Keywords

4.1 Introduction

In Operational Modal Analysis (OMA) of large civil structures, it is usually possible to identify consistent modal parameters in terms of natural frequencies and mode shapes but when it comes to damping of the structure, it appears to be much more difficult to quantify consistent estimates. If we consider an offshore wind turbine (OWT), the structure is exposed to a number of different operational conditions in terms of climatic changes, scour depth, the active system of the OWT itself and many other parameters. The variations in operational conditions influence the dynamic behavior of the structure and this is potentially the reason why it is so difficult to obtain stable damping estimates. Some of these challenges are described in [1, 2].

Today, a variety of modal parameter estimation (MPE) methods are available and while they all are capable of identifying and extract modal parameters of a given system, the results turn out to be slightly different. It is possible to establish a general formulation for many of the MPE methods based on either the Frequency Response Function (FRF) or Impulse Response Function (IRF) which makes it more intuitive to grasp the difference in modal parameters each method computes. A unified formulation for modal identification is presented in [3] which makes it possible to compare the most commonly used MPE methods.

The motivation for this paper comes from the experience that it is very difficult to obtain reliable damping estimates. If we include multiple MPE methods and investigate their characteristic equations, we expect that a confidence interval can be defined which bounds the errors, including both random and bias errors. In this paper, the pragmatic uncertainty bounds are defined by studying the modal parameters obtained by using the Mulitple-reference Ibrahim Time Domain (MITD) method and the Modified Multiple-reference Ibrahim Time Domain method. Among other MPE methods, the MITD and MMITD method are presented in [4].

In [5] and [6], numerical methods for calculating the uncertainty on the modal parameters have been presented with focus on simulated and measured vibration data.

4.2 Theory

The FRF matrix of a system, [H], can be described as the inverse of the system impedance matrix, [Z] which can be obtained by applying the Laplace transformation of Newton’s equation based on physical properties including the mass, [M], stiffness, [K], and damping matrices, [C]. For large systems, such as most civil structures, we rarely know the damping matrix and extraction of modal parameters based on the entire modal matrices is inefficient in terms of computation. Instead, the Laplace transformation of Newton’s equation can be rewritten using the mode shape matrix, [ Ψ], and pole matrix, [S]

$$\displaystyle \begin{aligned}{}[H(f)]= [\Psi][S^{-1}][\Psi^T] \end{aligned} $$
(4.1)

In time domain, this can be written as

$$\displaystyle \begin{aligned}{}[h(t)] = [\Psi]\lceil e^{s_rt} \rfloor[L]^T \end{aligned} $$
(4.2)

where Ψ is the eigenvector matrix, \( \lceil e ^{s_rt}\rfloor \) is the diagonal pole matrix and L is the modal participation factor (MPF) matrix.

4.2.1 Multiple-Reference Ibrahim Time Domain (MITD) Method

The MITD method is an extension of the Ibrahim time domain (ITD) method presented in [7]. The MITD method allows for multiple references to be included in the parameter estimation and is applicable for Operational Modal Analysis (OMA) as it works on free decay measurements and thus correlation functions.

The common equation given in Eq. (4.2) is also valid for the MITD method. Next step is to repeat this equation at different times, a total of m times in row direction and n times in column direction

$$\displaystyle \begin{aligned}{}[H_{mn}(t)]=\left[ \begin{array}{cccc} {[h(t)]} & {[h(t+\Delta t) ]} & \cdots & {[h(t + (n-1) \Delta t) ]} \\ {[h(t+\Delta t)]} & {[h(t+2\Delta t) ]} & \cdots & {[h(t + n\Delta t)]} \\ \vdots & \vdots & \cdots & \vdots \\ {[ h(t+(m-1)\Delta t)} ] & {[h(t+n\Delta t)]} & \ldots & {[ h(t + (m+n-2) \Delta t)]} \end{array} \right] {} \end{aligned} $$
(4.3)

This equation is known as the block Hankel matrix (referred to as the Hankel matrix). By using the Hankel matrix, Eq. (4.2) can be expanded into

$$\displaystyle \begin{aligned}{}[H_{mn}(t)]=[\tilde{\Psi}]\lceil e^{s_rt}\rfloor [\tilde{L}]^T \end{aligned} $$
(4.4)

