Structural health monitoring and damage detection using an intelligent parameter varying (IPV) technique

Most structural health monitoring and damage detection strategies utilize dynamic response information to identify the existence, location, and magnitude of damage. Traditional model-based techniques seek to identify parametric changes in a linear dynamic model, while non-model-based techniques focus on changes in the temporal and frequency characteristics of the system response. Because restoring forces in base-excited structures can exhibit highly non-linear characteristics, non-linear model-based approaches may be better suited for reliable health monitoring and damage detection. This paper presents the application of a novel intelligent parameter varying (IPV) modeling and system identification technique, developed by the authors, to detect damage in base-excited structures. This IPV technique overcomes specific limitations of traditional model-based and non-model-based approaches, as demonstrated through comparative simulations with wavelet analysis methods. These simulations confirm the effectiveness of the IPV technique, and show that performance is not compromised by the introduction of realistic structural non-linearities and ground excitation characteristics.     

On-line identification of non-linear hysteretic structures using an adaptive tracking technique

System identification and damage detection based on vibration data have received considerable attention recently because of their importance to structural health monitoring. Various technical approaches have been proposed in the literature; however, the on-line identification of the changes of parameters for non-linear structures due to damages is still a challenging problem. In this paper, we propose an on-line adaptive tracking technique, based on the least-square estimation, to identify the system parameters and their changes of non-linear hysteretic structures. The method proposed is capable of tracking abrupt or slow changes of the system parameters from which the damage event and the severity of the structural damage can be detected and evaluated. Simulation results for tracking the parametric changes of non-linear hysteretic structures are presented to demonstrate the application and effectiveness of the proposed technique in detecting the structural damages.     

Effect of non-linearities in the identification and fault detection of gear-pair systems

This study investigates issues related to parametric identification and health monitoring of dynamical systems with non-linear characteristics. In the first part, a gear-pair system supported on bearings with rolling elements is selected as an example mechanical model and the corresponding equations of motion are set up. This model possesses strongly non-linear characteristics, accounting for gear backlash and bearing stiffness non-linearities. Then, the basic steps of the parametric identification and fault detection procedure employed are outlined briefly. In particular, a Bayesian statistical framework is adopted in order to estimate the optimal values of the gear and bearing model parameters. This is achieved by combining experimental information from vibration measurements with theoretical information built into a parametric mathematical model of the system. In the second part of the study, characteristic numerical results are presented. First, based on the effect of the system parameters on its dynamics, a solid basis is created for explaining some of the peculiar results obtained by applying classical gradient-based optimization methodologies for the strongly non-linear system examined. Some serious difficulties, associated with the existence of irregular response or the coexistence of multiple motions, are first pointed out. A solution to some of these problems, through the application of a suitable genetic algorithm, is then presented. Special problems, related to more classical identification issues associated with the presence of measurement noise and model error, are also investigated.     

Application of particle swarm optimization and genetic algorithms to multiobjective damage identification inverse problems with modelling errors

Structural health monitoring has become an important research topic in conjunction with the damage assessment of structures. The use of system identification approaches for damage detection using inverse methods has become more widespread in recent years and their formulation in a multiobjective framework has become more usual. Inverse problems require the use of an initial baseline model of the undamaged structure. Modelling errors in the baseline model whose effects exceed the modal sensitivity to damage are critical and make an accurate estimation of damage impossible. Artificial intelligence techniques based on genetic algorithms are used increasingly as an alternative to more classical techniques to solve this kind of problem especially due to their feasibility for managing multiobjective problems. This paper outlines an understanding of how particle swarm optimization methods operate in damage identification problems based on multiobjective FE updating procedures and takes modelling errors into account. One experimental example is used to show their performance in comparison with genetic algorithms.     

A new method for detecting non-linear damping and restoring forces in non-linear oscillation systems from transient data

The purpose of this study is to recover the functional form of both non-linear damping and non-linear restoring forces in the non-linear oscillatory motions of an autonomous system. Using two sets of measured motion response data of the system, an inverse problem is formulated for recovering (or identification): the differential equation of motion is transformed into an equivalent integral equation of motion. The identification, which is non-linear, is shown to be one-to-one. However, the inverse problem formulated herein is concerned with the Volterra-type of non-linear integral equation of the first kind. This leads to numerical instability: solutions of the inverse problem lack stability properties. In order to overcome the difficulty, a regularization method is applied to the identification process. In addition, an L-curve criterion, combined with regularization, is introduced to find an optimal choice for the regularization parameter (i.e., the number of iterations), in the presence of noisy data. The workability of the identification is investigated for simultaneously recovering the functional form of the non-linear damping and the non-linear restoring forces through a numerical experiment.     

