Modal Parameters Research Articles

Robust model updating in structural dynamics using a new non-implausibility-motivated optimisation approach

Over the last few decades, model updating has become popular in structural dynamics, as it can be used to calibrate (finite element) models, with applications in assessing whether damage has occurred in a structural health monitoring context. Early approaches focused on determining the “best” fitting model in a deterministic manner. For example, mathematical optimisation was employed to minimise the discrepancy between measured and simulated modal parameters. More up-to-date approaches take uncertainties, e.g., due to measurement errors or model discrepancy, into account. In this context, Bayesian model updating has become increasingly popular. Recently, “likelihood-free” approaches have been proposed as an alternative to (exact) Bayesian model updating, with Bayesian history matching (BHM) being a promising “likelihood-free” technique. However, since BHM is based on an approximation of the simulation model using a Gaussian process regression (GPR), it can become inaccurate for highly non-linear and especially for (quasi-)discontinuous problems. Therefore, in this work, a new non-implausibility-motivated optimisation (NIMO) approach is proposed, which overcomes the non-linear space problem. The method is a combination of global optimisation and GPR. Global optimisation is used to accurately determine a non-implausible region in the design space, even for discontinuous problems. Subsequently, a GPR is fitted within the non-implausible region to efficiently approximate a posterior distribution. First, the NIMO approach is verified using test functions. Second, a validation is conducted by localising damage on a laboratory beam structure. It is demonstrated that the NIMO approach yields more robust results compared to BHM, while its computing times are manageable and – depending on the objective function – even smaller compared to BHM.

Static and Dynamic Response Characteristics of a Full-Scale Long-Cantilever Tower under Various Loading Conditions

Long-cantilever transmission towers are drawing more and more attention in ultrahigh-voltage (UHV) transmission projects. To meet the requirements of high-voltage safety clearance, the structural characteristics of long-cantilever transmission towers are significantly different from traditional ones. Aiming to ensure the safety of long-cantilever transmission towers during service, this paper investigates their static and dynamic characteristics under various extreme loading conditions based on full-scale tests and numerical simulation. With the strain data tested in seven typical cases, the bending moments and axial forces of the main members were back-calculated and the initial imperfections were calculated based on these results. Using a stochastic subspace identification algorithm, the evolution of the modal parameters during the full-scale test was revealed. The damage position of the main members was identified according to the mode shape changes, and the dynamic collapse process was analyzed via the measured dynamic responses and recorded videos. Numerical simulation was conducted in ANSYS using the calculated initial imperfections, and the results were in good agreement with the full-scale test. The results show that torsion action was not dominant for this tower and the failure mode was still the bending failure at the tower body. The proposed imperfection quantification method makes up for the deficiency of the empirical values, and using experimental results to guide numerical simulation proves practical. The modal parameters vary greatly under different loading conditions, which can be attributed to the damage after each loading case.

Automatic identification of modal parameters for high arch dams based on SSI incorporating SSA and K-means algorithm

Automatic identification of modal parameters for high arch dams based on SSI incorporating SSA and K-means algorithm

Structural Damping Ratio Identification through Iterative Frequency Domain Decomposition

The frequency domain decomposition method (FDD) is a commonly used method to identify structural modal parameters and analyze the structural dynamic performance. Although FDD has a high identification accuracy in natural frequencies and mode shapes, damping ratios cannot be identified accurately. The logarithmic decay method to identify the damping ratio in FDD needs the transformation from a narrow frequency band to the time domain, which leads to the loss of the corresponding mode outside the narrow band and add other modes in the narrow band. In this paper, an iterative frequency domain decomposition method (iFDD) is proposed to identify the structural damping ratio in frequency domain. Firstly, the natural frequencies and mode shapes are obtained by the FDD method, and the damping ratios during this identification process are assumed to be unknowns. Then, the natural frequencies, mode shapes, and unknowns are substituted into the output power spectrum formula to construct nonlinear equations. The damping ratios are obtained by solving the equations using iterative optimization methods. Finally, a numerical example and benchmark model are employed to validate the effectiveness of the proposed method. The results showed that the proposed iFDD method can identify the damping ratio precisely in the frequency domain, which is better than the traditional FDD performance.

