Identification of time-varying system and energy-based optimization of adaptive control in seismically excited structure

The combination of structural health monitoring and vibration control is of great importance to provide components of smart structures.While synthetic algorithms have been proposed,adaptive control that is compatible with changing conditions still needs to be used,and time-varying systems are required to be simultaneously estimated with the application of adaptive control.In this research,the identification of structural time-varying dynamic characteristics and optimized simple adaptive control are integrated.First,reduced variations of physical parameters are estimated online using the multiple forgetting factor recursive least squares(MFRLS)method.Then,the energy from the structural vibration is simultaneously specified to optimize the control force with the identified parameters to be operational.Optimization is also performed based on the probability density function of the energy under the seismic excitation at any time.Finally,the optimal control force is obtained by the simple adaptive control(SAC)algorithm and energy coefficient.A numerical example and benchmark structure are employed to investigate the efficiency of the proposed approach.The simulation results revealed the effectiveness of the integrated online identification and optimal adaptive control in systems.

Development of a software platform for bridge modal and damage identification based on ambient excitation

Modal and damage identification based on ambient excitation can greatly improve the efficiency of high-speed railway bridge vibration detection.This paper first describes the basic principles of stochastic subspace identification,peak-picking,and frequency domain decomposition method in modal analysis based on ambient excitation,and the effectiveness of these three methods is verified through finite element calculation and numerical simulation,Then the damage element is added to the finite element model to simulate the crack,and the curvature mode difference and the curvature mode area difference square ratio are calculated by using the stochastic subspace identification results to verify their ability of damage identification and location.Finally,the above modal and damage identification techniques are integrated to develop a bridge modal and damage identification software platform.The final results show that all three modal identification methods can accurately identify the vibration frequency and mode shape,both damage identification methods can accurately identify and locate the damage,and the developed software platform is simple and efficient.

Modal Parameter Identification Method of Jacket Platform Structure Based on AFDD and Optimized FBFFT

Offshore platforms are susceptible to structural damage due to prolonged exposure to random loads,such as wind,waves,and currents.This is particularly true for platforms that have been in service for an extended period.Identifying the modal parameters of offshore platforms is crucial for damage diagno sis,as it serves as a prerequisite and foundation for the process.Therefore,it holds great significance to prioritize the identification of these parameters.Aiming at the shortcomings of the traditional Fast Bayesian Fast Fourier Transform(FBFFT) method,this paper proposes a modal parameter identification method based on Automatic Frequency Domain Decomposition(AFDD) and optimized FBFFT.By introducing the AFDD method and Powell optimization algorithm,this method can automatically identify the initial value of natural frequency and solve the objective function efficiently and simply.In order to verify the feasibility and effectiveness of the proposed method,it is used to identify the modal parameters of the IASC-ASCE benchmark model and the j acket platform structure model,and the Most Probable Value(MPV) of the modal parameters and their respective posterior uncertainties are successfully identified.The identification results of the IASC-ASCE benc hmark model are compared with the identification re sults of the MODE-ID method,which verifies the effectivene ss and accuracy of the proposed method for identifying modal parameters.It provides a simple and feasible method for quantifying the influence of uncertain factors such as environmental parameters on the identification results,and also provide s a reference for modal parameter identification of other large structures.

IDENTIFICATION OF TIME-VARYING MODAL PARAMETERS USING LINEAR TIME-FREQUENCY REPRESENTATION

A new method of parameter identification based on linear time-frequencyrepresentation and Hubert transform is proposed to identity modal parameters of linear time-varyingsystems from measured vibration responses. Using Gabor expansion and synthesis theory, measuredresponses are represented in the time-frequency domain and modal components are reconstructed bytime-frequency filtering. The Hilbert transform is applied to obtain time histories of the amplitudeand phase angle of each modal component, from which time-varying frequencies and damping ratios areidentified. The proposed method has been demonstrated with a numerical example in which a lineartime-varying system of two degrees of freedom is used to validate the identification scheme based ontime-frequency representation. Simulation results have indicated that time-frequency representationpresents an effective tool for modal parameter identification of time-varying systems.

Modal Identification of Linear Time-varying Systems Using Continuous Wavelet Transform

A wavelet based identification method for linear time-varying systems is presented,and the ridge and skeleton of the continuous wavelet transform of free response is used to extract time-varying parameters. The stiffness and damping coefficients of single-degree-of—freedom systems,frequencies and damping ratios of multi-degree-of-freedom systems are estimated without any prior information of systems. The proposed method is applied to linear time-varying systems with both abrupt and smooth variation parameters. Gaussian white noise is added to the response to test the anti-noise performance of the algorithm. The simulation results show that the proposed method is capable of accurately tracking the variation of the systems.

