P-52 was measured by the waveradar MIROS MKII5 in the platform P-18. Similar to the current,it was not possible to have the wave measurement at the P-52exact location. However, P-18 is 50 km apart from P-52 and theyare under the same wave climate influence.

Theoretical Modeling

Wave Loads. The low-frequency second-order motion spectramight be evaluated through the following relation:

Sð�Þa ðxÞ ¼ HaðxÞj j2Sð�ÞF;a ðxÞ (1)

where Sð�ÞF;a ðxÞ indicates the second-order force spectra and Ha(x)

represents the transfer function of the a-motion for a force of uni-tary magnitude:

HaðxÞ ¼1

�x2ðM þ AðxÞÞ þ ixðBðxÞ þ BextÞ þ ðCþ CextÞ(2)

In Eq. (2), M is the system mass, A, B is the added mass andpotential damping matrixes, and C the hydrostatic restoring ma-trix. Bext and Cext represent the linearized external damping andrestoring forces and moments.

The second-order force spectra Sð�ÞF;a ðxÞ may be calculated from

the quadratic transfer functions (QTFs) taking into account differ-ent pairs of frequencies. Let’s then consider the pair (X;Xþ l),the parameter l representing a small frequency difference. Onemay then write (see, for instance, Ref. [3]):

SF;aðlÞ ¼ 8

ð10

SðXÞSðXþ lÞ Fð�Þa ðX; Xþ lÞ�� 2dX (3)

where S(x) represents the wave spectrum and Fð�Þa denotes the

low-frequency QTFs.The core of the problem concerns the calculation of the low-

frequency second-order wave loads, which shall be obtained bymeans of a numerical code; in this work WAMIT

VR

was used to solvethe complete second order problem (see Refs. [4,5]).

Wind Loads. Having estimated the wind velocity spectral den-sity function, it is possible to obtain the moment spectrum asexplained in Ref. [6]. Consider the drag force in a structure as:

Fig. 5 P-52 picture after the topside installation (deck mating)in Angra dos Reis

Fig. 6 Picture of the equipment OCTANSVR

installed at P-52

Table 1 P-52 characteristics

Length (m) 120.0Beam (m) 110.0Draft (m) 27.5Air gap (m) 11.0Displacement (t) 80985.8Oil production capacity (bod) 180.000Gas compression capacity (m3pd) 9.3 mm

1See www.falmouth.com/DataSheets/2ACMRev3.pdf.2See www.rdinstruments.com/datasheets/oceanobserver_ds_lr.pdf.3See www.falmouth.com/DataSheets/3DWAVE.pdf.4See www.youngusa.com. 5See www.miros.no/wave_and_current_radar.php.

Journal of Offshore Mechanics and Arctic Engineering AUGUST 2013, Vol. 135 / 031106-3


www.falmouth.com/DataSheets/2ACMRev3.pdf

www.rdinstruments.com/datasheets/oceanobserver_ds_lr.pdf

www.falmouth.com/DataSheets/3DWAVE.pdf

www.youngusa.com

www.miros.no/wave_and_current_radar.php

�Fd ¼1

2qarAVCd

�U210 (4)

being qar, the mass density of the air, AV, the projected area in thedirection of wind incidence, Cd, the drag coefficient and �U10, themean wind velocity at an elevation of 10 ms. In case the wind hasnot been measured at a height z of 10 ms, which is the case in thiswork, the following correction must be made:

U10 ¼ UZ10

z

� �1=7

(5)

Another important parameter that needs to be defined is the aer-odynamic admittance function that can be computed through:

vðxÞj j2¼ 1

1þ 2xffiffiffiffiffiffiAV

p

3 �U10

� � (6)

Note that Eq. (6) shows that the dynamic wind load willdecrease as the frequency increases. Finally, the wind momentspectral density function can be calculated as:

SMVðxÞ ¼ 4

�F2dZa

�U210

vðxÞj j2SVðxÞ (7)

where Za is the distance between the center of the projected windarea and the body center of gravity (see Ref. [6]). SV is the windvelocity spectral density function.

