Frequency Response Function Matrix Research Articles

Enhancing accuracy and robustness in FBS decoupling through Tikhonov regularization and unit balancing

Frequency-based Substructuring (FBS) enables the prediction of transfer functions for an assembled system based on those of a single-unit system. Conversely, FBS decoupling derives the frequency response function (FRF) of a single-unit system from the assembled system. This approach is valuable when obtaining the transfer functions of a system is challenging due to boundary conditions and deformation conditions. FBS decoupling can enhance accuracy by using additional indicator sensors. However, this often results in significant noise due to the ill-posed FRF matrix inversion. In this paper, Tikhonov regularization is employed to reduce noise-induced errors. Furthermore, this paper addresses the issue of inadequate Tikhonov regularization caused by the inherent scaling differences between the rotational and translational components within the FRF matrix. To rectify this challenge, a unit balancing method is proposed to mitigate the influence of scaling differences. This method enhances the accuracy and reliability of FRF matrix inversion processes, contributing to enhanced noise reduction and robustness in practical applications.

Quantification and prediction of error in MIMO tests

Dynamic testing of systems is most realistic of real-world conditions when multiple input and multiple output (MIMO) techniques are used. To replicate measured environmental conditions, a series of desired outputs (responses) on the system must be realized by inverting the frequency response functions (FRF) matrix for input estimation. System identification is dependent on the number of inputs and outputs, which affect the type of solutions. In input estimation, the auto spectra of the determined inputs are affected by both the auto spectra and cross spectra of the desired outputs. This research evaluates errors and investigates two sources in an experimental setting. The first significant error source is test-to-test variability in FRFs from system identification tests, quantified by magnitude variability from long-duration tests. The second significant error source is the realization of time histories from frequency domain inputs. The spectral content of the realized random process deviates from the desired content and is also quantified. The structure is a linear three-story frame fixed at the base; two inputs are provided uniaxially with the use of suspended electrodynamic shakers. The proposed model for error sources was found to be effective in predicting errors in a SISO (single-input single-output) and a square (2-input 2-output) MIMO test.

An accurate method for identifying hub dynamic loads by condition number of measured FRF matrix on helicopter fuselage

AbstractIt is a simple method to identify the hub dynamic loads of rotor by measuring the vibration responses on helicopter fuselage. However, the identification accuracy of the hub dynamic loads is related to the layout or placement of measuring points on the fuselage. The identification will be inaccurate if the layout of measuring points on the fuselage is unreasonable to result in the “ill-conditioned” frequency response function (FRF) matrix measured on the fuselage. In order to avoid the inaccurate identification due to the “ill-conditioned” measured FRF matrix, an accurate method for identifying the hub dynamic loads of rotor by vibration measurement on helicopter fuselage is proposed in this paper. In the proposed method, the reasonable layout of the measuring points on the fuselage for the “well-conditioned” measured FRF matrix can be obtained according to the condition number of the measured FRF matrix on the fuselage, and then the hub dynamic loads of rotor can be accurately identified. The simulation and experiment of the identification of the hub dynamic loads on a dynamically similar frame structure of a helicopter cockpit floor have verified the effectiveness and accuracy of the proposed method.

Locating structural nonlinearities using linear frequency response functions and nonlinear orthogonal projections

This paper addresses the identification of the spatial location of nonlinearities based on classical modal testing. Interpreting nonlinear systems as linear systems with a nonlinear feedback loop shows that the nonlinear contribution to the response and the nonlinear restoring forces are related through the linear frequency response function (FRF) matrix. Specifically, if the nonlinearity is localized, the response’s nonlinear part is intimately linked with the linear FRF at the nonlinearity location. Thus, processing input–output data measured under pseudorandom excitation, the nonlinearity location can be inferred from the comparison of the nonlinear part of the response with the different measured FRFs. A second approach, which provides a direct estimation of the nonlinear restoring forces at the different degrees of freedom, also utilizes the same relations. Both procedures are demonstrated numerically and experimentally.

