Identification Of Mode Shapes Research Articles

High-resolution frequency domain decomposition for modal analysis of bridges using train-induced free-vibrations

Modal parameters are structural inherent characteristics that can be applied for revealing performance of railway bridges. Free vibration signals generated by a passage of train are commonly utilized to estimate the modal parameters of railway bridges due to their higher signal-to-noise ratios compared to random vibrations caused by ambient loads. However, since free vibration signals rapidly decay over time, the available free-vibration data is typically short-time. When using the fast Fourier transform-based spectral estimation method for modal identification from short-time vibration data, a phenomenon known as spectral leakage occurs, leading to miss-identification of some structural modes. In this study, the classical frequency domain decomposition (FDD) is improved for modal identification of railway bridges, in which the higher resolution auto-power spectral density (PSD) and cross-PSD functions are calculated through the autoregressive (AR) model-based method. The AR model-based method improves both the smoothness and resolution of the PSD functions compared to the fast Fourier transform technique. These AR model-based PSD functions are then employed in the FDD process to facilitate frequency and mode shape identification while avoiding spurious noise modes. The proposed eigenvalue fitting technique is subsequently utilized to estimate damping ratios. Numerical simulation data as well as vibration data from an actual bridge are analyzed to validate the proposed method, with a comparison made to the Welch’s PSD-based method. The results demonstrate that the modified FDD approach enables more effective identification of structural modes, even in the presence of closely-spaced modes.

Drive-by bridge mode shape identification using novel reference-based component scaling method

This article proposes a novel method, entitled Reference-based Component Scaling (RCS) method, for estimation of bridge mode shapes using vehicle scanning methods. The proposed method uses two instrumented vehicles to measure the accelerations on the bridge. While one vehicle remains stationary on a pre-determined location, the other one travels over the bridge with a constant speed both exciting the bridge and recording its response. The accelerations recorded on both vehicles are then processed using variational mode decomposition method and Hilbert transform to obtain their modal components. Leveraging its comprehensive framework that incorporates modal components derived from both vehicles, the RCS method effectively tackles the primary challenges encountered by vehicle scanning methods. These challenges include: i) shifting and shrinking effects of bridge damping on mode shapes, ii) distortions in modal components due to incomplete elimination of the roughness effect and iii) distortions in modal components caused by acceleration peaks during the vehicle´s transition from a rigid platform to the bridge. To this end, the RCS method addresses adverse effects of bridge damping without a priori knowledge of the modal damping ratios. To assess the accuracy of the proposed method, numerical analyses were carried out on three different bridges with different number of spans and boundary conditions. The effect of crucial parameters such as road roughness, bridge damping, uncertainties in measurements, and existing traffic on the accuracy of the proposed method was investigated. The results of the numerical study show that the RCS method stands alone in providing accurate estimates of mode shapes of bridges that are seated on elastic bearings, that have realistic damping ratios and relatively high road roughness profiles.

Identification Uncertainties of Bending Modes of an Onshore Wind Turbine for Vibration-Based Monitoring

This study considers the identification uncertainties of closely spaced bending modes of an operating onshore concrete-steel hybrid wind turbine tower. The knowledge gained contributes to making mode shapes applicable to wind turbine tower monitoring rather than just mode tracking. One reason is that closely spaced modes make it difficult to determine reliable mode shapes for them. For example, the well-known covariance-driven stochastic subspace identification (SSI-COV) yields complex mode shapes with multiple mean phases in the complex plane, which does not allow error-free transformation to the real space. In contrast, the Bayesian Operational Modal Analysis (BAYOMA) allows the determination of real mode shapes. The application of BAYOMA presents a further challenge when quantifying the associated uncertainties, as the typical assumption of a linear, time-invariant system is violated. Therefore, validity is not self-evident and a comprehensive investigation and comparison of results is required. It has already been shown in a previous study that the significant part of the uncertainty in the mode shapes corresponds to their orientation in the mode subspace (MSS). Despite all the challenges mentioned, there is still a great need to develop reliable monitoring parameters (MPs) for Structural Health Monitoring (SHM). This study contributes to this by analysing metrics for comparing mode shapes. In addition to the well-known Modal Assurance Criteria (MAC), the Second-Order MAC (S2MAC) is also used to eliminate the alignment uncertainty by comparing the mode shape with a MSS. In addition, the mode shape identification uncertainties of BAYOMA are also considered. Including uncertainties is also essential for the typically used natural frequencies and damping ratios, which can be more appropriately used if the identification uncertainty is known.

