Signal denoising method for modal analysis of an offshore platform

Journal of Loss Prevention in the Process Industries 63 (2020) 104000

Contents lists available at ScienceDirect

Journal of Loss Prevention in the Process Industries journal homepage: http://www.elsevier.com/locate/jlp

Signal denoising method for modal analysis of an offshore platform Xingxian Bao a, *, Huihui Sun b, Gregorio Iglesias c, d, Teng Wang a, Chen Shi a a

School of Petroleum Engineering, China University of Petroleum (East China), Qingdao, 266580, China Development Center of Qingdao National Laboratory for Marine Science and Technology, Qingdao, 266237, China c MaREI Centre, Environmental Research Institute & School of Engineering, University College Cork, Cork, Ireland d School of Engineering, University of Plymouth, Plymouth, PL4 8AA, United Kingdom b

A R T I C L E I N F O

A B S T R A C T

Keywords: Modal analysis Signal denoising Sea test Jacket-type offshore platform

The modal analysis of offshore structures is a key element of structural health monitoring based on vibration. A difficulty encountered by practitioners and researchers is that an accurate modal analysis is challenging in noisy environments. The objective of this work is to develop and implement a signal denoising method based on solving the inverse singular value problem of a measured (noisy) data matrix with prescribed entries to recon­ struct a filtered data matrix. The measured (noisy) impulse response function (IRF) is used to build a square or nearly square Hankel matrix. The normalized rank determination indicator (RDI) of the Hankel matrix is introduced to determine the number of prescribed non-zero singular values in the prescribed entries inverse singular value problem (PEISVP). The reconstructed (filtered) matrix must maintain the original Hankel struc­ ture, and preserve the number of non-zero singular values. Once the filtered IRF has been obtained, the complex exponential (CE) method is applied for the modal analysis. To validate the proposed method, hereafter referred to as PEISVP-CE, we undertake the numerical simulation of a five-story shear building. Once successfully vali­ dated, we apply the PEISVP-CE method to an offshore field experiment of a jacket platform under the step relaxation. We find that the PEISVP-CE method is effective at eliminating noise and, therefore, appropriate for the modal analysis of offshore structures.

1. Introduction Offshore platforms are fundamental for offshore oil and gas exploi­ tation, and their structural health must be monitored during their ser­ vice life. The structural health monitoring of offshore structures plays an important role in guaranteeing the success of offshore operations and the integrity management of structures (Wang et al., 2018). Modal analysis based on vibration measurements is typically a key element of structural health monitoring of offshore platforms. Haeri et al. (2017) developed a simple two and three dimensional reference model for monitoring health of the offshore jacket platform based on the first few fundamental frequencies and mode shapes of the structure. Rizzo et al. (2018) analyzed the variation of main frequencies and mode shapes of the offshore platform VEGA-A in the Sicily channel in Italy, using over 30 years of monitoring activities, and proposed a modified Kernel-PCA method for long-term platform health monitoring. Shokr­ gozar and Asgarian (2018) identified the modal parameters of a scaled model of steel jacket type offshore platform through experimental and numerical simulation, considering the soil-pile-structure interaction. Li

et al. (2008) applied the cross-model cross-mode method for identifying the damage to individual members of offshore jacket platforms based on limited, spatially incomplete modal data. The modal parameters of large structures such as offshore platforms or bridges must typically be identified from output signals owing to the difficulties in measuring the input forces. A number of output data-only identification algorithms have been developed both in the time and frequency domain over the past few decades (Maia et al., 1997; Ewins, 2000). Operational modal analysis that utilizes the dynamic responses of a structure due to ambient excitation (e.g., wind, waves, traffic loading and operating machines) has recently been widely used in parameter identification of civil engineering structures (Brincker and Ventura, 2015; Brandt et al., 2017; Su et al., 2014). However, the response of large structures such as offshore platforms due to ambient excitation in operational conditions is usually weak, and may not include the enough modal information we concerned. The step relaxation method (McConnell and Varoto, 2008) become an acceptable alternative in field test of large structures, because we can obtain stronger vibration responses and more modal information of

* Corresponding author. E-mail address: [email protected] (X. Bao). https://doi.org/10.1016/j.jlp.2019.104000 Received 15 July 2019; Received in revised form 18 October 2019; Accepted 5 November 2019 Available online 13 November 2019 0950-4230/© 2019 Elsevier Ltd. All rights reserved.

