Output-only Data Research Articles

A new optimal sensor placement method for virtual sensing of composite laminate

Identifying modal coordinates from output-only data is a key link of virtual sensing technology based on the modal extension method. It is also one of the goals of optimal sensor placement (OSP). Traditional OSP methods are based on the underlying assumption that the estimated values of modal coordinates are unbiased estimates of real values. However, due to uncertainty and the characteristics of an inverse problem, the unbiased estimation obtained from the output-only data may seriously deviate from the true value. This study proposes a new OSP method for composite virtual strain sensing, which can obtain the global unbiased estimation of modal coordinates. First, a Bayesian probabilistic model for virtual sensing considering model uncertainty and measurement error is formulated. Then, the K-L divergence, which measures the reduction in utility by removing sensors from the full configuration, is used to obtain the unbiased estimation of modal parameters. Finally, using the regularization mechanism in the Bayesian method, a new variance determination method is proposed to improve the stability of the solution. Considering both unbiasedness and stability, NSGA-II multi-objective optimization algorithm and two OSP evaluation criteria are used to obtain the final optimal sensor placement. To illustrate the effectiveness of the proposed method, an example involving a laminate plate is considered, accompanied by comprehensive discussions.

MAntRA: A framework for model agnostic reliability analysis

We propose a novel model-agnostic data-driven reliability analysis framework for time-dependent reliability analysis. The proposed approach – referred to as MAntRA – combines Bayesian inference and stochastic differential equations to evaluate the reliability of stochastically-driven dynamical systems for which the governing physics is a priori unknown. A two-stage approach is adopted: in the first stage, an efficient variational Bayesian equation discovery algorithm is developed to determine the governing physics of an underlying stochastic differential equation (SDE) from measured output-only data. The developed algorithm is efficient and accounts for epistemic uncertainty due to limited and noisy data and aleatoric uncertainty because of environmental effects and external excitation. In the second stage, the discovered SDE is solved using a stochastic integration scheme, and the probability of failure is computed. The efficacy of the proposed approach is illustrated in four numerical examples. The results obtained indicate the possible application of the proposed approach for reliability analysis of in-situ and heritage structures from on-site measurements.

A new approach of ensemble learning in fully automated identification of structural modal parameters of concrete gravity dams: A case study of the Koyna dam

This study proposes a new automatic modal identification algorithm that identifies the system's modal parameters based on output-only data. The identification process is based on a stochastic subspace algorithm. Stochastic subspace algorithms only need the model degree to identify the modal parameters of the system. A model degree is an appropriate degree of the state-space model and it is estimated by using the Singular Value Criterion. Several clustering algorithms with different approaches have been used. The most powerful data fusion algorithm, which is called Dempster-Schaffer Theory is applied for integrating the results of the algorithms and preventing divergence of results. Furthermore, it focused on measuring the uncertainty of the identified modal parameters. In the proposed modal identification algorithm, only one data set is used for identification. In this situation, the variety of data is small and limited, and statistical resampling methods seem to be a good option. Establishing the above cycle enables the identification algorithm to identify modal parameters and quantify their uncertainty based only on a data set. This algorithm uses many fields, such as data collection, signal processing techniques, model reduction, feature extraction, clustering and data fusion. The Koyna gravity dam is selected as a case study to evaluate the performance of the developed algorithm. In the first step, a comparative study is carried out to verify the developed finite element model of the Koyna dam. Afterwards, the dam is excited by the Koyna earthquake, and dynamic responses of the dam are collected from the minimum number of points required for system identification. It is noteworthy the number of modes that are identified in this work is equal to 4 vibration modes. The maximum difference between the natural frequencies identified by the present algorithm and the finite element model does not exceed 4.33%. Also, the average difference is less than two percent, which indicates the high accuracy of the system identification algorithm. Also, all the vibration modes in the desired frequency range have been correctly identified. The results indicate that modal parameters with remarkable precision have been extracted by the modified algorithm.

