Mooring line tension observed through a maximum entropy spectrum

J Mar Sci Technol (1999) 4:68–75 Mooring line tension observed through a maximum entropy spectrum Svein Ivar Sagatun1, Finn Gunnar Nielsen1, Erling Handal2, and Nils Veland2 Norsk Hydro Exploration and Production ASA, Research Center, Department of Marine Science, N5020 Bergen, Norway Det norske Veritas,Oslo, Norway 1 2 Abstract: A method for measuring mooring line tension is proposed based on observation of the natural frequencies of the mooring line segment between the winch and the fairlead. The anchor line tension is observed through the string equation where an analytical expression for the line’s eigenfrequencies is obtained. The tension is observed on line by utilizing a nonparametric system identification approach in which the peaks of a maximum entropy spectrum of the transverse acceleration measures of the vibrating string are automatically identified. The method is verified against full-scale data from the Troll B floating concrete oil production platform operating in the North Sea. Key words: mooring line tension, observer, maximum entropy Introduction This article proposes a method for estimating mooring line tension based on accelerometer measurements. The anchor line tension is observed through the line’s string equation in which an analytical expression for the line’s eigen-frequencies is obtained. The tension is observed on line by utilizing a nonparametric system identification approach in which the peaks of a maximum entropy spectrum of the acceleration of the vibrating string is automatically identified. The tension corresponding to the found modal frequency is then calculated. The method is verified against full-scale data from the Troll B floating oil production platform operating in the North Sea west of Bergen, Norway (Fig. 1). Address correspondence to: S. Sagatun Received for publication on Feb. 24, 1999; accepted on July 26, 1999 Background Semisubmersible platforms are used for drilling and production of oil and gas. The design of the mooring system is an increasing challenge as the operations are moving toward deeper water. For operational reasons, as well as for design verification, it is important to have control of the tension in the mooring lines. A standard way of measuring tension is by use of force transducers mounted at the foundation of the mooring line winches. This is a robust way of measuring, but it has some drawbacks. The mean load can be difficult to measure accurately due to drift in the zero level. Experience has also proven that it is difficult to calibrate these cells accurately. A third important drawback of this approach is that it is expensive and difficult to repair these load cells offshore. The design lifetime of an oilproducing platform may exceed 20 years without going to dock. For very accurate measurements of line tension, instrumented links are introduced in the mooring chain. This solution, however, has serious practical drawbacks, first and foremost lack of reliability over a long time span. In the following an alternative method is presented. The method is a simple way of calibrating the load cells on the winch foundations with respect to mean load. We have also used the method to measure the dynamic variations of the mooring loads. The method is based on the fact that the natural frequencies of a vibrating string are functions of the tension in the string. We have measured the vibrations of the mooring line above the fairlead, since the length, mass, and boundary conditions are well known. The mooring line from the fairlead to the winch (Fig. 2) has a constant length, of which approximately half is submerged. The transverse vibrations of the line may be measured by an accelerometer attached to the line. Several of the eigenmodes of vibrations will be observed in the measurements. The eigenmodes will be time-varying because of the time- S.I. Sagatun et al.: Mooring line tension observed through a maximum entropy spectrum Fig. 1. The Troll B floating oil production platform operated by Norsk Hydro ASA on location west of Bergen, Norway 69 modes through the peaks of a fast Fourier transform (FFT) spectrum. To obtain an accurate resolution of the frequencies, we need long time histories, that is, the method may be used to identify mean tensions. However, the dynamic variations