Mode Shape Estimation Research Articles

Modal participation in multiple input Ibrahim time domain identification

The Ibrahim time domain (ITD) identification technique was one of the first techniques formulated for multiple output modal analysis based on impulse response functions or general free decays. However, the technique has not been used much in recent decades due to the fact that the technique was originally formulated for single input systems that suffer from well-known problems in case of closely spaced modes. In this paper, a known, but more modern formulation of the ITD technique is discussed. In this formulation the technique becomes multiple input by adding some Toeplitz matrices over a set of free decays. It is shown that a special participation matrix can be defined that cancels out whenever the system matrix is estimated. The participation matrix becomes rank deficient if a mode is missing in the responses, but if any mode is present in one of the considered free decays, the participation matrix has full rank. This secures that all modes will be contained in the estimated system matrix. Finally, it is discussed how correlation functions estimated from the operational responses of structures can be used as free decays for the multiple-input ITD formulation, and the estimation errors of the identification technique are investigated in a simulation study with closely spaced modes. The simulation study shows that the multiple-input formulation provides estimates with significantly smaller errors on both mode shape and natural frequency estimates.

Earthquake‐induced structural response output‐only identification by two different Operational Modal Analysis techniques

SummaryOutput‐only system identification is developed here towards assessing current modal dynamic properties of buildings under seismic excitation. Earthquake‐induced structural response signals are adopted as input channels for two different Operational Modal Analysis (OMA) techniques, namely, a refined Frequency Domain Decomposition (rFDD) algorithm and an improved Data‐Driven Stochastic Subspace Identification (SSI‐DATA) procedure. Despite that short‐duration, non‐stationary, earthquake‐induced structural response signals shall not fulfil traditional OMA assumptions, these implementations are specifically formulated to operate with seismic responses and simultaneous heavy damping (in terms of identification challenge), for a consistent estimation of natural frequencies, mode shapes, and modal damping ratios. A linear ten‐storey frame structure under a set of ten selected earthquake base‐excitation instances is numerically simulated, by comparing the results from the two identification methods. According to this study, best up‐to‐date, reinterpreted OMA techniques may effectively be used to characterize the current dynamic behaviour of buildings, thus allowing for potential Structural Health Monitoring approaches in the Earthquake Engineering range.

A refined Frequency Domain Decomposition tool for structural modal monitoring in earthquake engineering

Output-only structural identification is developed by a refined Frequency Domain Decomposition (rFDD) approach, towards assessing current modal properties of heavy-damped buildings (in terms of identification challenge), under strong ground motions. Structural responses from earthquake excitations are taken as input signals for the identification algorithm. A new dedicated computational procedure, based on coupled Chebyshev Type II bandpass filters, is outlined for the effective estimation of natural frequencies, mode shapes and modal damping ratios. The identification technique is also coupled with a Gabor Wavelet Transform, resulting in an effective and self-contained time-frequency analysis framework. Simulated response signals generated by shear-type frames (with variable structural features) are used as a necessary validation condition. In this context use is made of a complete set of seismic records taken from the FEMA P695 database, i.e. all 44 “Far-Field” (22 NS, 22 WE) earthquake signals. The modal estimates are statistically compared to their target values, proving the accuracy of the developed algorithm in providing prompt and accurate estimates of all current strong ground motion modal parameters. At this stage, such analysis tool may be employed for convenient application in the realm of Earthquake Engineering, towards potential Structural Health Monitoring and damage detection purposes.

