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The problem of parametric output-only identification of time-varying structures in a recursive manner is considered. A kernelized time-dependent autoregressive moving average (TARMA) model is proposed by expanding the time-varying model parameters onto the basis set of kernel functions in a reproducing kernel Hilbert space. An exponentially weighted kernel recursive extended least squares TARMA identification scheme is proposed, and a sliding-window technique is subsequently applied to fix the computational complexity for each consecutive update, allowing the method to operate online in time-varying environments. The proposed sliding-window exponentially weighted kernel recursive extended least squares TARMA method is employed for the identification of a laboratory time-varying structure consisting of a simply supported beam and a moving mass sliding on it. The proposed method is comparatively assessed against an existing recursive pseudo-linear regression TARMA method via Monte Carlo experiments and shown to be capable of accurately tracking the time-varying dynamics. Furthermore, the comparisons demonstrate the superior achievable accuracy, lower computational complexity and enhanced online identification capability of the proposed kernel recursive extended least squares TARMA 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 stability-preserving 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 experimental case. The proposed approach is capable of accurately estimating the time-varying modes, and provides a new way to determine the dynamic stability of LTV systems by using the estimated time-varying modes.

This paper proposes an extended modal identification method to extract the time-varying modes and poles of linear periodically time -varying (LPTV) systems. A set of poles defines a stabilitypreserving transformation relat ing the original LPTV state equation to a constant upper triangular state equation and the time-varying modes contain the informat ion regarding the stability of the original LPTV system. The relation between the time-varying modes and poles is presented. The time-varying modes and poles of LPTV systems are shown to be the extension of the linear time-invariant (LTI) case. The proposed identification method is finally illustrated by a numerical example.

This paper investigates the problem of output-only modal parameter estimation of time-varying structures. A two-stage least square method for modal parameter estimation of time-varying structures is presented, consisting of a least square complex frequency-domain stage for the modal parameter estimation of all discrete time samples and a weighted least square stage for the estimation of continuous-time-represented modal parameters. This paper introduces a fuzzy-clustering-based approach for validating and sifting the modal parameters and develops four modal assurance criterion-based distance functions to improve the quality of clustering of operational mode shapes. Furthermore, the proposed method has been validated by a simulation example and a laboratory experiment.

This paper investigates the problem of modal parameter estimation of time-varying structures under unknown excitation. A time–frequency-domain maximum likelihood estimator of modal parameters for linear time-varying structures is presented by adapting the frequency-domain maximum likelihood estimator to the time–frequency domain. The proposed estimator is parametric, that is, the linear time-varying structures are represented by a time-dependent common-denominator model. To adapt the existing frequency-domain estimator for time-invariant structures to the time–frequency methods for time-varying cases, an orthogonal polynomial and z -domain mapping hybrid basis function is presented, which has the advantageous numerical condition and with which it is convenient to calculate the modal parameters. A series of numerical examples have evaluated and illustrated the performance of the proposed maximum likelihood estimator, and a group of laboratory experiments has further validated the proposed estimator.

Modal parameters such as natural frequencies and mode shapes are sensitive indicators of structural damage. However, they are not only sensitive to damage, but also to the environmental conditions such as, humidity, wind and most important, temperature. For civil engineering structures, modal changes produced by environmental conditions can be equivalent or greater than the ones produced by damage. This article proposes a damage detection method which is able to deal with temperature variations. The objective function correlates mode shapes and natural frequencies, and a parallel genetic algorithm handles the inverse problem. The numerical model of the structure assumes that the elasticity modulus of the materials is temperature-dependent. The algorithm updates the temperature and damage parameters together. Therefore, it is possible to distinguish between temperature effects and real damage events. Simulated data of a three-span bridge and experimental one of the I-40 Bridge validate the proposed methodology. Results show that the proposed algorithm is able to assess the experimental damage despite of temperature variations.

