S Gres

P Andersen

C Hoen

L Damkilde

Orthogonal projection-based harmonic signal removal for operational modal analysis

Keywords: Operational Modal Analysis, Stochastic Subspace Identification, Rotating Machinery, Harmonic Removal, Nonstationary Harmonics

A presence of a high amplitude periodic signals in the output responses from operating structures often pose a challenge for output-only system identification and, in case of health monitoring, damage detection/localization methods. This paper introduces a pre-processing approach that removes the harmonic part from the output signals directly in the time domain. The new method uses orthogonal projections of the harmonic realization of the signal onto the raw time series within the stochastic subspace framework. Proposed algorithm is tested on two experimental examples. First, an aluminum plate excited with both random white and periodic excitations. Second, a full-scale industrial case of a ferry excited by a random environmental load with harmonic interference from a rotating machinery on-board. In both cases the proposed method removes the harmonics from the structural responses while leaving the random part of the output signal.

INTRODUCTION

Subspace-based system identification methods became practical tools for identification of the eigenstructure of linear vibrating systems, notably for capacity of solving large models and for the consistency in parameter estimates under non-stationary noise excitations [START_REF] Benveniste | Nonstationary Consistency of Subspace Methods[END_REF]. A family of output-only stochastic subspace-based algorithms (SSI) is deployed for Operational Modal Analysis (OMA) in fields such as aeronautic, mechanical and biomedical engineering, for examples see [START_REF] Basseville | Output-Only Subspace-Based Structural Identification: From Theory to Industrial Testing Practice[END_REF], and formulates basis for a number of Structural Health Monitoring (SHM) methods like in [START_REF] Balm És | Statistical model-based damage localization: A combined subspace-based and substructuring approach[END_REF], [START_REF] Mevel | Subspace-based damage detection under changes in the ambient excitation statistics[END_REF]. However, using subspace-based algorithms for OMA and SHM is problematic when applied to structures with rotating machinery, like wind turbines, diesel engines or helicopters, due to the presence of harmonic inputs. In context of a system subjected to a random load combined with steady state signal, the subspace-based methods identify the harmonics as very lightly damped modes that one can filter in the mode selection process. Difficulties emerge when the harmonic excitations are non-stationary, coincide with the structural modes or are of high energy that masks the system response to the random part of the input. In that case, a desirable solution is to discard the harmonic influence over the output signal prior to system identification and without additional knowledge of such, e.g. tachometer measurements. A development of such harmonic removal method in SSI framework is the focus of the present paper. In context of reducing the harmonic peaks from the response of the system prior to OMA a number of methods exists. Timesynchronous-averaging (TSA) is a signal processing technique that extracts periodic waveforms from signals by averaging synchronized blocks of the signal in the angular domain. In application to harmonic removal the averaged signal is subtracted from the raw measurements, what results in removal of the frequencies depicted to synchronize the blocks, for examples see [START_REF] Peeters | Removing Disturbing Harmonics in Operational Modal Analysis[END_REF]. Angle matching is often achieved via measurements of a signal with a frequency equivalent to harmonic excitation, which is not always available in some of the real-life applications and was attempted to overcome in [START_REF] Combet | An automated methodology for performing time synchronous averaging of a gearbox signal without speed sensor[END_REF]. A family of methods that does not require tachometer measurements are based on cepstrum (an inverse Fourier transform of the logarithm of the spectrum). A number of applications of the cepstral lifters to harmonic removal can be found in [START_REF] Randall | New cepstral techniques for operational modal analysis[END_REF]. A relatively new group of methods is based on estimating a parametric harmonic signal model fitted via e.g. numerical Gauss-Newton algorithm, see [START_REF] Bienert | A harmonic peak reduction technique for operational modal analysis of rotating machinery[END_REF], or non-parametric methods, see [START_REF] Pintelon | Continuous-time operational modal analysis in the presence of harmonic disturbancesThe multivariate case[END_REF], which estimate is later subtracted from the measurements.

The method proposed in this paper computes a harmonic realization of a modal state space model that is fitted to the recorded responses and projected orthogonally onto the raw signals. The paper is organized as follows: the next section contains derivation of the method, subsequently both the experimental and full-scale results are presented in the section 3 and section 4 concludes the paper.

THEORETICAL FRAMEWORK

First paragraph of this section recalls the main principles of the SSI algorithm, in particular the concept of orthogonal projections and innovation component of the data that lead to proposed harmonic reduction method, which description follows in the next paragraph.

State space modeling

Consider a second order differential equation of motion (1) to model a behavior of a d degree of freedom, viscously damped, linear time-invariant (LTI) structural system,

M ü(t) + D u(t) + Ku(t) = v(t) y(t) = Ca ü + Cv u(t) + C d u(t) (1)
where M , D, K ∈ R dxd are mass, damping and stiffness matrices respectively; Ca, Cv, C d ∈ R rxd are selection matrices for accelerations, velocities and displacements where r is a number of sensors; u(t), v(t) ∈ R dx1 denote a vector of displacements and a vector of external forces respectively; y(t) ∈ R rx1 is a output vector; dot expresses a derivative with respect to time t.

Providing that input v(t) is unmeasured, transform (1) to a stochastic state-space form and sample it with k discrete time steps ∆t, what writes,

x k+1 = Ax k + v k y k = Cx k + w k (2)
where the system states x k = [u(k∆t) T u(k∆t) T ] T ; zero mean white noise vectors v k and w k denote the process and output noise respectively; A ∈ R nxn , C ∈ R rxn are the state and observation matrices with,

A = exp( 0 I -M -1 K -M -1 D ∆t), C =   C d 0 0 Cv -Ca -1 M -1 K -Ca -1 M -1 D  
where n is a model order. While the system matrices {A, C} and subsequently the modal parameters-natural frequencies f , damping ratios ζ and mode shapes φ can be identified from (2) with covariance driven SSI (SSI-cov), see [10], we pursue data driven algorithm (SSI-data), see [START_REF] Van Overschee | Subspace Identification for Linear Systems[END_REF], with associated construction of Kalman filter to formulate the optimal prediction of the states and the output itself.

Let z k = E(x k |y k-1 ) be the approximation of x k given the past data vector y k-1 where E(x k |y k-1 ) denotes the image of x k orthogonally projected onto the subspace spanned by the stacked vector of past observations y k-1 with,

E(x|y) = E(xy T )E(yy T ) -1 y (3)
where E(•) denotes the expectation operator. Provided that both x k and y k-1 are jointly normally distributed the expression E(x k |y k-1 ) is the conditional expectation, see [START_REF] Aoki | State space modeling of time series[END_REF], and yields the best linear (least square) prediction of

x k given y k-1 . A step forward predictor of z k writes z k+1 = E(x k+1 |y k ) which, combined with the innovation component of the data e k = y k - y k|k-1,k-2,...,1 , gives z k+1 = E(x k+1 |y k-1 ) + E(x k+1 |e k ).
By definition, the residual e k and the past data sequence y k-1,k-2,...1 are not correlated. Substitute z k+1 with (2) to formulate the innovation state space model,

z k+1 = Az k + Ke k (4a) y k = Cz k + e k ( 4b
)
where K = E(x k e k )E(e k e k ) -1 is the steady state Kalman filter sequence. To predict a time response of the LTI system from the matrices {A, C, K} given inputs y k insert (4b) into (4a) to obtain,

z k+1 = (A -KC)z k + Ky k (5a) ŷk = Cz k (5b)
where ŷk is the realization of the response and the innovations e k = y k -ŷk are the prediction errors.

