Modal Parameters Of System Research Articles

Parameters identification of closely spaced modes with the covariance-drivern stochastic subspace and damping ratio dispersion method

Electromechanical equipment with quasi-symmetric and partially symmetrical structures is widespread in the industry, which not only makes the dynamic characteristics of the system more complex but also leads to severe closely spaced modes. The existence of closely spaced modes can lead to more confusion, because severe closely spaced modes have similar frequency eigenvalues and some singular eigenvalues are prone to generate spurious modes in matrix computations. In this paper, the covariance-drivern stochastic subspace and damping ratio dispersion (SSI-COV-DRD) method is proposed, which in comparison with previous methods, identification process directly is performed in the stochastic subspace by solving the system matrix of the time domain signal to overcome modal aliasing of the frequency-based method. Firstly, a Toeplitz matrix is constructed with a time series signal and then the system modal parameters are gained from the system matrix after performing singular value decomposition (SVD) to the Toeplitz matrix. Subsequently, the poles of the continuous model order are classified by a hierarchical clustering algorithm to obtain stabilization diagram and the real physical modes are automatically extracted from the stabilization diagram according to the choice criteria. Finally, the damping ratio dispersion index is calculated to express the nonlinear distribution characteristic of the damping ratio. Numerical examples for the five-order free vibration simulation signals with noise-free and noise indicate that the proposed method can simultaneously identify the modal frequency and damping ratio with high accuracy in severe closely spaced modes, especially for the repeated frequency modes. Since the bolt connection rotor of the aero-engine usually produces severe closely spaced modes in vibration due to the structural symmetry, the frequency and damping ratio are extracted and the damping ratio dispersion index is calculated from different bolt loosening states, which is helpful for the bolt loosening detection of aviation safety assurance and maintenance.

A new approach of ensemble learning in fully automated identification of structural modal parameters of concrete gravity dams: A case study of the Koyna dam

This study proposes a new automatic modal identification algorithm that identifies the system's modal parameters based on output-only data. The identification process is based on a stochastic subspace algorithm. Stochastic subspace algorithms only need the model degree to identify the modal parameters of the system. A model degree is an appropriate degree of the state-space model and it is estimated by using the Singular Value Criterion. Several clustering algorithms with different approaches have been used. The most powerful data fusion algorithm, which is called Dempster-Schaffer Theory is applied for integrating the results of the algorithms and preventing divergence of results. Furthermore, it focused on measuring the uncertainty of the identified modal parameters. In the proposed modal identification algorithm, only one data set is used for identification. In this situation, the variety of data is small and limited, and statistical resampling methods seem to be a good option. Establishing the above cycle enables the identification algorithm to identify modal parameters and quantify their uncertainty based only on a data set. This algorithm uses many fields, such as data collection, signal processing techniques, model reduction, feature extraction, clustering and data fusion. The Koyna gravity dam is selected as a case study to evaluate the performance of the developed algorithm. In the first step, a comparative study is carried out to verify the developed finite element model of the Koyna dam. Afterwards, the dam is excited by the Koyna earthquake, and dynamic responses of the dam are collected from the minimum number of points required for system identification. It is noteworthy the number of modes that are identified in this work is equal to 4 vibration modes. The maximum difference between the natural frequencies identified by the present algorithm and the finite element model does not exceed 4.33%. Also, the average difference is less than two percent, which indicates the high accuracy of the system identification algorithm. Also, all the vibration modes in the desired frequency range have been correctly identified. The results indicate that modal parameters with remarkable precision have been extracted by the modified algorithm.