Research on Digital Modeling Method of Triaxial Vibration System

To meet the demand of providing basis for improving triaxial vibration system, it is necessary to extract the overall Frequency Response Function (FRF) matrix of triaxial vibration system from its digital model. The mechanical parts of triaxial vibration system is connected tightly which satisfy rigid connection assumption. The Jetmudsen FRF method is used to construct the overall frequency response function matrix of triaxial vibration system. The electrodynamic shaker (referred to as the shaker) is an important component of the three-axis vibration system, and its function is to provide the test bench with an excitation force in a certain direction. Therefore, it is necessary to define a large number of internal nodes on the shaker. The FRF matrix of shaker (including origin FRF matrix of internal nodes and joints, cross-point FRF matrix of internal nodes and joints) should be obtain to carry out the construction of overall FRF matrix of triaxial vibration system. It is difficult because the size of FRF matrix of shaker is too large. So a way to simplify the distributed loads exerting on the structure into multi-point concentrated loads is proposed, it can be used to obtain the simplified digital model of the electrodynamic shaker. Based study above, the overall frequency response function matrix of the triaxial vibration system is obtained, and the digital modeling of the triaxial vibration system is completed.

Prediction of the Posture-Dependent Tool Tip Dynamics in Robotic Milling Based on Multi-Task Gaussian Process Regressions

Chatter vibration is one of the main factors that limit the productivity and quality of the robotic milling process. To predict the robotic milling stability, it is essential to obtain the tool tip frequency response function (FRF). The tool tip dynamics of a robot heavily depend on its postures and used tools. A state-of-art methodology of combining the regression model with the Receptance Coupling Substructure Analysis (RCSA) method is proved to be effective in predicting tool tip FRFs of machine tools for different positions and tools. However, for the milling robot, the cross coupling FRFs have an obvious influence on the dynamic property of the milling robot, thereby greatly affecting the milling stability boundary. It is of great challenge to directly integrate the effect of the cross coupling FRFs into the state-of-art approach to predict the tool tip dynamics. To tackle this challenge, in this paper, we propose an approach to predict the posture-dependent tool tip dynamics for different tools in robotic milling considering the cross coupling FRFs. First, a more comprehensive RCSA procedure is adopted to include the cross coupling FRFs. Then, the impact test is designed to measure the required FRF matrix. By fitting the measured FRF matrix with the multiple-degree-of-freedom (MDOF) model, the number of modal parameters is significantly reduced. Next, the Multi-Task Gaussian Process (MTGP) regression model is employed to mine the physical correlations between different modal parameters. Compared to the ordinary Gaussian Process regression model, the number of required regression models in MTGP is reduced and the prediction performance is improved in terms of accuracy and robustness. Furthermore, the effectiveness of the proposed approach is validated by the impact test and milling experiment on an industrial robot.

Towards a low cost method for measuring the airborne sound insulation of partitions

BS EN ISO 10140-2 outlines a method for estimating the airborne sound insulation of a partition using adjacent rooms. In this paper, a new method is presented by which the airborne sound insulation is obtained without the need for a transmission suite. Using a stand-alone partition, a representative blocked pressure field is applied numerically to a measured frequency response function matrix characterising the partition. The incident and transmitted powers are determined from the blocked pressure and velocity field on the source and receiver sides, respectively. Results of several partitions measured using the proposed methodology will be compared against estimates obtained according to BS EN ISO 10140-2.

Force Identification Based on Response Signals Captured with High-Speed Three-Dimensional Digital Image Correlation

Structural Health Monitoring (SHM) systems allow three types of diagnostic tasks to be performed, namely damage identification, loads monitoring, and damage prognosis. Only if all three tasks are correctly fulfilled can the useful remaining life of a structure be estimated credibly. This paper deals with the second task and aimed to extend state-of-the-art in load identification, by demonstrating that it is feasible to achieve it through the analysis of response signals captured with high-speed three-dimensional Digital Image Correlation (HS 3D-DIC). The efficacy of the proposed procedure is demonstrated experimentally on a frame structure under broadband vibration excitation. Full-field vibration displacement signals are captured with the use of two high-speed cameras and processed with 3D-DIC. Loads are identified with two different algorithms based on inverting the Frequency Response Function (FRF) matrix and modal filtration (MF). The paper discusses both methods providing their theoretical background and experimental performance.

