Frequency Response Function Estimation Research Articles

A Spatial-Frequency Approach to Point-Wise Frequency Response Function Estimation with Digital Image Correlation

The use of digital image correlation for modal analysis is becoming an appealing option thanks to its non-contact and full-field measurement process. However, frequency response function (FRF) estimation can be challenging due to the limited number of time domain data and heavy measurement noise. Thereby, the present work aims to propose a method which improves the estimation accuracy of point-wise FRFs. Firstly, a Gaussian-process-based spatial-frequency model is proposed, which makes use of the intrinsic properties of the FRF and the local spatial information of field measurements. Then, a Bayesian solution is developed, which is enforced by a stable and efficient numerical procedure. Finally, the effectiveness of the proposed method is verified by making a comparison with the spectral estimator through the use of simulated data, and it is further validated based on an experimental application.

Accurate frequency response function estimation using noise measurements in experimental modal analysis

Accurate frequency response function estimation using noise measurements in experimental modal analysis

An experimental investigation on the fluid–structure coupling in horizontal pipes conveying two-phase intermittent flow

Two-phase flow is typically found in several industrial applications, such as in the production and transportation of oil and gas in the petrochemical industry, in the catalytic cracking and microreactors in the chemical industry, and in nuclear reactor cooling pumps. Measurement of two-phase flow features is usually necessary and has been done in several ways, including pressure probes, resistive sensors, gamma-ray, wire-mesh sensor and many others. However, these are either intrusive or invasive techniques, which might be of challenging application in industrial environments, or rely on a hazardous radioactive source. Vibration-based measurement of two-phase flow in pipes stands out as a non-invasive/non-intrusive approach and, consequently, multiphase-flow induced vibration in pipes has receiving increasing attention in recent years. In this work, the dynamic behaviour of a horizontal tube conveying an intermittent two-phase gas-liquid flow is investigated based on indirect approaches. The phenomenon of fluid–structure coupling is investigated using acceleration and pressure measurement. Moreover, the bubble size distribution is estimated from high-speed camera images and a deep learning model for image segmentation, along with its spectral content and time modulation. Focus is given at frequency bands around the cut-on frequencies of the circumferential wave modes of the pipe. An approach based on the estimation of frequency response function of the pressure and vibration at the liquid slug and elongated bubble is proposed, such that the coherence function can be used as quantitative measure of the coupling. Two experimental conditions with intermittent flow are investigated as representative cases. It is shown that there is a great vibration amplification at the cut-on frequencies of circumferential wave modes in pipes due to the corresponding structural wave and pressure coupling. Consequently, the frequency of passage of bubble can be estimated from the demodulation of vibration response filtered at the cut-on modes. The experimental results pave the way for innovative vibration-based measurement approaches.

Local modeling for FRF estimation with noisy input measurements

The frequency response function (FRF) is an essential means by which dynamic systems are qualified. In recent years, local modeling approaches have been extensively researched and shown to significantly outperform traditional FRF estimators. However, the standard local modeling approach assumes a perfectly-known system input, which results in biased FRF estimates in the presence of input noise. This paper derives a simple adjustment that can be used to improve FRF estimation for systems subjected to random excitation with noisy input data. This improvement can be implemented with little modification to standard local modeling algorithms and with little additional computational burden. The adjustment is coupled with a model selection procedure to avoid underfitting and overfitting. The methods presented in this paper are validated on a simulation, and they are shown to reduce bias due to input noise.

A Local Gaussian Process Regression Approach to Frequency Response Function Estimation

Frequency response function (FRF) estimation is a classical subject in system Identification. In the past two decades, there have been remarkable advances in developing local methods for this subject, e.g., the local polynomial method, local rational method, and iterative local rational method. The recent concentrations for local methods are two issues: the model order selection and the Identification of lightly damped systems. To address these two issues, we propose a new local method called local Gaussian process regression (LGPR). We show that the frequency response function locally is either analytic or resonant, and this prior knowledge can be embedded into a kernel-based regularized estimate through a dot-product kernel plus a resonance kernel induced by a second-order resonant system. The LGPR provides a new route to tackle the aforementioned issues. In the numerical simulations, the LGPR shows the best FRF estimation accuracy compared with the existing local methods, and moreover, the LGPR is more robust with respect to sample size and noise level.

