Improved Bayesian Model Updating Method for Frequency Response Function with Metrics Utilizing NHBFT-PCA

Abstract

To establish a high-fidelity model of engineering structures, this paper introduces an improved Bayesian model updating method for stochastic dynamic models based on frequency response functions (FRFs). A novel validation metric is proposed first within the Bayesian theory by using the normalized half-power bandwidth frequency transformation (NHBFT) and the principal component analysis (PCA) method to process the analytical and experimental frequency response functions. Subsequently, traditional Bayesian and approximate Bayesian computation (ABC) are improved by integrating NHBFT-PCA metrics for different application scenarios. The efficacy of the improved Bayesian model updating method is demonstrated through a numerical case involving a three-degrees-of-freedom system and the experimental case of a bolted joint lap plate structure. Comparative analysis shows that the improved method outperforms conventional methods. The efforts of this study provide an effective and efficient updating method for dynamic model updating based on the FRFs, addressing some of the existing challenges associated with FRF-based model updating.

1. Introduction

Advancements in computational simulation technologies, such as finite element methods (FEM), have promoted the progress of computer-aided engineering (CAE). In the process of engineering modeling and simulation (M&S), the credibility of the numerical simulation results depends largely on the quality of the model structure, the precision of the numerical discretization, and more importantly, the reasonableness of the parameter values. With increasing reliability requirements, uncertainties such as material variability, form and location tolerances, and ambiguous boundaries during structural design need to be considered.

In response to the issues above, several probability-based methods have emerged in the domain of stochastic model updating to align the predictions with the experimental results [1,2,3,4,5]. It is worth mentioning that Bayesian theory is widely used to solve stochastic model updating problems due to its advantages in regards to statistical inference and parameter estimation [6]. Within the Bayesian framework, Mottershead et al. [7] introduced two prominent stochastic model updating techniques: sensitivity-based updating and Bayesian model updating. Ni et al. [8] proposed a novel likelihood-free Bayesian inference method for structural parameter identification. Fang et al. [9] employed a stochastic response surface model to obtain parameter variability at a reduced computational expense. Mondal et al. [10] used reduced dimensional parameterization in the Bayesian approach to obtain truncated posteriors. On the other hand, in the context of limited, insufficient, and ambiguous data, the construction of the necessary prior estimation of the joint probability density function of the non-deterministic parameter values is subjective. Thus, non-probabilistic methods, such as interval probability [11,12,13,14] and evidence theory [15], have garnered considerable interest in recent years. In non-probabilistic methods, parameters are assumed not as probability distributions, but as intervals, neglecting the possible dependence between the output parameters. As such, they are not capable of quantifying any dependence between the interval-uncertain parameters of the numerical model, modeled as an interval field.

Regardless of the methodology employed for model updating, appropriate comparison metrics must be established to provide a quantitative measure of the agreement between numerical predictions and experimental measurements [16,17,18]. Different metrics may be required for different engineering applications. The most typical output features in dynamic model updating are the modal quantities, such as intrinsic frequencies and vibration shapes. The modal assurance criterion (MAC) has traditionally been utilized to compare finite element analysis (FEA) results with experimental findings [19]. However, metrics based on modal feature quantities often fail to accurately assess the consistency of the entire dynamical system across its full range and may introduce errors during modal feature identification [20]. The FRF contains more information regarding structural dynamic characteristics than for modal quantities. Therefore, numerous researchers have attempted to measure the consistency of the frequency response function (FRF) rather than the modal frequencies and shapes during the model updating process [21,22,23,24].

The conventional approach involves comparing one FRF with another at each individual spectral line, with the sum of squared differences (SSD) being the most commonly used metric. As an alternative, the frequency domain assurance criterion (FDAC) was introduced to determine the linear-frequency shift between two FRFs [25]. Lee et al. [26] introduced a metric termed FRFSM, scaling the difference between two decibel-scaled FRFs using a normal probability density function (PDF). Manring et al. [27] highlighted the discrepancy between the point-to-point comparison of FRFs and an engineer’s visual inspection of FRFs, addressing this issue by providing a method for slicing and log frequency shifting (LFS) of a comparison FRF. However, when considering the uncertainties in both the experiments and simulations, the metrics used for comparing FRFs mentioned above may become ineffective.

Hegde [28] established a linear transmit relationship between the parameters and the response by segregating the amplitude and phase of the FRF in a single-degree-of-freedom (SDOF) system, employing maximum likelihood estimation methods to infer parameter mean and variance. Arora et al. [29] emphasized the difference between the mean FRF and FRF constructed from mean parameters to identify the stiffness and damping matrix of the real and imaginary parts of the frequency response data, respectively. Zhang et al. [30] proposed a generalized model updating method using the FRF to separate and reduce the model-form and parameter uncertainty, but the standard deviations of the parameters were overestimated. Wang et al. [31] proposed a model updating method based on the feature map of the FRF using a Bayesian convolutional neural network (BCNN), which eliminates the steps of modal identification and modal matching and can simultaneously adjust the structural parameters and damping parameters of the model.

