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

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.

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.

This paper provides data-driven tools to detect and quantify approximately the influence of the time variation of a system under test in classical frequency response function (FRF) measurements. To achieve this, the best linear time-invariant approximation of a linear time-varying system is defined and is estimated using existing FRF estimators. An analysis of the residuals of the latter estimation reveals the frequency band in which the contributions from the time variation dominates the disturbing measurement noise and, thus, is significant. All concepts are illustrated on a simulation and real measurement examples.

Indexes for performance evaluation of cameras applied to dynamic measurements

In modal testing, the force transducer and accelerometer mounted on the test structure will introduce extra mass effects to the system and then affect the measured Frequency Response Functions (FRFs). This paper proposed a method for assessing sensor mass effects on the measured FRFs in shaker modal testing. The assessment method offers some distinct advantages where very few FRFs measurements are required, and more importantly it does not require calculations involving several measured FRFs, hence avoiding further contaminations of the measured data. In view of two different ways of response measurements, two cases have been discussed: (1) shaker + laser Doppler vibrometer and (2) shaker + accelerometer. In case (1), only force transducer mass need be assessed. In case 2, however, both force transducer and accelerometer masses should be considered. Especially in Case 2, the overall as well as the individual mass effects of the two transducers were assessed. It is found that the assessment method is quite effective in the experimental validation for Case 1. A simple numerical example for Case 2 presented also illustrates the good theoretical performance of the method. The same example is extended to incorporate simulated noise, simulating an experimental situation, and it is shown that the accuracy of assessment results will be affected to some extent by the noise. It is suggested that the measured FRFs be preprocessed using the curve-fitting procedure before applying the proposed method.

Elimination of transducer mass loading effects from frequency response functions

The identification of the rigid body properties of a structure is an important matter in various structural dynamic applications, namely in structural modification (coupling/uncoupling), optimization and vibration control. In most situations, the experimental route is the only via to obtain the desired dynamic properties. This goal is achieved by an inverse process which starts from the measurement of Frequency Response Functions (FRFs) and ends on matrices describing the mass distribution of a rigid body.The objective of the present article is the evaluation and comparison of the performance of three different kinds of identification methods that share a common characteristic – they all use FRFs as inputs to the inverse process. One of them, the Inertia Restraint Method (IRM) makes use of the accelerance FRF (information condensed at zero frequency), while the other two methods make use of receptance FRFs as inputs, although in different manners; in one of them the properties are evaluated directly from the measurement data – Direct Physical Parameter Identification Method (DPPIM) – whereas the other type requires a modal identification procedure as an intermediate step – Modal Method (MM). An actual experimental structure forms the case study to illustrate the performance of the three different techniques.

In modal testing, the measured frequency response functions (FRFs) are often affected by transducer mass effects. Especially when multiple sensors are employed, transducers mass effects are more significant. This paper deals with removing sensors mass effects from the measured FRFs in hammer impact testing. 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. Firstly, correction method for single accelerometer case is discussed. Further, general correction formulation for multiple accelerometers case is derived. Then, validations of the proposed method are demonstrated using simulated and experimental data. It shows good effectiveness in experimental validation for single accelerometer case. Numerical simulation also demonstrates a good performance for multiple accelerometers case. However, further simulations with noisy-polluted data show that the accuracy of correction results will be affected to some extent by the noise. It is suggested that the measured FRFs be preprocessed using the curve-fitting procedure or noise reduction processing before applying the proposed method.

In many engineering applications the level of nonlinear distortions in frequency response function (FRF) measurements is quantified using specially designed periodic excitation signals called random phase multisines and periodic noise. The technique is based on the concept of the best linear approximation (BLA) and it allows one to check the validity of the linear framework with a simple experiment. Although the classical BLA theory can handle measurement noise only, in most applications the noise generated by the system -- called process noise -- is the dominant noise source. Therefore, there is a need to extend the existing BLA theory to the process noise case. In this paper we study in detail the impact of the process noise on the BLA of nonlinear continuous-time systems operating in a closed loop. It is shown that the existing nonparametric estimation methods for detecting and quantifying the level of nonlinear distortions in FRF measurements are still applicable in the presence of process noise. All results are also valid for discrete-time systems and systems operating in open loop.

Estimation of uncorrelated content in experimentally measured frequency response functions using three measurement channels

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.

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.

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.

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

Acording to the fact that the finite element model of electromagnetic vibration shaker for virtual experiment is not accurate enough to complete accurately spacecraft test, made a correlation analysis of the finite element output frequency response function and the measured frequency response function by their correlation coefficients. Analyzed the sensitivity of the materials for FRF and screened the parameters to update, made the correlation coefficient error of electromagnetic vibration shaker finite element model frequency response function and the measured as the optimization objective, the optimization and modification of shaker finite element model parameters were completed by iteration method. The frequency response function of the modified finite element model approximately agreed with the experimental frequency response function. It met the virtual experiments of electromagnetic vibration shaker.