From Frequency Response Functions Research Articles

Using incomplete FRF measurements for damage detection of structures with closely-spaced eigenvalues

In this paper, a new model updating-based damage detection method based on a modified sensitivity equation is proposed that can tackle the problem of damage detection in structures with closely-spaced eigenvalues. It is known that modal information, such as natural frequencies, in these structures can be of close proximity, making the procedure of damage detection hard if not impossible. The obtained sensitivity equation uses incomplete measurements from frequency response functions (FRFs) to conduct the challenging damage detection of structures with closely-spaced eigenvalues. Although using FRF for damage detection of this kind of structures can offer some advantages over modal data, there are still some challenges that need to be addressed. For instance, it is not possible to have the response of the structure measured in all of its degrees of freedom. As such, the proposed FRF-based model updating method is capable of overcoming these limitations and has the advantage of avoiding modal analysis errors. In order to evaluate the efficiency of the proposed method, two numerical examples of a 144-element three-layered laminated composite plate and a 120-element three-dimensional truss structure, as examples of structures with closely-spaced eigenvalues, are studied. The results demonstrate the capability of the proposed method in damage detection of structures with closely-spaced eigenvalues with incomplete measurement data. Moreover, the results of comparison between the proposed method with some other methods demonstrate the superiority of the proposed method in damage detection of structures with closely-spaced eigenvalues using noisy incomplete FRF data.

Wafer Stage Modal Model Identification by Modified Vector Fitting Method

Vector fitting is modified in this study for the parametric identification of a model with an undamped rigid body mode in the frequency domain. The modal model of a six-degree-of-freedom (6DOF) wafer stage is identified using the modified vector fitting method. The modal model is typically fitted from the matrix fraction description (MFD) form of the transfer function, which increases the cost function. The vector fitting in frequency domain provides an approach to fitting the modal model directly from frequency response functions(FRFs) via a partial rational basis function, the poles of which can be obtained by pole relocation technology. The traditional vector fitting method is not applicable in identifying systems containing rigid body modes. The method is reformulated here and effectively applied for wafer stage identification. The accuracy of the fitted model is validated by control loop simulation.

Electromechanical properties identification for groups of piezoelectric energy harvester based on Bayesian inference

A framework that allows the use of well-known dynamic estimators to infer the electromechanical properties in Piezoelectric Energy Harvesters (PEHs) is presented here. The framework is based on Bayesian inference applied over experimental results obtained from Frequency Response Functions (FRFs). The posterior probability density function is approximated adopting the Transitional Markov Chain Monte Carlo algorithm. A similar approach has been developed recently to perform the electromechanical properties updating for a single PEH. However, our results show that the former approach is not suitable to update the properties associated to a set of PEHs since it mismatches the normalized FRF. The proposed framework extends the previous formulation to solve this issue. The likelihood function is modified to account for a predictive model with three outputs obtained by manipulating the information available in the FRF. The proposed framework in this contribution can be used by manufacturers to update the nominal properties of groups of devices and, simultaneously, to identify the variability induced by the manufacturing process.

Visco-elastohydrodynamic lubrication of layered materials with imperfect layer-substrate interfaces

This paper reports a novel visco-elastohydrodynamic lubrication (VEHL) model for the lubrication interface formed by a rigid sphere and a viscoelastic layer imperfectly coated on an elastic half-space, aiming to study the effects of layer viscoelasticity and imperfect layer-substrate interface on the lubrication behavior. The surface viscoelastic deformation is calculated based on a linear viscoelastic model and the correspondence principle. The contact solution is formulated with the discrete convolution and fast Fourier transform (DC-FFT) algorithm by using the influence coefficients (ICs) converted from frequency response functions (FRFs). The model is validated by using the results from its degenerated versions through comparisons with available published data. The new VEHL model is implemented to explore the effects of surface speed, layer thickness, and layer-substrate elastic modulus ratio, as well as the degree of displacement discontinuity or stiffness defect at the layer-substrate interface.

