Search

Highlights • The derivation process of force-displacement relationship is presented for the Iwan beam element. Then, the linear beam element and Iwan beam element are used to simulate the bolted cantilever beam structure. • The linear natural frequency of the structure is obtained by low-amplitude random white noise test, and the linear parameters is updated based on the sensitivity analysis method. • The nonlinear model updating method based on the instantaneous characteristics of the principal response components is proposed to update nonlinear model of the bolted cantilever beam, and the parameters of Iwan beam element is obtained. Abstract Mechanical joints usually affect structural dynamic responses, leading to local nonlinear stiffness and damping. In this paper, hysteresis Iwan model is discussed in detail to describe nonlinear mechanical behaviors of contact surface. Based on adjusted Iwan model, the derivation process of force-displacement relationship is presented for Iwan beam element. Meanwhile, relevant experimental study on the dynamic behaviors of a cantilever beam with a bolted joint is performed. Linear beam element and Iwan beam element are used to simulate the bolted cantilever beam structure, and the corresponding structural dynamic responses are calculated by forward incremental displacement center difference method. Then, combined with the nonlinear structural model updating method based on the instantaneous characteristics of principal dynamic response components, the bolted cantilever beam structure is updated and the parameters of Iwan beam element are obtained. The updated results are in good agreement with the experimental results, that is, the feasibility and effectiveness of the proposed nonlinear structural model updating method are verified by comparing the updated responses and measured data. [ABSTRACT FROM AUTHOR]

This study applies the theoretical findings of circularly-symmetric complex normal ratio distribution Yan and Ren (2016) [1,2] to transmissibility-based modal analysis from a statistical viewpoint. A probabilistic model of transmissibility function in the vicinity of the resonant frequency is formulated in modal domain, while some insightful comments are offered. It theoretically reveals that the statistics of transmissibility function around the resonant frequency is solely dependent on ‘noise-to-signal’ ratio and mode shapes. As a sequel to the development of the probabilistic model of transmissibility function in modal domain, this study poses the process of modal identification in the context of Bayesian framework by borrowing a novel paradigm. Implementation issues unique to the proposed approach are resolved by Lagrange multiplier approach. Also, this study explores the possibility of applying Bayesian analysis in distinguishing harmonic components and structural ones. The approaches are verified through simulated data and experimentally testing data. The uncertainty behavior due to variation of different factors is also discussed in detail. [ABSTRACT FROM AUTHOR]

Recent advances in signal processing and structural dynamics have spurred the adoption of transmissibility functions in academia and industry alike. Due to the inherent randomness of measurement and variability of environmental conditions, uncertainty impacts its applications. This study is focused on statistical inference for raw scalar transmissibility functions modeled as complex ratio random variables. The goal is achieved through companion papers. This paper (Part I) is dedicated to dealing with a formal mathematical proof. New theorems on multivariate circularly-symmetric complex normal ratio distribution are proved on the basis of principle of probabilistic transformation of continuous random vectors. The closed-form distributional formulas for multivariate ratios of correlated circularly-symmetric complex normal random variables are analytically derived. Afterwards, several properties are deduced as corollaries and lemmas to the new theorems. Monte Carlo simulation (MCS) is utilized to verify the accuracy of some representative cases. This work lays the mathematical groundwork to find probabilistic models for raw scalar transmissibility functions, which are to be expounded in detail in Part II of this study. [ABSTRACT FROM AUTHOR]

In Part I of this study, some new theorems, corollaries and lemmas on circularly-symmetric complex normal ratio distribution have been mathematically proved. This part II paper is dedicated to providing a rigorous treatment of statistical properties of raw scalar transmissibility functions at an arbitrary frequency line. On the basis of statistics of raw FFT coefficients and circularly-symmetric complex normal ratio distribution, explicit closed-form probabilistic models are established for both multivariate and univariate scalar transmissibility functions. Also, remarks on the independence of transmissibility functions at different frequency lines and the shape of the probability density function (PDF) of univariate case are presented. The statistical structures of probabilistic models are concise, compact and easy-implemented with a low computational effort. They hold for general stationary vector processes, either Gaussian stochastic processes or non-Gaussian stochastic processes. The accuracy of proposed models is verified using numerical example as well as field test data of a high-rise building and a long-span cable-stayed bridge. This study yields new insights into the qualitative analysis of the uncertainty of scalar transmissibility functions, which paves the way for developing new statistical methodologies for modal analysis, model updating or damage detection using responses only without input information. [ABSTRACT FROM AUTHOR]