Targeted modal response control of structures with inerter-based vibration absorbers considering modal interaction effects

Inerter-based vibration absorbers (IVAs) have been widely studied for the vibration control of multi-degree-of-freedom (MDOF) structures. Common analytical optimization strategies of IVAs for controlling the modal response of a specific mode neglect the contributions from non-resonant modes (denoted as modal interaction effects hereafter), which could weaken the validity of the optimum solution, while directly employing numerical search methods usually implies expensive computational costs in repeatedly evaluating control objectives under different sets of IVA parameters. To aim at both accuracy and efficiency of optimum design solution of IVAs for targeted modal response control, the Sherman-Morrison inversion (SMI) is exploited in this paper to derive explicit expressions of frequency response functions (FRFs) of IVA-controlled structures based on an MDOF model established in modal coordinates. In this way, the non-zero off-diagonal elements and precise expressions of diagonal elements in the FRF matrix are explicitly incorporated in the design process, with full account for modal interaction effects. The SMI is further implemented to calculate the FRF matrix of structures controlled by multi-IVA (MIVA) in an iterative manner, and subsequently, design strategies for targeted modal control using (M)IVA are proposed with particular focuses on wind-resistance and seismic design. Based on a high-rise building taken as the case study, the accuracy and efficiency in calculating the FRF matrix are quantitively assessed by Monte Carlo simulations. Two design examples, i.e., using a single tuned inerter damper (TID) for wind-resistance design and using a multi-TID for seismic design, are presented to validate the precision and efficiency of proposed design strategies. The results indicate that modal interaction effects are increasingly important for the design of (M)IVAs having larger inertance or being tuned to higher modes.

MTPA- and MSM-based Vibration Transfer of 6-DOF Manipulator for Anchor Drilling

An anchor drilling for a coal mine support system can liberate an operator from heavy work, but will cause serious vibration, which willbe transmitted to the pedestal from the roof bolter along a manipulator. Based on the multi-level transfer path analysis (MTPA) and modal superposition method (MSM), a vibration transfer model for the subsystem composed of the joints of a manipulator with six degrees of freedom (DOF) was established. Moreover, its frequency response function matrix was also built. The 6-DOF excitation of the roof bolter was deduced. The exciting force on the roof bolter transmitted to the pedestal along the 6-DOF manipulator was analysed with a force Jacobian matrix, to identify the external loading on the pedestal. A case in engineering practice shows that the amplitude of each DOF of the pedestal from large to small is as follows: bending vibration (component 1), longitudinal vibration, torsional vibration, bending vibration (component 2), rotational vibration around z-axis, rotational vibration around y-axis. The pedestal is mainly in the form of bending vibration. The theory of vibration transfer along the 6-DOF manipulator for anchor drilling proposed in this article can provide a theoretical foundation for the development of vibration-damping techniques and the design of absorbers.

A Hilbert transform sensitivity-based model-updating method for damage detection of structures with closely-spaced eigenvalues

In this paper, a novel method is proposed for damage detection of structures with closely-spaced eigenvalues. The proposed method uses a transformed form of the condensed frequency response function matrix each of whose columns is obtained as the sum of the unwrapped instantaneous Hilbert phase of the corresponding decomposed column of the original matrix using Empirical Mode Decomposition (EMD) algorithm. A new sensitivity-based model updating equation is then developed, which uses the constructed new matrix as input. The constructed sensitivity-based equation is solved via the least squares method through iterations to update unknown structural damage indices in a finite element model of the structure. To demonstrate the capability of the proposed method, the problem of damage detection in a composite laminate plate and a spatial truss structure, as examples of structures with closely-spaced eigenvalues, is solved. Moreover, the results obtained from the proposed method are compared against two other methods from the literature. The results show that the proposed method is far more effective at updating damage indices when incomplete highly noisy data is available.

