Load identification of offshore platform for fatigue life estimation

Nevena Perisic1, Poul Henning Kirkegaard2, and Ulf T. Tygesen1

1Ramboll Oil & Gas, Willemoesgade 2, Esbjerg 6700, Denmark2Department of Civil Engineering, Aalborg University, Sohngaardsholsvej 57, Aalborg, 9000, DK

Abstract

The lifetime of an offshore platform is typically governed by accumulated fatigue damage. Thus, the load time historyis an essential parameter for prediction of the lifetime of the structure and its components. Consequently, monitoringofstructural loads is of special importance in relation to re-assessment of offshore platforms. Structural monitoring systems(SMSs) on offshore structures typically consist of a set of sensors such as strain gauges, accelerometers, wave radarsand GPSs, however direct measuring of the actual loading is usually not feasible. One approach is to measure the loadsindirectly by monitoring of the available dynamic responses of the structure. This work investigates the possibility for usingan economically beneficial, model-based load estimation algorithm for indirect measuring of the loading forces actingonthe offshore structure. The algorithm is based on the reduced order model of the structure and the discrete Kalman filterwhich recursively estimates unknown states of the system inreal time. As a test-case, the algorithm is designed to estimatethe equivalent total loading forces of the structure. The loads are estimated from noised displacement measurements ofasingle location on the topside of the offshore structure. The method is validated using simulated data for two wave loadingcases: regular and irregular wave loadings.

Keywords: Load identification; Kalman filter; Offshore oil platforms;State-space model; Structural monitoring systems

1 Introduction

Structural monitoring systems (SMSs) of offshore structures usually consist of a set of sensors, such as strain gauges,accelerometers, wave radars and GPSs. However direct measuring of the actual loading is usually not feasible as it wouldrequire a high number of sensors, if possible at all. Consequently the loading forces need to be extracted indirectly.Knowledge of the actual loading on the structure is crucial for: prediction of the remaining lifetime and lifetime extension[1], component design validation and application of the advanced control algorithms in case of the need for accumulatedfatigue monitoring. The fatigue loads have direct influenceon the lifetime of the structure, where as extreme loads cancause instantaneous failures. Sometimes, monitoring of response characteristics can also be used as part of a warningsystem to provide alerts even before damages are developed which is very important for improving structure performanceand increasing reliability.

Dynamic load estimation is a well-known problem, which represents a specific type of an inverse identification problem.The main challenge with the inverse problem is that it is ’ill-posed’ [2] in mathematical sense thus existence, stability anduniqueness of its solution are violated. Methods that are solving the problem of load estimation are based on the idea totransfer the ill-posed problem into a well-posed one. Basedon the type of the system model, they can be divided in threemain categories [3]:

1. Deterministic methods: Majority of the available load estimation algorithms belong to this group[4]. These methodsare based on the deterministic dependencies between data, and their results depend strongly on the accuracy of theidentified inverse model. The solutions can be find in the time[5], [6] or frequency domains [7], [8], [9], [10], [11],[12].

2. Stochastic methods: This group of methods is based on statistical, i.e. stochastic models between the input andthe output data [13], [14], [15]. Thus, these methods require the measurements of the input and the output data fromthe real, operational systems in order to identify stochastic models of the systems. However, for some systems is notpossible to measure input at all.

3. Artificial intelligence methods: This group of methods includes use of the artificial neural networks [16], [17], theevolutionary algorithms [18] and the fuzzy logics [19] for load identification. The artificial intelligence methods

1

are used when there is not enough physical knowledge about the structure or when the model is very complex so itcannot be processed in the real time. These methods assume that the model of the system is a black-box, so theyrequire a learning process in order to determine the relations between the inputs and the outputs.