Parameter, input and state estimation for linear structural dynamics using parametric model order reduction and augmented Kalman filtering

Tracking the evolution in time of parameters, input and states of a structural dynamic system is often difficult, since their direct measurement can be problematic or even impossible. It is of great interest to estimate these quantities based on output-only data from a limited set of sensors. This work proposes an estimation technique for states, inputs and material parameters for structural dynamics models based on an Augmented Extended Kalman Filter. A parametric Model Order Reduction technique is proposed to construct a Reduced Order Model which maintains an explicit dependency on material parameters, enabling the parameter estimation thanks to a low computational cost and an efficient derivation of the linearized system. The choice of sensor configurations that ensure the observability of unknown quantities is discussed as well. The proposed methodology shows highly promising results and could be employed for model refinement or condition monitoring. The methodology is validated both numerically and experimentally, using data acquired on a scaled wind turbine blade, with errors on the estimated parameters lower than 3.5% with respect to experimentally identified parameter values.

Structural damage identification by a sensitivity-based finite element model updating method using transmissibility function data

ABSTRACT This study presents a novel sensitivity-based finite element model updating approach to estimate structural parameters using output-only data. An exact sensitivity relation based on transmissibility data is introduced. Partially measured structural responses are included in mathematical formulations to address incomplete measurement problems and improve the accuracy of the sensitivity relationship. The least-squares algorithm is used to solve the normalised set of equations, and also a weighting approach is used to improve the parameter estimation results. The performance of the proposed method is evaluated based on numerical data of an offshore jacket platform model. In addition to taking the added mass in the calculations, appropriate approaches have been used to select the excitation points and optimal sensor placement. The stability of the proposed method against measurement and mass modelling errors and incompleteness of the measured responses are studied.

Proportional flexibility-based damage detection for buildings in unknown mass scenarios: The case of severely truncated modal spaces

Modal flexibility-based methods are recognized in the literature as effective methods for detecting damage in structures using vibration measurements. However, the application of such methods with output-only data (as is the case of civil structures tested under ambient excitations) is challenging, because the mass normalization factors required for assembling the modal flexibility matrix cannot be estimated directly from the data. To address this issue, a method for output-only damage detection and localization in building structures that can be applied with minimal or no a-priori information about the structural masses has been proposed in a previous work by the authors. In the method, to estimate the mass distribution and to assemble the proportional flexibility matrices, the modal orthogonality relationships that involve the mode shapes and the structural masses are used to form a system of equations, and through that system of equations, an inverse problem is solved for the estimation of the mass distribution. Reducing the number of available modes, however, reduces the number of available equations, while the number of unknowns (i.e., the number of degrees of freedom) stay the same; hence, the approach cannot be applied when the modal space is severely truncated. This paper proposes a technique for estimating the mass distribution and assembling proportional flexibility matrices from output-only vibration data that overcomes the mentioned limitations and that can be applied for any number of identified modes, including the case in which the modal space is severely truncated. The proposed technique is then integrated into the above-mentioned method for output-only damage detection in building structures, to verify the applicability of the whole procedure when considering the case of severely truncated modal spaces. Numerical simulations and experimental vibration tests were used to demonstrate the validity and effectiveness of the approaches proposed in this paper.

Machine‐learning‐based methods for output‐only structural modal identification

In this study, we propose a machine-learning-based approach to identify the modal parameters of the output-only data for structural health monitoring (SHM) that makes full use of the characteristic of independence of modal responses and the principle of machine learning. By taking advantage of the independence feature of each mode, we use the principle of unsupervised learning, making the training process of the deep neural network becomes the process of modal separation. A self-coding deep neural network is designed to identify the structural modal parameters from the vibration data of structures. The mixture signals, that is, the structural response data, are used as the input of the neural network. Then we use a complex loss function to restrict the training process of the neural network, making the output of the third layer the modal responses we want, and the weights of the last two layers are mode shapes. The deep neural network is essentially a nonlinear objective function optimization problem. A novel loss function is proposed to constrain the independent feature with consideration of uncorrelation and non-Gaussianity to restrict the designed neural network to obtain the structural modal parameters. A numerical example of a simple structure and an example of actual SHM data from a cable-stayed bridge are presented to illustrate the modal parameter identification ability of the proposed approach. The results show the approach's good capability in blindly extracting modal information from system responses.