Harmonic reduction

The procedure that follows consists of three steps: first a similarity transform of the innovation state space to a modal state space, second, a prediction of modes that correspond to the harmonic frequencies-selected prior to the reduction or detected with automated methods such as in [START_REF] Jacobsen | Eliminating the Influence of Harmonic Components in Operational Modal Analysis[END_REF] and last an orthogonal projection of raw time series onto the harmonic realization of the output.

A linear transformation of the Kalman states (4a) to modal coordinates, for reference see [START_REF] Hoen | Subspace Identification of Modal Coordinate Time Series[END_REF], is given by,

ẑt = V zt (6)
where V denotes a transformation matrix and yields right eigenvectors of A,

AV = λV, λ ∈ C nxn , V ∈ C rxn
Substituting the Kalman states from (4a) and (4b) with its similarity transforms, the innovation state space model takes a modal form and is represented by,

ẑt+1 = V -1 AV ẑt + V -1 Ket yt = CV ẑt + et (7)
where the transformed state matrices can be written as

A0 = V -1 AV ∈ C nxn , C0 = CV ∈ C rxn , K0 = V -1 K ∈ C nxr .
Notice that A0 = λ and C0 = φ, where the latter implies that a realization of the output in the modal form will render the response of the model due to identified modes. That manifests with,

ẑt+1 = (A0 -K0C0) ẑt + K0yt (8a) ŷt = C0Im ẑt (8b)
where Im ∈ R nxn is a diagonal selection matrix with ones on the diagonal entries that corresponds to the pairs of complex conjugate eigenvalues of A reflecting the response from selected modes.

Suppose that the state space system from (4a) is completely controllable and observable, that is CVn = 0 ∀ n and V -1 n K = 0 ∀ n and i har being a Boolean vector mapping the complex conjugate eigenvalues of A to the frequency of harmonic input. To remove the harmonic modes from the system response, first consider a prediction of the harmonic part given by,

ŷhar = C0diag(i har ) ẑt (9)
Subsequently, one can decouple the true response of the system from the harmonic-affected response by projecting the latter orthogonally onto harmonic realization ŷhar , what writes,

E(yt|ŷ har ) = yt -E(yt ŷT har )E(ŷ har ŷT har ) -1 ŷhar (10)
where the resultant residual is orthogonal to the subspace spanned by ŷhar and, given a complete identification of the harmonic modes, yields a reduced matrix,

ŷred = E(yt|ŷ har ) (11)
with the harmonics removed.

RESULTS

The efficiency of the orthogonal projection-based harmonic removal is investigated with two laboratory tests of a plate and a full-scale, in-operation, dynamic test of a ship. The performance of the method is assessed based on its capacity to remove the harmonic poles from the measurements while maintaining the natural modal properties of the structure unaffected. A variation of the method is implemented in ARTeMIS Modal Pro 5.2, for reference see [START_REF]Structural-Vibration-Solutions[END_REF], which is used here for signal processing and presentation of results.

Experimental results

The measurements were collected from a rectangular aluminum plate subjected to both random and harmonic excitation and equipped with a 16-channel acquisition system recording accelerations normal to the plate surface. The experimental tests were carried out by Niels-Jørgen Jacobsen from B&K Naerum in Denmark. The plate model and the experimental setup are illustrated on Figures 1 and2. The signals were sampled with 4096Hz over 120s intervals. To challenge the harmonic reduction algorithm two non-trivial input schemes are considered, namely:

• Random combined with swept sine of frequency varying in between 340-342Hz and a sweep rate of 10 Hz/s,

• Random combined with burst sine of frequency of 370Hz and 10s burst intervals.

A short-time Fourier transforms (STFT) of the recorded accelerations for the two scenarios are depicted on Figures 3 and4. The non-stationary harmonic frequencies in the measured accelerations are represented as a 5dB dash red line in Figure 3 and 20dB bold red line in Figure 4. A power difference between the steady state and the system response is 10-25dB, Figure 3 and Figure 4. Small fluctuations (2Hz) of the swept sine will become visible after zooming in the Figure 4. In both cases harmonics are in the vicinity, for the swept sine directly on top, of the 1 st natural frequency of the plate, where the latter is 341Hz, see Table 1. In context of OMA, performing the system identification on the two tested data sets even with the harmonic-robust methods is ineffective (high model orders, multiple spurious modes and unstable stabilization diagrams), hereby the reduction attempt is fully justified.

The analysis requires several user-defined parameters. First, the algorithm proposed in (9) and (10) operates in the premise that the harmonic frequencies are selected prior to the analysis and as such, we pick a sharp peak of the first singular values (SVs) of the power spectral densities (PSD) of the measured accelerations yielding the frequency of the harmonic input. Second, we define a number of block rows in the data Hankel matrix and a range of damping ratios to navigate the identification algorithm towards a state space model that predicts the harmonic modes in a optimal way. The subject of the optimum is not discussed in this work.

Both experimental tests are processed with one set of the user-defined inputs, namely: 45 block-rows of data Hankel matrix and 0.00-0.1% range of the harmonic mode damping ratios. The manually picked frequency of the harmonic response is 370Hz and 341Hz respectively for burst sine and swept sine. The example of a graphical user interface containing the input parameters and SVs of PSD from the data before and after the harmonic reduction, for the burst sine case, is illustrated on Figure 5. A visual inspection of Figure 5 shows a successful removal of harmonics at 370Hz, what is in agreement with the STFT of the modified data, see Figure 6, and confirmed by the comparison of the modal properties of the plate subjected to random vibrations and the OMA of the measurements after harmonic removal, see Table 1. In case of swept sine, harmonics are removed recursively in a second trial run with the same input parameters; the results are depicted on Figure 7 and Table 1. Figure 6 illustrates that the response of the plate to the burst sine input (dashed red line on Figure 3) is removed without leaving any additional harmonic artifacts in the modified signals. As for the swept sine test, the Figure 7 indicates that after the harmonic reduction the magnitude of the power spectrum in the 340-342Hz range dropped from 20dB to 6dB, although, since the frequency of the harmonics coincide with the 1 st natural frequency of the plate, it is not a direct sign that the harmonics are completely removed. To verify both cases, we refer to Table 1 containing an overview of the modal properties of the plate extracted both from random vibration measurements and the modified signals after the harmonic removal. The influence of the shaker weight has been included. Modal properties are estimated with an output-only SSI with Extended Unweighted Principal Component (UPCX) method, for reference see [START_REF] Andersen | Variance computation of modal parameter estimates from UPC subspace identification[END_REF]. Considering that the results of the random vibrations test are the true modal parameters of the plate, the natural frequency of the first three modes exhibits a marginal deviation of max. 0.5% between the different data sets. That is not the case for the damping ratio. For the swept sine test: the smallest error yields 1.42% in the 1 st mode and the highest error yields 9.92% in the 2 nd mode. For the burst sine test: the smallest error writes 4.25% in the 2 nd mode and the highest error writes 4.97% in the 1 st mode. The highest errors are significant, in particular when recalling that e.g. the swept sine (340-342Hz) input is mixed with the 1 st mode (341.01Hz) instead of the 2 nd (472.10Hz), see Figure 5. One possible reason for that bias can be a non-linear damping behavior of the plate while loaded by a high amplitude period input; other might be a non-optimal choice of the state space model to predict the harmonic response of the signal.

