The Frequency Domain Decomposition (FDD) technique is known as one of the most user friendly and powerful techniques for operational modal analysis of structures. However, the classical implementation of the technique requires some user interaction. The present paper describes an algorithm for automated FDD, thus a version of FDD where no user interaction is required. Such algorithm can be used for obtaining a default estimate of modal parameters in commercial software for operational modal analysis or even more important – it can be used as the modal information engine in a system for structural health monitoring. Introduction Frequency domain techniques have always been popular. Even among people who will state that they only use time domain techniques for modal identification, as soon as they get new data in their hands, the first thing they will normally do is to take a look at some frequency domain functions. For Operational Modal Analysis frequency domain techniques are based on spectral density functions, Bendat and Piersol [1]. Working directly with spectral density function has been popular and is still used a lot; see for instance Felber [2]. One of the problems working directly with the spectral density functions is the amount of data the user has to work with simultaneously. For instance in a case with 8 = m N channels of measurements, the user has to deal with different spectral density functions. Further, even though the spectral densities by directly depicting the modal peaks and thus gives a direct indication of the presence of modes, the spectral density function in itself does not provide the user with modal information since the spectral density function is linear combination of the modal responses. Therefore working directly with spectral density functions will limit modal identification to cases with well separated modes. 36 2 / ) 8 8 ( 2 = + The Frequency Domain Decomposition technique is a way to solve these two problems, Brincker et al [3], [4]. The technique simplifies the user interaction because the user has only to consider one frequency domain function the singular value plot of the spectral density matrix. This plot concentrates information from all spectral density functions. Further, if some simple assumptions are fulfilled, the technique directly provides a modal decomposition of the vibration information, and the modal information for each mode – even in the case of closely spaced modes and noise – can be extracted easily and accurately. The principle in the Frequency domain Decomposition (FDD) techniques is easiest illustrated by realizing that any response can by written in modal co-ordinates ) ( ... ) ( ) ( ) ( 2 1 1 1 t t q t q t Φq φ φ y = + + = (1) Now obtaining the covariance matrix of the responses { } T yy t t E ) ( ) ( ) ( y y C τ τ + = (2) and using equation (1) leads to { } T qq T T yy t t E Φ ΦC Φ q Φq C ) ( ) ( ) ( ) ( τ τ τ = + = (3) Then by taking the Fourier transform T qq yy f f Φ ΦG G ) ( ) ( = (4) Thus if the modal co-ordinates are un-correlated, the power spectral density matrix of the modal coordinates is diagonal, and thus, if the mode shapes are orthogonal, then Eq. (4) is a singular value decomposition (SVD) of the response spectral matrix. ) ( f qq S Therefore, FDD is based on taking the SVD of the spectral density matrix [ ] T i yy f s f f ) ( ) ( ) ( U U G = (5) The matrix is a matrix of singular vectors and the matrix [ L , , 2 1 u u U = ] [ ] i s is a diagonal matrix of singular values. As it appears from this explanation, plotting the singular values of the spectral density matrix will provide an overlaid plot of the auto spectral densities of the modal coordinates. Note here that the singular matrix is a function of frequency because of the sorting process that is taking place as a part of the SVD algorithm. A mode is identified by looking at where the first singular value has a peak, let us say at the [ L , , 2 1 u u U = ] frequency . This defines in the simplest form of the FDD technique the peak picking version of FDD the modal frequency. The corresponding mode shape is obtained as the corresponding first singular vector in . 0 f 1 u U ) ( 0 1 f u φ = (6) Introducing modal discrimination The process of findings peaks on a function is actually easy to automate. However, we need to define indicators that can help us distinguishing between different modes and between modes and noise. Let us say that we have identified a peak in the first singular value. The question is now if this is a liable modal peak or is if it just a noise peak. Calculating the correlation between the first singular vector at the peak – the mode shape vector at that point and the first singular vector at neighboring points defines the discriminator function called the modal coherence ) ( ) ( ) ( 0 1 1 0 1 f f f d T u u = (7) If the modal coherence is close to unity, then the first singular value at the neighboring point correspond to the same modal coordinate, and therefore, the same mode is dominating. This function is helpful in discriminating between points dominated by modal information and points dominated by noise. If the components of each of the vectors in Eq. (6) are random, then { } 0 ) ( ) ( 1 0 1 = f f E T u u (8) and since the length is unity { } m T N f f Var / 1 ) ( ) ( 1 0 1 = u u (9) Thus the more measurement channels we have the closer two points with random (non-physical) information will get to zero. A reasonable criterion for accepting the neighboring point as a point with similar physical information, and thus accepting the presence of physical information at that frequency, could be by introducing a threshold level and the requirement 1 Ω
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