Stochastic model updating (SMU) has been increasingly applied in quantifying structural parameter uncertainty from responses variability. SMU for parameter uncertainty quantification refers to the problem of inverse uncertainty quantification (IUQ), which is a nontrivial task. Inverse problem solved with optimization usually brings about the issues of gradient computation, ill-conditionedness, and non-uniqueness. Moreover, the uncertainty present in response makes the inverse problem more complicated. In this study, Bayesian approach is adopted in SMU for parameter uncertainty quantification. The prominent strength of Bayesian approach for IUQ problem is that it solves IUQ problem in a straightforward manner, which enables it to avoid the previous issues. However, when applied to engineering structures that are modeled with a high-resolution finite element model (FEM), Bayesian approach is still computationally expensive since the commonly used Markov chain Monte Carlo (MCMC) method for Bayesian inference requires a large number of model runs to guarantee the convergence. Herein we reduce computational cost in two aspects. On the one hand, the fast-running Gaussian process model (GPM) is utilized to approximate the time-consuming high-resolution FEM. On the other hand, the advanced MCMC method using delayed rejection adaptive Metropolis (DRAM) algorithm that incorporates local adaptive strategy with global adaptive strategy is employed for Bayesian inference. In addition, we propose the use of the powerful variance-based global sensitivity analysis (GSA) in parameter selection to exclude non-influential parameters from calibration parameters, which yields a reduced-order model and thus further alleviates the computational burden. A simulated aluminum plate and a real-world complex cable-stayed pedestrian bridge are presented to illustrate the proposed framework and verify its feasibility. [ABSTRACT FROM AUTHOR]

• The time–frequency representation of fractional Fourier transform is theoretically derived in term of the window. • The analytical ridge point and corresponding ridge line of fractional Fourier transform is defined and presented. • The fractional Fourier transform based instantaneous frequency identification algorithm is formulated. • The feasibility and effectiveness of proposed method is verified by numerical signal and experimental results. The fractional Fourier transform involving a rotation angle as an additional variable is capable to transform a signal to any fractional Fourier domain in between time and frequency domain. With insight into the important role of rotation angle and time–frequency projection of fractional Fourier transform, the time–frequency representation of fractional Fourier transform is in this paper theoretically derived in terms of the window. It is demonstrated that the fractional Fourier transform in time–frequency domain is an algorithm of classical Fourier transform combined the chirp window that is adaptive dilation and translation. The window parameter properties are investigated in details and compared with those of wavelet window. To realize the structural instantaneous frequency identification, the analytical ridge point and corresponding ridge line based on the time–frequency representation of fractional Fourier transform is defined and presented. The fractional Fourier transform based instantaneous frequency identification algorithm is established accordingly by extracting ridge line. The effectiveness of proposed method is verified by a numerical multi-component non-stationary signal, a three-story frame on the shaking table excited by non-stationary inputs with different signal-to-noise ratios and a cable experiment with the time-varying tensioned forces. The identified instantaneous frequency results are compared with those obtained from short-time Fourier transform and wavelet transform. Both numerical and experimental verifications have demonstrated that the proposed fractional Fourier transform based method is capable of identification of instantaneous frequencies involving non-stationary signals with a sound robustness to noise. The fractional Fourier transform can be an alternative tool to process the non-stationary signals in time–frequency domain. [ABSTRACT FROM AUTHOR]

Abstract: Stochastic model updating must be considered for quantifying uncertainties inherently existing in real-world engineering structures. By this means the statistical properties, instead of deterministic values, of structural parameters can be sought indicating the parameter variability. However, the implementation of stochastic model updating is much more complicated than that of deterministic methods particularly in the aspects of theoretical complexity and low computational efficiency. This study attempts to propose a simple and cost-efficient method by decomposing a stochastic updating process into a series of deterministic ones with the aid of response surface models and Monte Carlo simulation. The response surface models are used as surrogates for original FE models in the interest of programming simplification, fast response computation and easy inverse optimization. Monte Carlo simulation is adopted for generating samples from the assumed or measured probability distributions of responses. Each sample corresponds to an individual deterministic inverse process predicting the deterministic values of parameters. Then the parameter means and variances can be statistically estimated based on all the parameter predictions by running all the samples. Meanwhile, the analysis of variance approach is employed for the evaluation of parameter variability significance. The proposed method has been demonstrated firstly on a numerical beam and then a set of nominally identical steel plates tested in the laboratory. It is found that compared with the existing stochastic model updating methods, the proposed method presents similar accuracy while its primary merits consist in its simple implementation and cost efficiency in response computation and inverse optimization. [Copyright &y& Elsevier]

Abstract: Modal strain energy has been widely utilized as a damage-sensitive feature to detect structural damage. Based on the closed-form of element modal strain energy sensitivity, a statistic structural damage detection algorithm is proposed in this paper. The method is oriented to the ambient vibration measurements where only the operational mode shapes are available. The established element damage equations are incorporated with the modal scaling factors so that the damage and modal scaling factor can be solved simultaneously. The statistic analysis is implemented to take into account the uncertainties existing in the structural model and measured structural modal parameters. The sensitivity formulas of system matrix and residual vector with respect to noise are derived. The proposed statistics based element damage equations have a compact algebraic form, which does not require a large computation time and a high storage capacity. The method is verified by a numerical simply supported beam with different assumed damage. The effects of noise and damage levels on damage detection are investigated. It is demonstrated that the proposed method can identify most likely damaged elements. [Copyright &y& Elsevier]