Spectral Analysis. The spectral analysis used in this work isbased on the Welch method [7], considering a Hanning windowwith 50% of overlap. The degrees of freedom (DOF) represent thenumber of points for each segment divided by the number of seg-ments used by the subdivision of the data series. The aim of theprocedure described above is to smooth out the raw spectrum.

Through the spectral density functions, the significant ampli-tudes can be calculated by:

m0 ¼ð1

0

SaðxÞdx (8)

AS ¼ 2ffiffiffiffiffiffim0

p(9)

Measurement Results

This section presents the time series measured for the P-52motions and the incident environmental conditions for a specificevent. Figure 2 shows a scheme of the directional spectrum ofwaves, and the orientation of the Semisubmersible. As mentionedbefore, the waves were multidirectional coming from 150 to250 deg in relation to north (see Fig. 2). In all cases, the measuredcurrent velocities were very low, around 0.1 m/s. Therefore, it isevident that the current was not the environmental agent that causedthe resonant P-52 roll and pitch motions. The wind came from thesouth. Forces and moments caused by wind are small and not con-tribute to the resonant motion as shown later in the text.

First, the roll and pitch statistics, 1st and 2nd order significantamplitudes and natural periods for each hour of June 2008 arepresented.

All the spectral analysis was done with DOF-16 and the secondorder significant amplitudes were computed with Eq. (8) in theperiods range between 25 and 80 s. The natural periods were esti-mated by identifying the peak of the response spectral densityfunction for periods above 20 s.

The first observation that can be made from the graphics ofFigs. 7–9 is that the roll and pitch second order motions, in spiteof having lower amplitudes, are not negligible. In some cases,they are at the same level of the first order significant amplitudes.

Regarding the natural periods, the average obtained for roll andpitch are 50 and 41.2 s, respectively. It will be seen later that thesevalues are consistent with the metacentric heights (GMt e GMl)obtained by the stability calculations.

For a better understanding of the phenomenon, the case withlarger values of roll and pitch second order motions (1st hour ofJune 17) is presented next (see Fig. 10).

Note that for the 1st order motions, Eq. (8) is computed from 0to 20 s, and for the 2nd order ones, from 20 to 80 s.

A curious fact that can be observed in Fig. 10 is that the firstorder pitch motion is larger than the roll; on the other hand, the op-posite happens for the resonant behavior. Furthermore, in Fig. 10, itis possible to estimate the natural periods for roll and pitch throughthe peak of the spectrum. They are 49.3 and 41.7 s, respectively.

Regarding the environmental conditions, Fig. 11 presents thecurrent velocity measured close to the P-52 location. As it was notpossible to measure the current in the exact location of P-52,measurements in three different locations are presented and in aperiod larger than the one chosen for the motions (5 days insteadof 1 h). In all cases, one can see that the measured current veloc-ities were very low, around 0.1 m/s. Therefore, it is evident thatthe current was not the environmental agent that caused the reso-nant P-52 roll and pitch motions. It is also important to mentionthat this low value of current velocity is not expected to have an influ-ence on the resonant motions, as a high current velocity increases thedamping level, therefore decreasing the resonant motions.

As already mentioned, it was not possible to obtain a continu-ous time series for the wind. Figure 12 presents the time series forwind velocity considered in this work, including the gaps in theacquisition.

Fig. 7 Statistics of P-52 roll (a) and pitch (b) measured withOCTANS

VR

: June 2008



Figure 13 presents the wind velocity spectral density functionfor each portion of the time series. Also included is the spectrumconsidering all portions grouped. There is a considerable disper-sion in the spectral density functions in the periods of interest(around 40 and 50 s—roll and pitch natural periods). Trying toavoid underestimating the importance of the wind excitation, the

spectral density function of portion 3 (0 h 20 of June 17, 2008)was considered.

Figure 14 shows the directional wave spectrum measured at thesame time of the motions acquisition. It is a plot of the wave spec-trum map with respect to the wave frequency and direction (0 degindicates waves coming from the north direction). A radar plot ofthe same spectrum, with the radial coordinate representing thewave period is presented in Fig. 2, where a sketch of the platformheading (P-52 heads north) is presented, as mentioned above.

