Time-varying Modal Parameters Research Articles

Modal Parameter Identification of a Structure Under Earthquake via a Wavelet-Based Subspace Approach

This paper introduces a novel wavelet-based methodology for identifying the modal parameters of a structure in the aftermath of an earthquake. Our proposed approach seamlessly combines a subspace method with a stationary wavelet packet transform. By relocating the subspace method into the wavelet domain and introducing a weighting function, complemented by a moving window technique, the efficiency of our approach is significantly augmented. This enhancement ensures the precise identification of the time-varying modal parameters of a structure. The capacity of the stationary wavelet packet transform for rich signal decomposition and exceptional time-frequency localization is harnessed in our approach. Different subspaces within the stationary wavelet packet transform encapsulate signals with distinct frequency sub-bands, leveraging the fine filtering property to not only discern modes with pronounced modal interference, but also identify numerous modes from the responses of a limited number of measured degrees of freedom. To validate our methodology, we processed numerically simulated responses of both time-invariant and time-varying six-floor shear buildings, accounting for noise and incomplete measurements. Additionally, our approach was applied to the seismic responses of a cable-stayed bridge and the nonlinear responses of a five-story steel frame during a shaking table test. The identified modal parameters were meticulously compared with published results, underscoring the applicability and reliability of our approach for processing real measured data.

A lightweight stochastic subspace identification-based modal parameters identification method of time-varying structural systems

The application of stochastic subspace identification (SSI) on identifying structural time-varying modal parameters suffers from the shortcomings of poor computational efficiency and susceptibility to spurious modes. This work overcomes the shortcomings by the following procedures based on existing covariance-driven SSI. First, the covariance-driven SSI is embedded with techniques of randomized singular value decomposition and subspace iteration for reducing computation cost and avoiding accuracy loss, forming the lightweight SSI (lwSSI). Then, the lwSSI is combined with the technique of sliding window for eliminating spurious modes and continuously identifying time-varying modal parameters. The proposed lwSSI-based identification method is applied on the simulation of a gravity dam, and the identified time-varying modal parameters show the same trend as the assumed time-varying elastic modules. Furthermore, the characteristics of the lwSSI-based identification method, including noise resistance, computational efficiency, and identification accuracy, are investigated. Finally, the identification of a steel cantilever beam with the time-varying modal parameters caused by the mass distribution experimentally validates the effectiveness of the proposed lwSSI-based identification method.

Time-varying modal identification using multi-channel measurements based on multivariate variational mode decomposition

Structural modal parameters are essential to provide dynamic information for data analysis of structural health monitoring (SHM) system. Because of the variation of the physical properties or the influence of the environment, the structural modal parameters may change with time during the operation, showing time-varying characteristics. Therefore, accurate identification of time-varying modal parameters shows to be an important issue for SHM. In this paper, a time-varying modal identification method is proposed by improving the multivariate variational mode decomposition (MVMD) method with autoregressive power spectrum and windowed principal component analysis (PCA). Firstly, the method for determination of initial center frequency is proposed by autoregressive power spectrum to improve the efficiency of MVMD. Secondly, intrinsic mode function for each mode is extracted using multi-channel responses by MVMD. Subsequently, instantaneous frequencies are identified through detecting the ridgeline of the synchro-squeezed short-time Fourier transform (SSTFT). Moreover, identification method for time-varying mode shapes is proposed by using the windowed PCA of the multi-channel intrinsic modes. Finally, the proposed method is verified by the numerical and practical studies. The results of the numerical study show that the method is effective for continuously varying modal parameters under impulse and random excitations. Through the data analysis of practical bridge, the capability for practical application is demonstrated.

Recursive identification of the time-varying frequency parameters of a space solar power satellite using a robust subspace tracking method based on the variable forgetting factor

Considering changes in the structural configuration caused by the rotation of solar panel arrays in space solar power satellites, this study proposes an improved robust recursive subspace tracking algorithm to identify the time-varying modal frequency parameters of the system. In this algorithm, variable forgetting factor and robust weight coefficient are involved to improve the tracking adaptability of time-varying system and decrease the influence of impulsive disturbance. Then, the time-varying pseudo modal frequency parameters are identified using the recursive projection subspace tracking. In the numerical simulations, the influence of the different rotation speeds of the solar panel arrays on the frequency identification results is studied. The computational results for the different rotation speeds reveal that the tracking performance of the proposed algorithm is better than that of the original method for time-varying system. Under the influence of impulsive disturbances, the relative values of the average relative error under the proposed algorithm are approximately 65.9–75.2% less than those under the original algorithm. Additionally, the results reveal that when the rotation speed of the solar panel arrays is time-varying, the proposed algorithm achieves a better tracking performance than the conventional recursive methods by the adjustment of the variable forgetting factor. When the signal-to-noise ratio is 15 dB, the absolute and relative values of the average relative error under the proposed algorithm are less by approximately 6.4% and 64.8%, respectively, under the two reference methods with a fixed forgetting factor.