Despite the various methodologies proposed, several obstacles still exist in regards to FRF-based stochastic model updating, including the following:

• Transmission between parameters and FRFs near resonance peaks is complex, and the distribution forms of the FRFs and parameters can differ significantly. Consequently, previous stochastic model updating research has often circumvented the data near the peaks [32].
• The sensitivity of the amplitude difference between analytical and experimental FRFs varies greatly across frequencies, making model updating performance highly dependent upon frequency points selection.
• FRF information may constitute a high-dimensional feature matrix. High-dimensional and highly correlated data will lead to singularity in the covariance matrix, posing challenges for model updating using Bayesian methods. Hence, it is necessary to find an effective metrics to measure the agreement between the experimental FRFs data and the analytical FRFs data in the high-dimensional, strongly nonlinear region near the resonance points.

To address the challenges mentioned above, this paper proposes an improved Bayesian model updating method for FRFs. The improved Bayesian model updating method is integrated with novel validation metrics, based on normalized half-power bandwidth frequency transformation (NHBFT). This metric addresses the issue of modal data mismatch in model updating using FRF data, and it enables the transformed FRF data to conform to the assumption of normal distribution. The metric enables the effective quantification of disagreement between experimental and simulation data. Additionally, we introduce the principal component analysis (PCA) method for data dimensionality reduction, mitigating data correlation and redundancy in cases of excessive data points in FRFs. The NHBFT-PCA metric fully utilizes the information embedded in FRFs. The proposed metrics can be well combined with Bayesian theory, including traditional Bayesian and approximate Bayesian computations. By employing the metric, the complex transmission between the parameters and the frequency response function is described linearly, thereby making the model updating results more acceptable.

The remainder of this article is organized as follows. Section 2 describes the fundamental theory of the improved Bayesian model updating method for FRFs with metrics utilizing NHBFT-PCA. Section 3 employs a three-degrees-of-freedom mass-spring system as a numerical study to validate the proposed method. In Section 4, an experimental case is used to illustrate the applicability and advantages of the proposed methodology. It integrates experimental data to identify the structural uncertainty parameters and compare the results with those obtained using conventional methods. Section 5 summarizes the conclusions drawn from the study.

2. Theory and Methodology

2.1. Bayesian Model Updating

In this section, the probability-based Bayesian model updating method is introduced for parameter identification, employing the Markov chain Monte Carlo (MCMC) algorithm for random sampling. Given the prior distribution of parameters and observed data, the posterior probability of parameters can be updated according to Bayes’ theorem:

$$P(\theta |\tilde{Y})=\frac{\pi (\theta )P(\tilde{Y}|\theta )}{{\displaystyle {\int }_{\Theta }P(\tilde{Y}|\theta )\pi (\theta )d\theta }}$$

(1)

where $P(\theta |\tilde{Y})$ and $\pi (\theta )$ are the posterior and prior probabilities of $\theta$, respectively; the denominator ${\displaystyle {\int }_{\Theta }P(\tilde{Y}|\theta )\pi (\theta )d\theta }$ is used to normalize the posterior probability. $P(\tilde{Y}|\theta )$ represents the likelihood function, also known as $L(\theta |\tilde{Y})$; the posterior probability is proportional to the likelihood function; the greater the value of the likelihood function, the more likely the uncertain parameters are to be close to the target value. The multivariate Gaussian likelihood function is usually assumed to be the joint probability density function

$$L(\theta |\tilde{Y})={\displaystyle \prod _{i=1}^n\frac{1}{\sqrt{{(2\pi )}^d\left|\sum \right|}}\exp \left(-\frac{1}{2}\left({{\mathit{e}}_i}^{\mathsf{{\rm T}}}{\sum }^{-1}{\mathit{e}}_i\right)\right)}$$

(2)

where $\sum$ is the covariance matrix of the experimental data; $n$ is the number of the observed data; $d$ is the dimension of the response output; and ${\mathit{e}}_i$ represents the validation metrics between the analytical and measured data, which can be defined by:

$${\mathit{e}}_i={\tilde{\mathit{Y}}}_i-\mathit{Y}(\mathit{\theta })$$

(3)

Since the function of Y on v is usually relatively complex or implicit, it is difficult to directly utilize the expression for integration calculation, so the solution of the posterior distribution is generally sampled using MCMC methods. The DREAM algorithm, or differential evolutionary adaptive metropolis algorithm, serves as an enhanced MCMC method. It enables multi-chain parallel sampling and dynamically adjusts step size and direction during search operations. This adaptive mechanism proves advantageous for guiding samples from local optimal to global points, resulting in highly efficient computational capabilities [33]. In this paper, the DREAM-MATLAB toolbox [34] is used to calculate the posterior distribution of uncertainty parameters. However, the traditional Bayesian updating method assumes normal distribution of the data. However, in many engineering scenarios, the distribution does not always adhere to normality due to the limited number of test samples available. In situations where data are sparse and the distribution is unknown, it becomes necessary to introduce approximate Bayesian computational (ABC) theory [35] for addressing challenges in FRF-based stochastic model updating. ABC operates on the principle of likelihood. It accepts sampled parameter values if the difference between sampled and observed data falls below a certain threshold; otherwise, it rejects them. Consequently, the posterior probability in Equation (1) can be formulated as follows:

$$P(\theta |\tilde{Y})\approx P(\theta |\rho (\tilde{Y},Y)\le \epsilon )$$

(4)

where $\rho (\tilde{Y},Y)$ represents the distance between observed data and simulated data, and $\epsilon$ is a sufficiently small quantity, called the tolerance error. Combined with the DREAM algorithm, an approximate likelihood function can be express as:

$$L(\theta |\tilde{Y})=\left\{\begin{array}{cc}rand(a,b),& {\displaystyle \prod _{j=1}^m\left\{I(|S({\tilde{Y}}_j)-Y_j(\theta )|\le {\epsilon }_j\left.)\right\}>0\right.}\hfill \\ 0,\hfill & {\displaystyle \prod _{j=1}^m\left\{I(|S({\tilde{Y}}_j)-Y_j(\theta )|\le {\epsilon }_j\left.)\right\}=0\right.}\hfill \end{array}\right.$$

(5)

where $S({\tilde{Y}}_j)$ represents the statistical quantities of the observed data, and $Y_j(\theta )$ is the simulated data at specific parameters. $I(\cdot )$ is the indicator function. The indicator function takes on the value of 1 when the inequality within the brackets is satisfied, and it assumes the value of 0 when the inequality is not satisfied. $rand(a,b)$ represents a randomly generated number within the interval $(a,b)$. Both $a$ and $b$ are real numbers significantly greater than zero, and there is only minimal discrepancy between them. Equation (5) demonstrates that when the discrepancy between the test and simulation is acceptable, the likelihood function yields a relatively high value. Conversely, when this criterion is not met, the likelihood function evaluates to zero, facilitating the acceptance or rejection of the parameters. Additionally, to ensure the continuous exploration of parameters satisfying the convergence criteria, without premature termination, the likelihood function is assigned a random value close to a larger number rather than a fixed value.

It can be seen from the above theory that a suitable comparison metrics is very important in Bayesian model updating. In the following section, the NHBFT-PCA technique is employed to introduce a novel validation metric aimed at addressing challenges encountered in stochastic model updating utilizing FRF data. This approach aims to establish a robust and stable error matrix. Additionally, A new technique for model updating based on FRF data is developed based on the Bayesian approach, which can replace the traditional model updating method and effectively deal with the dynamic stochastic model updating based on FRFs. Section 2.3 discusses this improved model updating technique in detail.

2.2. Metrics Utilizing NHBFT-PCA

In this section, a metric for formulating the updating likelihood function is proposed. The FRF data is processed by introducing NHBFT and PCA dimensionality reduction technology. The NHBFT method and the PCA method are conducted independently, with PCA performed after NHBFT. First, the distance matrix is obtained using NHBFT to enhance the statistical nature of the frequency response data employed for model updating, and thereby, the degree of agreement between the numerical predictions and experimental measurements can be measured effectively. PCA is then applied to reduce the dimensionality of the distance matrix, avoiding differences in results caused by matrix singularity and point selection strategy. We will elaborate on how the NHBFT method and the PCA method are applied to FRF data in the following section.

2.2.1. Normalized Half-Power Bandwidth Frequency Transformation for FRF

Generally, for a multi-degrees-of-freedom vibration system, the differential equation under a force can be expressed as follows:

$$M\ddot{x}(t)+C\dot{x}(t)+Kx(t)=F(t)$$

(6)

where $M$, $K$, and $C$ are the mass, stiffness, and damping matrices respectively; $F(t)$ is the exciting force vector; $x(t)$ is the response of the system. The frequency response function for a multi-degrees-of-freedom system can be obtained by:

$$H(\omega )={(K-M{\omega }^2+\mathrm{j}\omega C)}^{-1}$$

(7)

where $\mathrm{j}$ is the complex symbol; $H(\omega )$ is an $n\times n$ matrix in which the element $H_{ij}(\omega )$ means the response at ith DOF, while a unit excitation is applied at the jth DOF.

For a specific position, the analytical and experimental FRFs are usually compared at each individual spectral line in the conventional model updating method. The metrics $e$ in Equation (3) can be defined by:

$$e(\omega )=d\left\{H_{\mathrm{e}}(\omega ),H_{\mathrm{a}}(\omega ;\theta )\right\}$$

(8)

where $e\left\{w\right\}$ denotes the distance calculation function, such as SSD, and $H_{\mathrm{e}}(\omega )$ and $H_{\mathrm{a}}(\omega ;\theta )$ represent the amplitude of the experimental and analytical FRFs at the determined frequency, respectively.

However, in Figure 1, it is evident that the key features of FRFs, such as natural frequencies and amplitudes, can shift and vary in the frequency and amplitude range due to uncertainties in regards to system stiffness, mass, and other parameters. When measuring the degree of agreement between analytical and experimental FRFs, the results obtained within the resonance region vary drastically compared to those in other regions due to the high nonlinear characteristics of FRF around the resonance point. Consequently, the metrics derived at each individual frequency point deviate from a normal distribution [36], posing challenges for uncertainty analysis and quantification under a Bayesian framework. Moreover, the high-dimensional nature of FRF data further complicates model updating within the Bayesian framework.

