Instantaneous Modal Parameters Research Articles

Identification of forced time-varying systems via intrinsic chirp component decomposition

Modal parameter identification is significant in the field of vibration and control for tracking structural vibrations and evaluating structural performances. A novel parameter identification method based on the intrinsic chirp component decomposition (ICCD) and the Hilbert transform is proposed for forced time-varying systems. The developed ICCD decomposes the measured response signal into a limited number of intrinsic chirp components, and instantaneous modal parameters of the system are identified by the Hilbert transform of each intrinsic chirp component. The effectiveness and the accuracy of the proposed approach are investigated in two case studies, that is, a discrete mass-spring-damper oscillator and a cantilever beam. Smooth and periodical parameter variations are discussed in each case. Comparison with the empirical mode decomposition based method validates that the proposed method has better identification ability on estimating the instantaneous natural frequency of the system with the presence of noise. Results demonstrate that the ICCD-based approach could be a reasonable alternative for identification of forced time-varying systems in a noise environment.

Operational Modal Analysis of a Rotating Structure Subject to Random Excitation Using a Tracking Continuously Scanning Laser Doppler Vibrometer via an Improved Demodulation Method

Abstract A new operational modal analysis (OMA) method is developed for estimation of modal parameters (MPs) of a rotating structure (RS) subject to random excitation using a nonuniform rotating beam model, an image processing method, and an improved demodulation method. The solution to the governing equation of a nonuniform rotating beam is derived, which can be considered as the response of the beam measured by a continuously scanning laser Doppler vibrometer (CSLDV) system. A recently developed tracking CSLDV system can track and scan the RS. The image processing method determines the angular position of the RS so that the tracking CSLDV system can sweep its laser spot along a time-varying path on it. The improved demodulation method obtains undamped mode shapes (UMSs) of the RS by multiplying its measured response by sinusoids whose frequencies are its damped natural frequencies (DNFs) that are obtained from the fast Fourier transform (FFT) of the measured response. Experimental investigation of the OMA method using the tracking CSLDV system is conducted, and MPs of a rotating fan blade (RFB), including DNFs and UMSs, with different constant speeds and its instantaneous MPs with a nonconstant speed are estimated. Estimated first DNFs and UMSs of the stationary fan blade and RFB are compared with those from the lifting method that was previously developed by the authors.

Seismic structural health monitoring using the modal assurance distribution

AbstractThanks to emerging techniques in the field of signal processing and due to improvements in smart sensing systems which enable the event‐triggered acquisition of high‐fidelity data at the occurrence of strong ground motion events, seismic structural health monitoring has grown considerably in the last few decades. In this paper, the modal assurance distribution, an alternative time‐frequency representation of the modal features of multivariate and multicomponent signals, is extended for application to short‐term nonstationary vibrational structural responses in which the system may manifest its nonlinear behavior. A general procedure for the extraction of the decoupled normal modes is presented, which allows the identification of instantaneous modal parameters in order to investigate in detail the structural behavior during earthquakes. Valuable information that cannot be recovered by means of traditional criteria can thus be exploited for accurate damage assessment. The results obtained for two case studies consisting of a numerical model with softening nonlinear behavior and a full‐scale experimental reinforced concrete benchmark show the potential and applicability of the method proposed for the integrity assessment of civil structures.

Damage Detection in Initially Nonlinear Structures Based on Variational Mode Decomposition

Nonlinear characteristics in the dynamic behaviors of civil structures degrade the performance of damage detection of the linear theory based traditional time- and frequency-domain methods. To overcome this challenge, this paper proposes a damage detection approach for nonlinear structures based on Variational Mode Decomposition (VMD). In this approach, the measured dynamic responses from nonlinear structures under earthquake excitations are adaptively decomposed into a finite number of monocomponents by using VMD. Each decomposed mono-component represents an amplitude modulated and frequency modulated (AMFM) signal with a limited frequency bandwidth. Hilbert transform is then employed to identify the instantaneous modal parameters of the decomposed monomodes, including instantaneous frequencies and mode shapes. Based on the identified modal parameters from the decomposed structural dynamic responses, two damage indices are defined to identify the location and severity of structural damage, respectively. To validate the effectiveness and accuracy of the proposed approach, a nonlinear seven-storey shear building model with four different damage cases under earthquake excitations is used in the numerical studies. In experimental verifications, data from shake table tests on a 12-storey scaled reinforced concrete frame structure with different earthquake excitations are analyzed with the proposed approach. The results in both numerical studies and experimental validations demonstrate that the proposed approach can be successfully applied for nonlinear structural damage identification.