Test [-] f1[Hz] f2[Hz] f3[Hz] ζ1[%] ζ2[%] ζ3[%]

Full-scale test

The ferry is excited by a random environmental load (wind and waves) with a harmonic interference from a rotating machinery on-board. The ferry docks at Flensburg shipyard, is 199.8m long and powered by 20MW 9 cylinder MAN B&W engine with nominal rotation speed of 123rpm. The dynamic test is undergone under fully operational conditions; the responses are recorded using a 16-channel acquisition system and sampled with 128Hz. The measurements were carried out by Dr. Sven-Erik Rosenow, Santiago Uhlenbrock and Prof. G ünter Schlottmann from University of Rostock in Germany and made available to analysis by Structural Vibration Solutions from Aalborg, Denmark. The same data set was analyzed in [START_REF] Rosenow | Parameter Extraction of Ship Structures in Presence of Stochastic and Harmonic Excitations[END_REF] and [START_REF] Jacobsen | Operational Modal Analysis on Structures with Rotating Parts[END_REF] with respect to harmonic detection. Model of the ship with the measured degrees of freedom is illustrated on Figures 8 and9. Prior to harmonic reduction measurements are decimated to 16Hz. During the test the rotational frequency of the engine is 2.05Hz, hence a family of four harmonic frequencies: 2.05Hz, 4.1Hz, 6.15Hz and 7.8Hz is present in the decimated response signals, see Figure 10. The harmonic reduction is prepared with user-defined parameters depicted on Figure 10, with two harmonic peaks at 2.05Hz and 4.1Hz selected to reduce. Figure 10 illustrates that along with peaks at 2.05Hz and 4.1Hz the algorithm automatically reduced the harmonics at 6.15Hz and 7.8Hz. A marginal remanence of the harmonic mode at 4.1Hz is still present in the second largest SV as well as an artifact of the 6.15Hz peak. That fact, however, is not expected to have influence on the system identification of the modified signals.

For more graphic overview Figures 11 and12 show the STFTs of the measurements before and after the harmonic reduction. Multiple stable modes are depicted on Figure 13. The harmonic modes previously present in the measurements are not detected nor they influence the identification algorithm after the harmonic reduction.

CONCLUSION

In this paper we presented both the theory and the application of the orthogonal projection-based harmonic reduction method. Proposed algorithm successfully removed the harmonics from the experimental response signals albeit introducing a bias varying in range 1.42-9.92% between the different modal damping ratios. The source of the error can be explained by a non-linear damping behavior of the plate while excited with strong periodic signals, thus larger than from the random vibration test or a suboptimal choice of a the state space model order for predicting the harmonic realization of the response. Similar results for the harmonic removal were achieved in context of a real-life application. Developed method removed the harmonic frequencies propagated from the engine of an operating ferry and enabled efficient OMA of the modified response signals.

The application of the method can be extended not only to harmonic reduction\removal but also to removing any arbitrary mode of the system. That might be useful to remove high energy modes present, for instance, in forced excitation during structural testing.

The bottleneck of the method is the choice of the state space model order such that it solely predicts the modal response that corresponds to selected harmonic frequencies. That, along with conducting more tests on a real measurements with strong non-stationary harmonic excitations and large number of channels will be the subject of our future work.

Figure 1 :

 1 Figure 1: The experimental setup: test plate with 16 acceleration channels, shaker, acquisition system.

Figure 2 :

 2 Figure 2: The test plate model with 16 acceleration channels attached.

Figure 3 :

 3 Figure 3: STFT channel 1-3 before the harmonic reduction.Burst sine: 370Hz, 10s.

Figure 4 :

 4 Figure 4: STFT channel 1-3 before the harmonic reduction.Swept sine: 340-342Hz, 10Hz/s.

Figure 5 :

 5 Figure 5: Removal harmonic at 370 Hz. Burst sine: 370Hz, 10s.

Figure 6 :

 6 Figure 6: STFT channel 1-3 after the harmonic reduction.Burst sine: 370Hz, 10s.

Figure 7 :

 7 Figure 7: STFT channel 1-3 after the harmonic reduction.Swept sine: 340-342Hz, 10Hz.

Figure 8 :

 8 Figure 8: The ferry at the Flensburg shipyard.

Figure 9 :

 9 Figure 9: The ferry model with 16 acceleration channels attached.

Figure 10 :

 10 Figure 10: Removal of predefined harmonics at 2.05Hz and 4.1Hz. Harmonics at 6.15Hz and 7.8Hz removed automatically.

Figure 11 :

 11 Figure 11: STFT channel 5-9 before the harmonic reduction. Harmonics at 2.05Hz, 4.1Hz, 6.15Hz and 7.8Hz.

Figure 12 :

 12 Figure 12: STFT channel 5-9 after the harmonic reduction.Harmonics at 2.05Hz,4.1Hz,6.15Hz and 7.8Hz removed.

Figure 13 :

 13 Figure 13: Stabilization diagram of the estimated state space models from the signals after the harmonic removal. SSI-UPCX.

TABLE 1 :

 1 Three first natural frequencies and damping ratios of the plate estimated with SSI-UPCX.

	Random vibration	341.01 472.10 688.60	0.56	0.49	0.88
	Random + swept sine 341.33 472.62 688.50	0.57	0.45	0.84
	Random + burst sine	341.18 472.89 688.88	0.59	0.47	0.81

ACKNOWLEDGMENTS

The authors would like to thank Niels-Jørgen Jacobsen from B&K Naerum in Denmark for conducting the experimental tests and for sharing its results.


S Gres

P Andersen

L Damkilde

Operational Modal Analysis of Rotating Machinery

Keywords: Operational Modal Analysis, Rotating Machinery, Harmonic Removal, Harmonic Detection, Stochastic Subspace Identification

Harmonic excitation of structures caused by rotating equipment is a problem faced by many engineers in the field of Operational Modal Analysis (OMA). Several methods to discard the influence of harmonic inputs over systems natural responses has been proposed in the literature and implemented in various software solutions. This paper recalls some of the most used techniques and uses a new time domain method for removing harmonics from measurements. Deployed method does not rely on filtering, statistical detection nor on non-linear fitting. Instead, it predicts the harmonic part of the time series and deploys an orthogonal projection of the latter onto the raw measurements to remove the harmonic part of the signal. The new technique is a part of an semi-automated framework for OMA of structures contaminated with harmonics, whose flow is presented in this paper. The merit of the framework is discussed in the context of OMA of a full scale operating ship with rotating machinery on-board.