Output-Only Time-Varying Modal Parameter Identification Method Based on the TARMAX Model for the Milling of a Thin-Walled Workpiece.

The process parameters chosen for high-performance machining in the milling of a thin-walled workpiece are determined by a stability prediction model, which needs accurate modal parameters of the machining system. However, the in-process modal parameters are different from the offline modal parameters and are difficult to precisely obtain due to material removal. To address this problem, an accurate time-dependent autoregressive moving average with an exogenous input (TARMAX) method is proposed for the identification of the modal parameters in the milling of a thin-walled workpiece. In this process, a TARMAX model considering external force excitation is constructed to characterize the actual condition in the milling of a thin-walled workpiece. Then, recursive method and sliding window recursive method are used to identify TARMAX model parameters under time-varying cutting conditions. Subsequently, a three-degree of freedom (3-DOF) time-varying structure numerical model under theoretical milling forces and white-noise excitation is established, and the computational results show that the predicted natural frequencies using the proposed method are in close agreement with the simulated values. Finally, several experiments are designed and carried out to validate the effectiveness of the proposed method. The experimental results show that the predicted accuracy of the proposed method using actual cutting forces is 95.68%. Good agreement has been drawn in the numerical simulation and machining experiments. Our further research objectives will focus on the prediction of the damping ratios, modal stiffness, and modal mass.

Stability enhancement and chatter suppression in continuous radial immersion milling

For continuous radial immersion milling operations, the dominant mode shape becomes difficult to determine when stiffness of the tool and workpiece are comparable, and this can pose a great challenge for ensuring machining processes stability. In this paper, we propose a rapid method to obtain time-varying modal parameters of the workpiece by combining experimental measurements with the receptance coupling method. Firstly, the contact parameters between the workpiece and vise were identified by a so-called dynamic coupling matrix. Then the mode shapes and the time-varying natural frequency of the workpiece were determined using the modal parameters of workpiece. Finally, the stability lobe diagrams (SLDs) were computed using the modal parameters and then were validated by undertaking immersion milling experiments. The experiments showed a more conservative and practical SLD for general workpiece under continuous radial immersion, where the workpiece mode had not always dominated the machining process. Based on the proposed method, we also explored two modifications in form of additional cylinder masses and passive support, to suppress chatter. Both modifications were effective in enhancing the minimum boundary of the conservative SLD, and the modification of passive support worked better. Although the modification of the workpiece could improve the stability boundary, it indirectly affected the dynamics of the milling tool through the interaction area between the workpiece and milling tool.

Iterative reference-driven S-transform time-varying parameter identification for bridges under moving vehicle

Traditional bridge health monitoring is both expensive and time-consuming because of the need for installing a large quantity of sensors on the bridge. The indirect monitoring method as a feasible alternative has become popular over the last decade, as useful information can be gathered from moving vehicles installed with sensors by considering vehicle-bridge interaction. This paper describes an indirect time-varying modal parameter identification method for bridges under a moving test vehicle. A novel reference-driven S-transform is proposed for the parametric modal identification of time-varying structural systems with non-stationary characteristics based on a multisensory arrangement, comprising a fixed reference sensor installed on the bridge and another movable sensor installed on a moving test vehicle. By the phase-preservation property of S-transform and the multisensory time-frequency analysis, the proposed iterative reference-driven S-transform can be used to extract the time-varying modal parameters of the vehicle-bridge system when the vehicle moves on the bridge. Based on the coherence between measurements from the sensors mounted on the bridge and vehicle, one can track the time-varying characteristics by combining the spatial and time-frequency information obtained by the multisensory array. The efficiency of the proposed reference-driven S-transform method is validated by numerical simulations and laboratory experiments using a simply supported beam with a moving vehicle. The effects of the vehicle property, driving speed, road roughness and measurement noise are discussed. The numerical and experimental results have demonstrated that the proposed method is useful for time-varying parameter extraction with a multisensory system. This will be useful to structural condition monitoring of bridges.