An hybrid real coded Genetic Algorithm with damage penalization is implemented to locate and quantify structural damage. Genetic Algorithms provide a powerful tool to solved optimization problems. With an appropriate selection of their operators and parameters they can potentially explore the entire solution space and reach the global optimum. Here, the set-up of the Genetic Algorithm operators and parameters is addressed, providing guidelines to their selection in similar damage detection problems. The performance of five fundamental functions based on modal data is studied. In addition, this paper proposes the use of a damage penalization that satisfactorily avoids false damage detection due to experimental noise or numerical errors. A tridimensional space frame structure with single and multiple damages scenarios provides an experimental framework which verifies the approach. The method is tested with different levels of incompleteness in the measured degrees of freedom. The results show that this approach reaches a much more precise solution than conventional optimization methods. A scenario of three simultaneous damage locations was correctly located and quantified by measuring only a 6.3% of the total degrees of freedom. ispartof: Mechanical Systems and Signal Processing vol:25 issue:5 pages:1559-1573 status: published

SUMMARY Antiresonances have become an attractive alternative in structural damage assessment. They can be identified easier and more accurately than mode shapes, and still providing the same information. Antiresonances are derived from point frequency response functions (FRFs) or from transfer FRFs. However, antiresonances from transfer FRFs are very sensitive to small structural changes, and the matching between numerical and experimental antiresonances is affected. This problem is solved if antiresonances from point FRFs are used. However, it implies an experimental procedure that differs from a common modal testing, which may become not practical or too expensive. This paper proposes a damage detection method able to deal with transfer antiresonances. The inverse problem is handled by a Parallel Genetic Algorithm. In this case, a perfect match between the antiresonances is not required because the optimization is not gradient based. Moreover, the matching can change at each step and the optimization is not affected. The algorithm is verified with two experimental cases: an exhaust system of a car with a single fatigue crack and a tridimensional space frame structure with single and multiple damage scenarios. Results are compared with the ones obtained using mode shapes. Damage detected is consistent with the experimental damage in both cases. Copyrightr 2010 John Wiley & Sons, Ltd. Received 23 January 2010; Revised 15 April 2010; Accepted 28 April 2010

The detection of damage with model-based methods is a constrained nonlinear optimization problem. Conventional optimization approaches usually lead to local minima. Furthermore, they are highly sensitive to experimental noise or numerical errors. Genetic algorithms (GAs) provide an attractive alternative since they can potentially explore the entire solution space and reach the global optimum. However, GAs are inherently slow when they work with complicated or time consuming objective functions. To overcome this problem parallel GAs are proposed, and they are particularly easy to implement and provide a superior numerical performance. In this study, a real-coded parallel GA is implemented to detect structural damage. The objective function is based on operational modal data; it considers the initial errors in the numerical model. False damage detection is avoided by using damage penalization. The algorithm is verified with two experimental cases. First, a test structure of an airplane subjected to three increasing levels of damage. Second, a multiple cracked reinforced concrete beam that is subjected to a nonsymmetrical increasing static load to introduce cracks. In both cases, the detected damage has a good correspondence with the experimental damage.

The resonant-based identification of the in-plane elastic properties of orthotropic materials implies the estimation of four principal elastic parameters: E 1 , E 2 , G 12 , and ν 12 . The two elastic moduli and the shear modulus can easily be derived from the resonant frequencies of the flexural and torsional vibration modes, respectively. The identification of the Poisson’s ratio, however, is much more challenging, since most frequencies are not sufficiently sensitive to it. The present work addresses this problem by determining the test specimen specifications that create the optimal conditions for the identification of the Poisson’s ratio. Two methods are suggested for the determination of the Poisson’s ratio of orthotropic materials: the first employs the resonant frequencies of a plate-shaped specimen, while the second uses the resonant frequencies of a set of beam-shaped specimens. Both methods are experimentally validated using a stainless steel sheet.