Sparse Bayesian learning with model reduction for probabilistic structural damage detection with limited measurements

A new variant of damage detection algorithm is proposed based on Sparse Bayesian Learning (SBL) and model reduction for probabilistic structural damage detection. By exploring recent developments in SBL, we aim to produce reliable damage identification even for high-dimensional model parameter spaces for higher-resolution damage localization. By introducing the system modal parameters, the matching of model and experimental modes and solving the nonlinear eigenvalue problem of a structural model are not required. However, one inherent difficulty is that, because only a small number of degrees of freedom (DOFs) can be measured in practice due to limited instrumentation and unmeasurable rotational DOFs, the computation of the system mode shapes has a minimum chance of success. The proposed algorithm extends the applicability of SBL-based damage detection to cases with limited measurements by exploiting model reduction techniques to avoid computation of the system mode shapes for the full model. To effectively incorporate the model reduction procedure, a two-stage SBL-based damage detection algorithm is proposed, in which the first stage updates the reduced system modal properties employing the modal data, and the second stage learns the stiffness parameters and their associated hyper-parameters to produce model sparseness of stiffness losses. The performance of the proposed algorithm is investigated through a series of numerical and experimental studies involving two beam structures and a long-span cable-stayed bridge. Both modeling errors and measurement noises are considered in these studies. The results show that, despite limited measured DOFs in conjunction with modeling error and measurement noise, the proposed algorithm based on SBL and model reduction can successfully locate and quantify the damage along with their posterior uncertainties, which give a sense of identification confidence.

Modal identification of concrete arch dam by fully automated operational modal identification

Modal identification is a type of system identification, which studies on the modal parameters of systems by using modal test. In the case of using operational or ambient modal analysis, there is no need to measure excitation, and the system output data are adequate for identification purposes. These modal parameters of system are of great importance from the engineering point of view particularly in the area of system identification, damage detection, and condition monitoring. In a fully-automated identification approach, the modal parameters are extracted without intervention of a specialized user. In this study, a Fully Automated Operational Modal Identification algorithm is developed to identify modal parameters of an arch dam. For this purpose, Morrow point arch dam is selected, and a 3-D finite element of the dam is established. In the first step, the computed natural frequencies of the dam based on the forced vibration tests are utilized to verify the developed finite-element model of the dam. Afterwards, the dam is excited by El-Centro earthquake, and its seismic response is recorded. The collected signals are presented to the proposed identification algorithm. The proposed algorithm presents a methodology to classify the physical mode shapes from spurious modes using the Stochastic Subspace framework. The results indicate acceptable identification via the proposed algorithm using a dataset and non-repetitive data acquisition. Moreover, the identification process is automated and it does not require any user interaction. It is noteworthy to state that the algorithm can identify any dam with any number of mode shapes, and it is practical and straightforward to apply.

Acoustics Source Identification of Diesel Engines Based on Variational Mode Decomposition, Fast Independent Component Analysis, and Hilbert Transformation

Diesel engines are widely used in railway systems, particularly in freight trains. Despite their high efficiency in energy conversion, they usually generate high levels of acoustics pollution during operation. In order to mitigate this problem, a series of active/passive acoustics control methods are used to reduce noise. Most of these methods are only effective if the prior knowledge of sources is given. In other words, it is essential to recognize the acoustics source. Variational mode decomposition (VMD) is a signal processing method that enhances the signal corrupted by background noise. However, the decomposed results of VMD depend on their mode parameter and penalty parameter. Therefore, an evaluation method based on system modal parameters (natural frequency and damping ratio) is proposed to select the mode parameter, and the penalty parameter can be selected from the power spectra of signals. In order to increase the accuracy of decomposition for diesel engines and find out the sources of acoustics, a method combining VMD, fast independent component analysis, and Hilbert transformation (VMD-FastICA-HT) is proposed for the separation and identification of different sources for diesel engines. The optimization results indicate that when the penalty parameter value is 1.5 to 16 times the maximum signal amplitude, better decomposition results can be achieved. Therefore, the separated independent acoustics are more accurate in source identification. Furthermore, both simulation data and in situ operational data of diesel engines for vehicles are used to validate the effectiveness of the proposed method.