The expanded mode shapes are given as

$$\displaystyle \begin{aligned} \left[\tilde{\Psi}\right]=\left[ \begin{array}{c} {[\Psi]} \\ {[\Psi]}\lceil e^{s_r\Delta t}\rfloor \\ \vdots\\ {[\Psi]}\lceil e^{s_r(n-1)\Delta t}\rfloor \end{array} \right] \end{aligned} $$
(4.5)

and the extended MPF matrix

$$\displaystyle \begin{aligned} \left[\tilde{L}\right]=\left[ \begin{array}{cccc} {[L]^T} & {\lceil e^{s_r\Delta t}\rfloor[L]^T} & \cdots & {\lceil e^{s_r(n-1)\Delta t}\rfloor[L]^T} \end{array} \right] \end{aligned} $$
(4.6)

It is now possible to rewrite Eq. (4.4) by post multiplying with the pseudo-inverse of the expanded mode shape matrix \( \left [\tilde {\Psi }\right ] \)

$$\displaystyle \begin{aligned}{}[\tilde{\Psi}]^+[H_{mn}(t)]=\lceil e^{s_rt}\rfloor [\tilde{L}]^T \end{aligned} $$
(4.7)

If a new Hankel matrix is defined, shifted Δt in time, the diagonal pole matrix becomes \( \lceil e^{s_r(t+\Delta t)}\rfloor \) which can be rewritten to \( \lceil e^{s_rt}\rfloor \lceil e^{s_r\Delta t}\rfloor \). Using Eq. (4.7) with the time shifted Hankel matrix, we get

$$\displaystyle \begin{aligned}{}[H_{mn}(t+\Delta t)] =[\tilde{\Psi}]\lceil e^{s_r\Delta t}\rfloor \tilde[{\Psi}]^+ [H_{mn}(t)] \end{aligned} $$
(4.8)

Defining a system matrix \( [A] = [\tilde {\Psi }]\lceil e^{s_r\Delta t}\rfloor \tilde {\Psi }]^+ \) results in

$$\displaystyle \begin{aligned}{}[A] [H_{mn}(t)] = [H_{mn}(t+\Delta t)]. {} \end{aligned} $$
(4.9)

The system matrix can be calculated as

$$\displaystyle \begin{aligned} &[A][H_{mn}(t)] = [H_{mn}(t+\Delta t)] \implies [A][H_{mn}(t)][H_{mn}(t)]^T = [H_{mn}(t+\Delta t) ][H_{mn}(t)]^T\\ &\implies [A_1] = [H_{mn}(t+\Delta t)][H_{mn}(t)]^T([H_{mn}(t)][H_{mn}(t)]^T)^{-1} \end{aligned} $$
(4.10)

From Equation (4.8) and (4.10) an eigenvalue problem for each column in \(\left [ \tilde {\Psi }\right ]\) can be established which needs to be solved in order to obtain the modal parameters

$$\displaystyle \begin{aligned}{}[A] \left[ \tilde{\Psi} \right]= \left[ \tilde{\Psi} \right] \lceil e^{s_r \Delta t} \rfloor {} \end{aligned} $$
(4.11)

Here, λ r are the eigenvalues and correspond to \(e^{s_r \Delta t}\). The poles of the system, s r can now be calculated as \(s_r = f_s \ln (\lambda _r)\).

4.2.2 Modified Multiple-Reference Ibrahim Time Domain (MMITD) Method

For the MMITD method, the transposed Hankel matrix at time t is required

$$\displaystyle \begin{aligned} \left[ H_{mn} (t) \right]^T = \left[ \tilde{L} \right] \lceil e^{s_r t} \rfloor \left[ \tilde{\psi} \right]^T {} \end{aligned} $$
(4.12)

The time shifted Hankel matrix then becomes

$$\displaystyle \begin{aligned} \left[ H_{mn} (t + \Delta t) \right]^T = \left[ \tilde{L} \right] \lceil e^{s_r \Delta t} \lceil e^{s_r t} \rfloor \left[ \tilde{\psi} \right]^T {} \end{aligned} $$
(4.13)