To simplify the calculation of Eq. (3), the corresponding unidir-ectional wave spectrum can be obtained from the directional wavespectrum by the following expression:

SðxÞ ¼XJ

j¼1

Sðx; hjÞ (10)

where j is the index of each wave direction of the spectrum. Themain wave direction is obtained according to a standard procedure.

Comparison of Full Scale Measurements

and Theoretical Computations

This section presents the wind and wave loads computation.Current loads were disregarded as it was shown that the currentvelocity was very low during the monitoring period.

Wind Loads Computation. The information about the P-52aerodynamic coefficients was obtained from the mooring design

Fig. 8 P-52 roll (a) and pitch (b) 1st and 2nd order significantamplitudes: June 2008

Fig. 9 P-52 roll, pitch and heave natural periods: June 2008

Fig. 10 P-52 time series for roll (a) and pitch (b)



report of this Unit (see Ref. [8]). The wind areas in the x and ydirections are, respectively, 2709 and 3007 m2. The centers ofpressure are, respectively, 22 and 23 m above the water level. Thewind force coefficients are presented in Fig. 15.

Thus, using Eq. (7), the aerodynamic information and the windspectral density function in the x and y directions, the windmoment spectral density functions, that can produce roll and pitchmotions, were calculated (see Fig. 16). It is important to mentionthat, in this case, the wind direction was almost aligned with theheading direction of the platform. Then it was already expectedthat the wind moment would be greater for pitch than for roll.

The next step is to compare the moment spectral density func-tions for wave and wind and verify their importance for the sec-ond order roll and pitch dynamics of P-52.

Wave Loads Computation. To compute the wave loads, itwas decided to use the WAMIT

VR

program that solves the completesecond order potential problem. To calibrate the model usingWAMIT

VR

, first it is necessary to obtain the geometric and mechani-cal properties of the platform P-52 during the period of this casestudy. The Barge Engineer from P-52 informs on daily basis theweight distribution and the tanks level using the software SSTAB(see Ref. [9])—an in house PETROBRAS program to evaluate thestability properties of a floating body (Fig. 17). Particularly, itestimates the center of gravity height and the unit’s draft, primor-dial input data for the hydrodynamic model. On June 17, 2008,

Fig. 11 Current velocity measured in (m/s) in the vicinity ofP-52 June 15–19, 2008

Fig. 12 Wind velocity measured in P-52

Fig. 13 Spectral density function for the time series showed inFig. 12

Fig. 14 Wave spectrum map: directional wave spectrum meas-ured in the first hour of June 17, 2008

Fig. 15 Force coefficients by direction of wind incidence



the draft of P-52 was T¼ 27.87 m and the center of gravity heightKG¼ 28 m. It is important to note that around this draft and thiscenter of gravity, the longitudinal metacentric height of P-52 isalways greater that the transversal one. As the natural period canbe calculated by:

Tn ¼ 2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI þ Ia

GMDþ C0

r(11)

being C0, the stiffness caused by the mooring lines and risers. As

it is not expected that roll and pitch added inertias should be muchdifferent due to an almost symmetric geometry of the platform(and the same way, also C

0), it can be concluded that the pitch nat-

ural period should be smaller than the roll natural period. In fact,this was observed in Figs. 9 and 18.

As far as the hydrodynamic model calibration is concerned, theinertias and external stiffness C

0caused by mooring line and risers

are still missing at this point. To avoid a very complex estimationprocess, it was decided to adopt the following method:

(1) The geometric model was constructed using the softwareMULTSURF

VR

(see Ref. [10] and Fig. 19).(2) Using the wave spectrum and running WAMIT

VR

, the firstorder Response Amplitude Operators (RAO) for roll andpitch were computed, varying the values of inertias.

(3) By spectral crossing between the RAO curves and the wavespectral density, the first order spectral response for roll andpitch was calculated.

(4) Inertial values were obtained by the best fit between com-puted and measured response spectra.