Novel tensor subspace system identification algorithm to identify time-varying modal parameters of bridge structures

Subspace-based system identification algorithms have been developed as an advanced technique for performing modal analysis. We introduce a novel tensor subspace-based algorithm to identify the time-varying modal parameters of bridge structures. A new time dimension is introduced in the traditional Hankel matrix, and a mathematical model of tensor subspace decomposition is established. Combined with the stabilization diagram, tensor parallel factor decomposition is used to estimate the frequencies, mode shapes, and modal damping ratios. The effectiveness of the proposed algorithm is validated by comparing it with the classical sliding-window–based stochastic subspace algorithm on a model cable-stayed bridge dynamic test. The proposed algorithm is further applied to process the dynamic responses of a real bridge health monitoring system to identify its time-varying modal frequencies. Our results demonstrated that the proposed algorithm significantly reduces computational efforts and extends the range of solution ideas for future out-only time-varying system identification problems.

Study of modal parameter estimation of time-varying mechanical system in time-frequency domain based on output-only method

With increasing mobile instrument inspection systems like mobile inspection cameras applied in modern industrial automation lines, their time-varying modal parameters need to be thoroughly studied since they are operating under various random vibration excitations which are mostly unpredictable and unrepeatable. This paper investigates modal parameter identification of time-varying mobile mechanical systems through a poly-reference expansion based on output-only method in time-frequency domain. The poly-reference parametric model is presented through a time-frequency-domain right matrix fraction description. Based on the poly-reference parametric model, a poly-reference least square estimator and a poly-reference maximum likelihood estimator of modal parameters of time-varying systems are developed. For the case of only one batch of measurements, a novel approach to estimate the covariance matrix for the poly-reference maximum likelihood estimator by non-repeatable measurement is also proposed in this paper. A numerical example demonstrates the advantageous performance of the proposed estimators, such as the estimation of the close-coupled modes. Finally, a laboratory experiment validates these proposed estimators via a comparative way.

Application of decoupled ARMA model to modal identification of linear time-varying system based on the ICA and assumption of “short-time linearly varying”

A new approach for time-varying (TV) modal parameters identification is proposed in this research. In the identification process, the entire signal is divided into successive short time windows, where the structure response under white noise excitation is transformed into modal coordinates by the Independent Component Analysis (ICA) method. The decoupled time-varying Auto-Regressive Moving-Average (ARMA) model based on the new assumption of “short-time linearly varying” (STLV) is established with the extracted modal coordinates. The TV parameters can be obtained by solving a nonlinear least squares problem. The main motivation of the proposed approach is to simplify the model and reduce the computational difficulty. The effectiveness and accuracy are validated via both the numerical example and experiment.

Time-Varying Parameter Identification of Bridges Subject to Moving Vehicles Using Ridge Extraction Based on Empirical Wavelet Transform

For a vehicle moving over a bridge, the vehicle-bridge interaction (VBI) embraces the time-varying modal parameters of the system. The identification of non-stationary characteristics of bridge responses due to moving vehicle load is important and remains a challenging task. A new method based on the improved empirical wavelet transform (EWT) along with ridge detection of signals in time-frequency representation (TFR) is proposed to estimate the instantaneous frequencies (IFs) of the bridge. Numerical studies are conducted using a VBI model to investigate the time-varying characteristics of the system. The effects of the measurement noise, road surface roughness and structural damage on the bridge IFs are investigated. Finally, the dynamic responses of an in-situ cable-stayed bridge subjected to a passing vehicle are analyzed to further explore the time varying characteristics of the VBI system. Numerical and experimental studies demonstrate the feasibility and effectiveness of the proposed method on the IF estimation. The identified IFs reveal important time-varying characteristics of the bridge dynamics that is significant to evaluating the actual performance of operational bridges in operation and may be used for structural health assessment.

Moving Window Differential Evolution Independent Component Analysis-Based Operational Modal Analysis for Slow Linear Time-Varying Structures

In order to identify time-varying transient modal parameters only from nonstationary vibration response measurement signals for slow linear time-varying (SLTV) structures which are weakly damped, a moving window differential evolution (DE) independent component analysis- (ICA-) based operational modal analysis (OMA) method is proposed in this paper. Firstly, in order to overcome the problems in traditional ICA-based OMA, such as easy to go into local optima and difficult-to-identify high-order modal parameters, we combine DE with ICA and propose a differential evolution independent component analysis- (DEICA-) based OMA method for linear time invariant (LTI) structures. Secondly, we combine the moving widow technique with DEICA and propose a moving window differential evolution independent component analysis- (MWDEICA-) based OMA method for SLTV structures. The MWDEICA-based OMA method has high global searching ability, robustness, and complexity of time and space. The modal identification results in a three-degree-of-freedom structure with slow time-varying mass show that this MWDEICA-based OMA method can identify transient time-varying modal parameters effectively only from nonstationary vibration response measurement signals and has better performances than moving window traditional ICA-based OMA.