Multi-damage identification based on joint approximate diagonalisation and robust distance measure

Mode shapes or operational deflection shapes are highly sensitive to damage and can be used for multi-damage identification. Nevertheless, one drawback of this kind of methods is that the extracted spatial shape features tend to be compromised by noise, which degrades their damage identification accuracy, especially for incipient damage. To overcome this, joint approximate diagonalisation (JAD) also known as simultaneous diagonalisation is investigated to estimate mode shapes (MS’s) statistically. The major advantage of JAD method is that it efficiently provides the common Eigen-structure of a set of power spectral density matrices. In this paper, a new criterion in terms of coefficient of variation (CV) is utilised to numerically demonstrate the better noise robustness and accuracy of JAD method over traditional frequency domain decomposition method (FDD). Another original contribution is that a new robust damage index (DI) is proposed, which is comprised of local MS distortions of several modes weighted by their associated vibration participation factors. The advantage of doing this is to include fair contributions from changes of all modes concerned. Moreover, the proposed DI provides a measure of damage-induced changes in ‘modal vibration energy’ in terms of the selected mode shapes. Finally, an experimental study is presented to verify the efficiency and noise robustness of JAD method and the proposed DI. The results show that the proposed DI is effective and robust under random vibration situations, which indicates that it has the potential to be applied to practical engineering structures with ambient excitations.

On the use of a passing vehicle for the estimation of bridge mode shapes

This paper presents a novel algorithm for the estimation of bridge mode shapes using the response measured on a passing vehicle. A truck-trailer system is assumed, equipped with an external excitation at a frequency close to one of the bridge natural frequencies. The excitation makes the bridge response dominant at its natural frequency. The acceleration responses are measured on two following axles of the vehicle. It is shown that the amplitude of the signal includes the operational deflected shape data which can be used to estimate the bridge mode shapes. The energy of the responses measured on two following axles is obtained using the Hilbert Huang Transform. It is shown that the bridge mode shape can be estimated with high resolution and accuracy using a rescaling process. The presence of road roughness introduces additional contributions to the response measured on the vehicle, in addition to the bridge response. The concept of subtraction of the responses measured from two identical axles is used to remove the effect of road roughness.

Seismic FDD modal identification and monitoring of building properties from real strong-motion structural response signals

In the present study, output-only modal dynamic identification and monitoring of building properties is attempted successfully by processing real earthquake-induced structural response signals. This is achieved through an enhanced version of a recently-developed refined Frequency Domain Decomposition (rFDD) approach, which in the earlier implementation was adopted to analyse synthetic seismic response signals only. Despite that short duration, nonstationary seismic response data and heavy structural damping shall not fulfil traditional Operational Modal Analysis assumptions, the present rFDD response-only algorithm allows for the effective estimation of strong-motion natural frequencies, mode shapes, and modal damping ratios, with real seismic response signals. The present rFDD enhancement derives from a preprocessing time-frequency analysis and from an integrated approach for Power Spectral Density matrix computation, which constitute crucial innovative issues for the treatment of real earthquake response data. A monitoring case study is analysed by taking the real strong-motion response records from a seven-storey reinforced concrete building in Van Nuys, California, from 1987 to the latest 2014 events (Center of Engineering Strong Motion Data database), as recorded before, during and after the 1994 Northridge earthquake, which severely damaged the building (then retrofitted). This paper proves the effectiveness of the proposed enhanced rFDD algorithm as a robust method for monitoring current structural modal properties under real earthquake excitations. This shall allow for identifying possible variations of structural features along experienced seismic histories, providing then a fundamental tool towards Earthquake Engineering and Structural Health Monitoring purposes.

Output only system identification based on synchrosqueezed transform

A new blind source separation formulation for modal identification has been developed based on synchrosqueezed transform. It is shown that by using acceleration measurements there is a greater chance of identifying higher modes than with displacement/velocity measurements. An elegant and robust procedure for estimation of mode shapes and modal damping is presented which avoids tedious computations required for other blind source separation formulations, such as the independent component analysis and second order blind identification. A numerical example study based on the recorded earthquake response of the UCLA Factor building has been used to validate the proposed formulation.