An Efficient Implementation for Regularized Frequency Response Function and Transient Estimation

In this paper, we propose an efficient implementation for the kernel-based regularized frequency response function (FRF) and transient estimation. In particular, we show that when the diagonal correlated (DC) kernel is used, the output kernel matrix in the frequency domain has a hierarchically of-diagonal low-rank (HODLR) structure, and by exploring this structure, the FRF and transient estimates can be computed very efficiently The overall computational complexity is O(y2Nb) log2Nb), where Nb is the number of data in the interested frequency band and 7 is the HODLR rank of the output kernel matrix that is often much smaller than N&. The efficacy of the proposed implementation is demonstrated by numerical simulations.

Empirical Transfer Function Estimation With Differential Filtering and Its Application to Fine Positioning Control of Galvano Scanner

Fast and precise point-to-point (PTP) positioning control is a critical technology for improving the throughput and product quality of industrial machines. This study aims to investigate the frequency response function (FRF) estimation method with differential filtering (ETFE-Diff: empirical transfer function estimation with differential filtering) that uses the input and output signals of PTP motion during processing operation and to apply ETFE-Diff to a feedforward (FF) compensation adjustment. In this study, the principle of ETFE-Diff for FRF estimation was clarified, compared to the existing FRF estimation methods, i.e., empirical transfer function estimation and local rational modeling. Moreover, the effectiveness of ETFE-Diff for FF compensation adjustment in improving positioning accuracy of a galvano scanner as an industrial servo system was experimentally demonstrated.

Development of automated impact system for modal analysis of micro-end mill

The micro cutting tool used in micromilling process is highly susceptible to instability attributed to its low strength. A catastrophic failure of micro-end mill can occur if the chatter is not controlled during micromilling process, especially for micromilling of difficult-to-machine materials, like Ti and Ni alloys. Consequently, the control of chatter in micromilling process is necessary to manufacture the components with required dimensional and geometrical accuracy. Chatter can be avoided by selecting the stable process parameters from stability lobe diagram or by changing the process parameters during machining process by chatter onset identification. The both ways of chatter avoidance require the reliable identification of the modal parameters of the micro-end mill. Experimental modal analysis is most accurate method for identification of modal parameters owing to its ability of consideration of machine tool compliance. Hence, in the present work, automated impact system has been developed to impact the micro-end mill under both stationary and rotating conditions. The developed impact system has been found to be impacting the micro-end with maximum deviation of 2.6% in force amplitude due to repeated impacts. A maximum variation of 1.7% has been found in the natural frequency due to different impacts of the micro-end mill under both stationary and rotating conditions. The accurate estimation of frequency response function is expected to predict the stability of micromilling process with high reliability.

A two-stage Polynomial Chaos Expansion application for bound estimation of uncertain FRFs

Polynomial Chaos Expansion (PCE) is a method for analysing uncertain vibratory structures with lower computational effort. It may simply be described as a curve fitting method with orthogonal basis terms, where the polynomial type, dimension and order are predefined for the uncertain responses. However, the polynomial order in PCE must be very high to accurately estimate statistical moments of the frequency response function in resonance regions of lightly damped and uncertain structures. To solve this issue different transformation techniques are reported in the literature, where implementations of PCE produce higher accuracy with a lower order polynomial. However, these transformations lose the attraction for using PCE, since they require some additional mathematical operations and, mostly, they present high accuracy if the higher orders of polynomials are again of interest. In this study, an efficient approach is presented for the upper bound estimation of the uncertain frequency response functions (FRFs) via PCE with lower order terms without performing any transformation. Rather than one-stage application of PCE for the desired response of an uncertain problem, the approach comprises a two-stage application of the classical PCE, i.e. first for the natural frequencies and then for the FRF calculations. As an example application of the approach, a thin beam for two different uncertainty cases is considered, namely local and global uncertainty. The local and global input uncertainties are generated by variability of lumped masses added at the boundary and Young's modulus, respectively. The FRF bounds are compared with extensive experimental and numerical Monte Carlo simulations, showing that low order polynomials are sufficient to calculate the bounds accurately with the technique described.