Lately, the focus of the research has been shifted towards combined deterministic-stochastic methods. These methodsare based on the Kalman filter [20] and their origins are in thecontrol theory. They are spread from the completelydeterministic to those where the stochastic noise is assumed to be present in all measurements and states of the system.The Kalman filter is an optimal, recursive, computationallyefficient, state-estimation algorithm. It requires storing onlythe previous estimation and it is very efficient in handling the measurement and process noise in real time. One of the firstapplications of the Kalman filters to the problem of the input-estimation are presented in [21], [22], [23], [24]. The methodconsists of two stages. In the first stage, the Kalman filter estimates states of the system assuming that the input force isnotavailable. In the next stage, a recursive least-square estimator provides estimation of the unknown loading force. In [25]an improved and changed method for the load reconstruction is presented. The new algorithm is simplified and it is basedon estimation of all unknown signals using only the first stage of the previously described method, i.e. the Kalman filter.

In a recent work [26], an augmented Kalman filter was used for direct estimation of the unknown loading force. Thestate of the system is augmented with the unknown load, and its dynamic evolution is assumed known, i.e. as a tuning factor.The augmented Kalman filter is tested on simulated data, and in a laboratory experiment on an instrumented steel beam.The augmented Kalman filter was also used for the wind turbineload estimation in [27], [28] for the online estimation ofthe drive-train shaft torque and the tower root bending moment from the standard turbine measurements. The limitationof the augmented Kalman filter approach is that it is suitablefor load identification in static environment, i.e. where thestructure operates in one operating regime. In [28] the method is expanded to allow for more dynamic environment andchanges of the operating points by applying the multi-modelestimation algorithm.

Problem statement and the scope of the workThis work investigates a possibility of using the augmentedKalman filterfor the load estimation on offshore platform structure. Thepresented method is model-based and it transforms aFE model of the structure into the state-space form, where all unknown signals are collected in the state vector andfurther estimated recursively using the Kalman filter. How well the distribution of the structural loads acting onthe structure can be estimated depends on the complexity of the used model, and its observability. The Kalmanfilter approach assumes that stochastic noise is present in all measurements of the system and it accounts for modeluncertainties.

A test-case for estimation of the total equivalent loading force on the platform from displacement measurements fora single node and simple two-degree-of-freedom (2DOF) model of the structure is presented. The total equivalentloading force is for simplicity in the present study defined as the point force acting on the topside of the offshoreoil and gas platform, which results in the measured displacements, i.e. the estimated loading force is equivalent tothe effect from the wave loading. In order to represent all ofthe frequencies observed in the simulated displacementsignal by a 2DOF system, the estimated equivalent loading force also includes fictive structural modes for the highermodes. From the estimated equivalent loading force, it is straightforward to calculate e.g. the total overturningmoment acting on the structure at the soil level. Additionally, the estimated signals will be post-processed by usinglow-pass filter, in order to validate their accuracy in the wave loading frequency range. All data used in this work aresimulated data and the measurement noise is artificially added in order to account for measurement uncertainties.

2 Load identification algorithm

The proposed load identification algorithm consists of the following steps that should be iterated until sufficient accuracyof the estimated signal is reached:

1. System modelingSpecify and identify a state-space model of the system that includes all relevant dynamics. Start withthe simplest possible, and if necessary increase complexity.

2. Design of the load observerDesign a linear or nonlinear state estimator for recursive estimation of the state vector ofthe model identified in the previous step.

3. Parameter tuning and algorithm validation Tune the parameters of the estimator by using simulated or measureddata. Compare the estimated signal with the referent signalif available.