Modeling and parameter identification of linear time-varying systems based on adaptive chirplet transform under random excitation

In this paper, a time–frequency algorithm based on adaptive chirplet transform for parameter modeling and identification of Linear Time-Varying (LTV) systems under random excitation is presented. It is assumed that the solution of responses of LTV structures is expressed as the sum of multicomponent Linear Frequency Modulated (LFM) signals in a short-time. Then the measured acceleration response is used to perform the adaptive chirplet transform, in which an integral algorithm is employed to reconstruct the velocity and displacement responses. The vibration differential equation with time-varying coefficients is transformed into a simple linear equation. Furthermore, for systems under random excitation, the input–output relation based on correlation function is also derived to estimate the parameters including physicals parameters and instantaneous modal parameters. The full procedure of the method is presented and validated by using simulated responses. The results show that the presented method is accurate and robust for various LTV systems under random excitation.

In-Process Monitoring of Changing Dynamics of a Thin-Walled Component During Milling Operation by Ball Shooter Excitation

During the milling of thin-walled workpieces, the natural frequencies might change radically due to the material removal. To avoid resonant spindle speeds and chatter vibration, a precise knowledge of the instantaneous modal parameters is necessary. Many different numerical methods exist to predict the changes; however, small unmodelled effects can lead to unreliable results. The natural frequencies could be measured by human experts based on modal analysis for an often interrupted process; however, this method is not acceptable during production. We propose an online measurement method with an automatic ball shooter device which can excite a wide frequency range of the flexible workpiece. The method is presented for the case of blade profile machining. The change of the natural frequencies is predicted based on analytical models and finite element simulations. The measurement response for the impulse excitation of the ball shooter device is compared to the results of impulse modal tests performed with a micro hammer. It is shown that the ball shooter is capable of determining even the slight variation of the natural frequencies during the machining process and of distinguishing the slight change caused by different clamping methods. An improved FE model is proposed to include the contact stiffness of the fixture.

Modal assurance distribution of multivariate signals for modal identification of time-varying dynamic systems

Most time–frequency representations (TFRs) and signal analysis methods used for the identification of dynamic systems through non-parametric techniques are based on univariate signals. However, combining the information obtained from different sensors to investigate the overall behavior of the monitored structure is not trivial, as different recordings may show different features. Moreover, methods based upon the analysis of the energy density distribution in the time–frequency plane generally suffer from problems related to crossing and closely-spaced modes. In this paper, a new time–frequency representation of multivariate and multicomponent signals based on the modal assurance criterion (MAC) is presented. The analysis of the modal assurance distribution (MAD) thus obtained enables the extraction of decoupled modal responses, which can then be used to evaluate the instantaneous modal parameters of time-varying systems. To this end, a decomposition algorithm based on modal assurance (DAMA) is proposed, employing the watershed segmentation of the MAD. The results for two case studies, a finite element model and a full-scale experimental benchmark, are shown, considering both the original MAD and two enhanced versions, here proposed to improve its readability. The results are compared with those obtained from modern and widely used techniques, showing the promising efficacy of the proposed method for signals with time-varying frequency and amplitude, even in the presence of narrow-band disturbances and white noise, as well as with vanishing modes.

Mono-Component Feature Extraction for Condition Assessment in Civil Structures Using Empirical Wavelet Transform.

This paper proposes a methodology to process and interpret the complex signals acquired from the health monitoring of civil structures via scale-space empirical wavelet transform (EWT). The FREEVIB method, a widely used instantaneous modal parameters identification method, determines the structural characteristics from the individual components separated by EWT first. The scale-space EWT turns the detecting of the frequency boundaries into the scale-space representation of the Fourier spectrum. As well, to find meaningful modes becomes a clustering problem on the length of minima scale-space curves. The Otsu’s algorithm is employed to determine the threshold for the clustering analysis. To retain the time-varying features, the EWT-extracted mono-components are analyzed by the FREEVIB method to obtain the instantaneous modal parameters and the linearity characteristics of the structures. Both simulated and real SHM signals from civil structures are used to validate the effectiveness of the present method. The results demonstrate that the proposed methodology is capable of separating the signal components, even those closely spaced ones in frequency domain, with high accuracy, and extracting the structural features reliably.