will not be captured. For this purpose the maximum entropy method (MEM) has been employed. This method has a much better frequency resolution than the FFT. The Norwegian Maritime Directorate3 and Det norske Veritas in its Position Mooring rules— POSMOOR4 require that mooring line tension be constantly monitored. However, it is not required that the anchors shall be actively used in the automatic feedback loop comprising the POSMOOR system. Hence, active control of the anchor tension by windlasses or winches may only be used to set new offset positions or to compensate for mean load in extreme weather conditions. These types of operations are done without tension feedback. The tension signals are normally monitored and used in on-line consequence analysis, alarm and warning systems, and simulations. Thus, the requirements for phase lag in the tension estimates are much less stringent than the requirements for position or thruster feedback signals. The tension observer has been successfully tested on full-scale data measured on a mooring line on the Troll B floating oil production platform operated by Norsk Hydro Exploration and Production ASA in the North Sea west of Bergen, Norway (Fig. 1). Concept Fig. 2. Fairlead, windlass, and chain (not drawn to scale) varying tension caused by external excitation, such as movements of the platform due to waves. The eigenmodes will be lightly damped, and hence the peaks in the spectrum are expected to be sharp. We propose that by finding the frequencies of these vibrations we would have an indirect measure of the tension. The amplitudes of the accelerations as such are not important for the present use. Similar approaches have previously been proposed to identify the modal damping ratios and natural frequencies of jacket structures.1,2 The most frequently used method to find the frequency content of a signal is by observing the eigen- We propose to observe the anchor line tension by measuring the transverse accelerations of the mooring line segment between the winch and the fairlead. The acceleration measurements are then converted to a frequency spectrum where the frequency peaks corresponding to the line’s eigenmodes are identified and then transferred to tension. The main challenges are first to make a reliable frequency spectrum with as short a measuring period (few measures) as possible to reduce lag, and second to identify the correct peaks that correspond to the anchor line’s (modeled as a string) eigenmodes. The first problem is solved by using the maximum entropy method for finding the frequency spectrum. The second problem is more complicated because of the oscillating disturbances on the platform and hence the anchor line tension. The first-order wave motions normally have a period between 5 and 15 s. The motions due to the second-order wave loads are also present, these with periods in the range of 25– 200 s. Third, we have the oscillating eigenmodes of the string, which are a direct function of the time-varying tension. 70 S.I. Sagatun et al.: Mooring line tension observed through a maximum entropy spectrum Verification of string assumption Anchor line model The anchor chain between the windlass and the fairlead (Fig. 2) is modeled as a tensioned beam with the following Euler-Bernoulli equation: r( x, t ) - ∂ 2w( x, t ) ∂t 2 + ∂2 ∂ x2 Ê ∂ 2w ˆ Á EI 2 ˜ ∂x ¯ Ë ∂w( x, t ) ˘ ∂ È ÍT ( x, t ) ˙ - f ( x, t ) = 0 ∂x Í ∂x ˙ Î ˚ (1) subject to 0 < x < L and w(0, t) = w(L, t) = 0 where w(x, t) and x denote transverse displacement and longitudinal position along the line, respectively; L is the length of the string; T(x, t) is axial tension; r(x, t) is mass per unit length; and t is time. This is a boundary value problem that cannot be analytically solved for abritrary r(x, t) and T(x, t). Neither of those is constant: r(x, t) has a contribution from added mass where and when it is submerged in water, and T(x, t) is time varying due to varying external loads on the platform. However, we will assume in the following that the added mass contribution is constant and incorporated in the r(x, t) = r term. We assume that the anchor chain can be treated as a string, and hence the term representing ∂ 2 Ê ∂ 2w ˆ EI 2 ˜ term is neglected. bending stiffness [the ∂ x 2 ÁË ∂x ¯ We will also use the quasi-static assumption that the tension T(x, t) varies much more slowly