X. Bao et al.

Journal of Loss Prevention in the Process Industries 63 (2020) 104000

structures using step relaxation than ambient excitation. In addition, the vibration responses from step relaxation are free decay data, which can be directly used to identify modal parameters by some time domain identification methods, such as complex exponential (CE) (Maia et al., 1997; Ewins, 2000), Ibrahim time-domain scheme (ITD) (Ibrahim and Mikulcik, 1973), and eigensystem realization algorithm (ERA) (Juang, 1994). Hu et al. (2011) employed Prony’s method to identify multiple modal parameters of a numerical model of a jacket-type offshore plat­ form from step relaxation data. Li et al. (2012) implemented an extended Prony’s method to identify modal parameters of a jacket-type offshore platform under step relaxation in field test. Difficulties arise, however, no matter ambient excitation or step relaxation is applied in noisy environments, the response of large structures in field test is easily polluted by noise (Amezquita-Sanchez and Adeli, 2016; Cao et al., 2016; Li et al., 2012). The traditional method to estimate modal parameters in noisy en­ vironments usually absorbs false modes first. Subsequently, the stability diagram is applied to separate the genuine from the false modes (Alle­ mang and Brown, 1998). However, the stability diagram would be affected by noise in the measured response. If the noise level is high, the false modes would also tend to be stable when the order of the model increases. Therefore, it is important to develop a signal denoising method for identifying modal parameters with high precision. In recent years, a number of studies dealt with the signal denoising algorithm for modal analysis and fault diagnosis. Jiang et al. (2007) proposed a noise removal method using the Bayesian discrete wavelet packet transform for system estimation. Hazra et al. (2012) presented time-frequency blind source separation methods for the identification of weak and close modes in noisy situations. Hu et al. (2010, 2012) and Bao et al. (2015, 2019) considered the noise reduction from measured vi­ bration signals as a problem of low-rank approximation of a Hankel matrix in linear algebra, and introduced Cadzow’s algorithm (Cadzow, 1988), a suboptimal method in the L2-norm criterion (De Moor, 1994), to conduct signal denoising. Cai et al. (2014, 2016, 2017) proposed a signal denoising and fault diagnosis methodology using Bayesian networks. This study develops a novel signal denoising algorithm for the esti­ mation of modal parameters. The proposed algorithm solves the inverse singular value problem of a measured (noisy) data matrix with pre­ scribed entries (Hu and Li, 2008) to reconstruct a filtered data matrix. In this algorithm, the original (noisy) impulse response function (IRF) is used to build a Hankel matrix. The reconstructed (filtered) data matrix should also be Hankel structured and preserve the number of non-zero singular values. The prescribed entries inverse singular value problem (PEISVP) here is also a problem of approximating a Hankel matrix with lower rank. After solving the PEISVP, the filtered IRF is obtained, and the modal analysis is carried out using the output data-only method in the time domain. Nowadays, several output data-only methods to identify modal pa­ rameters in the time domain have been implemented in commercial software, such as the CE, ERA and SSI method (Overschee and De Moor, 1996; Peeters and De Roeck, 2000). However, from the point of view of sensitivity to noise in output data, the CE method, compared with the other two, has a major disadvantage, namely that no denoising pro­ cedure is included (Braun and Ram, 1987). In order to show the effec­ tiveness of the proposed signal denoising algorithm, in this study the CE method is utilized for modal analysis from measured and filtered IRFs. This article is structured as follows. The signal denoising method based on solving the PEISVP is presented in Section 2, and the validity of the PEISVP-CE procedure is demonstrated in Section 3 by means of a numerical example – a five-story shear building with two cases of noise level. In Section 4, the performance of the method proposed is shown with a field experiment, more specifically, with field test acceleration data of a real jacket-type offshore platform under the step relaxation by pulling one leg of the test structure using a boat, which can trigger much stronger vibration responses than ambient excitation.

2. Solution of the inverse singular value problem with prescribed entries for signal denoising The inverse singular value problem (ISVP) consists in reconstructing a matrix from some given singular values and/or singular vectors. The inverse singular value problem with prescribed entries (PEISVP) is a special class of ISVP in mathematical terms. In this study, the prescribed entries in PEISVP are Hankel structured, and the number of non-zero singular values is preserved. The signal denoising of the measured IRF is carried out by solving the PEISVP of the Hankel matrix (obtained from the measured IRF) with the number of prescribed non-zero singular values. 2.1. PEISVP For a Hankel matrix Z 2 Rm�n , there exists a singular value decom­ position (SVD) of Z: Z ¼ UΣVT

(1)

Where U ¼ ½u1 ; …; um � 2 Rm�m and V ¼ ½v1 ; …; vn � 2 Rn�n are unitary matrices, Σ ¼ diagðσ1 ; ⋯; σ g ; ⋯; σ p Þ 2 Rm�n , with p ¼ minðm; nÞ, is a di­ agonal matrix and σ 1 � ⋯ � σ g � ⋯ � σ p � 0. The PEISVP here is to find a Hankel matrix Z 2 Rm�n , with the sin­ gular values σ1 � ⋯ � σ g > σgþ1 ¼ ⋯ ¼ σp ¼ 0, such that Zvi ¼ σ i ui ; i ¼ 1; …; p;

(2)

and ZT ui ¼ σi νi ; i ¼ 1; …; p;

(3)

The Hankel matrix Z can be expressed as: ns X

Z¼

bs Zs

(4)

s¼1 s Where bs are unknown parameters, fZs gns¼1 2 Rm�n are a series of basis of the linear space of Z, ns ¼ m þ n 1, and 2 3 2 3 1 0 ⋯ 0 0 1 ⋯ 0 60 0 ⋯ 07 61 0 ⋯ 07 7 6 7 Ζ1 ¼ 6 4 ⋮ ⋮ ⋱ ⋮ 5; Ζ2 ¼ 4 ⋮ ⋮ ⋱ ⋮ 5; ⋯Ζmþn 1 0 0 ⋯ 0 0 0 ⋯ 0 2 3 0 0 ⋯ 0 60 0 ⋯ 07 7 ¼6 4 ⋮ ⋮ ⋱ ⋮ 5: 0 0 ⋯ 1

Inserting Eq. (4) into Eq. (2) leads to ns X

bs Zs vi ¼ σ i ui

(5)

s¼1

Pre-multiplying Eq. (5) with eTj , the transpose of the standard basis vector ej 2 Rm , we obtain: ns X

bs eTj Zs vi ¼ σ i eTj ui

(6)

s¼1

Denoting the scalars eTj Zs vi and σ i eTj ui to be Pij;s and cij , respectively, and replacing the index ij by y, yields: ns X

bs Py; s ¼ cy

(7)

s¼1

Obviously, Eq. (7) is a linear equation with bs to be determined. By the same token, inserting Eq. (4) into Eq. (3) yields

2

Journal of Loss Prevention in the Process Industries 63 (2020) 104000

X. Bao et al. ns X

of the Hankel matrix Zð0Þ to determine the rank of Zð0Þ , which is also the number g. Theoretically, the number g is double the number of modes included in the measured IRF (Hu et al., 2010). A normalized rank determination indicator is introduced in this study to determine the number g. The details are demonstrated in Section 3.1.1. Note that, in

(8)

bs ZTs ui ¼ σi vi

s¼1

Pre-multiplying Eq. (8) with eTj , the transpose of the standard basis vector ej 2 Rn , we have: ns X

the iteration procedure, the number of non-zero singular values in σ i

ð0Þ

(9)

bs eTj ZTs ui ¼ σi eTj vi

e., g), rather than the values of non-zero singular values in

s¼1

ð0Þ σ ð0Þ 2 ; ⋯; σ g ), is preserved.

