Measured Frequency Response Function Data Research Articles

Improved denoising methods for smoothing frequency response functions

Measurement errors, including but not limited to noise, persist regardless of the sophistication of the experimental measurement system. Measurements contaminated with noise have proved challenging to work with from a post-processing standpoint, especially when the measurements are used directly in the analytical processing. Noise contamination in frequency response functions limits the accuracy of many applications, such as impedance modeling, requiring the direct use of measured frequency response function data. Existing approaches, such as Modal Parameter Estimation (curve-fitting), that address the de-noising of frequency response functions also suffer from various deficiencies. The errors associated with the estimation process, including modal truncation along with user interaction and interpretation, are often challenging to mitigate.Two alternative methods of frequency response function de-noising were developed in this work. The first method involves a direct approach that smooths measured frequency response functions without Modal Parameter Estimation or significant user interaction. The second approach works to alleviate the truncated effects of unmeasured modes in frequency response function synthesis more effectively than the existing upper and lower residual compensation terms. The proposed frequency response function de-noising methods are validated analytically and experimentally using various component finite element models and experimental tests.

An iterative identification method of joint properties and its applications in an end-toothed connection structure

Accurate identification of mechanical connections plays a significant role in predicting the dynamic behaviors of assembled structures. Traditional identification methods based on frequency response function (FRF) decoupling may not be suitable for the joints with many interface degrees of freedom (DOFs), especially in the case of insufficient measured FRF data and large experimental noise. In this paper, a new iterative method is proposed to solve this problem. First, the incremental basic formulas on the measured DOFs and pseudo-measured DOFs are derived, respectively. The former avoids the measurements of rotational or interface DOFs whereas the latter is suitable for the identification of the joint with many interface DOFs. Then, the joint properties expressed in terms of mass, stiffness and damping matrices are updated from the solution of the linear equation system combined with the iterative basic identification equations at different frequencies. Thus, the original problem is transformed into estimating the parameters iteratively by minimizing the differences between predicted responses and measured ones, which effectively improves the identification accuracy and numerical stability. Moreover, the iterative method is extended to the identification of nonlinear joints based on describing function theory. The validity and superiority of the proposed method are verified by a simple lumped parameter system. Then, a numerical example of a structure connected with bolted joints, which has a large number of interface DOFs, is simulated to verify the convergence and the robustness of the method to noise. Finally, the joint dynamic properties of an end-toothed connection under different contact states are identified, and the assembly responses are also accurately predicted. The effectiveness and the applicability of the proposed iterative identification method to the real structures are well validated.

Prediction of the Physical Properties of a Structural Member by the Impact Hammer Test.

The frequency response function (FRF) in the frequency domain is a black box used to collect physical information and to indicate the modal characteristics of a dynamic system. Analyzing the collected FRF data through the impact hammer test, dynamic parameters, such as stiffness, mass, and the damping matrix, can be estimated. By extracting and analyzing the FRFs within certain ranges of the lowest few resonance frequencies, this study presents a nondestructive method to estimate the dynamic parameters and the material properties. Updating of the dynamic parameters and material properties is a crucial process for the subsequent design and maintenance. This study presents a method to estimate the physical properties of structural members using measured FRF data and generalized inverse. By extracting and analyzing the FRFs within certain ranges of the lowest few resonance frequencies, the dynamic parameters were predicted. It was observed in numerical experiments that the proposed method could properly estimate the elastic modulus and dynamic parameters of steel beams, although the results were affected by the extracted FRF ranges. The physical properties were close to more accurate values in taking the FRFs at more resonance frequencies, as the member was flexible. The proposed method was also extended to a nondestructive test for an estimation of the compressive strength of concrete. However, it faced difficulty due to the external noise contained in the measured data. It was found sin the nondestructive test that the proposed technique was affected by external noise, unlike a simple steel beam. The concrete strength could be predicted by taking the FRFs in a wide frequency range containing the lowest two resonance frequencies and by averaging the repeated test results.

Structural damage detection in plates using a deep neural network–couple sparse coding classification ensemble method

The present study aims at identifying damages in plate structures by applying a pattern recognition–based damage detection technique using the frequency response function. The large number of degrees of freedom is one of the crucial obstacles in the way of accurately identifying damages in plate structures. On the other hand, frequency response functions include many details that dramatically lower the computing speed and enlarge the memory needed for storing data, hampering the application of this method. Furthermore, this study performs principal component analysis as an authoritative feature extraction method with the purpose of reducing the dimensions of the measured frequency response function data and generating distinct feature patterns. Also, because there has been no individual optimal classifier applicable to all problems, an ensemble comprising two powerful classifiers containing couple sparse coding classification and deep neural networks is used to predict the structure damage. This study evaluates the accuracy of damage detection by the proposed method in square-shaped structural plates with the lengths of 1 m and 2 m under different damage scenarios, namely, single and multiple element.