Automated operational modal analysis for civil engineering structures with compressed measurements

Introducing compressive sensing (CS) to modal analysis enables modal identification approaches to process compressed measurements, which enhances the efficiency in both data transmission and analysis for long-term monitoring of civil engineering structures. Nevertheless, the existing methods normally require users’ judgment in mode selection, and thus they cannot realize automated modal analysis. To bridge this gap, this study proposes an automated CS-based operational modal analysis (OMA) method, namely Automated Sparse Decomposition (ASD). A mode selection strategy is proposed to gradually eliminate noise and false modes and thus realize automated modal identification based on compressed measurements. A search strategy with both extensive and intensive search stages is proposed to realize higher frequency search resolution with low computational costs. Additionally, an automated mode shape identification process is designed to enhance accuracy. To demonstrate the effectiveness of the proposed method, experimental data on an offshore wind turbine model and real-world monitoring data from an in-service cable-stayed bridge are analyzed. The results show that the identification accuracy of the proposed ASD is comparable to state-of-art CS-based methods. Due to its superior performance in both efficiency and accuracy, the proposed method has the potential to become a promising online modal analysis tool.

Multi-Objective Optimization of Thin-Walled Composite Axisymmetric Structures Using Neural Surrogate Models and Genetic Algorithms.

Composite shells find diverse applications across industries due to their high strength-to-weight ratio and tailored properties. Optimizing parameters such as matrix-reinforcement ratio and orientation of the reinforcement is crucial for achieving the desired performance metrics. Stochastic optimization, specifically genetic algorithms, offer solutions, yet their computational intensity hinders widespread use. Surrogate models, employing neural networks, emerge as efficient alternatives by approximating objective functions and bypassing costly computations. This study investigates surrogate models in multi-objective optimization of composite shells. It incorporates deep neural networks to approximate relationships between input parameters and key metrics, enabling exploration of design possibilities. Incorporating mode shape identification enhances accuracy, especially in multi-criteria optimization. Employing network ensembles strengthens reliability by mitigating model weaknesses. Efficiency analysis assesses required computations, managing the trade-off between cost and accuracy. Considering complex input parameters and comparing against the Monte Carlo approach further demonstrates the methodology's efficacy. This work showcases the successful integration of network ensembles employed as surrogate models and mode shape identification, enhancing multi-objective optimization in engineering applications. The approach's efficiency in handling intricate designs and enhancing accuracy has broad implications for optimization methodologies.

A General Approach to Construct Full-Field Responses and High-Resolution Mode Shapes of Bridges Using Sparse Measurements

The full-field responses and high-resolution mode shapes are of paramount importance in the detection of bridge damage and finite element model updating. As enlightened by the Fourier series expansion of the mode shapes for a bridge, this study proposes an approach for compressive sensing-based reconstruction of the full-field responses and identification of high-resolution mode shapes, in which a general basis matrix is newly formulated, with particular application to beam-like continuous or non-uniform bridges. The merits of the proposed approach are model-free, output-only, and easy to implement, especially the need of a few sensors. The effectiveness of the approach is validated through both numerical and experimental examples by considering various bridge boundary conditions and non-uniform cross-sections, from which the potential application of this general approach is foreseeable.

