Continuous-time operational modal analysis in the presence of harmonic disturbances—The multivariate case

Recently [R. Pintelon, B. Peeters, P. Guillaume, Continuous-time operational modal analysis in the presence of harmonic disturbances, Mechanical Systems and Signal Processing 22 (5) (2008) 1017–1035] a single-output algorithm for continuous-time operational modal analysis in the presence of harmonic disturbances with time-varying frequency has been developed. This paper extends the results of Pintelon, et al. [Continuous-time operational modal analysis in the presence of harmonic disturbances, Mechanical Systems and Signal Processing 22 (5) (2008) 1017–1035] to multi-output signals. The statistical performance of the proposed maximum likelihood estimator is illustrated on simulations and real helicopter data.     

An average inverse power ratio method for the damping estimation from a frequency response function

The well-known bandwidth method, especially the half-power bandwidth method [1], [3], [4] (Bishop and Gladwell, 1963; Nashif et al., 1985; Yin, 2008) is probably the simplest method for the damping estimation from a frequency response function (FRF). This method is quite efficient for simple model test cases when modes are well separated. However, the determination of the two frequencies corresponding to a given power ratio value requires numerical interpolations of experimental discrete data of frequency response functions. In this paper, an average inverse power ratio method is proposed. The average value is computed at two frequencies symmetrically located from a peak amplitude frequency. Then the damping ratio can be estimated by using a formula similar to that of the bandwidth method. In this way, the damping estimation becomes straightforward from the FRF data. This new method could be useful for engineers in practice due to its extreme simplicity.     

An operational modal analysis approach based on parametrically identified multivariable transmissibilities

In this contribution the approach to identify modal parameters from output-only (scalar) transmissibility measurements [C. Devriendt, P. Guillaume, The use of transmissibility measurements in output-only modal analysis, Mechanical Systems and Signal Processing 21 (7) (2007) 2689–2696] is generalized to multivariable transmissibilities. In general, the poles that are identified from (scalar as well as multivariable) transmissibility measurements do not correspond with the system's poles. However, by combining transmissibility measurements under different loading conditions, it is shown in this paper how model parameters can be identified from multivariable transmissibility measurements.     

Substraction technique and finite difference formulas for modal parameter estimation

This paper deals with the use of the substraction technique and the very simple finite difference formulas for modal parameter estimation. For closely spaced modes, the estimation of the formulas is not accurate and depends strongly on the frequency-response data involved in computations. An iterative procedure based on the substraction technique combined with the finite difference formulas is proposed. Application to experimental frequency response data shows that after the iterative procedure is applied, the estimation depends much less on the data involved, and the estimation reliability and accuracy are significantly improved, especially for closely spaced modes.     

PARAMETRIC TIME-DOMAIN METHODS FOR THE IDENTIFICATION OF VIBRATING STRUCTURES—A CRITICAL COMPARISON AND ASSESSMENT

A critical comparison of four stochastic (PEM, 2SLS, LMS, IV) and three deterministic (LS, Prony, ERA) methods for the parametric time-domain identification of vibrating structures from random excitation and noise-corrupted response signals is presented. Concise summaries of the methods, highlighting their principles and realisations, are provided, while the study is based upon a six-degree-of-freedom structural model characterised by two closely spaced modes, two weak modes and a wide range of modal damping. Monte-Carlo experiments under two different (wideband/narrowband) noise environments are performed, along with comparisons with non-parametric frequency domain identification.The stochastic methods—most notably PEM, LMS and IV—are, at the price of increased complexity, shown to lead to potential advantages in non-negligible noise cases, while deterministic methods—most notably Prony and ERA—may suffice under negligible noise. In addition: (a) Model order estimation is shown not to be straightforward, and significant overdetermination is required (especially by the LS). (b) A weak closely spaced mode is hard to identify, while being completely missed by the deterministic methods and the 2SLS. (c) A highly damped mode presents some difficulty as well (mainly for the LS). (d) False modes are exhibited, primarily by the LS (wideband noise) and the ERA (narrowband noise). (e) The achievable estimation accuracy is generally high for the natural frequencies, lower for the damping ratios, and even more so for the residues (mode shapes). Furthermore, accuracy is somewhat lower for the closely spaced modes and significantly lower for the two highly damped modes. (f) PEM, LMS and IV achieve lower bias errors and good overall accuracy, followed by the 2SLS, Prony, ERA and, finally, LS. (g) Unstable modes are mainly exhibited by the IV and ERA. (h) All methods appear sensitive to the selected model order and design parameters, and user expertise is necessary.     