Blind identification of full-field vibration modes of output-only structures from uniformly-sampled, possibly temporally-aliased (sub-Nyquist), video measurements

Enhancing the spatial and temporal resolution of vibration measurements and modal analysis could significantly benefit dynamic modelling, analysis, and health monitoring of structures. For example, spatially high-density mode shapes are critical for accurate vibration-based damage localization. In experimental or operational modal analysis, higher (frequency) modes, which may be outside the frequency range of the measurement, contain local structural features that can improve damage localization as well as the construction and updating of the modal-based dynamic model of the structure. In general, the resolution of vibration measurements can be increased by enhanced hardware. Traditional vibration measurement sensors such as accelerometers have high-frequency sampling capacity; however, they are discrete point-wise sensors only providing sparse, low spatial sensing resolution measurements, while dense deployment to achieve high spatial resolution is expensive and results in the mass-loading effect and modification of structure's surface. Non-contact measurement methods such as scanning laser vibrometers provide high spatial and temporal resolution sensing capacity; however, they make measurements sequentially that requires considerable acquisition time. As an alternative non-contact method, digital video cameras are relatively low-cost, agile, and provide high spatial resolution, simultaneous, measurements. Combined with vision based algorithms (e.g., image correlation or template matching, optical flow, etc.), video camera based measurements have been successfully used for experimental and operational vibration measurement and subsequent modal analysis. However, the sampling frequency of most affordable digital cameras is limited to 30–60Hz, while high-speed cameras for higher frequency vibration measurements are extremely costly. This work develops a computational algorithm capable of performing vibration measurement at a uniform sampling frequency lower than what is required by the Shannon-Nyquist sampling theorem for output-only modal analysis. In particular, the spatio-temporal uncoupling property of the modal expansion of structural vibration responses enables a direct modal decoupling of the temporally-aliased vibration measurements by existing output-only modal analysis methods, yielding (full-field) mode shapes estimation directly. Then the signal aliasing properties in modal analysis is exploited to estimate the modal frequencies and damping ratios. The proposed method is validated by laboratory experiments where output-only modal identification is conducted on temporally-aliased acceleration responses and particularly the temporally-aliased video measurements of bench-scale structures, including a three-story building structure and a cantilever beam.

On the application of correlation function matrices in OMA

In this paper the theoretical solution for the correlation function matrix of the random response of a structural system is re-visited. It is shown that using the classical definition of the correlation functions, the row space is defined by the mode shapes of the system, whereas the column space is defined by the modal participation vectors. This means that only the rows can be used for unbiased modal identification in operational modal analysis and if the columns are used for identification, then bias will be introduced on the mode shape estimates. It is pointed out that the mode shape bias is strongly dependent on the frequency distance between the modes, i.e. bias will significantly increase in case of closely spaced modes. The identification errors on the estimated biased and unbiased mode shapes are studied in a simulation example.

705 質量変更法を用いた正規化固有モードと固有振動数の推定(機械力学・計測制御VII)

705 質量変更法を用いた正規化固有モードと固有振動数の推定(機械力学・計測制御VII)

EMD-based output-only identification of mode shapes of linear structures

The Hilbert-Huang transform (HHT) consists of empirical mode decomposition (EMD) and Hilbert spectral analysis. EMD has been successfully applied for identification of mode shapes of structures based on input-output approaches. This paper aims to extend application of EMD for output-only identification of mode shapes of linear structures. In this regard, a new simple and efficient method based on band-pass filtering and EMD is proposed. Having rather accurate estimates of modal frequencies from measured responses, the proposed method is capable to extract the corresponding mode shapes. In order to evaluate the accuracy and performance of the proposed identification method, two case studies are considered. In the first case, the performance of the method is validated through the analysis of simulated responses obtained from an analytical structural model with known dynamical properties. The low-amplitude responses recorded from the UCLA Factor Building during the 2004 Parkfield earthquake are used in the second case to identify the first three mode shapes of the building in three different directions. The results demonstrate the remarkable ability of the proposed method in correct estimation of mode shapes of the linear structures based on rather accurate modal frequencies.