(5) Once inertia was estimated, the external stiffness C0

wasfound by matching the roll and pitch natural periods to theones measured (Tn4¼ 49.3 and Tn5¼ 41.7 s).

As a result, the mass and stiffness matrixes used as input datafor the WAMIT

VR

program in this case study are given in Tables 2–5.

Fig. 16 Wind moment spectral density functions

Fig. 17 SSTAB model prepared for P-52

Fig. 18 Spectral density function for the time series presentedin Fig. 10: roll (a) and pitch (b)

Fig. 19 Geometric model of P-52 constructed with MultSurfVR



The hydrostatic stiffness for roll and pitch are around 50% ofthe external stiffness for this case study.

Another important definition is the range of frequencies for theanalysis. As presented in Eq. (3), the second order force spectrumfunction is computed in relation to the difference-frequenciesl ¼ ðxi � xjÞ. For that reason, in this study case, it was chosen:21< 2p/l< 53 s.

Finally, as WAMITVR

computes the second order potential func-tion including the free surface forcing terms, it was necessary togenerate a mesh in the free surface. For this purpose, a mesh of2344 panels was considered, in a circular surface with a radiusof 80 m.

With the numerical model prepared, it was possible to calculatethe second order wave moments and compare them with the windloads.

Fig. 20 Spectral density functions of second order roll andpitch moments caused by waves (smwaroll and smwapitch):first hour of June 17, 2008

Table 2 Mass matrix considered for the case study

Surge Sway Heave Roll Pitch Yaw

Surge 81032.7 0.0 0.0 0.0 0.0 0.0Sway 0.0 81032.7 0.0 0.0 0.0 0.0Heave 0.0 0.0 81032.7 0.0 0.0 0.0Roll 0.0 0.0 0.0 63776242.2 0.0 0.0Pitch 0.0 0.0 0.0 0.0 120127021.9 0.0Yaw 0.0 0.0 0.0 0.0 0.0 120127021.9

Table 3 Units used for the mass matrix

1 2 3 4 5 6

1 t t.m23

4 t.m t.m2

56

Table 4 External stiffness matrix considered for the casestudy

Surge Sway Heave Roll Pitch Yaw

Surge 0.0 0.0 0.0 0.0 0.0 0.0Sway 0.0 0.0 0.0 0.0 0.0 0.0Heave 0.0 0.0 0.0 0.0 0.0 0.0Roll 0.0 0.0 0.0 379232.9 0.0 0.0Pitch 0.0 0.0 0.0 0.0 721246.0 0.0Yaw 0.0 0.0 0.0 0.0 0.0 0.0

Table 5 Units used for the external stiffness matrix

1 2 3 4 5 6

1 KN/m KN23

4 KN KN.m56

Fig. 21 Comparison of resonant second order roll (a) and pitch(b). P-52 theoretical and measured movements in the first hourof June 17, 2008 (DOF—degrees of freedom used in the spectralanalysis).



One may see in Fig. 20 that the moment spectra caused by windhave the magnitude of 106, while the ones caused by waves are oforder 108 and 109. Consequently, it can be considered that the sec-ond order wind loads are also negligible compared to the waves.As the current was also discarded as a possible environmentalagent causing the resonant second order motions, this indicatesthat the resonant roll and pitch motions observed were indeedexcited primarily by waves.

Damping Evaluation. Once the second order loads were com-puted and using Eqs. (1) and (2), it was possible to calculate thelevel of damping that adjusted the numerical and measured signif-icant amplitudes in the resonant range.

The graphics presented in Fig. 21 indicate that the externaldamping values Ze, the fraction of critical damping, necessary tocalibrate the roll and pitch motions are, respectively, 2% and 10%.

One should observe that the damping is caused not only by theP-52 hull, but also by the mooring lines and risers. For an estima-tion of the damping caused only by the hull, it would be necessaryto compare the measured results with a coupled dynamic simula-tion including the lines and its drag coefficients.

Note also that the viscous damping estimated for roll was con-siderably lower than the one for pitch. This is a result of using theunidirectional wave spectrum (see Fig. 22) in the computations. Abetter evaluation of this effect is discussed next.

