Expansion Technique of Frequency Response Function Data and its Applications

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.

Summary An efficient approach to identify multiple cracks in a slender beam using the frequency response function data is presented. It is formulated in a general form from the dynamic stiffness equation of motion for a structure and then applied to a slender beam. The cracks are modeled by rotational springs and the frequency response function is computed based on a spectral element model by the spectral finite element. The procedure gives a linear relationship explicitly between the changes of the measured frequency response function and crack parameters. The inverse problem is solved iteratively for the depths and locations of the cracks through the sensitivity-based model updating. Some numerically simulated tests on beam examples are provided for validating the feasibility of the method to identify the cracks. The results are generally agreement with the target values. Finally, the effect of noise on the damage identification is discussed in the numerical examples.

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.

Sparse polynomial chaos expansions of frequency response functions using stochastic frequency transformation

System equivalent model mixing

Most of the mechanical systems are composed of different subsystems coupled by several links. Any excitation acting on the system is divided into several internal forces which propagate through these links or so called transfer paths. This paper presents the use of experimental transfer path analysis in identifying the transmission paths for a heavy duty truck in order to estimate the vibration and noise transmitted from the cabin and engine mounts. The most challenging part of the TPA analysis is estimation of the internal operational forces so that the total response can be predicted accurately. At the circumstances where direct measurement of the operational forces is impossible, especially for complex structures, a common approach to address the problem is based on a measured frequency response function (FRF) matrix and a set of operational responses. The main problem of this approach is the inversion of the FRF matrices which can be ill-conditioned. Once the internal operational forces are estimated, the vibration or noise response for the selected location in the truck can be calculated. To validate the predicted results, coherence of the collected data and the condition numbers of FRF matrices are investigated so that the accuracy of the predicted results can be quantified for the frequency band of interest. The predictions of the total response are compared with the experimentally measured data such that the coherence and condition number related observations are validated.

Vibration characteristics of a powerplant, an important parameter to evaluate the booming noise and the durability of each parts of an automobile is a major field of simulation analysis using the finite element method (FEM). Overall vibration characteristics, however, depend heavily on the joint properties when it comes to the analysis of complicated structures like a powerplant where a number of parts are incorporated. Among other characteristics, it is extremely difficult to conduct accurate identification of the damping characteristics of the joint area, which had been the result of inaccurate analyses. This paper reports a newly established method to solve these problems through the application of frequency response function (FRF) data and FEM models. Also described is the result of application of the method to the evaluation of a transversely-mounted in-line four-cylinder engine in a front-wheel drive layout. The application revealed that the joint properties can be identified fairly accurately, which in turn allows reliable prediction of vibration level.

In experimental modal analysis, the use of characterized excitation and the choice of window functions are fundamental to the estimation of system frequency response function (FRF) data. It greatly influences the quality of the resulting FRF measurements. Discussed are combination methods of several commonly-used excitation signals with window function, which is relative to lightly damped systems. Study shows that burst random signal with Rectangular works well for lightly damped structures, and pure random with Hanning will bring about higher error. A modal testing of white car body was done. Experimental parameters analysis are given to demonstrate the study result presented above. So it poses a practicable method for improving the accuracy of frequency response function (FRF) measurements of lightly damped structure by using proper excitation signal with appropriate window function.

Damage detection in an offshore platform using incomplete noisy FRF data by a novel Bayesian model updating method

The evaluation of the structural performance and durability using only measurement data without any baseline data is practical. One measurement data type is the frequency response function (FRF) data set, which does not provide enough information about the health state because of the noise included in the data. Proper orthogonal modes (POMs) extracted from the FRF data in a prescribed frequency range are utilized as an index to recognize the existence of damage. The POMs represent the principal axes of inertia formed by the distribution of data on the modal coordinate curve. This work proposes a damage detection method to trace damage based on the POM energy curvature at the element of the finite element model. The validity of the proposed method is illustrated by a numerical experiment and an experimental test.

Linear systems are good approximations of the input-output relationship for many real systems. In practice, ensembles of nominally identical systems can seldom be adequately represented by a single linear system. Some accommodation needs to be made for representing inherent ensemble randomness. One approach would be to use a parameterized model of the system, with the random nature captured in discrete parameters of the parameterized model. Another approach is to use data measured from each system in the ensemble to represent the random nature of the ensemble in a non-parametric form. This is the approach described in this paper. There are several different types of data that could be collected from each system in the ensemble, each type capable of capturing the input-output behavior of the linear system: input-output time domain data, perhaps with specific excitation sequences; Markov parameters; and Frequency Response Functions (FRFs) to name a few. We choose to work with FRF data because many system attributes can be easily interpreted by inspection of the FRF. FRF data can also be used directly for control design. This paper develops a Karhunen-Loeve expansion representation for linear system behavior based on FRF data to develop a compact representation of the uncertainty inherent in an ensemble of systems. This non-parametric, compact, representation of the distribution of linear systems can then be used to characterize the performance and stability of a given feedback control law, as well as for control law design.

Modal parameters, i.e., modal frequencies, damping ratios, and mode shapes of a two disk-shaft system are estimated in Multiple Input/Multiple Output (MIMO) scheme. The response at the output degrees of freedom (dof) of the considered structure due to the excitation at the input dofs is estimated from theoretical analysis. The generated theoretical data is used to estimate the Frequency Response Function (FRF) matrix, which relates the output (vibration) at response points with the input at the excitation points in the frequency domain. The corresponding Impulse Response Function (IRF) matrix, which relates response and excitation in the time domain, is obtained by Inverse Fast Fourier Transform (IFFT) of the FRF matrix. The unified Matrix Polynomial Approach (UMPA) is employed in the frequency and time domains with the estimated FRF and IRF matrices, respectively, to estimate the desired modal parameters. The obtained results are compared with results from Theoretical Modal Analysis (TMA).

A DUAL FITTING ALGORITHM FOR MODAL PARAMETERS IDENTIFICATION

Reduction of large frequency response function data sets using a robust singular value decomposition

A ‘Sub-PolyMAX’ method is proposed in this paper not only for estimating modal parameters, but also for identifying structural flexibility by processing the impact test data in narrow frequency bands. The traditional PolyMAX method obtains denominator polynomial coefficients by minimizing the least square (LS) errors of frequency response function (FRF) estimates over the whole frequency range, but FRF peaks in different structural modes may have different levels of magnitude, which leads to the modal parameters identified for the modes with small FRF peaks being inaccurate. In contrast, the proposed Sub-PolyMAX method implements the LS solver in each subspace of the whole frequency range separately; thus the results identified from a narrow frequency band are not affected by FRF data in other frequency bands. In performing structural identification in narrow frequency bands, not in the whole frequency space, the proposed method has the following merits: (1) it produces accurate modal parameters, even for the modes with very small FRF peaks; (2) it significantly reduces computation cost by reducing the number of frequency lines and the model order in each LS implementation; (3) it accurately identifies structural flexibility from impact test data, from which structural deflection under any static load can be predicted. Numerical and laboratory examples are investigated to verify the effectiveness of the proposed method.