A New Bayesian Method for Dynamic System Identification Using FFT Data

Abstract

Dynamic system identification is an important field of research focused on identifying accurate system models of structures for predicting dynamic behaviors. This field finds widespread application in downstream research such as response prediction, structural failure and reliability analysis, and related areas of structural health monitoring. Conventional methods update structural finite element models (FEMs) using experimental modal parameters, because excitations are difficult to measure for full-scale structures and measured responses cannot be used as data in model updating. One challenge of conventional methods is thus that additional time is required for modal analysis, and by packing response data into modal parameters, original information in response data may be lost and cannot be used for model updating. Concerning this issue, this paper develops a dynamical system identification method to directly update an FEM using experimental fast Fourier transform (FFT) data following a Bayesian approach. The modeling of FFTs combing FEM and an efficient algorithm for processing the large amount of FFTs are not available for conventional methods. In this paper, the posterior probability density function (PDF) of the model parameters is derived assuming that FFTs at different frequency. instances follow independent and identically distributed complex Gaussian distributions under the long-data condition. One contribution of this work is that the sub-structure FEM analysis is integrated into the formulation of the posterior PDF to make the direct use of FFT data possible, and increase computational efficiency. The most probable values (MPVs) of the model parameters are obtained by maximizing the posterior PDF. By making use of the special mathematical structure of the posterior PDF, a novel algorithm that iterates among the model parameters is developed to efficiently search parameter space for the maximization. A numerical case has demonstrated that the proposed method can accurately identify the FEM of the target structure together with the unmeasured excitation parameters and damping ratios.

1 Introduction

Bayesian modal identification and model updating based on ambient vibration data have garnered increasing attention among engineers and researchers working in the field of structural health monitoring (SHM) [1,2,3]. Since there is a one-to-one correspondence between the time-domain data and its FFT, transforming responses into the frequency domain signals enables the full utilization of well-separated modal information [4]. Modeling FFT of response using modal parameters make it convenient to construct the covariance matrix of FFT [5], and thus its PDF [6, 7]. It is practical to assume that around a peak of the plot of response auto-power spectral densities (PSDs) only one structural mode is dominant [8]. In this situation, it is convenient to model FFTs around a spectral peak, and FFTs around different spectral peaks contain main dynamic information of a structure, so only the FFTs around the spectral peaks of interest are used as data for Bayesian inference in this work. The modal parameters calculated by a FEM are used to construct the PDF of FFTs. Model updating needs modal parameters calculated with different sets of FEM parameters, which means many times of finite element analysis. The excitation data is challenging to measure directly, for large structures with thousands of degrees of freedom (DOFs), this will consume a substantial amount of computational resources. To alleviate computational burden, the sub-structure method is employed in this study to reduce FEMs. This method has been widely used for dynamic analysis [9]. By integrating the sub-structure method into the formulas of Bayesian model updating [10,11,12], the computation time is reduced.

This paper proposes a new method to update dynamic system models of structures using FFTs of responses data without measuring excitations. This approach bypasses modal analysis, allowing for the direct update of the finite element model of a full-scale structure.

2 Methodology

2.1 Posterior PDF

The FFTs of measured acceleration data ${\hat{\mathbf{x}}}_{j} \in R^{n}$ at $n$ measurement channels are defined as:

$$ {\mathbf{\mathcal{F}}}_{k} = {\mathbf{F}}_{{\text{k}}} + {\mathbf{iG}}_{{\text{k}}} = \sqrt {\frac{2\Delta t}{N}} \mathop \sum \limits_{j = 1}^{N} {\hat{\mathbf{x}}}_{j} {\text{exp}}\left\{ { - 2{\uppi }{\mathbf{i}}\left[ {\left( {k - 1} \right)\left( {j - 1} \right)/N} \right]} \right\} $$

(1)

where ${\mathbf{F}}_{k}$ and ${\mathbf{G}}_{k}$ represent the real and imaginary parts of the FFTs, respectively; $\Delta t$ is the sampling time interval; $k = 2,3, \ldots ,N_{q}$ is the frequency index; $N_{q}$ is the FFT ordinate at the Nyquist frequency; $j = 1,2, \ldots ,N$ is the time index; $N$ is the total number of samples.