This yields another equation in the form of Eq. (7): ns X s¼1

ij is replaced by y. Now, solving the PEISVP with g prescribed non-zero singular values (i.e., σ1 ; σ2 ; ⋯; σ g ) of Z, we can form gðm þ nÞlinear equations based on Eq. (7) and Eq. (10) to solve for ns unknown parameters. Therefore, a matrix form combining Eq. (7) and Eq. (10) is written as:

N X

bðtÞ ¼

(11)

Where X

Introducing ωrþN ¼ n Eq. (13) as

is the generalized inverse of X.

2.2. Procedure of signal denoising

2N X

bðtÞ ¼

ðkÞ

2N X

bl ¼

L2-norm, bð0Þ IRF.

kb

Ar ðV r Þl

(16)

r

where V r ¼ es Δt . Once V r (i.e. modal frequencies and damping ratios) are known, Eq. (16) becomes linear with respect to the remaining un­ known modal parameters Ar . Because an N DOF dynamic system is mathematically equivalent to a linear differential equation of order 2N, the IRF sequence fbl g must satisfy a linear difference equation:

(3) Form X and f ðkÞ based on ui ; vi and σ p , using Eqs. (7), (10) and (11). t t (4) Solve bðkþ1Þ from Eq. (12), i.e., bðkþ1Þ ¼ XðkÞ f ðkÞ , where XðkÞ is ðkÞ the generalized inverse of X . (5) Form the Hankel matrix Zðkþ1Þ based on bðkþ1Þ , using Eq. (4). ðkÞ

2X N 1

b2Nþn þ

βm bmþn ¼ 0; ​ ​ ​ ​ n ¼ 0; 1; …

(17)

m¼0

b k � 10 14 , where k⋅k refers to the kbð0Þ k denotes the measured IRF and bðkþ1Þ denotes the filtered

The stopping criterion is

(15)

r¼1

ðkÞ

ð0Þ

(14)

Where Δt is the time interval. From Eqs. (14) and (15), we have

values in σ i , and set other singular values to zeros). ðkÞ

r

Ar es t

bl ¼ bðlΔtÞ; ​ ​ l ¼ 0; 1; 2; ⋯

(2) Form based on and the number of prescribed non-zero singular values (i.e., preserve the first g non-zero singular ðkÞ

ωrd , ArþN ¼ Ar� for r ¼ 1;…;N, we can rewrite

ðkÞ

σ ðkÞ i

ðkÞ

(13)

If the IRF has been obtained in a discrete form, it can be expressed as

(1) Perform the SVD of ZðkÞ to obtain σi ; ui and vi , where the superscript (k) denotes the quantity associated with the k-th iteration, k ¼ 0; 1; 2; ⋯ , and Zð0Þ is the initial Hankel matrix based on the measured IRF. ðkÞ

�

r¼1

The signal denoising procedure based on solving the PEISVP includes the following five steps, which are repeatedly taken at each iteration:

σ ðkÞ p

r�t

where Ar is the complex amplitude, and the superscript ‘‘*’’ denotes the complex conjugate operator. sr is the system pole, sr ¼ ωrn ξr þ iωrd , and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωrd ¼ ωrn 1 ðξr Þ2 .

(12) t

r

Ar es t þ Ar� es

r¼1

Where X ¼ ½Py; s ; Qy; s � 2 RgðmþnÞ�ns , b 2 Rns , f ¼ ½cy ; dy � 2 RgðmþnÞ . The SVD is used to solve for b: b ¼ X tf

ð0Þ

The basic principle of the CE method is that any free vibration response function, including IRF, can be expressed as a series of complex exponential components, and each of them contains the eigenvalue and eigenvector properties of one mode (Maia et al., 1997; Ewins, 2000). The brief derivation of the essential of CE method is given as below. The IRF of an N DOF system can be expressed as:

where Qij;s and dij denote the scalars eTj ZTs ui and σ i eTj vi , respectively, and

Xb ¼ f

(i.

(i.e., σ1 ;

2.4. PEISVP-CE procedure for modal analysis

(10)

bs Qy; s ¼ dy

σð0Þ i

ðkþ1Þ

where βm ; ​ ​ ​ m ¼ 0; …; 2N 1 are real constant coefficients. Inserting Eq. (16) into Eq. (17) leads to: 2N X

2.3. Key issues in the procedure of signal denoising

βm ðV r Þm ¼ 0

(18)

m¼0

where β2N is equal to 1, and the roots of the polynomial are V 1 ; V 2 ; …; V 2N . Thus, the values of βm determine the values of V r , and hence the modal frequencies and damping ratios. The original CE method is solving βm from the IRF data based on the successive applications of Eq. (17) for 2N times. Taking n ¼ 0; …; 2N 1 with Eq. (17) results in a full set of 2N equations 9 8 9 2 38 b0 β0 > b2N > b1 b2 ⋯ b2N 1 > > > > > > < = < = 6 b1 b2 b3 ⋯ b2N 7 β1 b2Nþ1 6 7 ¼ (19) 4 ⋮ ⋮ ⋮ ⋯ ⋮ 5> ⋮ ⋮ > > > > > > > ; : ; : b2N 1 b2N b2Nþ1 ⋯ b4N 2 β2N 1 b4N 1

The dimension of a Hankel matrix Zð0Þ is an important issue in the proposed noise reduction procedure. It has been proved that the computational time to solve the PEISVP increases with the size of Zð0Þ (Hu et al., 2010). However, if we choose a smaller size ofZð0Þ , some modes in the IRF may be lost. Without consideration of computer time and to ensure that no modes are lost, a square or nearly square Hankel matrix Zð0Þ is a better choice.