Time and frequency domain nonlinear system characterization for mechanical fault identification

Mechanical systems are often nonlinear with nonlinear components and nonlinear connections, and mechanical damage frequently causes changes in the nonlinear characteristics of mechanical systems, e.g. loosening of bolts increases Coulomb friction nonlinearity. Consequently, methods which characterize the nonlinear behavior of mechanical systems are well-suited to detect such damage. This paper presents passive time and frequency domain methods that exploit the changes in the nonlinear behavior of a mechanical system to identify damage. In the time domain, fundamental mechanics models are used to generate restoring forces, which characterize the nonlinear nature of internal forces in system components under loading. The onset of nonlinear damage results in changes to the restoring forces, which can be used as indicators of damage. Analogously, in the frequency domain, transmissibility (output-only) versions of auto-regressive exogenous input (ARX) models are used to locate and characterize the degree to which faults change the nonlinear correlations present in the response data. First, it is shown that damage causes changes in the restoring force characteristics, which can be used to detect damage. Second, it is shown that damage also alters the nonlinear correlations in the data that can be used to locate and track the progress of damage. Both restoring forces and auto-regressive transmissibility methods utilize operational response data for damage identification. Mechanical faults in ground vehicle suspension systems, e.g. loosening of bolts, are identified using experimental data.     

Data-based Identification of nonlinear restoring force under spatially incomplete excitations with power series polynomial model

One of the present barriers to the realization of structural health monitoring is the lack of efficient and general identification methodologies for dealing with nonlinearity, because a priori knowledge of the nature and mathematical form of the nonlinearities of typical engineering structures are usually unknown. The studies on the identification of restoring force, which can be considered as a direct indicator of the extent of the nonlinearity, have received increasing attention in recent years. In this paper, the nonlinear restoring force (NRF) was estimated by using a power series polynomial, and each coefficient of the polynomial was identified by means of standard least-square techniques. No information about the system was needed, and only the applied excitations and the corresponding response time series were used for the identification. Two different cases, in which the system was under complete and incomplete excitations, were investigated. Moreover, the effect of noise level was also taken into consideration. The feasibility and robustness of the proposed approach were verified via a 2-degree-of-freedom (DOF) lumped-mass numerical model, and experimental tests on a 4-story shear building with magneto-rheological (MR) dampers which served to simulate nonlinear behavior. The results show that the proposed data-based method is capable of identifying the NRF in a chain-like multi-degree-of-freedom engineering structures without any assumptions on the structural parameters, and provides a promising way for damage detection in the presence of structural nonlinearities.     

Development of data-based model-free representation of non-conservative dissipative systems

A general procedure is presented for developing data-based, non-parametric models of non-linear multi-degree-of-freedom, non-conservative, dissipative systems. Two broad classes of methods are discussed: one relying on the representation of the system restoring forces in a polynomial-basis format, and the other using artificial neural networks to map the complex transformations relating the system state variables to the needed system outputs. A non-linear two-degree-of-freedom system is used to formulate the approach under discussion and to generate synthetic data for calibrating the efficiency of the two methods in capturing complex non-linear phenomena (such as dry friction, hysteresis, dead-space non-linearities, and polynomial-type non-linearities) that are widely encountered in the applied mechanics field. Subsequently, a reconfigurable test apparatus was used to generate experimental measurements from a physical non-linear “joint” involving two-dimensional motion (translation and rotation) and complicated interaction forces between the different motion axes, among its internal elements. Both the polynomial-basis approach and the neural network method were used to develop high-fidelity, non-parametric models of the physical test article. The ability of the identified models to accurately “generalize” the essential features of the non-linear system was verified by comparing the predictions of the models with experimental measurements from data sets corresponding to different excitations than those used for identification purposes. It is shown that the identification techniques under discussion can be useful tools for developing accurate simulation models of complex multi-dimensional non-linear systems under broadband excitation.     

