Estimated Frequency Response Function Research Articles

Turbulent Airwake Estimation from Helicopter–Ship Wind-Tunnel Data

This paper presents a stochastic approach for modeling the turbulent airwake suitable for real-time simulation of the helicopter–ship dynamic interface. This approach relies on the measurements of unsteady loads collected during a wind-tunnel test campaign with a scaled helicopter operating over the deck of simple frigate shape 1. Power spectral densities of the measured aerodynamic loads combined with the estimated frequency response functions are used to find, through an optimization algorithm, a model of airwake spectra over the range of frequencies which mainly affects the pilot workload during shipboard operations. Then, a set of autoregressive filters is designed for every particular rotor position and wind-over-deck condition, so that when driven by white noise, the spectrum of the output will reproduce those obtained from the optimization. This approach is applied to three different tested wind directions and three rotor positions by implementing the autoregressive filters into the multibody model of the experimental rotor. Frequency response analysis of the aerodynamic loads demonstrates that the turbulent airwake model obtained from the experimental data can predict the unsteadiness of loads comparable to those measured in the wind tunnel across the bandwidth of interest for pilot activities. The identified airwake models could be applied to a full-scale model to simulate the unsteady loads effectively experienced by the helicopter during a ship landing flight.

Estimating frequency response functions by using transient responses of systems to simple excitations

This study develops an efficient method using input–output data on estimating non-parametric and then parametric frequency response functions (FRFs) associated with a dynamic system. Contrary to most existing FRF estimation methods that have been theoretically based on the steady-state or stationary responses, the proposed method uses the transient responses that are induced by simple, such as sinusoidal or two-exponential, excitations instead. Its numerical procedure for obtaining a non-parametric FRF is nothing more than solving a set of linear equations in which the unknown variables are individual components of the sought FRF. Furthermore, from the non-parametric FRF, its corresponding poles and residues are extracted through a procedure that involves a usage of the inverse discrete Fourier transform and the Prony-SS method; afterwards, a partial fraction FRF in terms of poles and residues can be achieved. Three numerical examples, including computer simulations and lab experiment, are provided to demonstrate the superior performance of the developed method.

Accuracy improvement method for dynamic substructuring models in vehicle systems

This study proposes an improved dynamic substructuring model using the estimated frequency response function information at coupling points between substructures. An assembled system generally consists of two or more substructures connected by a bolt. Individual substructure evaluation excluding the effects of other components is important in the development stage of a general mechanical system because the vibro-acoustic performance of the system depends on the specific combination of substructures. Therefore, this study predicted the final coupling system performance using information from the initial evaluation of the individual substructures. Accurate measurements of the joint properties are required to accurately estimate the dynamic assembled system characteristics; however, physical constraints typically limit such measurements at actual coupling points. Accordingly, a method that utilizes generalized coupling properties to estimate the dynamic characteristics of a new coupling system based on the characteristics of an original substructure is proposed. Virtual point transformation is then used to estimate accurate frequency response functions at the coupling points of the assembled system based on convenient measurements. The proposed method was validated using a vehicle suspension that was hard mounted in a test jig and onto an actual vehicle body to estimate the vibration characteristics of the assembled system. The findings of this study contribute to the accurate estimation of the dynamic properties of many real-world bolt-assembled systems.

An inconsistent phase selection assessment for harmonic peaks elimination in operational modal testing

An improvement on modal analysis technique is always exacerbated by the limitation on both operational modal analysis (OMA) and experimental modal analysis (EMA). In a recent year, a novel method was introduced named impact-synchronous modal analysis (ISMA) which represents a magnificent achievement in this field. The efficiency of this method as a viable option for EMA and OMA is proven in previous research. However, a quick and straightforward real-time ISMA method is desired as the current procedure is labour-intensive and time-consuming due to the lack of control on the impact timing with respect to phase angle of the disturbances. Thus, the aim of this paper is to identify the significance of phase difference information between acceleration response and cyclic load component in eliminating the disturbances through impact-synchronous time averaging. The paper presented a phase selection assessment, and the results showed that a few averages, (i.e. four averages) are sufficient to filter out the disturbances by 72–80% of dominant periodic response due to cyclic load and over 50% reduction for second harmonic, when the phase angles with respect to the impact are inconsistent for each impact applied. A better modal identification result is obtained through a straightforward way of eliminating the periodic response. Thus, the estimated frequency response function is strongly enhanced and good correlation is observed between modal extraction data and benchmark EMA result.