Development and Validation of a LabVIEW Automated Software System for Displacement and Dynamic Modal Parameters Analysis Purposes

The structural health monitoring (SHM) technique is a highly competent operative process dedicated to improving the resilience of an infrastructure by evaluating its system state. SHM is performed to identify any modification in the dynamic properties of an infrastructure by evaluating the acceleration, natural frequencies, and damping ratios. Apart from the vibrational measurements, SHM is employed to assess the displacement. Consequently, sensors are mounted on the investigated framework aiming to collect frequent readings at regularly spaced time intervals during and after being induced. In this study, a LabVIEW program was developed for vibrational monitoring and system evaluation. In a case study reported herein, it calculates the natural frequencies as well as the damping and displacement parameters of a cantilever steel beam after being subjected to excitation at its free end. For that purpose, a Bridge Diagnostic Inc. (BDI) accelerometer and a displacement transducer were parallelly mounted on the free end of the beam. The developed program was capable of detecting the eigenfrequencies, the damping properties, and the displacements from the acceleration data. The evaluated parameters were estimated with the ARTeMIS modal analysis software for comparison purposes. The reported response confirmed that the proposed system strongly conducted the desired performance as it successfully identified the system state and modal parameters.

Low-rank approximation of Hankel matrices in denoising applications for statistical damage diagnosis of wind turbine blades

Model order selection is a fundamental task in subspace identification for estimation of modal parameters, uncertainty propagation and damage diagnosis. However, the true model order and the related low-rank structure of the dynamic system are generally unknown. In this paper, a statistical methodology to actively select the dynamic signal subspace in covariance-driven subspace identification is developed on the basis of statistical analysis of the eigenvalue condition numbers of the output covariance Hankel matrix. It is shown that the condition numbers highly sensitive to random perturbations characterize the noise subspace. The signal subspace is separated from the noise subspace by analyzing two statistical parameters associated with the condition number sensitivity, whose thresholds are user-defined. A practical algorithm to retrieve the system dynamics is designed and demonstrated on a running example of a simulated wind turbine blade benchmark. The resultant framework is then applied in the context of damage detection on a medium-size wind turbine blade. It is demonstrated that the detectability of small damage is enhanced compared to the classic approaches and robustness of damage diagnosis is increased by reducing the number of false alarms.

Experimental quasi-static out-of-plane test of a U-shaped brick masonry wall

Past earthquakes demonstrated the vulnerability of brick masonry façade walls even against moderate earthquakes. The present work aims at providing an insight into the out-of-plane response of clay brick masonry walls, especially focusing on a traditional Portuguese building typology, namely the ‘placa’, towards a correct interpretation and assessment of their seismic behaviour. A thorough experimental campaign in laboratory environment was conducted, including material characterisation through destructive and non-destructive techniques and the quasi-static test with airbag on a U-shaped clay brick masonry specimen. Main experimental results are discussed with specific attention to the analysis of the masonry wall cyclic response, the damage pattern, the seismic performance, and the evolution of the modal parameters at distinct stages of the test. The material characterisation demonstrates the low mechanical properties of the masonry associated with such buildings. The specimen presents a stiff and almost linear behaviour until the peak, followed by an initial softening until about 60% of the peak. The damage pattern evidences a failure governed by an out-of-plane rotation combined with shear sliding.

UAV-based operational modal analysis method using improved homography-based perspective rectification method

In this paper, a new dynamic photogrammetry method using an unmanned aerial vehicle (UAV) is proposed and the UAV-based operational modal analysis method is established. The presented dynamic photogrammetry method can measure the dynamic displacements of structure using one or two cameras. However, it is difficult to use fixed cameras or handheld cameras to measure the dynamic responses for some outdoor structures, because it is hard to reach for measurement or maintenance sometimes. Therefore, UAV is an effective alternative and the homography-based perspective rectification method can be used to correct the UAV images. However, the presented methods use only four calibration points. Additionally, there is still obvious signal drift in the extracted dynamic responses. In this paper, a new time varying homography matrix, calculated using n calibration points, is proposed here for the first time to correct the perspective distortion for image at different time and the relative displacements are used to eliminate the influence of UAV in-plane motion. The influence analysis of UAV vibration and the method to eliminate the UAV vibration are the novelty of this paper. And then, the UAV-based operational modal analysis method is proposed. The proposed method is verified using a plane frame structure in laboratory. The results show that the proposed method can effectively eliminate the vibration of UAV and identify the structural modal parameters. Under the impulse excitation and white noise excitation, when the calibration points n is larger than 60, the relative errors of natural frequency and damping ratio are less than 0.4% and 5.45%, respectively. And the MAC values are all greater than 0.99. Meanwhile, the influence of the number of adopted calibration points on modal parameters identification is considered in the experiment and the advised value is given.