Operational modal analysis using symbolic regression for a nonlinear vibration system

The existence of nonlinearity is an inevitable frequent occurrence that should be considered to accurately identify the modal parameters of a vibration system using operational modal analysis. A problem is that the traditional operational modal analysis method based on the linear modal theory is not applicable to modal parameter identification of vibration systems with nonlinearity. A solution is as follows: this paper is aimed at solving the problem by proposing a new operational modal analysis method to carry out modal parameter identification for a nonlinear vibration system. The new operational modal analysis method, based on the forced response and symbolic regression method without assuming any pre-existing information and only using mathematical symbols, is introduced to solve the problem by automatically searching for the expression structure and modal parameters of a system in nonlinear normal modes. The simulation result of a three-degrees-of-freedom nonlinear system reveals the high accuracy of the proposed operational modal analysis method in extracting the modal parameters. Then, a rod fastening rotor model is considered, and the capability of the proposed operational modal analysis method to precisely extract its modal parameters is further evaluated.

A comparative study on the effectiveness of system parameters in monitoring pre-load loss in bolted joints

This article presents the monitoring of bolt loosening in a single bolted lap joint between two steel beams. The bolt loosening is initially studied using system modal parameters for various boundary conditions. Natural frequency and modal damping ratio in lower modes were not found to be reliable. Transmissibility function was seen to have a more reliable trend to identify the state of the joint. An in-house setup was developed to produce loosening in the bolted joint. Testing configuration was optimized by structural analysis. Strain measurements were carried out using Fiber Bragg-grating sensors while sinusoidal load was applied on the test structure. The results have been used to discuss a viable approach to identify the health of, or recognize a loosening trend, in a bolted joint.

Adjustable-Smooth Polynomial Command-Shaping Control With Linear Hoisting

Great amount of work has been dedicated to eliminate residual vibrations in rest-to-rest motion. Considerable amount of these methods is based on convolving a general input signal with a sequence of timed impulses. These impulses usually have large jumps in their profiles and are chosen depending on the system modal parameters. Furthermore, classical input shaping methods are usually used for constant cable length and are sensitive to any change in the system parameters. To overcome these limitations, polynomial command shapers with adjustable maneuvering time are proposed. The equation of motion of a simple pendulum with the effect of hoisting is derived, linearized, and solved in order to eliminate residual vibrations in rest-to-rest maneuvers. Several cases including smooth, semi-smooth and unsmooth continuous shapers are simulated numerically and validated experimentally on an experimental overhead crane. Numerical and experimental results show that the proposed polynomial command shaper eliminates residual vibrations effectively. The effect of linear hoisting is also included and discussed. To enhance the shaper performance, extra parameters are added to the polynomial function to reduce shaper sensitivity. Results show that the effect of adding these parameters greatly enhances the shaper performance.

Optimal Determination of Structural Dynamical Parameters Using Ambient Vibration

By using ambient vibration, a new approach based on improvement and correction of system characteristic matrix in modal vibration is provided. The result is that actual system characteristic matrices are accurately made such that the error is minimized at great extent. This clearly shows how the system parameters can be updated in a more reliable way. Firstly, by approximation, the actual system characteristic matrices are determined using the singular value decomposition of block Hankel matrix that is built from response correlation matrix. Secondly, by black-box modeling approximation, the input-output relation of the system through Kalman theory is made in order to make the system characteristic matrices optimal definite. Furthermore, by expressing Hankel matrix’s multiplicities from Eigen solution of the system state matrix obtained in previous iteration, it is possible to determine both the covariance of non-measurable process noise and measurement noise matrixes which are present in the Riccati equation. This means that both measurement and process covariance noises’ matrixes are indirectly built only from measured out-put data. The repetition of iterations is done until the error is sufficiently minimized. And then system modal parameters are extracted from these obtained system characteristic matrices. This system is used for modal update of the system in which modal parameters are applied directly and iterative methods. The code supporting this algorithm can be interfaced with the codes of the finite elements.