Linear viscoelastic behavior of asphalt mixes from dynamic frequency response functions

SummaryIn the small strain domain, asphalt mixes (AM) have a linear viscoelastic (LVE) behavior that is strongly dependent on frequency and temperature. The maximum ratio of modulus values can be up to 1000, and traditional elastic analyses are not pertinent. The possibility to characterize AM from frequency response functions (FRFs) was studied. A new optimization process using the finite element method (FEM) has been developed to back‐calculate the LVE properties of AM from FRFs. The numerical optimization process was applied to a reference material with averaged LVE properties determined from tension‐compression tests performed on a wide variety of AM types. The LVE properties were modeled considering the 3‐Dim version of the model 2S2P1D (2 Springs, 2 Parabolic elements, and 1 Dashpot). Reference FRFs for the considered reference material were obtained from FEM simulations. Three different configurations that may be of interest for practical tests were studied at five different temperatures. The proposed numerical optimization method consists in performing separate optimizations at each temperature to obtain the LVE properties for the considered temperature. Then values obtained at each temperature are considered to optimize 2S2P1D and Williams Landel Ferry (WLF) Equation constants to simulate the global LVE behavior of the material. The accuracy of the process was assessed regarding both the calculated FRFs and the complex modulus evaluation. Results indicate that the proposed optimization process converges almost perfectly towards the reference FRFs. The simulated complex modulus values are also in very good agreement with the values of the reference material.

An alternative representation of the receptance: The ‘elliptical plane’ and its modal properties

Abstract Modal Identification from Frequency Response Functions (FRFs) has been extensively investigated up to the point its research reached a stagnation state. Yet, a new approach to determine the modal damping factors from FRFs was recently proposed, showing that there still is scope for new findings in the field. Contrary to other modal identification methods which are based on the dynamic motion governing equations, the method used the dissipated energy per cycle of vibration as a starting point. For lightly damped systems with conveniently spaced modes, it produced quite accurate results, especially when compared to the well-known method of the inverse. The method used a plot of the sine of the phase of the receptance against its amplitude, whereby damping was determined from the slope of a linear fit to the resulting plot. In this paper, it is shown that this plot has other (perhaps more important) special properties that were not explored before. Near resonant frequencies, its shape is elliptical, whereby the real and imaginary parts of the modal constants can be determined from numerical curve-fitting. This finding allowed developing a new method which formulation is presented in this paper. The method is discussed through numerical and experimental examples. Although the intention is not to present a new modal identification method that is superior to other existing ones (like the method of the inverse or those based on the Nyquist plot), the authors believe that this new representation of the receptance and its properties may bring valuable insights for other researchers in the field.

Evaluation of the Performance of Different Damage Indicators in Railway Bridges

This article describes a damage identification methodology based on a genetic algorithm and its application to the numerical model of a railway bridge. The identification method is based on an iterative process that in each iteration compares the values of the bridge's dynamic response for a test damage scenario with the response values for a reference damage scenario. The introduction of damage to the test numerical model is based on the modification of a preselected numerical parameter values. The genetic algorithm allows the identification of damage by minimizing the residue of an objective function constructed from the responses of the test and reference model, and based on the so-called damage indicators. In this study are defined and tested damage indicators based on modal parameters (frequencies and vibration modes) in parameters built from the modal parameters (modal curvatures and flexibility curvatures), and in dynamic responses under action of impulsive loads from frequency response functions (FRFs). The proposed methodology for identifying damage was tested based on a two-dimensional numerical finite element model of a railway bridge including the railway track. The results allowed evaluating the efficiency and reliability of the methodology and of the damage indicators which, in most situations, showed potential in the detection, location and quantification of the severity of the damage.