Let an augmented vector ${\mathbf{Z}}_{k} = \left[ {{\mathbf{F}}_{k}^{{\text{T}}} ,{\mathbf{G}}_{k}^{{\text{T}}} } \right] \in R^{2n} ,$ denotes the measured data. It has been proven that $\left\{ {{\mathbf{Z}}_{k} } \right\}$ asymptotically follows a zero-mean Gaussian distribution when the number of response data points is large enough, which is easy to fulfill in practice for measurement with a high sampling rate and long duration. This result can be used to formulate the posterior PDF of the uncertain model parameters θ conditional on the experimental FFTs. Assuming that the prior PDF is a uniform PDF, according to Bayes’ theorem, the posterior PDF $p\left( {{{\varvec{\uptheta}}}{|}\left\{ {{\mathbf{Z}}_{k} } \right\}} \right)$ is proportional to the likelihood function $p\left( {\left\{ {{\mathbf{Z}}_{k} } \right\}{|}{{\varvec{\uptheta}}}} \right)$:

$$ p\left( {{{\varvec{\uptheta}}}{|}\left\{ {{\mathbf{Z}}_{k} } \right\}} \right) \propto p\left( {\left\{ {{\mathbf{Z}}_{k} } \right\}{|}{{\varvec{\uptheta}}}} \right) $$

(2)

where the uncertain parameters ${{\varvec{\uptheta}}} = \left\{ {{{\varvec{\upalpha}}},\left\{ {S_{m} ,{\upzeta }_{m} :,m = 1,2, \cdots ,N_{m} } \right\},\sigma^{2} } \right\}$ include the stiffness parameters ${{\varvec{\upalpha}}}$ for FEM, PSD of modal excitation $S_{m}$ and damping ratio ${\upzeta }_{m}$ of the $m$-th mode for $N_{m}$ modes, and PSD of prediction error $\sigma^{2}$. Using the result that FFTs at different frequency instances follow independent and identically distributed complex Gaussian distributions, the likelihood function can be constructed as:

$$ \begin{gathered} p\left( {\left\{ {{\mathbf{Z}}_{k} } \right\}|{{\varvec{\uptheta}}}} \right) \hfill \\ = \mathop \prod \limits_{m = 1}^{{N_{m} }} \mathop \prod \limits_{k = 2}^{{N_{q,m} }} \left( {2\pi } \right)^{{ - \frac{{N_{q,m} - 1}}{2}}} {\text{det}}\left( {{\mathbf{C}}_{k,m} \left( {{\varvec{\uptheta}}} \right)} \right)^{\frac{1}{2}} {\text{exp}}\left( { - \frac{1}{2}\mathop \sum \limits_{k = 2}^{{N_{q,m} }} {\mathbf{Z}}_{k}^{T} {\mathbf{C}}_{k,m} \left( {{\varvec{\uptheta}}} \right)^{ - 1} {\mathbf{Z}}_{k} } \right) \hfill \\ \end{gathered} $$

(3)

where the covariance matrix can be constructed using the modal parameters calculated by FEM:

$$ \begin{aligned} {\mathbf{C}}_{k,m} \left( {{\varvec{\uptheta}}} \right) = \left( {\frac{{S_{m} D_{k,m} }}{2}} \right) & \left[ {\begin{array}{*{20}c} {{\mathbf{\varphi }}_{m} {\mathbf{\varphi }}_{m}^{{\text{T}}} } & 0 \\ 0 & {{\mathbf{\varphi }}_{m} {\mathbf{\varphi }}_{m}^{{\text{T}}} } \\ \end{array} } \right] + \left( {\frac{{\sigma^{2} }}{2}} \right){\mathbf{I}}_{2n} \\ = \frac{1}{2}\left[ {\begin{array}{*{20}c} {{\mathbf{E}}_{k,m} } & 0 \\ 0 & {{\mathbf{E}}_{k,m} } \\ \end{array} } \right] \\ \end{aligned} $$