Sequential non-linear least-square estimation for damage identification of structures

An early detection of structural damage is an important goal of any structural health monitoring system. In particular, the ability to detect damages on-line, based on vibration data measured from sensors, will ensure the reliability and safety of the structures. In this connection, innovative data analysis techniques for the on-line damage detection of structures have received considerable attentions recently, although the problem is quite challenging. In this paper, we proposed a new data analysis method, referred to as the sequential non-linear least-square (SNLSE) approach, for the on-line identification of structural parameters. This new approach has significant advantages over the extended Kalman filter (EKF) approach in terms of the stability and convergence of the solution as well as the computational efforts involved. Further, an adaptive tracking technique recently proposed has been implemented in the proposed SNLSE to identify the time-varying system parameters of the structure. The accuracy and effectiveness of the proposed approach have been demonstrated using the Phase I ASCE structural health monitoring benchmark building, a non-linear elastic structure and non-linear hysteretic structures. Simulation results indicate that the proposed approach is capable of tracking on-line the changes of structural parameters leading to the identification of structural damages.     

A fractional non-linear creep model for coal considering damage effect and experimental validation

Accurate prediction of coal׳s creep behavior is of great significance to coalbed methane extraction. In this study, taking into account the visco-elastic–plastic characteristics and the damage effect, a fractional non-linear model is proposed to describe the creep behavior of coal. The constitutive and creep equations of the proposed fractional non-linear model are derived via the Boltzmann superposition principle and discrete inverse Laplace transform. Furthermore, uniaxial creep tests under different axial stress conditions were carried out to validate the proposed model. It is found that the present model can describe the experimental data from creep tests with better accuracy than classical models. Particularly, the present model can predict the accelerating creep deformation of coal which classical models fail to reproduce. Finally, the parametric sensitivity analysis is performed to investigate the effects of model parameters on the creep strain. It is verified that the introduction of fractional parameters and damage factor in the present model is essential to accurate prediction of the full creep stage of coal.     

Restoring force and dynamic loadings identification for a nonlinear chain-like structure with partially unknown excitations

Most of the currently employed vibration-based identification approaches for structural damage detection are based on eigenvalues and/or eigenvectors extracted from dynamic response measurements, and strictly speaking, are only suitable for linear system. However, the inception and growth of damage in engineering structures under severe dynamic loadings are typical nonlinear procedures. Consequently, it is crucial to develop general structural restoring force and excitation identification approaches for nonlinear dynamic systems because the restoring force rather than equivalent stiffness can act as a direct indicator of the extent of the nonlinearity and be used to quantitatively evaluate the absorbed energy during vibration, and the dynamic loading is an important factor for structural remaining life forecast. In this study, based on the instantaneous state vectors and partially unknown excitation, a power series polynomial model (PSPM) was utilized to model the nonlinear restoring force (NRF) of a chain-like nonlinear multi-degree-of-freedom (MDOF) structure. To improve the efficiency and accuracy of the proposed approach, an iterative approach, namely weighted adaptive iterative least-squares estimation with incomplete measured excitations (WAILSE-IME), where a weight coefficient and a learning coefficient were involved, was proposed to identify the restoring force of the structure as well as the unknown dynamic loadings simultaneously. The response measurements of the structure, i.e., the acceleration, velocity, and displacement, and partially known excitations were utilized for identification. The feasibility and robustness of the proposed approach was verified by numerical simulation with a 4 degree-of-freedom (DOF) numerical model incorporating a nonlinear structural member, and by experimental measurements with a four-story frame model equipped with two magneto-rheological (MR) dampers mimicking nonlinear behavior. The results show the proposed approach by combining the PSPM and WAILSE-IME algorithm is capable of effectively representing and identifying the NRF of the chain-like MDOF nonlinear system with partially unknown external excitations, and provide a potential way for damage prognosis and condition evaluation of engineering structures under dynamic loadings which should be regarded as a nonlinear system.     

