IDENTIFICATION OF NON-PROPORTIONAL MODAL DAMPING MATRIX AND REAL NORMAL MODES

Equations of motion for non-proportionally damped structures cannot be decoupled by using the real normal modes. For such structures, the complex normal modes are in common use for this purpose, but for the validation of finite element mass and stiffness matrices where physical damping matrices are not available, the related experimental real normal modes must be known. In previous publications, an identification theory using the real normal modes and the non-diagonal modal damping matrix for the non-proportionally damped system and some applications with the computer code ISSPA were presented. However, the theory cannot assure the symmetry of the identified modal damping matrix, which must be theoretically symmetric. In this paper, a method for identifying the symmetric non-proportional modal damping matrix using undamped modal parameters obtained from ISSPA is presented and the validity of the method is demon-strated through both numerical and experimental examples.     

Model updating of damped structures using FRF data

Due to the important contribution of damping on structural vibration, model updating of damped structures becomes significant and remains an issue in most model updating methods developed to date. In this paper, the frequency response function(FRF) method, which is one of the most frequently referenced model updating methods, has been further developed to identify damping matrices of structural systems, as well as mass and stiffness matrices. In order to overcome the problem of complexity of measured FRF and modal data, complex updating formulations using FRF data to identify damping coefficients have been established for the cases of proportional damping and general non-proportional damping. To demonstrate the effectiveness of the proposed complex FRF updating method, numerical simulations based on the GARTEUR structure with structural damping have been presented. The updated results have shown that the complex FRF updating method can be used to derive accurate updated mass and stiffness modelling errors and system damping matrices.     

Identification of modal parameters of unmeasured modes using multiple FRF modal analysis method

Current modal analysis methods seek to identify the modal parameters of some or all of the modes in the measured frequency range of interest. In many applications however, it will be very useful if modal parameters of some of the out-of-range modes can be identified during modal analysis. Such a goal is obviously theoretically possible since the raw measured frequency response functions (FRFs), upon which modal analysis is performed, do contain adequate information about the out-of-range modes in the form of residue contributions. In this paper, a new method for the estimation of modal parameters using multiple FRFs analysis is presented. In the process of modal identification, the proposed method not only presents accurate modal parameters of the modes which are present in the measurement frequency range, but also quite accurately identifies some of the modes which are not measured. The method calculates the required modal parameters by solving eigenvalue problem of an equivalent eigensystem derived from those measured FRF data. All measured FRFs are used simultaneously to construct the equivalent eigensystem matrices from which natural frequencies, damping loss factors and modeshape vectors of interest are solved. Since the identification problem is reduced to an eigenvalue problem of an equivalent system, natural frequencies and damping loss factors identified are consistent. 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 have shown the method is capable of accurately identifying modal parameters of out-of-range modes.     

Output-only modal identification based on hierarchical Hough transform

Sparse component analysis (SCA) has been introduced to the output-only modal identification for several years. This paper proposes a new method based on hierarchical Hough transform to extract the modal parameters of mechanical structures. First, the measured system responses are transformed to Time-frequency (TF) domain using Short time Fourier transform (STFT) to get a sparse representation. Then, Hough transform is applied to the TF coefficients hierarchically to identify the hyperplanes and the mixing matrix is calculated. Finally, the modal responses are recovered by using l ₁ -optimization and inverse STFT. From the recovered modal responses, natural frequencies and damping ratios are extracted. Numerical simulation of a 4 Degree-of-freedom (DOF) spring-mass system verifies the validity of the method. Free vibration of a steel cantilever beam is captured by a high-speed camera and then analyzed by the proposed method. The comparison of the estimated natural frequencies and damping ratios illustrates the good performance of the proposed algorithm.     