Mobile crowdsensing framework for drive-by-based dense spatial-resolution bridge mode shape identification

Moving vehicles equipped with various types of sensors can efficiently monitor the health conditions of a population of transportation infrastructure such as bridges. This paper presents a mobile crowdsensing framework to identify dense spatial-resolution bridge mode shapes using sparse drive-by measurements. The proposed method converts mode shape identification into a physical-informed optimization problem with two objective function terms. The first objective minimises the mode shape identification error based on the fact that the ratio of a specific order mode shape value at any two locations is time-invariant. Since the bridge mode shape should be globally smooth even when the local stiffness is discontinuous, the smoothness of the identified mode shape is introduced as the second objective. The feasibility and advantages of the proposed model are verified numerically and through large-scale experimental studies. Numerical results demonstrate that the proposed method can efficiently identify bridge mode shapes with a desirable accuracy. The adverse effects of road roughness and measurement noise on the mode shape identification accuracy are substantially suppressed by introducing crowdsensing and making use of collected responses over multiple trips. The applicability of the proposed method for bridges having varying cross sections and multiple spans is also studied. A series of drive-by tests with different vehicle masses and speeds are conducted on a large-scale footbridge. The experimental results verify that the proposed method can accurately identify the bridge mode shapes and is robust to vehicle mass and speed variation. The identification accuracy of large-scale bridge mode shapes using crowdsensing drive-by measurements is demonstrated in this study.

A Bayesian approach to global mode shape identification using modal assurance criterion-based discrepancy model

Modal identification of large-scale engineering structures may require multiple measurements obtained from different locations, due to the limitations in the instrumentation equipment. In such cases, the global values for modal frequencies and damping ratios can be estimated over the measurement setups by simply obtaining their statistical expected values. Estimation of global mode shape vectors, however, arises as a much more challenging problem since the local mode shapes confine at different degrees of freedom of the measured structure. To overcome this problem, a novel Bayesian methodology is presented in this study using a modal assurance criterion-based discrepancy model. By making use of this model, a novel solution procedure is presented to the existing literature, and mode shape normalization-based problems arising in the estimation of covariance matrix are solved. The computational efficiency of the proposed method is verified by two numerical and one real data example. The conducted investigations reveal that the presented methodology shows a good performance in terms of computational accuracy.

Geometric imperfections in CFS structural members, Part II: Data-driven modeling and probabilistic validation

This paper is the second part in a two-part series that examines the role of geometric imperfections in load–displacement response and collapse behavior of cold-formed steel (CFS) structural members. In part I (Farzanian et al., 2023) a review of the basics and a summary of efforts geared towards modeling geometric imperfections was provided. This included the ingredients needed for systematic identification of elastic buckling mode shapes, due to their frequent use in approximating geometric imperfections, and the development of high fidelity shell finite element (FE) nonlinear collapse analysis of CFS members seeded with different geometric imperfection models. Part II is the companion to the analysis in part I that revealed a noticeable variability in the collapse behavior and load carrying capacity of CFS members as a result of adopting different imperfection modeling strategies. In this part we provide the necessary steps to build a probabilistic framework that rests on machine learning, polynomial approximation, and parametric and non-parametric statistical inference and uses imperfection data to build a consistent stochastic field model of geometric imperfections. This data-driven model is then used to generate the much needed realizations of geometric imperfections that are faithful to the underlying statistics of measured data and therefore can be seeded, directly, in nonlinear collapse analysis of CFS members. Nonlinear shell FE analysis, within a stochastic simulation framework, is finally used to (statistically) quantify the impact of geometric imperfections and to enable probabilistic validation of geometric imperfection models.

An effective procedure for extracting mode shapes of simply-supported bridges using virtual contact-point responses of two-axle vehicles

This paper proposed an effective procedure using virtual contact-point responses to extract mode shapes of a simply-supported bridge. The theoretical closed-form solution of virtual contact-point response using two connected two-axle vehicles is derived based on the vehicle-bridge interaction theory. Variational mode decomposition and band-pass filter (VMD-BPF) method is adopted to decompose the virtual contact-point responses and extract the bridge mode shapes. And a mode shape splicing method is proposed to reduce the influence of vehicle length on the bridge mode shape identification. The accuracy of the identified mode shapes is verified by numerical simulations, indicating that the virtual contact-point responses of the front axle or rear axle in the two-axle vehicle can be used for mode shape extraction. Furthermore, parametric studies such as track irregularity, vehicle speed, and noise are conducted to analyze the parameters that affect the identification accuracy of the mode shapes. The results show that the first four order natural frequencies and corresponding mode shapes of bridges can be well identified by the virtual contact-point responses even when the track irregularity of grade 4 is used. It is also demonstrated that the proposed procedure has a good performance and robustness in extracting the mode frequencies and mode shapes of simply-supported bridges.