2

2.1 State-space model of the reduced order system

Usually, the responses of structures exposed to wind or wavedynamic loading are driven by the first few fundamentalmodes so reduced order, linear models can be used. For the estimation of the total equivalent loading forces acting onthe offshore platform, the simplest possible model consists of the first north-south (N-S) and the first east-west (E-W)fundamental modes, thus it can be represented by a linear twodegrees-of-freedom model (2DOF). In this way, onlytranslation is considered where it is assumed that rotational motion is not present. Also, in this work it is assumed thatelevation is negligible and that motions in N-S and E-W directions are independent and uncoupled, thus they can beapproached independently. These assumptions are valid forthe simulated data, where in practice potential coupling needto be checked. 2DOF model is given by:

q(t)+2ξnwnq(t)+w2nq(t) = F(t)m−1 (1)

wherem is the equivalent mass matrix andF(t) corresponds to the total equivalent loading force applied to the toppoint of the structure, which would have the equivalent effect on the platform as wind and wave loading forces acting onthe whole structure. The displacement vectorq(t), the natural frequencies vectorwn(t), the damping ratio vectorξ (t) anddriving force vectorF(t) are given by:

q(t) =[

x(t)y(t)

]

, wn =

[wn,x 00 wn,y

]

, ξn =

[ξx 00 ξy

]

, F(t) =[

Fx(t)Fy(t)

]

, m =

[mx 00 my

]

(2)

By defining the displacement and the velocity of the platformtower top as states of the system, equation (1) can berepresented in a state-space form as follows:

x(t)y(t)x(t)y(t)

︸ ︷︷ ︸

X(t)

=

0 0 1 00 0 0 1

−w2n,x 0 −2ξxwn,x 0

0 −w2n,y 0 −2ξywn,y

︸ ︷︷ ︸

Ac

x(t)y(t)x(t)y(t)

︸ ︷︷ ︸

X(t)

+

0 00 0

m−1x 00 m−1

y

︸ ︷︷ ︸

Bc

[Fx(t)Fy(t)

]

︸ ︷︷ ︸

U(t)

(3)

The measurements assumed available are positions, thus themeasurement equation is specified by:

[Y1(t)Y2(t)

]

︸ ︷︷ ︸

Y(t)

=

[1 0 0 00 1 0 0

]

︸ ︷︷ ︸

Cc

x(t)y(t)x(t)y(t)

︸ ︷︷ ︸

X(t)

(4)

In practice, the measurements of the system are obtained at discrete time intervals. Thus, it is more feasible to usea model of the system that directly maps discrete input to discrete output data sets. Such a discrete-time model can beobtained from the continuous-time model by performing discretisation and it is determined by the sampling timeTs. Thediscrete-time model has the following form:

Xk = AdXk−1+BdUk−1 (5)

Yk = CdXk−1 (6)

The system matrices in the discrete time are defined by:

Ad = eAcTs, Bd = A

[

I −e−AcTs]

A−1c Bc, Cd = Cc (7)

The state-space model in a discrete-time defined by equations 5 and 6 represents deterministic model of the system. Inorder to account for uncertainties, unknown process and measurement noises are included in the model:

Xk = AdXk−1+BdUk−1+Wk−1 (8)

Yk = CdXk−1+Vk−1 (9)

whereWk andVk are the stationary, stochastic system and measurement noise vectors with appropriate dimensions.For the sake of clarity, it is assumed that they are zero mean with covariance matricesQ andR respectively. In the case thatnoise is not zero mean, it is possible to transform the colored noise to the white noise [29], hence use the same approach as

3

described in the following. In order later to allow the stateestimator to identify the unknown input signals, the state vectoris augmented as follows:

XAk =

Xk

Fx,k

Fy,k

(10)

It is assumed that the unknown input vectors can be represented as the outputs of the first order dynamic systems indiscrete time:

Fx,k = αxFx,k−1+Ψx,k−1 (11)

Fy,k = αyFy,k−1+Ψy,k−1 (12)

The covariances of theΨx andΨy areSx andSy. Sx, Sy, αx andαy represent tuning parameters that should be experi-mentally determined for a specific problem and they are part of the augmenter process noise covariance matrix:

QA =

[Q 00 S

]

, S=

[Sx 00 Sy

]

(13)

The augmented system matrices are consequently:

A =

[Ad Bd

0 D

]

, C =[

Cd 0]