To execute the calculation with the directional wave elevationspectrum, first, step 4 of the numerical model calibrationexplained before was redone, considering the first order motionscomputed by:

AS;a ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i¼1

Xm

j¼1

Sðxi; hjÞ � RAOaðxi; hjÞ�� 2DxDh

vuut (12)

being n and m, the number of wave frequencies and directions,respectively. Using Eq. (12), the radii of gyration found to cali-brate the model were Kxx¼ 37.75 m and Kyy¼ 39.05 m, respec-tively. Before, when using the unidirectional wave spectrum,these values were Kxx¼ 28.95 m and Kyy¼ 37.23 m. Note thatthe roll and pitch radii of gyration estimated using the directionalwave spectrum are closer than in the previous method. Consider-ing the symmetric geometry of the semisubmersible, this seemsmore consistent.

Finally, the same procedure was applied to obtain the signifi-cant amplitudes for the roll and pitch resonant motions, varyingthe external damping to fit the full scale measurements. The equa-tion used is as follows:

AS;a ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8Xn

i¼1

Xm

j¼1

Xn

k¼1

Xm

l¼1

Sðxi; hjÞ � Sðxk; hlÞ � RAOð2�Þa ðxi; hj;xk; hlÞ

�� 2DxDh

vuut (13)

The graphics presented in Fig. 23 indicate that the external damping(Ze) values necessary to calibrate the roll and pitch movements arenow around 10% and 12%, therefore both of the same level. Thisresult is expected as the level of pitch and roll motions are similar.

The above results show that the use of wave directional spec-trum instead of the unidirectional spectrum plays an importantrole in the resonant dynamics and should be taken into accountwhen trying to calibrate or compare second order motions in the

Fig. 22 Unidirectional wave spectrum density function meas-ured in the first hour of June 17, 2008

Fig. 23 Comparison of resonant second order roll (a) and pitch(b). P-52 theoretical and measured movements in the first hourof June 17, 2008: theoretical calculation with directional wavespectrum (DOF—degrees of freedom used in the spectralanalysis).



vertical plane of a semisubmersible platform. In Ref. [11], it canbe also found an interesting evaluation of QTF computation forTLPs considering multidirectional waves.

Conclusions

A study with full scale measurements was performed for thesemisubmersible unit PETROBRAS P-52. This platform is operat-ing at Roncador field in Campos Basin. The measurement cam-paign was carried out in June 2008. From spectral analysis of thesignals measured with a MRU installed at P-52, it was possible toestimate the significant amplitudes in the wave frequency rangeand also for resonant motions. During June 2008, it was evaluatedthat P-52 experienced significant roll and pitch second ordermotions. The identification of the peaks in the response spectraldensity function also permitted the identification of the roll andpitch natural periods.

Aiming at evaluating the origin of these resonant motions, acase study was performed for the first hour of June 17, 2008:the moment when higher pitch and roll second order motionswere identified by analyzing the environmental conditions atthe time (wave, wind, and current) and confronting theoreticalcomputations with the measurements, it was possible to verifythat the waves were the primary reason for these motions. Fur-thermore, with the support of theoretical calculations, the exter-nal damping of the P-52 platform (including mooring lines andrisers) could be estimated. It was also shown that the directionalwave spectrum may play an important role in the second-orderdynamics and should be taken into account instead of the uni-directional wave spectrum in order to guarantee consistentestimations.

Acknowledgment

Authors wish to thank PETROBRAS for supporting thisresearch project. The third and fourth authors acknowledge thesupport given by CNPq for their research work. The authorswould also like to thank Fabiana Cavinatto for her support in thepreparation of the figures.

References[1] MARINTEK REPORT, 2002, “Roncador FPU Model Tests—Main Report,”

Norwegian Marine Technology Research Institute, Trondheim, Norway.[2] Ogilvie, T. F., 1983, “Second-Order Hydrodynamic Effects on Ocean

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http://dx.doi.org/10.2118/4837-PA

http://dx.doi.org/10.1109/TAU.1967.1161901

http://dx.doi.org/10.1016/0141-1187(92)90040-Q

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