INTRODUCTION

In the context of OMA of rotating machinery, that is estimation of natural frequencies, damping ratios and mode shapes of structures subjected to both harmonic and random excitations based only on the response measurements, a couple of schemes has been proposed. As such, each strategy that deals with harmonic disturbances in OMA consists of a common triad: harmonic detection, harmonic removal and system parameter computation.

In that context Peeters et al. [START_REF] Peeters | Removing Disturbing Harmonics in Operational Modal Analysis[END_REF] presented a scheme of time-synchronous-averaging (TSA) and polyreference least-squares complex frequency-domain method (PolyMAX) that was used for OMA of an steady-flight helicopter and an operating diesel engine. Deployed duo enabled OMA of harmonic-affected signals by removing the harmonic peaks from an auto spectral densities of the measurement channels what improved clarity of the estimated stabilization diagrams. For a non-stationary harmonics the TSA required an tachometer measurements from rotating components for angular resampling, which is most often not recorded in real-life application; that was attempted to overcome by Combet and Gelman [START_REF] Combet | An automated methodology for performing time synchronous averaging of a gearbox signal without speed sensor[END_REF]. An alternative strategy for OMA of rotating machinery is based on cepstral methods. Cepstrum is an inverse Fourier transform of the logarithm of the spectrum which real part is deployed in the recent studies by Randall and Smith [START_REF] Randall | New cepstral techniques for operational modal analysis[END_REF] and Randall et al. [START_REF] Randall | Repressing the effects of variable speed harmonic orders in operational modal analysis[END_REF] to remove both stationary and varying harmonics from the time signals. Cepstral techniques can coincide with OMA by computing the pole and zero part of system transfer function by curve-fitting the liftered response measurements, what can also lead to estimation of frequency response functions, shown in [START_REF] Randall | Repressing the effects of variable speed harmonic orders in operational modal analysis[END_REF]. Otherwise, cepstrally edited measurements can be used with a standard system identification methods for modal parameter computation.

The trio for a semi-automated OMA of rotating machinery proposed in this paper comprise of harmonic detection using a measure of signals kurtosis, harmonic reduction with orthogonal projections of a harmonic realizations of the signal onto the raw signals and estimation of modal parameters of the system with their statistical uncertainties propagated from the measurements. The paper is organized as follows: the next section recalls the methods, subsequently the results are presented in the section 3 and section 4 contains concluding remarks.

Proposed framework for OMA of rotating machinery contains well-known methods and the concepts familiar to OMA community in general. Following section recalls their main assumptions and provides a brief description of the novel harmonic reduction technique. Examples illustrating the concepts listed in this section are from an experimental setup of a rectangular aluminum plate subjected to both random and harmonic excitations. More description about the experimental data can be found in [START_REF] Jacobsen | Using enhanced frequency domain decomposition as a robust technique to harmonic excitation in operational modal analysis[END_REF].

Harmonic detection

Since a distribution of a response of a structural system subjected to a random input is asymptotically Gaussian, the presence of deterministic (harmonic) components in the random inputs can be detected by examining its moments. The fourth moment, namely kurtosis γ (1), is a measure of the heaviness of the tail of the distribution and yields γ = 3 for a normally distributed samples x with zero mean µ and a unit variance σ,

γ(x|µ, σ) = E[(x -µ 4 )] σ 4 (1)
where E(•) is an expectation operator. An illustration of the probability density function from the modal response from an experimental plate subjected to both random and harmonic loads is depicted on Figure 1. The fact that γ ≈ 1.5 for a sinusoidal signals with µ = 0 and σ = 1 is deployed by Jacobsen et al. [START_REF] Jacobsen | Using enhanced frequency domain decomposition as a robust technique to harmonic excitation in operational modal analysis[END_REF] in an extended kurtosis check, which is based on extensive calculation of kurtosis of a signal yi that is bandpass filtered around each examined frequency fi. Frequencies fj for which the kurtosis differs from the median computed up to Nyquist frequency are denoted as harmonics.

A similar, yet less exhaustive test was proposed by Andersen et al. [START_REF] Andersen | Estimating Modal Parameters of Civil Engineering Structures subject to Ambient and Harmonic Excitation[END_REF] where only the fi that exhibits an abrupt change in corresponding singular values (SVs) of power spectral densities (PSDs) of measured channels are tested. The computation of the abrupt change of the SVs is based on a logarithmic metric between current and a median of windowed SVs.

Harmonic reduction

An application of the orthogonal projection-based algorithm to harmonic reduction was described by Gres et al. [START_REF] Gres | Orthogonal projection-based harmonic signal removal for operational modal analysis[END_REF]. The method follows three steps, namely: a similarity transform of the innovation state space to a modal state space, a prediction of modes that correspond to the harmonic frequencies and an orthogonal projection of raw time series onto the harmonic realization of the output. The algorithm exploits the fact that given a state space system that is completely controllable and observable the subspace-based methods identify all the harmonics as lightly damped modes that can be well predicted and removed by a orthogonal projection onto the subspace spanned by the raw measurements.

The algorithm begins with a similarity transform of the innovation state space model to a modal state space that writes (2a),

ẑt+1 = (A0 -K0C0) ẑt + K0yt (2a) ŷt = C0Im ẑt (2b)
where A0, C0 and K0 are the state, observation and Kalman gain matrices in a modal form, ẑt are the modal states and yt denotes raw time series. The prediction of the modal state space model ŷt is given by (2b) where Im is the diagonal selection matrix with ones on a diagonal entries corresponding to harmonic modes. The projection itself is given by (3),

E(yt|ŷ har ) = yt -E(yt ŷT har )E(ŷ har ŷT har ) -1 ŷhar ( 3
)
where ŷhar is the predicted harmonic response. The projection yields the reduced, harmonic-free response y red (4),

y red = E(yt|ŷ har ) (4)
To illustrate the concept behind the algorithm, we selected the harmonic frequencies from the experimental plate measurements, namely 374Hz, 748Hz and 1496Hz, for which the prediction of an optimally selected state space model and the projection onto raw measurements were computed, see Figure 2. The SVs of the PSDs from the predicted harmonic modes (blue line) match well with the harmonic peaks in the raw measurements, as depicted on Figure 2. The SVs from the data after projection do not contain any artifacts from the orthogonally projected harmonics, hence concluded as harmonic-free.

In general, the principal given by (3) and (4) can be extended not only to harmonic reduction but also to removal of any arbitrary mode from the measurements. That is illustrated based on the same experimental data set from a plate on Figure 3. The top subplot of Figure 3 illustrates the SVs of the PSDs from the plate measurements reduced by mode 4 at 868Hz, and mode 8 at 1660Hz. The realization of the modes at 868Hz and 1660Hz is depicted on the bottom part of Figure 3 (thick black lines), along with the complete modal decomposition of the structural modes from the plate measurements. That suggests that any mode can be selected to be reduced in a similar manner.

Modal parameters computation

The identification of the system matrices of a state space model using both data and covariance driven stochastic subspace identification (SSI-data, SSI-cov) algorithms is subjected to variance errors due to e.g. a finite number of data samples used to calculate the Hankel matrix, unmeasured inputs, measurement noise. A scheme to propagate the sample covariances through all the steps of modal parameter computation for the SSI-cov was proposed by Reynders et al. [START_REF] Reynders | Uncertainty bounds on modal parameters obtained from stochastic subspace identification[END_REF] and D öhler and Mevel [START_REF] Mevel | Fast multi-order computation of system matrices in subspace-based system identification[END_REF]. That was extended with SSI-data and input-output algorithms in Mellinger et al. [START_REF] Mellinger | Variance estimation of modal parameters from output-only and input/output subspace-based system identification[END_REF].