Full-Field Mode Shape Identification Based on Subpixel Edge Detection and Tracking

Most research on computer vision (CV)-based vibration measurement is limited to the determination of discrete or coarse mode shapes of the structure. The continuous edge of the structure in images has rich optical features, and thus, by identifying and tracking the movement of the structure’s edge, it is possible to determine high-resolution full-field mode shapes of the structure without a preset target. The present study proposes a CV-based method of full-field mode shape identification using the subpixel edge detection and tracking techniques. Firstly, the Canny operator is applied on each frame of the structure vibration video to extract the pixel-level edges, and the improved Zernike orthogonal moment (ZOM) subpixel edge detection technique is adopted to relocate the precise structure edges. Then, all the detected edge points are tracked to obtain the full-field dense displacement time history that is subsequently used to determine the structure frequencies and compute full-field mode shapes by combining the covariance driven stochastic subspace identification (SSI-COV) with the hierarchical cluster analysis. Finally, the proposed method is verified on the aluminum cantilever beam in the laboratory and the Humen Bridge in the field. The results show that the proposed method is able to detect more precise structure edges and identify the full-field displacement and mode shapes of structures without the need for installing artificial targets on the structure in advance, which provides valuable information for the structural condition assessment, especially for structures with small-amplitude vibrations.

Mode Shape Identification and Damage Detection of Bridge by Movable Sensory System

Indirect method to identify dynamic properties of bridge by a passing vehicle is attractive recently since it does not require pre-installed sensors on bridge. However, the extracted dynamic properties are not accurate enough to identify local damages due to road surface roughness. To overcome this issue, a method to identify mode shapes and detect local damages in a simply supported bridge by movable sensory system is proposed in this study. Instead of using passing vehicles, two stationary vehicles are adopted in this study to perform as movable sensory system. They stay stationary at two different locations on the bridge subject to ambient environmental excitation, and their vertical accelerations are collected to calculate the vertical displacement of the two contact points between vehicles and bridge. Then they move to another two locations and the same procedures are repeated. After all locations along the bridge are tested, a matrix similar to frequency response matrix can be constructed and SVD method is used to extract the mode shapes. The element stiffness is thereafter evaluated by the fundamental mode shape, which can be then used to detect local stiffness reduction in the bridge. Numerical study has been conducted to validate the proposed method, and the effect of bridge damping, vehicle damping, measurement noise, and traffic flow is investigated. Moreover, field testing was conducted on Li-Zi-Wan Bridge and the results showed that the proposed method can re-construct the mode shapes and evaluate element bending stiffness as accurately as conventional direct approach. The proposed method is convenient and easy to implement in practice; and it is more robust because it is not affected by road surface roughness.

A Novel Approach for Cable Tension Monitoring Based on Mode Shape Identification

Estimation and monitoring of cable tension is of great significance in the structural assessment of cable-supported bridges. For short cables, the traditional cable tension identification method via frequency measurement has large errors due to the influence of complex boundaries, which affect the accuracy of estimation. A new cable tension estimation method based on mode shape identification with a multiple sensor arrangement on the cable can take the influence of boundary conditions into account and its accuracy has been verified. However, it requires more sensors compared to the traditional frequency-based method, which will significantly increase the cost of long-term monitoring in practice. Therefore, a novel approach for cable tension monitoring considering both cost and accuracy is further proposed in this study. The approach adopts multiple sensors to measure the influence of boundary conditions. Then, only a single sensor is required for long-term monitoring of the cable. In this paper, an analytical model of the cable is firstly established. The influence of boundary conditions is calculated, which ensures the accuracy of mode shape identification. Furthermore, a field experiment is carried out to verify the effectiveness of the new approach. The results have demonstrated the effectiveness and accurateness of the proposed method in long-term short cable tension monitoring.