(4)

where it has been assumed that around each spectral peak only one mode is dominant ${\mathbf{E}}_{k,m} = S_{m} D_{k,m} {\mathbf{\varphi }}_{m} {\mathbf{\varphi }}_{m}^{{\text{T}}} + \sigma^{2} {\mathbf{I}}_{n}$ represents the power spectral density (PSD) of the response data, $D_{k,m} = \left[ {\left( {\beta_{k,m}^{2} - 1} \right)^{2} + \left( {2\zeta_{m} \beta_{k,m} } \right)^{2} } \right]^{ - 1}$ is the dynamic amplification factor where $\beta_{k} = f_{m} /f_{k}$; $f_{m}$ and ${\mathbf{\varphi }}_{m}$ are the natural frequency and mode shape of the $m$-th mode calculated by FEM; $f_{k} = \left( {k - 1} \right)/\left( {N{\Delta }t} \right)$.

The most probable value of ${{\varvec{\uptheta}}}$ is obtained by maximizing the posterior PDF, which is equivalent to minimize the negative logarithmic likelihood function (NLLF) as follows:

$$ \begin{aligned} L\left( {{\varvec{\uptheta}}} \right) = - nN_{f} {\text{ln}}2 & + \left( {n - 1} \right)N_{f} {\text{ln}}\sigma^{2} + \mathop \sum \limits_{m = 1}^{{N_{m} }} \mathop \sum \limits_{k} {\text{ln}}\left( {S_{m} D_{k,m} + \sigma^{2} } \right) \\ & + \sigma^{ - 2} \left( {d - {\mathbf{\varphi }}_{m}^{{\text{T}}} {\mathbf{A}}_{m} {\mathbf{\varphi }}_{m} } \right) \\ \end{aligned} $$

(5)

where ${\mathbf{A}}_{m} = \mathop \sum \limits_{k} \left[ {1 + S_{m} /S_{m} D_{k,m} } \right]^{ - 1} {\mathbf{D}}_{k} ;{\mathbf{D}}_{k} = {\mathbf{F}}_{k} {\mathbf{F}}_{k}^{{\text{T}}} + {\mathbf{G}}_{k} {\mathbf{G}}_{k}^{{\text{T}}} .$

It is known from the above equation that the natural frequencies needed for $D_{k,m}$ and mode shapes can be obtained from FEM. A new iterative algorithm was developed. To facilitate the iteration among different model parameters, analytical formulations of $S_{m}$ and $\sigma^{2}$ were derived with the assumption of high signal-to-noise ratio for asymptotic approximation. Given each stiffness parameter ${{\varvec{\upalpha}}}$, minimization is done separately for $S_{m}$ and $\sigma^{2}$ with their analytical formulations as the initial values. Minimization is also done for damping ratios. Iteration is conducted among $S_{m}$, $\sigma^{2}$ and $\zeta_{m}$ until convergence. We minimize the NLLF until the difference between two iteration steps $L\left( {{{\varvec{\uptheta}}}_{i} } \right)$ and $L\left( {{{\varvec{\uptheta}}}_{i - 1} } \right)$ is small enough.

2.2 Sub-Structure Finite Element Analysis

Repeated finite element analysis is needed for minimizing the NLLF. This could be computationally intensive if a full FEM is used. We propose to reduce the order of FEM with the sub-structure method. With an FEM divided into s sub-structures, the mass matrix ${\mathbf{M}}^{s}$ and stiffness matrix ${\mathbf{K}}^{s}$ for sub-structure s are partitioned as

$$ {\mathbf{M}}^{s} = \left[ {\begin{array}{*{20}c} {{\mathbf{M}}_{ii}^{s} } & {{\mathbf{M}}_{ib}^{s} } \\ {{\mathbf{M}}_{bi}^{s} } & {{\mathbf{M}}_{bb}^{s} } \\ \end{array} } \right] \in R^{{n^{s} \times n^{s} }} ,{\mathbf{K}}^{s} = \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{ii}^{s} } & {{\mathbf{K}}_{ib}^{s} } \\ {{\mathbf{K}}_{bi}^{s} } & {{\mathbf{K}}_{bb}^{s} } \\ \end{array} } \right] \in R^{{n^{s} \times n^{s} }} ,s = 1,{ } \ldots { },N_{s} $$