The numerical results from a finite element (FE) model often differ from the experimental results of real structures. FE model updating is often required to identify and correct the uncertain parameters of FE model and is usually posed as an optimisation problem. Setting up of an objective function, selecting updating parameters and using robust optimisation algorithm are three crucial steps in FE model updating. In this paper, a multiobjective optimisation technique is used to extremise two objective functions simultaneously which overcomes the difficulty of weighing the individual objective function of more objectives in conventional FE model updating procedure. Eigenfrequency residual and modal strain energy residual are used as two objective functions of the multiobjective optimisation. Only few updating parameters are selected on the basis of the prior knowledge of the dynamic behaviours of the structure and eigenfrequency sensitivity study. The proposed FE model updating procedure is first applied to the simulated simply supported beam. This case study shows that the methodology is robust with an effective detection of assumed damaged elements. The procedure is then successfully applied to the updating of a precast continuous box girder bridge that was tested on field under operational conditions. [Copyright &y& Elsevier]

• ISCT modifies window function by adding three parameters, which can be obtained by energy CM and optimization algorithm. • IMSSSCT further improves energy concentration and effectively improves IF identification performance and robustness. • The convincing numerical and experimental recognition results confirm the superior performance of IMSSSCT. To improve the accuracy of structural instantaneous frequency (IF) identified by the spline-kernelled chirplet transform, a combination of an improved spline-kernelled chirplet transform and a multi-synchrosqueezing technique is proposed for analyzing nonstationary signals and structural IF identification in this paper. Firstly, an improved spline-kernelled chirplet transform (ISCT) is put forward by introducing a three parameter Gaussian window function for an existing spline-kernelled chirplet transform. These three parameters are successfully obtained through an optimization algorithm and the energy concentration measure (CM). Then, combined with a multi-synchrosqueezing technique, a so-called improved multi-synchrosqueezing spline-kernelled chirplet transform (IMSSSCT) approach is further presented to improve the time–frequency energy concentration. Finally, the accuracy and practicability of the proposed IMSSSCT method are validated by analytical signals, a single-degree-of-freedom Duffing system, a three-layer shear frame, as well as a cantilever beam experiment. The research results indicate that the IMSSSCT technique can effectively and accurately identify the IFs of nonlinear and time-varying structures. [ABSTRACT FROM AUTHOR]

Transfer learning-based damage identification is widely explored as it can utilize prior damage knowledge from finite element model (FEM) to identify damage without real structure damage labels. However, lacking a substantial amount of real structure data and overlooking the variances in conditional (local) distributions between source domain (data generated by FEM) and target domain (real structure data) severely restrict its practical application. This paper proposes a transfer learning-based damage identification method by combining Wasserstein Generative Adversarial Network with Gradient Penalty (WGAN-GP) and Dynamic Adversarial Adaptation Network (DAAN). Firstly, WGAN-GP generative model is employed to augment real structure samples to address the issue of insufficient data. Then, an optimization transfer model with the capability of balancing the importance of marginal and conditional distributions between FEM data and real structure data is proposed based on DAAN transfer model. It dynamically learns damage-invariant features for damage identification. Thirdly, damage identification procedures are provided by combining the generative model WGAN-GP and the transfer model DAAN with moving load induced responses as inputs. Numerical and experimental examples indicate that the proposed method achieves superior transfer effect with limited real structure data and maintains high damage identification accuracy compared to the traditional transfer learning-based methods. [ABSTRACT FROM AUTHOR]

• The aPCE-based method is used for analytical UQ and GSA of dynamic characteristics. • The aPCE-based method can handle arbitrary probability distributed random variables. • The MEP scheme is adopted to analytically derive the PDF of dynamic characteristics. • The effectiveness of the aPCE-based method is verified by the brute-force MCS. Uncertainty quantification (UQ) and global sensitivity analysis (GSA) of dynamic characteristics of complex systems subjected to uncertainty are jointly investigated in this paper. An efficient approach based on arbitrary polynomial chaos expansion (aPCE) is presented for analytical, unified implementation of UQ and GSA in structural dynamics. For UQ of dynamic characteristics, statistical moments and probability distributions of dynamic characteristics are analytically derived. Specifically, the aPCE is used to analytically calculate the statistical moments, and then the maximum entropy principle (MEP) is adopted to derive the closed-form expressions of the probability distributions using the obtained statistical moments. As an extension of UQ, GSA, which aims to assess the quantitative contributions of different structural parameters to the resultant variations of dynamic characteristics, is also analytically achieved by simply post-processing the aPCE coefficients. The present aPCE UQ and GSA method is highly computationally efficient for large-scale, complex structures, and it is also generally applicable independent of parameter distributions. The proposed aPCE-based approach for UQ and GSA is validated through a numerical truss bridge by the brute-force Monte Carlo simulation (MCS), and then is applied to a long-span steel arch bridge. [ABSTRACT FROM AUTHOR]

Where R and I denote the real and imaginary parts of the complex correlation coefficient ;u1R and u1I denote the real and imaginary parts of u1. 1 R.J. Baxley, B.T. Walkenhorst, G. Acosta-Marum, Complex Gaussian ratio distribution with applications for error rate calculation in fading channels with imperfect CSI, Proceedings of IEEE Global Telecommunications Conference (GLOBECOM), 2010, 1-5R. J. Baxley, B. T. Walkenhorst, G. Acosta-Marum, Complex Gaussian ratio distribution with applications for error rate calculation in fading channels with imperfect CSI, Proceedings of IEEE Global Telecommunications Conference (GLOBECOM) (2010) 1-5. [Extracted from the article]