Similar to Equation (4.7), we get

$$\displaystyle \begin{aligned} \lceil e^{s_r t} \rfloor \left[ \tilde{\psi} \right]^T = \left[ \tilde{L} \right]^+ \left[ H_{mn} (t) \right]^T {} \end{aligned} $$
(4.14)

Inserting this into Eq. (4.13), the equation becomes

$$\displaystyle \begin{aligned} \left[ H_{mn} (t + \Delta t) \right]^T = \left[ \tilde{L} \right] \lceil e^{s_r \Delta t} \rfloor \left[ \tilde{L} \right]^+ \left[ H_{mn} (t) \right]^T {} \end{aligned} $$
(4.15)

Post multiplication of \(\left [ H_{mn} (t) \right ]\) into Eq. (4.15) gives

$$\displaystyle \begin{aligned} \left[ H_{mn} (t + \Delta t) \right]^T \left[ H_{mn} (t) \right] = \left[ \tilde{L} \right] \lceil e^{s_r \Delta t} \rfloor \left[ \tilde{L} \right]^+ \left[ H_{mn} (t) \right]^T \left[ H_{mn} (t) \right] {} \end{aligned} $$
(4.16)

A system matrix is defined as

$$\displaystyle \begin{aligned} \left[ A \right] = \left(\left[ H_{mn} (t + \Delta t) \right]^T \left[ H_{mn} (t) \right] \right) \left( \left[ H_{mn} (t) \right]^T \left[ H_{mn} (t) \right] \right) ^{-1} {} \end{aligned} $$
(4.17)

Finally, in order to obtain the eigenvalue problem, post multiplying Eq. (4.16) with \(\left (\left [ H_{mn} (t) \right ]^T \left [ H_{mn} (t) \right ] \right )^{-1}\) and \(\left [ \tilde {L} \right ]\) is needed

$$\displaystyle \begin{aligned} \left[ A \right] \left[ \tilde{L} \right] = \left[ \tilde{L} \right] \lceil e^{s_r \Delta t} \rfloor {} \end{aligned} $$
(4.18)

It is important to note that the MMITD method gives us the modal participation factors ([L]) and not directly the mode shapes, in contrary to the MITD method. The mode shapes can be estimated in a second step, by using the least squares time domain method [4].

Relevant for both MITD and MMITD is data compression. In this paper, Singular Value Decomposition (SVD) is applied which condenses the frequency response matrix so that the resulting matrix contains the same information as the original.

When working with MPE methods, a number of decisions have to be made on how to solve the system equation for the modal parameter estimation. Some of the decisions are described below.

4.2.3 Low Order Versus High Order Method

Instead of using a physically-based mathematical model to describe the various MPE methods, it is also possible to consider a matrix coefficient polynomial model where the common characteristics can be more intuitively identified. By considering a particular response point p and reference point q, Eq. (4.2) can be written as

$$\displaystyle \begin{aligned} h_{pq}(t_i) = \frac{u_p(t_i)}{f_q(t_i)}=\frac{\beta_n(z)^n+\beta_{n-1}(z)^{n-1}+\cdots+\beta_0(z)^0}{\alpha_m(z)^m+\alpha_{m-1}(z)^{m-1}+\cdots+\alpha_0(z)^0} \end{aligned} $$
(4.19)

Where z = e s Δt and in this case, t i denotes an arbitrary time – this can be seen as the measured time. By collecting the terms in Eq. (4.19), the polynomial can be written as

$$\displaystyle \begin{aligned} h_{pq}(t_i) =\dfrac{u_p(\omega_i)}{f_q(\omega_i)} = \dfrac{\sum_{l=0}^{n}\beta_l(z)^l}{\sum_{k=0}^{m}\alpha_k(z)^k} \end{aligned} $$
(4.20)

If the MPE method considered is based on the original FRF or IRF, the polynomial order describing the denominator (α k) in Eq. (4.20) will in most cases be low (one or two) and the MPE method will be described as a low order method, accordingly. The resulting dimensions of α k is [N SxN S] and β k is [N LxN L]. Here, S denotes the short dimension which refers to the number of references (inputs) and L denotes the long dimension which refers to the number of responses (outputs). If the MPE method is based on the transposed FRF or IRF, instead, the dimensions of α k and β k are reversed and thus the polynomial order of the denominator (that is α k) will become higher. The method is now defined as a high order method.