Indirect Spatial Mode Shape Identification of T-Girder Bridges Using Contact-Point Response of a Moving Test Vehicle: A Theoretical Study

Indirect bridge modal parameter identification using vehicle responses has been widely studied for two-dimensional bridge model, while rare work is devoted to the indirect identification of bridge spatial mode shape (BSMS). This paper proposes an effective procedure for BSMS identification of the widely used T-girder bridges, in which the closed-form solution is derived based on the vehicle contact-point response. The BSMS is identified using the bridge related components extracted from contact-point responses of a vehicle running at different transverse positions on the bridge. The detailed procedure for the identification of BSMS is introduced, and the feasibility of the method is analyzed numerically. To verify its applicability, the influences of some key factors in the vehicle–bridge system, such as the vehicle speed, the driving position, and the road surface roughness, on the identification results are analyzed. The effect of the bridge damage conditions on the BSMS identification is also investigated. The results show that the spatial mode shapes of T-girder bridges can be identified using the proposed method with high accuracy and efficiency.

Simultaneous Identification of Fundamental Frequency, Mode Shape, and Damping of a Bridge by a Passing Vehicle-Trailer System

A theoretical framework is proposed to identify fundamental frequency, mode shape, and damping of a bridge simultaneously by a passing vehicle-trailers system with only accelerometers. The framework is the first attempt to use indirect method to identify all fundamental modal properties of a bridge simultaneously without shaker, laser sensors and pre-known information. The vehicle performs as both “tractor” and “excitor” and the trailers perform as “message carriers”. The residual response defined as the response difference of one trailer to another one was usually adopted in the previous studies to remove the adverse effect of road roughness; however, it lacks theoretical support. Instead, the residual response of one trailer–bridge contact point to another one is used in this study, and theoretical analysis proves that it can effectively eliminate the adverse effect of road roughness. Different from previous studies, the residual response of contact point in this study is obtained in the frequency domain rather than the time domain. Frequency and damping can be evaluated by applying single degree-of-freedom (SDOF) model-based curve fitting technique to several discrete values from the Fourier transform of the residual response. Mode shapes can be reconstructed by dividing the Fourier transform of one residual response at the natural frequency by another one without time lag. The proposed framework is validated by independent finite element (FE) analysis and it shows great performance. Parametric study including vehicle speed, measurement noise, calibration error, road surface roughness and boundary condition has also been conducted to check its feasibility in various scenarios.

Output-only complete mode shape identification of bridges using a limited number of sensors

Mode shape is an important structural property that needs be identified in structural condition monitoring and analysis. The conventional modal identification methods usually can only provide sparse and low-spatial resolution mode shape measurements because of the limited number of sensors, which could seriously affect the effectiveness and accuracy of the mode shape-based structural damage detection methods. In this paper, a new output-only method is proposed for identifying the complete mode shapes for beam-like bridge structures under a moving load using a limited number of sensors based on proper orthogonal decomposition (POD). Firstly, the physical interpretation of the POD of the responses of a beam-like bridge under a moving load is derived. It is proved that each component of the POD consists of a mode shape of the bridge and a dynamic component with the frequency corresponding to this mode. The contribution ratio of the component in the POD gives an excellent estimation of the modal contribution ratio (CR) in terms of the vibration energy. The mode shape can be identified by filtering out the dynamic information in each component of the POD. The identified mode shapes have a high spatial resolution that they can be regarded as the complete mode shapes of the bridge. To demonstrate the proposed method, a beam bridge with different boundary conditions is numerically modelled and a simply supported beam bridge model is tested in the lab. Both the numerical and experimental results demonstrate that this method can successfully identify the complete mode shapes with high resolution and modal contribution ratios with a limited number of sensors. This method breaks through the challenge on the requirement of a large number of sensors in vibration tests for full mode shape identifications in traditional methods.

