A frequency–spatial domain decomposition (FSDD) method for operational modal analysis

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Abstract

Following a brief review of the development of operational modal identification techniques, we describe a new method named frequency–spatial domain decomposition (FSDD), with theoretical background, formulation and algorithm. Three typical applications to civil engineering structures are presented to demonstrate the procedure and features of the method: a large-span stadium roof for finite-element model verification, a highway bridge for damage detection and a long-span cable-stayed bridge for structural health monitoring.

Introduction

Experimental modal analysis (EMA) has been widely applied to trouble shooting, structural dynamics modification, computational model updating, optimal dynamic design; and passive and active vibration control, as well as to vibration-based structural health monitoring in the aerospace, mechanical and civil engineering. An array of single-input/single-output (SISO), single-input/multi-output (SIMO) and multi-input/multi-output (MIMO) modal identification algorithms in the time, frequency and spatial domains have been developed in the past three decades.

However, the traditional EMA has a number of limitations. For example, (1) artificial excitation is normally conducted in the laboratory to measure the frequency response function (FRF) and impulse response function (IRF),which are typically used as primary data for subsequent modal parameter extraction. Unfortunately, FRF and IRF are very difficult, or even impossible, to measure in the field testing, especially for large structures; (2) in many industrial applications, the real operation conditions may differ significantly from those for lab testing with artificial excitation; and (3) components rather than complete systems are tested in the lab environment, and the boundary conditions need to be reasonably simulated.

Operational modal analysis (OMA) of mechanical systems subject to ambient or natural excitations under operational conditions has attracted a lot of attention in civil engineering since the early 1990s. It is also of great interest in aerospace and mechanical engineering due to its many advantages. For example, (1) OMA is cheap and fast to conduct, it needs no elaborate excitation equipment and boundary condition simulation. The traditional modal testing is therefore reduced to response measurement; (2) the dynamic characteristics of whole structural systems and not simply individual components, can be obtained under real operational conditions; (3) system characteristics under real loading can be linearized due to broadband random excitations; (4) all or part of the measurement coordinates can be used as references and the identification algorithm used for OMA must be MIMO-type. As a consequence, closely spaced modes or modes with repeated modal frequencies can be easily handled and hence it is suitable for real-world complex structures; and (5) operational modal identification with output-only measurements can be utilized not only for structural dynamic design and control, but also for in-situ vibration-based structural health monitoring of the structures.

In the early 1990s a natural excitation technique (NExT) was proposed [1]. NExT is based on the principle that correlation functions (COR) measured under natural excitations, which are assumed as white noise processes, can be expressed as the sum of exponentially decayed sinusoids. Modal parameters, i.e. natural frequency, damping ratio and mode shape coefficient of each decaying sinusoid can be identified from a measured COR subjected to natural excitations, just as IRF can be utilized for modal extraction in EMA. According to this principle, major time domain (TD) MIMO MID algorithms such as polyreference complex exponential (PRCE) [2], extended Ibrahim time domain (EITD) [3], eigen realization algorithm (ERA) [4] and their extensions [5], [6], which are successfully and widely employed for traditional EMA, can be adopted for OMA by using COR instead of IRF.

Many sophisticated TD operational modal identification methods have been proposed in the last 15 years based on modern system identification theory. NExT-type PRCE and EITD found better theoretical explanation based on a multi-dimensional, or vector autoregressive moving average, model (ARMAV) via an instrument-variable method (IVM). NExT-type ERA is, in essence, an implementation of a stochastic realization-based method [7]. The stochastic subspace technique (SST) was proposed as an extension of the subspace state-space system identification method [8] for output-only measurements [9]. SST is preferred to classical prediction-error methods, which are typical computationally intensive non-linear methods, because these are linear procedures.