Virtual sensor for monitoring the geometric and kinematic accuracy of machine tools

The performance of machine tools depends on their working accuracy. The paper presents a new approach for in-process monitoring the geometric and kinematic accuracy of a machine tools. This approach utilizes a virtual sensor for capturing displacements at the tool center point (TCP). The virtual sensor is set up by using a data-driven approach based on extended transmissibility functions. The input data for monitoring are the NC internal data. The transmissibility functions are established via standard experimental procedures such as double ball bar (DBB) tests tests and estimation of frequency response functions (FRF). The data from the virtual sensor can be used for process monitoring and troubleshooting when launching a new process.

A Wavelet-Based Approach to FRF Identification From Incomplete Data

Frequency Response Function (FRF) estimation from measured data is an essential step in the design, the control, and the analysis of complex dynamical systems, including thermal and motion systems. Especially for systems that require long measurement time, missing samples in the data record, e.g., due to measurement interruptions, often occur. The aim of this paper is to achieve accurate identification of non-parametric FRF models of periodically excited systems from noisy output measurements with missing samples. An identification framework is established that exploits a wavelet-based transform to separate the effect of the missing samples in the time domain from the system characteristics in the frequency domain. The framework encompasses both a time-invariant and a time-varying wavelet-based estimator, which provide different mechanisms to address the missing samples. Experimental results from a thermo-dynamical system confirm that the estimators enable accurate identification.

Data-driven fractional order feedback and model-less feedforward control of a XY reluctance-actuated micropositioning stage.

This paper proposes a compound data-driven control method to solve the problems of low damping resonance, different dynamic properties, and hysteresis in the large-range compliant micropositioning stage driven by a Maxwell reluctance actuator. First, in order to verify the proposed control algorithm, a reluctance-actuated, XY compliant micropositioning stage is constructed according to the principle of operation of a reluctance actuator. Second, in order to eliminate the influence of complex dynamics on the controller design, a fractional order proportional-integral feedback controller is designed using a data iterative feedback turning algorithm. Third, the finite impulse response feedforward filter is optimized using experimental data, and the on-line inverse estimation of the system frequency response function and its iterative feedforward compensation are carried out to further eliminate the influence of light damping resonance. Finally, the proposed control method is used for tracking the experiment and compared with other methods. The experimental results show that the proposed control method can better meet the requirements of high precision, fast speed, and strong anti-interference ability for large stroke micro/nanopositioning and tracking.

Vehicle parameter identification based on vehicle frequency response function

Accurate vehicle parameter information plays an important role in assessing the conditions of roads and bridges, along with the corresponding maintenance. This study considered a vehicle parameter identification method based on a vehicle frequency response function (FRF). First, the vehicle FRF was deduced with respect to the displacements of the vehicle-road contact points, thereby building the relationships among the FRF, vehicle responses, and road profile in the frequency domain. Next, using the responses of vehicles passing over on-road bumps of known size, a direct estimation of the vehicle FRF was described. Then, a combination of Tikhonov regularization and a shape function method was used to update the estimated vehicle FRF by removing the singular data owing to the direct computation of the vehicle FRF. Subsequently, the modifying factors of the vehicle parameters were iteratively identified based on a sensitivity analysis of the estimated FRF to the vehicle parameters. A numerical simulation for vehicle parameter identification was performed to test the effectiveness of the proposed methods, considering a 5% Gaussian noise pollution and the influences of different driving speeds. At last, field tests of a vehicle passing over bumps were performed for the verification of vehicle parameter identification.

