Frequency-domain approach for the parametric identification of structures with modal interference

Performing forced vibration tests on full-scale structures is the most reliable way of determining the relevant modal parameters in structural dynamics, such as modal frequencies, mode shapes, modal damping, and modal masses. This study describes the modal identification of a double-level curved cable-stayed bridge with separate deck systems for pedestrians and vehicles via forced vibration tests. The steady-state structural responses to sinusoidal excitations produced by an electrodynamic shaker are recorded under varying excitation frequencies, and the frequency response functions are established. The measured frequency response functions are curve fitted to estimate the modal parameters. The numerical simulation of frequency response function–based modal parameter identification of an elastically multi-supported continuous beam structure is carried out, and the emphasis has been placed on the evaluation of the effect of an additional shaker mass, excitation frequency step and range, multi-mode vibration, and noise on identification results. Finally, the modal parameters for the first lateral mode of a double-level curved cable-stayed bridge are identified by forced vibration experiments, and the results are compared with those from ambient vibration tests and free vibration tests. The effect of the unmeasured wind excitation on identification is discussed. It is shown that the effect of ambient vibration is minor for wind velocity of 3–5 m/s. The damping ratios identified by forced and free vibration tests are comparable, while those from ambient vibration are subject to large variations. The modal mass obtained from forced vibration tests is in good agreement with finite element prediction, which provides design basis for mass-type dampers.

Output-only modal dynamic identification of frames by a refined FDD algorithm at seismic input and high damping

High-speed digital image correlation is a camera-based displacement measurement technique that has experienced increasing popularity for modal analysis. It is due to a high spatial density of measurement that provides an outstanding interpretation of mode shapes or operational deflection shapes. However, its sensitivity is typically lower than other techniques and transducers such as accelerometers or laser vibrometry. With these measurements, the estimation of any transfer function in the frequency domain is noisier in valley regions of the response, including anti-resonances. Therefore, an accurate analytical model of the response should be used for modal identification. Modal analysis algorithms perform an identification of the modal parameters and a reconstruction of the response based on the transfer function between force excitation and displacement response. This is called the frequency response function. However, force measurements are not available in many experiments, where the motion excitation is recorded instead. This is the case when the vibration is applied to the specimen through a motion in its base. This transfer function, also known as the transmissibility function, relates the motion of excitation and response, whose shape differs from the frequency response function. This difference in the response function is relevant in a relatively noisy measurement like using digital image correlation. Hence, in this work, a transformation of the transmissibility function into equivalent frequency response function, based on theoretical models, is performed prior to modal identification in base motion tests. More suitable experimental response functions are obtained, increasing the accurateness of the modal parameter estimation.KeywordsModal analysisTransmissibility functionsFull-field measurement

Modal parameter identification via ambient vibration is popular but faces challenges from uncertainties due to unknown inputs and low signal‐to‐noise ratio. Bayesian methods are gaining increasing attention for operational modal identification due to their ability to quantify uncertainties. However, improvements in computational efficiency are needed, particularly when addressing numerous modes and degrees of freedom. To address this challenge, this study proposes an innovative approach, termed the “Bayesian spectral decomposition” method (BSD), employing the decompose‐and‐conquer strategy. This novel method, operating within the frequency domain, identifies each mode individually by exploiting their inherent separated modal characteristics. For each mode, the response spectrum matrix undergoes an eigenvalue decomposition, yielding crucial eigenvalues (incorporating frequency and damping information) and eigenvectors (containing mode shape information). Subsequently, statistical properties of the eigenvalues and eigenvectors are utilized to establish likelihood functions for Bayesian parameter identification. By combining prior information, the posterior probability distribution functions of modal parameters are derived. The optimal solution is then obtained by resolving the maximum posterior probability distribution function problem. To further quantify the uncertainty of modal parameters, Gaussian distributions are employed to approximate the posterior probability distribution functions. The adoption of the decomposition approach circumvents the joint identification of all modal parameters, substantially reducing the parameter dimensions for optimization. Consequently, this strategy leads to decreased computational complexity and significantly improved computational stability. The effectiveness of the BSD is confirmed through simulated data generated from an 8‐story shear building as well as measured data collected from both an experimental shear frame and the Canton Tower. The results demonstrate that the proposed method achieves high accuracy in identifying modal parameters, greatly improves computational efficiency, and effectively quantifies the uncertainties in modal parameters.