Dynamic Stability Analysis of Linear Time-varying Systems via an Extended Modal Identification Approach

The problem of linear time-varying（LTV） system modal analysis is considered based on time-dependent state space representations, as classical modal analysis of linear time-invariant systems and current LTV system modal analysis under the ＂frozen-time＂ assumption are not able to determine the dynamic stability of LTV systems. Time-dependent state space representations of LTV systems are first introduced, and the corresponding modal analysis theories are subsequently presented via a stabilitypreserving state transformation. The time-varying modes of LTV systems are extended in terms of uniqueness, and are further interpreted to determine the system＇s stability. An extended modal identification is proposed to estimate the time-varying modes, consisting of the estimation of the state transition matrix via a subspace-based method and the extraction of the time-varying modes by the QR decomposition. The proposed approach is numerically validated by three numerical cases, and is experimentally validated by a coupled moving-mass simply supported beam exper- imental case. The proposed approach is capable of accurately estimating the time-varying modes, and provides anew way to determine the dynamic stability of LTV systems by using the estimated time-varying modes.

Identification of Time-Varying Modal Parameters for Thermo-Elastic Structure Subject to Unsteady Heating

A time-varying modal parameter identification method combined with Bayesian information criterion(BIC)and grey correlation analysis(GCA)is presented for a kind of thermo-elastic structures with sparse natural frequencies and subject to an unsteady temperature field.To demonstrate the method,the thermo-elastic structure to be identified is taken as a simply-supported beam with an axially movable boundary and subject to both random excitation and an unsteady temperature field,and the dynamic outputs of the beam are first simulated as the measured data for the identification.Then,an improved time-varying autoregressive(TVAR)model is generated from the simulated input and output of the system.The time-varying coefficients of the TVAR model are expanded as a finite set of time basis functions that facilitate the time-varying coefficients to be time-invariant.According to the BIC for preliminarily determining the scope of the order number,the grey system theory is introduced to determine the order of TVAR and the dimension of the basis functions simultaneously via the absolute grey correlation degree(AGCD).Finally,the time-varying instantaneous frequencies of the system are estimated by using the recursive least squares method.The identified results are capable of tracking the slow time-varying natural frequencies with high accuracy no matter for noise-free or noisy estimation.

Modal Identification of Offshore Platforms Using Statistical Method Based on ERA

Identification of modal parameters of a linear structure with output-only measurements has received much attention over the past decades. In the paper, the Natural Excitation Technique (NExT) is used for acquisition of the impulse signals from the structural responses. Then Eigensystem Realization Algorithm (ERA) is utilized for modal identification. For disregarding the fictitious ‘computational modes', a procedure, Statistically Averaging Modal Frequency Method (SAMFM), is developed to distinguish the true modes from noise modes, and to improve the precision of the identified modal frequencies of the structure. An offshore platform is modeled with the finite element method. The theoretical modal parameters are obtained for a comparison with the identified values. The dynamic responses of the platform under random wave loading are computed for providing the output signals used for identification with ERA. Results of simulation demonstrate that the proposed method can determine the system modal frequency with high precision.

Output only modal identification and structural damage detection using time frequency & wavelet techniques

The primary objective of this paper is to develop output only modal identification and structural damage detection. Identification of multi-degree of freedom （MDOF） linear time invariant （LTI） and linear time variant （LTV--due to damage） systems based on Time-frequency （TF） techniques--such as short-time Fourier transform （STFT）, empirical mode decomposition （EMD）, and wavelets--is proposed. STFT, EMD, and wavelet methods developed to date are reviewed in detail. In addition a Hilbert transform （HT） approach to determine frequency and damping is also presented. In this paper, STFT, EMD, HT and wavelet techniques are developed for decomposition of free vibration response of MDOF systems into their modal components. Once the modal components are obtained, each one is processed using Hilbert transform to obtain the modal frequency and damping ratios. In addition, the ratio of modal components at different degrees of freedom facilitate determination of mode shape. In cases with output only modal identification using ambient/random response, the random decrement technique is used to obtain free vibration response. The advantage of TF techniques is that they arc signal based; hence, can be used for output only modal identification. A three degree of freedom 1：10 scale model test structure is used to validate the proposed output only modal identification techniques based on STFT, EMD, HT, wavelets. Both measured free vibration and forced vibration （white noise） response are considered. The secondary objective of this paper is to show the relative ease with which the TF techniques can be used for modal identification and their potential for real world applications where output only identification is essential. Recorded ambient vibration data processed using techniques such as the random decrement technique can be used to obtain the free vibration response, so that further processing using TF based modal identification can be performed.