Mechanically, the majority of the FRF information is contained in the neighborhood of resonant frequencies. Therefore, a natural consideration is to normalize each modal frequency in the analytical model and experimental model and carry out the model updating using the information within the normalized half-power bandwidth of each mode. The kth order half-power bandwidth of the FRF takes the form of:

$$\mathsf{\Delta }{\omega }_k=2{\zeta }_k{\omega }_k$$

(9)

where ${\zeta }_k$ and ${\omega }_k$ are the relative modal damping ratio and the natural frequency of the kth order mode, respectively. As shown in Figure 2, only the data within the half-power bandwidth are used for comparing and model updating. After extracting the half-power bandwidth of each FRF, each band can be expressed as a frequency interval:

$$\mathsf{\Omega }=[{\omega }^l,{\omega }^r]$$

(10)

where ${\omega }^l$ and ${\omega }^r$ are the left and right boundaries of the band, respectively. Each band can then be divided into two monotonic parts, as follows:

$$\mathsf{\Omega }=[{\omega }^l,{\omega }^p]\cup [{\omega }^p,{\omega }^r]$$

(11)

where ${\omega }^p$ denotes the peak frequency in the frequency band. In both intervals obtained, a frequency resampling task is conducted by following the rule that the same number of frequencies should be ensured for the corresponding order of interval for all FRFs. For a group of FRFs with uncertainty, the equivalent frequency of the kth order half-power bandwidth in the qth FRF can be expressed as:

$${\mathsf{\Omega }}_q^k=\left({\omega }^l\cdots {\omega }^i\cdots {\omega }^p\cdots {\omega }^j\cdots {\omega }^r\right)\left(q=1,2\cdots m\right)$$

(12)

where ${\omega }^i(i=1,2,\cdots n_1)$ denotes the corresponding resampling frequency between ${\omega }^l$ and ${\omega }^p$, ${\omega }^i(i=1,2,\cdots n_1)$ is the corresponding part between ${\omega }^p$ and ${\omega }^r$, and $n_1$ and $n_2$ are the number of frequency interpolation points taken in two parts, respectively. This idea comes from the method of random frequency transformations, which was used by Refs. [37,38] to perform the modeling of polynomial chaotic expansions (PCE) of the frequency response functions. According to this way of thinking, we can sample the amplitude within the corresponding bandwidth using the sampling method given in Equation (12) to obtain the equivalent amplitude of the kth order half-power bandwidth in the qth FRF:

$$H_q^k=\left(H^l\cdots H^i\cdots H^p\cdots H^j\cdots H^r\right)\left(q=1,2\cdots m\right)$$

(13)

Thus, for the comparison of analytical and experimental frequency response functions, the new metric can be defined as:

$$\left\{\begin{array}{l}{e}_{\mathsf{\Omega }}=d({\mathsf{\Omega }}_a(\mathsf{\Theta }),{\mathsf{\Omega }}_e)\\ e_H=d(H_a({\mathsf{\Omega }}_a,\mathsf{\Theta }),H_e({\mathsf{\Omega }}_e))\\ e=\left[e_{\mathsf{\Omega }},e_H\right]\end{array}\right.$$

(14)

where $d()$ denotes the distance calculation function, such as Euclidean distance. ${\mathsf{\Omega }}_a$ and ${\mathsf{\Omega }}_e$ are the equivalent frequencies in the analytical and experimental FRFs, respectively. $\mathsf{\Theta }$ represents the uncertainty parameters of the simulation model, such as the stiffness and damping parameters.$e_{\mathsf{\Omega }}$ is the metric-described consistency between the experimental and analytical equivalent frequencies; $H_a({\mathsf{\Omega }}_a,\mathsf{\Theta })$ and $H_e({\mathsf{\Omega }}_e)$ show the corresponding analytical and experimental amplitude in the equivalent frequencies. $e_H$ is the metric-described consistency between the experimental and analytical amplitude at the equivalent frequencies. $e$ is the total metric of the experimental and analytical FRFs. Figure 2 illustrates the methodology employed for comparing analytical and experimental FRFs utilizing the NHBFT method. This method defines the comparison of two FRFs as the evaluation of multiple resampling points within a half-power bandwidth across both frequency and amplitude dimensions, as shown in Figure 3. For convenience, we introduce a new coordinate by normalizing each interval to [0, 1], and it can be seen that $e_{\mathsf{\Omega }}$ and $e_H$ are less dependent on frequency after normalization, and the two metrics have a common variation interval, which aids in the data selection for model updating.

