Mode Shape Estimation Research Articles

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.

Improving mode extraction with physics-informed neural network

This study aims to enhance conventional mode extraction methods in ocean waveguides using a physics-informed neural network (PINN). Mode extraction involves estimating mode wavenumbers and corresponding mode depth functions. The approach considers a scenario with a single frequency source towed at a constant depth and measured from a vertical line array (VLA). Conventional mode extraction methods applied to experimental data face two problems. First, mode shape estimation is limited because the receivers only cover a partial waveguide. Second, the wavenumber spectrum is affected by issues such as Doppler shift and range errors. To address these challenges, we train the PINN with measured data, generating a densely sampled complex pressure field, including the unmeasured region above the VLA. We then apply the same mode extraction methods to both the raw data and the PINN-generated data for comparison. The proposed method is validated using data from the SWellEx-96, demonstrating improved mode extraction performance compared to using raw experimental data directly.

Subspace-based modal identification and uncertainty quantification from video image flows

Vibration measurements for structural health monitoring (SHM) by operational modal analysis (OMA) are classically obtained from sensors that are embedded in or physically attached to the monitored structure, like accelerometers or strain gauges. However, the setup time of these sensors and their restricted number and space coverage limit their monitoring capabilities. Video image-based sensing methods can overcome these shortcomings. With adequate image processing methods, motion signals are extracted from video image flows, which are then processed by system identification methods to estimate modal parameters. In this way, the pixels in selected regions of interest within the images act as a dense network of contactless sensors distributed over the whole structure. In this paper, the efficiency of this video-based approach is demonstrated with laboratory experiments on a cantilever beam, in particular, by evaluating its capability for detecting weak damages mimicked by slight mass modifications. To this end, the steerable filter-based method (ST), that recovers displacements from local phase, is first extended to overcome its motion limitation of one pixel size. Then, the performance of the improved motion extraction method is compared with two other well-established methods in the context of OMA, where natural frequencies, damping ratios and mode shapes with high spatial resolution are estimated together with their uncertainty bounds using covariance-driven subspace identification. The compared methods are evaluated with the help of reference laser displacement measurements as well as a finite element model of the beam, revealing differences in the accuracy of the estimated mode shapes depending on the chosen method for motion extraction. Finally, aiming to investigate early structural damage detection, experiments are carried out under small structural changes and the results are compared to a reference state with the help of estimated uncertainties. Small but statistically significant changes in the modal parameters are detected, showing the potential of the vision based framework for SHM.

Efficacy of Vehicle Scanning Methods in Estimating the Mode Shapes of Bridges Seated on Elastic Supports.

This study systematically assesses the efficacy of the vehicle scanning methods (VSM) in accurately estimating the mode shapes of bridges seated on elastic supports. Three state-of-the-art VSM methods are employed to obtain the mode shapes of bridges using the vehicle data during its travel. Two of the evaluated methods use a signal decomposition technique to extract the modal components of the bridge from the contact point of the response while the third one relies on the segmentation of the measured signals along the bridge deck and applying an operational modal analysis tool to each segmented signal to estimate the mode shapes. Numerical analyses are conducted on one single- and one two-span bridge, considering smooth and rough road profiles, different vehicle speeds, and presence of lead vehicle. The accuracy of the numerical models used in developing and assessing vehicle scanning models is tested, and the results of the study demonstrate the method using a half-car vehicle model and signal decomposition technique shows robustness against increasing vehicle speeds and road roughness while the method applying the segmentation of the measured signals provides relatively accurate mode shape estimates at the bridge edges at low speed, although the three methods have their limitations. It is also observed that simplified bridge and vehicle models can hide potential challenges that arise from the complexity of actual vehicle and bridge systems. Considering that a significant number of bridges worldwide are built on elastic supports, the practical success of vehicle scanning methods depends on their ability to handle elastic boundary conditions with reliability. Therefore, the article provides valuable insights into the capabilities and limitations of the current vehicle scanning methods, paving the way for further advancements and refinements in these techniques.

