Prediction of Radiated Structural Noise Using Decomposed Vibro-Acoustic Frequency Response Functions

In industry, the use of blocked forces for the characterisation of structure borne sound sources is now common practice, whereas the sound power is typically used for airborne sound sources. In principle, it should also be possible to use blocked forces (or blocked pressures) to predict airborne sound radiation using a similar approach and measurement methodology as the in-situ blocked force method. This is achieved by discretising the surface of the source into small unit cells, and measuring the vibration velocity whilst operational. In order to convert these velocities to blocked forces, the mobility of the source for each of the positions must also be known: and to predict the radiated sound, a measured vibro-acoustic frequency response function is also required. Typically vibro-acoustic FRFs are measured by exciting the structure with an instrumented hammer or shaker, but they may also be measured reciprocally using a volume velocity source. Described in the paper is an experiment conducted in a fully anechoic chamber to investigate the feasibility of the above approach, i.e. using blocked pressure and reciprocally measured vibro-acoustic FRFs to characterise and predict airborne sound. The method is validated by comparing the directly measured radiated sound pressure to that which is predicted according to the above.

This study proposes an analytical method for building a vibro-acoustic model of a panel-cavity system induced by a turbulent boundary layer (TBL) excitation, which is used to study the influences of boundary conditions on the vibro-acoustic responses of the system. In the theoretic model, the elastic restraints at the plate’s edges and the wall impedance of the cavity wall are considered at the same time. Based on the mode expansion, the vibro-acoustic response function of the vibro-acoustic model is built to present the coupling between the TBL and the system constructed by the panel and the cavity. Then, the model is validated by comparing it with the previous study, which shows good agreements. Finally, the effects of the boundary conditions, such as stiffness, damping, and impedance, on the mean quadratic velocity spectra and radiated sound power of the panel-cavity system are studied. It is demonstrated that the decrease of the boundary stiffness makes the radiated sound power of the panel drop above the frequency of the panel third mode. The boundary damping of the panel decreases the vibro-acoustic response of the panel-cavity system by working as an attenuation factor of the boundary. Furthermore, the increase of the radiated sound power with different wall impedances is due to the coupling between the structural mode of the panel and the acoustic mode of the cavity, or the resonance of acoustic mode in the vibration propagation direction.

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.

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.

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.

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.

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.

High-speed digital image correlation is a camera-based displacement measurement technique that has experienced increasing popularity for modal analysis. It is due to a high spatial density of measurement that provides an outstanding interpretation of mode shapes or operational deflection shapes. However, its sensitivity is typically lower than other techniques and transducers such as accelerometers or laser vibrometry. With these measurements, the estimation of any transfer function in the frequency domain is noisier in valley regions of the response, including anti-resonances. Therefore, an accurate analytical model of the response should be used for modal identification. Modal analysis algorithms perform an identification of the modal parameters and a reconstruction of the response based on the transfer function between force excitation and displacement response. This is called the frequency response function. However, force measurements are not available in many experiments, where the motion excitation is recorded instead. This is the case when the vibration is applied to the specimen through a motion in its base. This transfer function, also known as the transmissibility function, relates the motion of excitation and response, whose shape differs from the frequency response function. This difference in the response function is relevant in a relatively noisy measurement like using digital image correlation. Hence, in this work, a transformation of the transmissibility function into equivalent frequency response function, based on theoretical models, is performed prior to modal identification in base motion tests. More suitable experimental response functions are obtained, increasing the accurateness of the modal parameter estimation.KeywordsModal analysisTransmissibility functionsFull-field measurement

Errors involved in computing impulse response functions via frequency response functions

This paper proposes an estimation method of the loaded static transmission error (STE) waveform by vibration measurement under operating load conditions of a gearbox. The proposed method is based on the understanding that vibration is determined by the loaded STE and frequency response function (FRF) of the gearbox. The loaded STE is calculated by dividing the measured vibration by the FRF. The FRF is derived by continuously combining the measured frequency response curves of mesh fundamental and harmonic frequency components with the calculated FRF by using a dynamic model of the gearbox. The proposed method was performed on a single-stage helical gear vibration test rig. The loaded STE was accurately estimated by using parameter optimization that constructs a dynamic model with a best fit to the measured frequency response curve. The estimated results were compared to the calculated results by tooth contact analysis and showed good agreement. Our findings confirmed that the effectiveness of the proposed method was verified experimentally.

The concept of higher-order frequency response functions(FRFs), which is based on Volterra series expansion of nonlinearfunctions, is used for analysis of kinetics of nonlinear adsorptionsystems. Four different kinetic mechanisms: Langmuir kinetics, filmresistance control, micropore diffusion control and pore-surfacediffusion control were analyzed and the results were compared. It wasshown that, contrary to the linear frequency response characteristicfunctions, the higher-order FRFs corresponding to different mechanismsdiffer in shape. This result offers great potential for theidentification of the adsorption-diffusion mechanism governing theprocess. It is shown that the second order FRFs give sufficientinformation for distinguishing different mechanisms.

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.