However, all TD modal identification algorithms have a serious problem in model order determination and structural mode discrimination. Noise (spurious or computational) modes are always generated when extracting structural (physical) modes. These computational modes are even necessary to accommodate unwanted effects, such as measurement noise, leakage, residuals, non-linearity and un-modeled characteristics. The computational modes play an important role in permitting more accurate modal estimation by supplying statistical DOF to absorb these effects. In the traditional modal identification for EMA, IRF can be obtained via inverse FFT of FRF, and might need fewer computational modes. For operational modal identification, which utilizes correlation functions calculated from random response data, the problems with model order determination and structural modes discrimination become much more significant. Many modal validation techniques have been developed for distinguishing between structural and noise modes. An array of modal indicators has been developed for this purpose. A graphical approach making use of a stability diagram is a more widely adopted measure. However, up to now, there has been no guarantee of being able to distinguish structural modes from spurious ones when dealing with complex structures with noisy measurements.

However, classical frequency domain (FD) techniques, such as PSD peak-picking, have been applied for OMA. The peak-picking technique gives reasonable modal estimates if the modes are well separated [10]. Its main advantages compared to the TD techniques are that it has no bother of computational modes and is much faster and simpler to use. However, it is normally inaccurate. The accuracy of modal frequency estimation is limited to the frequency resolution of the PSD spectrum, operational deflection shapes are obtained instead of natural mode shapes, and damping ratio estimation via half-power point is biased or even impossible. Moreover, it is very difficult, if not impossible, to use when dealing with closely spaced modes, which are often encountered with OMA in real-world complex structures.

The challenge was to develop a new FD method that has most of the advantages without the major disadvantages of the PSD peak-picking technique. A new FD operational modal identification technique, named frequency domain decomposition (FDD), was developed in the year 2000 to meet this challenge [11]. The first generation FDD technique was proposed for estimation of modal frequencies and mode shapes. An enhance FDD (EFDD) was then developed to extend to damping ratio extraction [12]. A frequency–spatial domain decomposition (FSDD) has been proposed to further improve the FDD performance [13].

The theoretical background with relevant formulation and an algorithm for the FSDD method is introduced in this paper. Modal decomposition of output full PSD and half PSD is presented in detail, which acts as theoretical background of OMA in frequency domain under the assumption that the excitation spectrum is flat, or the input is white noise, i.e. its PSD is a constant matrix. The relevant approximate formulation is then derived for narrow band modal identification. The major procedure of the FSDD method is described based on spectrum decomposition of the output PSD via singular value or eigenvalue decomposition under the approximation that mode shapes are orthogonal. Three applications of FSDD to the OMA of typical civil engineering structures are then presented to demonstrate its procedures and features. These three typical applications are a large-span stadium roof for verifying the finite-element model, a highway bridge for damage detection and a long-span cable-stayed bridge for structural health monitoring.

Section snippets

Modal decomposition of full PSD

The frequency domain decomposition (FDD) technique is based on the formula of output power spectral density (PSD) with respect to stochastic input PSD.Gyy(iω)=Η(iω)Gxx(iω)Η(iω)Hwhere Gxx(), Gyy() are N×N input and PSD matrices, respectively, N is the number of measurements, and H() is the frequency response function (FRF) matrix, which can be expressed in partial fractions form asH(iω)=m=1M(Rriωλm+Rm*iωλm*)where λm=σm+iωdmis the mth pole andσm,ωdm are the damping factor and damped

Applications of FSDD to OMA of civil engineering structures

Three applications of the FSDD method to civil engineering structures are presented: a. large-span stadium roof, a highway bridge and a long-span cable-stayed bridge. The procedures and features are shown in the applications. It can be seen that structural modes are clearly located by the modal indication functions (MIF) with the ability to deal with very closely spaced or even repeated modes. Enhanced PSD, obtained via a modal filter making use of a singular vector corresponding to a relevant

Conclusion

The development and theoretical background of a new operational modal identification method, frequency–spatial domain decomposition (FSDD) is described in this paper. The theoretical background with relevant formulation and algorithm has been presented in detail. Three applications to typical civil engineering structures are presented: a. large-span stadium roof for verifying finite-element model, a highway bridge for damage detection and a long-span cable-stayed bridge for structural health

Acknowledgement

The support from the Global COE Program at the Tokyo Polytechnic University, sponsored by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, is gratefully acknowledged. The first two authors would also like to acknowledge the support of the Aeronautical Science Foundation of China for the research project No. 04I52065.

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