FDD based modal identification of structures using least squares approach

In the conventional frequency domain modal identification procedures, modal parameters are attempted to be identified using power spectral density matrix decomposition. As it is well stated in the literature, the quality of estimated mode shapes significantly increases when an average PSD matrix over a selected frequency band is utilized in the identification procedure. In this context, this study aims at deriving a weighted average for the PSD matrix to increase the quality of the estimated mode shapes by using an FDD-based modal model. Then, the remaining modal parameters comprising modal frequency, damping ratio and spectral density of modal exciting force are calculated through a least squares equation. Moreover, the presented FDD based approach is also adapted to the solution of multiple setup problem. The proposed method for single and multiple setup cases is validated by making use of one numerical and two field data examples. The obtained results reveal that the presented frequency domain technique can successfully achieve high quality of modal estimates in the presence of measurement noise, and it can be easily implemented to field applications.

Spectral optimization-based modal identification: A novel operational modal analysis technique

In this work, we introduce a novel method, namely Spectral Optimization-based Modal Identification (SOMI) for estimating the modal parameters of structures. SOMI is developed to estimate highly accurate modal damping ratios and mode shapes using the Frequency Domain Decomposition (FDD) principles. The FDD method (and most of the present modal identification techniques), encounters some problems such as modal complexity, lack of accuracy in the case of large damping ratios, and noticeable error in mode shape estimation. In the FDD formulation, the Singular Value Decomposition (SVD) estimation is precise only for lightly damped structures for obtaining the mode shapes and the Power Spectral Density (PSD) function of the modal coordinates. Moreover, the complexity of the mode shapes obtained from the FDD method is significantly biased and should not be used for physical interpretations. The main idea behind the proposed SOMI method is the elimination of the SVD estimation and obtaining the PSD function of the modal coordinates and the mode shapes with the least approximations. Moreover, the modal damping ratio is directly estimated in the frequency domain without any transformation into time domain using two equations derived in this paper. SOMI method uses optimization to make the PSD function of the modal coordinates follow the theoretical equations and estimates highly accurate mode shapes. In this work, the derivations are given for the displacement, velocity, and acceleration signals. The detailed formulation of the proposed method is presented, it is validated with some examples, and its competitive accuracy is compared with the FDD method. The accuracy and robustness of SOMI in noisy conditions are also assessed, with results indicating its effectiveness in handling noise and maintaining accuracy.

Reliability Assessment of a Vision-Based Dynamic Displacement Measurement System Using an Unmanned Aerial Vehicle.

In recent years, many studies have been conducted on the vision-based displacement measurement system using an unmanned aerial vehicle, which has been used in actual structure measurements. In this study, the dynamic measurement reliability of a vision-based displacement measurement system using an unmanned aerial vehicle was examined by measuring various vibrations with a frequency of 0 to 3 Hz and a displacement of 0 to 100 mm. Furthermore, free vibration was applied to model structures with one and two stories, and the response was measured to examine the accuracy of identifying structural dynamic characteristics. The vibration measurement results demonstrated that the vision-based displacement measurement system using an unmanned aerial vehicle has an average root mean square percentage error of 0.662% compared with the laser distance sensor in all experiments. However, the errors were relatively large in the displacement measurement of 10 mm or less regardless of the frequency. In the structure measurements, all sensors demonstrated the same mode frequency based on the accelerometer, and the damping ratios were extremely similar, except for the laser distance sensor measurement value of the two-story structure. Mode shape estimation was obtained and compared using the modal assurance criterion value compared with the accelerometer, and the values for the vision-based displacement measurement system using an unmanned aerial vehicle were close to 1. According to these results, the vision-based displacement measurement system using an unmanned aerial vehicle demonstrated results similar to those of conventional displacement sensors and can thus replace conventional displacement sensors.