A robust hyper-elastic beam model under bi-axial normal-shear loadings

In this paper, a simple and robust constitutive model is proposed to simulate mechanical behaviors of hyper-elastic materials under bi-axial normal-shear loadings in the finite strain regime. The Mooney–Rivlin strain energy function is adopted to develop a two-dimensional (2D) normal-shear constitutive model within the framework of continuum mechanics. A motion field is first proposed for combined normal and shear deformations. The deformation gradient of the proposed field is calculated and then substituted into right Cauchy–Green deformation tensor. Constitutive equations are then derived for normal and shear deformations. They are two explicit coupled equations with high-level polynomial non-linearity. In order to examine capabilities of the developed hyper-elastic model, uniaxial tensile responses and non-linear stability behaviors of moderately thick straight and curved beams undergoing normal axial and transverse shear deformations are simulated and compared with experiments. Fused deposition modeling technique as a 3D printing technology is implemented to fabricate hyper-elastic beam structures from soft poly-lactic acid filaments. The printed specimens are tested under tensile/compressive in-plane and compressive out-of-plane forces. A finite element formulation along with the Newton–Raphson and Riks techniques is also developed to trace non-linear equilibrium path of beam structures in large defamation regimes. It is shown that the model is capable of predicting non-linear equilibrium characteristics of hyper-elastic straight and curved beams. It is found that the modeling of shear deformation and finite strain is essential toward an accurate prediction of the non-linear equilibrium responses of moderately thick hyper-elastic beams. Due to simplicity and accuracy, the model can serve in the future studies dealing with the analysis of hyper-elastic structures in which two normal and shear stress components are dominant.     

Non-linear parametric modelling of a high-speed rail hydraulic yaw damper with series clearance and stiffness

At high operational speeds, the train system becomes very sensitive to parameter variations of components. Therefore, it is imperative to incorporate more accurate component models in the vehicle dynamics studies. This study addresses a more subtle and comprehensive non-linear parametric model of a high-speed rail hydraulic yaw damper. A new concept of a hydraulic yaw damper model is suggested, in which the small mounting clearance, the series stiffness, and the viscous damping are built in. The series stiffness is the tandem result of the dynamic oil stiffness, the rubber attachment stiffness, and the mounting seat stiffness. A dynamic oil property model is established and coupled to the entire modelling process, in which the density, the dynamic viscosity, the volumetric elastic modulus, and the stiffness of the oil are all changeable in terms of the instantaneous working pressure, the oil temperature, and the entrapped air ratio of the oil. The dynamic flow loss and the valve system dynamics are also incorporated. Experiments validated that the established non-linear parametric model is accurate and robust in predicting the damping characteristics within an extremely wide speed range. The validated damper model was then successfully applied to a thorough parameter sensitivity analysis and damping nature prediction under practical, in-service conditions. The established damper model couples all the main influential factors that are not or are insufficiently considered in normal-speed problems; thus, it will be more accurate and appropriate for furthering high-speed problem studies.     

Damage localization using transient non-linear elastic wave spectroscopy on composite structures

To reduce the costs related to maintenance of aircraft structures, there is the need to develop new robust, accurate and reliable damage detection methods. A possible answer to this problem is offered by newly developed non-linear acoustic/ultrasonic techniques, which monitor the non-linear elastic wave propagation behaviour introduced by damage, to detect its presence and location.In this paper, a new transient non-linear elastic wave spectroscopy (TNEWS) is presented for the detection and localization of a scattered zone (damage) in a composite plate. The TNEWS analyses the uncorrelations between two structural dynamic responses generated by two different pulse excitation amplitudes by using a time-frequency coherence function. A numerical validation of the proposed method is presented. Damage was introduced and modelled using a multi-scale material constitutive model (Preisach-Mayergoyz space).The developed technique identified in a clear manner the faulted zone, showing its robustness to locate and characterize non-linear sources in composite materials     

On the damping decrement for non-linear oscillations

The logarithmic damping decrement is obtained as a function of arbitrary non-linear restoring forces and arbitrary, but small, non-linear damping forces. General expressions are obtained for both amplitude-dependent and speed-dependent damping. The special case of a cubic restoring force with quadratic amplitude-dependent damping and the special case of a cubic restoring force with quadratic speed-dependent damping are considered in detail. The results of the analysis suggest how experimental data can be utilized to identify and evaluate the damping parameters for a given non-linear oscillator.     

On the use of non-linear vibrations and the anti-resonances of Higher-Order Frequency Response Functions for crack detection in pipeline beam

The identification of new scientific challenges, as well as the increasing high-performance computing support, indicates that the benefits of applying novel nonlinear techniques for crack detection will continue to grow. So, significant effort has been invested in recent years to develop effective techniques to detect crack in mechanical structures. The objective of this paper is to discuss and propose a robust diagnostic of damage based on non-linear vibrational measurements with particular regard to the Higher-Order Frequency Response Functions. An important observation is that the appearances of the non-linear harmonic components and the emerging anti-resonances in Higher-Order Frequency Response Functions can provide useful information on the presence of cracks and may be used on an on-line crack monitoring system for small levels of damage. Efficiency of the proposed methodology is illustrated through numerical examples for a pipeline beam including a breathing crack.     