(14)

where0 is (2 x 2) zero matrix andD is (2 x 2) diagonal matrix where elements on diagonal areαx andαy, selectedto be 1 in this work. The system matricesAd, Bd andCd are defined by the first natural frequencies, the damping ratiosand the sampling time. Modal parameters are identified from the dynamic responses of the structure and the sampling ratiocorresponds to the measurement sampling ratio, thus completely defining the grey-box reduced order model of the system.Thus, the augmented state-space model in discrete-time is:

XAk = AXA

k−1+WAk−1 (15)

Yk = CXAk−1+Vk−1 (16)

2.2 Load estimation algorithm

In order to estimate unknown values of the augmented vector,Equations (15) and (16), the Kalman filter [20] is used as anoptimal, linear state estimator. The idea is to estimate states of the system by using recursive algorithm. The structure ofthe Kalman filter is given in Figure 1, [30], whereKk represents the Kalman gain calculated at time instantk. One way todescribe the Kalman filter is as a filter that whitens the measurements and extract the maximal information from them. TheKalman filter is the optimal filtering solution for the discrete-time filtering model, when the dynamic and measurementsmodels are linear Gaussian, uncorrelated, unbiased and independent white noises. If they are uncorrelated, unbiased andindependent white noises but not Gaussian, the Kalman filterrepresents the best linear filter. The filter is optimal in a sensethat the Kalman gainKk is calculated so that it minimizes a weighted 2-norm of the expected value of the estimation errore= Y − Y = Y −CX.

The complete procedure for deriving the Kalman filter equations can be found in [29]. The tuning parameters of theKalman filter are covariance matrices of the measurement andprocess noises. By increasing the covariance of the processnoise, more emphasis is given to the measurements where by increasing the covariance of the measurement noise, moreconfidence is placed on the model than on the measurements.

3 Results

3.1 Simulation setup

In order to generate simulated data for testing of the load estimation algorithm, the RAMBOLL Offshore Structure AnalysisProgram package, ROSAP is used. The ROSAP package is a finite element program for static, dynamic and non-linearanalysis of frame and truss structures for wind and wave loadconditions. The Valdemar offshore platform is a tripodplatform, installed in 2006 in the Danish North Sea 250 km west of Denmark. The platform is designed as a Not-Normally-Manned Platform (NNMP). The operator of the platform is Maersk Oil. Ramboll Oil & Gas (ROG) is the engineering

4

PROCESS��������

DAQ

A

C

k��

���

����

�

��

K

�

�

��

Figure 1: Structure of the Kalman filter in parallel with the real process

consultant preparing technical specification for the SMS, data post-processing, analysis and documentation. The FE modelof the Valdemar platform installed with the sensor set-up isshown in Figure 2.

Simulations are performed for both regular and irregular wave loading. Regular waves are characterized by singleharmonic sinus with wave height, frequency, phase and direction. Irregular waves are characterized by a sum of harmonicsinus with different wave heights, frequencies, phases anddirections. Figure 3 shows simulated time series of the normal-ized total wave loading forces acting on the structure for the case of regular and irregular waves. The total loading force iscalculated as a sum of all the forces acting on each of the elements in the FE model of the structure.

In order to compare the quality of the estimations, the normalized mean square error (NMSE) function is calculated forthe estimated equivalent loading force time-series, see Figure 17. The NMSE compares the simulated total loading forcetime seriesF with the estimatedF signal, whereσ2

F is the variance of the simulated force. Generally, a NMSE of under5% indicates good estimations, where the values under 1% correspond to perfect estimations.

NMSE[%] =100

Nσ2F

N

∑i=1

(F− F

)2(17)

The location on the platform where the displacement is assumed to be measured corresponds to the location of one ofthe GPS sensors installed on the Valdemar offshore platform. Thus, in the following simulations, it is also tested if thissingle point sensor location is sufficient for reconstruction of the total equivalent loading force on the structure.