An Innovative Structural Dynamic Identification Procedure Combining Time Domain OMA Technique and GA

In this paper an innovative and simple Operational Modal Analysis (OMA) method for structural dynamic identification is proposed. It combines the recently introduced Time Domain–Analytical Signal Method (TD–ASM) with the Genetic Algorithm (GA). Specifically, TD–ASM is firstly employed to estimate a subspace of candidate modal parameters, and then the GA is used to identify the structural parameters minimizing the fitness value returned by an appropriately introduced objective function. Notably, this method can be used to estimate structural parameters even for high damping ratios, and it also allows one to identify the Power Spectral Density (PSD) of the structural excitation. The reliability of the proposed method is proved through several numerical applications on two different Multi Degree of Freedom (MDoF) systems, also considering comparisons with other OMA methods. The results obtained in terms of modal parameters identification, Frequency Response Functions (FRFs) matrix estimation, and structural response prediction show the reliability of the proposed procedure.

Vibration-based damage identification in composite plates using 3D-DIC and wavelet analysis

High-speed three-dimensional digital image correlation (3D-DIC) techniques can acquire full-field vibration responses under a single excitation, which is not restricted by the number of degrees of freedom (DOFs) in measurement compared with accelerometers and laser Doppler vibrometers (LDVs). Therefore, 3D-DIC exhibits compelling capacity in dynamic testing, especially for nondestructive damage detection. Based on experimental modal analysis (EMA), this study presents a damage detection method without using a baseline model, which combines the advantages of 3D-DIC with two-dimensional continuous wavelet transform (2D-CWT) for damage detection and recognition. The carbon fiber reinforced plastic (CFRP) composite plates with prefabricated artificial damages, including spatial notches and local crack, were investigated. The mode shapes of damaged CFRP composite plates were obtained by singular value decomposition (SVD) from the matrix of frequency response functions (FRFs). Then the mode shapes were analyzed by using wavelet transform, and the damage index was derived from the wavelet coefficients. For comparison, two other damage indices, namely mode shape curvature and polynomial fitting difference, were also derived. It is found that 3D-DIC could provide different measurement DOFs in line with the damage form for post-processing; and the Mexican-hat wavelet analysis can accurately detect the location and size of damage by suppressing the effects of measurement noise.

Influence of skew angle on the random vibration response of propeller-shafting system induced by turbulent inflow

The influence of skew angle on the random vibration response of propeller induced by turbulent inflow is investigated. A three-step method is applied to compute the broadband force loaded on the propeller face and its random vibration response. Firstly, the broadband force spectrum induced by the turbulent inflow is calculated based on the strip theory. Then the transfer frequency response function matrix between face nodes and the transmitted forces are formed based on the finite element model. Finally, the transmitted forces to the bearings are computed by multiplying the broadband forces matrix to the transfer functions matrix. The random response of David Taylor Model Basin (DTMB) propellers 4381–4384 are calculated and compared. The influences of skew angle on the vibration response are discussed. It is shown that the influence of skew angle on the random response of propeller lies in two aspects. On the one hand, the increase of skew angle will slightly increase the broadband force near the blade passing frequency when the turbulence integral length is greater than blade spacing. On the other hand, the propeller rigidity is decreased with the increase of skew angle. Thus the force transmitted to bearings and blade vibration intensity is enhanced, which has the potential to increase structural noise radiated by the propeller and hull.