2.2.2. PCA for Dimension Reduction

However, FRFs often contain high-dimensional discrete data points, and how to choose the appropriate number of frequency interpolation points remains a problem. When the settings of $n_1$ and $n_2$ are too large, the total metric $e$ shows high-dimensional characteristics, increasing the correlation and redundancy between the metrics and leading to an incorrect Bayesian likelihood function solution. If the number of settings is too small, it does not entirely cover the original FRF and misses the information contained in the frequency response function. In this section, the PCA technique [39] is introduced to extract all the information carried by the high-dimensional FRF data. PCA is the most widely used and stable data dimensionality reduction method, and it is relatively simple to implement, effectively representing the frequency response information of high-dimensional nonlinearities using less data.

Consider a set of FRF data after normalized half-power bandwidth frequency transformation $Y_i=\left\{{\omega }_k^1,{\omega }_k^2,\dots ,{\omega }_k^n,H_k^1,H_k^2,\dots ,H_k^n\right\}$, which represents the frequency and amplitude of the n data points resampled in the kth order half-power band. The dimension of $Y_i$ can be expressed as $2*n*k$. When there is an FRFs group $Y_e={\left(Y_1,Y_2,\dots ,Y_r\right)}^T$, the covariance matrix $C\in {ℝ}^{m\times m}$ of $Y$ can be expressed as:

$$(C-\lambda E)U=0$$

(15)

where ${\lambda }_k(k=1,2,\dots m)$ represents the eigenvalue, and $u_k(k=1,2,\dots m)$ refers to the eigenvector. ${\lambda }_k$ is ordered in descending order ${\lambda }_1>{\lambda }_2>{\lambda }_3>\cdots >{\lambda }_m$. The corresponding eigenvector is denoted as $\{u_1,u_2,\dots ,u_m\}$, considering that the $p(p\ll m)$ order principal components are sufficient to represent the original information. The following transformations can be applied to the original FRFs using the first $p$-order eigenvectors:

$$Z=YU$$

(16)

where $U$ denotes the first $p$-order eigenvector matrix, and $Z$ denotes the transformation matrix containing the first $p$-order principal components of the original data. If the transformation matrix of the analytical FRF is $Z_a(\mathsf{\Theta })$ and the experimental FRF is $Z_e$, the metric for comparing the two FRFs can be redefined as:

$$E=d\left\{Z_e,Z_a(\mathsf{\Theta })\right\}$$

(17)

2.3. Improved Bayesian Model Updating Method Interated with Metrics Using NHBFT-PCA

In this section, we will integrate the above-defined metric obtained by NHBFT-PCA into the Bayesian model updating framework to demonstrate the generality of the metric.

A key development of the Bayes’ theorem focuses on the likelihood function $L(\theta |\tilde{Y})$, which needs to be customized to adapt to different updating objectives. In this paper, the likelihood function integrating the metric using NHBFT-PCA can be expressed as:

$$L(\theta |\tilde{Y})={\displaystyle \prod _{i=1}^n\frac{1}{\sqrt{{(2\pi )}^d\left|\sum \right|}}\exp \left(-\frac{1}{2}\left({\mathit{E}}_i{\left(Z_{\mathrm{e}},Z_a(\theta )\right)}^{{\rm T}}{\sum }^{-1}{\mathit{E}}_i\left(Z_{\mathrm{e}},Z_a(\theta )\right)\right)\right)}$$

(18)

${\mathit{E}}_i$ is the comparison metric directly integrated in the likelihood function, which ensures that when the uncertainty coefficient $\theta$ is updated, ${\mathit{E}}_i$ takes the minimum value, while the likelihood takes the maximum probability.

The use of the multivariate Gaussian likelihood function in the MCMC method causes the acceptance of the small and medium probability of the parameter to be updated to be significantly decreased, which affects the estimation of posterior distributions of the parameter; therefore, this effect can be improved by writing the logarithmic form of the likelihood function in the following form [36]:

$$L(\theta |Y_e)=-\frac{d}{2}\log (2\pi )-\frac{1}{2}\log (|\mathsf{\Sigma }|)-\frac{1}{2n}\sum _{i=1}^{\pi }{E}_i^{\mathrm{T}}{\mathsf{\Sigma }}^{-1}{E}_i$$

(19)

Specifically, when dealing with sparse experimental data and an unknown distribution, the approximate likelihood function described in Equation (5) can be formulated using the NHBFT-PCA metrics as follows:

$$L(\theta |Y_e)=\left\{\begin{array}{cc}rand(a,b),\hfill & {\displaystyle \prod _{j=1}^m\left\{I(|{\mathit{E}}_j\left(Z_{\mathrm{e}},Z_a(\theta )\right)|\le {\epsilon }_j\left.)\right\}>0\right.}\hfill \\ 0,\hfill & {\displaystyle \prod _{j=1}^m\left\{I(|{\mathit{E}}_j\left(Z_{\mathrm{e}},Z_a(\theta )\right)|\le {\epsilon }_j\left.)\right\}=0\right.}\hfill \end{array}\right.$$

(20)

A complete flowchart of the improved Bayesian model updating method with metrics using NHBFT-PCA is illustrated in Figure 4.

Step 1: Data preprocessing—This is a crucial step in the entire model updating process. First, resample each piece of experimental FRF data using the NHBFT method to obtain transformed FRF data, thereby constructing a stable distance matrix $e$; then use PCA to process the distance matrix and obtain the corresponding feature matrix $Z$ after dimensionality reduction. Finally, calculate the statistical characteristics, such as mean and variations of the processed FRF data.