A Contribution to Improve the Accuracy of Chatter Prediction in Machine Tools Using the Stability Lobe Diagram

The chatter phenomenon can severely limit the power available for milling. The stability lobe diagram (SLD) is a very fast and simple method to predict the chatter free zone, allowing the selection of the most adequate spindle speed and depth of cut for higher productivity. However, the data used to calculate the SLD, coming from frequency response functions (FRFs), must be acquired adequately to improve the predictability. FRFs result differently depending on the activation of the spindle electronic control. The present work uses SLDs to investigate these differences and experimental end milling tests to assess the accuracy of SLDs curves. Results indicate that the inclusion of spindle electronic control provides better accuracy in predicting the chatter in milling.

Two Dimensional Damage Localization Using the Interpolation Method

This paper presents a vibration based procedure for locating reductions of stiffness in two-dimensional structures that can be modeled as plates. This procedure is a generalization to the two-dimensional case of the previously published Interpolation Damage Detection Method (IDDM). The method is based on the definition of a damage sensitive feature in terms of the accuracy of a spline function in interpolating the operational displacement shapes of the structure. These latter are recovered from frequency response functions (FRFs) measured at different locations of the structure during vibrations. At the i-th location, the FRF is calculated through spline interpolation using the FRF’s recorded at the all the instrumented locations but the i-th. For two-dimensional structures a spline surface is defined to interpolate the operational shapes. The accuracy of the spline interpolation is measured by an error function defined as the difference between the measured and interpolated operational mode shapes. At a certain location an increase (statistically meaningful) of the interpolation error, with respect to a reference configuration, points out a localized variation of the operational shapes thus revealing the existence of damage. The two dimensional IDDM algorithm is checked herein through numerical simulations, using the FE model of a plate and modeling local reductions of stiffness through a reduction of the elastic modulus of the material of one or more elements of the model.

The three-stage artificial neural network method for damage assessment of building structures

AbstractBuilding structures are often huge and composed of a number of elements. It may not be possible to make modal measurements along the large number of degrees of freedom. Structural damage detection therefore becomes much more challenging both in terms of measurement and subsequent analyses. Accordingly, a problem in structural damage detection is requirement of a systematic and effective method. Among the developed damage detection techniques, artificial neural networks (ANNs) have become promising tools recently. The main drawback of using ANNs in structural condition monitoring is the requirement of enormous computational effort. To address this issue, a novel technique is proposed using “damage index” derived from frequency response functions (FRFs) with the three-stage ANN method to detect damage. The basic idea of this method is to establish features of damaged structure using FRFs from different measurement points. Then using these features, damage indices of damage cases of the structure are...

The Three-Stage Artificial Neural Network Method for Damage Assessment of Building Structures

Building structures are often huge and composed of a number of elements. It may not be possible to make modal measurements along the large number of degrees of freedom. Structural damage detection therefore becomes much more challenging both in terms of measurement and subsequent analyses. Accordingly, a problem in structural damage detection is requirement of a systematic and effective method. Among the developed damage detection techniques, artificial neural networks (ANNs) have become promising tools recently. The main drawback of using ANNs in structural condition monitoring is the requirement of enormous computational effort. To address this issue, a novel technique is proposed using “damage index” derived from frequency response functions (FRFs) with the three-stage ANN method to detect damage. The basic idea of this method is to establish features of damaged structure using FRFs from different measurement points. Then using these features, damage indices of damage cases of the structure are identified. Damage indices corresponding to different damage locations and severities are introduced to ANNs. The effectiveness of the proposed method is validated using the finite element model of a 10-storey framed structure. The results show that the principal component analysis based damage index is suitable for structural damage detection.