Scheduled Controller Design for Systems With Varying Sensor Configurations: A Frequency-Domain Approach

A novel parameter-dependent scheduled controller design is proposed for systems which exhibit switched dynamics. Switching is a result of varying sensor configurations available in the measurement system. The control architecture is based on a priori designed local linear time-invariant controllers, which are designed using frequency-domain loop-shaping techniques. Moreover, conditions on measured frequency response function data of the plant are provided under which the closed-loop system is input-to-state stable for arbitrarily fast parameter variations based on a generalized version of the circle criterion. By presenting a design and analysis framework for scheduled control systems that is easily implementable in existing control software, and does not require parametric plant models, this paper connects well to the industrial control practice. The effectiveness of the proposed scheduling technique, as a way to improve both transient and steady-state performance, is demonstrated by means of a case study which includes measurement results obtained from an industrial wafer stage system.

Model Updating of a Concrete Beam with Extensive Distributed Damage Using Experimental Frequency Response Function

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.

Practical tuning guide to mixed feedback and feedforward control of soft-mounted vibration isolators

Abstract: This paper proposes a practical tuning guide for mixed feedback and feedforward control design of active soft-mounted vibration isolators. A model is derived based on measured frequency response function data obtained from an industrial vibration isolator. Using this model, feedback and feedforward controllers are designed and evaluated. The mixed-sensor feedback controller uses a combination of relative position feedback and absolute acceleration feedback. By mixing the sensor signals, amplification of sensor noise is minimized. It is shown that feedback control can greatly enhance the disturbance rejection properties of disturbances that act directly on the payload of the machine. In addition, a feedforward controller is designed that improves the rejection of indirect disturbances on the machine, i.e. disturbances that are transmitted via the vibration isolator.

Structural Damage Detection in a Truss Bridge Model Using Fuzzy Clustering and Measured FRF Data Reduced by Principal Component Projection

This study deals with vibration-based damage detection in a truss bridge model and suggests a novel methodology based on fuzzy clustering and measured frequency response function (FRF) data reduced by principal component projection. A six-bay truss bridge model is designed and fabricated in laboratory, various connection damages are simulated by loosening the end connecter bolts, and the environmental effects are taken into account by changing in excitation force levels of a shaker. The FRFs of the healthy and the damaged structure are used as initial data. The FRF data normalization is performed for eliminating the effects caused by the environmental and operational variability. Two data projection algorithms, namely principal component analysis (PCA) and kernel principal component analysis (KPCA) are adopted for data compression and the median values of principal components are defined for damage feature extraction. The fuzzy c-means (FCM) clustering algorithm is used to categorize these features for structural damage detection. The illustrated results show that the proposed method can effectively identify the bridge damages simulated by loosening the bolted joints of the truss bridge structure. It is sensitive to the structural damage but it is non-sensitive to the effect of the environmental and operational variations. This makes it quite generic and permits its potential development for real and complex truss bridges in site.

Embedded sensitivity functions for characterizing structural damage

The use of embedded sensitivities in vibration-based damage identification for structuralhealth monitoring is discussed. Embedded sensitivity functions are algebraic combinationsof measured frequency response function data that determine how the forced responses ofstructural systems change with perturbations in mass, damping, or stiffness. After thetheory of embedded sensitivity functions is reviewed, they are applied to characterizedamage in an analytical three-degree-of-freedom system, a bench top two-degree-of-freedomsystem, and a full-scale exhaust system. By comparing embedded sensitivity functions withfinite difference functions using undamaged and damaged frequency responsefunctions, damage is shown to be properly detected, located, and quantified intheory and practice assuming that structures are only damaged in one location.Demonstrations on the bench top system and exhaust system indicate that thetechnique is most effective when certain frequency ranges in the data are analyzed toavoid ranges of low signal-to-noise ratios and when changes to frequency responsefunctions are not excessive to avoid distortions in the estimated perturbations.

A two-level neural network approach for dynamic FE model updating including damping

This paper presents a two-level neural network scheme for finite element (FE) model updating in which both the structural parameters and the damping ratios are updated. Considering the fact that in a lightly damped system the damping has only negligible influence on the resonance and antiresonance frequencies of the system, in the first-level updating the model is assumed to be free of damping and the structural parameters are updated using the natural and antiresonance frequencies as the response data. With the updated structural parameters from the above first-level updating, the second-level updating procedure deals only with the damping ratios, using the integrals of frequency response function (FRF) as reference responses. For the selection of a proper response configuration, a sensitivity analysis scheme is proposed, taking into account the carry-over error during the first-level updating in addition to the anticipated error in the measured FRF data. Through a numerical example it is shown that the approach is effective and efficient. It is also shown that by means of a noise injection learning the neural network can acquire considerable noise-resisting ability, resulting in about 50% reduction of the errors in the updated parameters as compared to the anticipated errors from the sensitivity analysis.