A modal‐based Kalman filtering framework for mode extraction and decomposition of damped structures

AbstractThe mode shape is one of the important modal parameters that enables to visualize the intrinsic behavior of a structure as well as the quantity of interest by extracting or separating modal response from measurements. In this study, a new output‐only framework is proposed to extract modes using a modal‐based Kalman filter defined in the modal space and identify the mode shape by manipulating the correlation between the separated modes and the measured responses. It is also shown that the proposed framework can be extended to estimate the mode shapes of a non‐classically damped structure in state space when the state variable is constructed from the measured responses and applied to the modal‐based Kalman filter. The mode shape estimation framework proposed in this study was verified by numerical simulations and full‐scale measurements. From the verification examples and their results, it was noted that the proposed modal identification framework is not influenced by the presence of noise, and it can be applied to identify the state‐space mode shapes of non‐classically damped systems as well as systems with very closely distributed modes such as buildings equipped with tuned mass dampers.

Investigation of long-term modal properties of a supertall building under environmental and operational variations

The information on long-term modal properties of civil structures under environmental and operational variations is generally required in structural health monitoring and vibration control. In operational modal analysis (OMA), damping estimates usually present larger fluctuations and uncertainties than natural frequency and mode shape estimates, which affects the precise determination of the long-term damping characteristics of civil structures. To this end, a combined OMA scheme is adopted in this paper based on the variational mode extraction technique and covariance-driven stochastic subspace identification algorithm to reduce errors in modal identification, especially for damping estimates. Through numerical simulation study of a three-story frame structure, the effectiveness and accuracy of the combined scheme for modal estimations are validated. Moreover, the combined approach is further utilized to evaluate the long-term modal properties of a 600 m high skyscraper under environmental and operational variations from the measured response records of 183 days. Based on the reliable modal estimates by the combined method, the effects of environmental factors (air temperature and relative humidity), wind conditions, and building response amplitudes on the long-term modal properties and their statistical characteristics are investigated in detail. This paper aims to provide an effective tool for analyzing the long-term modal properties of civil structures and further reveal the time evolution of the dynamical characteristics of supertall buildings under environmental and operation variations. • Present a combined scheme based on variational mode extraction technique and covariance-driven stochastic subspace identification algorithm to reduce errors in structural modal estimates. • Suggest an empirical formula for determining minimum length of responses required for accurate modal estimates. • Validate effectiveness and accuracy of the combined scheme via numerical simulation study. • Investigate long-term evolution of dynamical characteristics of a 600 m high skyscraper under environmental and operational variations.

Accelerating convergence in Bayesian operational modal analysis with Fisher information matrix

Bayesian operational modal analysis (BAYOMA) has been increasingly applied to ambient vibration tests, providing a fundamental means for estimating modal properties as well as quantifying their identification uncertainty consistent with probability rules and modelling assumptions. Computing the most probable value of modal parameters in BAYOMA involves a high-dimensional numerical optimisation of the negative log-likelihood function (NLLF). Brute-force optimisation using generic algorithms treating the NLLF as a black box is computationally prohibitive and often non-converging. Efficient iterative algorithms have been developed over the past decade, but challenges still exist, e.g., for very close modes that may require significantly more iterations or (in some cases) may not be converging. Leveraging on recent advance in the mathematics and understanding of the Fisher Information Matrix (FIM) of modal parameters, an efficient method based on Newton-type iterations is proposed in this work to improve computational efficiency and convergence robustness. The MPV estimate in each iteration is updated based on two characteristic types of principal directions of the FIM. Move of the first type involves mode shapes only and is orthogonal to the subspace spanned by the current mode shape estimates. The second type involves updates of all modal parameters, where the mode shapes move within the subspace. Compact analytical expressions are derived for the gradient of NLLF to ensure accurate and efficient use in determining each iteration move. The performance of the proposed method is investigated with a comprehensive study based on synthetic, laboratory and field data. Results reveal that the proposed algorithm generally outperforms existing ones in terms of computational time by an order of magnitude, and it has better convergence robustness especially for challenging cases with very close modes and high modal force coherence.