A procedure to restore measurement induced violations of reciprocity and passivity for FRF-based substructuring

Frequency-based substructuring is a very popular approach to predict the vibroacoustic behaviour of built-up mechanical systems. Even if extensively used since a long time for sundry purposes, yet this method may encounter difficulties when attempting to include test-based substructure models. The reason is that the characteristics of the experimentally obtained subsystems are corrupted by unavoidable measurement inaccuracies. Such inaccuracies often drastically affect the prediction of the combined system behaviour, especially when the problem is badly conditioned. Here we will focus on two types of measurement induced inaccuracies: reciprocity and passivity violations. A new approach that enforces these two physical properties upon a test-based frequency response function (FRF) model is proposed. A well-behaving system is created while affecting the original corrupted FRF matrix as little as possible. Besides guaranteeing a physically consistent behaviour, the approach considerably reduces the impact of noise for coupling applications and hence decreases the quality requirements for experimental data.In this work the method is presented and applied to a poorly-conditioned hybrid substructuring approach: a test-based plate model coupled through its rotational degrees of freedom with an analytical model of a damping layer. This is a particularly critical test case as the noise and physical anomalies already present in the experimentally obtained plate data turn out to be further amplified by the finite difference procedure needed to estimate its rotational degrees of freedom. It is shown that the removal of the physical inconsistencies from the plate data using the novel procedure, indeed yields more stable and reliable results after coupling.

Substructural damage detection using frequency response function based inverse dynamic substructuring

In this paper, a substructural damage detection approach is presented using frequency response function (FRF)-based inverse dynamic substructuring and FRF-based model updating. In practice, often only one subsystem of the global structure is critical and susceptible to damage and therefore needs monitoring. In the proposed method, one subsystem is the “main” subsystem and is susceptible to damage and the “residual” subsystem(s) are considered undamaged and the damage identification is only applied to the main subsystem. The FRF matrix of the main subsystem is obtained as a “standalone” component that is completely decoupled from the residual subsystem(s) without the need of interface virtual support or identifying interface forces. The FRF measurement is performed on the global damaged structure but the FRFs of the main substructure (damaged) is obtained by “decoupling” the known FRF(s) of the residual subsystem from the global system FRFs and the damage detection is performed only on the main substructure. An FRF-based model updating method using numerical sensitivities is then used for damage identification of the main subsystem. The FRF obtained from the finite element model of the main substructure is updated using the FRF of the damaged main substructure (obtained from decoupling) and the location and quantity of the damage is identified. Numerical and experimental examples are presented to illustrate the damage detection procedure and the damage in the main substructure is detected, located and quantified with good accuracy.

Load identification with regularized total least-squares method

Load identification in structural dynamics is an ill-conditioned inverse problem, and the errors existing in both the frequency response function matrix and the acceleration response have a great influence on the accuracy of identification. The Tikhonov regularized least-squares method, which is a common approach for load identification, takes the effect of the acceleration response errors into account but neglects the effect of the errors of the frequency response function matrix. In this article, a Tikhonov regularized total least-squares method for load identification is presented. First, the total least-squares method which can minimize the errors of the frequency response function matrix and acceleration response simultaneously is introduced into load identification. Then Tikhonov regularization is used to regularize the total least-squares method to improve the ill-conditioning of the frequency response function matrix. The regularization parameter is selected by the L-curve criterion. To validate the performance of the regularized total least-squares method, a load identification simulation with two excitation loads is studied on a plate based on the finite element method and a load identification experiment with two excitation loads is conducted on an aluminum plate. Both simulation and experiment results show that the excitation loads identified by the regularized total least-squares method match the actual loads well although there are errors existing in both the frequency response function matrix and acceleration response. In experiment, the average relative error of the regularized total least-squares method is 13.00% for excitation load 1 and 20.02% for excitation load 2, whereas the average relative error of the regularized least-squares method is 35.86% and 53.09% for excitation load 1 and excitation load 2, respectively. This result reveals that the regularized total least-squares method is more effective than the regularized least-squares method for load identification.