密集模态分离及其参数识别方法研究

针对密集模态参数识别精度差的问题,分析造成识别精度低的原因.根据影响模态混叠程度的固有频率和阻尼比两大因素,定义衡量模态间混叠程度的模态密集度因子.基于化多模态参数识别问题为单模态参数识别问题的思想,提出加逆衰减指数窗与带通滤波相结合的密集模态分离方法,先将各模态近似提取出来,再用时频分析方法识别模态参数.研究相关处理去噪方法,理论证明小阻尼情况下相关处理去噪的模态参数不变性.通过仿真和实验验证方法的有效性和优越性.     

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.     

Frequency-domain approach for the parametric identification of structures with modal interference

Journal of Mechanical Science and Technology - This study aims to improve the accuracy of the parametric estimation of systems with modal interference in the frequency domain. The theory of modal...     

Application of polynomial control to design a robust oscillation-damping controller in a multimachine power system

This paper addresses the application of a static Var compensator (SVC) to improve the damping of interarea oscillations. Optimal location and size of SVC are defined using bifurcation and modal analysis to satisfy its primary application. Furthermore, the best-input signal for damping controller is selected using Hankel singular values and right half plane-zeros. The proposed approach is aimed to design a robust PI controller based on interval plants and Kharitonov׳s theorem. The objective here is to determine the stability region to attain robust stability, the desired phase margin, gain margin, and bandwidth. The intersection of the resulting stability regions yields the set of k_p–k_i parameters. In addition, optimal multiobjective design of PI controller using particle swarm optimization (PSO) algorithm is presented. The effectiveness of the suggested controllers in damping of local and interarea oscillation modes of a multimachine power system, over a wide range of loading conditions and system configurations, is confirmed through eigenvalue analysis and nonlinear time domain simulation.     

Exact vibration results for stepped circular plates with free edges

A pontoon-type, very large floating structure (VLFS) is often modeled as a huge plate with free edges when performing a hydroelastic analysis under the action of waves. The analysis consists of separating the hydrodynamic analysis from the dynamic response analysis of the VLFS. The deflection of the plate is decomposed into vibration modes where as many higher modes as possible should be used to capture the actual deflection shapes and the stresses. It is generally accepted that finite element method and the Ritz-type energy method fail to model zones with steep gradients which are encountered in, for instance, the stress resultants near the free edges of plates [Journal of Engineering Mechanics 1983;109(2):537–56]. Moreover, the natural boundary conditions are not satisfied completely because they are not enforced a priori [International Journal of Solids and Structures 2001;38:6525–58, Journal of Computational Structural Engineering 2001;1(1):49–57, Journal of Structural Engineering ASCE 2002;128(2):249–57, Computers and Structures 2002:80(2):145–54]. Exact solutions for frequencies, mode shapes and modal stress resultants are thus very important as they provide valuable benchmarks for assessing the convergence, accuracy and validity of numerical results obtained using the finite element method. To this end, we present the exact vibration results for stepped circular plates with free edges. When employed in a hydroelastic analysis, these exact vibration solutions yield accurate deflections and stress resultants (stresses) for circular VLFSs with stepped drafts.     