Equation (4.2) directly shows that the MITD method is a low order method since it is built on the original FRF whereas the MMITD method is built on the transposed FRF seen from Eq. (4.12) making it a high order method.

4.2.4 Normalization to the Lowest or Highest Coefficient of the Matrix Polynomials

The roots of the matrix coefficients can be determined by solving for the eigenvalues of the companion matrix. This requires normalization of either the lowest order coefficient (α 0 = 1) or the highest order coefficient (α(m) = 1). It can be worth to consider the influence of the normalization since it may influence the results to some extent. A study comparing low order and high order normalization has been carried out in [8] which, however, generally showed an insignificant difference in the estimated modal parameters.

4.2.5 Calculating the System Matrix with Respect to Either the Original Hankel Matrix or the Time Shifted Hankel Matrix

If we consider Eq. (4.10), the system matrix [A] can also be calculated using the time shifted Hankel matrix ([H mn(t +  Δt)]). If this is the case, [A] will become

$$\displaystyle \begin{aligned}{}[A_2]= [H_{mn}(t+\Delta t) ] [H_{mn}(t+\Delta t) ]^T([H_{mn}(t)] [H_{mn}(t+\Delta t) ]^T)^{-1} \end{aligned} $$
(4.21)

It turns out, that the resulting modal parameters change slightly depending on which Hankel matrix is used for calculating the system matrix. According to [9], it is good practice to estimate the modal parameters by considering both cases and average the extracted modal parameters.

4.3 Methodology

The modal parameters are extracted from simulated acceleration time data based on an FE-model representing an OWT with 9 references located evenly spaced along the structure. In the simulation, only the first 4 non-symmetrical bending modes are included and the damping of each mode has been set to 1%. In Table 4.1, the settings for simulation of time data are listed. The natural frequencies of the first 4 non-symmetrical bending modes are 0.309, 1.197, 2.006 and 4.300 Hz, respectively.

Table 4.1 Settings for simulation of acceleration time data
Full size table

For most MPE methods, it is necessary to determine the optimal model order which yields better results in terms of modal parameters. Here, it is very popular to utilize the so-called stabilization diagram which visualizes the calculated poles as a function of increasing model order. In order to select poles that represent the actual natural modes of the system, certain engineering skills and judgment is required. An algorithm for automated operational modal analysis (AOMA) is presented in [10] that successfully has estimated modal parameters from different data sets including experimental measurements from a bridge test. The algorithm utilizes a statistical representation of modal parameters and is complemented by a number of decision rules based on the modal assurance criterion. The estimated modal parameters are only dependent on a few specified tolerances which reduces any bias that may have been introduced by the user. A similar approach is carried out in this paper to estimate the modal parameters of the simulated data.

In this paper, a low order and a high order method are considered which refers to MITD and MMITD, respectively, but the coefficient matrices of the system are only normalized to the highest order. Both cases of either using the first Hankel matrix or the time shifted Hankel matrix are also taken into account. In order to truly investigate the outcome of these considerations, it has been important to ensure that as many input parameters to the MPE method as possible remain unchanged. One of the more important input parameters in the MPE methods, is the first time lag and the total number of time lags considered in the correlation functions. Since the correlation function of the noise (typically broadband noise) dies out fast, the first few time lag values should be skipped – in this case, the first 10 were skipped. The total number of time lags is fixed but for the modal parameter estimation, the start value will vary for each computation.

In Table 4.2, the input parameters for the estimation of modal parameters are given.

Table 4.2 Input parameters for modal parameter estimation
Full size table

4.4 Results and Discussion

During modal parameter estimation, a total of 4 separate calculations are carried out. Results of the first two are visualized in Fig. 4.1a and b which are based on the MITD method with changes to the system matrix [A] as to how it is calculated – that is either [A 1] from Eq. (4.10) or [A 2] from Eq. (4.21). The results of the last two are visualized in Fig. 4.1c and d which are based on the MMITD method, similarly with changes to the system matrix. The Mode Indicator Function (MIF) overlay shown in the figures, is an average of the Power Spectral Density using Welch method with a 50% overlap Hanning window for all output channels. From the figures, it is very difficult to tell any difference between the MITD method and MMITD method whereas the largest difference can be seen when switching between [A 1] and [A 2]. Nevertheless, the estimates for all 4 calculations are very similar.