This work introduces a resonant-based, mixed numerical–experimental method for the determination of the in-plane elastic properties of the constituent materials of laminates. This non-destructive method identifies elastic properties from the resonant frequencies of beam-shaped layered specimens, using a set of finite element models. The method is demonstrated on a thermal barrier coating system made of NiCoCrAlY bondcoat and yttria-stabilised zirconia topcoat deposited by air-plasma spraying on stainless steel. The stainless steel was found to be elastically anisotropic, while both bondcoat and topcoat exhibited in-plane isotropy. Moreover, the topcoat Poisson's ratio approached zero, and the bondcoat properties varied with the coating thickness. Scanning electron microscopy was used to correlate the identified elastic properties with the coating microstructure.

In the design of rotating electrical machines, the vibro-acoustic behaviour becomes an important criterion. In an electrical machine, electromagnetically induced forces may cause vibrations of, for instance, the stator. These vibrations may radiate annoying acoustic energy, especially at resonances of the stator assembly. This paper describes some techniques to analyse the vibro-acoustic behaviour of a (brushless DC-) motor. However, the technique is quite general and can be used for all types of electrical machines. The “inverse-frequency-response-function” technique (I FRF) allows the identification of the actual vibration behaviour of the motor frame from the measurement of the acoustic field around the motor. Experimental modal analysis yields the resonant behaviour of the motor. A combination of both techniques indicates the critical resonances, causing the undesired acoustic emissions. This may lead to an improved, quieter design of the electrical motor.

The elastic and damping properties of two different soft magnetic composite (SMC) materials processed by a powder metallurgy route (Somaloy ™ 500 + 0.5 wt.% Kenolube and Somaloy ™ 500 + 0.6 wt.% LB1) have been characterised by means of non-destructive resonant vibration analysis, using the impulse excitation technique (IET). Microstructural characterisation by means of X-ray diffraction (XRD) and scanning electron microscopy (SEM) was carried out in order to understand the structural heterogeneity revealed with IET. A simple structural finite element model confirmed the link between microstructural heterogeneity of the SMC bodies and their vibration behaviour.

Mixed numerical-experimental techniques (MNETs) combine experimental test results and numerical modelling techniques with the goal of identifying physical properties. Experimental results always come with a level of uncertainty. This input uncertainty will migrate through the MNET routine, and will result in an uncertainty on the identified parameters. Therefore, MNET procedures should not only provide an estimated value for the physical properties, they should also provide additional information about the reliability of the results obtained. This paper presents a routine that is able to transform the uncertainty on the input parameters of a MNET into the uncertainty on the output parameters. The approach does not require any change of the initial MNET routine, it is just an additional computational step that has to be performed after the MNET has identified the deterministic values of the unknown physical properties. The routine is demonstrated on a vibration based MNET used to identify elastic material properties.

Cylindrical specimens made of the Ni-based super-alloy Inconel 625 (IN 625) were coated with (a) NiCoCrAlY, or (b) NiCoCrAlY and yttria-stabilised zirconia (YSZ: in this case, zirconia with 7-8 wt% yttria), using the electron beam - physical vapor deposition (EB-PVD) technique. In the bi-layer coatings, the YSZ layer is the thermal barrier coating (TBC) and the NiCoCrAlY layer is the metallic bond coat (BC). The BC improves the bonding between the substrate and the ceramic TBC, while the low thermal conductivity of the TBC oers high-temperature protection to the substrate. This paper focuses on the determination of the elastic moduli of the substrate and the coating layers of the test samples. The elastic moduli of the three dierent materials (IN 625, NiCoCrAlY and YSZ) were determined by means of a mixed numerical - experimental technique (MNET). The employed MNET was based on the comparison of the experimentally measured resonant frequencies of the rst bending mode of the test samples to the numerically calculated ones. The unknown elastic properties were determined by ne-tuning the elastic material parameters of the numerical models so as to enable the reproduction of the experimentally measured resonant frequencies.