A method to compute the (co)variance of the modal parameters of the global model order with a data-driven Unweighted Principal Component (UPC) algorithm was developed by D öhler et al. [START_REF] Andersen | Variance computation of modal parameter estimates from UPC subspace identification[END_REF]. Method deploys the fast uncertainty computation from [START_REF] Mevel | Fast multi-order computation of system matrices in subspace-based system identification[END_REF] and involves computation of the modal parameters of the global mode as a weighted mean of different model orders where the (co)variance of the global model is computed subsequently as a covariance of the weighted mean. The implementation of this algorithm, denoted as SSI with Extended Unweighted Principal Component (UPCX), is used here for the computation of modal parameters and corresponding uncertainties.

RESULTS

In the following, we apply presented framework for dealing with harmonics in OMA to an experimental measurements of a full scale ferry during its operation. The analysis is conducted in semi-automatic chain and is illustrated based on the results from a commercial software ARTeMIS Modal Pro 5.2 [START_REF]Structural-Vibration-Solutions[END_REF].

The ferry with its geometrical and mechanical data is presented on Figure 4. The dynamic tests are conducted under operational conditions where the response measurements are recorded with piezoelectric accelerometers using a 16-channel B&K Dyn-X acquisition system. The ferry is excited by a combination of a random environmental (wind and waves) and harmonic (operating engine and propeller) loads. The response is measured over a period of 90min and sampled with 128Hz. The measurements were carried out by Dr. Sven-Erik Rosenow, Santiago Uhlenbrock and Prof. G ünter Schlottmann from University of Rostock in Germany and made available to analysis by Structural Vibration Solutions A/S from Aalborg, Denmark. The same data set was analyzed in [START_REF] Jacobsen | Operational Modal Analysis on Structures with Rotating Parts[END_REF] and [START_REF] Rosenow | Parameter Extraction of Ship Structures in Presence of Stochastic and Harmonic Excitations[END_REF] with respect to harmonic detection and in [START_REF] Gres | Orthogonal projection-based harmonic signal removal for operational modal analysis[END_REF] for harmonic reduction.

Prior to harmonic detection the measurements are detrended, normalized by the standard deviation of each channel and decimated to 21.33Hz. The families of the harmonic frequencies caused by the engine operating at the frequency of 2.05Hz and the propeller blade at 8.2Hz are identified with fast kurtosis check, what is illustrated on the Figures 5 and 6 In total five frequencies with the kurtosis below 2.65 are denoted as harmonics, see Figure 5, what is in good agreement with the sharp peaks of SVs depicted on Figure 6. Selected frequencies are one of the input parameters for the orthogonal projectionbased harmonic reduction. Besides that, user defined settings boil down to setting a number of block rows of the data Hankel matrix and a maximum damping ratio of the modes to reduce. Selected parameters navigate the method in finding the optimal state space model to predict the harmonic responses. The input settings with the SVs of PSDs of the measurements before and after the reduction are illustrated on Figure 7. Figure 7 illustrates that the harmonic peaks at 2.05Hz, 4.1Hz and 6.15Hz are removed completely. The first SV shows a remanence of the propeller mode at 8.2Hz and the engine mode at 10.2Hz, however both are marginal and not identified during system identification, see Figure 10. The harmonic removal uncovered a mode at 8.14Hz, which being reasonably damped, 1.62%, and with a low complexity, 7.19%, indicates a structural mode close to a resonance with the propeller frequency.

To prove the merit of removing the harmonics prior to system identification we conducted a OMA of two data sets before and after harmonic reduction. That is illustrated by the results of the 95% confidence ellipsoids of the 1 st mode frequency and damping ratios and the modal assurance criteria (MAC) between the identified sets of mode shapes, see Figures 8 and9 respectively. The modal parameters from both sets are computed with SSI-UPCX with max. model order of 140 and 38 eigenvalues in the state matrix. For the global mode estimate, the pairs of the natural frequency and the damping ratio of the 1 st mode (unaffected by the harmonics) yields 1.33Hz with 1.99% and 1.34Hz with 2.02%, respectively for the measurements before and after the harmonic reduction. After examining the uncertainty ellipsoids depicted on Figure 8 it becomes apparent that the difference between the modal properties of two modes described above is acceptable hence the mean values from both test are encapsulated within a common uncertainty interval. The diagonal MAC values illustrated on Figure 9 are varying in between 0.75-0.99 suggesting that the modes with similar shapes are computed in both data sets. The modes at 3.8Hz, 5.35Hz, 6.35Hz, 6.86Hz, 8.4Hz and 8.7Hz were not identified before harmonic reduction, see Figure 9. For the completeness of the results a stabilization diagram with corresponding natural frequencies estimated after the harmonic removal is presented on Figure 10. One can notice a mode present at 2.29Hz which is denoted as a noise mode-that is a result of a high coefficient of variation of the frequency of the corresponding mode, namely 0.015. Otherwise, the remaining modes present in the measurements exhibit low uncertainties for both the frequency and damping ratio below the threshold being 0.01 and 1 respectively. The harmonic modes previously present in the recorded data are not detected.

CONCLUSION

In this paper we presented a semi-automated framework for OMA of rotating machinery. Proposed strategy combines the wellknown methods for harmonic detection and modal parameters computation with a novel harmonic reduction technique. Deployed methods operate in time domain, do not require additional inputs for reduction of non-stationary harmonics such as a tachometer measurements and estimate the variance of a global modal parameters, which provides more precise identification of the modal properties in case of realistic applications.

Figure 1 :

 1 Figure 1: PDF of a predicted structural mode response (left) and a predicted harmonic mode response (right).

Figure 2 :

 2 Figure 2: SVs of PSDs from raw, predicted harmonic and projected harmonics-free measurements.

Figure 3 :

 3 Figure 3: SVs of PSDs from the projected response with mode 4 and 8 removed and predicted responses from each structural mode.

Figure 4 :

 4 Figure 4: Full-scale ship docking at Flensburg shipyard.

  .