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

For a structural dynamics engineer assessing a system’s vibration characteristics, the frequency response function (FRF) is indispensable, irrespective of whether the setup being tested is experimental, numerical or analytical. This paper outlines the different FRF estimation techniques that have been developed over the years. Algorithms that compute an ordinary least squares (OLS) estimate of the FRF, assuming uncorrelated measurement noise to be present either on the output (H1) or the input (H2) signals, have been compared with those that employ total least squares (TLS) equations by making use of an augmented input–output auto and cross power (GFFX/GXFF) matrix at every frequency such as the Hv (using eigenvalue decomposition) and HSVD (using singular value decomposition) algorithms. Another FRF estimation method based upon Cholesky decomposition (HCD) is also discussed. Further discussion has been included of TLS algorithms computing the FRFs for one output at a time, as historically presented in the development of the Hv algorithm, with a case that evaluates the FRF matrix computed for all the outputs simultaneously. The development of the corresponding coherence functions has been presented, highlighting the method dependent paradigms that led to concepts such as virtual coherence and partial coherence while underscoring their equivalence with ordinary and multiple coherence calculations. It has been shown that some of the existing conditioned coherence metrics are inconsistent from an input–output standpoint, for which the corrected interpretations have been subsequently described. This article is unique in that no previous work summarizes all of the previously developed FRF estimation and coherence algorithms and no previous paper explains the inconsistencies in the conditioned coherence functions with respect to multiple coherence.

Modal Identification of a Soil-subway System with Emphasis on Scattering of Seismic Waves Induced by Uniform and Non-uniform Support Excitations

ABSTRACT Subways, as one of the critical structures in urban design for solving traffic problems, require continuous monitoring during operation. Therefore, it is imperative to study the scattering of seismic waves caused by the presence of these underground structures. System identification is one of the most efficient methods for investigating structural behavior during operation and identifying potential problems that may be encountered for an in-service underground structure. In this study, through extending an advanced system identification method called Frequency Response Function Matrix and Singular Value Decomposition (FRFM-SVD) and wavelet transform methods, modal parameters of a soil-subway system were identified. Further, using a random vibration method based on coherence functions, non-uniform accelerations under site condition were generated and the seismic behavior of soil-subway model under uniform and non-uniform seismic excitations were studied. The obtained results indicated that the FRFM-SVD method developed in this study shows a reasonable accuracy for identifying the seismic behavior of underground structures. The maximum amplification of the responses will occur in the proximity of the subway to the amount of 1.75 under non-uniform excitation, which is almost 20% greater than that from uniform excitation. Thus, to increase the accuracy of the estimation of ground surface accelerations, the effect of the presence of underground structures should be accompanied by considering non-uniform excitation in the analysis.

Frequency-based decoupling and finite element model updating in vibration of cable–beam systems

Interactions between cable and structure affect the modal properties of cabled structures such as overhead electricity transmission and distribution line systems. Modal properties of a single in-service pole are difficult to determine. A frequency response function of a pole impacted with a modal hammer will contain information about not only the pole but also the conductors and adjacent poles connected thereby. This article presents a generally applicable method to extract modal properties of a single structural element, within an interacting system of cables and structures, with particular application to electricity poles. A scalable experimental lab-scale pole-line consisting of a cantilever beam and stranded cable and a more complex system consisting of three cantilever beams and a stranded cable are used to validate the method. The frequency response function of a cantilever (“pole”) is predicted by substructural decoupling of measured cable dynamics (known frequency response function matrix) from the measured response of the assembled cable–beam system (known frequency response function matrix). Various amounts of sag can be present in the cable. Comparison of the estimated and directly obtained pole frequency response functions show good agreement, demonstrating that the method can be used in cabled structures to obtain modal properties of an individual structural element with the effects of cables and adjacent structural elements filtered out. A frequency response function–based finite element model updating is then proposed to overcome the practical limitation of accessing some components of the real-world system for mounting sensors. Frequency response functions corresponding to inaccessible points are generated based on the measured frequency response functions corresponding to accessible points. The results verify that the frequency response function–based finite element model updating can be used for substructural decoupling of systems in which some essential points, such as coupling points, are inaccessible for direct frequency response function measurement.