Step 2: Parameter initialization—Initialize and assign the parameters to be updated according to the previously obtained information; set the number of iterations. In addition, we can assign an interval to each parameter to ensure the rationality of the parameter value.

Step 3: Modeling and analyzing—Construct the correct FEM model based on the real object, and substitute the initial parameters into the model to obtain FRF data.

Step 4: Metrics calculation and evaluation—Compute the likelihood function using the processed FRF data from the experiment and the simulation; evaluate the model confidence according to the likelihood.

Step 5: Iterations—Use the DREAM algorithm to generate new parameters based on the result of the likelihood function; repeat steps (3)–(4) by substituting the newly generated parameters into the model until the iterations reach the maximum setting number.

Step 6: Postprocessing—Once the loop terminates, estimate the posterior distributions of the parameters according to the chains produced during the iterations.

3. Numerical Study

In this section, the performance of the previously described metrics is examined using the model updating results of a three-degrees-of-freedom numerical case. This numerical case is commonly employed to assess the effectiveness of dynamic stochastic model updating methods [40,41], as depicted in Figure 5. The nominal values of the parameters are $m_1=m_2=m_3=1\mathrm{kg}$, $k_3=k_4=1\mathrm{N}/\mathrm{m}$, and $k_6=3\mathrm{N}/\mathrm{m}$; $k_1$, $k_2$, and $k_5$ are the uncertain parameters, following the Gaussian distribution, with the mean value of ${\mu }_{k_1}={\mu }_{k_2}={\mu }_{k_5}=1\mathrm{N}/\mathrm{m}$ and the standard deviation of ${\sigma }_{k_1}={\sigma }_{k_2}={\sigma }_{k_5}=0.2\mathrm{N}/\mathrm{m}$. For convenience, the system’s damping is assumed by introducing Rayleigh damping with $C=\alpha M+\beta K$, where $\alpha$ and $\beta$ are the damping ratios, with Gaussian distributions of $\alpha$~N(0.005, 0.001²) and $\beta$~N(0.005, 0.001²), respectively. A uniform prior distribution is assumed by setting $k_i\in [0,3]\mathrm{N}/\mathrm{m}(i=1,2,5)$ and $\alpha ,\beta \in [0,0.01]$.

3.1. Bayesian Model Updating with Metrics Using FRFSM

The FRFSM metric was chosen for comparison with the metrics using NHBFT-PCA. The normal PDF, $f$, is defined as:

$$f(x;\mu ,{\sigma }^2)=\frac{1}{\sigma \sqrt{2\pi }}{\mathrm{e}}^{-\frac{1}{2}{(\frac{x-\mu }{\sigma })}^2}$$

(21)

where $x$, $m$, and $s$ refer to the variable, mean, and standard deviation, respectively. Thus, the FRFSM is defined as:

$$FRFSM=\frac{1}{N}{\displaystyle \sum _{j=1}^N\frac{f\left(\epsilon (j),0,{\sigma }_0^2\right)}{f\left(0,0,{\sigma }_0^2\right)}}$$

(22)

$${\epsilon }_j=\left|10{\log }_{10}{‖H_a({\omega }_j)‖}^2-10{\log }_{10}{‖H_e({\omega }_j)‖}^2\right|,j=1,\cdots ,N$$

(23)

Here, $N$ and ${\sigma }_0$ are the number of frequencies and a reference value, respectively. Equation (20) introduces ${\epsilon }_j$, which is the magnitude of the difference between decibel-scaled versions of the FRFs. A likelihood FRFSM that assumes a normal distribution of structural responses performs well when applied to complex structures because it identifies similarities, even at high modal densities and in high-damping environments.

Through Monte Carlo simulation, 500 groups of experimental FRFs were obtained by sampling from the target parameter distribution. Figure 6 shows the use of conventional metrics for comparison of two frequency response functions, i.e., taking a certain number of points evenly within the frequency domain and comparing the amplitude at each individual spectral line to obtain the validation metrics between the analytical FRF and the experimental FRF. Figure 7 shows the Markov chains for the five uncertain parameters after 5000 iterations, and the poor convergence of the parameters is obtained using this method, particularly for the damping parameters. As shown in Figure 8, by comparing the true distributions, it can be found that the posteriori distributions of the parameters are still very different from the target distributions. Overall, the model updating results obtained by the FRFSM metric are poor, and the metric does not provide a good measure of the consistency between the analytical FRFs and the experimental FRFs. In addition, when using conventional metrics for model updating, the selection of points is limited, and the results of model updating also vary with the selection of points.

3.2. Bayesian Model Updating with Metrics Using NHBFT-PCA

The NHBFT-PCA-based metric is used to solve the problems noted in this section and to improve the model updating results. NHBFT-PCA firstly extracts three half-power bandwidths of each frequency response function, which restricts the boundaries of the selected points so that the points selected for comparison contain more system vibration features. With the dimensionality reduction technique, we can select any number of points within each half-power bandwidth to update the model. Assuming that the total number of points used for model updating is $n$ ($n=15$, $n=36$, and $n=180$), in the following example, we will discuss the results of model updating when n takes different values to verify the robustness of the method.