Bayesian modal identification of non-classically damped systems using time-domain data

In this work, we present a Bayesian time–domain method for identifying the complex modal parameters of a linear dynamic system from ambient vibration data. Complex modal parameter identification is valid for both classically and non-classically damped systems, in contrast to real modal parameter identification, which is exclusively applicable to classically damped systems. A Bayesian method is adopted here which makes it possible to determine the joint posterior Probability Density Function (PDF) of the modal parameters for given measured data and modeling assumptions. For sufficient data, it is seen that the modal parameters are globally identifiable, and the posterior PDF is approximated by a Gaussian distribution with mean at the optimal parameters that maximize the posterior PDF. The uncertainty in the identified modal parameters is given by the covariance matrix of the posterior PDF. Analytical expressions are derived for computing the gradient for efficient optimization and the Hessian for determining the covariance matrix. Further, it is shown that the assumption of classical damping during analysis results in poor prediction of damping ratios in the case of non-classically damped systems. The application of the proposed approach is illustrated by means of simulated examples and an experimental study.

A novel physics-informed framework for real-time adaptive modal parameters estimation of offshore structures

Singular value decomposition provides a rigorous mathematical foundation for these monitoring methods, such as modal analysis, damage detection, etc. As substantial limitations including computational efficiency and adaptive capabilities have been identified in the matrix process, data-driven algorithms with enhanced efficiency and accuracy to adapt the rapid development of information and technologies have been highly required in offshore structures monitoring. To meet this demand, a novel physics-informed frame work enabling the real-time adaptive monitoring has been proposed in this paper. The working principle of the developed framework has been represented by the smart conversion of physics-informed modal identification into the optimal process of fast and accurate solving an eigensystem governed by differential equations via recurrent neural network. The ingenious design of the proposed framework has complied with the rule of singular value decomposition used for modal identification and therefore, learning capabilities of the physics-informed framework have been remarkably enhanced by successfully addressing two bottleneck problems including proper initialized input values and the optimal time increment of the developed recurrent neural network. Both numerical simulations and the field-data based study of a mono-pile offshore wind turbine structure have been presented to examine the superior performance of the proposed framework. Results have shown that the proposed framework has the ability to adaptively identify modal parameters with a higher level of computational efficiency as compared with traditional methods. Furthermore, the computational advantage of the develop framework has demonstrated the potential to integrate with sensor networks and edge computing for smart monitoring and maintenance in various engineering subjects.

Recursive variational mode decomposition enhanced by orthogonalization algorithm for accurate structural modal identification

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.

Application of Underdetermined Blind Source Separation Algorithm on the Low-Frequency Oscillation in Power Systems

The timely discovery of low-frequency oscillations in power systems and accurate identification of their modal parameters is critical in numerous applications. Therefore, we investigated the feasibility of using multi-channel signals and established a relative theory. An algorithm based on the underdetermined blind source separation (UBSS) algorithm was proposed using this theory. First, the energy ratio function was used to determine the fault occurrence time. Then, the Bayesian information criterion was used to estimate the number of fault sources, and the boundary conditions were set to determine the number of fault sources. Next, the UBSS algorithm was used to analyze raw data, extract individual components that characterize faults, and subsequently measure low-frequency oscillation modal parameters through the Hilbert transform. Finally, the fast independent component analysis (FastICA) algorithm was used to separate noise signal from raw data. This separation considerably reduced noise disturbance and ensured the stability of the proposed method. Model simulation was conducted in MATLAB and experimental measurement revealed that the proposed method effectively reduced noise disturbance and could be applied to conditions with considerable disturbance.