Uncertainty quantification in data-driven stochastic subspace identification

A crucial aspect in system identification is the assessment of the accuracy of the identified system matrices. Stochastic Subspace Identification (SSI) is a widely used approach for the identification of linear systems from output-only data because it combines a high computational robustness and efficiency with a high estimation accuracy. Practical approaches for estimating the (co)variance of system matrices that are identified using SSI exist for the case where the system matrices are obtained from the shift-invariant structure of the extended observability matrix. However, in data-driven SSI, the system matrices are often obtained in a different way, using identified state sequences. This case is treated in the present work, for three common types of weighting. First, it is shown that the estimated system matrices depend entirely on sample output correlation estimates, the covariance of which can be straightforwardly estimated. Subsequently, a linear sensitivity analysis of the data-driven SSI algorithm is performed, such that the covariance of the identified system matrices can be also computed. A memory efficient implementation is obtained by computing the related Jacobian only implicitly. An extensive numerical validation, covering a range of parameter choices, demonstrates the accuracy of the estimated variance of the identified system description. Finally, the practical use of the method in the context of operational modal analysis is demonstrated in an experimental case study.

Weighted Transmissibility-Based Operational Modal Analysis for Identification of Structures Using Seismic Responses

Transmissibility-based operational modal analysis is an effective approach to identify the modal parameters using the output-only data from the ambient vibrations. This approach, based on the SVD technique, has a good ability to determine the vibrational modes with high certainty. However, the results of the present research indicate that the transmissibility approach has not an appropriate performance in identifying the vibrational modes using the short random records such as seismic records. These records have a high spectral leakage and dispersion due to the non-periodic harmonic components caused by the vibrational characteristics of the system and the strong ambient excitations (e.g., earthquake). In this paper, an efficient approach called weighted transmissibility-based operational modal analysis is proposed to properly identify the natural frequencies and mode shapes using the short seismic records. The proposed approach combines the single-reference weighted transmissibility functions associated with different transferring outputs obtained from single loading condition in a unique matrix formulation. It extracts the vibrational modes using the efficient $${\text{PIS}}\left( \omega \right)$$ function calculated by the singular values of the weighted transmissibility matrices. In this research, several theoretical examples, including 2-DOF, 3-DOF, 5-DOF and 10-DOF systems, have been provided to indicate the capabilities of the proposed approach in identifying the modal parameters of structural systems.

Dynamic behaviour of a 13-story reinforced concrete building under ambient vibration, forced vibration, and earthquake excitation

This paper presents a unique study of dynamic behaviour of a full-scale 13-story reinforced concrete building under forced vibration, ambient vibration, and distal earthquake-induced excitation. Initially, an eccentric mass shaker located on the upper floor of the building was used to excite the building while instrumented with eight accelerometers. Then, ambient vibrations induced by traffic and wind were recorded over a period of two weeks, utilizing over 40 tri-axial accelerometers. During the sampling period, the building was excited by a M6.5 earthquake and high-quality vibration data sets were recorded. Modal parameters of the building were identified with a range of frequency and time domain system identification (SI) methods, and the influence of excitation force characteristics on performance of various techniques was also investigated. A finite element model of the building was developed using SAP2000 to compare modal properties to the identified counterparts from experiments. Results showed a strong correlation between modal parameters identified by different SI methods, and all techniques provided accurate estimation of modal parameters when used with output-only data. Modal parameters determined from ambient vibrations were comparable to their forced vibration counterparts, providing evidence that ambient vibration testing can be as effective as forced vibration testing in determining modal parameters for similar size structures. The near stationary vibration data can produce accurate estimation of modal parameters. However, further study is required to facilitate application of output-only SI techniques with non-stationary data. There was a good match between peak displacement amplitudes obtained by the numerical model and those experimentally measured at different excitation frequencies. This match demonstrated the accuracy of the finite element model to predict realistic dynamic behaviour of the building under various harmonic excitation forces.

응답의존 모달 데이터를 이용한 고층건물의 FE모델 향상

응답의존 모달 데이터를 이용한 고층건물의 FE모델 향상

A novel mode shape reconstruction method for damage diagnosis of cracked beam

Abstract This paper presents a novel mode shape reconstruction method for damage diagnosis of cracked beam. The proposed method uses output-only data of damaged structure to identify and reconstruct the mode shape of healthy structure. Following that, the curvature mode shapes (CMS) is employed as a damage sensitive feature to locate the damage. In order to identify the mode shape of a target structure with measurement noises, EMD with intermittency criteria is applied to separate dynamic responses into individual modal responses of all grid points on the target structure. A statistical analysis is proposed to evaluate mode shapes from obtained modal responses. Then the Fourier series fitting is employed to mathematically represent the mode shape of the healthy structure. After that, the curvature mode shapes (CMS) which has been investigated as a promising feature for damage identification are acquired by a central difference approximation. Numerical studies are conducted for methodology illustration and validation. Effects of broader damage, multiple damage and measurement noise level are investigated in detail. The accuracy and effectiveness of the proposed method are demonstrated and verified by comparison with conventional method. Finally, a two-span continuous steel I-girder bridge is used to demonstrate the proposed method for more practical applications.