Detection of multiple damage using modal curvature in noisy environments has become a research focus of considerable challenge and great significance over the last few years. However, a noticeable deficiency of modal curvature is its susceptibility to noise, which usually results in a noisy modal curvature with obscured damage signature. To address this deficiency, this study formulates a new concept of complex-wavelet modal curvature. Complex-wavelet modal curvature features the ability to reveal and delineate damage under noisy conditions. The effectiveness of the concept is analytically verified using cracked beams with various types of boundary conditions. The applicability is further experimentally validated by an aluminum beam with a single crack and a carbon-fiber-reinforced polymer composite beam with three cracks in the laboratory with mode shapes measured by a scanning laser vibrometer. Both analytical and experimental results have demonstrated that the complex-wavelet modal curvature is capable of revealing slight damage by eliminating noise interference, with no need for prior knowledge of either material properties or boundary conditions of the beam under inspection. [ABSTRACT FROM AUTHOR]

This paper proposes a new nonlinear joint model updating method for shear type structures based on the instantaneous characteristics of the decomposed structural dynamic responses. To obtain an accurate representation of a nonlinear system׳s dynamics, the nonlinear joint model is described as the nonlinear spring element with bilinear stiffness. The instantaneous frequencies and amplitudes of the decomposed mono-component are first extracted by the analytical mode decomposition (AMD) method. Then, an objective function based on the residuals of the instantaneous frequencies and amplitudes between the experimental structure and the nonlinear model is created for the nonlinear joint model updating. The optimal values of the nonlinear joint model parameters are obtained by minimizing the objective function using the simulated annealing global optimization method. To validate the effectiveness of the proposed method, a single-story shear type structure subjected to earthquake and harmonic excitations is simulated as a numerical example. Then, a beam structure with multiple local nonlinear elements subjected to earthquake excitation is also simulated. The nonlinear beam structure is updated based on the global and local model using the proposed method. The results show that the proposed local nonlinear model updating method is more effective for structures with multiple local nonlinear elements. Finally, the proposed method is verified by the shake table test of a real high voltage switch structure. The accuracy of the proposed method is quantified both in numerical and experimental applications using the defined error indices. Both the numerical and experimental results have shown that the proposed method can effectively update the nonlinear joint model. [ABSTRACT FROM AUTHOR]

In cable-stayed bridges, cables are susceptible to fatigue damage due to cyclic traffic loads. The accurate monitoring of cable tension is critical in assessing fatigue damage and ensuring the service performance of cables. The vibration method establishes a formula between the cable tension and its vibration frequency. Therefore, the main challenge in identifying the time-varying cable tension is to determine the instantaneous frequencies (IFs) of the cable. Variational nonlinear chirp mode decomposition (VNCMD) exhibits promising advantages in identifying IFs from measured vibration signals, owing to its robustness against noise interference. However, hyperparameter selection significantly affects the IF identification accuracy of VNCMD, limiting its applicability. In this study, an improved VNCMD, termed variational nonlinear chirp mode extraction (VNCME), is proposed to address this limitation and is used to identify the time-varying cable tension. First, VNCME constructs a novel algorithm framework by modifying the optimisation function of VNCMD, eliminating the need for hyperparameter presetting and enabling the identification of a specific-order IF for the cable. The identified IF is then incorporated into the formula of the vibration method to calculate the time-varying cable tension. Examples of nonstationary signals of a sinusoidal function, finite element model of a cable, and laboratory experiments of scaled cables are used to illustrate the advantages of the proposed method, which proves to be successful in time-varying cable tension identification. In addition, the proposed method is implemented on an actual highway cable-stayed bridge to identify the time-varying cable tension over a 24-h duration. The results demonstrate that the proposed method is practical and reliable, providing a new path for the real-time monitoring of cable tension. [ABSTRACT FROM AUTHOR]

Stochastic model updating provides an effective way of handling uncertainties existing in real-world structures. In general, probabilistic theories, fuzzy mathematics or interval analyses are involved in the solution of inverse problems. However in practice, probability distributions or membership functions of structural parameters are often unavailable due to insufficient information of a structure. At this moment an interval model updating procedure shows its superiority in the aspect of problem simplification since only the upper and lower bounds of parameters and responses are sought. To this end, this study develops a new concept of interval response surface models for the purpose of efficiently implementing the interval model updating procedure. The frequent interval overestimation due to the use of interval arithmetic can be maximally avoided leading to accurate estimation of parameter intervals. Meanwhile, the establishment of an interval inverse problem is highly simplified, accompanied by a saving of computational costs. By this means a relatively simple and cost-efficient interval updating process can be achieved. Lastly, the feasibility and reliability of the developed method have been verified against a numerical mass–spring system and also against a set of experimentally tested steel plates. [ABSTRACT FROM AUTHOR]