A Loewner-Based System Identification and Structural Health Monitoring Approach for Mechanical Systems

Data-driven structural health monitoring (SHM) requires precise estimates of the target system behaviour. In this sense, SHM by means of modal parameters is strictly linked to system identification (SI). However, existing frequency-domain SI techniques have several theoretical and practical drawbacks. This paper proposes using an input-output system identification technique based on rational interpolation, known as the Loewner framework (LF), to estimate the modal properties of mechanical systems. Pioneeringly, the Loewner framework mode shapes and natural frequencies estimated by LF are then applied as damage-sensitive features for damage detection. To assess its capability, the Loewner framework is validated on both numerical and experimental datasets and compared to established system identification techniques. Promising results are achieved in terms of accuracy and reliability.

Multiple damage prediction in tubular rectangular beam model using frequency response-based mode shape curvature with back-propagation neural network

In the civil infrastructure, the structures made up of tubular sections played an important role due to an economic point of view, lower self-weight, and stability against functional loads. The periodical maintenance or inspection of the structure is mandatory nowadays to achieve the potential utilization of the structural system. In this study, an experimental, numerical, and analytical study is carried out to investigate the structural fault and its severity in a tubular rectangular beam made up of structural steel. Modal parameters are extracted with the help of a dynamic data logger (B&K) by exciting an impact hammer on model structure and extracting a data with a set of unidirectional accelerometers. First, two displacement mode shapes are extracted using obtained modal parameters. The modal parameters are expected to contain environmental noise during experimentation, so, de-noising is must to obtain noise-free data. Artificial neural network training is utilized to reduce the noise from experimental modal parameters. Using the modified modal parameters, the mode shape curvature is obtained, and so-called modified mode shape curvature (MMSC) is used to calculate the curvature damage index. The curvature damage index is appropriate to investigate multiple fault locations with different fault levels in tubular rectangular beam structures.

Identification of Closely Spaced Modes of a Long-Span Suspension Bridge Based on Bayesian Inference

Closely spaced modes commonly observed in long-span suspension bridges can greatly increase the difficulty of identifying and tracking modal parameters. Most existing studies generally focus on identifying the closely spaced modes and quantifying the uncertainties based on numerical and experimental models. Further research focusing on full-scale long-span bridges is still required. A case study on identifying the closely spaced modes of the Qixiashan Yangtze River Bridge, a long-span suspension bridge with a main span of 1 418 m, is conducted in this paper. The effectiveness of the generalized fast Bayesian fast Fourier transform (GFBFFT) method is verified by both the simulated and monitoring data. The results show that a larger coefficient of variation (COV) and higher uncertainty is typically contained in the closely spaced modes than the separated modes. Compared with the FDD and SSI methods, the GFBFFT method guarantees higher identification accuracy of modal parameters and can serve as a reliable tool to identify the closely spaced modes.

Influence of system changes on closely spaced modes of a large-scale concrete tower for the application to structural health monitoring

AbstractConcrete steel towers are increasingly being used for onshore wind turbines. The lower part consists of separated segmented concrete rings connected with dry joints. Due to slight deviations from the axisymmetric cross-section, closely spaced modes occur. Therefore, the influences of small system changes on closely spaced modes, particularly the mode shapes, should be investigated to enable reliable vibration-based monitoring. In this context, the influence of imperfections due to the waviness of the dry joints requires attention. As no acceleration measurements on concrete towers considering small system changes have been performed so far, this has not yet been investigated. Therefore, an experiment is carried out using a large-scale laboratory model of a prestressed concrete segment tower. The system modifications are introduced by changing the preload. This changes the influence of imperfections of the surfaces of the horizontal dry joints, estimated by measuring strain and displacement at the lowest joint. An increasing preload causes the first two pairs of bending modes to move closer together. This enables to study the effect of the closeness of natural frequencies on the related mode shapes based on the same structure. Thus, the known effects of increasing uncertainty of the alignment and a rotation of the mode shape in the mode subspace with closer natural frequencies can be shown experimentally. In this work, the operational modal analysis (OMA) methods Bayesian-OMA (BAYOMA) and Stochastic Subspace Identification (SSI) are used. Local imperfections can significantly affect modal parameters, so these should be considered for vibration-based monitoring.