Modal testing and detection of pretension deviation in a cable dome structure

Pretension deviation may cause stiffness degradation and overstress that can compromise the safety of tensile structures, which can be diagnosed by modal identification. This article presents modal tests on a 1:10 scaled model of a herringbone-ribbed cable dome structure. An optimal sensor placement scheme is proposed to observe the geometric stiffness change induced by pretension deviation. Based on the tests, different output-only modal identification techniques were implemented. A substructure strategy was adopted to overcome the limited measurement quantity and provide localized diagnoses. The experimental results show that operational modal analysis methods based on output-only data can effectively identify major modes of massive structures. The sensitivity of modal characteristics to pretension deviation is also evaluated via experimental comparisons, and modifications are implemented in an analytical finite element model to approximate the test model. The identified modal information can help locate stiffness degradation and thereby pretension loss in tensile structures. A modified modal strain energy method is proposed to detect pretension loss from decentralized testing and is verified by the test results.

Structural scaling factor identification from output-only data by a moving mass technique

Abstract Ambient vibration testing is a convenient and cost-effective alternative to impact testing as it doesn’t require artificial excitation devices and traffic control. The lack of input force measurement in ambient vibration testing, however, does impede the identification of structural scaling factors, which are extremely important for Frequency Response Functions (FRFs) estimation and flexibility identification. A moving mass technique of identifying structural scaling factors from output-only data is proposed in this article. By this method, a vehicle passing over a bridge is regarded as a moving mass with spatial and time variation on the structure, inducing a vehicle-bridge system with time-varying modal properties. Theoretical derivation will be performed to identify scaling factors from the measured vibration responses of the bridge with and without the moving vehicle. Magnitudes of FRFs and the structural flexibility matrix can be obtained once the scaling factors are identified. The proposed method delivers comparable results to impact testing but is considerably more convenient. Numerical and experimental examples are studied to verify the effectiveness of proposed method. The effects of vehicle speed, mass value and measurement noise are also investigated.

Output-only damage detection in buildings using proportional modal flexibility-based deflections in unknown mass scenarios

Methods based on the evaluation of structural deflections from experimental modal flexibility matrices are important tools in vibration-based damage detection, especially for building structures. Modal flexibility matrices, however, can not be directly estimated when dealing with output-only vibration data (e.g. in the important case of structures tested under ambient vibrations). One can use an a-priori estimate of the mass matrix of the structure or the added mass method to obtain the modal normalization constants needed to assemble the flexibility matrix, but these operations can be challenging for real-life civil structures. To address this problem, a modal flexibility-based approach for output-only damage detection and localization in building structures that can be applied with minimal or no a-priori information on the structural masses is proposed in this paper. This approach is also able to deal with the general case in which mass modifications, e.g. due to operational variability in structures, are present before and after damage. The approach is based on the estimation (directly from output-only data) of modal flexibility-based deflections of building structures that are proportional with respect to the corresponding true deflections. From analytical investigations it was found that the missing scaling factor between the two deflections can be made theoretically equal to the total mass of the structure, as proposed in the paper. Then, interstory drifts evaluated from the proportional deflections are used for damage detection and localization according to two proposed strategies. The first strategy can be applied with minimal a-priori information on the masses (i.e. using a parameter that quantifies an eventual relative modification of the total mass of the structure before and after damage). The second one is a more advantageous strategy that can be applied from output-only data without any a-priori information on the masses, even in the case in which the masses are varied before and after damage. The effectiveness of the proposed approach was demonstrated using both numerical simulations and experimental vibration tests on frame building structures.

Operational modal analysis and fatigue life estimation of a chisel plow arm under soil-induced random excitations

Chisel plow is an important implement for primary tillage when the soil is dry and firm. In this research, fatigue life of a prototype of chisel plow arm under soil induced random excitation is analyzed. For this purpose, first, the field test is carried out on the chisel plow arm under soil-induced random excitations and operational modal analysis of the chisel plow arm is performed using stochastic subspace identification approach. In addition, a finite element model is constructed and successfully validated by stochastic subspace identification results. Then, the validated finite element model and power spectral density results of the measured random responses are used to achieve a process of the Mises stress in the arm needed for fatigue analysis. Finally, the fatigue life estimation of chisel plow arm is carried out in frequency domain using Wirsching-Light and Dirlik methods for different values of geometrical variables. The results of operational modal analysis demonstrate the possibility of identifying modal model of chisel plow arm in its operating condition on a basis of output-only data and the results of fatigue life estimation show that Wirsching-Light method provides a marginally safer prediction for chisel plow.