System Identification of a Base-Isolated Bridge by Ambient and Forced Vibration Tests

Abstract Sciarapotamo Bridge, located in Reggio Calabria Province, Italy, has been recently retrofitted by replacing its deteriorated concrete bridge deck with a composite steel- concrete deck, and isolating the new deck with eight high damping rubber, and four multi-directional sliding bearings. Full-scale ambient and forced vibration dynamic commissioning tests were performed on the bridge just before the bridge has become operational. Three different output-only system identification methods with ambient vibration measurements were used for modal parameter identification, namely: (1) multiple reference natural excitation technique in conjunction with eigen-system realization algorithm, (2) enhanced frequency domain decomposition, and (3) data driven stochastic subspace identification methods. Modal parameter estimation results using different output-only identification methods were presented and compared within the paper. In addition, relatively higher level forced vibration test data was used for estimating frequency response functions, identifying isolated natural frequencies, and damping ratios of the retrofitted bridge in as-built condition. Estimated damping coefficients under different excitation levels are of great value, showing the total damping in the system at low level ambient and relatively higher level forced vibration tests. Experimentally obtained modal parameters of the bridge set the undamaged benchmark state of the bridge for future potential condition assessment studies.

Issues with Combining Road Elevation Spectral Models and Vehicle Vibration Response to Estimate Vehicle Dynamic Characteristics

The research presented in this paper focuses on issues associated with the development of an experimental technique to estimate the dynamic characteristics of wheeled vehicles (namely, the frequency response function) using only in-service response data. To validate the approach and eliminate complexities associated with multi-wheel vehicles, a single-wheeled prototype vehicle was designed and commissioned. The vertical vibration acceleration of the prototype vehicle's sprung mass was measured during normal operation. The power spectral density function was computed and used to estimate the frequency response function of the vehicle. A number of experiments using various configurations of the single-wheeled prototype vehicle were undertaken, along with a series of vibration table experiments to provide a comparison with the estimated frequency response functions. The results show that the best estimate of the frequency response function using the vehicle response data provides reasonable agreement with the measured laboratory experiments when the value of the slope of the spectral function is not set to the value suggested by the International Organisation for Standardisation. Another technique was further developed to estimate the value of the pavement spectral slope using only in-service response data; however, this technique does not yield consistent and accurate estimates. Interestingly, the main resonance of the vehicle is in agreement between the vibration table and response data around the sprung mass of all three vehicle configurations when inspected using linear scales (regardless of the variation in the spectral shape of the excitation), although the additional modes (including the unsprung mass) differs for all vehicles. Copyright © 2014 John Wiley & Sons, Ltd.

Harmonic and modal frequency discrimination in time domain operational modal analysis

An algorithm of operational modal analysis in time domain is presented for the discrimination between structural modes, harmonic excitations and spurious modes. A classification factor is developed and based on the construction of the damped signal- to- noise ratio which separates the physical modes from the numerical ones. The harmonic excitations are then recognized at very high power indexes and zero damping. Furthermore, after the modal discrimination, the harmonic spectrum and the estimated frequency response function can be derived. The method can be updated in the time domain either by a short time sliding window or a recursive technique to provide the modal monitoring in both stationary and non-stationary process. An industrial application on a hydro-electrical generator is discussed to identify the machine frequencies under operational working conditions.

Driver's Arms' Time-Variant Neuromuscular Admittance During Real Car Test-Track Driving

Attempts to measure and model driver steering behavior have been so far mainly performed with driving simulators and time-invariant techniques. The goal of this paper was to quantify the driver's arms' time-variant admittance in real driving and to provide a range of parametrically fitted values on the estimated frequency response functions. The human arms' neuromuscular (NMS) admittance was estimated by applying torque disturbances on the steering wheel during real car test-track driving. To capture the time-variant behavior, the admittance was estimated using a 1.28-s sliding time window. The results showed that drivers adapt their admittance while cornering, exposing a variant behavior during different corners and driving speeds. The frequency response function (FRF) of the admittance while cornering has the properties of a second-order system. During cornering, drivers have increased stiffness values, whilst in straight driving, the FRFs resemble a second-order system ( -40 dB/decade gain drop; double pole at low frequencies) for low frequencies, with a zero for frequencies above 6 Hz (on average). The FRFs during cornering were parametrically fitted to a second-order inertia-spring-damper model. The fitted parameter values can be used for NMS driver models and motivate the stability analysis of the combined closed-loop driver steering system.