Atomic Norm Minimization for Modal Analysis From Random and Compressed Samples

Modal analysis is the process of estimating a system's modal parameters such as its natural frequencies and mode shapes. One application of modal analysis is in structural health monitoring (SHM), where a network of sensors may be used to collect vibration data from a physical structure such as a building or bridge. There is a growing interest in developing automated techniques for SHM based on data collected in a wireless sensor network. In order to conserve power and extend battery life, however, it is desirable to minimize the amount of data that must be collected and transmitted in such a sensor network. In this paper, we highlight the fact that modal analysis can be formulated as an atomic norm minimization (ANM) problem, which can be solved efficiently and in some cases recover perfectly a structure's mode shapes and frequencies. We survey a broad class of sampling and compression strategies that one might consider in a physical sensor network, and we provide bounds on the sample complexity of these compressive schemes in order to recover a structure's mode shapes and frequencies via ANM. A main contribution of our paper is to establish a bound on the sample complexity of modal analysis with random temporal compression, and in this scenario we prove that the samples per sensor can actually decrease as the number of sensors increases. We also extend an atomic norm denoising problem to the multiple measurement vector (MMV) setting in the case of uniform sampling.

Bayesian system identification based on hierarchical sparse Bayesian learning and Gibbs sampling with application to structural damage assessment

Bayesian system identification has attracted substantial interest in recent years for inferring structural models based on measured dynamic response from a structural dynamical system. The focus in this paper is Bayesian system identification based on noisy incomplete modal data where we can impose spatially-sparse stiffness changes when updating a structural model. To this end, based on a similar hierarchical sparse Bayesian learning model from our previous work, we propose two Gibbs sampling algorithms. The algorithms differ in their strategies to deal with the posterior uncertainty of the equation-error precision parameter, but both sample from the conditional posterior probability density functions (PDFs) for the structural stiffness parameters and system modal parameters. The effective dimension for the Gibbs sampling is low because iterative sampling is done from only three conditional posterior PDFs that correspond to three parameter groups, along with sampling of the equation-error precision parameter from another conditional posterior PDF in one of the algorithms where it is not integrated out as a “nuisance” parameter. A nice feature from a computational perspective is that it is not necessary to solve a nonlinear eigenvalue problem of a structural model. The effectiveness and robustness of the proposed algorithms are illustrated by applying them to the IASE-ASCE Phase II simulated and experimental benchmark studies. The goal is to use incomplete modal data identified before and after possible damage to detect and assess spatially-sparse stiffness reductions induced by any damage. Our past and current focus on meeting challenges arising from Bayesian inference of structural stiffness serves to strengthen the capability of vibration-based structural system identification but our methods also have much broader applicability for inverse problems in science and technology where system matrices are to be inferred from noisy partial information about their eigenquantities.

Vibration-based damage detection for structural connections using incomplete modal data by Bayesian approach and model reduction technique

Most of the existing damage detection methods focused on damage along members of the structure without considering possible damage at its connections. Under the Bayesian framework, the finite element (FE) model reduction technique and the system mode concept, this paper presents a practical method for structural bolted-connection damage detection using noisy incomplete modal parameters identified from limited number of sensors. Based on the incomplete modal identification results, the most probable structural model parameters, the most probable system eigenvalues and partial modes shapes together with the associated uncertainties can be identified simultaneously. There are several significant features of the proposed method: (1) it does not require computation of the system mode shapes for the full model due to the FE model reduction technique; (2) matching between measured modes and model predicted modes is avoided in contrast to most existing methods in the literature; and (3) an efficient iterative solution strategy is also proposed to resolve the difficulties arisen from the high-dimensional nonlinear optimization problem for the structural model parameters and the incomplete system modal parameters. Numerical simulations and experimental verifications of a four-storey two-bay bolt-connected steel frame and a two-storey laboratory bolted frame, respectively, are utilized to demonstrate the validity and efficiency of proposed methodology.