This paper compares results of three different methods to determine the in-plane elastic properties of sheet materials. Results obtained with standard resonant beam and tensile tests are used to assess a mixed numerical–experimental technique developed to determine the in-plane elastic properties of orthotropic plates from the resonance frequencies of rectangular plate samples (the so-called ‘Resonalyser’ technique). Test materials were selected on the basis of an expected low degree of elastic anisotropy in order to put the accuracy and sensitivity of the different techniques to assess anisotropic materials to a test. Therefore, aluminium alloy and stainless steel samples were prepared from hot-rolled plates, deliberately avoiding pronounced cold-rolling textures. The differences between the results obtained with the three experimental approaches are critically evaluated. In the case of very thin plates, the existing mixed numerical–experimental Resonalyser procedure succeeded in accurately identifying the elastic material properties. A slightly adapted procedure is proposed to extend the applicability of the Resonalyser procedure to thicker plates.

Output-only modal analysis allows the estimation of modal models for structures in their real in-operation conditions. One of the main drawbacks of this approach is that a part of the modal model can no longer be estimated. As a result, the operational mode shape estimates remain incorrectly scaled, dependent on the unknown level of ambient excitation. This incompleteness somewhat restricts the applicability of in-operation modal models. No techniques were previously available for the normalisation of operational mode shapes purely based on experimental output-only data. Recently, a sensitivity-based normalisation approach, that can be used for output-only data, was proposed and successfully tested on simple laboratory structures. The principle behind the normalisation technique is the interpretation of the shifts in modal parameters induced by well-known structural modifications. In this contribution, the sensitivity-based normalisation approach was experimentally validated on a basis of vibration experiments performed on a tractor-sprayer combination. (C) 2003 Silsoe Research Institute. All rights reserved Published by Elsevier Science Ltd.

It is a well-known fact that the linear dynamic behaviour of mechanical structures can be studied by modelling the relation between force(s) input(s)], acting on the structure, and their resulting structural vibration response(s) [output(s)]. For industrial structures, in their real in-operation conditions, it often becomes hard (or impossible) to experimentally measure the excitation. For this reason, attention has been paid to the development of system identification techniques that work on a basis of response data only. The use of such techniques allows the identification of modal models for structures excited by unknown ambient noise and vibration. In this contribution, controlled vibration experiments were conducted on a sprayer boom that was mounted on a tractor. A comparison was made between the results of a classic input-output and output-only modal analysis based on maximum likelihood system identification techniques. Due to the interaction between the dynamic artificial excitation device and the test structure, the output-only approach modelled the excitation device together with the test structure. The identified output-only modal parameters were validated by direct comparison to the identification results obtained on a classic input-output transfer function data set between the generator input signal and the structural responses. (C) 2003 Silsoe Research Institute. All rights reserved Published by Elsevier Science Ltd.

For rotating electric machines, the reluctance forces (Maxwell stresses) acting on the stator teeth are a major cause of noise emission. Next to the reluctance forces, magnetostriction is a potential cause of additional noise from electric machines. First, a thermal stress analogy is used to introduce magnetostriction in the finite-element framework. Next, we present the computation and comparison of the stator vibration spectra caused by these two effects separately, by example of a 45 kW induction machine. Moreover, two kinds of magnetostriction characteristics of the stator yoke material are compared: a quadratic /spl lambda/(B) curve and a /spl lambda/(B) curve with zero-crossing around 1.5 Tesla.

The problem for measurement point selection in damage detection procedures is addressed. The concept of average mutual information is applied in order to find the optimal distance between measurement points. The idea is to select the measurement points in such a way that the taken measurements are independent, i.e. the measurements do not `learn' from each other. The average mutual information can be utilized as a kind of an autocorrelation function for the purpose. It gives the average amount of information that two points `learn' from each other. Thus the minimum of the average mutual information will provide the distance between measurement points with independent measurements. The idea to use the first minimum of the average mutual information is taken from nonlinear dynamics. The proposed approach is demonstrated on a test case. The results show that it is possible to decrease significantly the number of measurement points, without decreasing the precision of the solution.

