Nonlinear Structural Identification Research Articles

Identification of a Duffing oscillator using particle Gibbs with ancestor sampling

The Duffing oscillator remains a key benchmark in nonlinear systems analysis and poses interesting challenges in nonlinear structural identification. The use of particle methods or sequential Monte Carlo (SMC) is becoming a more common approach for tackling these nonlinear dynamical systems, within structural dynamics and beyond. This paper demonstrates the use of a tailored SMC algorithm within a Markov Chain Monte Carlo (MCMC) scheme to allow inference over the latent states and parameters of the Duffing oscillator in a Bayesian manner. This approach to system identification offers a statistically more rigorous treatment of the problem than the common state-augmentation methods where the parameters of the model are included as additional latent states. It is shown how recent advances in particle MCMC methods, namely the particle Gibbs with ancestor sampling (PG-AS) algorithm is capable of performing efficient Bayesian inference, even in cases where little is known about the system parameters a priori. The advantage of this Bayesian approach is the quantification of uncertainty, not only in the system parameters but also in the states of the model (displacement and velocity) even in the presence of measurement noise.

A modified time domain subspace method for nonlinear identification based on nonlinear separation strategy

Nonlinear factors existing in engineering structures have drawn considerable attention, and nonlinear identification is a competent technique to understand the dynamic characteristics of nonlinear structures. Therefore, in this paper, a novel nonlinear separation subspace identification (NSSI) algorithm based on subspace algorithm and nonlinear separation strategy is proposed to conduct nonlinear parameter identification of nonlinear structures. For the proposed NSSI algorithm, the low-level excitation test is firstly conducted to obtain the transfer matrix in the linear response formula. Then, the obtained transfer matrix is used in the high-level excitation test to calculate the nonlinear response part by the proposed nonlinear separation strategy, and the subspace algorithm is utilized to identify the nonlinear parameter on the modified state-space model including only the nonlinear part. The proposed NSSI algorithm can reduce the coupling error caused by simultaneously processing both the large number part (corresponding to the linear part) and small number part (corresponding to the nonlinear part) in the traditional nonlinear subspace identification (NSI) algorithm. At last, two numerical experiments are given to validate the effectiveness of the developed novel nonlinear identification method. Furthermore, some influence factors are discussed to show the stability of the identification algorithm, and some comparisons between the proposed NSSI method and traditional NSI method are also conducted to demonstrate the advantages of the novel method.

Extended State Dependent Parameter modelling with a Data-Based Mechanistic approach to nonlinear model structure identification

A unified approach to Multiple and single State Dependent Parameter modelling, termed Extended State Dependent Parameters (ESDP) modelling, of nonlinear dynamic systems described by time-varying dynamic models applied to ARX or transfer-function model structures. Crucially, the approach proposes an effective model structure identification method using a novel Information Criterion (IC) taking into account model complexity in terms of the number of states involved. In ESDP, model structure involves not only the model orders, but also selection of the states driving the parameters, which effectively prevents the use of most current IC. This leads to a powerful methodology for investigating nonlinear systems building on the Data-Based Mechanistic (DBM) philosophy of Young and expanding the applications of the existing DBM methods. The methodologies presented are tested and demonstrated on both simulated data and on high frequency hydrological observations, showing how structure identification leads to discovery of dynamic relationships between system variables.

Sensitivity Analysis for Global Parameter Identification. Application to Aerodynamic Coefficients

The parameter identification of grey-box nonlinear model structures remains a challenging task. Analytical models, constructed from physical laws, allow a complete description of the system behavior, with however a high complexity of the mathematical structure of the system. This paper presents an identification procedure adapted to nonlinear models and able to cope with the difficulties related to the model structure. This procedure, composed of several steps from model development to its validation, takes into account the identifiability of model parameters. It highlights the importance of performing sensitivity analysis of the output variables w.r.t. unknown parameters - input conditions and/or model parameters - to give insights on the choice of the experimental conditions able to provide sufficient information for the estimation step. These results are illustrated in the case of the identification of aerodynamic coefficients of a space probe based on free flight measurements.

Improved decentralized structural identification with output-only measurements

This paper proposes an improved decentralized structural identification approach with output-only measurements. The improved approach can be used for system identification of both linear and nonlinear structures. A large-scale structure is divided into a number of smaller zones according to its finite element configuration. Each zone is dynamically tested in sequence with its own set of sensor placement. The external excitation forces in each zone are identified using the Kalman filter technique. Structural parameters of the whole structure are divided into several subsets and then updated by using the Newton-SOR method. Both the external excitations and structural parameters are iteratively updated until a defined convergence criterion is met. The proposed technique is then applied to two numerical examples: a six floor building and a planar truss structure. The nonlinear system parameters of the building are correctly identified. The unknown excitation force, damage location, and damage severity in the plane truss structure are successfully identified. The effect of measurement noise on the identified results is also studied. An eight floor shear type structure is finally tested in the laboratory. The experimental results further verify the effectiveness and efficiency of the proposed technique in damage identification using output-only measurements.