3.2 Load estimation results: Regular waves

Table 1 presents the NMSE values calculated for the estimated total equivalent loading force, in N-S and E-W directions.Estimations are obtained for different measurement noise levels. Noise is characterized by signal-to-noise ratio (SNR)meaning the ratio between the simulated signal and noise powers. In the case where no measurement noise is present, SNRtends to infinity. High noise level is characterized by a low SNR value.

5

Figure 2: Valdemar offshore platform (left) and FE model (right). In the FE model, the installed sensors for the SMS arealso shown.

SNR Direction NMSE[%]

Inf N-S 1.0021000 N-S 1.941500 N-S 2.572200 N-S 3.974100 N-S 4.63220 N-S 5.19410 N-S 5.783Inf E-W 0.588

1000 E-W 0.842500 E-W 1.491200 E-W 3.013100 E-W 4.63320 E-W 4.99810 E-W 5.525

Table 1: NMSE for the total equivalent loading force estimations for different signal to noise ratios, for the case of regularwaves .

As it can be seen from the Table 1, very good estimations are achieved even in the presence of high measurement noise.Taking in consideration that only the 2DOF model of the platform was used, it is observed that the selected simulatedmeasurement of the displacement placed at the top side of theoil platform contains sufficient information for estimation ofthe total platform reaction forces/ moments when the structure is exposed to regular wave loading. Thus, the Kalman filtercan extract this information from the measurement signal and estimate the unknown loading even in the presence of highnoise.

6

200 250 300 350 400 450 500 550 600 650 700−1

−0.5

0

0.5

1

Time [s]

Nor

mal

ized

Tot

alW

ave

Load

ing

For

ce [k

N]

200 250 300 350 400 450 500 550 600 650 700−1

−0.5

0

0.5

1

Nor

mal

ized

Tot

al

Wav

e Lo

adin

g F

orce

[kN

]

Irregular Waves

Regular Waves

Figure 3: Total normalized loading force of the regular and irregular waves acting on the structure, time domain.

300 350 400 450

−200

−100

0

100

200

Time [s]

Tot

al r

eact

ion

forc

e [k

N]

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210

−5

100

105

1010

Frequency [Hz]

Pow

er s

pect

rum

of

the

tota

l rea

ctio

n fo

rce

ReferenceEstimations

Figure 4: Estimated vs. simulated total equivalent loadingforce for the case of very high measurement noise (SNR=20)and N-S motion of the platform

7

3.3 Load estimation: Irregular waves

In the case when the structure is exposed to irregular waves,the power spectrum of the waves contains more frequenciesas can be seen from the wider spectrum compared to the spectrum for the regular waves. By comparing the normalizedpower spectrums for the simulated displacement, the wave loading and the overtuning moment at sea bed level, see Figure5, it can be seen that the irregular waves excite more modes ofthe structure than the linear waves, as expected. However,the wave loading at high frequencies does not cause large displacements of the platform.

Also, in the simulated displacement in the N-S direction, see Figure 5, lower plot, torsional modes of the structure arepresent at 1.219 Hz and 1.816 Hz. However, torsional modes donot affect the overturning moments of the structure at seabed level. As these peaks will also be present in the estimated equivalent loading force, the estimation in the N-S directionwill be of lower quality than in the E-W direction.

The NMSEs of the total equivalent loading force estimationsfor different measurement noise levels, for the case ofirregular waves are presented in Table 2, NMSE1 values. As itcan be seen from the results, estimations are of lowerquality than for regular waves. The NMSE values are around 30% for the E-W case, and around 42% for the N-S case.In order to understand the observed higher NMSEs, the reference signal (the original simulated data) and the estimationsfor medium and high measurement noise levels, in both time and frequency domain, are presented in Figures 6 and 7. Itis seen that the wave loading and motion of the structure is dominant in the E-W direction (wave load amplitude range isapprox. [0-300kN]), where only very little motion is present in the N-S direction (wave load amplitude range is approx.[0-5kN]). This explains much higher values of the NMSE in N-Sdirection. For determination of the resulting loadingforce, estimation of the force in E-W direction is more relevant. Figure 6 corresponds to displacement in the E-W directionand from this plot it can be seen that the estimation error is aresult of considering a high frequency range from 0Hz andup to 5Hz. When estimating the quasi-static wave loading only the frequency range from 0Hz to say 0.4Hz is of interest.Figure 7 corresponds to N-S motion and one can conclude that the wave and the structural frequencies except the first mode(around 0.5Hz) are slightly underestimated, but this is probably due to very low wave loading, i.e. almost no wave loadingin this direction, hence more noise content. Also, the torsion frequency content around 1.2Hz is present in the estimations,but of cause not present in the reference signal (overtuningmoment at sea bed level). This explains for the higher NMSElevels.