Another critical issue is how many non-zero singular values in σi should be preserved, i.e., how to determine the number g in Step (2) of the procedure of signal denoising. An effective method is using the SVD ðkÞ

3

X. Bao et al.

Journal of Loss Prevention in the Process Industries 63 (2020) 104000

However, it is easy to see the difference between the clean and noisy signals in the frequency domain. These results indicate that it is possible to evaluate the effectiveness of noise reduction intuitively in the fre­ quency domain.

or Bβ ¼

~ b

(20)

Where B 2 R2N�2N is a square matrix with Hankel structured, which is referred to as a matrix with constant skew-diagonals. From Eq. (20), we can obtain the unknown polynomial coefficients β¼

~ B 1b

3.1.1. Signal denoising A segment of simulated IRF with 1024 time steps (0–20.48 s) is applied for the following analysis. A nearly square Hankel matrix Hm�n is constructed with dimensions 513 � 512. It is worth noting that m þ n 1 is equal to 1024, and we assume m � n without loss of the generality. In the PEISVP method, the number of prescribed non-zero singular values, i.e., the rank of H513�512 , can be determined by means of SVD. A normalized rank determination indicator (RDI) is introduced in this study:

(21)

Once the filtered IRF has been obtained by implementing the PEISVP solving steps, the CE method is used for modal analysis in this study. The proposed PEISVP-CE procedure is illustrated concisely in Fig. 1. 3. Numerical verification A five-story shear building numerical model was applied to validate the proposed PEISVP-CE method. The five uniform mass, stiffness and damping coefficients are taken to be 200 kg, 2.0 � 105 N/m and 200 N/ m, respectively. The 1st floor is at the fixed end, and the 5th floor at the free end. The theoretical modal frequencies of the numerical model are 1.4325, 4.1815, 6.5917, 8.4679, and 9.6583 Hz. The theoretical damping ratios are 0.45004%, 1.3137%, 2.0709%, 2.6603%, and 3.0342%. The acceleration responses of the numerical model under an impulse excitation on the 1st floor, also called clean IRF in this study, with a duration of 100 s are simulated using the “impulse” function on Matlab with a sampling frequency of 50 Hz. There are 5 IRFs corre­ sponding to as many floors. The IRF on the 1st floor is retained for later analysis without loss of generality. The noise is simulated as additive Gaussian white noise. The noise level is quantified by a stated percent­ age, defined as the ratio of the standard deviation of the white noise to that of the clean IRF. 3.1. Case 1: 10% noise level The comparison between the clean and noisy (10% noise level) signal in the time and frequency domains is shown in Fig. 2. It can be observed that the clean signal almost coincides with the noisy signal in the time domain due to the fact that the relative amplitude of the noise is small.

Fig. 2. Comparison of clean and noisy signals in the time and fre­ quency domains.

Fig. 1. Flowchart of the proposed PEISVP-CE procedure. 4

X. Bao et al.

RDIðiÞ ¼

σi

Journal of Loss Prevention in the Process Industries 63 (2020) 104000

σiþ1 ; i ¼ 1; …; n σ iþ1

normalizedRDIðiÞ ¼

RDIðiÞ ; i ¼ 1; …; n maxðRDIÞ

The noise reduction is implemented by solving the PEISVP based on the fact that the number of prescribed non-zero singular values is 10 for the Hankel matrix H513�512 . The clean and filtered signals in the fre­ quency domain are in excellent agreement (Fig. 4), proving that the noise has been effectively removed from the noisy signal.

(22)

1;

1

(23)

where σi is the singular value in descending order of Hm�n and maxðRDIÞ is the maximum RDI value among RDIðiÞ. The maximum normalized RDI (i.e., 1) corresponds to the rank of Hm�n . The normalized RDI values related to the clean and noisy IRFs are plotted in Fig. 3. For a five-story shear building, the rank of the clean data matrix H513�512 should theo­ retically be 10, i.e., double the number of modes contained in the clean IRF. This is verified in Fig. 3 (a), because the index of singular values corresponding to the maximum normalized RDI value is 10. For a noisy data matrix H513�512 (seen in Fig. 3 (b)), the index of singular values corresponding to the maximum normalized RDI value is still 10, although other normalized RDI values are larger than those in a clean data matrixH513�512 due to the noise. This means that all the 5 modes are identifiable from the noisy IRF; however, the accuracy of the estimation may be affected by noise.

3.1.2. Modal analysis In the traditional estimation of modal parameters, over-determined models are usually applied and “noise modes” (false modes) are absor­ bed first; subsequently, the stability diagram method is used to separate the true modes from the false modes. When producing a stability dia­ gram, the modal frequencies, the damping ratios and the modal vectors are compared between two consecutive model orders with preset sta­ bility standards. If all the differences of the modal frequencies, damping ratios and modal vectors do not exceed the thresholds set by the stability standards, the mode is plotted as a stable one. In this study, the stability standards include three thresholds: 1% for modal frequencies, 2% for the modal vector, and 5% for damping ratios. The stability diagrams for the estimation results using the CE method from the noisy and filtered IRFs are presented in Fig. 5. The modal estimation is conducted repeatedly with increasing model order, starting with 5 (determined from the normalized RDI values in Section 3.1.1). Note that the symbols “o” and “*” represent non-stable and stable modes, respectively. As can be seen in Fig. 5 (a), the second, third, and fourth modes are approximately stable in the stability diagram for the noisy signal. However, many poles for the first and fifth modes are un­ stable when the model order is less than 30, because they do not meet the stability standards. All the 5 modes are nearly stable when the model order exceeds 60; however, some false modes between 6 Hz and 12 Hz also become stable with increasing model order. It is worth noting that only 3 modes can be identified using the noisy signal when the model order is 5, as can be observed in Fig. 5 (a); what is worse, they are not stable. By contrast, 5 very consistent and stable modes occur from model order 5 and above (Fig. 5 (b)). This indicates that it is possible to identify all the 5 modes just using model order 5 from the filtered signal, and it is not necessary to repeat the modal estimation process with higher model orders. The identification of the modal parameters is carried out using the clean, filtered and noisy data, respectively, just with model order 5. Tables 1 and 2 list the estimated modal frequencies and damping ratios, respectively, with model order 5. It may be seen that the frequencies and damping ratios identified using the clean data agree well with the theoretical values. All the five frequencies identified using the filtered signal are almost identical to those obtained from the clean signal, with the maximum error, 0.387%, related to the fifth mode. Similarly, the first four damping

Fig. 3. Normalized RDI values related to (a) clean and (b) noisy IRFs.