As demonstrated in Figure 9, five points were selected within the three half-power bandwidths of each experimental and simulated FRF curve, when $n=15$. These points include bandwidth boundary points, resonance points, and off-peak points within the frequency domain. Figure 10 illustrates how the NHBFT-PCA-based metrics are obtained. Using the method introduced in Section 2.2, we transform the difference in frequency and amplitude between two FRFs into a feature matrix $E_1,E_2,E_3\in R^{N\times p}$; here, $N$ denotes the number of experimental FRFs, and $p$ represents the dimension of the matrix after the dimensionality reduction process. The feature matrix $E=\left\{E_1,E_2,E_3\right\}$ is then the metric obtained by NHBFT-PCA, which contains the vibration characteristics of the selected points at each order of the modes, allowing for more accurate model updating.

After 2000 iterations, the posterior distribution of the updating parameters with the selection of different numbers of points is shown in Figure 11, from which we can see that the posterior distribution of parameters for all sub-cases ($n=15$, $n=36$, and $n=180$) show good agreement with the target PDF. Compared to traditional Bayesian methods, all posterior probabilities of parameters obtained through improved Bayesian methods are closer to the target distribution of the parameters. Although there is still some gap in the damping parameter $\beta$ compared to that of the target distribution, the results obtained have already shown significant improvement over those obtained by traditional methods. To better validate the methods, we will quantitatively compare the Bayesian method using NHBFT-PCA measurement and the Bayesian method using FRFSM; see below.

The results of model updating using conventional metrics (FRFSM) are compared with the results of model updating using NHBFT-PCA-based metrics, as illustrated in Figure 12. Regardless of the number of selected points, the model updating results using the NHBFT-PCA-based metrics performed better than did the FRFSM, and the parameter CDF curves are closer to the target values. Among them, the NHBFT-PCA-based metric performs well in different sub-cases and shows good robustness. The mean and standard deviation errors of the parameters obtained by two different metrics are shown in Figure 13. Regardless of the stiffness parameter or the damping parameter, the errors obtained by the NHBFT-PCA-based metric are generally smaller than those of the FRFSM metric, especially for the damping parameter.

The performance of the model updating based on the NHBFT-PCA metrics is significantly improved compared to that of the model updating results using the FRFSM. These advances are mainly reflected in the improvement of convergence speed (especially damping) and accuracy. In addition, the method no longer restricts the choice of data points, effectively utilizing the resonance data while ensuring good stability of the results for FRF-based model updating.

4. Experimental Case

4.1. Simulation and Experimental Test

In this section, an experimental case of a bolted structure is designed to apply the proposed improved Bayesian model updating method, as shown in Figure 14. The bolted structure considered in this work includes two plates connected by a bolted joint, the material of both plates is aluminum, where the interface of the two plates is 60 mm long and 50.8 mm wide, and the rest of the dimensions are indicated in Figure 14.

Hierarchical thinking [42] is considered to update the bolted assembly structure, where the structure is disassembled into two aluminum plates, along with the bolted lap joints, and the finite element modeling of the bolted structure is shown in Figure 15. The two beams are modeled using shell elements, and the contact interface between the two plates is modeled by a continuum with a very small, but finite, thickness [43]; this continuum is referred to as the thin layer, and the modulus parameter of the thin layer unit is used to characterize the stiffness of the bolted joints [44]. CHEXA elements are used in Patran 2020 to establish the above bolted structure, and the bolt entities and bolt holes are ignored. Node recombination is performed on the contact interface, in which the length-to-thickness ratio of the thin-layer elements is taken to be 10.0. Figure 16 illustrates the experimental setup, where the bolted structure is suspended using two rubber bands on a rigid frame to simulate free–free boundary conditions. Then, a force hammer was used to induce impulse excitation for the collection of vibration response data via the accelerometer. The locations of the excitation and the measurement points are illustrated in Figure 16. In order to fully consider the experimental uncertainty, as well as the uncertainty in the contact interface, this experiment was repeated with twenty sets of replicates at different bolt preloads.

In the following section, the material properties of the two plates will be updated separately, which helps to eliminate the uncertainties associated with inherently unknown material parameters. Secondly, the joint parameters are identified based on the FRFs measured from the bolted beam assembly in order to calibrate the stiffness and damping characteristics of the bolted lap joints. Finally, the parameters obtained by model updating are combined to obtain an accurately updated finite element model to validate the applicability of the proposed model updating method.

4.2. Identification of Material Properties

The identification of inherent material parameters is first carried out for two beams; the initial values of the material parameters are taken as the modulus of elasticity $E$ = 69,000 Mpa, Poisson’s ratio $\mu$ = 0.33, and density $\rho$ = 2700 kg/m³. The optimal material parameters are obtained using the sensitivity-based model updating method integrated with the metrics based on the NHBFT-PCA method.