Operational modal analysis for linear time-varying continuous dynamic structure based on LMPCA

Based on the theory of time-frozen, this paper proposes a novel operational modal analysis (OMA) for linear time-varying structure using limited memory principal component analysis (LMPCA). Compared with modal coordinate de- composition of non-stationary response signals for linear time-varying continuous beam structure, it is found that PCA can be decomposed to extract time-varying instantaneous modal parameters, which reveal structure characteristics in limited memory window length L. Numerical simulations results on a cantilever beam with slow time-varying mass are presented to verify the TV instantaneous natural frequencies and modal shapes identification capability of the LMPCA.

Nonlinear model calibration of a shear wall building using time and frequency data features

This paper investigates the effects of different factors on the performance of nonlinear model updating for a seven-story shear wall building model. The accuracy of calibrated models using different data features and modeling assumptions is studied by comparing the time and frequency responses of the models with the exact simulated ones. Simplified nonlinear finite element models of the shear wall building are calibrated so that the misfit between the considered response data features of the models and the structure is minimized. A refined FE model of the test structure, which was calibrated manually to match the shake table test data, is used instead of the real structure for this performance evaluation study. The simplified parsimonious FE models are composed of simple nonlinear beam-column fiber elements with nonlinearity infused in them by assigning generated hysteretic nonlinear material behaviors to uniaxial stress–strain relationship of the fibers. Four different types of data features and their combinations are used for model calibration: (1) time-varying instantaneous modal parameters, (2) displacement time histories, (3) acceleration time histories, and (4) dissipated hysteretic energy. It has been observed that the calibrated simplified FE models can accurately predict the nonlinear structural response in the absence of significant modeling errors. In the last part of this study, the physics-based models are further simplified for casting into state-space formulation and a real-time identification is performed using an Unscented Kalman filter. It has been shown that the performance of calibrated state-space models can be satisfactory when reasonable modeling assumptions are used.

Parametric Identification of Nonlinear Vibration Systems Via Polynomial Chirplet Transform

The response of a nonlinear oscillator is characterized by its instantaneous amplitude (IA) and instantaneous frequency (IF) features, which can be significantly affected by the physical properties of the system. Accordingly, the system properties could be inferred from the IA and IF of its response if both instantaneous features can be identified accurately. To fulfill such an idea, a nonlinear system parameter identification method is proposed in this paper with the aid of polynomial chirplet transform (PCT), which has been proved a powerful tool for processing nonstationary signals. First, the PCT is used to extract the instantaneous characteristics, i.e., IA and IF, from nonlinear system responses. Second, instantaneous modal parameters estimation was adopted to extract backbone and damping curves, which characterize the inherent nonlinearities of the system. Third, the physical property parameters of the system were estimated through fitting the identified average nonlinear characteristic curves. Finally, the proposed nonlinear identification method is experimentally validated through comparing with two Hilbert transform (HT) based methods.

Application of Cauchy wavelet transformation to identify time-variant modal parameters of structures

This work proposes a procedure for accurately identifying instantaneous modal parameters of a linear time-varying system using a time-varying autoregressive with exogenous input (TVARX) model with the continuous Cauchy wavelet transform (CCWT). An appropriate TVARX model is established using the velocity and displacement responses of the system under consideration. The time-varying coefficients of the TVARX are expanded as piecewise polynomial functions. CCWTs with various scale parameters are then applied to the TVARX model to evaluate the instantaneous modal parameters of different modes. The CCWTs of the velocity and displacement responses are analytically obtained from the CCWT of the measured acceleration responses. The effectiveness and accuracy of the proposed procedure are validated by numerical simulations of single and multiple degrees of freedom systems that have periodically varying and sharply varying stiffness and damping coefficients. The effects of noise, the Cauchy wavelet function and the order of the polynomial on the evaluation of the modal parameters are explored in processing the numerically simulated acceleration responses of systems with a single degree of freedom subjected to base excitation. Finally, the proposed procedure is adopted to determine the modal parameters of a five-story symmetric steel frame from its measured acceleration responses in a shaking table test. The measured strains reveal the yielding of columns in the first story. The variations of the identified instantaneous natural frequencies and modal damping ratios with time are consistent with the physical phenomena that are observed from the measured strains and base excitation acceleration.