Accurate and timely identification of modal parameters of long-span bridges is important for bridge health monitoring and wind tunnel tests. Wavelet analysis is one of the most advantageous methods for identification because of its good localization characteristics in both time and frequency domain. In recent years, the wavelet method has been applied more frequently in parameter identification of linear and nonlinear systems. In this article, based on wavelet ridges and wavelet skeleton, the improved modal parameter identification method was studied. To find the appropriate time-frequency resolution, an optimal wavelet basis design principle based on minimum Shannon entropy was proposed. Aiming at endpoint effect in wavelet transform, a prediction continuation method based on support vector machine (SVM) was proposed, which can effectively suppress the endpoint effect of the extended samples. In view of the fact that the ridges of metric matrices obtained by the traditional crazy climber algorithm cannot fully reflect the distribution of ridges of modulus value matrices of wavelet coefficients, an improved high-precision crazy climber algorithm was put forward to accurately identify the position of the ridge of wavelet coefficients. Finally, taking a long-span cable-stayed bridge and a long-span suspension bridge as the engineering background, improved continuous wavelet transform (CWT) was applied to modal parameter identification of bridge under ambient excitation. The modal parameters such as modal frequency, damping ratio, and mode shape were obtained. Compared with the calculation value of the numerical simulation of long-span cable-stayed bridge and wind tunnel test of long-span suspension bridge, the reliability of CWT for modal parameter identification of long-span bridges under ambient excitation was verified.

A DUAL FITTING ALGORITHM FOR MODAL PARAMETERS IDENTIFICATION

Modal Identification from response data only is studied for structural systems under nonstationary ambient vibration. The topic of this paper is the estimation of modal parameters from nonstationary ambient vibration data by applying the random decrement algorithm with time-varying threshold level. In the conventional random decrement algorithm, the threshold level for evaluating randomdec signatures is defined as the standard deviation value of response data of the reference channel. The distortion of randomdec signatures may be, however, induced by the error involved in noise from the original response data in practice. To improve the accuracy of identification, a modification of the sampling procedure in random decrement algorithm is proposed for modal-parameter identification from the nonstationary ambient response data. The time-varying threshold level is presented for the acquisition of available sample time history to perform averaging analysis, and defined as the temporal root-mean-square function of structural response, which can appropriately describe a wide variety of nonstationary behaviors in reality, such as the time-varying amplitude (variance) of a nonstationary process in a seismic record. Numerical simulations confirm the validity and robustness of the proposed modal-identification method from nonstationary ambient response data under noisy conditions.

Modal parameters, i.e., modal frequencies, damping ratios, and mode shapes of a two disk-shaft system are estimated in Multiple Input/Multiple Output (MIMO) scheme. The response at the output degrees of freedom (dof) of the considered structure due to the excitation at the input dofs is estimated from theoretical analysis. The generated theoretical data is used to estimate the Frequency Response Function (FRF) matrix, which relates the output (vibration) at response points with the input at the excitation points in the frequency domain. The corresponding Impulse Response Function (IRF) matrix, which relates response and excitation in the time domain, is obtained by Inverse Fast Fourier Transform (IFFT) of the FRF matrix. The unified Matrix Polynomial Approach (UMPA) is employed in the frequency and time domains with the estimated FRF and IRF matrices, respectively, to estimate the desired modal parameters. The obtained results are compared with results from Theoretical Modal Analysis (TMA).

Structural modal parameter identification with the Power-Exponential window function

As all experimental procedures, Experimental Modal Analysis (EMA) is subject to a wide range of potential testing and processing errors. The modal identification methods are sensitive to these errors, yielding modal results which are uncertain up to certain error bounds. The question hence is what these error bounds on test data and modal parameters are. In this paper, the studied source of uncertainty is related to the variance (noise) on the Frequency Response Function (FRF) measurements. Under the H1 assumptions and in single-input cases, the FRF variances can be computed from the coherences and the FRFs. In multiple-input cases, some more measurement functions are required. Advanced system identification methods like the Maximum Likelihood Estimator (MLE) and PolyMAX Plus have the possibility to take the uncertainty on the measurement data into account and to propagate the data uncertainty to (modal) parameter uncertainty. This paper will review FRF variance estimation techniques, including some pragmatic approaches. The basic concepts of Maximum Likelihood Estimation and the calculation of confidence bounds will be discussed. Some typical structural testing and modal analysis cases will be used as illustration of the discussed concepts.

Modal parameter identification is an important content for the research on dynamic behavior of civil engineering structures. For the conventional modal parameter identification techniques, excitation signal and response signal are necessary. But excitations can not be measured for structures under natural loading conditions, or excited by random forces. So, the operational modal parameter identification approaches are developing to identify structural modal parameters by using vibration response signal directly. In this paper, a new method for modal parameter identification depending on ambient excitation response signal is proposed based on independent component analysis (ICA). For modal coordinates independent of each other, ICA can extract modal coordinates of structure and estimate mode shape matrix using vibration response signal directly. The free vibration responses can be obtained by random decrement technique (RDT) from modal coordinates which are random processes, and then the natural frequencies and damping ratios are calculated by Hilbert transform. Numerical simulation results indicate that this method is effective and suitable for output-only structural modal parameter identification.