Fig. 4. Comparison of the clean and filtered signals in the frequency domain. 5

X. Bao et al.

Journal of Loss Prevention in the Process Industries 63 (2020) 104000

ratios identified using the filtered data are very consistent with those obtained from the clean data, with less than 4% error. The fifth mode being the weakest, and easily corrupted by noise (Fig. 2), the relative difference of the fifth damping ratio is greater (24.161%). However, it is not possible to estimate the first and fifth modal pa­ rameters using the noisy signal, just with model order 5 (as is readily seen in Fig. 5 (a)). Even worse, the identified second, third, and fourth modes contain greater errors – in particular, the damping ratios. The relative differences of the three damping ratios are 2206.16%, 1475.50% and 203.37%, respectively. It is worth noting that it is also possible to estimate all the 5 modes from the noisy signal using the stability diagram method, when the model order exceeds 60 and all the 5 modes are stable. This means that the process to identify the modal parameters needs to be repeated at least 60 times. Obviously, the modal analysis from the measured signal using the stability diagram method is a time-consuming operation. 3.2. Case 2: 20% noise level The normalized RDI values related to the noisy IRF (with 20% noise level) are plotted in Fig. 6. The index of singular values corresponding to the maximum normalized RDI value is 8, implying that it is possible to identify only 4 modes from the noisy IRF, and one mode cannot be estimated due to the noise. The noise elimination is carried out by solving the PEISVP for the Hankel matrix H513�512 . Fig. 7 shows the clean, noisy and filtered signals in the frequency domain. It may be seen that the fifth mode is totally buried in the noise, then eliminated altogether with the noise by the PEISVP solving procedure. However, there is a good agreement between the filtered and clean curves related to the first four modes. The identification of the modal parameters is carried out using filtered and noisy data. Fig. 8 (a) is the stability diagram for the noisy signal. It is found that the true modes cannot be easily identified, for many poles are non-stable, although there are some nearly consistent poles for the first four modes. By contrast with Fig. 8 (a), there are four stable and consistent modes in Fig. 8 (b) for model order 4 and above. It follows that the first four modes may be readily identified from the filtered signal using solely model order 4, with no need to repeat the modal analysis calculation for higher model orders. The estimated modal frequencies and damping ratios with model order 4 from the filtered signal are listed in Table 3. It is found that the maximum relative difference between the identified frequencies from the filtered and clean signals is 0.54%, and between the damping ratios, 5.97%. In comparison

Fig. 5. Stability diagrams related to (a) noisy and (b) filtered IRFs. Symbols “o” and “*” represent non-stable and stable modes, respectively. Table 1 The frequencies identified using different signals with model order 5 (Hz). Mode

Clean fc

Filtered ff

Noisy fm

1 2 3 4 5

1.4325 4.1815 6.5917 8.4679 9.6581

1.4324 4.1814 6.5923 8.4528 9.6207

– 3.1339 5.1698 8.6754 –

Relative difference � � � � � �fm fc �=fc �ff 0.007% 0.002% 0.009% 0.178% 0.387%

� fc �=fc

– 25.05% 21.57% 2.450% –

Table 2 The damping ratios identified using different signals with model order 5 (%). Mode

Clean ξc

Filtered ξf

Noisy ξm

1 2 3 4 5

0.45004 1.3137 2.0709 2.6603 3.0342

0.44127 1.3265 1.9894 2.6876 2.3011

– 30.296 32.627 8.0706 –

Relative difference � � �ξf jξm ξc �=ξc 1.949 0.974 3.935 1.026 24.161

ξc j=ξc

– 2206.16 1475.50 203.37 –

Fig. 6. Normalized RDI values related to the noisy IRF (with 20% noise level). 6

X. Bao et al.

Journal of Loss Prevention in the Process Industries 63 (2020) 104000

with Case 1, the relative difference in Case 2 is slightly larger due to the higher noise level. However, the results are satisfactory considering the poor identification results using only model order 4 from the noisy signal (Fig. 8 (a)). Only two false modes can be identified from the noisy signal using model order 4. These results show that satisfactory estimation results can be ob­ tained using the PEISVP-CE approach based solely on the determined model order, with no need to repeat the trial calculation with higher model orders. The model order, which is half the number of prescribed non-zero singular values, is determined in advance in the PEISVP solving procedure. On these grounds, it is clear that the PEISVP-CE approach can improve the precision and efficiency of modal identification significantly. 4. Field test of a jacket-type offshore platform Since the exploitation of the Shengli Oilfield (Bohai Bay) began in 1994, more and more offshore platforms have been built and put to use in China. So far, more than 100 jacket-type offshore platforms have been commissioned in Bohai Bay, China. The design life of these platforms is generally 20 years, so some of them are approaching, or have already reached, the end of their design life. However, these platforms are poised to remain in service in order to meet the requirements for development of the Shengli Oilfield. Therefore, it is important to monitor the structural health of these aging platforms, and to assess their capacity to remain in service. Modal analysis is a crucial element in vibration-based structural health monitoring. The approach developed in the previous sections is illustrated here through a case study: an aging jacket-type offshore platform (Fig. 9). The elevation of the deck is 10.5 m above mean sea level, and the elevations of the three layers of the platform are þ5.5 m, 1.5 m and 8.5 m. The dimensions of the jacket legs, horizontal and vertical braces of the platform are shown in Table 4. In this test, eight triaxial accelerometers were mounted on four legs at elevations of 5 m and 6 m, to record the vibration response signals of each output point. Although great progress has been made in the field of operational modal analysis for large and complex structures, the vibration responses of the platform under ambient excitation (i.e., winds, waves and cur­ rents) in this study are too weak to be used for identifying the modal parameters. In order to obtain stronger free vibration responses while ensuring the security of the platform – the collision between a ship and the platform is obviously not an option – the sea test was conducted under the step relaxation by pulling one leg of the test platform using a boat. In the test, leg no. 2 of the platform was pulled at point 1 by a boat in the –x direction. Fig. 10 shows the tugboat used to pull the platform. To control the maximum tension of the rope, a tension limiter was set at 60 kN. The platform was released when the tension of the rope reached 60 kN, and the vibration acceleration response signals were then measured for about 40 s by the eight accelerometers. The sampling frequency was 200 Hz. In total, 24 acceleration response records were obtained from the eight accelerometers in the three Cartesian directions (x, y, and z). As the excitation force was exerted along the –x direction, the acceleration response signals in the x and –x directions with am­ plitudes exceeding those in the y and z directions were retained for

