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

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

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

Impulse response functions (IRFs) and frequency response functions (FRFs) are bases for modal parameter identification of single-input, single-output (SISO) and multiple-input, multiple-out (MIMO) systems, and the two functions can be transformed from each other using the fast Fourier transform and the inverse fast Fourier transform. An efficient iterative algorithm is developed in this work to directly and accurately calculate the IRFs of SISO and MIMO systems in the time domain using relatively short input and output data series. The iterative algorithm can avoid the time-consuming inversion of a large matrix in the conventional least-square method for calculating an IRF, greatly reducing the computation time. In addition, a fitting index and an error energy decreasing coefficient are introduced to evaluate the accuracy in calculating an IRF and to provide the termination criterion for the iterative algorithm. A new coherence function is also introduced to evaluate the accuracy of calculated IRFs and FRFs at different spectral lines. Two examples are given to illustrate the effectiveness and efficiency of the methodology.

New Coherence Function with Measurements of One Sampling Period

The nonlinear frequency response of a chromatographic column for the adsorption of two dissolved components is analyzed using the concept of higher order frequency response functions (FRFs) which is based on the Volterra series and generalized Fourier transform. By applying this concept a nonlinear model of a system is replaced by an infinite series of the FRFs of the first, second, etc. order. The FRFs up to the third order are derived theoretically starting from the equilibrium-dispersive model, which is used for description of a chromatographic column, and applying the harmonic probing method. The functions that relate outlet concentration changes of each component to the corresponding inlet concentration changes are derived. At the inlet of a chromatographic column, it is considered: (a) the concentration change of one of the components keeping the concentration of the other component constant and (b) the concentration change of both components keeping their ratio constant. The FRFs are calculated numerically for different steady-state concentrations and relative mixture compositions. It has been found that, despite certain differences in initial conditions, the FRFs exhibit similar behavior. For higher frequencies, the amplitudes of the FRFs tend to zero and phases to −∞. In the low frequency range, which is of interest for investigation of equilibrium parameters, these functions have similar behavior, but tend to different asymptotic values. Correlations between coefficients of competitive adsorption isotherms, i.e. partial isotherm derivatives, and the derived FRFs are established. This theoretical result offers the potential to use the analysis of the nonlinear frequency response of a chromatographic column for estimation of competitive adsorption isotherms.

Frequency response functions (FRFs) can be calculated from a composite function of other FRFs composition and decomposition problems. Considering that an experimentally measured FRF has an uncertainty in its magnitude and phase, what would be the uncertainty of the magnitude and phase of an FRF calculated from a composite function of experimentally measured FRFs? This work derives analytical expressions for the uncertainty of magnitude and phase of FRFs calculated from functions composed of the basic arithmetic operations involving measured FRFs and their inherent uncertainties. The derivation of the uncertainty values comes from the application of multivariable error analysis to the basic arithmetic operations considering the real and the imaginary parts of the FRFs involved. The resultant uncertainty of the composite FRF is obtained by applying the methodology to every operation in the composite function. Hence, an analytical method is proposed for applying to both composition and decomposition problems. An example of a decomposition problem is presented, and the calculated uncertainty of the magnitude and the phase of the composite FRF showed good agreement with results obtained from Monte Carlo simulation. In addition, it is shown that the blind application of the root-mean-square rule to the composite function written in terms of magnitude is not an appropriate procedure. The proposed methodology is accurate in predicting the uncertainty of composite FRFs.

Inverse substructuring method has been recently proposed and applied for inverse analysis of the dynamical response of product transport system. The component-level frequency response functions (FRFs) and the coupling dynamic stiffness for facilitating the cushioning packaging design are all predicted from only the system-level FRFs. However, the system-level FRFs from coupling degree of freedoms may not be measured accurately because of the difficulties of vibration excitation and response measurement for the coupled interface between packaged product and vehicle within the limited accessible space. The aim of this paper is to develop a new FRF-based indirect inverse substructuring method for the analysis of the dynamic characteristics of a three-substructure coupled product transport system without measuring system-level FRFs at the coupling degree of freedoms. By enforcing the dynamic equilibrium conditions at the coupling coordinates and the displacement compatibility conditions, a closed-form analytical solution to inverse sub-structuring analysis of multi-substructure coupled product transport system is derived based on the relationship of easy-to-monitor component-level FRFs and the system-level FRFs at the coupling coordinates.. The proposed method is validated by a lumped mass-spring-damper model, and the predicted coupling dynamic stiffness is compared with the direct computation, showing exact agreement. Then, the FRF tests of a physical prototype of multi-substructure coupled product transport system are performed to further check the accuracy of the suggested method. The method developed offers an approach to predict the unknown coupling dynamic stiffness from measured FRFs purely. The proposed method may help to obtain the main controlling factors and contributions from the various structure-borne paths for product transport system. Copyright © 2014 John Wiley & Sons, Ltd.

This paper presents frequency response function (FRF) expansion techniques that minimize the difference between the analytical and predicted FRF matrices to satisfy the FRF constraints. The measured FRF relationships at a small number of locations were used as constraints. The expansion method is useful for estimating the rotational FRFs that are difficult to measure or apply using frequency-based substructuring (FBS) techniques. The validity of the proposed method, including the effects of external noise, was confirmed using numerical examples. An FBS algorithm was also derived by incorporating the FRFs of each substructure, and the compatibility conditions were transformed into FRFs with pseudomasses at the joint nodes. A discrepancy between the FRFs of the synthesized system estimated using the proposed method and the analysis results of the entire system was observed in the numerical example, and the reasons for the discrepancy were investigated and discussed. It is estimated that the proposed FBS method can be enhanced by combining it with other dynamic substructuring techniques and supplementing additional information, such as measured FRFs.