Dynamic Response Of Tower Structures

The present study focuses on the tower type structures response to the dynamic loads. The study analyzes the possible mode shapes regarding to tower structure. The estimation of mode shapes and their dependence from structural changes was made for an existing tower structure. To get an acceptable tower’s vibration level and avoid possibility of resonance effect from usual serviceability loads it was evaluated options to change natural frequencies of the structure. It is performed existing 36m high sightseeing tower dynamic analysis and proposed potential solutions to increase critical natural frequencies of the structure. In this study to obtain dynamic parameters of the sightseeing tower structure have been used finite element models and calculation techniques.

Improvements for electromechanical oscillation mode estimation via subspace identification methods

Two improvements of power system electromechanical oscillation estimation via subspace identification are proposed in this paper. Singular value-based criteria are introduced not only to improve the computation efficiency for determining the model order of the estimated system but also to improve the accuracy of the estimates. A mode matching method based on the characteristics of accurate mode shape estimates is proposed to ensure the correctness of the estimates. Numerical results from three different scale power systems illustrate the feasibility of the proposed methods.

Optimal sensor configuration for flexible structures with multi-dimensional mode shapes

A framework for deciding the optimal sensor configuration is implemented for civil structures with multi-dimensional mode shapes, which enhances the applicability of structural health monitoring for existing structures. Optimal sensor placement (OSP) algorithms are used to determine the best sensor configuration for structures with a priori knowledge of modal information. The signal strength at each node is evaluated by effective independence and modified variance methods. Euclidean norm of signal strength indices associated with each node is used to expand OSP applicability into flexible structures. The number of sensors for each method is determined using the threshold for modal assurance criterion (MAC) between estimated (from a set of observations) and target mode shapes. Kriging is utilized to infer the modal estimates for unobserved locations with a weighted sum of known neighbors. A Kriging model can be expressed as a sum of linear regression and random error which is assumed as the realization of a stochastic process. This study presents the effects of Kriging parameters for the accurate estimation of mode shapes and the minimum number of sensors. The feasible ranges to satisfy MAC criteria are investigated and used to suggest the adequate searching bounds for associated parameters. The finite element model of a tall building is used to demonstrate the application of optimal sensor configuration. The dynamic modes of flexible structure at centroid are appropriately interpreted into the outermost sensor locations when OSP methods are implemented. Kriging is successfully used to interpolate the mode shapes from a set of sensors and to monitor structures associated with multi-dimensional mode shapes.

Modal Parameters Estimation of Building Structures from Vibration Test Data Using Observability Measurement

The load distribution to each mode of a structure under seismic loading depends on the modal participation factors and mode shapes and thus the exact estimation of modal participation factors and mode shapes is essential to analyze the seismic response of a structure. In this study, an identification procedure for modal participation factors and mode shapes from a vibration test is proposed. The modal participation factors and mode shapes are obtained from the relationship between observability matrices realized from the system identification. Using the observability matrices, it is possible to transform an arbitrarily identified state space model obtained from the experimental data into a state space model which is defined in a domain with physical meaning. Then, the modal participation factor can be estimated based on the transformation matrix between two state space models. The numerical simulation is performed to evaluate the proposed procedure, and the results show that the modal participation factor and mode shapes are estimated from the structural responses accurately. The procedure is also applied to the experimental data obtained from the shaking table test of a three-story shear building model.

Application of Modal Parameter Estimation Methods for Continuous Wavelet Transform-Based Damage Detection for Beam-Like Structures

This paper presents a hybrid damage detection method based on continuous wavelet transform (CWT) and modal parameter identification techniques for beam-like structures. First, two kinds of mode shape estimation methods, herein referred to as the quadrature peaks picking (QPP) and rational fraction polynomial (RFP) methods, are used to identify the first four mode shapes of an intact beam-like structure based on the hammer/accelerometer modal experiment. The results are compared and validated using a numerical simulation with ABAQUS software. In order to determine the damage detection effectiveness between the QPP-based method and the RFP-based method when applying the CWT technique, the first two mode shapes calculated by the QPP and RFP methods are analyzed using CWT. The experiment, performed on different damage scenarios involving beam-like structures, shows that, due to the outstanding advantage of the denoising characteristic of the RFP-based (RFP-CWT) technique, the RFP-CWT method gives a clearer indication of the damage location than the conventionally used QPP-based (QPP-CWT) method. Finally, an overall evaluation of the damage detection is outlined, as the identification results suggest that the newly proposed RFP-CWT method is accurate and reliable in terms of detection of damage locations on beam-like structures.