Fig. 7. Clean, noisy and filtered signals in the frequency domain.

Table 3 Frequencies and damping ratios identified using clean and filtered signals. Mode

1 2 3 4

Fig. 8. Stability diagrams related to: (a) noisy (with 20% noise level), and (b) filtered IRFs. Symbols “o” and “*” represent non-stable and stable modes, respectively. 7

Clean fc (Hz)

ξc (%)

Filtered ff (Hz)

ξf (%)

Relative difference � � � � � �ξf ξc �=ξc � fc �=fc �ff

1.4325 4.1815 6.5917 8.4679

0.45004 1.3137 2.0709 2.6603

1.4321 4.1814 6.5940 8.4221

0.42316 1.3565 1.9822 2.7034

0.03% 0.002% 0.03% 0.54%

5.97% 3.26% 4.28% 1.62%

X. Bao et al.

Journal of Loss Prevention in the Process Industries 63 (2020) 104000

Fig. 11. Measured acceleration response signals.

Fig. 9. Jacket-type offshore platform tested in the case study.

implying that there are 3 identifiable modes embedded in the 2 measured IRFs. The noise reduction is carried out through the PEISVP solving pro­ cedure for the two measured data matrices H701�700 , respectively. The measured and filtered signals from Sensors 1 and 2 in the frequency domain are shown in Fig. 13. As expected, in both cases the filtered signals are smoother and have three spectral peaks between 0 and 5 Hz. Although the measured frequency-magnitude curve peaks seem to match the filtered frequency-magnitude curve peaks between 0 and 5 Hz, this is not enough to ensure the accuracy of the modal estimation using the measured signals due to the noise. In particular, the identifi­ cation of the damping ratio is easily contaminated by noise. The modal analyses using the measured and filtered signals are conducted in this section. The stability diagrams are also plotted to help estimate the true modes. As can be seen in Fig. 14 (a), the stable modes cannot be obtained until the model order is above 58. However, when the model order is above 80, there are 8 stable modes between 0 and 10 Hz. Evidently, some “noise modes”, such as the mode between 0 and 2 Hz, and those between 4 Hz and 10 Hz, are also stable. This indicates that the stability diagram cannot distinguish the true from the false modes effectively, when the signal measured in a noisy situation is directly used for modal analysis. On the contrary, the stability diagram pertaining to the filtered signal (Fig. 14 (b)) has 3 consistent and stable modes from model order 3 to 30, which means that the modal param­ eters can be correctly identified using the filtered signal with a model order of only 3 – additional identification with higher model orders is not necessary. Similarly, it is difficult to directly identify the true modes from the measured x direction IRF related to Sensor 2, because several false modes are also stable when the model order is higher than 60 (Fig. 15 (a)). However, it is straightforward to identify the true modes from the filtered signal using only model order 3, as the three modes remain consistent and stable from model order 3 to 30 (Fig. 15 (b)). The same procedures of signal denoising and modal parameters estimation are carried out for the other six measured IRFs in the x and –x directions. As expected, the consistent and stable modes can be obtained from the six filtered signals, respectively. Tables 5 and 6 list the iden­ tified frequencies and damping ratios, respectively, from all the eight filtered IRFs. As the modal parameters identified directly from the measured signals are inefficient and inaccurate, the estimation results from the measured signals are not presented here. Theoretically, as the modal parameters are global, the modal parameters identified from different sensors should be identical only if the modes can be excited and the sensors are not mounted on the nodes of a certain mode. However, due to the noise and the different response locations, there are always

Table 4 Dimensions of the legs and braces of the jacket-type offshore platform tested. Dimension (m)

Legs

Horizontal braces

Outer diameter Wall thickness

1.34

0.6

0.6

0.6

0.4

0.025

0.02

0.02

0.02

0.014

8.5 m

1.5 m

þ5.5 m

Vertical braces (between 8.5 m and 1.5 m)

Fig. 10. Boat used in the sea test.

analysis. For simplicity, here is presented the performance of the proposed approach for the IRF measured in the x direction by Sensors 1 and 2. The measured acceleration response signals in the x direction are shown in Fig. 11. The relatively stable time segments from 8 s to 15 s were retained for analysis. Two nearly square Hankel matrices H701�700 are constructed using 1400 time steps (8–15 s) from Sensors 1 and 2. To determine the number of prescribed non-zero singular values of H701�700 , the normalized RDI values are presented in Fig. 12. It may be seen that 6 is a reasonable number of non-zero singular values to be preserved for both matrices, 8

X. Bao et al.

Journal of Loss Prevention in the Process Industries 63 (2020) 104000

Fig. 12. Normalized RDI values related to measured IRF in x direction from Sensors 1 and 2.

Fig. 14. Stability diagrams pertaining to: (a) measured, and (b) filtered signals of Sensor 1. Symbols “o” and “*” represent non-stable and stable modes, respectively.

Fig. 13. Measured and filtered signals related to: (a) Sensor 1, and (b) Sensor 2.