The influence of shock absorber characteristics’ nonlinearities on suspension Frequency Response Function estimation and possibilities of simplified characteristics modelling

The paper shortly presents shock absorber design evolution and resulting achieved characteristics. The way in which suspension performance is evaluated is described giving information about models used for suspension parameter tuning during simulation testing of suspension transmissibility (FRF - Frequency Response Function) for most important suspension assessment criteria. More detailed information about models of shock absorber (damper) nonlinearities of characteristics allows for description of methods of linear and nonlinear suspension models FRF estimation. Testing linear suspension model is possible with the use of analytical transfer function formulas which were used to verify methods for estimation FRF using estimated power spectral density functions of excitation and response signals. Designing appropriate input signal allowing to get useful response signals was necessary to for the success of this research. Proposed FRF estimation method was used for linear estimation of nonlinear suspension for a given range of working condition. It was demonstrated that there is no single value of a damping coefficient which would make the linear model responses similar to the responses of the nonlinear one. Then the bilinear model was proposed, giving good damper static nonlinear characteristic.

A frequency domain approach for local module identification in dynamic networks

In classical approaches of dynamic network identification, in order to identify a system (module) embedded in a dynamic network, one has to formulate a Multi-input-Single-output (MISO) identification problem that requires identification of a parametric model for all the modules constituting the MISO setup including (possibly) the noise model, and determine their model order. This requirement leads to model order selection steps for modules that are of no interest to the experimenter which increases the computational complexity for large-sized networks. Also, identification using a parametric noise model (like BJ method) can suffer from local minima, however neglecting the noise model has its impact on the variance of the estimates. In this paper, we provide a two-step identification approach to avoid these issues such that the complexity of the problem is independent of the complexity of the network. The first step involves performing a non-parametric indirect approach for a MISO identification problem to get the non-parametric frequency response function (FRF) estimates and its variance as a function of frequency. In the second step, the estimated non-parametric FRF of the target module is smoothed using a parametric frequency domain estimator with the estimated variance from the previous step as the non-parametric noise model. The developed approach is practical with weak assumptions on noise, uses the available toolbox, requires a parametric model only for the target module of interest, and uses a non-parametric noise model to reduce the variance of the estimates. Numerical simulations illustrate the potentials of the introduced method in comparison with the classical identification methods.

First Results on Modelling of a Plate Heat Exchanger of a District Heating System

This paper describes the first results of the modelling of a plate heat exchanger which is installed in the VUB district heating network. A properly controlled energy exchange through the heat exchanger improves the economy of the entire system, minimizes the pollutant emissions and fossil fuel consumption, in accordance with the main goals of the European energy plan till 2030. To achieve these goals, an accurate model of the plate heat exchanger is required. In this article two approaches are considered to accurately model the heat exchanger: a physics-based white-box and a data-based black-box modelling technique. The white-box modelling approach uses the unsteady energy balance and heat transfer equations to predict the temperature evolutions in the plate heat exchanger. For the (data-driven) black-box modelling approach, there are three interrelated steps. Step 1 is the (semi-automatic) preprocessing of the measurements and the building of a nonparametric frequency response function (FRF) model. Step 2 is the building of a linear parametric (state-space) model, based on the FRF estimate. Step 3 is building a polynomial nonlinear state-space model that relies on the original linear state-space model. The proposed black-box method outperforms the white-box techniques in terms of prediction capabilities and computational needs.

Frequency response function estimation techniques and the corresponding coherence functions: A review and update