Structural health monitoring of offshore wind turbines using automated operational modal analysis

This article will present and discuss the approach and the first results of a long-term dynamic monitoring campaign on an offshore wind turbine in the Belgian North Sea. It focuses on the vibration levels and modal parameters of the fundamental modes of the support structure. These parameters are crucial to minimize the operation and maintenance costs and to extend the lifetime of offshore wind turbine structure and mechanical systems. In order to perform a proper continuous monitoring during operation, a fast and reliable solution, applicable on an industrial scale, has been developed. It will be shown that the use of appropriate vibration measurement equipment together with state-of-the art operational modal analysis techniques can provide accurate estimates of natural frequencies, damping ratios, and mode shapes of offshore wind turbines. The identification methods have been automated and their reliability has been improved, so that the system can track small changes in the dynamic behavior of offshore wind turbines. The advanced modal analysis tools used in this application include the poly-reference least squares complex frequency-domain estimator, commercially known as PolyMAX, and the covariance-driven stochastic subspace identification method. The implemented processing strategy will be demonstrated on data continuously collected during 2 weeks, while the wind turbine was idling or parked.

Modal estimation by FBG for flexible structures attitude control

This work investigates an online mode shape estimation method to estimate the time-varying modal properties and correct IMU readings in real time using distributed strain measurements of FBG sensor arrays. Compared to the notch filter approach, the proposed method removes structural vibration information without adding phase lag or transient dynamics. Numerical simulations on a flexible rocket attitude control and the experiments on a vertical test beam with configurable mass distribution are used for demonstration.

Damage assessment of a bridge based on mode shapes estimated by responses of passing vehicles

In this study, an indirect approach is developed for assessing the state of a bridge on the basis of mode shapes estimated by the responses of passing vehicles. Two types of damages, i.e., immobilization of a support and decrease in beam stiffness at the center, are evaluated with varying degrees of road roughness and measurement noise. The assessment theory's feasibility is verified through numerical simulations of interactive vibration between a two-dimensional beam and passing vehicles modeled simply as sprung mass. It is determined that the damage state can be recognized by the estimated mode shapes when the beam incurs severe damage, such as immobilization of rotational support, and the responses contain no noise. However, the developed theory has low robustness against noise. Therefore, numerous measurements are needed for damage identification when the measurement is contaminated with noise.

Implementing physical constraints for noise-only mode shape estimation on real data

Many underwater acoustic tasks sample a narrowband pressure field with a vertical line array. In the absence of strong local sources, the noise sampled by an array includes both a spatially correlated and a spatially uncorrelated component. The acoustic waveguide's modes form a basis for the spatially correlated noise component generated by distant sources. This basis can be estimated from the eigenvectors of the noise sample covariance matrix. Propagation physics constrain the mode shapes to be real, but the eigenvectors are generally complex. These complex vectors require a phase rotation for each eigenvector prior to taking the real part. Previous work found that the eigenvector mode estimates' phases include spatially correlated noise. This noise correlation degrades the performance of spatial averaging and minimum variance estimates of the unknown phase rotation. The best estimate is the phase of the array element with the maximum magnitude for each mode, which is robust to the noise's spatial correlation. This research evaluates the previously developed phase rotation techniques using a real data set. The data was collected off the north coast of Elba Island by the SACLANT Centre in 1993. [This work was supported by ONR.]