Modal Identification of Structures from Seismic Response Data via Amplitude-Dependent Time Series Model

The present work develops a novel procedure of establishing a amplitude-dependent time series model for a nonlinear system and estimating the instantaneous modal parameters of the system from the dynamical responses. The undetermined coefficient in a amplitude-dependent autoregressive with exogenous input (amplitude-dependent ARX) model are assumed as function s of amplitude and are expanded by shape functions constructing by moving least-squares with polynomial basis functions. The amplitude of dynamical responses could be obtained by Hilbert transform. The instantaneous modal parameters of the system are directly estimated from the coefficient in the amplitude-dependent ARX model. Finally, the proposed approach is applied to process measured data for a frame specimen subjected to a series of base excitations in shaking table tests. The specimen was damaged during testing. The identified modal parameters are consistent with observed physical phenomena.

Identification of Instantaneous Modal Parameter of Time‐Varying Systems via a Wavelet‐Based Approach and Its Application

AbstractThis work presents an efficient approach using time‐varying autoregressive with exogenous input (TVARX) model and a substructure technique to identify the instantaneous modal parameters of a linear time‐varying structure and its substructures. The identified instantaneous natural frequencies can be used to identify earthquake damage to a building, including the specific floors that are damaged. An appropriate TVARX model of the dynamic responses of a structure or substructure is established using a basis function expansion and regression approach combined with continuous wavelet transform. The effectiveness of the proposed approach is validated using numerically simulated earthquake responses of a five‐storey shear building with time‐varying stiffness and damping coefficients. In terms of accuracy in determining the instantaneous modal parameters of a structure from noisy responses, the proposed approach is superior to typical basis function expansion and regression approach. The proposed method is further applied to process the dynamic responses of an eight‐storey steel frame in shaking table tests to identify its instantaneous modal parameters and to locate the storeys whose columns yielded under a strong base excitation.

Deterministic-stochastic subspace identification method for identification of nonlinear structures as time-varying linear systems

This paper proposes the use of the deterministic-stochastic subspace identification (DSI) method, an input-output parametric linear system identification method, for characterization of nonlinear dynamic structural systems based on their time-varying amplitude-dependent instantaneous (i.e., based on short time-windows) modal parameters. Performance of the DSI method for estimation of instantaneous modal parameters of nonlinear systems is investigated using numerical as well as experimental data. In this study, DSI is used for extracting instantaneous modal parameters of single degree-of-freedom (SDOF) as well as 7-DOF systems with different hysteretic material behavior. Nonlinear responses of the SDOF and 7-DOF systems are simulated due to different seismic excitations using the OpenSees structural analysis software. Modal identification results are compared with those obtained using wavelet transform and the exact values. Effects of four input factors are studied on the variability of identified instantaneous modal parameters: (1) type of material nonlinearity, (2) level of nonlinearity, (3) input excitation, and (4) length of data windows used in the identification. The accuracy of the identified instantaneous modal parameters is evaluated along the response time history while varying the above mentioned input factors. Overall, DSI outperforms the wavelet transform for short-time/instantaneous modal identification of nonlinear structural systems and provides reasonably accurate results especially when the material hysteretic behavior is smooth such as the considered Giuffré-Menegotto-Pinto hysteretic model. Finally, DSI has been applied for short-time modal identification of a full-scale seven-story reinforced concrete shear wall structure based on its measured response to different seismic base excitations on a shake table. The identified instantaneous natural frequencies of the first vibration mode can accurately track the variation in the structure's effective stiffness along its response.

Frequency modulated empirical mode decomposition method for the identification of instantaneous modal parameters of aeroelastic systems

Instantaneous modal parameters are effective indicators of nonlinearity in an aeroelastic system. The Identification of them requires reliable algorithms. A frequency modulated empirical mode decomposition (FM-EMD) method is proposed in this paper to track the variations in the modal parameters of a multi-DOF bridge–air system. The proposed method is a lifted version of the traditional EMD method. It overcomes one of the major shortcomings of its traditional version, i.e. the inability to separate closely spaced modes. Numerical verification was provided to show the effectiveness of the FM-EMD method. Wind tunnel testing data of a partially streamlined box girder sectional model of a suspension bridge was processed subsequently with the proposed method. Instantaneous frequencies and damping ratios of the aeroelastic system during the free decay vibration were realized. Nonlinearity in the transient aeroelastic vibration was identified.