Full‐field out‐of‐plane vibration displacement acquisition based on speckle‐projection digital image correlation and its application in damage localization

AbstractStereo‐digital image correlation (Stereo‐DIC) has been widely explored for modal analysis in plate‐type structures due to its noncontact and full‐field advantages. However, when the traditional stereo‐DIC is adopted to capture the out‐of‐plane displacements, several challenging issues exist such as the development of surface speckles, asynchronous camera recording, and efficiency and accuracy degradation due to high computation costs. Moreover, with the captured out‐of‐plane displacements, effective and efficient evaluation of the high spatial resolution mode shapes and their application to damage localization are also critical problems. To tackle these issues, a speckle‐projection DIC technique using a single high‐speed camera is proposed to obtain the out‐of‐plane vibration displacements. Moreover, an enhanced peak‐picking modal analysis method is adopted to enhance the estimation accuracy and efficiency of mode shapes. In addition, the low‐rank property of mode shapes in an intact state and the spatial sparse property of damage locations are harnessed for the detection of damage positions without requiring reference data on the healthy state. Finally, the modal analysis and damage localization results based on the proposed speckle‐projection DIC are compared with those of the traditional two‐camera stereo‐DIC technique to verify its feasibility and effectiveness. It is found that the differences in the identified resonant frequencies between these two methods are smaller than 1% for higher modes. Moreover, the proposed speckle‐projection DIC has the same accuracy as the traditional two‐camera stereo‐DIC in terms of measurement accuracy, mode shape estimation, and damage localization.

Blind modal identification for decentralized sensor network by modified sparse component analysis in frequency-domain subspace

It is practical to develop a modal identification method using a decentralized sensor network for field tests. Sparse component analysis (SCA) is a potential method in the framework of blind source separation to overcome the difficulty of using limited sensors. Nevertheless, it is still challenging for SCA to identify a larger number of modes than the number of measurements using the decentralized sensor network. This is because the partial mode shapes may be similar. It is convenient to perform SCA in frequency-domain subspaces to improve SCA further for decentralized sensor implementation; this is owing to the distinguishability of different partial mode shapes. In this study, a modal identification method applied to a largely underdetermined case caused by a decentralized sensor network is proposed by modifying SCA by automatic segmentation of frequency-domain subspaces. First, a method for determining the segmenting point is developed by identifying the valleys of the energy fluctuation spectrum using the scale-space peak detection method. Second, single-mode-points in the time–frequency plane are detected to facilitate mode shape estimation in the frequency-domain subspaces. Third, density peak clustering with the clustering number determination method is developed to estimate mode shapes, particularly for closely spaced modes. Finally, the modal frequencies and damping ratios are identified through the inverse operator in the frequency-domain subspaces and the curve fitting of the expected instant spectrum. The proposed method is verified by a 10 degree-of-freedom system, a structure with close modes, and an actual cable-stayed bridge. The investigations verified the effectiveness of the proposed method for decentralized sensor networks and practical applications.

Acoustic modal analysis of room responses from the perspective of state-space balanced realization with application to field interpolation.

Despite its importance in structural dynamics and vibration, modal analysis is rarely performed in acoustics due to the high modal density of sound fields. A novel acoustic modal analysis (AMA) approach is proposed in this paper for enclosed acoustic fields, such as a room, from the perspective of state-space formulation of control systems. A single-input-multiple-output (SIMO) state-space model is established in light of the balanced realization (BR), given impulse response measurements. The BR model is then converted to a modal form such that the modal parameters, including natural frequencies, damping ratios, and mode shapes, can be estimated. To reconstruct mode shapes, plane wave decomposition (PWD) and compressive sensing (CS) techniques are exploited to solve the underdetermined problem for a spatially sparse representation of mode shapes under the Schroeder frequency. As a result, a model of a continuous system can be "interpolated" for any arbitrary source-receiver positions on the basis of the estimated mode shapes. With the identified modal parameters, the low-frequency and early reflection part of room impulse responses (RIRs) can be synthesized for arbitrary source-sensor pairs. The proposed AMA acoustic field interpolation is validated by extensive simulations and experiments.