Modal analysis on tire with respect to different parameters

This paper presents experimental modal analysis of non-rotating tires under different boundary conditions. A test rig with four guides in vertical (radial) direction and two guides in axial direction was designed to support the tire-rim assembly with a free support. The setup permits to carry out the experiments on the grounded supported tire-rim assembly while changing the value of the static load acting on the wheel axis. Under static load condition, it is found that, tire deflection depends on the applied static radial force in a hysteresis manner and a third-order polynomial was used to fit the data during loading and unloading conditions. The relationship between static stiffness in radial direction and tire deflection is nonlinear and depends on loading/unloading conditions for different tire pressures. The response of the tire is quite similar to the response of viscously damped mass system for impulse force which is provided by an impact hammer. The results show that the system modal parameters can be obtained respective of loading or unloading conditions with a maximum difference of 1.992% for frequency values and 3.66% for damping values. This study has a practical value for the description of mechanical properties of tires.

HIERARCHICAL SPARSE BAYESIAN LEARNING FOR STRUCUTRAL HEALTH MONITORING WITH INCOMPLETE MODAL DATA

For civil structures, structural damage due to severe loading events such as earthquakes, or due to long-term environmental degradation, usually occurs in localized areas of a structure. A new sparse Bayesian probabilistic framework for computing the probability of localized stiffness reductions induced by damage is presented that uses noisy incomplete modal data from before and after possible damage. This new approach employs system modal parameters of the structure as extra variables for Bayesian model updating with incomplete modal data. A specific hierarchical Bayesian model is constructed that promotes spatial sparseness in the inferred stiffness reductions in a way that is consistent with the Bayesian Ockham razor. To obtain the most plausible model of sparse stiffness reductions together with its uncertainty within a specified class of models, the method employs an optimization scheme that iterates among all uncertain parameters, including the hierarchical hyper-parameters. The approach has four important benefits: (1) it infers spatially-sparse stiffness changes based on the identified modal parameters; (2) the uncertainty in the inferred stiffness reductions is quantified; (3) no matching of model and experimental modes is needed, and (4) solving the nonlinear eigenvalue problem of a structural model is not required. The proposed method is applied to two previously-studied examples using simulated data: a ten-story shear-building and the three-dimensional braced-frame model from the Phase II Simulated Benchmark problem sponsored by the IASC-ASCE Task Group on Structural Health Monitoring. The results show that the occurrence of false-positive and false-negative damage detection is clearly reduced in the presence of modeling error. Furthermore, the identified most probable stiffness loss ratios are close to their actual values.

Estimation of the Dynamic Properties of Non-linear Packaging Materials Using a Reverse Multiple Input/Single Output Based Approach

During the distribution phase, packaged consignments are exposed to a variety of environmental hazards (such as vibrations) that, if excessively severe, may cause damage to or even destroy the product. Structural deterioration can be tracked by monitoring variations in the packaging system's modal parameters, particularly its natural frequency (stiffness). Natural frequency estimates are often extracted using a least-squares regression curve fit, applied to an estimate of the system's frequency response function (FRF). FRF estimates are generally obtained using the Fourier transform with a single input and single output (SISO). This approach is suitable for many applications; however, as the non-linearity of the system under analysis increases, the ability of this technique to accurately monitor changes in the system will decrease. In addition, when the excitation to the non-linear system is varied (increased or decreased amplitude), a SISO-based approach may indicate a shift in natural frequency (as a result of the varied input) even though no change in the condition of the system has occurred. This paper discusses an approach that is designed to separate the linear component of the system's FRF using a reverse multiple input/single output (RMISO) algorithm. Such separation will allow traditional modal parameter extraction (curve fitting) techniques to be used to monitor the condition of non-linear systems. The paper presents the results of experiments in which expanded polystyrene samples were subjected to broad-band random base excitation with a free-moving load placed atop the cushion sample. Continuous acceleration measurements of the vibration table and the free-moving load were used to compute the FRFs of the cushions, and the differences between a conventional (SISO) approach and the proposed RMISO-based parameter extraction technique were evaluated. Copyright © 2014 John Wiley & Sons, Ltd.