Stiffness matrix of thin-walled curved beam for spatially coupled stability analysis

The exact stiffness matrix for the spatially coupled stability analysis of thin-walled curved beam with non-symmetric cross-section subjected to uniform compression is newly presented. For this, the elastic strain energy including the axially/flexurally/torsionally coupled terms and the potential energy due to the initial stress resultant are introduced. Then, equilibrium equations and force–deformation relationship are derived from the energy principle and explicit expressions for displacement parameters are derived based on power series expansions of displacement components. Next, the exact element stiffness matrix is determined using force–deformation relationships. In order to verify the accuracy of this study, the numerical solutions are presented and compared with the finite element solutions using the Hermitian curved beam elements. In addition, the coupling of symmetric and anti-symmetric modes at the buckling load crossover with change in curvature of beam is investigated.     

Output-only analysis of structures with closely spaced poles

In the framework of the operational modal analysis, several approaches have been developed for estimating the modal parameters, i.e., natural frequencies, damping ratios, and mode shapes. Specifically, a technique capable to evaluate the biased (i.e., unscaled by a constant or an almost constant function) frequency response functions, FRFs, has been proposed. Assuming that only the responses of the structure are disposable, the technique allows one to estimate biased FRFs starting from the power spectral densities, PSDs, and applying the Hilbert transform. This paper deals with the estimates of the modal analysis parameters mentioned above. It is possible to obtain each single mode shape, from the singular vectors achieved by applying the singular value decomposition to the FRF matrix evaluated at the spectral line corresponding to the selected natural frequency. A special attention will be devoted to structures with coupled modes, i.e., closely spaced modes. Once the FRFs have been obtained, the natural frequencies and damping ratios could be achieved either in the frequency domain or in the time domain. Experimental tests, carried out on beams, plates and on the AB-204 helicopter blade, will be presented.     

基于奇异值分解的频响函数降噪方法

为了提高外场测试中频响函数的信噪比,提出了一种基于奇异值分解的频响函数降噪方法。该方法首先对脉冲响应函数在相空间内进行重构;然后对重构轨道矩阵进行奇异值分解达到降噪的目的。其中,降噪阶次通过奇异熵增量进行确定。采用GARTEUR飞机模型建立具有密集模态的仿真算例进行验证。结果表明,在噪声干扰较大时,该降噪方法能够显著改善模态参数的识别精度,尤其是阻尼的识别精度。     

Measurement, identification and modelling of damping in pneumatic tyres

This paper deals with a research approach specifically designed for the measurement, identification and modelling of damping in pneumatic tyres. As a less rigorous and more global approach of characterizing the engineering property of tyre damping, experimental modal analysis and methodologies based on both real and complex modes are introduced. In the context of the real modal analysis, the flexible ring tyre model is also investigated. A standard testing procedure is employed for the collection of vibration data in pneumatic tyres, and the linearity and reciprocity of the tyre structure and the orthogonality of its modes is proven at the initial stage of investigation. After demonstrating the disadvantages of the proportional damping assumption in describing the tyre vibration, the first-order complex modal interpretation with general viscous damping assumption is implemented and discussed, including an appropriate normalization of the complex modes. As a further extension to the analysis, the second-order Rayleigh's small-damping approximation is introduced into the complex modes framework, in order to obtain an accurate global estimation of the damping distribution.As a natural evolvement of the applied damping models in this paper, from the simplest single-valued proportional damping in flexible ring tyre model, proportional viscous damping in real modal interpretation, up to the generally distributed viscous damping in second-order approximation of the complex modal interpretation, the finally identified damping matrices, both in modal and physical coordinates, present a reasonable explanation of damping effect in pneumatic tyres. Agreement between the theory and experimentally identified results also proves the suitability of this approach for damping identification in complex structures.     