Automatic modal parameter identification of high arch dams:feasibility verification

Modal parameters, including fundamental frequencies, damping ratios, and mode shapes, could be used to evaluate the health condition of structures. Automatic modal parameter identification, which plays an essential role in realtime structural health monitoring, has become a popular topic in recent years. In this study, an automatic modal parameter identification procedure for high arch dams is proposed. The proposed procedure is implemented by combining the densitybased spatial clustering of applications with noise(DBSCAN) algorithm and the stochastic subspace identification(SSI). The 210-m-high Dagangshan Dam is investigated as an example to verify the feasibility of the procedure. The results show that the DBSCAN algorithm is robust enough to interpret the stabilization diagram from SSI and may avoid outline modes. This leads to the proposed procedure obtaining a better performance than the partitioned clustering and hierarchical clustering algorithms. In addition, the errors of the identified frequencies of the arch dam are within 4%, and the identified mode shapes are in agreement with those obtained from the finite element model, which implies that the proposed procedure is accurate enough to use in modal parameter identification. The procedure is feasible for online modal parameter identification and modal tracking of arch dams.

Modal identification based on Hilbert-Huang Transform of structural response with S VD preprocessing

In recent years, Empirical mode decomposition and Hilbert spectral analysis have been combined to identify system parameters. Singular-Value Decomposition is pro- posed as a signal preprocessing technique of Hilbert-Huang Transform to extract modal parameters for closely spaced modes and low-energy components. The proposed method is applied to a simulated airplane model built in Automatic Dynamic Analysis of Mechanical Systems software. The results demonstrate that the identified modal parameters are in good agreement with the baseline model.

Modal identification of multi-degree-of-freedom structures based on intrinsic chirp component decomposition method

Modal parameter identification is a mature technology.However,there are some challenges in its practical applications such as the identification of vibration systems involving closely spaced modes and intensive noise contamination.This paper proposes a new time-frequency method based on intrinsic chirp component decomposition(ICCD)to address these issues.In this method,a redundant Fourier model is used to ameliorate border distortions and improve the accuracy of signal reconstruction.The effectiveness and accuracy of the proposed method are illustrated using three examples:a cantilever beam structure with intensive noise contamination or environmental interference,a four-degree-of-freedom structure with two closely spaced modes,and an impact test on a cantilever rectangular plate.By comparison with the identification method based on the empirical wavelet transform(EWT),it is shown that the presented method is effective,even in a high-noise environment,and the dynamic characteristics of closely spaced modes are accurately determined.

Physical Parameter Identification Method Based on Modal Analysis for Two-axis On-road Vehicles:Theory and Simulation

Physical parameters are very important for vehicle dynamic modeling and analysis.However,most of physical parameter identification methods are assuming some physical parameters of vehicle are known,and the other unknown parameters can be identified.In order to identify physical parameters of vehicle in the case that all physical parameters are unknown,a methodology based on the State Variable Method（SVM） for physical parameter identification of two-axis on-road vehicle is presented.The modal parameters of the vehicle are identified by the SVM,furthermore,the physical parameters of the vehicle are estimated by least squares method.In numerical simulations,physical parameters of Ford Granada are chosen as parameters of vehicle model,and half-sine bump function is chosen to simulate tire stimulated by impulse excitation.The first numerical simulation shows that the present method can identify all of the physical parameters and the largest absolute value of percentage error of the identified physical parameter is 0.205%;and the effect of the errors of additional mass,structural parameter and measurement noise are discussed in the following simulations,the results shows that when signal contains 30 d B noise,the largest absolute value of percentage error of the identification is 3.78%.These simulations verify that the presented method is effective and accurate for physical parameter identification of two-axis on-road vehicles.The proposed methodology can identify all physical parameters of 7-DOF vehicle model by using free-decay responses of vehicle without need to assume some physical parameters are known.

A Modal Identification Algorithm Combining Blind Source Separation and State Space Realization

A modal identification algorithm is developed, combining techniques from Second Order Blind Source Separation (SOBSS) and State Space Realization (SSR) theory. In this hybrid algorithm, a set of correlation matrices is generated using time-shifted, analytic data and assembled into several Hankel matrices. Dissimilar left and right matrices are found, which diagonalize the set of nonhermetian Hankel matrices. The complex-valued modal matrix is obtained from this decomposition. The modal responses, modal auto-correlation functions and discrete-time plant matrix (in state space modal form) are subsequently identified. System eigenvalues are computed from the plant matrix to obtain the natural frequencies and modal fractions of critical damping. Joint Approximate Diagonalization (JAD) of the Hankel matrices enables the under determined (more modes than sensors) problem to be effectively treated without restrictions on the number of sensors required. Because the analytic signal is used, the redundant complex conjugate pairs are eliminated, reducing the system order (number of modes) to be identified half. This enables smaller Hankel matrix sizes and reduced computational effort. The modal auto-correlation functions provide an expedient means of screening out spurious computational modes or modes corresponding to noise sources, eliminating the need for a consistency diagram. In addition, the reduction in the number of modes enables the modal responses to be identified when there are at least as many sensors as independent (not including conjugate pairs) modes. A further benefit of the algorithm is that identification of dissimilar left and right diagonalizers preclude the need for windowing of the analytic data. The effectiveness of the new modal identification method is demonstrated using vibration data from a 6 DOF simulation, 4-story building simulation and the Heritage court tower building.