Adaptive constrained unscented Kalman filtering for real-time nonlinear structural system identification

The unscented Kalman filter (UKF) is often used for nonlinear system identification in civil engineering; nevertheless, the application of the UKF to highly nonlinear structures could not provide accurate results. In this paper, an improvement of the UKF algorithm has been adopted. This methodology can consider state constraints, and it can estimate the measurement noise covariance matrix. The results obtained adopting a modified UKF have been compared to the ones obtained using the UKF for parameter estimation of a single degree of freedom nonlinear hysteretic system. The second part of this work shows results of an experimental activity on a base-isolated prototype structure. Both numerical and experimental results underline that the adopted algorithm produces better state estimation and parameter identification than the UKF, being capable of taking into account parameter boundaries. The adopted algorithm is more robust than the standard UKF in the case of measuring noise variation.

Bayesian nonlinear structural FE model and seismic input identification for damage assessment of civil structures

A methodology is proposed to update mechanics-based nonlinear finite element (FE) models of civil structures subjected to unknown input excitation. The approach allows to jointly estimate unknown time-invariant model parameters of a nonlinear FE model of the structure and the unknown time histories of input excitations using spatially-sparse output response measurements recorded during an earthquake event. The unscented Kalman filter, which circumvents the computation of FE response sensitivities with respect to the unknown model parameters and unknown input excitations by using a deterministic sampling approach, is employed as the estimation tool. The use of measurement data obtained from arrays of heterogeneous sensors, including accelerometers, displacement sensors, and strain gauges is investigated. Based on the estimated FE model parameters and input excitations, the updated nonlinear FE model can be interrogated to detect, localize, classify, and assess damage in the structure. Numerically simulated response data of a three-dimensional 4-story 2-by-1 bay steel frame structure with six unknown model parameters subjected to unknown bi-directional horizontal seismic excitation, and a three-dimensional 5-story 2-by-1 bay reinforced concrete frame structure with nine unknown model parameters subjected to unknown bi-directional horizontal seismic excitation are used to illustrate and validate the proposed methodology. The results of the validation studies show the excellent performance and robustness of the proposed algorithm to jointly estimate unknown FE model parameters and unknown input excitations.

A Two-stage Parametric Identification of Strong Nonlinear Structural Systems with Incomplete Response Measurements

Compared with the identification of linear structural parameters, it is more difficult to conduct parametric identification of strong nonlinear structural systems, especially when only incomplete structural responses are available. Although the extended Kalman filter (EKF) is useful for structural identification with partial measurements of structural responses and can be extended for the identification of nonlinear structures, EKF approximates nonlinear system through Taylor series expansion and is therefore not effective for the identification of strong nonlinear structural systems. Other approaches such as the unscented Kalman filter (UKF) have been proposed for the identification of strong nonlinear problems. Based on the fact that nonlinearities exist in local areas of structures, a straightforward two-stage identification approach is proposed in this paper for the identification of strong nonlinear structural parameters with incomplete response measurements. In the first stage, the locations of nonlinearities are identified based on the EKF for the identification of the equivalent linear structures. In the second stage, the UKF is utilized to identify the parameters of strong nonlinear structural systems. Therefore, the parametric identification of strong nonlinear structural parameters is simplified by the proposed approach. Several numerical examples with different nonlinear models and locations are used to validate the proposed approach.

Identification of Nonlinearities in Joints of a Wing Structure

Nonlinear structural identification is essential in engineering. As new materials are being used and structures become slender and lighter, nonlinear behaviour of structures becomes more important. There have been many studies into the development and application of system identification methods for structural nonlinearity based on changes in natural frequencies, mode shapes and damping ratios. A great challenge is to identify nonlinearity in large structural systems. Much work has been undertaken in the development of nonlinear system identification methods (e.g. Hilbert Transform, NARMAX, and Proper Orthogonal Decomposition), however, it is arguable that most of these methods are cumbersome when applied to realistic large structures that contain mostly linear modes with some local nonlinearity (e.g. aircraft engine pylon attachment to a wing). In this paper, a multi-shaker force appropriation method is developed to determine the underlying linear and nonlinear structural properties through the use of the measurement and generation of restoring force surfaces. One undamped mode is excited in each multi-shaker test. Essentially, this technique is a derivative of the restoring surface method and involves a non-linear curve fitting performed in modal space. A reduced finite element model is established and its effectiveness in revealing the nonlinear characteristics of the system is discussed. The method is demonstrated through both numerical simulations and experiments on a simple jointed laboratory structure with seeded faults, which represents an engine pylon structure that consists of a rectangular wing with two stores suspended underneath.