(6)

where $n^{s}$ represents the number of DOFs of sub-structure s; $N_{s}$ denotes the number of sub-structures; the subscripts i and b represents internal and boundary DOFs, respectively. The coordinate set for internal and boundary DOFs is denoted as ${\mathbf{u}}_{i}^{s} \left( t \right) \in R^{{n_{i}^{s} }}$ and ${\mathbf{u}}_{b}^{s} \left( t \right) \in R^{{n_{b}^{s} }}$, respectively. Consequently, the physical coordinates of sub-structure s are expressed as ${\mathbf{u}}^{s} \left( t \right) = {\mathbf{u}}_{i}^{s} \left( t \right) + {\mathbf{u}}_{b}^{s} \left( t \right) \in R^{{n^{s} \times 1}}$, where $n^{s} = n_{i}^{s} + n_{b}^{s}$.

According to the Craig–Bampton fixed-interface mode method, the Ritz coordinate transformation matrix can be used for reduction and is expressed as:[11]

$$ {{\varvec{\Psi}}}^{s} = \left[ {\begin{array}{*{20}c} {{{\varvec{\Phi}}}_{ik}^{s} } & {{{\varvec{\Psi}}}_{ib}^{s} } \\ {0_{bk}^{s} } & {{\mathbf{I}}_{bb}^{s} } \\ \end{array} } \right] \in R^{{n^{s} \times \hat{n}^{s} }} $$

(7)

where $\hat{n}^{s} = n_{ik}^{s} + n_{b}^{s}$ and ${{\varvec{\Phi}}}_{ik}^{s} \in R^{{n^{s} \times n_{ik}^{s} }}$ represents the low-order dominant mode shapes ($n_{ik}^{s} < n_{i}^{s}$) of the internal DOFs in each sub-structure. ${{\varvec{\Psi}}}^{s}$ that is comprised of the k-th mode coordinates of internal DOFs and all physical coordinates at interface DOFs is used to convert the physical coordinates into the generalized coordinates. The mass and stiffness matrix of the reduced sub-structure s are represented as

$$ {\hat{\mathbf{M}}}^{s} = {{\varvec{\Psi}}}^{{s}{\text{T}}} {\mathbf{M}}^{s} {{\varvec{\Psi}}}^{s} \in R^{{\hat{n}^{s} \times \hat{n}^{s} }} ,{\hat{\mathbf{K}}}^{s} = {{\varvec{\Psi}}}^{{s}{\text{T}}} {\mathbf{K}}^{s} {{\varvec{\Psi}}}^{s} \in R^{{\hat{n}^{s} \times \hat{n}^{s} }} $$

(8)

Finally, the mass matrix and stiffness matrix of the reduced model are assembled through all reduced sub-structure, denoted as ${\hat{\mathbf{M}}}$ and ${\hat{\mathbf{K}}}$, respectively.