Modal parameter identification of time-varying systems via wavelet-based frequency response function

Modal parameter identification of systems, structures and machines with variable physical parameters is an important task for their damage diagnosis, maintenance and repair, and life cycle management, especially when time-varying vibration modes are involved. The paper proposes a new combined, two-step, i.e., estimated wavelet-based frequency response function (FRF) and least-squares iterative algorithm, modeling approach in order to determine time-varying vibration modes based on Gaussian white noise input excitation. The time-varying wavelet-based FRF is estimated based on output noise estimation model $$H_1$$ ; the second step identifies time-varying modal parameters based on estimated wavelet-based FRF with the use of least-squares iterative algorithm. Estimated wavelet-based FRF can reveal the dynamic characteristics of the system correctly similar to the theoretical FRF by single degree of freedom (SDOF). The combined method is demonstrated using system identification analysis based on the simulated stiffness-varying and mass-varying multiple-degree-of-freedom (MDOF) system subjected to random Gaussian white noise excitation. The results show that the proposed combined method accurately identifies modal parameter of the time-invariant analyzed structure, and correctly captures modal parameter of the time-varying analyzed structure. The modal frequency and shape agree with theoretical results well. The analysis results also indicate that the proposed combined method is sensitive to the position of input excitation. The estimation accuracy wavelet-based FRF can be improved by selecting the average number of times and wavelet parameters.

Structural flexibility identification via moving-vehicle-induced time-varying modal parameters

To overcome the limitation of existing methods only identifying basic dynamic properties of bridge structures, a moving vehicle strategy was proposed to identify more meaningful parameters including mass-normalized mode shapes and structural flexibility matrix, which can be further adopted for static deflection prediction and condition assessment of bridge structures. The contribution of this article is to investigate the intrinsic relationship between mass-normalized scaling factor and time-varying modal parameters identified from moving vehicle-induced responses. In the proposed method, the variational mode decomposition (VMD) algorithm was firstly utilized to decompose vehicle-induced responses into several intrinsic mode function (IMFs), then time-varying frequencies of the vehicle-bridge coupled system are identified by performing Hilbert transform of each IMF; Second, the relationship between mass-normalized scaling factor and time-varying dynamic properties of the coupled system was theoretically derived, and structural flexibility matrix was computed by combining calculated scaling factor and dynamic properties of the bridge without moving vehicle; Finally, a numerical example of simply supported bridge was employed to verify the correctness of proposed method, in which influences of predefined parameters (i.e. penalty parameter and number of decomposed IMFs) in VMD algorithm, road roughness, measurement noise, vehicle weight, vehicle spring and driving speed on identification results were investigated. To further verify the correctness of this method, an experimental model of steel beam was designed in the laboratory for structural flexibility identification and static deflection prediction. A good agreement between predicted deflections using the proposed method and directly measured values by using displacement transducers has successfully verified the proposed method.

A hybrid method of multi-objective particle swarm optimization and k-means clustering and its application to modal parameter estimation in the time–frequency domain

Modal parameters provide an insight into the dynamical properties of structures. In the time–frequency domain–based methods, time–frequency ridges contain crucial information on the characteristics of multicomponent signals, and manually extracting time–frequency ridges is a huge burden, especially when long-time time-varying modal parameters are focused on. In this study, time–frequency ridge extraction is converted into a multi-objective optimization problem, and a new hybrid method of multi-objective particle swarm optimization and k-means clustering is proposed to solve such a multi-objective optimization problem. In the hybrid method, the particle swarm is partitioned into sub-swarms by k-means clustering, and the sub-swarms are used to search new solutions for updating a finite-sized external archive, which is used as the exclusive centroids of the k-means clustering. Simultaneously, the finite-sized external archive serves as global best positions of sub-swarms. Both simulated and experimental cases are applied to validate the hybrid method. With the aid of the hybrid method, the influence of varying temperatures on modal parameters of a column beam is experimentally analyzed in detail.

