New Coherence Function with Measurements of One Sampling Period

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.

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.

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.

A metric on the similarity between two frequency response functions

An explicit bi-variate stochastic model of the El Nino - Southern Oscillation system (ENSO) is built for the first time directly from observations of the Southern Oscillation Index (SOI) and sea surface temperature (SST) in the Nino region 3.4 (1876-2011). According to the resulting stochastic difference equation, the system remembers its past for two years, has the eigenfrequency 0.24 year -1 and the damping factors 0.5 and 0.05. Statistics of SST variations, including properties of El Nino and La Nina, are also estimated through a Monte Carlo experiment. The time-domain model is used to describe ENSO in the frequency domain, including maximum entropy estimates of spectra, coherent spectra, coherence function, and frequency response function. The spectra are peaked at 0.22 year -1 ; the coherence increases from 0.6 at the lowest frequencies to over 0.9 between 0.18 year -1 and 0.33 year -1 exceeding 0.8 at all frequencies above 0.12 year -1 . This means that SST and SOI determine each other by up to 85% everywhere but at the lowest frequencies. The gain factor is peaked at 0.24 year -1 while the phase factor's magnitude is close to π everywhere. Therefore, the statistical predictability of the bi-variate ENSO time series does not improve much over the scalar case predictability.

Active vibration control is an effective method for vibration control of a helicopter fuselage. However, when there are large dynamic modeling errors of a control system, the effectiveness of active vibration control may decrease, or even the control may diverge by using the existing harmonic control algorithms. Large modeling errors will result in inaccurate frequency response functions in the control system. In this paper, a new frequency response function updating-harmonic identification algorithm and its convergence condition have been proposed for effective active vibration control of a helicopter fuselage with large modeling errors through identifying the frequency response function of the control system in real time. In addition, to improve the accuracy of identifying frequency response function and further improve the effect of active vibration control, a dynamic harmonic control weight matrix algorithm is also proposed in this paper. The active vibration control simulations and experimental validation on a dynamically similar and geometrically scaled structure of a helicopter fuselage verify that the proposed algorithm in this paper can achieve faster speed of convergence and better effect of vibration control when there are large dynamic modeling errors over the existing harmonic control algorithms.

In modal testing, the measured Frequency Response Functions (FRFs) are often affected by transducer mass effects. Especially in Multi-Input-Multi-Output (MIMO) modal test, multiple sensors are employed and the effects of additional masses are more significant. This paper deal with removing masses effects of multiple sensors from the measured FRFs. The proposed method offers some advantages over the available techniques in that extra FRFs measurements with different configurations are not required during the correction process. First, a general correction formulation is derived theoretically. Then, validations of the proposed methods are demonstrated using simulated data. It demonstrates a good performance in numerical simulation. To investigate the performance of the proposed method in practical application, further simulations are presented by employing noise-polluted data. And it is shown that the accuracy of correction results will be affected to some extent by the noise. It is thus suggested that the measured FRFs be preprocessed using the curve-fitting procedure or noise reduction processing before applying the proposed method. Since the FRFs measurements required for corrections refer to different excitation points, it requires moving the exciter to different locations. This is usually easy to achieve in hammer impact testing. However, moving the exciter is usually inconvenient in shaker modal testing. Further work is necessary to extend this proposed method so as to make it also applicable for shaker modal test case.

The frequency response function (FRF) method has been well used to determine the worst spindle speeds and their critical limiting chip width for turning operation by finding the maximum negative real part of the FRF. In this study, a modified FRF concept is adapted for a 2 DOF milling system of planar isotropic dynamics to determine the worst spindle speeds and the critical limiting axial depth of cut in explicit, analytic formulas. Analogous to the formulation of worst spindle speeds, similar expression for the best spindle speeds is also obtained. The modified FRF is obtained by multiplying the original FRF of the structure with a complex scaling factor, corresponding to a scaling and a rotation of its original Nyquist plot. The scaling factor is determined analytically from the system characteristic equation with the radial cutting constant and radial immersion angle as the major system parameters. Through the presented method, it is also shown that the worst spindle speeds for a milling operation can be found without the prior knowledge of modal dynamics and stability lobe diagram. The proposed analytical expressions are confirmed by the existing stability models and experimentally verified.

<p><em>The main objective of this study is to detect gearbox fault signals through vibration and noise signal analysis. Experimental tests were carried out using several sets of Healthy and faulty Delrin gears under different speeds and different loads. Initially, Healthy Delrin gears were assembled in the gearbox test rig and signals were sensed using accelerometer and microphone and recorded using data acquisition systems (DAQ) to know the standard behavior of normal gears. Then, the Healthy Delrin gears were replaced by faulty Delrin gears and signals were recorded for each one of the cases separately under the same condition. After that, five proposed techniques of measuring vibrations and noise, namely; Time domain analysis, Frequency Response Function(FRF), Fast Fourier Transform(FFT), Angular domain analysis, and Octave Band Analysis</em> <em>are employed to evaluate the gearbox defects. As of experimental results, it was establish that the vibration and noise signals play a significant role in evaluation of defects in the gearbox.</em> <em>Finally, it is found that the vibration amplitudes of the gearbox are increased resulting in increased tooth breakage under different speeds and different loads.</em></p>