Structural damage detection using measured FRF data

The underlying principle behind structural damage detection techniques is that vibration signature, e.g. modal properties or frequency response function (FRF) data, is a sensitive indicator of structural physical integrity and thus can be used to detect damage. Since indirectly-measured modal data contain accumulative errors incurred in modal parameter extraction and provide much less information than FRF data, it is more reasonable and reliable to use directly-measured FRF data for structural damage detection. In this paper, a new damage detection algorithm is formulated to utilize an original analytical model and FRF data measured prior and posterior to damage for structural damage detection. Based on nonlinear perturbation equations of FRF data, an algorithm has been derived which can be used to determine a damage vector indicating both location and magnitude of damage from perturbation equations of FRF data. An additional development with respect to the proposed technique is an effective technique introduced for weighting perturbation equations of FRF data at selected locations and frequencies so as to reduce influence of measurement errors on accuracy of damage detection to the minimum. For extension of the proposed algorithm to cases of incomplete measurement in terms of coordinates, an iterative version of the proposed algorithm has been introduced. The validity and applicability of the proposed damage detection algorithm have been demonstrated through numerical and experimental studies on a practical plane 3-bay frame structure, respectively.

New method for accurate and consistent identification of modal parameters

A new method for the estimation of modal parameters is presented. Unlike the majority of existing methods, which involve complicated curvefitting and interpolative procedures, the proposed method calculates the modal parameters by solving the eigenvalue problem of an equivalent eigensystem derived from measured frequency response function (FRF) data. It is developed based on the practical assumption that only one incomplete column of the FRF matrix of the test structure has been measured in a frequency range of interest. All measured FRFs are used simultaneously to construct the equivalent eigensystem matrices from which natural frequencies, damping loss factors, and mode shape vectors of interest are solved. Because the identification problem is reduced to an eigenvalue problem of an equivalent system, natural frequencies and damping loss factors identified are consistent. Further procedures for normalizing the identified eigenvectors so that they become mass normalized are developed. Applications of the method to both numerically simulated and practically measured FRF data are given to demonstrate the practicality of the proposed method, and the results obtained are indeed very promising.

Identification of Damping Characteristics of Viscoelastically Damped Structures Using Vibration Test Results

Damping plays an important role in structural stability and vibration control and so frequently special efforts are taken to increase structural damping, of which the most effective measure to date is the viscoelastic damping treatment. However, damping properties, especially those of viscoelastic damping, are the most difficult to model analytically but can only be revealed by vibration test. This paper examines the feasibility of identifying damping characteristics of viscoelastically damped structures whose damping and stiffness characteristics are frequency dependent by correlating finite element modelling with vibration test results and presents an effective method for achieving such an objective. Measured frequency response function data are used instead of conventional modal data, since in the case of viscoelastically damped structures that are frequency dependent modal data are not defined while frequency response functions can be easily measured. The method is developed based on the practical assumption that the measured coordinates are incomplete and the identified damping matrix is physically meaningful and possesses the same physical connectivity as that of the stiffness matrix. Computational aspects regarding the practical application of the proposed method are discussed. Several numerical case studies are given to demonstrate the practical applicability of the proposed method.

Finite element model updating using frequency response function data: II. Case study on a medium-size finite element model

This work reports findings from a case study which is aimed at updating a medium-size finite element model using frequency response function (FRF) data. The numerical stability of the method and its applicability to true measured FRF data were investigated in the case of a plate-beam structure which was modeled using about 500 dof. The initial finite element model of the structure was found to be in fairly good agreement with measured data, a feature which allowed a systematic evaluation of various factors affecting the quality of model updating. After several attempts, it was possible to update the initial FRFs so that they matched the measured ones within engineering accuracy. However, the non-uniqueness of the updated model became apparent since it was possible to find several such models, the FRFs of each of which satisfied the basic matching criterion.

Improving finite element models in the higher frequency range using modified frequency response function sensitivity method

Finite element analysis has been widely used in structural design and analysis. Nevertheless, there usually exist inconsistencies between finite element analysis and dynamic test data. In order to improve finite element models, a number of different methods have been developed in recent years and most of them assume that measured modal data are available. While accurate modal parameters may be obtained from measured frequency response function (FRF) data in the lower frequency range, this becomes very difficult when FRF data measured in the higher frequency range are concerned. For applications related to the prediction of noise, an accurate analytical model in the higher frequency range becomes essential and can only be established by using measured FRF data directly. In this paper, a modified frequency response function sensitivity method is formulated and implemented to the validation of finite element models in the higher frequency range. The method is developed based on the practical assumption that both measured modes and coordinates are incomplete. Numerical case studies are carried out to demonstrate the practical applications of the proposed method.