Abstract: From a practical point of view, uncertainties existing in structural parameters and measurements must be handled in order to provide reliable structural condition evaluations. At this moment, deterministic model updating loses its practicability and a stochastic updating procedure should be employed seeking for statistical properties of parameters and responses. Presently this topic has not been well investigated on account of its greater complexity in theoretical configuration and difficulty in inverse problem solutions after involving uncertainty analyses. Due to it, this paper attempts to develop a stochastic model updating method for parameter variability estimation. Uncertain parameters and responses are correlated through stochastic response surface models, which are actually explicit polynomial chaos expansions based on Hermite polynomials. Then by establishing a stochastic inverse problem, parameter means and standard deviations are updated in a separate and successive way. For the purposes of problem simplification and optimization efficiency, in each updating iteration stochastic response surface models are reconstructed to avoid the construction and analysis of sensitivity matrices. Meanwhile, in the interest of investigating the effects of parameter variability on responses, a parameter sensitivity analysis method has been developed based on the derivation of polynomial chaos expansions. Lastly the feasibility and reliability of the proposed methods have been validated using a numerical beam and then a set of nominally identical metal plates. After comparing with a perturbation method, it is found that the proposed method can estimate parameter variability with satisfactory accuracy and the complexity of the inverse problem can be highly reduced resulting in cost-efficient optimization. [Copyright &y& Elsevier]

• A novel MFI method based on learning dictionary with double sparsity is proposed. • Dictionary learning technique is employed to design a better fit force dictionary based on the measured response data. • The force dictionary is assumed to be expressed sparsely over the base dictionary, and the moving force is sparse over the force dictionary simultaneously. • Sparse K-SVD algorithm is employed to realize the learning process through alternatively updating between double sparse codes. Moving force identification (MFI) is essential for the bridge safety as it is one of the major loads acting on the bridge deck. MFI techniques based on force dictionary are promising owing to their prominent performance in solving ill-posed problems and calculation efficiency. Since the specific forms of authentic moving forces are complex and unknown, a fixed dictionary normally adopted tends to fail in expressing moving forces sparsely enough. In this study, dictionary learning (DL) is introduced into the field of MFI to design a better fit force dictionary based on the measured response data. A novel MFI method based on learning dictionary with double sparsity is proposed. Firstly, the MFI equation in time domain which describes the relationship between moving force and measured structural responses is established. Then a sparse dictionary model is designed in which the force dictionary is assumed to be expressed sparsely over the base dictionary, and the moving force is sparse over the force dictionary simultaneously. Moreover, the sparse K-singular-value-decomposition (K-SVD) algorithm is employed to realize the learning process through alternatively updating between double sparse codes. Finally, the learned force dictionary and moving forces are estimated through base force dictionary and double sparse codes. Numerical simulations and experimental studies are carried out to investigate the performance of the proposed method, and the results clearly certify its effectiveness and robustness. [ABSTRACT FROM AUTHOR]

• An S-transform-based time-varying characteristic analysis method is proposed to extract the frequency fluctuations. • The approximation description between structural instantaneous frequency and S-transform is deducted. • A novel Gaussian window function is proposed for S-transform to improve the time-frequency resolution. • The laboratory vehicle-bridge dynamic interaction experiment is conducted for verification. • Parametric study involving road surface roughness are discussed in the numerical simulation. The health monitoring of bridge under moving vehicles demands a high-efficient time-varying characteristic analysis method to extract the frequency fluctuations caused by vehicle-bridge dynamic interaction effect. Characteristics identification of structures becomes a challenging task especially when time-varying vibration modes are involved. For this purpose, an instantaneous frequency identification technique based on modified S-transform reassignment is presented in this paper. Firstly, the general approximation between structural instantaneous frequency and S-transform is deduced. The Gaussian window is employed in S-transform and a novel first-order frequency function with two parameters is added into the Gaussian window function to improve the time–frequency resolution. The added function makes the Gaussian window scalable-sliding and the optimal values of the parameters are selected based on time–frequency concentration criterion. Considering the Rényi entropy as the measurement of criterion, the information performance extracted from S-transform spectrogram is evaluated. Then the time–frequency reassignment technique is applied to improve the readability of S-transform spectrogram to assure the accuracy of the extracted instantaneous characteristics, which enhances the time–frequency concentration efficiently. The proposed method is verified by numerical simulation, the results obtained by the spectrograms derived from short-time Fourier transform, wavelet transform, modified S-transform and reassigned modified S-transform are compared. Afterwards, the time-varying frequency components related to the vehicle-bridge interaction are extracted from the response signal. Numerical simulations involving different parameter analysis and laboratory experiments are conducted to validate the flexibility of the technique in vehicle-bridge system. The effect of vehicle weight, speed and road surface roughness on the identified results are discussed. Both numerical and experimental results have demonstrated that a higher time–frequency concentration in the spectrogram can be observed of the proposed modified S-transform reassignment method than other time–frequency methods, and subsequently a higher accuracy of instantaneous frequency identification is achieved which provides a good tool for time-varying characteristic analysis in bridge health monitoring. [ABSTRACT FROM AUTHOR]