Online Modal Identification of Concrete Dams Using the Subspace Tracking-Based Method

To investigate the time-varying dynamic characteristics of concrete dams under the excitation of large earthquakes for online structural health monitoring and damage evaluation, an online modal identification procedure based on strong-motion records is proposed. The online modal identification of concrete dams is expressed as a subspace tracking problem, and a newly developed recursive stochastic subspace identification (RSSI) method based on the generalized yet another subspace tracker (GYAST) algorithm, which exploits both the accuracy of the subspace identification and fast computational capability, is used to extract the time-varying modal parameters of concrete dams during earthquakes. With the simulated vibration response records, a numerical example is used to verify the accuracy, robustness, and efficiency of the proposed GYAST-based, time-varying modal identification method. Then, the realistic strong-motion records of the Pacoima arch dam are analysed using the proposed modal identification procedure, and the time-varying characteristics of the concrete arch dam during three different earthquakes are analysed.

Identification of the time-varying modal parameters of a spacecraft with flexible appendages using a recursive predictor-based subspace identification algorithm

This study focuses on the recursive identification of the time-varying modal parameters of on-orbit spacecraft caused by structural configuration changes. For this purpose, an algorithm called recursive predictor-based subspace identification is applied as an alternative method to improve the computational efficiency and noise robustness, and to implement an online identification of system parameters. In the existing time-domain identification methods, the eigensystem realization algorithm and subspace identification methods are usually applied to obtain the on-orbit spacecraft modal parameters. However, these approaches are designed based on a time-invariant system and singular value decomposition, which require a significant amount of computational time. Thus, these methods are difficult to employ for online identification. According to the adaptive filter theory, the recursive predictor-based subspace identification algorithm can not only avoid the singular value decomposition computation but also provide unbiased estimates in a general noisy framework using the recursive least squares approach. Furthermore, in comparison with the classical projection approximation subspace tracking series recursive algorithm, the recursive predictor-based subspace identification method is more suitable for systems with strong noise disturbances. By establishing the dynamics model of a large rigid-flexible coupling spacecraft, three cases of on-orbit modal parameter variation with time are investigated, and the corresponding system frequencies are identified using the recursive predictor-based subspace identification, projection approximation subspace tracking, and singular value decomposition methods. The results demonstrate that the recursive predictor-based subspace identification algorithm can be used to effectively perform an online parameter identification, and the corresponding computational efficiency and noise robustness are better than those of the singular value decomposition and projection approximation subspace tracking series approaches, respectively. Finally, the applicability of this method is also verified through a numerical simulation.

Time-Varying Modal Parameters Identification by Subspace Tracking Algorithm and Its Validation Method

This article presents a time-varying modal parameter identification method based on the novel information criterion (NIC) algorithm and a post-process method for time-varying modal parameter estimation. In the practical application of the time-varying modal parameter identification algorithm, the identified results contain both real modal parameters and aberrant ones caused by the measurement noise. In order to improve the quality of the identified results as well as sifting and validating the real modal parameters, a post-process procedure based on density-based spatial clustering of applications with noise (DBSCAN) algorithm is introduced. The efficiency of the proposed approach is first verified through a numerical simulation of a cantilever Euler-Bernoulli beam with a time-varying mass. Then the proposed approach is experimentally demonstrated by composite sandwich structure in a time-varying high temperature environment. The identified results illustrate that the proposed approach can obtain real modal frequencies in low signal-to-noise ratio (SNR) scenarios.

Vibration Control of a Time-Varying Modal-Parameter Footbridge: Study of Semi-Active Implementable Strategies

This paper explores different vibration control strategies for the cancellation of human-induced vibration on a structure with time-varying modal parameters. The main motivation of this study is a lively urban stress-ribbon footbridge (Pedro Gomez Bosque, Valladolid, Spain) that, after a whole-year monitoring, several natural frequencies within the band of interest (normal paring frequency range) have been tracked. The most perceptible vibration mode of the structure at approximately 1.8 Hz changes up to 20%. In order to find a solution for this real case, this paper takes the annual modal parameter estimates (approx. 14000 estimations) of this mode and designs three control strategies: a) a tuned mass damper (TMD) tuned to the most-repeated modal properties of the aforementioned mode, b) two semi-active TMD strategies, one with an on-off control law for the TMD damping, and other with frequency and damping tuned by updating the damper force. All strategies have been carefully compared considering two structure models: a) only the aforementioned mode and b) all the other tracked modes. The results have been compared considering human-induced vibrations and have helped the authors on making a decision of the most advisable strategy to be practically implemented.

Scour Depth Determination of Bridge Piers Based on Time-Varying Modal Parameters: Application to Hangzhou Bay Bridge

Scour Depth Determination of Bridge Piers Based on Time-Varying Modal Parameters: Application to Hangzhou Bay Bridge