EVALUATION OF DAMPING NON-PROPORTIONALITY USING IDENTIFIED MODAL INFORMATION

This paper focuses on quantification of damping non-proportionality present in a discrete vibratory system. The study assumes that the information available is a set of identified system eigenvalues and eigenvectors and that the system parameters such as mass, stiffness, and damping matrices are unknown a priori. This set of modal parameters may be incomplete. The investigation is concentrated on how two existing analytical indices can be utilised when the modal damping matrix is not available. The quantification procedure starts with extraction of normal modes using a known algorithm. It is shown that two matrices, by-products of the normal model extraction, can be used to study damping non-proportionality. The first matrix is a scaled modal damping matrix. The paper shows that the indices developed from the scaled modal damping matrix preserves the properties of the indices based on the analytical modal damping matrix. The second matrix is a complex matrix which is obtained by expanding complex modes into the subspace of real modes. The off-diagonal elements of the complex matrix indicate coupling between modes due to damping non-proportionality. Based on this characteristic, three new indices are proposed. Numerical examples are presented to illustrate the use of the new indices and to compare them with the indices that are described in literature.     

Improved independent component analysis based modal identification of higher damping structures

Structural modal parameter identification under ambient excitation has strong engineering value and theoretical significance. As the most popular tool for solving Blind Source Separation (BSS) problems, Independent Component Analysis (ICA) is able to directly extract the time-domain modal parameters, including frequencies, damping ratios and modal shapes. ICA, however, has a fatal flaw of failing to identify structures with higher damping. To overcome the flaw above, the paper proposes a new method named “ICA + IDT”. Firstly, free vibration response of a structure is obtained from structural outputs under ambient excitation. Inverse damping transfer (IDT) is employed to turn a highly damped signal into a low damping response signal without changing of frequencies and mode shapes. Then, structural modal parameters are extracted from the low damping response signal by ICA. Finally, the identified damping ratios are adjusted to eliminate the impact of IDT. To verify the effectiveness and applicability of IDT + ICA proposed herein, two numerical simulations—mass-spring model and simply supported concrete beam—and an experiment model of three-story steel frame are built, and the analysis results reveal that presented method can identify structures with higher damping effectively.     

A framework for blind modal identification using joint approximate diagonalization

In this paper, a second-order statistical method employed in blind source separation (BSS) is adapted for use in modal parameter identification. Modal responses and mode shapes are estimated by the use of second-order blind identification (SOBI) on an expanded and pre-treated dataset. Frequency and damping can be obtained from the modal responses by simple single degree of freedom methods. Using this approach, a class of new non-parametric output-only modal identification algorithms is proposed and examples of its use are provided. It is demonstrated that the proposed methodology provides a novel and robust approach to modal identification. For the example shown, it is deduced that quality of the modal parameters produced by the method is competitive with the state of the art parametric methods.     

A theoretical experimental method based on modal analysis for estimating the damping capacity of vibrating structures

The purpose of the present paper is to suggest a methodology to study quantitatively the damping features of vibrating structures. These features are given as function of the time necessary for the structure to dissipate a pre-fixed percentage of the energy provided by an ideal external impact excitation. The dissipation can derive either from internal friction of the material or from eventual dampers really existing in the system. In this paper the proposed computation, based on the parameters obtained by experimental modal analysis is used only for proportional damping but it can be extended to other types of damping.     

变动非比例阻尼结构的动态分析及其在汽车舒适性设计中的应用

提出了具有变动阻尼结构的动态分析新方法,直接利用非比例阻尼矩阵计算位移函数和脉冲响应函数,求得功率谱和时间历程。另外提出了对应于随机激励的结构可靠性分析、可靠度随阻尼变动的灵敏度分析方法,并把该方法成功地应用于行驶在不平整道路上车辆的舒适性分析。     

The determination of modal damping ratios and natural frequencies from bispectrum modeling

We present a new signal processing and testing technique by using a higher statistical moment, the bispectrum, to determine the damping ratio and natural frequency of offshore structures excited by both unexpected Gaussian forces and known non-Gaussian driving forces. Due to unexpected exciting forces, such as turbulence, in the ocean, environment, the transfer functions of offshore structures are not determined through operating a known driving force and measuring its response. In order to overcome this problem, some of the existing techniques try to model the unexpected forces as white Gaussian forces or almost white Gaussian forces and determine the modal parameters from the response only. Others try to average the input and output to suppress unexpected parts. Our method uses third-order moments to keep the influence of the unexpected Gaussian forces away from the determination of the transfer function of the structure which has linear properties. We model the third-order moment property of the response function with a bispectral model. The modal parameters can be calculated from the estimated model's coefficients. The method has been proven by a number of simulations.     