9

X. Bao et al.

Journal of Loss Prevention in the Process Industries 63 (2020) 104000

estimated damping ratios are between 2.56% and 8.78%, i.e., there is good agreement between the frequencies and damping ratios identified from all the eight filtered IRFs. The mean values of the identified modes from the eight filtered signals can be regarded as the true modes, although one cannot get the genuine modes of the platform. It is also found that the first and fourth modes can be estimated from each of the eight filtered signals. The second mode can only be estimated from Sensors 2, 3, 6 and 7, and the third mode only from Sensors 1, 4, 5 and 8. The reason is that the weak modes in different response signals cannot be identified, because they are regarded as noise and eliminated during the signal denoising procedure in this study. Additionally, the modal parameters identified using filtered signals from two sensors mounted on the same leg of the platform are remarkably consistent. Considering, for instance, leg no. 4 of the platform (with Sensors 4 and 8), the COVs of the modal frequencies identified from the two filtered signals are between 0 and 0.20%, and those of the damping ratios are between 0.23% and 4.77% (Table 7) – in all cases, lower that those obtained using all the eight filtered signals. 5. Conclusion The estimation of the modal parameters of offshore structures based on field tests is challenging due to the noisy environments. The CE method is widely used in modern modal analysis software; however, it is sensitive to the noise in the measured data. In this study a signal denoising algorithm based on the PEISVP method was developed and implemented together with the CE method to solve the problem of noise contamination in the estimation of modal parameters. The approach consists in reconstructing a filtered Hankel matrix with a number of prescribed non-zero singular values. This number is also the rank of the data matrix, which can be determined from the normalized RDI values. First, the PEISVP-CE method was validated with a numerical model of a five-story shear building. Two cases with noise levels of 10% and 20% are studied. The stability diagrams related to noisy and filtered data are applied in the modal analysis to help demonstrate the advantage of the proposed method. In the case with 10% noise, it was found that the false modes tend to be stable when the model order increases, implying that it is difficult to identify the true modes directly from the measured data, even using the stability diagram. However, it is possible to obtain all the five estimation results using the filtered data based solely on model order 5, with no need to repeat the model analysis for higher model orders. In the case with 20% noise, the fifth mode is buried in the noise and eliminated with it by the noise reduction procedure; conse­ quently, the first four modes can be easily identified using the filtered data based on model order 4, in contrast with the poor identification results from the noisy signal. Subsequently, the proposed method was applied to process the eight acceleration responses of a jacket-type offshore platform measured in a

Fig. 15. Stability diagrams related to: (a) measured, and (b) filtered signals of Sensor 2. Symbols “o” and “*” represent non-stable and stable modes, respectively.

deviations in the results obtained from different sensors. The identified modal frequencies and damping ratios present low coefficient of varia­ tions (COVs) (Tables 5 and 6, respectively). The COVs of the estimated modal frequencies are between 0.32% and 0.71%, and the COVs of the

Table 5 Modal frequencies of the tested platform identified from eight filtered signals (S1-S8) (Hz). Mode

S1

S2

S3

S4

S5

S6

S7

S8

Mean

COV(%)

1 2 3 4

2.186 – 3.195 4.178

2.202 3.098 – 4.151

2.199 3.078 – 4.202

2.185 – 3.181 4.169

2.189 – 3.201 4.185

2.199 3.048 – 4.186

2.202 3.089 – 4.179

2.186 – 3.181 4.181

2.194 3.078 3.190 4.179

0.35 0.71 0.32 0.35

Table 6 Damping ratios of the tested platform identified from eight filtered signals (S1-S8) (%). Mode

S1

S2

S3

S4

S5

S6

S7

S8

Mean

COV

1 2 3 4

1.515 – 1.392 5.016

1.616 3.431 – 4.127

1.635 4.023 – 4.561

1.530 – 1.478 5.092

1.576 – 1.413 4.712

1.645 3.355 – 4.123

1.601 3.419 – 4.325

1.525 – 1.427 4.760

1.580 3.557 1.428 4.590

3.28 8.78 2.56 8.16

10

X. Bao et al.

Journal of Loss Prevention in the Process Industries 63 (2020) 104000

Table 7 Modal estimation using two filtered signals (S4 and S8) from leg no. 4 of the platform. Mode 1 2 3 4

Frequencies (Hz)

Damping ratios (%)

S4

S8

Mean

COV (%)

S4

S8

Mean

COV

2.185 – 3.181 4.169

2.186 – 3.181 4.181

2.186 – 3.181 4.175

0.03 – 0 0.20

1.530 – 1.478 5.092

1.525 – 1.427 4.760

1.528 – 1.453 4.926

0.23 – 2.48 4.77

field test under step relaxation. As expected, the estimation of the modal parameters from the filtered signals was found to be more effective and accurate than that from the measured signals. Although the exact modes of the platform were not available, the performance of the PEISVP-CE method could be assessed by comparing the identification results from different signals. It was found that the results identified from the filtered signals agree well with one another, and the mean values can be regarded as true modes. The results of both the numerical and field (sea) test studies proved that the proposed PEISVP-CE method can improve the precision and efficiency of the modal analysis. It is noted that the mode shapes are not identified in this study, as the CE method is a single-input and single-output modal analysis method. The mode shapes can be estimated using a single-input multi-output modal analysis method (e.g., the least-squares complex exponential method) or a multi-input multi-output modal analysis method (e.g., the polyreference complex exponential method). In future work we will consider combining the signal denoising method proposed here with other modal analysis methods to identify the modal frequencies, damping ratios and mode shapes.