Bias Errors due to Leakage Effects When Estimating Frequency Response Functions

Frequency response functions are often utilized to characterize a system's dynamic response. For a wide range of engineering applications, it is desirable to determine frequency response functions for a system under stochastic excitation. In practice, the measurement data is contaminated by noise and some form of averaging is needed in order to obtain a consistent estimator. With Welch's method, the discrete Fourier transform is used and the data is segmented into smaller blocks so that averaging can be performed when estimating the spectrum. However, this segmentation introduces leakage effects. As a result, the estimated frequency response function suffers from both systematic (bias) and random errors due to leakage. In this paper the bias error in theH1andH2-estimate is studied and a new method is proposed to derive an approximate expression for the relative bias error at the resonance frequency with different window functions. The method is based on using a sum of real exponentials to describe the window's deterministic autocorrelation function. Simple expressions are derived for a rectangular window and a Hanning window. The theoretical expressions are verified with numerical simulations and a very good agreement is found between the results from the proposed bias expressions and the empirical results.

Bias Errors due to Leakage Effects When Estimating Frequency Response Functions

Frequency response functions are often utilized to characterize a system's dynamic response. For a wide range of engineering applications, it is desirable to determine frequency response functions for a system under stochastic excitation. In practice, the measurement data is contaminated by noise and some form of averaging is needed in order to obtain a consistent estimator. With Welch's method, the discrete Fourier transform is used and the data is segmented into smaller blocks so that averaging can be performed when estimating the spectrum. However, this segmentation introduces leakage effects. As a result, the estimated frequency response function suffers from both systematic (bias) and random errors due to leakage. In this paper the bias error in the H1 and H2-estimate is studied and a new method is proposed to derive an approximate expression for the relative bias error at the resonance frequency with different window functions. The method is based on using a sum of real exponentials to describe the window's deterministic autocorrelation function. Simple expressions are derived for a rectangular window and a Hanning window. The theoretical expressions are verified with numerical simulations and a very good agreement is found between the results from the proposed bias expressions and the empirical results.

Improved (non-)parametric identification of dynamic systems excited by periodic signals

The steady state response of a system to a periodic input is still subject to noise transients. For lightly damped systems these noise transients can significantly increase the variance of the estimated frequency response function (FRF). This paper presents a method that suppresses the influence of the noise transients (leakage errors) in nonparametric FRF and noise (co-)variance estimates of dynamic systems excited by periodic signals. The method is based on a local polynomial approximation of the noise leakage errors on the FRF. Compared with the classical approaches, the proposed procedure is more robust and needs less measurement time (two signal periods are sufficient). The theory is supported by simulation and real measurement examples.

Nonlinear Induced Variance of Frequency Response Function Measurements

This paper analyzes the variance of the estimated frequency response function (FRF) GBLA of the best linear approximation GBLA to a nonlinear system that is driven by random excitations. GBLA varies not only due to the disturbing measurement and process noise but also over different realizations of the random excitation because the nonlinear distortions depend on the input realization. It will be shown that the variance expression σGBLA2 that is obtained in the linear framework can also be used to calculate the variance that is induced by the nonlinear distortions. This validates the common engineering practice, where the linear FRF methodology is often used under nonlinear conditions.

Optimal Settings for Measuring Frequency Response Functions With Weighted Overlapped Segment Averaging

This paper investigates the measurement errors involved in estimating frequency response functions (FRFs) - and related quantities such as the coherence function - from weighted overlapped segment averaging, a technique that has become a standard in modern data analyzers due to its computational advantages. Particular attention is paid to leakage errors, for which this technique has frequently been criticized. Our main result is that a half-sine or diff window with about 2/3 overlap achieves the best compromise to reduce leakage errors in the case of stationary random excitations, and this is independently of the system FRF. This conclusion is to be contrasted with the customary habit of using a Hanning window with 1/2 overlap. The same reasoning confirms that a rectangular window without overlap minimizes measurement errors in the special case of multisine excitations. Moreover, practical formulas are provided to the reader for computing the bias and variance of the frequency response and coherence functions in the general case.

Modeling Human Multichannel Perception and Control Using Linear Time-Invariant Models

This paper introduces a two-step identification method of human multichannel perception and control. In the first step, frequency response functions are identified using linear time-invariant models. The analytical predictions of bias and variance in the estimated frequency response functions are validated using Monte Carlo simulations of a closed-loop control task and contrasted to a conventional method using Fourier coefficients. For both methods, the analytical predictions are reliable, but the linear time-invariant method has lower bias and variance than Fourier coefficients. It is further shown that the linear time-invariant method is more robust to higher levels of pilot remnant. Finally, both methods were successfully applied to experimental data from closed-loop control tasks with pilots.