基于随机减量技术的模态参数识别方法探讨

大型基础工程结构的特征参数识别通常是通过对环境载荷激励的结构响应进行分析来实现,随机减量(Random Decrement,RD)技术是环境激励下的模态参数识别方法中应用较广的方法。在实际应用中受环境、测量等条件的限制,信号常为含有某些优势频率的非平稳信号,常常导致随机减量技术在识别结构参数尤其是系统阻尼时带来较大误差。为提高随机减量技术在环境激励作用下识别结构参数的准确性,文中从分析随机减量信号频谱中的频率分布特性入手,结合随机减量函数产生的触发条件,给出了一种利用信号频谱的统计特征进行模态参数识别的方法。数值仿真结果表明该函数能准确识别在含有优势频率环境载荷作用下的结构参数。     

Data processing in subspace identification and modal parameter identification of an arch bridge

A data-processing method concerning subspace identification is presented to improve the identification of modal parameters from measured response data only. The identification procedure of this method consists of two phases, first estimating frequencies and damping ratios and then extracting mode shapes. Elements of Hankel matrices are specially rearranged to enhance the identifiability of weak characteristics and the robustness to noise contamination. Furthermore, an alternative stabilisation diagram in combination with component energy index is adopted to effectively separate spurious and physical modes. On the basis of identified frequencies, mode shapes are extracted from the signals obtained by filtering measured data with a series of band-pass filters. The proposed method was tested with a concrete-filled steel tubular arch bridge, which was subjected to ambient excitation. Gabor representation was also employed to process measured signals before conducting parameter identification. Identified results show that the proposed method can give a reliable separation of spurious and physical modes as well as accurate estimates of weak modes only from response signals.     

Output-only modal analysis of wind turbine tower based on vibration response under emergency stop

The identification technique of output-only modal parameters is proposed for the large wind turbine tower under emergency stop. Compared with the response of regular operating conditions, the immediate tower structural response under emergency stop much more resembles a state of free vibration, which is more appropriate for the modal identification of the wind turbine tower. The vibration response is measured in the nacelle, which is easy to perform in the field modal test. The variational mode decomposition (VMD) is applied to decompose the vibration response into several band-limited intrinsic mode functions. The free responses of decomposed functions are extracted by applying the random decrement technique (RDT). Finally, the modal damping ratio and natural frequency are identified from each free modal response by using the Hilbert transform method. Simulations and a 1.5 MW wind turbine field modal test results verify the effectiveness of the proposed identification method. The main modal parameters of wind turbine, including weak modes, are effectively extracted by using output-only vibration responses under emergency stop. The modal parameter identification method is provided for the large wind turbine structure under the engineering condition.     

Structural parameters identification using improved normal frequency response function method

An improved method that is based on a normal frequency response function (FRF) is proposed in this study in order to identify structural parameters such as mass, stiffness and damping matrices directly from the FRFs of a linear mechanical system. This paper demonstrates that the characteristic matrices may be extracted more accurately by using a weighted equation and by eliminating the matrix inverse operation. The method is verified for a four degrees-of-freedom lumped parameter system and an eight degrees-of-freedom finite element beam. Experimental verification is also performed for a free–free steel beam whose size and physical properties are the same as those of the finite element beam. The results show that the structural parameters, especially the damping matrix, can be estimated more accurately by the proposed method.     

Damping estimation using free decays and ambient vibration tests

This paper studies the quality of modal damping ratios estimates based on ambient and free vibration tests using both numerical simulations and data collected on large Civil Engineering structures where both tests were performed. The simulated data allowed to study the influence of factors like non-proportional damping or proximity of natural frequencies on the quality of the estimates and also to illustrate the influence of the identification algorithms parameters on the accuracy of the results. The analysis of data collected on a cable-stayed bridge, on the suspended roof of a stadium and on a footbridge permitted to compare the results of both testing approaches in real applications where some factors that cannot be realistically included in the numerical simulations can play an important role. The processing of ambient vibration responses is performed with two output-only identification approaches: frequency domain decomposition and stochastic subspace identification methods. The numerically simulated and the measured free decays were analyzed with a simple method based on filters and fitting of exponential decays and also with the use of subspace models.     