• Uncertainty of raw FRFs are modelled by multivariate complex-valued Gaussian ratio distribution. • Likelihood function is formulated by embedding the model parameters into the probabilistic model of FRFs. • Vectorization processing is proposed to avoid time-consuming loop operations involved in likelihood function. • Distributed parallel computing scheme is utilized to allow TMCMC stochastic simulation. • Stochastic model updating can achieve much better results than deterministic approaches. This study was devoted to investigating stochastic model updating in a Bayesian inference framework based on a frequency response function (FRF) vector without any post-processing such as smoothing and windowing. The statistics of raw FRFs were inferred with a multivariate complex-valued Gaussian ratio distribution. The likelihood function was formulated by embedding the theoretical FRFs that contained the model parameters to be updated in the class of the probability model of the raw FRFs. The Transitional Markov chain Monte Carlo (TMCMC) used to sample the posterior probability density function implies considerable computational toll because of the large batch of repetitive analyses of the forward model and the increasing expense of the likelihood function calculations with large-scale loop operations. The vectorized formula was derived analytically to avoid time-consuming loop operations involved in the likelihood function evaluation. Furthermore, a distributed parallel computing scheme was developed to allow the TMCMC stochastic simulation to run across multiple CPU cores on multiple computers in a network. The case studies demonstrated that the fast-computational scheme could exploit the availability of high-performance computing facilities to drastically reduce the time-to-solution. Finally, parametric analysis was utilized to illustrate the uncertainty propagation properties of the model parameters with the variations of the noise level, sampling time, and frequency bandwidth. [ABSTRACT FROM AUTHOR]

Vehicle on-board monitoring methods use responses of running vehicles to identify track irregularity in real time. But the parameters of both the bridges and the vehicles in such processes may have a non-negligible degree of uncertainty or randomness that will inevitably lead to uncertainty in the track irregularity identification. Quantifying such uncertainty is a great challenge as the vehicle-bridge system (VBS) to be treated with is time-dependent and random. This paper proposes an on-board track irregularity identification algorithm that realizes an inverse random dynamic analysis of the VBS to quantify estimation uncertainty by using a Bayesian Kalman filter technology. The proposed algorithm utilizes sigma point sets to simulate the random parameters and the noises, and is therefore capable of implementing high-fidelity probability density propagation in the nonlinear state transfer and observation functions describing the VBS. The proposed algorithm is validated by numerical examples with various running states and parameter uncertainty levels. [ABSTRACT FROM AUTHOR]

• A dynamic force identification approach based on composite trigonometric wavelet. • The challenge task of determining the time unit length in advance can be avoided. • The trigonometric wavelet performances better than the polynomial functions. • The identification accuracy can be enhanced progressively according to actual needs. The shape function-based methods are promising in dynamic force identification. However, the selection of shape function is still an issue as it determines the identification accuracy and efficiency to a large extent. This paper proposes a dynamic force identification approach by using the composite trigonometric wavelet as shape function which takes advantage of its 'wave' property in dynamic force expression. The whole domain of the dynamic force time history is segmented into different time units following the thought of finite element discretization and the local force is approximated by linear shape functions. Subsequently the force-response equation is established by assembling the calculated responses induced by shape function forces and the corresponding measured responses of all time units. Then trigonometric wavelet shape function is added to enhance the approximation capability of shape function and improve the identification accuracy progressively. Examples are employed to illustrate the effectiveness and superiority of the proposed method. [ABSTRACT FROM AUTHOR]

Vehicle and bridge systems (VBSs) are subjected to stochastic excitations. Significant research effort has been devoted to the stochastic response analysis of VBSs. In most previous studies, VBSs have been treated as deterministic systems with certain parameters. However, the real VBSs always have uncertain parameters, such as the mass and spring stiffness of the vehicles and elastic modulus of the bridge material, which significantly affect the response power spectrum (RPS) estimation. Moreover, the equations of motion of the VBSs are essentially second-order differential equations with time-dependent and random coefficient matrices, and the excitation-response spectral relation cannot be directly described using the frequency response functions. These properties render the available methods unsuitable for the RPS estimation of the VBSs when both structural uncertainty and stochastic excitations are considered. In the present study, a harmonic-wavelet approach is proposed to estimate the time-varying RPSs of the VBSs with random parameters. The second-order differential equations of motion of VBSs with random parameters are first transformed into linear algebraic equations with random coefficient matrices using a harmonic-wavelet analysis technique; then, the excitation-response spectral relation of VBSs is derived based on these linear algebraic equations. Finally, the RPSs of VBSs with uncertain parameters are solved using the perturbation technique. The accuracy of the proposed method was verified by comparing it with the Monte Carlo method using a numerical example of railway VBS. The numerical simulation results show that the parameter uncertainties significantly affect the RPSs of the VBSs and increase the amplitude of the time-varying RPSs. The uncertainty of vehicle stiffness was the most sensitive factor for vehicle RPSs, and the bridge elastic modulus was the most sensitive factor for the bridge RPSs. [ABSTRACT FROM AUTHOR]