The instrumented Canton Tower is a 610 m high-rise structure, which has been considered as a benchmark problem for structural health monitoring (SHM) research. In this paper, an improved automatic modal identification method is presented based on a natural excitation technique in conjunction with the eigensystem realization algorithm (NExT/ERA). In the proposed modal identification method, damping ratio, consistent mode indicator from observability matrices (CMI_O) and modal amplitude coherence (MAC) are used as criteria to distinguish the physically true modes from spurious modes. Enhanced frequency domain decomposition (EFDD), the data-driven stochastic subspace identification method (SSI-DATA) and the proposed method are respectively applied to extract the modal parameters of the Canton Tower under different environmental conditions. Results of modal parameter identification based on output-only measurements are presented and discussed. User-selected parameters used in those methods are suggested and discussed. Furthermore, the effect of environmental conditions on the dynamic characteristics of Canton tower is investigated.

An accurate prediction for the response of civil and mechanical engineering structures subject to ambient excitation requires the information of dynamic properties of these structures including natural frequencies, damping ratios and mode shapes. Since the excitation force is not available as a measured signal, we need to develop techniques which are capable of accurately extracting the modal parameters from output-only data. This article presents the results of modal parameter identification using two time-domain methods as follows: the autoregressive moving average vector (ARMAV) method and the state–space method. These methods directly work with the recorded time signals and allow the analysis of structures where only the output is measured, while the input is unmeasured and unknown. The equivalence between ARMAV and state–space approaches for the problem of modal parameter identification of vibrating systems is shown in the article. Using only the singular value decomposition of a block Hankel matrix of sample covariances, it is shown that these two approaches give identical modal parameters in the case where the block Hankel matrix has full row rank. The time-domain modal identification algorithms have a serious problem of model order determination: when extracting structural modes these algorithms always generate spurious modes. A modal indicator to differentiate spurious and structural modes is presented. Numerical and experimental examples are given to show the effectiveness of the ARMAV or state–space approaches in modal parameter identification using response data only.

Modal analysis is a fundamental and essential research direction in the field of structural engineering. The ultimate goal is to determine the modal parameters of the structures. However, the existing modal analysis algorithms often require a large number of parameter adjustments and manual intervention during operation, which cannot be fully automated. In order to realize the automatic identification of modal parameters, the automatic operational modal identification method (AOMI) is proposed based on the interpolated power spectral density estimation (IPSE). To achieve more precise spectrum analysis in the low-frequency band, IPSE is employed to perform Fourier transform on the original frequency domain segment with optimized frequency resolution. This enhances the sharpness of the obtained spectrum in the low-frequency range, making peak frequencies more discernible. Subsequently, the scale-space peak picking algorithm is used to automatically obtain the peak of the power spectral density (PSD), thus enabling the automatic identification of the natural frequency. Finally, the frequency domain decomposition method (FDD) is used to identify modal parameters based on the natural frequencies. The effectiveness of AOMI is verified through the modal identification of the old steel truss bridge and the three layer framework. Under the environmental excitation, the frequencies identified by the IPSE method is close to that of FDD, Bayesian fast fourier transform (FFT) and covariance driven stochastic subspace identification (SSI-COV). Furthermore, the PSD obtained through IPSE has sharper peak than that of FDD and the Welch’s method. The damping ratio identification accuracy and modal assurance criterion (MAC) are satisfactory in AOMI, which can improve the automatic performance.

In recent years, many efforts have been made to exploit full-field measurement optical techniques for modal identification. Three-dimensional digital image correlation using high-speed cameras has been extensively employed for this purpose. Modal identification algorithms are applied to process the frequency response functions (FRF), which relate the displacement response of the structure to the excitation force. However, one of the most common tests for modal analysis involves the base motion excitation of a structural element instead of force excitation. In this case, the relationship between response and excitation is typically based on displacements, which are known as transmissibility functions. In this study, a methodology for experimental modal analysis using high-speed 3D digital image correlation and base motion excitation tests is proposed. In particular, a cantilever beam was excited from its base with a random signal, using a clamped edge join. Full-field transmissibility functions were obtained through the beam and converted into FRF for proper identification, considering a single degree-of-freedom theoretical conversion. Subsequently, modal identification was performed using a circle-fit approach. The proposed methodology facilitates the management of the typically large amounts of data points involved in the DIC measurement during modal identification. Moreover, it was possible to determine the natural frequencies, damping ratios and full-field mode shapes without requiring any additional tests. Finally, the results were experimentally validated by comparing them with those obtained by employing traditional accelerometers, analytical models and finite element method analyses. The comparison was performed by using the quantitative indicator modal assurance criterion. The results showed a high level of correspondence, consolidating the proposed experimental methodology.