In order to see how well the estimations perform in the lower frequency range that corresponds to the quasi-static waveloading force, estimated and reference signals are post-processed using low-pass filter. The NMSE values are calculatedagain, see NMSE2 in Table 2, and top plots in Figures 6 and 7. Asit can be seen from the results, much better NMSEvalues are achieved when the contribution of the fictive structural frequencies to the response is filtered. In the main wavedirection, E-W, the NMSE of the filtered estimations are around 5% which is very good, especially taking in considerationthe simplicity of the used model and the assumptions that were made in the work. It is important to highlight that filteringis allowed as the first structural mode is around 0.5Hz whereas the dominating wave loading frequencies are in the rangeof 0Hz and up to around 0.4 Hz.

8

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

10−10

10−5

100

105

Frequency [Hz]

Normalized reaction bending moment, E−W directionScaled loading force, E−W directionScaled displacement, E−W direction

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

10−10

10−5

100

105

Frequency [Hz]

Normalized reaction bending moment, N−S directionScaled loading force, N−S directionScaled displacement, N−S direction

Figure 5: Comparison of the power spectrums of the normalized displacement measurement, the total wave loading forceand the total reaction moment.

965 970 975 980−400

−200

0

200

400

Time [s]

Tot

al r

eact

ion

forc

e [k

Nm

]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

100

105

Frequency [Hz]

Pow

er s

pect

rum

of t

he

tota

l rea

ctio

n fo

rce

965 970 975 980

−100

0

100

200

300

Time [s]

Tot

al w

ave

fo

rce

[kN

m]

ReferenceEstimation, SNR=200Estimation, SNR=20

Figure 6: Estimated vs. simulated wave loading force and total equivalent loading force in time and frequency domains,for the case of low (SNR=200) and high measurement noise (SNR=20) and E-W motion of the platform.

9

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

100

Frequency [Hz]

Pow

er s

pect

rum

of t

he

tota

l rea

ctio

n fo

rce

956 958 960 962 964 966 968 970 972 974 976 978−5

0

5

Time [s]

Tot

al w

ave

forc

e [k

N]

956 958 960 962 964 966 968 970 972 974 976 978−20

0

20

Time [s]

Tot

al r

eact

ion

forc

e [k

N]

ReferenceEstimation, SNR=20Estimation, SNR=200

Figure 7: Estimated vs. simulated wave loading force and total equivalent loading force in time and frequency domains,for the case of low (SNR=200) and high measurement noise (SNR=20) and N-S motion of the platform.

4 Conclusions and future work

SNR Direction NMSE1[%] NMSE2[%]

Inf E-W 28.50 5.501000 E-W 28.58 5.51500 E-W 28.37 5.48200 E-W 28.59 5.55100 E-W 29.23 5.6520 E-W 31.64 5.8310 E-W 35.31 6.92Inf N-S 41.56 10.59

1000 N-S 42.27 10.80500 N-S 41.45 10.48200 N-S 43.57 10.65100 N-S 41.58 10.7920 N-S 42.55 15.3110 N-S 46.15 21.15

Table 2: NMSE values of the total equivalent loading force estimations (NMSE1) and total wave force estimations(NMSE2) for different signal to noise ratios, for the case ofirregular waves.