基于贝叶斯运行模态分析法的砂轮架动态特性分析

为了研究砂轮架的固有模态和在工作环境中的谐波模态,运用贝叶斯理论和运行模态分析相结合的模态参数识别技术--分步测试的贝叶斯运行模态分析法对环境状态下的砂轮架进行模态识别,获得了结构的动态特性,包括固有频率、振型以及阻尼特性等,并对外激励和预测误差水平以及信噪比进行了评估;将该模态识别结果与传统试验模态分析结果进行对比,验证了贝叶斯运行模态分析法的应用可行性;进一步在工作环境下对砂轮架进行了振动测试,对比开机前的模态识别结果和开机后砂轮架的功率谱密度图,成功地区分出砂轮架的固有模态和实际工作环境中由结构周期激励引起的谐波模态。     

基于Hilbert-Huang变换与理想带通滤波器的系统识别

针对经验模态分解存在模态混叠现象,提出基于Hilbert-Huang变换与理想带通滤波器的系统识别方法。该方法利用傅里叶变换得到结构加速度响应频响函数,粗略估计固有频率范围,通过半功率带宽法设计理想带通滤波器,定量化确定通带带宽,使信号在经过滤波器后频域内零相移,同时不改变其幅值谱。结构响应通过指定频带的理想带通滤波器产生若干窄带信号,利用经验模态分解获取结构模态响应,经Hilbert变换构造模态响应解析信号,并通过线性最小二乘拟合提取结构模态参数与物理参数。结果表明:半功率带宽法可实现带通滤波器频带的定量化设计,理想带通滤波器的零相移特点较好契合Hilbert-Huang变换用于系统识别的要求,两者结合可有效地解决模态混叠现象,减少虚假模态,大大提高结构系统识别精度。     

共轭子结构解法

论述了机床结构在一般阻尼情况下的复模态分析中,采用共轭子结构解法来求解机床动态特性的模态参数表达式。共轭子结构解法以共轭复模态矩阵作为模态变换矩阵,整机相应分解为两个共轭子结构。利用振型叠加原理,求得整机的动态特性的模态参数表达式。共轭子结构解法简化了复杂的计算过程,也适用于其它复模态分析计算。     

Novel cyclostationarity-based blind source separation algorithm using second order statistical properties: Theory and application to the bearing defect diagnosis

Among signal processing techniques, blind source separation (BSS) and the underlying mathematical tool of independent component analysis (ICA) are of continuously growing interest in the scientific community of various research domains. Vibration analysis is a potential application field of this quite recent technique.Actually, BSS methods aim to retrieve unknown source signals from a set of their observations coming to a matrix of sensors, without necessarily having any prior knowledge about the sources. In monitoring and diagnosis purposes, bearing defects constitute a problem for manufacturers who aim at predicting those faults as well as potential engines breakdowns. These defects may be the unknown sources one wants to estimate from a set of recorded signals by a matrix of accelerometers placed close to the rotating machine.It has been shown that these vibration signals are wide-sense cyclostationary [[11] R.B.Randall, J. Antoni, S. Chobsaard, The relationship between spectral correlation and envelope analysis in the diagnostics of bearing faults and other cyclostationary machine signals, Mechanical Systems and Signal Processing 15 (5) (2001) 945–962]. The new algorithm of BSS proposed in this work is based, precisely, on that property. Second-order statistics of such processes led us to a new separation criterion for blind source separation. The theoretical results of this study, simulation and experimental analysis are presented in here. Perspectives for future research conclude this paper.     

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.     

基于跑车余振的混凝土梁桥阻尼比估算方法

分析了混凝土梁桥跑车余振信号的特点,提出桥梁自由衰减振动分段幅值估算阻尼比的方法,以提高阻尼比的识别精度。该方法通过对结构余振信号分段截取做FFT变换,利用相邻两块数据的幅值求得其阻尼比。仿真分析比较了该方法与传统半功率点法估算结构阻尼比的精度,结果表明该方法在有限衰减数据长度内精度优于传统半功率点法。利用该方法对某混凝土连续刚构桥的跑车余振信号进行阻尼比识别,得到前两阶竖弯频率对应的阻尼比,识别结果稳定,可作为该桥的实际阻尼比。