Damage Identification under Incomplete Mode Shape Data Using Optimization Technique Based on Generalized Flexibility Matrix

A generalized flexibility–based objective function utilized for structure damage identification is constructed for solving the constrained nonlinear least squares optimized problem. To begin with, the generalized flexibility matrix (GFM) proposed to solve the damage identification problem is recalled and a modal expansion method is introduced. Next, the objective function for iterative optimization process based on the GFM is formulated, and the Trust-Region algorithm is utilized to obtain the solution of the optimization problem for multiple damage cases. And then for computing the objective function gradient, the sensitivity analysis regarding design variables is derived. In addition, due to the spatial incompleteness, the influence of stiffness reduction and incomplete modal measurement data is discussed by means of two numerical examples with several damage cases. Finally, based on the computational results, it is evident that the presented approach provides good validity and reliability for the large and complicated engineering structures.

Nonlinear System Identification with Unknown Piecewise Time-Varying Delay

Identification of nonlinear systems with unknown piecewise time-varying delay is concerned in this paper.Multiple auto regressive exogenous(ARX) models are identified at different process operating points,and the complete dynamics of the nonlinear system is represented by using a combination of a normalized exponential function as the probability density function with each of the local models.The parameters of the local ARX models and the exponential functions as well as the unknown piecewise time-varying delays are estimated simultaneously under the framework of the expectation maximization(EM) algorithm.A simulation example is applied to demonstrating the proposed identification method.

A MODEL IDENTIFICATION METHOD OF VIBRATING STRUCTURES FROM INCOMPLETE MODAL INFORMATION

The accurate mathematical models for complicated structures are verydifficult to construct.The work presented here provides an identification method for estimating the mass, damping , and stiffness matrices of linear dynamical systems from incompleteexperimental data. The mass, stiffness, and damping matrices are assumed to be real,symmetric, and positive definite. The partial set of experimental complex eigenvalues and corresponding eigenvectors are given. In the proposed method the least squaresalgorithm is combined with the iteration technique to determine systems identified matrices and corresponding design parameters. several illustrative examples, are presented to demonstrate the reliability of the proposed method .It is emphasized thatthe mass, damping and stiffness martices can be identified simultaneously.

New modal identification method under the non-stationary Gaussian ambient excitation

Based on the multivariate continuous time autoregressive （CAR） model, this paper presents a new time-domain modal identification method of linear time-invariant system driven by the uniformly modulated Gaussian random excitation. The method can identify the physical parameters of the system from the response data. First, the structural dynamic equation is transformed into a continuous time autoregressive model （CAR） of order 3. Second, based on the assumption that the uniformly modulated function is approximately equal to a constant matrix in a very short period of time and on the property of the strong solution of the stochastic differential equation, the uniformly modulated function is identified piecewise. Two special situations are discussed. Finally, by virtue of the Girsanov theorem, we introduce a likelihood function, which is just a con- ditional density function. Maximizing the likelihood function gives the exact maximum likelihood estimators of model parameters. Numerical results show that the method has high precision and the computation is efficient.

A MODEL IDENTIFICATION METHOD OF VIBRATING STRUCTURES FROM INCOMPLETE MODAL INFORMATION

The accurate mathematical models for complicated structures are very difficult to construct.The work presented here provides an identification method for estimating the mass.damping,and stiffness matrices of linear dynamical systems from incomplete experimental data.The mass,stiffness and damping matrices are assumed to be real,symmetric,and positive definite The partial set of experimental complex eigenvalues and corresponding eigenvectors are given.In the proposed method the least squares algorithm is combined with the iteration technique to determine systems identified matrices and corresponding design parameters.Seeveral illustative examples,are presented to demonstrate the reliability of the proposed method .It is emphasized that the mass,damping and stiffness matrices can be identified simultaneously.

Identification of Modal Parameters with Linear Structure under Non-stationary Ambient Excitation

Empirical mode decomposition (EMD) is proposed to identify linear structure under non-stationary excitation,and non-white noise coefficient is introduced under the assumption of random signals consisting of white noise and non-white noise signals. The cross-correlation function of response signal is decomposed into mode functions and residue by EMD method. The identification technique of the modal parameters of single freedom degree is applied to each mode function to obtain natural frequencies, damping ratios and mode shapes. The results of identification of the five-degree freedom linear system demonstrate that the proposed method is effective in identifying the parameters of linear structures under non-stationary ambient excitation.