For a structural dynamics engineer assessing a system’s vibration characteristics, the frequency response function (FRF) is indispensable, irrespective of whether the setup being tested is experimental, numerical or analytical. This paper outlines the different FRF estimation techniques that have been developed over the years. Algorithms that compute an ordinary least squares (OLS) estimate of the FRF, assuming uncorrelated measurement noise to be present either on the output (H1) or the input (H2) signals, have been compared with those that employ total least squares (TLS) equations by making use of an augmented input–output auto and cross power (GFFX/GXFF) matrix at every frequency such as the Hv (using eigenvalue decomposition) and HSVD (using singular value decomposition) algorithms. Another FRF estimation method based upon Cholesky decomposition (HCD) is also discussed. Further discussion has been included of TLS algorithms computing the FRFs for one output at a time, as historically presented in the development of the Hv algorithm, with a case that evaluates the FRF matrix computed for all the outputs simultaneously. The development of the corresponding coherence functions has been presented, highlighting the method dependent paradigms that led to concepts such as virtual coherence and partial coherence while underscoring their equivalence with ordinary and multiple coherence calculations. It has been shown that some of the existing conditioned coherence metrics are inconsistent from an input–output standpoint, for which the corrected interpretations have been subsequently described. This article is unique in that no previous work summarizes all of the previously developed FRF estimation and coherence algorithms and no previous paper explains the inconsistencies in the conditioned coherence functions with respect to multiple coherence.

Diagnostics of Turbine Blades, Based on Estimation of Frequency Response Function

Abstract The aim of the study is to propose a diagnostics method based on blade’s vibration model in frequency domain. Amplitude spectrum A(ω) and phase spectrum φ(ω) have been experimentally identified which define transfer function G(jω) = P(ω) + jQ(ω) and their real P(ω) and imaginary Q(ω) parts. Transfer function G(jω) and resulting frequency, amplitude, phase characteristics and real P(ω) and imaginary Q(ω) parts are parts of blade’s diagnostics model. Changes in those characteristics are directly related to changes in the technical condition of the blades. It has been noticed that small changes in the technical condition cause large changes in frequency characteristics, which proves their usefulness in the diagnosis process.

Error model-based iterative compensation method for improving the estimation of frequency response function

For the estimation of frequency response function (FRF) using the window-based spectral estimation method, the error models of the FRF estimation are introduced, including window interpolation error and transient error models, and an error model-based iterative compensation (EMIC) method is then proposed to improve the FRF estimation accuracy. The specific iteration steps of the proposed EMIC method are presented, fast computations are adopted in the steps as far as possible; and that, iteration convergence and noise uncertainty are analyzed. The analyses reveal that, the proposed EMIC method is almost always convergent as long as the error of the FRF pre-estimation is small, intercepting enough data and using a window which has smaller and fewer side DFT bins as well as smaller front-end value for FRF estimation can result in better iteration convergence; whereas, the proposed EMIC method will slightly increase the noise uncertainty of FRF estimation compared with that without iterative compensation because it removes the window interpolation effects on noise just like removing the error. To verify the effectiveness and practicability of the proposed EMIC method, model simulation and actual sensor data verification are implemented, and multi existing methods are employed for comparison. The results show that, the proposed EMIC method can largely decrease the error of the FRF estimated by the window-based spectral estimation method, it has the best maximum error restraining capabilities especially in the resonance frequency band and relatively better comprehensive error restraining capabilities especially when the data is not long enough compared with the mentioned existing methods, also, the time consumption of the proposed EMIC method is acceptable and is usually seconds or less; and that, the theoretical analyses, model simulation results and actual data verification results are consistent with each other. Consequently, the proposed EMIC method is proven to be effective and practical for improving the FRF estimation of a stable LTI system from its step or impulse response data.

An expansion technique for the estimation of unmeasured rotational frequency response functions

Often numerical and measured data are not compatible both in type and number of degrees of freedom (DOFs). While finite element modelling can provide a full data set, the conventional vibration testing is usually limited, allowing only the measurement of part of the receptance matrix, namely the responses at translational DOFs due to applied forces. Therefore, the experimental frequency response function (FRF) matrix usually does not contain any information regarding rotational DOFs, especially those related to pure moment excitations. In this work, the authors propose a process that starts with a set of translational FRFs, conventionally measured, and expand it to estimate the entire receptance matrix. A generalized version of Kidder’s method, together with the principle of reciprocity, is used in the expansion process, which has the advantage of avoiding modal identification. The accuracy of the estimated FRFs is assessed by comparing the results with those of experimental data collected at a simple structure. It is worth mentioning that the use of a T-block is considered to obtain experimental rotational FRFs with the simple purpose of accuracy assessment.