Multi-channel Active Vibration Isolation for the Control of Underwater Sound Radiation From A Stiffened Cylindrical Structure: A Numerical Study

Numerical simulation of vibration control of a submerged stiffened cylindrical structure with active vibration isolators is presented. Vibration transmission from vibrating machinery to the cylindrical structure through the active vibration isolators is analyzed by a numerical model synthesized from frequency response functions (FRFs) and impedances. The coupled finite element/boundary element (FE/BE) method is employed to study the vibro-acoustic behavior of the fluid-loaded cylindrical structure. Sound pressure in the far-field is calculated in terms of the pressure and normal acceleration of the outer surface of the cylindrical shell. An adaptive multichannel control based on the filtered-x least mean squares (FxLMS) algorithm is used in the active vibration isolation. Simulation results have demonstrated that suppression of vibration of the four elastic foundations attached to the cylindrical shell will reduce the spatial-average mean-square velocity and the instantaneous radiated power of the cylindrical shell. As a result, suppression of vibration of the foundations leads to attenuation of sound radiation in the far-field induced by the radial displacement dominant mode of the shell. Moreover, vibration suppression is greatly influenced by the strong couplings among control channels. According to these results, it can be concluded that the proposed method is effective in the analysis of underwater sound radiation control of cylindrical structures.

Damage localization in linear-form structures based on sensitivity investigation for principal component analysis

This paper addresses the problem of damage detection and localization in linear-form structures. Principal component analysis (PCA) is a popular technique for dynamic system investigation. The aim of the paper is to present a damage diagnosis method based on sensitivities of PCA results in the frequency domain. Starting from frequency response functions (FRFs) measured at different locations on the structure; PCA is performed to determine the main features of the signals. Sensitivities of principal directions obtained from PCA to structural parameters are then computed and inspected according to the location of sensors; their variation from the healthy state to the damaged state indicates damage locations. It is worth noting that damage localization is performed without the need of modal identification. Influences of some features as noise, choice of parameter and number of sensors are discussed. The efficiency and limitations of the proposed method are illustrated using numerical and real-world examples.

The Use of Transmissibility Properties to Estimate FRFs on Modified Structures

When deriving an experimental model from Frequency Response Functions (FRFs), it may happen that the measurement of certain FRFs is impossible. This may be an important issue, mainly in the field of condition monitoring and damage detection, since some points of interest may become inaccessible in operational conditions. In this circumstance, it is useful to have some tools that can provide the prediction of such dynamic information. The transmissibility concept, extended to a general multiple degree-of-freedom system, can play an important role to circumvent these situations. The authors have shown in previous works that the estimation of such FRFs can be made possible by invoking important properties associated with the transmissibility function. The objective of this work is to evaluate different sets of FRFs, estimated by using the transmissibility concept and its associated properties, in an actual continuous structure to which different patterns of structural modification are applied. A supplementary study in this work shows that some of the sets for applied forces/known responses can better estimate the FRF data.

The Use of Transmissibility Properties to Estimate FRFs on Modified Structures

When deriving an experimental model from Frequency Response Functions (FRFs), it may happen that the measurement of certain FRFs is impossible. This may be an important issue, mainly in the field of condition monitoring and damage detection, since some points of interest may become inaccessible in operational conditions. In this circumstance, it is useful to have some tools that can provide the prediction of such dynamic information. The transmissibility concept, extended to a general multiple degree-of-freedom system, can play an important role to circumvent these situations. The authors have shown in previous works that the estimation of such FRFs can be made possible by invoking important properties associated with the transmissibility function. The objective of this work is to evaluate different sets of FRFs, estimated by using the transmissibility concept and its associated properties, in an actual continuous structure to which different patterns of structural modification are applied. A supplementary study in this work shows that some of the sets for applied forces/known responses can better estimate the FRF data.

Frequency-domain subspace identification using FRF data from arbitrary signals

Recently an extension of the frequency-domain subspace identification methods was developed that can cope with arbitrary excitation signals and transient effects. In this paper the implications of this extended model are investigated for the H 1 estimator. It is shown that an extended model makes it possible to start from frequency response functions (FRFs) without suffering from leakage, by modelling the initial conditions of each data block used in the averaging procedure of the H 1 estimator. In this way the measurements can be averaged in order to reduce both the data set and the noise levels without introducing bias errors caused by leakage phenomena. It is shown by both simulations and real-life measurements that the presented approach improves the modal parameter estimates and in particular the damping estimates.