The magnetic and mechanical finite element systems are combined into one magnetomechanical system. Investigating the coupling terms results in a finite element expression for the magnetic forces (Lorentz force and reluctance force) for both the linear and nonlinear case. The material deformation caused by magnetostriction is represented by an equivalent set of mechanical forces, giving the same strain to the material as magnetostriction does. The resulting magnetostriction force distribution is superposed onto other force distributions (external mechanical forces, magnetic forces) before starting the mechanical deformation or vibration analysis. This procedure is incorporated into a weakly‐coupled cascade solving of the magnetomechanical problem.

The magnetic material's deformation caused by magnetostriction is represented by an equivalent set of mechanical forces, giving the same deformation to the material as magnetostriction does. This is done in a way similar to how thermal stresses are usually incorporated into stress analysis. The resulting magnetostriction force distribution can be superposed onto other force distributions, like the magnetic force distribution. These two force distributions are the key ingredients of a numerically strong coupling of the magnetic and the mechanical finite element systems.

The magnetic material's deformation caused by magnetostriction is represented by an equivalent set of mechanical forces, giving the same deformation to the material as magnetostriction does. This is done in a way similar to how thermal stresses are usually incorporated. The resulting magnetostriction force distribution is summed to other force distributions (external mechanical forces, magnetic forces) before starting the mechanical deformation or vibration analysis. This procedure is incorporated into a weakly coupled cascade solving of the magnetomechanical problem.

Under natural stochastic excitations, responses of time-varying structures are always nonstationary stochastic signals, of which spectra change with time. This paper studies the time-dependent power spectrum density based on time-frequency analysis. Based on the time-dependent power spectrum density, a mathematical model of the time-frequency-domain two-step least square modal parameter estimator for time-varying structures is presented. In the first-step estimation, the modal parameters at each time instant are estimated using the least square complex frequency-domain method. Furthermore, the estimated modal parameters are sifted and sorted. Based on the sifted and sorted modal parameters, the time-varying resonance frequency, damping ratio and operational mode shapes are estimated in the second-step estimation. A numerical simulation example and a group of experiments validate this two-step least square modal parameter estimator for time-varying structures.

A strong coupling between the magnetic and the mechanical finite element model is presented. The two coupling terms represent magnetic forces and magnetostriction, respectively. The coupled system is solved using a fixed point iteration (successive substitution) with relaxation. The influence of physical parameters (low or high magnetostriction) and numerical parameters (scaling factors) on the convergence is investigated.

In recent years several methods have been developed for updating finite element models with regard to their dynamic characteristics, by using vibration information. Many techniques use modal characteristics (resonance frequencies, damping factors, modal displacement etc.) for tuning physical parameters of the model (elasticity moduli, mass density, plate thicknesses, cross sectional characteristics etc.) in order to obtain a finite element model which shows the same dynamic features as the real structure. One of these techniques is the User Acceptance model updating technique. Since this technique not only uses physically interpretable model parameters, but also allows an easy selection and/or weighting of those parameters and the dynamic characteristics to be matched, it is also applicable to dynamic structural optimisation. If used as such, the desired vibration parameters serve as a goal for the structural optimisation procedure. The algorithm determines modifications of the structure, within the imposed boundary conditions, that will match the desired dynamic behaviour, an example illustrates this process.

The modal assurance criterion (MAC) in general measures the degree of proportion between two (modal) vectors, in the form of a correlation coefficient of a least squares ratio estimate. The MAC principle can be extended in several ways, thus increasing its field of applications. The partial MAC (PMAC) correlates parts of (modal) vectors. The spatial MAC (SMAC) allows to compare different vector spaces. Furthermore this paper suggests a way of calculating the MAC sensitivities to model changes. All those extensions are illustrated by their possible uses in correlating measured dynamic data with (finite element) matrix models and in the area of model updating. Those applications might be helpful tools to indicate regions of poor measurement-model correlation, to complete measured vectors, to judge approximate eigenvalue solvers, or to improve model updating procedures.