The stability lobe diagrams predicted using the tool frequency response function (FRF) at the idle state usually have discrepancies compared with the actual stability cutting boundary. These discrepancies can be attributed to the effect of spindle rotating on the tool FRFs which are difficult to measure at the rotating state. This paper proposes a new tool FRF identification method without using noncontact sensor for the rotating state of the spindle. In this method, the FRFs with impact applied on smooth rotating tool and vibration response tested on spindle head are measured for two tools of different lengths clamped in spindle–holder assembly. Based on those FRFs, an inverse receptance coupling substructure analysis (RCSA) algorithm is developed to identify the FRFs of spindle–holder–partial tool assembly. A finite-element modeling (FEM) simulation is performed to verify the validity of inverse RCSA algorithm. The tool point FRFs at the spindle rotating state are obtained by coupling the FRFs of the spindle–holder–partial tool and the other partial tool. The effects of spindle rotational speed on tool point FRFs are investigated. The cutting experiment demonstrates that this method can accurately identify the tool point FRFs and predict cutting stability region under spindle rotating state.

Measurement of Frequency Response Functions (FRFs) on the Millennium Bridge in London was a key part in the validation of the retrofit solution for excessive sway vibration caused by walking pedestrians during the bridge opening in June 2000. The measured FRFs were employed to verify the analytical response analysis and to estimate and confirm analytical predictions of natural frequencies, modal damping ratios, mode shapes and modal masses corresponding to various configurations of prototype viscous and tuned mass dampers. These results were used to experimentally assess the effectiveness of the proposed retrofit solution. Although FRF measurement technology is fairly standard in many engineering disciplines, its application on the Millennium Bridge was quite problematic and required some unique solutions which are described in this paper. Modal testing of the Millennium Bridge clearly demonstrated that it was possible to measure good quality FRFs around a frequency as low as 0·5 Hz, which was the natural frequency of the first lateral mode of vibration. This was achieved by designing, constructing and installing a bespoke hydraulic shaker featuring a translatory moving mass of 1000 kg. As the shaker was not able to produce perfectly sinusoidal excitation for stepped-sine testing, FRFs were successfully estimated using the so-called Resolver method in conjunction with an Hc estimator. The standard H1 FRF estimator, which is used in the vast majority of modern digital spectrum analysers nowadays, was demonstrably unable to produce meaningful FRF estimates when using the Resolver approach. The FRF estimation and curve-fitting performed in situ proved to be crucial for the success of the modal testing of the Millennium Bridge. By evaluating the estimates of the experimental FRFs and the corresponding modal properties during the testing programme on site, a number of test errors were spotted. This, in turn, enabled immediate repetition of measurements and/or evaluation of the structural behaviour which would have been impossible if the FRF data analysis had been performed after the testing had been completed.

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

In this study, the influence of inerters on the dynamic characteristics of mass-spring chain systems is systematically investigated through both theoretical analysis and numerical simulations. Specifically, the effects of the inerter coefficient on the natural frequencies, mode shape vectors, and frequency response functions (FRFs) are examined. A novel concept, termed the critical inerter coefficient, is introduced to identify a threshold beyond which the dynamic behavior of a vibration system undergoes fundamental change. At this critical value, both the mode shapes and FRFs exhibit significant deviations from those observed in conventional mass-spring chain systems. Moreover, by introducing a new pole-cancellation mechanism, resonance peaks in the FRFs can be effectively eliminated, leading to essential modifications in the dynamic response. The simulation results further reveal that the combined action of dampers and inerters gives rise to frequency response behaviors that are qualitatively different from those in systems employing traditional damping mechanisms. In particular, not only is the resonance peak suppressed, but distinctive dips or bulges emerge in the FRFs near the natural frequency. Building on these findings, the analysis is extended to an n -degree-of-freedom ( n -DOF) mass-spring chain system incorporating inerters. A theoretical criterion for determining the critical inerter coefficients is derived, providing analytical insights into the influence of inerters on the natural frequencies, mode shape vectors, and FRFs of n -DOF systems. The theoretical predictions are corroborated through numerical simulations, which also uncover the intricate interaction mechanisms between damping and inerters in these systems. This work advances the understanding of how inerters modify the dynamic behavior of vibration systems and offers a theoretical framework for their utilization in structural vibration control and mitigation strategies.

A metric on the similarity between two frequency response functions

Reducing the impact of measurement errors in FRF-based substructure decoupling using a modal model

The flexion-extension motion of the head-neck complex (HNC) in the trunk horizontal (fore-and-aft) vibration was investigated. It was assumed that the motion of the HNC only occurs in mid-sagittal plane. An electro-hydraulic vibrator was used as a source of vibration. The input acceleration and HNC angular velocity were measured whereas the trunk of the seated subject was fixed to the seatback. Accordingly, the acceleration of the vibrator was considered to be equal to that of the trunk. Six subjects took part in the experiment. They were exposed to the random motion at a magnitude of 1.60ms-2rms (root-mean-square) for 50 seconds. The coherence and frequency response function between the HNC angular velocity and the trunk horizontal acceleration were then obtained on 0.5Hz to 10Hz. Averaging the obtained transmissibilities allowed to represent an optimized frequency response corresponding to each of the subjects. It was found that the HNC behavior above 0.8Hz was quasi-linear with a dominant resonance frequency between 0.8Hz and 1.6Hz. The variabilities of the results within and between subjects were investigated. It was further concluded that linear models could describe the HNC so that their orders should be selected in accordance with the frequency response function of the system as well as the objectives of modeling.