Detection and parametric identification of structural nonlinear restoring forces from partial measurements of structural responses

Compared with the identification of linear structures, it is more challenging to conduct identification of nonlinear structure systems, especially when the locations of structural nonlinearities are not clear in structural systems. Moreover, it is highly desirable to develop methods of parametric identification using partial measurements of structural responses for practical application. To cope with these issues, an identification method is proposed in this paper for the detection and parametric identification of structural nonlinear restoring forces using only partial measurements of structural responses. First, an equivalent linear structural system is proposed for a nonlinear structure and the locations of structural nonlinearities are detected. Then, the parameters of structural nonlinear restoring forces at the locations of identified structural nonlinearities together with the linear part structural parameters are identified by the extended Kalman filter. The proposed method simplifies the identification of nonlinear structures. Numerical examples of the identification of two nonlinear multi-story shear frames and a planar nonlinear truss with different nonlinear models and locations are used to validate the proposed method.

Locating and identifying model-free structural nonlinearities and systems using incomplete measured structural responses

Structural nonlinearity is a common phenomenon encountered in engineering structures under severe dynamic loading. It is necessary to localize and identify structural nonlinearities using structural dynamic measurements for damage detection and performance evaluation of structures. However, identification of nonlinear structural systems is a difficult task, especially when proper mathematical models for structural nonlinear behaviors are not available. In prior studies on nonparametric identification of nonlinear structures, the locations of structural nonlinearities are usually assumed known and all structural responses are measured. In this paper, an identification algorithm is proposed for locating and identifying model-free structural nonlinearities and systems using incomplete measurements of structural responses. First, equivalent linear structural systems are established and identified by the extended Kalman filter (EKF). The locations of structural nonlinearities are identified. Then, the model-free structural nonlinear restoring forces are approximated by power series polynomial models. The unscented Kalman filter (UKF) is utilized to identify structural nonlinear restoring forces and structural systems. Both numerical simulation examples and experimental test of a multi-story shear building with a MR damper are used to validate the proposed algorithm.

Identification of Nonlinear Structures by the Conditioned Reverse Path Method

The modeling method and identified method adapted to multi-degree-of-freedom structures with structural nonlinearities are established. The free-interface component mode synthesis method is used to establish the nonlinear governing equations by extending the connected relationships. Based on the modeling method, the conditioned reverse path method is used to identify frequency response functions of the nominal linear system without the influence of nonlinearities, and the parameters of nonlinearities are identified in the frequency domain. Nonlinear analysis and identification of a typical folding wing configuration with three free-play hinges are investigated. The nonlinear governing equation is established based on present methods, and the computational results are checked by finite-element programming. Based on coherence function, the effects of a nonlinear function are evaluated. The frequency response funtion of the nominal linear system and parameters of the nonlinear function are identified by the conditioned reverse path method within the noise signal. A ground vibration test of an all-movable fin is performed. The results show that the extending component mode synthesis method in the present work could be used to establish governing equations with structural nonlinearities. Based on the modeling method, the conditioned reverse path method can be used to identify the structural nonlinearities accurately.

An output-only stochastic parametric approach for the identification of linear and nonlinear structures under random base excitations: Advances and comparisons

In this paper a time domain output-only Dynamic Identification approach for Civil Structures (DICS) first formulated some years ago is reviewed and presented in a more generalized form. The approach in question, suitable for multi- and single-degrees-of-freedom systems, is based on the statistical moments and on the correlation functions of the response to base random excitations. The solving equations are obtained by applying the Itô differential stochastic calculus to some functions of the response. In the previous version ([21] Cavaleri, 2006; [22] Benfratello et al., 2009), the DICS method was based on the use of two classes of models (Restricted Potential Models and Linear Mass Proportional Damping Models) while its generalization for use with different models from the ones mentioned above is discussed. In the paper the new class of models to which the DICS method is applicable are described. Further, the advantages and disadvantages of the approach in question are examined, also by a comparison with some techniques available in the literature.