Figure 5 :

 5 Figure 5: Kurtosis of the signal calculated for the selected frequencies

Figure 6 :

 6 Figure 6: SVs of PSDs with harmonic indicators computed with the fast kurtosis check.

Figure 7 :

 7 Figure 7: Removal of harmonics at 2.05Hz, 4.1Hz, 6.15Hz, 8.2Hz and 10.2Hz.

Figure 8 :

 8 Figure 8: 95% confidence ellipsoids for 1 st mode frequency and damping before and after harmonic reduction.

Figure 9 :

 9 Figure 9: MAC diagram obtained from mode shapes of the system before and after harmonic reduction.

Figure 10 :

 10 Figure 10: Stabilization diagram of models obtained from data after the harmonic reduction. SSI-UPCX


S Gres

M Fejerskov

L B Ibsen

L Damkilde

Experimental damping assessment of full scale offshore Mono Bucket foundation

Fatigue loads induced by the wind and waves are usually the drivers for the design of modern offshore structures. The magnitude of the fatigue load is dependent on the site-specific environmental conditions and the dynamic properties of the system, i.e. damping ratios. In order to provide a cost-optimized design and accurately assess the fatigue lifetime of the structure, a proper description of the damping contribution from the foundation is required. This paper quantifies the system damping of a offshore meteorological mast supported by a Mono Bucket foundation based on a long-term experimental campaign. The structure is located at Dogger Bank west, North Sea, and equipped with a measurement system monitoring acceleration, strain, inclination and sea surface elevation. Natural frequencies and corresponding damping ratios are assessed using different operational modal analysis techniques, enhanced frequency domain decomposition and stochastic subspace identification. Application and results from both methods are compared and discussed. Research shows that the total damping ratio of the lowest eigenmode is normally distributed with mean value of 1.11% of critical damping. Linear correlation between the damping ratio and the significant wave height is observed.

Introduction

The high demand for clean energy production has led to a great development in the field of the offshore and onshore wind energy in the past years. Offshore wind generates higher design, installation and maintenance costs compared to onshore wind, however it is more effective in the power production due to the higher and more constant wind speeds and no limitations regarding the turbine sizes on the offshore sites. Despite the strive to deploy the increasingly larger turbines in the deeper waters with challenging soil conditions, the most frequently used support for the offshore wind turbines is still a monopile foundation. The size of the modern monopiles reaches up to the 7.8 m in diameter and the weight of 1300 tons, which generate significant costs as well as challenges during the production. The global imperative to reduce the cost of the offshore wind has result in a new support concepts based on the suction technique-Mono Bucket and Suction Bucket Jacket foundations.

The Mono Bucket (MB) originates from the oil and gas suction anchors [START_REF] Houlsby | Suction Caisson Foundations for Offshore Wind Turbines and Anemometer Masts[END_REF]. The concept merges the advantages of the suction technology, the bearing resistance of gravity based foundation and the skirt resistance of the monopiles, resulting in a foundation which is light and easy to install/decommission. Hence the MB is a relatively light offshore support with a high stiffness in the embedded part [START_REF] Houlsby | Suction Caisson Foundations for Offshore Wind Turbines and Anemometer Masts[END_REF], the fatigue wind and wave loads, which often are close to the natural frequency of the structure, are the design drivers for the foundation. The complex nature of the offshore loads that dynamically interact with the structural system pose a challenge towards the designers and call for the close identification of the parameters that influence the overall design. As an example, results from [START_REF] Seidel | Wave induced fatigue loads[END_REF] state that for a offshore monopile the increase in the system damping ratio by 50% results in the decrease of the fatigue loads by 20%. This substantial decrease in the loads justifies the interest of the industry towards accurate assessment of the dynamic properties of the system and supports the idea behind full scale offshore experimental campaigns.

The damping of the offshore system, or so-called offshore damping, consists of aerodynamic, hydrodynamic, soil and structural damping. The aerodynamic damping reflects the motion of the tower counteracted by the increase/decrease of the aerodynamic force from the turbine blades [START_REF] Cerda Salzmann | Aerodynamic damping in the design of support structures for offshore wind turbines[END_REF] and is a major source of the damping for the operating turbine. The hydrodynamic damping is the sum of the two contributions -wave radiation and viscous damping of the fluid. Both components are proportional to the relative velocity between the structure and the water. The hydrodynamic damping ratios recommended by the Germanischer Lloyd, GL, in [START_REF] Gujer | Overall Damping for Piled Offshore Support Structures, GL RC Guideline for the Certification of Offshore Wind Turbines[END_REF] are 0.11% and 0.15% for the radiation and the viscous damping respectively. The soil damping is divided in geometric, material and pore pressure components. The geometric damping describes the wave radiation into the soil volume and is neglected for frequencies below 1 Hz [START_REF] Tarp-Johansen | Comparing Sources of Damping of Cross-Wind Motion[END_REF]. The pore pressure part is related to the permeability and the water dissipation in the pores of the soil. Hysteretic material damping of the soil, as well as the steel, is described as an absorption of the motion due to the internal friction of the material. For the piled supports GL recommends the soil damping ratios between 0.53%-0.88% and the steel damping between 0.2%-0.3% [START_REF] Gujer | Overall Damping for Piled Offshore Support Structures, GL RC Guideline for the Certification of Offshore Wind Turbines[END_REF]. The best estimates for the overall added offshore damping subtracted with the aerodynamic damping, are stated as 0.9% and 1.2% depending on the simulation type [START_REF] Gujer | Overall Damping for Piled Offshore Support Structures, GL RC Guideline for the Certification of Offshore Wind Turbines[END_REF].

The offshore damping is not as extensively researched topic for the MB as for the monopile supports. A single experimental campaign regarding the cross-wind damping of Vestas V90 -3MW offshore wind turbine supported by the prototype MB foundation was presented in [START_REF] Damgaard | Dynamic Properties of Offshore Wind Turbine Foundations[END_REF]. The results provided the correlation between the structural accelerations and the soil damping with the maximum damping ratio of 0.6% for an acceleration level of 2.1 m s 2 . Due to the fact that the tests were conducted in near shore laboratory the site conditions were not fully reflecting the offshore environment what influences the components of the system damping.

The main purpose of this study is to determine the long term dynamic characteristics, eigenfrequencies and damping ratios, of the offshore meteorological mast at the Dogger Bank (DB) wind farm supported by the MB foundation. For this purpose the time and frequency domain OMA techniques are used with the long-term operational data from the structure. The results are correlated with the measured sea states and compared with existing publications. The single components of system damping ratio are separated. A database of the damping ratios for the analyzed location is created as a reference for future designs.

Structure and site conditions

The meteorological mast is located in the west of the site and is one of the two masts supported by the MB installed at the DB. On-site photo from the installation vessel is presented in Figure 1.

Structure

The MB is a modular welded steel foundation, divided into a shaft, webs, lid, and skirt modules. The webs and the shaft form a main frame of the foundation. The lid and the skirt create a horizontal and vertical base respectively. The MB is installed by the combination of self-weight and applied suction. The base is designed with 9 radial webs, diameter of 15 m, 7.5 m skirt length and 42.5 m shaft length, illustrated in

Structural monitoring

The structural monitoring sensor system setup consists of the accelerometers (Acc), inclinometers (Inc), strain gauges (SG), wave gauges (WR) and pressure transducers (PT). The configuration of all the sensors is presented in Figure 3. This study uses the accelerations and the surface elevations measurements to determine the relation between the modal characteristics of the structure and the environmental conditions. Accelerations are measured by the AS-5TG bi-axial and tri-axial accelerometers on the top of the shaft and the lid of the foundation. Two VegaPlus 61 wave radars measuring the instantaneous sea surface elevations are installed on the platform of the mast. Data is collected by the mobile, self sustainable SOMAT eDAQ acquisition system and saved on the industrial PC located in the shaft of the foundation. E 50 , E ur , φ and c ref are the secant modulus in drained triaxial test, the unloading/reloading modulus, the friction angle and the cohesion of the soil respectively. Regarding scour, bathymetry survey using the multi-beam echo sounder conducted in October 2014 concluded local 1m scour hole.