• Life-cycle monitoring of high-strength bolt connections is studied using AE data. • A physics-guided deep learning framework is developed based on EWT, CNN, and GMM. • Manifold damage mechanisms overlapped among different loading stages are diagnosed. • Four loading stages are identified according to proportions of damage mechanisms. Aiming at life-cycle condition monitoring of high-strength bolt connections, a physics-guided deep learning framework integrating supervised and unsupervised learning was developed to diagnose manifold damage mechanisms and identify different loading stages using acoustic emission (AE) data. For data preprocessing, empirical wavelet transform (EWT) was introduced to provide adaptive and compact time–frequency representations of AE signals where diverse damage mechanisms behaved with various energy distributions. A convolutional neural network (CNN) model was first built to extract damage-related features from multiscale EWT matrices of AE data labeled by the loading stages I, II-III, and IV that were redefined according to their dominant damage mechanisms. Stage I (static friction) could be identified directly, which was characterized by the distinct damage mechanism of plastic deformation of asperities. After that, Gaussian mixture model (GMM) was applied to cluster the extracted feature vectors of AE data induced by multiple damage mechanisms overlapped among stages II (sliding), III (confined), and IV (failure). With the help of Bayesian information criterion (BIC), three damage mechanisms were recognized, including abrasive wear of asperities, adhesive wear of asperities, and plastic deformation and cracking of screw, which were validated by their EWT components. Finally, based on the proportions of AE data corresponding to diverse damage mechanisms, the rules were established for identifying the four loading stages of high-strength bolt connections. Experimental results demonstrated that the three-class CNN model with data labeled by the redefined loading stages achieved a higher accuracy than the four-class CNN model with data straightforwardly labeled by the original loading stages. Not only the critical damage states of static friction failure (looseness) and shear failure, but the four loading stages of high-strength bolt connections and their overlapped damage mechanisms were successfully identified by the physics-guided CNN-GMM framework. [ABSTRACT FROM AUTHOR]

• Field-test data analysis emphasizes the possibility of non-Gaussianity of FFT. • Pronounced high-kurtosis feature with heavy tails is detected. • The complex t distribution is proposed for the robust statistical modeling. • The PDFs of multivariate and univariate complex t random variables are derived analytically. • The performance of complex Gaussian and t models are evaluated using field test data. Frequency-domain operational vibration responses have been widely applied in structural health monitoring (SHM). Characterizing and quantifying their uncertainties are of fundamental importance for enhancing the robustness of SHM technologies. The classic complex Gaussian distribution is being increasingly used to model the distributions of fast Fourier transform (FFT) coefficients due to its elegant and convenient mathematical nature. However, the field-test data analysis of engineering structures under operational vibrations in this study emphasize the possibility of non-Gaussianity of some observations. The higher peaks and heavier tails than those of a Gaussian distribution emerge as observable features. As a member of the general family of elliptically symmetric distributions, the t distribution has been widely recognized as a useful extension of the Gaussian distribution for the robust statistical modeling of data sets with heavier-than-normal tails. In this paper, we consider using the complex-valued t distribution to characterize the FFT coefficients with high kurtosis and heavy tails. The probability density function (PDF) of multivariate proper complex t random variables are derived based on the equivalent counterparts in the real-valued domain. The marginal PDFs of the real part, the imaginary part, the magnitude, and the phase of a univariate complex-valued t random variable are also derived analytically based on advanced integration techniques. The field test data of different engineering structures provide an illustration of the performance of complex Gaussian and complex t probabilistic models at different frequencies evaluated via the K-S test, goodness-of-fit test, and probability plots. [ABSTRACT FROM AUTHOR]

Highlights • First review paper focused on transmissibility-based system identification. • Fundamental concepts of transmissibility and its basic properties are summarized. • The literature review categorizes transmissibility functions as local and global. • Application of transmissibility function in modal analysis, damage detection and model updating are reviewed. • Gaps requiring additional work and possible future trends are highlighted. Abstract The difficulty of achieving controlled input has led to the development of new output-only structural health monitoring (SHM) approaches. Without measuring the input or assuming a specific input model, a transmissibility function (TF) as a mathematical representation of the output-to-output relationship has proven to be an attractive tool in SHM. The state of the art and challenges to TF analysis that cast SHM in the context of a system identification (SI) paradigm are reviewed and discussed in this study. This review starts with an overview of the fundamentals of TFs by illustrating its categories, connections with frequency response functions (FRFs), and basic properties. By categorizing TFs as local and global, this literature review limits the practice of various methodologies to the following key aspects: modal analysis, damage detection, and model updating. A brief treatment of the basic ideas is presented while a comprehensive and critical explanation of their contents is not attempted. Based on the review, existing studies are discussed, highlighting gaps requiring additional work and possible future trends for TF-based system identification. [ABSTRACT FROM AUTHOR]

• A hierarchical clustering algorithm enhanced by multivariate probabilistic distance is proposed for damage detection. • Multivariate probabilistic distance between different TF vectors is analytically derived by Laplace approximation. • A function vectorization scheme is developed to facilitate data fusion and improve computational efficiency. • The new hierarchical clustering can accommodate the uncertainty and correlation of multiple TFs. • Case studies validate the advantages of the method over hierarchical clustering with deterministic distance. This paper proposes a new damage detection method by integrating the advantage of transmissibility function (TF) as a health index sensitive to damage but robust to excitation and agglomerative hierarchical clustering (AHC) with intuitive explanation and visualization but avoiding specifying the number of clusters. Different from conventional AHC-based damage detection methods utilizing deterministic distance as a similarity metric and ignoring the distribution of structural features, a multivariate probabilistic distance-based similarity metric is proposed in this study to account for the uncertainty and correlation of multiple TFs following multivariate complex-valued Gaussian ratio distribution. To realize this, an analytically tractable approximation of the multivariate probabilistic distance is derived by Laplace's asymptotic expansion to avoid high-dimensional numerical integration. To accelerate the computation of probabilistic distances over a wide frequency band that are fused to formulate the similarity metric in AHC, a function vectorization scheme is proposed to avoid the time-consuming loop operation among different frequency points. A threshold is established via bootstrapped Monte Carlo simulation to cut the dendrogram produced by AHC. Two case studies are used to validate the performance of the proposed method, indicating that, compared to the damage detection methods based on the deterministic distance of the TF, the proposed method exhibits better performance due to improving the similarity metric based on multivariate probabilistic distance properly accommodating the correlation of different TFs. [ABSTRACT FROM AUTHOR]