Cai, B., Liu, Y., Fan, Q., Zhang, Y., Liu, Z., Yu, S., Ji, R., 2014. Multi-source information fusion based fault diagnosis of ground-source heat pump using Bayesian network. Appl. Energy 114, 1–9. Cai, B., Liu, H., Xie, M., 2016. A real-time fault diagnosis methodology of complex systems using object-oriented Bayesian networks. Mech. Syst. Signal Process. 80, 31–44. Cai, B., Zhao, Y., Liu, H., Xie, M., 2017. A data-driven fault diagnosis methodology in three-phase inverters for PMSM drive systems. IEEE Trans. Power Electron. 32 (7), 5590–5600. Cao, M., Xu, W., Ren, W.X., Ostachowicz, W., Sha, G.G., Pan, L.X., 2016. A concept of complex-wavelet modal curvature for detecting multiple cracks in beams under noisy conditions. Mech. Syst. Signal Process. 76–77, 555–575. De Moor, B., 1994. Total least squares for affinely structured matrices and the noisy realization problem. IEEE Trans. Signal Process. 42, 3104–3113. Ewins, D.J., 2000. Modal Testing: Theory, Practice and Applications, second ed. Research Studies Press Ltd., Baldock, Hertfordshire, England. Haeri, M.H., Lotfi, A., Dolatshahi, K.M., Golafshani, A.A., 2017. Inverse vibration technique for structural health monitoring of offshore jacket platforms. Appl. Ocean Res. 62, 181–198. Hazra, B., Sadhu, A., Roffel, A.J., Narasimhan, S., 2012. Hybrid time-frequency blind source separation towards ambient system identification of structures. Comput. Aided Civ. Inf. 27, 314–332. Hu, S.-L.J., Bao, X.X., Li, H.J., 2010. Model order determination and noise removal for modal parameter estimation. Mech. Syst. Signal Process. 24, 1605–1620. Hu, S.-L.J., Bao, X.X., Li, H.J., 2012. Improved polyreference time domain method for modal identification using local or global noise removal techniques. Sci. China Phys. Mech. Astron. 55, 1464–1474. Hu, S.-L.J., Li, H.J., 2008. A systematic linear space approach on solving partially described inverse eigenvalue problems. Inverse Probl. 24, 035014. Hu, S.-L.J., Li, P., Vincent, H.T., Li, H.J., 2011. Modal parameter estimation for jackettype platforms using free-vibration data. J. Waterw. Port, Coast. Ocean Eng. 137, 234–245. Ibrahim, S.R., Mikulcik, E.C., 1973. A time domain modal vibration test technique. Shock Vib. Bull. 43, 21–37. Jiang, X., Mahadevan, S., Adeli, H., 2007. Bayesian wavelet packet denoising for structural system identification. Struct. Control Health Monit. 14, 333–356. Juang, J.N., 1994. Applied System Identification. PTR Prentice Hall, Englewood Cliffs, NJ, USA. Li, H.J., Li, P., Hu, S.-L.J., 2012. Modal parameter estimation for jacket-type platforms using noisy free-vibration data: sea test study. Appl. Ocean Res. 37, 45–53. Li, H.J., Wang, J.R., Hu, S.-L.J., 2008. Using incomplete modal data for damage detection in offshore jacket structures. Ocean Eng. 35, 1793–1799. Maia, N., Silva, J., He, J., Lieven, N., Lin, R.-M., Skingle, G., To, W., Urgueira, A., 1997. Theoretical and Experimental Modal Analysis. Research Studies Press, Taunton, Somerset, England. McConnell, K.G., Varoto, P.S., 2008. Vibration Testing: Theory and Practice, second ed. John Wiley & Sons, Inc., NJ, USA. Peeters, B., De Roeck, G., 2000. Reference based stochastic subspace identification in civil engineering. Inverse Probl. Eng. 8, 47–74. Rizzo, M., Betti, M., Spadaccini, O., Vignoli, A., 2018. Structural dynamic monitoring of VEGA-A platform: 30 years of data. In: 9th European Workshop on Structural Health Monitoring, July 10-13, Manchester, United Kingdom. Shokrgozar, H.R., Asgarian, B., 2018. System identification of steel jacket type offshore platforms using vibration test. In: 9th European Workshop on Structural Health Monitoring, July 10-13, Manchester, United Kingdom. Su, W.C., Huang, C.S., Chen, C.H., Liu, C.Y., Huang, H.C., Le, Q.T., 2014. Identifying the modal parameters of a structure from ambient vibration data via the stationary wavelet packet. Comput. Aided Civ. Inf. 29, 738–757. Van Overschee, P., De Moor, B., 1996. Subspace Identification for Linear System: Theory, Implementation, Application. Kluwer Academic Publishers, Dordrecht. Wang, P., Tian, X., Peng, T., Luo, Y., 2018. A review of the state-of-the-art developments in the field monitoring of offshore structures. Ocean Eng. 147, 148–164.

Declaration of competing interest The authors declared that they have no conflicts of interest to this work. Acknowledgements This research was supported by the National Key R&D Program of China (Grant No. 2016YFC0303800), the National Natural Science Foundation of China (Grant No. 51979283) and the Shandong Provin­ cial Natural Science Foundation, China (Grant No. ZR2018MEE053). References Allemang, R.J., Brown, D.L., 1998. A unified matrix polynomial approach to modal identification. J. Sound Vib. 211, 301–322. Amezquita-Sanchez, J.P., Adeli, H., 2016. Signal processing techniques for vibrationbased health monitoring of structures. Arch. Comput. Methods Eng. 23, 1–15. Bao, X.X., Li, C.L., Xiong, C.B., 2015. Signal denoising algorithm for modal analysis. Appl. Phys. Lett. 107, 041901. Bao, X.X., Shi, C., 2019. Ambient vibration responses denoising for operational modal analysis of a jacket-type offshore platform. Ocean Eng. 172, 9–21. Brandt, A., Berardengo, M., Manzoni, S., Cigada, A., 2017. Scaling of mode shapes from operational modal analysis using harmonic forces. J. Sound Vib. 407, 128–143. Braun, S., Ram, Y.M., 1987. Determination of structural modes via the Prony method: system order and noise induced poles. J. Acoust. Soc. Am. 81, 1447–1459. Brincker, R., Ventura, C., 2015. Introduction to Operational Modal Analysis. John Wiley and Sons Ltd, Hoboken, United States. Cadzow, J.A., 1988. Signal enhancement—a composite property mapping algorithm. IEEE Trans. Acoust. Speech Signal Process. 36, 49–62.

11