Calibration of the numerical model of a track section over a railway bridge based on dynamic tests

Progressive updating strategies are commonly used for the calibration of numerical models such as those of a track-bridge system. These strategies involve, first, updating the track model based on dedicated dynamic tests, and second, the calibration of the bridge model based on ambient vibration tests considering the updated track parameters. This subsequent procedure is suggested as some track parameters are, in relation to the modal responses of the bridge, generally not sufficiently sensitive for a successful parameter estimation. This work describes the calibration of a specific finite element (FE) model of a track section over a bowstring-arch railway bridge. The model includes the rails, rail pads, sleepers, the ballast layer and the interfaces with the bridge and neighbouring track sections. A track receptance test allowed for the identification of natural frequencies and mode shapes of the track by applying two different techniques, an output-only and an input-output. Several modes of vibration related to the movement of the sleepers on the ballast were identified using the SSI-DATA method, at frequencies between 83.8 Hz and 142.2 Hz. The calibration of the track FE model was performed using an iterative methodology by means of a genetic algorithm, which proves efficiency and robustness in estimating several track numerical parameters, besides a significant upgrade in relation to the numerical model before calibration.

Experimental verification of indirect bridge mode shape identification using transmissibility of a passing vehicle

The use of structural vibration is one of the bridge health monitoring approaches which is often costly, time-consuming. The response of a passing vehicle includes the bridge response so that it is used for extracting the bridge modal parameters. In this paper, the transmissibility measurement of a passing vehicle is employed in order to identify the bridge mode shape indirectly. As the sensor is embedded on the axle of the vehicle, recording the signal is fulfilled during the vehicle passage and there is no need to stop the vehicle. On the other hand, there is white noise assumption in most other techniques, but excitation characteristics are not considered in the proposed method which is another advantage. In the numerical simulation, the bridge is assumed to be Euler Bernoulli and the vehicle is modeled as a 4DOF system. Solving the vehicle-bridge interaction equations, the acceleration of all degrees of freedom are obtained. Afterwards, the mode shape is identified by applying the short time transmissibility measurement on the acceleration. Since the performance of the indirect methods in real cases is associated with many obstacles and challenges, a setup for experimental verification of the proposed method has been designed and constructed in a laboratory. Numerical simulations and experimental results indicate the capability of the proposed method for indirect identification of bridge mode shape.

Spatially Resolved Experimental Modal Analysis on High-Speed Composite Rotors Using a Non-Contact, Non-Rotating Sensor.

Due to their lightweight properties, fiber-reinforced composites are well suited for large and fast rotating structures, such as fan blades in turbomachines. To investigate rotor safety and performance, in situ measurements of the structural dynamic behaviour must be performed during rotating conditions. An approach to measuring spatially resolved vibration responses of a rotating structure with a non-contact, non-rotating sensor is investigated here. The resulting spectra can be assigned to specific locations on the structure and have similar properties to the spectra measured with co-rotating sensors, such as strain gauges. The sampling frequency is increased by performing consecutive measurements with a constant excitation function and varying time delays. The method allows for a paradigm shift to unambiguous identification of natural frequencies and mode shapes with arbitrary rotor shapes and excitation functions without the need for co-rotating sensors. Deflection measurements on a glass fiber-reinforced polymer disk were performed with a diffraction grating-based sensor system at 40 measurement points with an uncertainty below 15 rad and a commercial triangulation sensor at 200 measurement points at surface speeds up to 300 m/s. A rotation-induced increase of two natural frequencies was measured, and their mode shapes were derived at the corresponding rotational speeds. A strain gauge was used for validation.

Identification of Mode Shapes of a Composite Cylinder Using Convolutional Neural Networks.

The aim of the following paper is to discuss a newly developed approach for the identification of vibration mode shapes of multilayer composite structures. To overcome the limitations of the approaches based on image analysis (two-dimensional structures, high spatial resolution of mode shapes description), convolutional neural networks (CNNs) are applied to create a three-dimensional mode shapes identification algorithm with a significantly reduced number of mode shape vector coordinates. The CNN-based procedure is accurate, effective, and robust to noisy input data. The appearance of local damage is not an obstacle. The change of the material and the occurrence of local material degradation do not affect the accuracy of the method. Moreover, the application of the proposed identification method allows identifying the material degradation occurrence.