Development of adaptive modeling techniques for non-linear hysteretic systems

Adaptive estimation procedures have gained significant attention by the research community to perform real-time identification of non-linear hysteretic structural systems under arbitrary dynamic excitations. Such techniques promise to provide real-time, robust tracking of system response as well as the ability to track time variation within the system being modeled. An overview of some of the authors’ previous work in this area is presented, along with a discussion of some of the emerging issues being tackled with regard to this class of problems. The trade-offs between parametric-based modeling and non-parametric modeling of non-linear hysteretic dynamic system behavior are discussed. Particular attention is given to (1) the effects of over- and under-parameterization on parameter convergence and system output tracking performance, (2) identifiability in multi-degree-of-freedom structural systems, (3) trade-offs in setting user-defined parameters for adaptive laws, and (4) the effects of noise on measurement integration. Both simulation and experimental results indicating the performance of the parametric and non-parametric methods are presented and their implications are discussed in the context of adaptive structures and structural health monitoring.     

Non-periodic motions of a Jeffcott rotor with non-linear elastic restoring forces

The conditions that give rise to non-periodic motions of a Jeffcott rotor in the presence of non-linear elastic restoring forces are examined. It is well known that non-periodic behaviours that characterise the dynamics of a rotor are fundamentally a consequence of two aspects: the non-linearity of the hydrodynamic forces in the lubricated bearings of the supports and the non-linearity that affects the elastic restoring forces in the shaft of the rotor. In the present research the analysis was restricted to the influence of the non-linearity that characterises the elastic restoring forces in the shaft, adopting a system that was selected the simplest as possible. This system was represented by a Jeffcott rotor with a shaft of mass that was negligible respect to the one of the disk, and supported with ball bearings. In order to check in a straightforward manner the non-linearity of the system and to confirm the results obtained through theoretical analysis, an investigation was carried out using an experimental model consisting of a rotating disk fitted in the middle of a piano wire pulled taut at its ends but leaving the tension adjustable. The adopted length/diameter ratio was high enough to assume the wire itself was perfectly flexible while its mass was negligible compared to that of the disk. Under such hypotheses the motion of the disk centre can be expressed by means of two ordinary, non-linear and coupled differential equations. The conditions that make the above motion non-periodic or chaotic were found through numerical integration of the equations of motion. A number of numerical trials were carried out using a 4th order Runge-Kutta routine with adaptive stepsize control. This procedure made it possible to plot the trajectories of the disk centre and the phase diagrams of the component motions, taken along two orthogonal coordinate axes, with their projections of the Poincaré sections. On the basis of the theoretical results obtained, the conditions that give rise to non-periodic motions of the experimental rotor were identified.     

Real-time parameter estimation for degrading and pinching hysteretic models

Many civil and mechanical structures exhibit hysteresis with degradation and/or pinching when subject to severe cyclic loadings such as earthquakes, wind, or sea waves. The modeling and identification of non-linear hysteretic systems with degradation and pinching is therefore a practical problem encountered in the engineering mechanics field. On-line identification of degrading and pinching hysteretic systems is quite a challenging problem because of its complexity. A recently developed technique, the unscented Kalman filter (UKF) which is capable of handling any functional non-linearity, is applied to the on-line parametric system identification of hysteretic differential models with degradation and pinching. Simulation results show that the UKF is efficient and effective for the real-time state estimation and parameter identification of highly non-linear hysteretic systems with degradation and pinching.     

Digital Image Mechanical Identification (DIMI)

A continuous pathway from digital images acquired during a mechanical test to quantitative identification of a constitutive law is presented herein based on displacement field analysis. From images, displacement fields are directly estimated within a finite element framework. From the latter, the application of the equilibrium gap method provides the means for rigidity field evaluation. In the present case, a reconditioned formulation is proposed for a better stability. Last, postulating a specific form of a damage law, a linear system is formed that gives a direct access to the (non-linear) damage growth law in one step. The two last procedures are presented, validated on an artificial case, and applied to the case of a biaxial tension of a composite sample driven up to failure. A quantitative estimate of the quality of the determination is proposed, and in the last application, it is shown that no more than 7% of the displacement field fluctuations are not accounted for by the determined damage law.