5 Conclusions and future work

This paper has tested a model-based, load estimation algorithm on the problem of online estimation of the total equivalentloading force on an oil and gas platform based on simulated displacement measurement from a single location on the topside of the platform. The algorithm is using the augmented Kalman filter as a recursive state estimator and a simple, 2DOFmodel of the platform. The method is tested on simulated data, assuming two loading scenarios: regular and irregular

10

wave loading. In order to test robustness of the algorithm onthe measurement noise, white noise is artificially added tothe displacement data. In the case of irregular waves, estimated forces are post processed using a low-pass filter in order tovalidate their goodness in the wave loading frequency range. Conclusions from this work are given in the following:

Regular waves For the case of regular wave loadings, the estimations of theequivalent loading force are almost perfectfor low values of the SNR (NMSE values are under 1%). For very high SNR, estimation results are still very goodboth for low and high frequencies with the NMSE values around5% for the highest simulated measurement noiselevel. As the estimation of the equivalent loading force is very good, the estimation of the quasi-static wave load isalso very good.

Irregular waves In the case of irregular waves, the equivalent loading estimation degrades significantly. The simulatedwave loading power spectrum is wider than in the case of the regular waves. As a consequence, more structural modesof the structure are excited at higher frequencies. However, the high frequency content present in the simulateddisplacement measurement corresponds to very small displacement amplitudes with no structural impact. Also,some fictive structural frequencies are present in the estimated total equivalent loading, which is not representedin the reference overtuning moment (torsion cannot be detected in the overturning moment), hence contributing tothe higher NMSE values. When estimating only the quasi-static wave loading forces by filtering processes andcomparing to the reference wave loading very good results are obtained. In the main wave direction (E-W), NMSEvalues in the range of 5% is achieved.

Overall results are very good and promising, especially taking in consideration the model simplicity. In the future workmore complex models of the platform will be considered so that the estimation of the total equivalent loading force couldbe further improved. The future work will consider use of multi DOF systems, instead of the presented simple 2DOF modelof the structure in order to estimate and represent the distribution of the wave loading forces on the offshore structure.

6 Acknowledgements

Cooperation of Maersk Oil in preparing this paper and providing data is gratefully acknowledged.

References

[1] Johannesson P. (1999)Rainflow analysis of switching Markov loads. PhD thesis, Lund Inst. of Technology, LundUniversity, Sweden.

[2] Hadamard J, (1923)Lectures on coughy’s problem in linear partial differential equationsYale University Press,NewHaven, (1923).

[3] Uhl T. (2007)The inverse identification problem and its technical application, Jurnal: Archive of AppliedMechanics,Springer-Verlag,vol. 77, pp 325-337

[4] Gladwell G. M. L, (1986)Inverse Problems in Vibration, Kluwer Academic Publishers, Dordrecht, Netherlands

[5] Klinkov M, Fritzen C.P,An updated comparison of the force reconstruction methods, Key Eng. Mater, 347 (2007),pp. 461466

[6] Nordstroem L. J. L, Nordberg T P,A critical comparison of time domain load identification methods, in: Proceedingsof the Sixth International Conference on Motion and Vibration Control, Saitama, Japan, 2002, pp. 11511156.

[7] Fabunmi J.A. (1986)Effects of structural modes on vibratory force determination by the pseudoinverse technique,AIAA J., 24 (3) (1986), pp. 504509

[8] Hillary B, Ewins D.J.The use of strain gauges in force determination and frequency response function measurements,in: Proceedings of IMAC II International Modal Analysis Conference, Orlando, 1984, pp. 627634.

[9] Stevens K.K,Force identification problems: an overview, in: Proceedings of SEM Spring Meeting, Houston, Texas,1987, pp. 838844.

