Embedded sensitivity functions for characterizing structural damage

The effectiveness of many structural health monitoring techniques depends on the placement of sensors and the location of input forces. Algorithms for determining optimal sensor and forcing locations typically require data, either simulated or measured, from the damaged structure. Embedded sensitivity functions provide an approach for determining the best available sensor location to detect damage with only data from the healthy structure. In this video and manuscript, the data acquisition procedure and best practices for determining the embedded sensitivity functions of a structure is presented. The frequency response functions used in the calculation of the embedded sensitivity functions are acquired using modal impact testing. Data is acquired and representative results are shown for a residential scale wind turbine blade. Strategies for evaluating the quality of the data being acquired are provided during the demonstration of the data acquisition process.

An experimental verification of a parameter estimation algorithm for finite-element (FE) model updating for damage quantification of a concrete beam with extensive damage is presented using the measured frequency response function (FRF) data. The intact stage of the beam and four progressive damage stages are investigated to update the FE model to quantify the damage across the beam. The method uses a subset of measured FRF data for FE model updating via a quasi-linear sensitivity equation of structural response. A least-squares algorithm with an appropriate weighting technique is used for parameter estimation. Proper selection of excitation frequency ranges for model updating is addressed. Predicted equivalent stiffnesses demonstrate the success and robustness of the method in damage quantification, even when using only a subset of the measured FRF data.

<div class="htmlview paragraph">In the development and manufacture of vehicle components and systems, it is often necessary to quickly identify optimal design modifications for mitigating noise and vibration problems to meet the production schedule. To address this need, experimental techniques for determining the sensitivity of forced vibration response to changes in mass, damping or stiffness properties are of great use. In order to distinguish physical changes in the system from nonlinear input-output distortion, experimental techniques for identifying nonlinear input-output models in mechanical systems are also needed. The use of experimental sensitivity measurements and analyses for studying linear and nonlinear forced vibration data is examined in this work. Embedded sensitivity functions are first used to identify design modifications for reducing a vibration resonant problem. These sensitivity functions are then applied to characterize and identify a nonlinear mechanical system and the accuracy of estimated nonlinear parameters with respect to several factors is examined. A vehicle exhaust system is used to experimentally demonstrate the results achieved using the diagnostic and system identification methods.</div>

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.

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.

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.

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 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 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.

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.

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.

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

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.

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.