than the string’s vibrating mode, and hence we can solve the above boundary value problem for constant T with separation of variables.5 Hence, we obtain the following expression for the eigen frequencies of the string: ] w n = np T rL2 (2) This section discusses the robustness of the assumptions made above; that is, we consider the string model’s sensitivity of the natural frequencies to bending stiffness, variation in tension, and variation in mass. For a string of length L with uniform mass distribution, r, and constant tension, T, the natural frequencies are given by Eq. 2. In our test case we have L = 62.7 m, r = 506 kg/m and T = 2000 kN. From Eq. 1 we then obtain the natural frequencies as given at the first line of Table 1. To account for the hydrodynamic added mass on the lower half of the string, we have modeled the chain by two parallel rods of diameter 0.152 m and computed the added mass using an added mass coefficient of 1.0. The obtained additional mass has been evenly distributed along the string in estimating the natural frequencies. If the r in Eq. 2 is increased to account for hydrodynamic mass, the results shown in the second line of the table are obtained. We observe that all of the natural frequencies are shifted 1.8% downward. The above chain model is used to investigate the effect of the chain’s bending stiffness (in tension). It is straightforward to use other added mass representations. However, Table 1 clearly shows that the eigen frequencies vary little with respect to perturbations of the added mass coefficients within realistic values. Next we will illustrate the importance of variation in tension. Assuming a top tension of 2000 kN and the weight of the line equal to 311 kN, the average tension becomes 1845 kN. Using this tension in Eq. 2, we obtain the natural frequencies at the third line of Table 1. A 4% reduction of all natural frequencies is obtained. In considering the importance of the bending stiffness, we approximate the chain by two parallel circular rods with diameter equal to the chain diameter. The moment of inertia with respect to bending is thus taken as twice the moment of inertia for one rod. The rigid rod case is an upper limit for the bending stiffness. If there is some sliding between the links, the stiffness will be considerably reduced. We have therefore computed the natural frequencies for this upper limit as well as 10% of this value. Using the highest stiffness, we observe less than Table 1. Sensitivity of the five first eigen-frequencies (Hz) of a vertical chain due to variation in mass, tension, and bending stiffness r [kg/m] T [kN/m] EI/5.46 ¥ 106 [N] n=1 n=2 n=3 n=4 n=5 506 524 524 524 524 524 2000 2000 1845 2000 2000 2000–1689 0 0 0 1 0.1 0 0.5014 0.4927 0.4732 0.4970 0.4940 0.4740 1.0027 0.9853 0.9464 0.9920 0.9860 0.9500 1.5041 1.4780 1.4196 1.4880 1.4790 1.4250 2.0054 1.9707 1.8928 1.9870 1.9750 1.9030 2.5068 2.4633 2.3659 2.4800 2.4650 2.3800 S.I. Sagatun et al.: Mooring line tension observed through a maximum entropy spectrum 1% increase in the natural frequency for the fifth mode, while the first mode increases less than 0.1%. Using a linear distribution of tension (2000–1689 kN) versus the average tension (1845 kN) causes a change in the frequencies of less than 1%. This is of the same order of magnitude as the difference between the analytical results and the finite element results. In summary, it can be concluded that the simple oscillating string model, using the average tension and an average mass including added mass, will provide very accurate results for the natural frequencies for the first four modes. The effect of bending stiffness seems insignificant. The relative significance of bending will increase as the mean tension is reduced. The maximum entropy method The maximum entropy method (MEM) is, compared to Fourier-based methods, a relatively new method for forming frequency spectra where some of the main inconveniences of the Fourier-based spectra are avoided, i.e., the windowing functions and the zero padding assumption at the start and end of the time series. The MEM was selected for this purpose, since it is particularly useful for short data series where its resolution is much better than that of any other spectrum. The method is derived in detail in Burg.6 A more available reference is Kanasewich.7 The rest of this