[10] Parloo E, Verboven P, Guillaume P, Van Overmeire M, (2003) Force identification by means of in-operational modalmodelsJ. Sound Vib., 262 (1) (2003), pp. 161173

11

[11] Vanlanduit S, Guillaume P, Cauberghe B, Parloo E, De Sitter G, Verboven P,On-line identification of operationalloads using exogenous inputsJ. Sound Vib., 285 (2005), pp. 267279

[12] Nordstrm L.J.L, (2006)A dynamic programming algorithm for input estimation on linear time-variant systemsCom-put. Methods Appl. Mech. Eng., 195, pp. 64076427

[13] Zion L, (1994),Predicting fatigue loads using regression diagnostics, in Proceedings of The American HelicopterSociety Annual Forum, Washington, DC, May 1113, pp. 13371358.

[14] Trujillo D.M, Busby H.R,Practical Inverse Engineering, CRC Press,London(1997)Uhl

[15] Stelzner D. A. and Krammer D. C, (1999),Input force estimation using an inverse structural filter, in Proceedings of17th IMAC, Orlando, FL, February 811.

[16] Uhl T. and Petko M, (2006),Load identification with the use of neural networks design and application, MechanicalSystems and Signal Processing.

[17] Uhl T, (2002)Identification of loads in mechanical structures-helicopter case study, Computer Assisted Mechanicsand Engineering Sciences, CAMES (Vol. 9), pp. 151-160.

[18] Uhl T. and Pieczara J, (2003)Identification of operational loading forces for mechanical structures, Arch. Transp.(Vol. 16), Nr. 2, pp. 109-126.

[19] Goral G, Zbydon K. and Uhl T, (2002)Intelligent transducers of in-operational loads in construction fatigue moni-toring, Mach. Dyn. Probl (Vol. 2-3), pp. 73-78.

[20] Kalman R.E, A new approach to linear filtering and prediction problemsJournal of Basic Engineering 82 (1): pp35-45. Retrieved 2008-05-03.

[21] Chan Y.T, Hu A.G.C and Plant, J.B, (1979)A Kalman Filter Based Tracking Scheme with Input Estimation, Aerospaceand Electronic Systems, IEEE Transactions on (Volume:AES-15 , Issue: 2 )

[22] Bogler P. L.Tracking a maneuvering target using input estimation, IEEE Trans. Aerosp. Electron. Syst., vol. AES-23,pp. 298-310, May 1987

[23] Hou M. and Xian S, (1989)Comments on ”Tracking a Maneuvering Target Using Input Estimation” , Aerospace andElectronic Systems, IEEE Transactions on (Volume:25 ,Issue:2 )

[24] Tuana P.C, Jia C.C, Fonga L.W. and Huanga W.T. (1996)An input estimation approach to on-line two-dimensionalinverse heat conduction problemsNumerical Heat Transfer, Part B: Fundamentals: An International Journal of Com-putation and Methodology, Volume 29, Issue 3, 1996

[25] Hwang J.S, Kareem A. and Kim W.J. (2009)Estimation of modal loads using structural responseJ. Sound Vib., 326, pp. 522539

[26] Lourens E, Reynders E, De Roeck G, Degrande G. and Lombaert G. (2012).An augmented Kalman filter for forceidentification in structural dynamicsMechanical Systems and Signal Processing, 27, 446-460.

[27] Perisic N, Pedersen B.J. and Kirkegaard P.H.(2012)Gearbox Fatigue Load Estimation for Condition Monitoring ofWind Turbines, Proceedings of International Conference on Noise and Vibration Engineering ISMA2012: Interna-tional Conference on Uncertainty in Structural Dynamics. KU Leuven, Department Werktuigkunde, pp. 4381-4392.

[28] Perisic N. and Kirkegaard PH. (2013)Low-Cost Tower Root Fatigue Load Estimation for StructuralHealth Moni-toring of Grouted Connections in Offshore Wind Turbines. In Damage Assesment of Strucure X: 10th InternationalConference on Damage Assessment of Structures (DAMAS 2013), July 8-10, 2013, Dublin, Ireland. Schweiz: TransTech Publications Ltd. pp. 676. (Key Engineering Materials, Vol. 569).

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