Site conditions

Applied system identification techniques

The dynamic properties of the meteorological mast is highly affected by the magnitude of ambient excitations. The convenient method to determine the modal parameters of such structure is to use output only system identification techniques with the responses analyzed for the similar loading conditions. In this paper Enhanced Frequency Domain Decomposition [START_REF] Brincker | Modal Identification from ambient responses using frequency domain decomposition[END_REF] (EEFD) and Stochastic Subspace Identification [START_REF] Overschee | Subspace Identification for Linear Systems[END_REF] (SSI) techniques are applied.

Enhanced Frequency Domain Decomposition

The EFDD, proposed in [START_REF] Brincker | Modal Identification from ambient responses using frequency domain decomposition[END_REF], is a modification of the Frequency Domain Decomposition technique that originates from a basic frequency domain approach. Both methods utilize that any lightly damped mode influence the response of the structure in the vicinity of its natural frequency. The difference is that (E)FDD uses singular value decomposition (SVD), of the response power spectral density matrices (PSD). The EFDD method was chose and implemented in MATLAB. Below is a general description of the method.

The dynamic response of a linear system can be presented as a sum of a product of the mode shapes, φ i , and the modal coordinates q i . The covariance function of a system response yields (1), [START_REF] Brincker | Modal Identification from ambient responses using frequency domain decomposition[END_REF].

C yy (τ ) = E[{y(t + τ )} {y(t)} H ] = E[[φ] {q(t + τ )} {q(t)} H [φ] H ] = [φ][C qq (τ )][φ] H (1)
where subscript H denotes a complex conjugate and transpose. Auto and cross spectral densities are calculated using a Welch averaged periodogram with the 50% signal overlapping and Hanning windowing (2), [START_REF] Bendat | Random data: Analysis and measurement procedures. 4th edn United Kingdom[END_REF].

G yy (f ) = [φ][G qq (f )][φ] H (2)
The key procedure in EFDD is to decompose the PSD matrices using the SVD and describe them as a set of the spectral density functions each corresponding to a SDOF system (3) illustrated in Figure 4. Frequency, Hz SVs G yy ( f)= Figure 4: Picking the peak singular value and corresponding mode shape vector, modified after [START_REF][END_REF] where Σ(f ) is a diagonal matrix holding the singular values in descending order and U (f ) is a matrix holding the singular vectors. Dominating s 1 (f i ) peak indicates the presence of the i-th mode mode. {u 1 (f i )} is a mode shape vector matching selected eigenfrequency [START_REF] Brincker | Introduction to operational modal analysis United States[END_REF]. Singular values corresponding to the singular vectors collinear with the singular vector of the peak singular value belong to the considered mode. Damping ratio can be obtained by taking the selected s 1 (f ) to the time domain and calculating the logarithmic decrement of the decaying auto-correlation function [START_REF] Brincker | Modal Identification from ambient responses using frequency domain decomposition[END_REF].

G yy (f ) = [U (f )][Σ(f )][V (f )] H (3)
[U( f)] s 1 ( f) 0 • • • 0 0 s 2 ( f) • • • 0 . . . . . . . . . . . . 0 0 0 s n ( f) [U( f)] H U( f) = {u 1 ( f)} { u 2 ( f)} • • • {u n ( f)} 0 0.5 1 1.5 2 2.5 10

Stochastic Subspace Identification

The SSI is a well-known component in system identification methods first introduced in [START_REF] Overschee | Subspace Identification for Linear Systems[END_REF] and [START_REF] Ho | Effective construction of linear state-variable models from input/output data[END_REF] that combine a state-space realization models with different numerical techniques (QR and SV decompositions). In this work the modal parameters of the mast are assessed by the data driven SSI with unweighted principal component (SSI-UPC) included in the ARTeMIS software [START_REF][END_REF]. Herein the brief description of the general data driven SSI method is presented.

The behavior of structural system is modeled by the linear time-invariant equation of motion [START_REF] Gujer | Overall Damping for Piled Offshore Support Structures, GL RC Guideline for the Certification of Offshore Wind Turbines[END_REF].

M ü(t) + C u(t) + Ku(t) = f (t) y(t) = B a ü(t) + e(t) (4)
where M , C, K are the mass, damping and stiffness matrix, u represents the continuous time displacement vector and f denotes the external force which is not measured but represented as white noise. Vector y contains measured operational responses, in this case the accelerations, B a is the output location matrix for the accelerometers used and e represents the measurement noise. The discrete time state-space formulation of (4) in a time step kτ is written as [START_REF] Tarp-Johansen | Comparing Sources of Damping of Cross-Wind Motion[END_REF].

x k+1 = A d x k + v k y k = C d x k + w k (5)
where A d is the discrete time state matrix describing the dynamics of the system and C d is the output observation matrix. Vectors v k and w k denote the process and measurement noise. The idea in data driven SSI is to relate the response formulation from (5) to a transformation of block Hankel matrix H formulated based on y k . The size of the block Hankel matrix yields N -2k columns and 2k block rows where N , k and m are the number of the data points, time shift and number of blocks respectively. The transformation is referred to as projection of the split H matrix such as H [:,1:k] block rows are projected to H [:,k+1:2k] resulting in a projection matrix O. When assuming that the matrix H contains a zero mean Gaussian signals, the projection is expressed as a conditional mean of the recorder random responses and the matrix O contains free decays of the system. Relation between the projection matrix and ( 5) is described in [START_REF] Damgaard | Dynamic Properties of Offshore Wind Turbine Foundations[END_REF].

O =        C d C d A d C d A d 2 .
. .

C d A d k-1        X = ΓX ( 6
)
where Γ is the observability matrix and X is the matrix of initial conditions of the free decays [START_REF] Brincker | Understanding Stochastic Subspace Identification[END_REF]. The projection matrix can be truncated by using the SVD of the initial O and constructing the new matrix from the non-zero singular values, Σ 1 see [START_REF] Brincker | Modal Identification from ambient responses using frequency domain decomposition[END_REF]. The observability matrix and the state vector are formulated by [START_REF] Overschee | Subspace Identification for Linear Systems[END_REF].

O = U 1 U 2 Σ 1 0 0 Σ 2 V 1 T V 2 T (7) Γ = U 1 Σ 1 0.5 X = Σ 1 0.5 V 1 T (8)
Finally, the discrete time state matrix is found from the regression of Γ. The discrete time poles are obtained from the eigenvalue decomposition of (A d ) matrix, where the eigenvalues and the damping ratios are calculated by the classical modal analysis formulas like in [START_REF] Brincker | Introduction to operational modal analysis United States[END_REF].

Data Interpretation

This section presents the overview of data gathered during the measurement period of 09/2013 -12/2013. First, the general overview of the measurements is presented. The modal properties of the mast are obtained by EFFD routines programmed in MATLAB and SSI-UPC techniques in ARTeMIS [START_REF][END_REF]. Section ends with separation of the system damping components using best-practice from the standards [START_REF] Gujer | Overall Damping for Piled Offshore Support Structures, GL RC Guideline for the Certification of Offshore Wind Turbines[END_REF] and the existing knowledge regarding the MB foundation [START_REF] Damgaard | Dynamic Properties of Offshore Wind Turbine Foundations[END_REF].