chapter is somewhat technical from a signal-processing point of view, and thus it may be skipped by a reader not familiar with and interested in this field. Notice that observation of a time-varying process through a frequency spectrum requires that we assume a “time-varying stationary” process. This contradiction in terms is used to characterize the fact that during one observation, using M samples, we assume a stationary process, whereas we do not require the process to be stationary from one observation (using M samples) to the next one, even if only one sample is different. That is, the first sample is deleted and a new sample is added as sample M + 1. The basic idea behind the concept of a maximum entropy spectrum is as follows. Design an optimal Norder AR (autoregressive) filter that whitens the input signal. Optimize the filter with respect to maximizing the entropy (see Eq. 3). The AR filter contains all the information in the input signal, since white noise is the only output from the filter. Thus, the power spectrum density of the transfer function of the AR filter spectrum S(z) represents all information from the input signal in an optimal sense according to Eq. 3. Note also that we have not made any assumptions on the values of the time series before and after its start and end (no zero padding and windowing). 71 The spectrum and the corresponding AR filter parameters can be derived in several ways,6 but the easiest and most straightforward way is to use the variational approach. The problem is to maximize the entropy: max Ú W -W ln S(w )dw (3) subject to r (n) = Ú W -W S(w )e i 2pnwDt dw -N £n£N where r(n) is the autocorrelation function with lag n, S(w) is the optimal predicted spectrum, Dt is the sam1 w . The term e-Wln S(w) dw is pling interval, and W = 2Dt the entropy of the time series in the frequency domain (see Burg6 or Kanasewich7). The solution of this variational problem is given as ( ) Sw = Pm Dt 1 - Ân =1 dn e N (4) 2 - i 2 pnwDt where Pm is the variance of the resulting white noise after passing the input time series through the filter. Equation 4 is derived by taking the power spectrum density of the transfer function of the AR filter: S(z) = Pm Dt 1 - Ân =1 dn zn N where z = e-i2pwDt. The Pm value and the filter coefficients in Eq. 4 are found by solving the following Wiener-Hopf equation: È r (0) r (1) L r (N ) ˘ È 1 ˘ Í ˙Í ˙ r (0) L r (N - 1)˙ Í - d1 ˙ Í r (1) ˙Í M ˙ = Í M M ˙Í Í ˙ L r (0) ˙˚ ÍÎ - dN ˙˚ ÍÎr (N ) r (N - 1) È Pm ˘ Í ˙ Í0˙ Í M ˙ Í ˙ ÍÎ 0 ˙˚ using the recursive Levinson-Durbin algorithm. This algorithm is described in most digital filter text books, e.g., Haykin.8 The observer The peaks of the spectrum can be determined analytidS w = 0. The identity cally by solving dw ( ) 2 N 1 -  dn e n =1 - i 2 pnwDt N ( = r o + 2 r n cos 2pnwD n =1 ) 72 S.I. Sagatun et al.: Mooring line tension observed through a maximum entropy spectrum where rn = N -n Âdd d0 = 1 i +n i i =0 is substituted in Eq. 4 before Eq. 4 is differentiated with respect to w, which yields9  N n =1 ( nr n sin 2pnwDt ) r o + 2Ân =1 r n cos(2pnwDt ) N =0 (5) Hence, we have an analytical expression of the frequency of the maximum or minimum peaks, that is, when the nominator in Eq. 5 becomes zero. We can look on the second derivatives of S(w) to check if the extrema found in Eq. 4 are maxima or minima. The second derivatives of the spectrum are given by ( ) d 2S w dw = 2 w =w o ( ) 4p 2 Pm2 Dt 3 N n =1 ( n 2 r n cos 2pnw 0 Dt ( ) Ê r + 2 N r cos 2pnwDt 2 ˆ Ân =1 n Ë o ¯ ) 2 Ï< 0 fi local maxima Ô Â n2 rn cos 2pnw 0 Dt = Ì= 0 fi indeterminate n =1 Ô< 0 fi local minima Ó ( ) dw will not be passed through the observer. 2 The time bandwidth of the observer is reduced with increased M, and hence, an optimal selection of M results from a compromise between the requirements for resolution of observed tension (which increases M) and the required bandwidth. In practice, variations in tension must be slower than <–81 dw to be observed through the observer. The sampling frequency is selected on the basis of the frequency of the highest eigen-mode we want to observe in the spectrum. We have chosen to include the p first modes in the observer. Hence, the sampling faster than Consequently, the test becomes N Fig. 3. Typical maximum entropy spectrum (db) from acceleration measurements plotted versus frequency (Hz) (6) Campbell9 also contains expressions for the mean and variance of the estimated extreme frequencies. An a priori estimate of the first eigen-frequency for the anchor line is important, since there are peaks in the spectrum that do not represent eigenmodes. These are due to first- and second-order wave disturbances (see Fig. 3). The a priori estimate is obtained by using Eq. 2 with a lower bound tension. The observer parameters Observer bandwidth and resolution in tension are two critical parameters for the observer. These parameters are controlled by the following three parameters: the 2p , the number of samples sampling frequency w s = Dt used in the spectrum M, and the filter order N. The number of samples M controls the resolution 2p Ê rad ˆ on the frequency axis according to dw = Á ˜ . In MDt Ë s ¯ other words, dw is an expression of the observer’s dw as an bandwidth. Filter designers normally use 2 expression for bandwidth. That is, all tension variations T (H ) . rL2 z Measurements from real world mooring lines show that the eigen-frequency does not linearly increase with mode number, and therefore a safety margin is added to the calculated ws. This is due to damping and the fact that the anchor chain is not a perfect string. The resolution of the observed tension T is a nonlinear function of the resolution of the frequency dw and the frequency w according to the expression 2 L2 Ê dT = 2wdw + (dw ) ˆ¯ r 2 ª 2kwdw . Thus, we notice Ë p that the resolution of tension is best for lower frequencies and lower mode numbers. frequency ws must be selected as w s > 2pp Tuning of the observer performance The classification rule4 has no requirements for time lag, but it contains a requirement that states that on-line consequence analysis must take place automatically at S.I. Sagatun et al.: Mooring line tension observed through a maximum entropy spectrum least every 5 min. As a compromise between lag and resolution, we have selected M = 128 samples. The full-scale data were originally sampled with 64 Hz and then resampled to 2–32 Hz. M = 128 and a sampling frequency of 2–32 Hz yields a delay of 48 s. Optimal selection of filter order (order of the AR filter comprising the MEM spectrum) has been extensively discussed in the literature; see, for instance Søderstrøm and Stoica.10 We have chosen to use the AR filter’s final prediction error measure (FPE), as given in Akaike11: M N P N FPE N = M m 1N 1+ ( ) (7) as our optimal criterion. Eq. 7 together with M = 128 results in an observer order N = 10 with the full-scale data presented in Fig. 4. We have chosen to track p = 2 eigenmodes as a compromise between reducing the variance of the estimated frequency and the belief in the string model. Full-scale test The proposed observer is tested on both full-scale data and simulated data for validation purposes. 73 April 1997. The proposed observer was first tested with the above-mentioned filter parameters on a 320 s time series. The time series was measured on the anchor chain 10 m below one of the windlasses taken from the Troll B platform (see Fig. 1). There are 62.7 m of freely vibrating mooring chain between the windlass and the fairlead (see Fig. 2). The mass per unit length of the chain is r = 524.5 kg/m. The results are presented in Fig. 4. The tracking of tension is rather good, and we can observe the platform motion due to the second-order wave loads. No wave motion is present in the tension estimate, since the bandwidth of the observer is less than the typical wave frequencies. Second, we tested the method for the purpose of identifying the platform’s low-frequency surge motion. We used a 3-h period and compared this with position data logged during the same period with a differential GPS system. This result is presented in Fig. 5. The primary spectrum peak on the position data-based spectrum corresponds very well with the primary peak on the spectrum composed of the mooring line accelerometer measurements. Both spectra have a peak frequency corresponding to a period of approximately 225 s. Note the difference between the maximum entropy spectrum and the corresponding Fourier-based spectrum in Fig. 6. It would be very hard to identify the correct signal peaks with an FFT-based spectrum with only 128 samples available. Tests on simulated data Tests using measures from Troll B mooring chain Measurements