Ambient vibration test-based deflection prediction of a posttensioned concrete continuous box girder bridge

Ambient vibration test generally only outputs basic modal parameters including frequencies, damping ratios, and unscaled mode shapes, which cannot directly support decision making of structural maintenance and management. In this article, structural deep-level parameters including unscaled and scaled flexibility identification of a posttensioned concrete continuous box girder bridge are studied by performing ambient vibration test, which is able to predict structural deflections by multiplying the static load with the identified flexibility. Ambient vibration test of the bridge and basic modal identification are firstly performed. Then, the method of unscaled flexibility identification is proposed by investigating the relationship between the frequency response function estimated from ambient test data and the analytical one. Finally, a mass-changing strategy is utilized to identify the scaled flexibility from output-only data, in which the key issue is to identify mass-normalized scaling factors by performing ambient vibration tests of tested structure before and after changing its mass. Numerical simulation of a simply supported beam and field test of a three-span bridge has been conducted to verify the capability and reliability of the proposed method. The good agreement between the predicted deflections from the identified flexibility and those measured from the static test successfully illustrates the effectiveness of the proposed method.

System Identification of One Historical Bridge Using Dynamic Test Data

Obtaining accurately estimation of modal parameters from dynamic test data is an essential part for analyzing civil engineering structures. A set of dynamic field tests were conducted at a historic bridge. These tests provided an opportunity to identify the modal properties of the bridge in current condition. The system identification of the bridge was performed by using two different methods: (1) peak picking method and (2) eigensystem realization algorithm. By analyzing the output-only data, the natural frequencies in the vertical, lateral and torsional directions were identified. After comparing the data between measurements and the finite element model, the identified parameters using these system identification methods are found to be considerable for each type of tests. And through eigensystem realization algorithm, much more comprehensive results were obtained than peak picking method.

An adaptive filtering-based solution for the Bayesian modal identification formulation

The online identification of the dynamic properties of large structures is an integral part of structural health monitoring. The success of any vibration-based condition monitoring system requires a computationally efficient identification procedure with the potential to be used online. Computational efficiency and automation are the two fundamental requirements of an effective identification procedure. In this study, we propose a solution of a Bayesian-based formulation for the online extraction of structural modal parameters using output-only data. The proposed algorithm uses a Bayesian maximum a posteriori (MAP) estimation; the objective function is defined as a posterior probabilistic density estimation of the dynamic parameters obtained from observations. The conditional probability distribution function (PDF) is computed using a Bayesian approach to minimum mean-square estimation. To reduce the computational costs, we propose to employ an adaptive filtering technique where expansion and truncation of the conditional PDF are implemented to consider the measurement time series as finite-order Markov processes. The methodology is validated through analyses of two documented examples. Comparison with traditional identification methods clearly shows the superiority of the proposed algorithm. The findings demonstrate that the Bayesian-based methodology provides reliable modal parameter identification in dynamic systems, even for data with high noise content.

Scaling Mode Shapes in Output-Only Structure by a Mass-Change-Based Method

A mass-change-based method based on output-only data for the rescaling of mode shapes in operational modal analysis (OMA) is introduced. The mass distribution matrix, which is defined as a diagonal matrix whose diagonal elements represent the ratios among the diagonal elements of the mass matrix, is calculated using the unscaled mode shapes. Based on the theory of null space, the mass distribution vector or mass distribution matrix is obtained. A small mass with calibrated weight is added to a certain location of the structure, and then the mass distribution vector of the modified structure is estimated. The mass matrix is identified according to the difference of the mass distribution vectors between the original and modified structures. Additionally, the universal set of modes is unnecessary when calculating the mass distribution matrix, indicating that modal truncation is allowed in the proposed method. The mass-scaled mode shapes estimated in OMA according to the proposed method are compared with those obtained by experimental modal analysis. A simulation is employed to validate the feasibility of the method. Finally, the method is tested on output-only data from an experiment on a five-storey structure, and the results confirm the effectiveness of the method.