A large number of vibration measurements are performed on the stator of a standard 5.2 kW electric motor. The rotor and the end-caps are removed. First, vibration measurements are performed on the stator without coils, i.e. consisting of the stator yoke (ferromagnetic iron) and the motor frame (cast iron) only. Second, vibration measurements are carried out on an identical stator with a standard coil system in the stator slots. Subsequently, a statistical energy analysis (SEA) is performed using these experimental data, in order to quantify the internal losses in the stator yoke and in the coils. The SEA also allows us to quantify the transmission of vibrations from stator yoke to motor frame (coupling loss factors), and to consider the influence of the presence of the coil system.

This paper presents the result of using an experimental spatial matrix identification method to predict the dynamics of a frame structure under a different boundary condition. The single-input-multiple-output frequency response functions of the test structure under the free-free boundary condition are measured by hammer testing. Using the FRFs, a set of spatial matrices is determined to represent its structural dynamic characteristics by the method. Then, using the identified spatial matrices, the dynamic characteristics of the test structure under the boundary condition of clamping 4 points is predicted. The prediction is practically accurate. The result of the prediction demonstrates that the spatial matrices identified by the method can be used for structural modification and substructure synthesis in the field of computer aided mechanical engineering.

In order to numerically predict the stator vibration spectrum, the mechanical behaviour of the stator (for low modal density) is correlated with the force distribution inside the machine. This force distribution is found considering the coupling between the magnetic and the mechanical finite element systems. This coupling results in a finite element based force expression covering both Lorentz forces and Maxwell forces. Using a weak coupling approach, magnetostriction forces are added to the force distribution. The numerically predicted stator vibration spectrum of a synchronous machine is compared to stator surface measurements. (4 pages)

This paper discusses the application of a damage detection methodology to monitor the location and extent of partial structural damage. The methodology combines, in an iterative way, the model updating technique based on frequency response functions (FRF) with monitoring data aiming at identifying the damage area of the structure. After the updating procedure reaches a good correlation between the models, it compares the parameters of the damage structure with those of the undamaged one to find the deteriorated area. The influence of the FEM mesh size on the evaluation of the extent of the damage has also been discussed. The methodology is applied using real experimental data from a spatial frame structure.

This paper presents two applications of the RADSER model updating technique (Lammens et al. (1995) and Larsson (1992)). The RADSER technique updates finite element model parameters by solution of a linearised set of equations that optimise the Reduced Analytical Dynamic Stiffness matrix based on Experimental Receptances. The first application deals with the identification of the dynamic characteristics of rubber mounts. The second application validates a coarse finite element model of a subframe of a Volvo 480.

This paper presents a model updating method based on experimental receptances. The presented method minimises the so called ‘indirect receptance difference’. First, the reduced analytical dynamic stiffness matrix is expressed as an approximate, linearised function of the updating parameters. In a numerically stable, iterative procedure, this reduced analytical dynamic stiffness matrix is changed in such a way that the analytical receptances match the experimental receptances at the updating frequencies. The updating frequencies are a set of selected frequency points in the frequency range of interest. Some considerations about an optimal selection of the updating frequencies are given. Finally, a mixed static-dynamic reduction scheme is discussed. Dynamic reduction of the analytical dynamic stiffness matrix at each updating frequency is physically exact, but it involves a great computational effort. The presented mixed static-dynamic reduction scheme is a simple strategy to reduce the computational cost with a minor loss of accuracy.

In recent years several methods have been developed for updating finite element models with regard to their dynamic characteristics, by means of experimentally obtained vibration information. Many techniques use modal characteristics (resonance frequencies, damping factors, modal displacements, ...) for tuning physical parameters of the model (elasticity moduli, mass density, plate thicknesses, cross sectional characteristics, ...) in order to obtain a finite element model which shows the same dynamic features as the real structure. One of those techniques is the U.A. Model Updating Technique.

Model optimization stands for all techniques using experimental modal parameters to modify an existing analytical model in order to improve the correlation bet\:een analytical and measured eigenvectors and eigenvalues. An iterative technique is proposed which combines theadvantages of the methods based upon orthogonality c onditions and those based upon the use of sensitivity values. In this method, a first modification of mass, stiffness and damping matrices is performed based upon the o rthogonality of the measured modes with respect to the analytical