Identification of nonlinear structures using discrete-time Volterra series

Mathematical modeling of mechanical structures is an important research area in structural dynamics. The goal is to obtain a model that accurately predicts the dynamics of the system. However, the nonlinear effects caused by gaps, backlash, joints, as well as large displacements are not as well understood as the linear counterpart. In this sense, the Volterra series is an interesting tool for the analysis of nonlinear systems, since it is a generalization of the linear model based on the impulse response function. This paper applies the discrete-time Volterra series expanded in orthonormal Kautz functions to identify a model of a nonlinear benchmark system represented by a Duffing oscillator. The input and output data are used to identify the Volterra kernels of the structure. After the identification of the model, the linear and nonlinear components of the response of the system can be analyzed separately. The paper concludes by indicating the main advantages and drawbacks of this technique to model the behavior of nonlinear systems.

Time-Varying Linear and Nonlinear Structural Identification with Analytical Mode Decomposition and Hilbert Transform

AbstractAnalytical mode decomposition (AMD) of a time series concerning any preselected bisecting frequency with Hilbert transform has been developed for closely spaced multicomponent signal decomposition. For this class of structures, it is often challenging, if not impossible, to apply empirical mode decomposition. In this study, the instantaneous structural frequencies are directly derived from the decomposed modal responses for systems with single and multiple degrees of freedom with both free and force vibrations, based on AMD combined with Hilbert transform analysis. The results show that the slow varying component of the instantaneous frequency of the signal is approximately equal to the instantaneous frequency of the systems for slowly time varying linear or weakly nonlinear structures. Both numerical simulations and experimental tests show that the proposed method is capable of tracking the frequency variations with high accuracy for time varying linear structures or weakly nonlinear structures.

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.

Fractal features of structures of gas flow in selected gas stations of I order

This paper presents results of research focused on identification of nonlinear structures and deterministic chaos which take place in gas flows in provinces Małopolska and Podkarpackie. Taking into account time series of gas flows at stations Solec Zdrój, Czechówka, Miłocin, Głogów and Huta Sendzimira from time period between January 2007 and September 2011, the authors detected by mean of respective identifications methods the existence of nonlinear structures in these time series. However the results are not unique. Therefore, the preliminary results should be checked in further research on the basis of other stations located in other provinces of Poland and by mean of more advance methods. These future research should definitely confirm or reject the existence of chaotic structure of gas flows at the stations of first order.

Fractal features of structures of gas flow in selected gas stations of I order

This paper presents results of research focused on identification of nonlinear structures and deterministic chaos which take place in gas flows in provinces Małopolska and Podkarpackie. Taking into account time series of gas flows at stations Solec Zdrój, Czechówka, Miłocin, Głogów and Huta Sendzimira from time period between January 2007 and September 2011, the authors detected by mean of respective identifications methods the existence of nonlinear structures in these time series. However the results are not unique. Therefore, the preliminary results should be checked in further research on the basis of other stations located in other provinces of Poland and by mean of more advance methods. These future research should definitely confirm or reject the existence of chaotic structure of gas flows at the stations of first order.

Nonlinear structural identification by the “Reverse Path” spectral method

When dealing with nonlinear dynamical systems, it is important to have efficient, accurate and reliable tools for estimating both the linear and nonlinear system parameters from measured data. An approach for nonlinear system identification widely studied in recent years is “Reverse Path”. This method is based on broad-band excitation and treats the nonlinear terms as feedback forces acting on an underlying linear system. Parameter estimation is performed in the frequency domain using conventional multiple-input–multiple-output or multiple-input–single-output techniques. This paper presents a generalized approach to apply the method of “Reverse Path” on continuous mechanical systems with multiple nonlinearities. The method requires few spectral calculations and is therefore suitable for use in iterative processes to locate and estimate structural nonlinearities. The proposed method is demonstrated in both simulations and experiments on continuous nonlinear mechanical structures. The results show that the method is effective on both simulated as well as experimental data.

Real-time nonlinear structural system identification via iterated unscented Kalman filter

Structural system identification has attracted much attention in the structural dynamics field over the past decades. The unscented Kalman filter (UKF) is often used to deal with nonlinear system identification in civil engineering field. In practices, applying a UKF to highly nonlinear structural systems is not a trivial task, particularly those subject to severe loading. Recently, a new technique, the iterated unscented Kalman filter (IUKF) is applicable to highly nonlinear systems. In this paper, the IUKF is applied for nonlinear structural system identification (NSSI). Experimental results show that the IUKF produces better state estimation and parameter identification than the UKF, and the IUKF is also more robust to measurement noise levels.