of acceleration of the anchor chain were logged in the period from 22 : 38 on 2 April to 07:38 on 3 A set of simulated data was also used to validate the filter performance. Simulated accelerometer measurements were generated, simulating a slowly varying Fig. 4. Top panel: raw unscaled acceleration data (sm/s2) plotted versus time (s). Middle panel: a typical spectrum (s2m/s3) used by the observer plotted versus frequency (Hz), (M = 128). Bottom panel: observed line tension (kN) plotted versus time (s) 74 S.I. Sagatun et al.: Mooring line tension observed through a maximum entropy spectrum Fig. 5. Left plot: observed nondimensional tension plotted versus frequency (Hz). The peak frequency corresponds to a period of 185 s. Right plot: measured nondimensional surge motion plotted versus frequency (Hz). The peak frequency corresponds to a period of 191 s Fig. 6. A Fourier-based spectrum using Welch’s method with a Hanning window, no overlap, and M = 128 compared with a maximum entropy-based spectrum with M = 128 and N = 10 tension time series shaped as a half-phase sinusoid with a period of 2000 s. The sampling time and the observer parameters are similar to the ones used with full-scale measures. Two modes are tracked. The results from these tests are shown in Fig. 7. Note that the observer tracks the slowly varying tension within the observer’s resolution boundaries, which are marked with uncertainty envelopes. Conclusion A mooring line tension observer using a string model and a maximum entropy spectrum was proposed and Fig. 7. Simulated anchor tension (half sine), observed tension (kN), and uncertainty envelopes representing the resolution (due to M) of the observer plotted versus time (s) tested against full-scale data from the Troll B platform. The MEM was successfully used to identify the mooring line’s vibrating modes. The eigen-frequencies of the mooring line modes were used to observe mooring line tension. It was shown that the maximum entropy spectrum has many favorable features compared with the traditional frequency spectrum (Fig. 6). The most important feature is the optimal resolution of spectrum peaks. The observer’s bandwidth does not permit its signal to be used together with positioning data in active position mooring, but the observer performance is well within the class society’s requirements for tension logging, consequence analysis, and simulation.4 That is, dw all tension variations faster than will not be passed 2 dw through the observer. Notice that wWAVE >> . The 2 time bandwidth of the observer is reduced with increased M, and hence, an optimum selection of M results from a compromise between the requirements for resolution of observed tension (which increases M) and the required bandwidth. References 1. Campbell RB, Vandiver JK (1982) The determination of modal damping ratios from maximum entropy spectral estimates. Dynamic Syst Measurement Control 104 2. Vandiver JK, Campell RB (1979) Estimation of natural frequencies and damping ratios of three similar offshore platforms using the maximum entropy spectral analysis. In: ASCE spring convention. April 6, Boston, MA 3. Directorate, The Norwegian Maritime (1987) J- regulations of 4 September 1987 no. 857 concerning anchoring/positioning systems on mobile offshore units. Norwegian Maritime Directorate 4. Veritas, Det Norske (1996) Rules for classification of mobile offshore units—position mooring (posmoor). Part 6, Chapter 2. DnV—Det Norske Veritas, Veritasveien 1, 1322 Høvik S.I. Sagatun et al.: Mooring line tension observed through a maximum entropy spectrum 5. Meirovitch L (1997) Principles and techniques of vibrations. Prentice-Hall, Upper Saddlenues 6. Burg JP (1975) Maximum entropy spectral analysis. Ph.D. thesis, Stanford University, Department of Geophysics 7. Kanasewich ER (1973) Time sequence analysis in geophysics. 3nd edn (1981) University of Alberta Press, Edmonton, Canada 8. Haykin S (1991) Adaptive filter theory. Prentice-Hall International, Englewood Cliffs, NJ 75 9. Campbell RB (1979) The estimation of natural frequencies and damping ratios of off-shore structures. Ph.D. thesis, MIT, Department of Ocean Engineering 10. Søderstrøm T, Stoica P (1989) System identification. PrenticeHall International, Hemel Hempstead 11. Akaike H (1969) Fitting autoregressive models for prediction. Ann Inst Statist Math 21:243–247