Structural response reconstruction based on the modal superposition method in the presence of closely spaced modes

An approach for structural response reconstruction based on the modal superposition method in the presence of closely spaced modes is proposed in this paper. In this method, the entire mode-set of a structure is divided into closely spaced modes and the rest of the modes. The rest of the modal responses whose response is known will be separated into individual modal response by using the empirical mode decomposition method with intermittency criteria. Starting from the mode shapes, derived from these modal responses, the rest of the modal responses at the unavailable locations can be acquired. Furthermore, the contribution of the closely spaced modal responses at the unavailable locations can be obtained based on these responses. This proposed method is valid if there is no periodic excitation. However, in practice it is common that a structure might be excited by transient, stochastic, periodic forces or a combination thereof, i.e. hybrid excitations. A hybrid approach for solving the reconstruction problem of hybrid excitations is also developed, which is based on the proposed method and transmissibility concept in response reconstruction. Numerical studies are conducted and compared with theoretical predictions for validation. Effects of background noise level, high damping ratio and multiple forces are investigated in detail.     

用凝聚技术综合时序分析法和有限元法识别机床的结构参数

机床接触面刚度和阻尼的确定是对机床进行动态分析和优化设计的关键问题之一。本文提出了一种识别机床接触刚度和阻尼的新方法,它利用一种新的凝聚技术把时序分析法和有限元法结合起来,从而只要利用一、二个不完全的振型就可以识别机床接触面的结构参数。这方法由两大部分组成。首先利用时序分析法从实验数据序列建立随机的自回滑动平均向量（ARMAV）模型并进而确定机床的模态参数。然后把机床结构有限元模型在某一复频下进行精确凝聚。并根据从时序分析法和凝聚后的有限元模型得出的模态参数必须相等的条件,就可以识别未知的机床结构参数。利用计算机仿真技术对新提出的方法进行了验证,证明它具有很高的识别精度。最后进行了立柱模型实验,对立柱底部的接触刚度和阻尼成功地进行了识别。     

Comparative study of damped FE model updating methods

Most of finite element (FE) model updating techniques do not employ damping matrices and hence, cannot be used for accurate prediction of complex frequency response functions (FRFs) and complex mode shapes. In this paper, a detailed comparison of two approaches of obtaining damped FE model updating methods are evaluated with the objective that the FRFs obtained from damped updated FE models is able to predict the measured FRFs accurately. In the first method, damped updating FE model is obtained by complex parameter-based updating procedure, which is a single-step procedure. In the second method, damped updated model is obtained by the FE model updating with damping identification, which is a two-step procedure. In the first step, mass and stiffness matrices are updated and in the second step, damping matrix is identified using updated mass and stiffness matrices, which are obtained in the previous step. The effectiveness of both methods is evaluated by numerical examples as well as by actual experimental data. Firstly, a study is performed using a numerical simulation based on fixed–fixed beam structure with non-proportional viscous damping model. The numerical study is followed by a case involving actual measured data for the case of F-shaped test structure. The updated results have shown that the complex parameter-based FE model updating procedure gives better matching of complex FRFs with the experimental data.     

Modal identification of linear non-proportionally damped systems by wavelet transform

A time-frequency identification technique based on wavelet transform is formulated and applied to free-decay responses of linear systems with non-proportional viscous damping. The Cauchy mother wavelet is used. Frequencies, modal damping ratios and complex mode shapes are identified from output-only free vibration signals. This identification technique has also shown to be effective when the (non-proportional) damping is significant.     

Identification of modal parameters from ambient vibration data using eigensystem realization algorithm with correlation technique

An effective identification method is developed for the determination of modal parameters of a structure based on the measured ambient response data. In this study, modification to Eigensystem Realization Algorithm with Data Correlation is proposed for modalparameter identification of structural systems subjected to stationary white-noise ambient vibration. By setting up a correlation -function matrix of stationary responses, as well as by introducing an appropriate matrix factorization, modal parameters of a system can be identified effectively through singular -value decomposition and eigenvalue analysis. Numerical simulations using practical excitation data confirm the validity and robustness of the proposed method in identifying modal parameters from stationary ambient vibration data under noisy conditions.