Dynamic displacement estimation and modal analysis of long-span bridges integrating multi-GNSS and acceleration measurements

Compared with acceleration-based modal analysis, displacement can provide a more reliable and robust identification result for output-only modal analysis of long-span bridges. However, the estimated displacements from acceleration records are frequently unavailable due to unrealistic drifts. Aiming at obtaining more accurate and stable results for determining the modal parameters, this study develops a multi-rate weighted data fusion approach for estimating displacement responses in dynamic monitoring of structures based on global navigation satellite system (GNSS) and acceleration measurements. The approach initially derives the local estimations from displacement and acceleration sensors via a Kalman filter algorithm with colored measurement noise, and later uses a weighted fusion criterion of scalar linear minimum variance to fuse the results of local estimations. Then, structural modal pamameters are identified by employing data-driven stochastic subspace identification (SSI) algorithm. The proposed approach is validated in a four degree-of-freedom numerical model and then applied to a long-span bridge in engineering practice. The results illustrate that the proposed approach can reduce the error of GNSS-obtained displacement and expand recognizable frequency range by introducing dynamic displacement component from acceleration measurement.

Modal parameter identification in civil structures via Hilbert transform ensemble with improved empirical wavelet transform

To overcome the inaccurate frequency band division in empirical wavelet transform (EWT) induced by high noise and modulation edge band, this study proposes an improved EWT (IEWT) and applies it to operational modal analysis (OMA) in civil structures. The IEWT segments the frequency band using spectral trend determined with removing the high-frequency components of the Fourier spectrum. The synchroextracting transform is employed to denoise the frequency components obtained from IEWT combined with the random decrement technique and Hilbert transform to identify modal parameters. The effectiveness of IEWT and the proposed modal analysis method is comprehensively verified by a full-scale cable-stayed footbridge. The results demonstrate that the proposed IEWT, random decrement technique, and Hilbert transform can accurately identify the modal parameters from multi-setup ambient vibration measurements.

Pose-Dependent Cutting Force Identification for Robotic Milling

AbstractCutting force identification is critical to improving industrial robot performance and reducing machining vibration. However, most indirect identification methods of cutting force are not applicable since the modal parameters of the robotic milling system vary with the robot pose. This paper presents a novel pose-dependent method to identify the cutting force using the acceleration signal generated by robotic milling. First, the modal parameters at different machining points are employed as a training dataset to develop the Gaussian Process Regression (GPR) model. Next, the modal parameters predicted by the GPR model are employed to optimize the cutting force estimation based on the minimum variance unbiased estimate method. Then, the Kalman filter method is employed to update the covariance matrix of the cutting force identification error and the state estimation error. Lastly, the effectiveness of the proposed method is verified with robotic milling experiments, and the results show that the identification error and time are acceptable under the condition of variable robot pose.

Assessment of 2D Digital Image Correlation for Experimental Modal Analysis of Transient Response of Beams Using a Continuous Wavelet Transform Method

The modal parameters of structures, and in particular their mode shapes, are generally determined based on the measurement of accelerometers or laser vibrometers. However, these sensors do not allow the performance of full-field measurements. In this study, the free vibration of a beam triggered by a shock is investigated using a high-speed camera with high image definition. With the help of digital image correlation (DIC), the beam displacement fields are deduced from the images. To analyse the DIC measurement quality, different tests and analyses are performed. First, the systematic errors and uncertainties in the DIC calculation for a simple translation are analysed considering different speckles. Then, tests on two configurations of a vibrating beam are filmed and full-field displacement measurements are computed. The modal parameters of the beam are deduced from these measurements using a continuous wavelet transform method. Particular care was taken to adapt the method to the post-processing of the numerous and noisy signals obtained for these experiments. All the steps of the post-processing are detailed in this paper. Finally, the modal parameters obtained with the proposed method are compared with those obtained in a more classical way using accelerometers and from the beam theory. In particular, the comparison of the signal-to-noise ratio of the different measurement methods is discussed.