Modal identification is critical for structural condition monitoring. Variational mode decomposition (VMD) has been widely applied to identify modal parameters and has achieved excellent performance. It is crucial for VMD to predefine the decomposition parameters, that is, the mode number and balance factor. However, in practical engineering, abnormal impulses and heavy noise render it difficult to preset the mode number and balance factor. Therefore, a novel method, termed orthogonal and recursive VMD (ORVMD), is proposed to overcome the difficulty of setting decomposition parameters in advance. ORVMD consists of two components: recursive VMD (RVMD) and a rough-to-precise decomposition scheme based on an orthogonal algorithm. RVMD is an iterative method of VMD that is used to circumvent the difficulty of predefining the mode number. A rough-to-precise decomposition scheme based on an orthogonal algorithm is proposed to address the difficulty of setting the balance factor. Furthermore, the proposed ORVMD in combination with the Hilbert transform (HT) is employed to estimate the modal parameters of the structures. The raw signals are pre-processed by using the random decrement technique (RDT) to obtain its random decrement signature (RDS) and then the proposed method is applied to the RDS to identify the modal parameters of a simulated system and a real arch bridge. The obtained results show that the proposed method outperforms other existing methods in separating multicomponent signals; thus, it is an efficient method for identifying the natural frequencies and damping ratios of structures. [ABSTRACT FROM AUTHOR]

Accurate identification of moving load is one of the main tasks in the field of bridge health monitoring. Due to the uncertainty in practical bridge, moving load identified via deterministic method is bound to be inconsistent with reality. As statistical information on the uncertain parameters is usually limited for actual bridges, this paper proposes a time-domain moving load identification method considering the uncertainty in the finite element model. Firstly, the moving load identification is divided into nominal value and interval radius identification. Then the deterministic methods based on shape function and second-order interval analysis are presented to identify the nominal value and interval radius, respectively. Finally, the effectiveness and superiority of the proposed method are verified through numerical and experimental examples, and the influence of the measurement point number, measurement noise, and vehicle velocity is investigated. The outcomes indicate that the results obtained by the deterministic method may deviate seriously from the actual one, and the calculation efficiency can be enhanced markedly compared with the probability method. The identification accuracy of the proposed method is higher than that of the first-order method at the same computational cost. [ABSTRACT FROM AUTHOR]

This paper presents finite element (FE) model updating method for real bridge structure under operational condition using modal flexibility. The theoretical background of the updating procedure is presented. The case study of a simulated simply supported beam demonstrates an effectiveness of modal flexibility in objective function. This objective function is then implemented in case study of a real concrete-filled tubular arch bridge. The bridge was tested under operational condition. Followed by the three-dimensional FE modeling of the bridge, an eigenvalue sensitivity study is carried out to select the most sensitive parameters to the concerned modes. Guyan technique is used to the mass normalization of the mode shapes extracted from ambient modal test to calculate the modal flexibility. The updated FE model of the bridge is able to produce a sufficient improvement on modal parameters of the concerned modes, which is in close agreement with the experiment results and updated parameters still preserve the physical meaning in practice. [Copyright &y& Elsevier]

• A unified scheme is proposed to solve complex-valued ratio distribution without closed-form solution. • A unified formula is derived by reducing the problem into multi-dimensional integrals. • Fast sparse-grid quadrature scheme is utilized to improve the efficiency of numerical integrals. • The unified method enables uncertainty quantification for FRFs and TFs. • The unified scheme is applicable to statistical inference for arbitrary ratio random variables. Complex-valued ratio distributions arises in many real applications such as statistical inference for frequency response functions (FRFs) and transmissibility functions (TFs) in structural health monitoring. As a sequel to our previous study, a unified scheme to solving complex ratio random variables is proposed in this study for the case when it is highly non-trivial or impossible to discover a closed-form solution such as the complex-valued t ratio distribution. Based on the probability transformation principle in the complex-valued domain, a unified formula is derived by reducing the concerned problem into multi-dimensional integrals, which can be solved by advanced numerical techniques. A fast sparse-grid quadrature (SGQ) scheme by constructing multivariate quadrature formulas using the combinations of tensor products of suitable one-dimensional formulas is utilized to improve the computational efficiency by avoiding the problem of curse of integral dimensionality. The unified methodology enables the efficient calculation of the probability density function (PDF) of a ratio random variable with its denominator and nominator specified by arbitrary probability distributions including Gaussian or non-Gaussian ratio random variables, correlated or independent random variables, bounded or unbounded ratio random variables. The unified scheme is applied to uncertainty quantification for raw FRFs and TFs without any post-processing such as averaging, smoothing and windowing, and the efficiency of the proposed scheme is verified by using the vibration test field data from a simply supported beam and from the Alamosa Canyon Bridge. [ABSTRACT FROM AUTHOR]