2.3 Parameterization of Model Updating Based on Sub-Structure Finite Element Analysis

The Craig-Bampton transformation can be done separately [11]. Therefore, the parameterization of the reduced-order FEM can be achieved conveniently using the reduced-order matrices of sub-structures:

$$ {\hat{\mathbf{K}}}\left( {\mathbf{x}} \right) = {\hat{\mathbf{K}}}^{0} + \mathop \sum \limits_{s = 1}^{{N_{s} - 1}} {\hat{\mathbf{K}}}^{s} {\text{x}}_{s} $$

(9)

where the superscript ‘^’ denotes the reduced-order stiffness matrix of the FEM. Consequently, the frequencies and mode shapes can be computed by solving the eigenvalue problem:

$$ \left( {{\hat{\mathbf{K}}}\left( {\mathbf{x}} \right) - \omega_{n} \left( {\mathbf{x}} \right)^{2} {\hat{\mathbf{M}}}} \right){\mathbf{\varphi }}_{n} \left( {\mathbf{x}} \right) = 0 $$

(10)

where $\omega_{n} \left( {\mathbf{x}} \right)$ and ${\mathbf{\varphi }}_{n} \left( {\mathbf{x}} \right)$ represent $nth$ modal frequency and mode shape, respectively. By continuously updating the stiffness coefficients ${\text{x}}_{s}$ of the sub-structures, the natural frequencies and mode shapes can be updated.

3 Case Study

A simulated 12-story building structure shown in Fig. 1 is used to validate the proposed method. Each two stories (eight columns and two floors) were considered as one sub-structure, so there are six sub-structures in this case. Accelerations were measured along the x direction at 6 stories, i.e., story 2, 4, 6, …, 12. Table 1 lists the basic information of the structure.

Fig. 1

The 12-story shear building

Table 1 The basic information of the structure

Accelerations of the structure were simulated with the PSD of modal excitation $0.25^{2} {\text{N}}^{2} /{\text{Hz}}$ and the PSD of prediction error $10^{ - 10} \left( {{\text{m}}/s^{2} } \right)^{2} /{\text{Hz}}$. The sampling frequency is 200 Hz and of the length of data is 300 s. The singular value (SV) spectrum is depicted in Fig. 2. The FFT data around the four peaks marked in the graph were used for updating the FEM.

Fig. 2

The SV spectrum of response data

The MPVs of the model parameters were identified. The MPVs of the stiffness parameters were input the FEM to calculate the natural frequencies and mode shapes. The calculated mode shapes are compared with the exact ones in Fig. 3. The errors between the calculated and exact natural frequencies are also shown above each sub-plot. Figure 3a shows the comparison for the four modes whose FFT data were included in model updating, i.e., the four selected spectral peaks correspond to these four modes. It is seen that the calculated and exact modal parameters match well. Figure 3b and c show the comparison for the modes whose FFT data were not included for model updating. The calculated modal parameters by the updated FEM can still well match with the exact ones. These results verify the feasibility of the proposed method.

Fig. 3

The comparison of the calculated and exact modal parameters

The successful validation of this method provides an efficient and accurate approach for dynamic system identification of full-scale structures. It yields a model that accurately predicts the dynamic behaviors of the structure under different conditions, while the identified results also provide reasonably accurate modal excitations. The proposed method can be used to continuously update the structural model when monitoring data are continuously measured, and the continuously updated model can be used to assess the structural performance in a timely fashion. This evaluation is crucial for ensuring the safety and stability of structures, and it plays an important role in SHM by enhancing the accuracy of analysis and design.

4 Conclusions

A new Bayesian method for dynamic system identification has been developed. The proposed method can use the original FFTs of measured responses to update the FEM of a target structure and statistical properties of modal excitations without measuring excitations. System identification is achieved by identifying the posterior PDF of uncertain model parameters conditional experimental FFT data. The posterior PDF is constructed using the fact that FFT at each frequency instance asymptotically follows a complex Gaussian distribution. Sub-structural finite element analysis is integrated into the formulation of the posterior PDF. By utilizing the Craig-Bampton transformation matrix for reducing the full FEM, convenient parameterization can be done. The sub-structure method also eliminates the need for re-analysis of fixed-interface orthogonal modes and interface constraint modes. A new iteration algorithm has been developed to efficiently search one local area of the parameter space at a time. The proposed method was verified numerically by 12-story building structure. The updated FEM could accurately predict the modal parameters, even for the ones whose experimental information was not included for model updating.