Cyclostationarity and the cepstrum for operational modal analysis of MIMO systems—Part II: Obtaining scaled mode shapes through finite element model updating

This paper presents a new technique for scaling mode shapes, obtained from cepstrum-based operational modal analysis (OMA) techniques, such as that described in the companion paper, using finite element model updating. This OMA technique estimated frequency response functions (FRFs) between a cyclostationary input and response measurements. If the input is frequentially white, the resulting FRFs can be obtained up to an overall scaling constant using the in-band poles and zeros identified in the OMA process and employing the response autospectrum as a reference to correct for the effect of out of band modes. In this way, the mode shapes would be scaled correctly relative to each other but would still have arbitrary overall magnitude. If the input is not white, then no reference is available to correct FRF regenerated from in-band poles and zeros, and so these FRFs will exhibit both an overall slope resulting from the effect of out-of-band poles and zeros, and an arbitrary magnitude. This overall slope will differ between measurement locations so even the relative scaling between the mode shapes will be lost. This paper describes a simple technique for recovering both the relative and overall scaling of the FRFs, and hence the mode shapes, based on finite element model updating.

FRF MEASUREMENTS WITH THE WEIGHTED OVERLAPPED SEGMENT AVERAGING TECHNIQUE

This paper investigates the measurement errors involved in estimating frequency response functions from weighted overlapped segment averaging, a technique that has become a standard in modern data analysers due to its computational advantages. A particular attention is paid to leakage errors, for which this technique has been frequently criticised. Our main result is that a Half-sine or Diff window with 67% overlap achieves the best compromise to reduce leakage errors, and this is independently of the system frequency response function. This conclusion is to be contrasted with the customary habit of using a Hanning window with 50% overlap.

APPLICATION OF THE CONDITIONED REVERSE PATH METHOD

Conventional frequency response estimations often give results contaminated by the presence of non-linearities and the extraction of underlying linear system properties is thus difficult. To overcome this problem, the implementation of the Conditioned Reverse Path spectral method to the identification of single- and multi-degree-of-freedom (sdof and mdof) non-linear systems is considered, showing that conventional methods such as ‘H 1’ and ‘H 2’ lead to estimated frequency response functions which are often completely inadequate, even in absence of measurement noise. This spectral method allows to estimate the coefficients of the non-linearities away from the location of the applied excitations and also to identify the linear dynamic compliance matrix when the number of excitations is smaller than the number of response locations. This paper presents some guidelines to successfully apply this method to real non-linear systems; furthermore, the results from an experimental test over a sdof suspension system are shown.

FREQUENCY RESPONSE FUNCTION ESTIMATE AND COHERENCE: PART II

This is but one example of how the looks of the estimated frequency response function can be deceptive. The evaluation of data requires the use of all the plots, the autospectrums of the input and the output, the coherence function and the estimated FRF. In addition, an understanding of how the data was collected, the method used for estimating the FRF and the meaning of all of these plots is needed. In the next installment, we will discuss the use of averaging to acquire quality estimates. There were 5 averages taken to acquire the data seen above. Was this enough or should I have taken more?

Linearity of sound transmission through the human skull in vivo

The linearity of sound propagation through the human skull was investigated. One male subject, equipped with bilateral skin-penetrating titanium fixtures for attachment of bone-anchored hearing aids, was studied thoroughly. Three different methods were used: comparison of the frequency response functions estimated at different signal levels (using stepped sine as well as random noise), comparison of the coherence function at different signal levels (using random noise), and the Hilbert transform of the estimated frequency response function. Frequencies from 0.1 to 10 kHz and signal levels up to 77 dB HL at discrete frequencies were used. No indication of any significant nonlinear behavior was found with the three methods used.

Modal analysis based on multiple input testing 1st Report: Frequency response function estimates using drive signals for exciters.

This paper discusses a methodology for estimating frequency response functions of structures. Firstly, causes of random errors associated with the frequency response function estimates are analyzed. Secondly, a method for improving the accuracy of the estimates is proposed. Finally, that technique is verified experimentally by multiple input testing for an actual structure. Consequently, the following results are obtained. A linear dependence between the input forces at the natural frequencies of the test structure increases the confidence intervals of the estimates. However, that estimating problem is improved by the proposed method in which the drive signals for exciters are considered to be the inputs for the testing system and both of the exciting forces and the response displacements are considered to be the outputs of the testing system.