Modal analysis and damage localization in plate-type structures via TDD and PE methods based on the data of an integrated highspeed camera system

With the fast development of full-field vibration measurements by combining digital image correlation (DIC) technique with the integrated highspeed camera system, the structural characteristic deflection shapes (CDS’s) can be readily captured, which are sensitive for damage localization. Naturally, the mode shapes are preferred due to their higher signal to noise ratio. However, traditional modal analysis methods are inefficient in processing the large volumes of data that acquired by high-speed camera system for mode shape-based damage localization. Moreover, how to extract the damage features and enhance the damage localization results without the baseline-data of the intact state is another challenging issue. To address these problems, a novel damage localization method is proposed by enhancing the mode shape estimation and optimizing the damage feature extraction. Firstly, the dominant singular vector at each interested resonant frequency is individually evaluated as the mode shapes in the proposed time domain decomposition (TDD) method. Moreover, based on the extracted mode shapes, a pseudo-excitation (PE) method is proposed to reveal the damage locations without baseline-data by examining the perturbation of local dynamic equilibrium. To circumvent the effects of measurement noise on PE method, an adaptive denoising technique is proposed based on Gaussian smoothing. Furthermore, the local contiguity and spatial sparsity of the damage-caused features are harnessed by a hierarchical clustering to optimize the damage localization performance via tuning the scale parameter of Gaussian smoothing. Finally, numerical and experimental studies of crack damaged plates are conducted to validate the feasibility and effectiveness of the proposed mode shape estimation and damage localization methods. During the experiment, the full-field vibration measurements are acquired by using an integrated binocular highspeed camera system. In addition, a comparison study based on benchmark data is carried out to demonstrate the performance and reliability of the proposed damage localization method.

Operational modal analysis of a catamaran using a limited set of accelerometers

ABSTRACT Experimental modal analysis tests were performed during the catamaran’s normal operation and using only 06 accelerometers on the main deck. This work focuses on estimating and comparing modal parameters using experimental (time and frequency domain techniques) and numerical analyses. Ten (10) experimental natural frequencies and (incomplete) mode shapes were estimated, in the range of 4–50 Hz, using Balance Realisation, Frequency-Spatial Domain Decomposition and Enhanced Frequency Domain Decomposition techniques. As only 06 accelerometers had been used for modal testing, a finite element (FE) model was used to augment the spatial resolution of the experimental mode shapes. Experimental and numerical natural frequencies showed good correlation with only the 3rd bending mode shape exhibiting 13.5% deviation. It was concluded that the experimental techniques can: (i) identify natural frequencies in the presence of harmonic components and, (ii) estimate mode shapes using a limited set of accelerometers, aided by a FE model.

Modeling the effects of modulated radiation pressure using a new acoustic lens

Modulated radiation pressure (MRP) is a method of utilizing the acoustic radiation force by using a double-sided suppressed carrier modulated (DSB-SCM) signal. This produces a modulated signal that oscillates at twice the modulation frequency. This process facilitates target detection and classification by using a high-carrier frequency to probe the target and shake it with the low-modulation frequency, which can be designed to be commensurate with the vibrational modes of the target. A frequency sweep can estimate the target resonant frequencies, while a physical scan along its length can estimate its mode shapes. Recently, a series of experiments were conducted at Washington State University with a newly designed and fabricated acoustic lens with a one-meter focal length. These experiments demonstrated that the first few bending modes of scaled targets like a solid aluminum cylinder and an open-ended aluminum cylindrical shell could be excited by MRP at a standoff distance of one meter and target mode shapes were measured using a physical scan. In this presentation, we show modeling results that confirm experimental findings: (1) estimation of target resonant frequencies using a frequency sweep and (2) estimation of target mode shapes using a physical scan.

Full-field mode shape estimation of a rotating structure subject to random excitation using a tracking continuously scanning laser Doppler vibrometer via a two-dimensional scan scheme

A two-dimensional (2D) scan scheme is developed for a tracking continuously scanning laser Doppler vibrometer (CSLDV) system to scan the whole surface of a rotating structure subject to random excitation. A tracking CSLDV system is developed to track a rotating structure and sweep its laser spot on its surface. The measured response of the structure using the 2D scan scheme of the tracking CSLDV system is considered as the response of the whole surface of the structure subject to random excitation. The measured response can be processed by an operational modal analysis (OMA) method called the improved demodulation method based on a rigorous nonuniform rotating plate model to obtain modal parameters of the rotating structure, such as damped natural frequencies and undamped full-field mode shapes. Damped natural frequencies of the rotating structure are estimated from the fast Fourier transform of the measured response. Undamped full-field mode shapes are estimated by multiplying the measured response by sinusoids whose frequencies are estimated damped natural frequencies. Experimental investigation of the 2D scan scheme of the tracking CSLDV system and OMA method is conducted, and damped natural frequencies and undamped full-field mode shapes of a rotating fan blade with different constant speeds are estimated. It is theoretically and experimentally shown that damped natural frequencies of the rotating fan blade increases with its rotation speed.