As depicted in Figure 17, the elastic modulus of the two beams reaches convergence after six iterations. This convergence indicates that the identified material properties have been successfully calibrated, leading to the generation of a more precise bolted beam assembly. Subsequently, as illustrated in Figure 18, the predicted frequency response function (FRF), after updating the inherent parameters, demonstrates closer alignment with the measured FRF when compared to the initial prediction. However, the model predictions still differ from the experimental results, and the model credibility cannot be guaranteed, so the effect of the joint parameters must be considered.

4.3. Identification of Joint Parameters

In this section, the identification of joint parameters, using orthotropic materials to describe the constitutive relationship of the thin-layer elements, is carried out. The values of the material parameters for the thin-layer elements are directly related to the bolted lap joints stiffness. After sensitivity analysis, it is found that the parameters ($E_{11},G_{22}$ and $G_{12}$) for the thin-layer elements have basically no effect on the responses, so the updating parameters include three material properties ($E_{33},G_{23}$, and $G_{31}$), as well as the modal damping ratio (${\zeta }_1,{\zeta }_2$ and ${\zeta }_3$) corresponding to the first three modes. Three different metrics were integrated with Bayesian theory for comparison: (1) model updating using FRFSM metrics; (2) model updating using NHBFT metrics; and (3) model updating using NHBFT-PCA-based metrics.

Figure 19 compares the predicted FRFs obtained through three different metrics with the experimental FRFs. Obviously, the model obtained using FRFSM performs poorly, with significantly higher resonance peak amplitudes compared to those in the experimental data. This discrepancy can be attributed to the non-convergence of the damping parameters, consistent with the findings in Section 3.1. The metrics obtained based on the NHBFT methodology, without PCA, were integrated with Bayesian theory for comparison. The prediction results improved greatly compared to those obtained using the FRFSM metrics. However, a phenomenon of frequency shift is observed in the predicted FRFs, which is most likely attributed to potential limitations in capturing frequency response information, resulting from an imperfect strategy in selecting data points. The prediction results using NHBFT-PCA metrics are closest to the experimental results. The frequencies and amplitudes of the FRFs are in good agreement with the experimental FRF, and the obtained model is the most accurate. In contrast, the prediction results obtained through NHBFT-PCA metrics closely align with experimental data, and the metric based on NHBFT-PCA can overcome the limitation of data points selection strategy. The frequencies and amplitudes of the FRFs show excellent agreement with experimental data, indicating that the model derived from the third method is the most accurate among the three strategies.

In order to fully demonstrate the universal applicability of the FRF comparison metrics proposed in this article, the cumulative distribution function (CDF) curves of predicted FRF obtained by using different strategies are compared with the experimental CDF curves at some individual spectral lines (156.25 Hz, 440.625 Hz, 821.875 Hz), as shown in Figure 20.

Overall, the CDF derived using the NHBFT-PCA method closely resembles the experimental CDF, whereas the prediction outcomes using the NHBFT method are inferior to those of the NHBFT-PCA method due to the incomplete utilization of frequency response information. Furthermore, Figure 21 illustrates the relative error of FRFs prediction across the entire frequency domain. The mean values of the relative errors are delineated by blue dashed lines. The results indicate superior performance of the NHBFT-PCA method compared to both the NHBFT strategy and the conventional approach. The proposed metrics have shown good adaptability with the Bayesian framework in this project case, and the evaluation results, such as relative error, from the metrics are consistent with conventional metrics. This suggests the universal applicability of the introduced metrics.

5. Conclusions

In this paper, an improved Bayesian model updating method is introduced to tackle key challenges encountered in dynamic model updating using frequency response data. We first propose a Bayesian model updating framework and then integrate a novel metric, based on the NHBFT-PCA method, into Bayesian theory to develop an improved Bayesian model updating technique. Finally, the detailed process of the improved Bayesian model updating method is summarized. The effectiveness of the proposed method is demonstrated through both numerical study and an experimental case. The conclusions are drawn as follows:

• The NHBFT-PCA-based metric effectively captures FRF information and accurately quantifies the difference between the experimental and analytical FRFs. The model updating results are significantly better than the conventional FRF validation metrics, and the metrics based on NHBFT-PCA are free from the limitation of multi-dimensional data samples.
• The Bayesian theory, combined with the metrics utilizing NHBFT-PCA, effectively addresses challenges regarding dynamic model updating, leading to enhanced performance and improving the singularity issue of the covariance matrix in traditional Bayesian methods caused by data correlation. Furthermore, the proposed metrics are adaptable to various posterior probability computation methods within the Bayesian framework, facilitating corresponding model updating techniques. Improvement strategies for conventional Bayesian and approximate Bayesian computations are described in this article.

The proposed metrics and model updating method exhibit good applicability in linear systems. Future development of this study will focus on extending the proposed improved model updating method to nonlinear systems. It is difficult to update the nonlinear properties in conjunction with the normal parameters in a uniform framework, and the nonlinearity is difficult to capture using the experimental measurement. It is challenging, but promising, to explore the FRF-based model updating by considering the issues addressed above, which is the priority of additional study in the future.