Hydrodynamic In-Line Force Coefficients of Oscillating Bluff Cylinders (Circular and Square) at Low Reynolds Numbers

An attempt to measure indirectly the hydrodynamic drag (cD) and inertia (cM) coefficients on oscillating bluff cylinders (circular and square) in quiescent fluid at low Reynolds numbers (low Stokes number) is presented. The Keulegan–Carpenter number was below 15. The experimental approach is based on performing free-decay tests of a spring-mounted cylinder submerged in a water tank. The identification of the instantaneous modal parameters (damping and frequency), via Hilbert transform, of the decaying oscillations allows the determination of (cD) and (cM) by direct comparison with the damping and natural frequency of the system in still air (tank without water). Advantages and shortcomings of this novel experimental approach are presented along the paper.

Identification of MDOF Non-Linear Uncoupled Dynamical Systems Using Hilbert Transform and Empirical Mode Decomposition Method

A non-linear dynamical system identification method using Hilbert transform (HT) and empirical mode decomposition (EMD) is proposed. For a single-degree-of-freedom (SDOF) nonlinear system, the Hilbert transform identification method is good at identifying the instantaneous modal parameters (natural frequencies, damping characteristics and their dependencies on a vibration amplitude and frequency). For the multi-degree-of-freedom (MDOF) non-linear uncoupled dynamical systems, the EMD method is attempting for the decomposition of response signals into a collection of mono-components signals, termed intrinsic mode functions (IMFs). Considering the IMFs admit a well-behaved Hilbert transform, the HT identification method has been applied for the identification of nonlinear properties. The numerical simulation of a 2-dof shear-beam building model with nonlinear stiffness illustrated the proposed technique.

Identification of Time‐Variant Modal Parameters Using Time‐Varying Autoregressive with Exogenous Input and Low‐Order Polynomial Function

Abstract: This work presents an approach that accurately identifies instantaneous modal parameters of a structure using time‐varying autoregressive with exogenous input (TVARX) model. By developing the equivalent relations between the equation of motion of a time‐varying structural system and the TVARX model, this work proves that instantaneous modal parameters of a time‐varying system can be directly estimated from the TVARX model coefficients established from displacement responses. A moving least‐squares technique incorporating polynomial basis functions is adopted to approximate the coefficient functions of the TVARX model. The coefficient functions of the TVARX model are represented by polynomials having time‐dependent coefficients, instead of constant coefficients as in traditional basis function expansion approaches, so that only low orders of polynomial basis functions are needed. Numerical studies are carried out to investigate the effects of parameters in the proposed approach on accurately determining instantaneous modal parameters. Numerical analyses also demonstrate that the proposed approach is superior to some published techniques (i.e., recursive technique with a forgetting factor, traditional basis function expansion approach, and weighted basis function expansion approach) in accurately estimating instantaneous modal parameters of a structure. Finally, the proposed approach is applied to process measured data for a frame specimen subjected to a series of base excitations in shaking table tests. The specimen was damaged during testing. The identified instantaneous modal parameters are consistent with observed physical phenomena.

Evaluation of earthquake-induced structural damages by wavelet transform

The dynamic behavior of inelastic structures during an earthquake is a complicated non-stationary process that is affected by the random characteristics of seismic ground motions. The conventional Fourier analysis describes the feature of a dynamic process by decomposing the signal into infinitely long sine and cosine series, which loses all time-located information. However, both time and frequency localizations are necessary for the analysis of an evolutionary spectrum of non-stationary processes. In this paper, an analytical approach for seismic ground motions is developed by applying the wavelet transform, which focuses on the energy input to the structure. The procedure of identification of the instantaneous modal parameters based on the continuous wavelet transform (CWT) is given in detail. And then, a novel method using the auto-regressive moving average (ARMA), called “prediction extension”, is presented to remedy the edge effect during the numerical computation of the CWT. The effectiveness of the method is verified by the use of the benchmark model developed by the American Society of Civil Engineers (ASCE). Finally, a scale model with three-storey reinforced concrete frame-share wall structure is made and tested on a shaking table to investigate the relation between the dynamic properties of structures and energy accumulation and its change rates during the earthquake. The results have shown that the wavelet transform is able to provide a deep insight into the identity of transient signals through time-frequency maps of the time variant spectral decomposition.