Fig. 4.1
figure 1

Histograms of calculated poles. ([-,red,line width = 1.0pt](0,0) – (5mm,0);) shows the MIF overlay. ([-,black,line width = 3.0pt](0,0) – (5mm,0);) shows the bins of the histogram and ([-,black,dashed,line width = 1.0pt](0,0) – (5mm,0);) indicates the minimum number of estimates required in a bin to be considered in the further analysis. (a) MITD and [A 1]. (b) MITD and [A 2]. (c) MMITD and [A 1]. (d) MMITD and [A 2]

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Estimates for all 4 calculations are collected into one set of parameters for each mode which has been identified as one of the FE modes. In order to ensure that each mode from all calculations have similar modal properties, a MAC criterion of 0.99 is applied. In Fig. 4.2, histograms of the collected damping estimates for the 4 modes (which match the FE model) are shown together with the 5%-fractile and 95%-fractile (90% confidence interval).

Fig. 4.2
figure 2

Histograms of damping estimates from identified modes. ([-,green,dashed,line width = 1.0pt](0,0) – (5mm,0);) indicates the lower boundary (5%-fractile) and ([-,red,dashed,line width = 1.0pt](0,0) – (5mm,0);) indicates the higher boundary (95%-fractile). (a) Mode 1 at 0.308 Hz. (b) Mode 2 at 1.197 Hz. (c) Mode 3 at 2.005 Hz. (d) Mode 4 at 4.307 Hz

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From the plots in Fig. 4.2a–d, there is no apparent distribution that fits them all. This is somehow unexpected since many of the studies on uncertainty bounds of modal parameters assume that the distribution of measured or simulated data is Gaussian. Mode 3 comes closest to a Gaussian distribution but the remainder seems far from – especially mode 2 and 4. For the first 3 modes, the majority of damping estimates are within a small range and the target damping of 1% lies within the 90% confidence interval. The 1% target damping is also within the boundaries for mode 4 but based on the lower boundary (5%-fractile), the damping is generally estimated lower than expected. It is also evident to see from the histograms in Fig. 4.1 that the number of damping estimates within one bin for the fourth mode is much lower than the number of estimates for the other modes. The fourth mode also has multiple bins very close in frequency with a significant number of estimates whereas the estimates lie in a single bin for each of the first 3 modes. This is possibly a result of a small bin width which is based on the first frequency, in fact, the frequency of the first mode divided by 50 (0.006 Hz). In Table 4.3, the 90% confidence interval is listed for all 4 modes. The total number of estimates for each mode is also listed and here, this number also tells that there are significantly fewer estimates of the fourth mode compared to the rest. The last column of the table displays the diagonal cross MAC value between the mode shape extracted from the simulated data and the mode shape derived from the FE model. In Fig. 4.3, the entire cross MAC matrix is visualized.

Fig. 4.3
figure 3

Cross MAC between mode shapes from simulated data and mode shapes from the FE model

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Table 4.3 Results from modal parameter extraction including uncertainty bounds
Full size table

In this case, the diagonal values are of most interest, since the MAC plot shows that the natural modes extracted from the simulated data follow the order of the FE modes. The diagonal cross MAC values also show, that there is almost no difference in the mode shapes extracted from the simulated data and the modes from the FE model.

4.5 Conclusions

This paper presented pragmatic uncertainty bounds for modal parameters obtained from simulated data in terms of a statistical confidence interval. In the first step, acceleration time data were simulated based on an FE model which represents an offshore wind turbine. In the second step, modal parameters of the system were extracted using the Multiple-reference Ibrahim Time Domain method and Modified Multiple-reference Ibrahim Time Domain method and finally, the results of both methods were compared. A statistical representation in terms of histograms showed that the distribution of the damping estimates is not Gaussian. It was shown that a 90% confidence interval bounded the true damping for all modes.