Theoretical Study on Cable’s Vibration Control by Single TMD

The commonly used viscous dampers for cable’s vibration mitigation have some unfavorable factors, such as the damping effect is not obvious for super long stay cable, the limitation of installation position, coupling vibration, etc. The cable-tuned mass damper system vibration model is put forward to solve this problem. The optimal cable-tuned mass damper system modal damping ratio and optimum design parameters, including cable vibration order, TMD’s stiffness, TMD’s mass, and TMD’s damping, were obtained by the method of complex models. The results can provide important reference for the design of TMD for stay cable.

Numerical and Experimental Investigation on Parameter Identification of Time-Varying Dynamical System Using Hilbert Transform and Empirical Mode Decomposition

This paper proposes an approach to identifying time-varying structural modal parameters using the Hilbert transform and empirical mode decomposition. Definition of instantaneous frequency and instantaneous damping ratio based on Hilbert transform for single-degree-of-freedom (SDOF) system is first introduced. The following is the definition of Hilbert damping spectrum from which the time-varying damping ratio of multi-degree-of-freedom (MDOF) system can be calculated. Identification procedures for both instantaneous frequency and damping ratios based on their definition are then introduced. Applicability of the proposed identification algorithm has been validated through several numerical examples. The instantaneous frequency and damping ratios of SDOF system under free vibration and under sinusoidal and white noise excitation have been identified. The proposed method is also applied to MDOF system with slow and sudden changing structural parameters. The results demonstrate that when the system modal parameters are slowly changing, the instantaneous frequency could be easily and well identified with satisfied accuracy for all cases. However, the instantaneous damping ratio could be extracted only when the system is lightly damped. The damping results are better for free vibration situation than for the forced vibration cases. It is also shown that the suggested method can easily track the abrupt change of system modal parameter under free vibration. The proposed method is then applied to a 12-story short-lag shear wall structure model tested on a shaking table. The instantaneous dynamic properties of the structure were identified and were then introduced as known parameters into a finite element model. Comparisons with the numerical results using constant structural parameters demonstrate that the calculated structural responses using the identified time-varying parameters are much closer to the experimental results.

A Research of Shaft Modal Simulation Method Based on the Preloaded Angular Contact Ball Bearing

In order to analysis modal characteristic of bearing with pre-tightening force on main spindle of numerical control lathe, this paper proposes a model of spindle modality analysis. This model is used to simulate the preloaded bearing shaft system modal, and the simulation results are verified by modal experiment. This paper takes 7005c as the research object to establish the equivalent-spring model based on the Hertz theory considering the pre-tightening force, whose focus is dealing with the contact between bearings rolling element and raceway. Then the model will be used to get the bearing stiffness for finite element simulation analysis. The shafting modal with preloaded bearing test platform is structured to get the shaft system modal parameters, which is compared with and verified the simulation analysis.

SIGNAL PROCESSING TECHNIQUES FOR ESTIMATING POWER SYSTEM MODAL PARAMETERS

Smart Grid is expected to provide a reliable power supply with fewer and briefer outages, cleaner power, and self-healing power systems through advanced Power Quality (PQ) monitoring, analysis and diagnosis of the PQ measurements and identification of the root cause, and timely automated controls. It is important to understand that signal processing has been an integral part of advancing and expanding the horizons of this PQ research significantly and the capabilities and applications of signal processing for PQ are continually evolving. This paper thus presents a survey on the proven and emerging signal applications for enhancing PQ, focusing on algorithms for estimating system modal parameters because resonant frequencies and their damping information are critical signatures in evaluating the PQ. In particular, we discuss the need for investigating time-varying and nonlinear characteristics of the modal parameters due to dynamic changes in system operating conditions and introduce promising signal processing techniques for this purpose.

Setting Parameters of the Analytic Wavelet Transforms for Estimating Electric Power System Damping Parameters

This letter presents a systematic guide to determine parameters of the Gabor wavelet often used as a mother wavelet in implementing the Analytic Wavelet Transform (AWT). The transform was proven to be a powerful tool in estimating power system modal parameters. Its performance however, depends on the wavelet parameters. They are challenging to determine and often require a time-consuming tuning process. Thus, this letter develops an algorithm to systematically determines AWT parameters, with its practicality and accuracy.