Measurement description

In total 50 days of the continuous measurements is sampled with the frequency of 20 Hz and analyzed.

During that period the structure experienced several significant storms including hurricane winds of Christian and Bodil [START_REF] Bodil | [END_REF].

The structural responses illustrated in Figure 5 are in good correlation with the significant wave heights (H s ). The structural monitoring system is not equipped with anemometer however two samples with the highest error in the Figure 5 indicate the presence of strong wind events. The design basis for the location states that the main wave load direction is N-W which is in good agreement with the rose obtained from the accelerometers, see Figure 6. The S-W is the main direction of the wind load what is also visible in Figure 6. Based on both Figure 5 and 6 it can be clearly stated that the response of the structure is driven by the wave loads, which will be correlated to the modal parameters in the next sections.

Ambient vibration analysis

Raw accelerometer data is detrended and high-pass filtered with the frequency cut-off of 0.01 Hz. For OMA accelerations were divided into 100 clusters reflecting a similar loading conditions with the long lengths varying from 2-12 h. The singular values of PSD for a single event are presented in Figure 7. The singular values for the measurement campaign are illustrated in Figure 8. The ratio between the peak SV of the 2 nd and the 1 st mode.

Figure 9 illustrates the ratio between the mean of the singular values for the first and second mode correlated with the H s . Assuming that the left and right singular vectors in [START_REF] Seidel | Wave induced fatigue loads[END_REF] are not subjected to any change through the measurement period, than the magnitude of the peak singular values (highlighted green and purple singular values for the 2 sets of modes at the 0.34 Hz and 1.01 Hz) is changing only due to the loading conditions.

For the calm sea the first mode response is dominant. With the increasing wave heights the second mode is more significant, reaching a half of the 1 st mode magnitude for the typical fatigue wave height in the North Sea, [START_REF] Seidel | Wave induced fatigue loads[END_REF], and doubled the contribution for the storm events. In practice this result illustrate that for load calculations like presented in [START_REF] Seidel | Wave induced fatigue loads[END_REF], the second mode shape should be accounted for in the response of the mast in order not to underestimate the load.

The eigenfrequencies and damping ratios of the 1 st mode are estimated using the SSI-UPC and EFDD techniques. The comparison of the two techniques with correlation to the H s and the histogram of system damping estimated using SSI-UPC are presented in Figures 10,[START_REF] Brincker | Introduction to operational modal analysis United States[END_REF] Regarding the 1 st mode eigenfrequency, the result obtained from both methods agree very well and illustrate the frequency close to 0.295 Hz throughout the measurements. By the visual inspection, the EFDD results are more scattered then SSI-UPC however the mean error between EFDD and SSI-UPC is 0.2% with the standard deviation of 0.16%. According to Figure 11, the results obtained from the both techniques illustrate the linear correlation between the 1 st mode damping ratio and the H s . The maximum added offshore damping ratios in the range of 2% is reached for the two storm events where the highest acceleration measurements peaks at 1.8 m s 2 . The minimum damping of 0.45% is observed during the calm sea states and the accelerations of 0.03 m s 2 .

The histogram presented in Figure 12 illustrates that the damping ratio of the 1 st mode is close to be normally distributed over the measurement campaign, with a 5% quantile and a 95% quantile of 0.57% and 1.61% respectively. Having analyzed the general behavior of the 1 st mode damping of the mast, the separation of its main components is addressed.

Starting from the minimum system damping, based on [START_REF] Gujer | Overall Damping for Piled Offshore Support Structures, GL RC Guideline for the Certification of Offshore Wind Turbines[END_REF] the viscous hydrodynamic damping component for the relative velocities during the calm sea is negligible and the radiation part of the hydrodynamic damping for the cylindrical structure of the 1 st natural frequency close to the 0.3 Hz and diameter of 4.7 m is equal to 0.12%. Accounting for the structural steel damping stated in [START_REF] Gujer | Overall Damping for Piled Offshore Support Structures, GL RC Guideline for the Certification of Offshore Wind Turbines[END_REF] the soil damping contribution in the minimum system damping is 0.05-0.15%. Regarding the maximum damping during the storm conditions, based on the soil damping-acceleration correlation for the MB in sand presented in [START_REF] Damgaard | Dynamic Properties of Offshore Wind Turbine Foundations[END_REF], the soil damping for measured acceleration levels oscillates at 0.6-0.8%. By subtracting the latter and accounting for the structural damping the hydrodynamic damping contribution to the system damping during the storm event would oscillate in the range of 0.9-1.2%.

Conclusion

This paper presents an experimental study of the long-term dynamic properties of the meteorological mast supported by the Mono Bucket foundation using the OMA methods. Based on the results and supported with the existing knowledge the paper addresses the contributions of the hydrodynamic, soil and structural damping to the added offshore damping. Linear correlation between the system damping and the wave heights measured on the site is obtained, with a substantial contribution of the hydrodynamic damping component during the storm events. The results expands the knowledge about the new foundation concept and can find the use in the fatigue limit state calculations. The future work will consider comparison between the experimental data with a numerical boundary element model of soil and the foundation.

Figure 2 .

 2 Figure 2. Three so-called clay chambers assist and provide active inclination control during the installation process.

Figure 1 :

 1 Figure 1: The Mono Bucket foundations for the masts at the DB.

Figure 2 :

 2 Figure 2: The geometry of the foundation. All the dimensions in millimeters.

Figure 3 :

 3 Figure 3: Arrangement of the structural monitoring system at the DB mast.

Figure 5 :

 5 Figure 5: The correlation of the RMS acceleration with the H s through the measurement period.

Figure 6 :

 6 Figure 6: The acceleration response rose through the measurement campaign, m s 2 .

Figure 7 :

 7 Figure 7: The singular values of the PSD matrix of the accelerations with the highlighted modes.

Figure 8 :Figure 9 :

 89 Figure 8: The singular values of the PSD matrices of the accelerations.

and 12 .EFDDFigure 10 :

 1210 Figure 10: The eigenfrequency the 1 st mode for the measurement period. Comparison between the EFDD and SSI-UPC techniques.

Figure 11 :

 11 Figure 11: The correlation of the 1 st mode damping ratios and H s for the measurement period. Comparison between the EFDD and SSI-UPC techniques.

Figure 12 :

 12 Figure 12: The histogram of the 1 st mode damping ratio throughout the measurement campaign.

Table 1 :

 1 Dogger Bank west is located at North Sea, 150 km to the east coast of UK. The water depth for the considered position is 24.8 m. The soil profile based on the in-situ testing consists of layers of dense sand and stiff clay and is listed in the Table1. Soil parameters based on the CPT.

	Depth [m]	Soil type	E 50 [MPa] E ur [MPa] φ c ref [kPa]
	2.0	Sand, very dense	90.8	27.2	-	42.1
	5.0	Clay, soft	6.3	15	40	-
	5.5	Sand, very dense	90.8	27.2	-	42.1
	13.6	Clay, firm	11.9	28.5	40	-
	26.0	Clay, stiff	15.6	37.5	40	-