Dynamic Deformation Measurement of Bridge Structure Based on GB-MIMO Radar

Dynamic deformation measurement is an important approach to monitor the structure stability. Due to its rapid imaging capabilities, ground-based multi-input multi-output (GB-MIMO) radar has shown great application potential in bridge structure monitoring. This paper proposes a dynamic deformation estimation method based on radar image series. Firstly, an improved clutter suppression method is utilized to overcome the estimation error in complex working status. Secondly, a two-step pixel extraction method is adopted to ensure both quantity and quality of sample pixels. Finally, frequency- and time-domain environmental stimulating methods are jointly considered for stable mode shape estimation. This paper systematically measured the dynamic deformations of bridge structures, including two suspension bridges and a cable-stayed bridge. The time-series deformation, deflection, vibration frequency and amplitude on the bridge structure are obtained by image domain measurement. And for the first time, the mode shapes of bridges are obtained through GB-MIMO radar. The correctness of structure parameter measurement is verified with the finite element model (FEM). Experimental results prove that with the proposed method, stable results could be acquired against weak and complex stimulation conditions.

Fundamental mode shape estimation and element stiffness evaluation of girder bridges by using passing tractor-trailers

A comprehensive indirect method is proposed in this study to estimate the dynamic characteristics of the fundamental mode and evaluate the element stiffness of a girder bridge using a passing tractor with two identical trailers. The novelty of this study is the use of the dynamic interaction between the two identical trailers and the bridge rather than that between the truck and the bridge. One of the advantages of this is that the dynamic characteristics of trailers are easier to model and analyse. Under the proposed method, the residual acceleration of the two trailers is first used to eliminate the adverse effects of road surface roughness. The fundamental natural frequency of the bridge is obtained from the Fourier transform of the resulting signal. Next, the component corresponding to this fundamental frequency was extracted using bandpass filtering. The corresponding damping ratio and mode shape can be reconstructed through a short-time Fourier transform (STFT) under the assumption that the fundamental mode shape amplitude peak is located at the midspan. Finally, the element stiffness of the bridge was evaluated using the reconstructed fundamental mode shape. In this study, the proposed method is investigated via numerical examples and performs well in different scenarios, for example, in the presence of high-level measurement noise and low-level road surface roughness. Furthermore, the method was validated through field testing on the Li-Zi-Wan bridge in Chongqing. The extracted fundamental mode shape agrees well with that obtained using the stochastic subspace identification (SSI) method. The evaluated element stiffness matches very well with both the original design and the static deflection test results.

Analytical investigation into error propagation of power spectral density transmissibility (PSDT) based on coherence function

Power spectral density transmissibility (PSDT) has attracted widespread interests because it is insensitive to the nature of excitation and has no demand for measuring the input. The classical spectral estimation procedure is usually subject to measurement and spectral leakage errors, which might cause distortion in the estimation of PSDT. This study investigates how the error of PSD estimation propagates into PSDT estimation based on analytical approximation formulas. Based on perturbation theory in tandem with statistical moments of the ratio of random variables, both first- and second-order asymptotic expressions of the mean and variance of PSDT estimation are derived in terms of coherence functions, PSDT measurements, and the number of averages. Given these theoretical findings, it is further revealed that the variance of PSDT estimation around the system poles approaches zero, and the variances of the magnitude and phase of the mode shape decrease to local minima. The performance of the analytical error propagation formulas is validated through numerical and field test data. Parametric studies into the effects of recording durations, window types, and reference outputs on the statistics of PSDT estimation and mode shape estimation are also conducted.