THE NEW HIGHER ORDER SPECTRAL TECHNIQUES FOR NON-LINEARITY MONITORING OF STRUCTURES AND MACHINERY. L. Gelman. Cranfield University, UK

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1 The 2 th International Conference of the Slovenian Society for Non-Destructive Testing Application of Conteporary Non-Destructive Testing in Engineering Septeber 4-6, 203, Portorož, Slovenia More info about this article: THE NEW HIGHER ORDER SPECTRAL TECHNIQUES FOR NON-LINEARITY MONITORING OF STRUCTURES AND MACHINERY L. Gelan Cranfield University, UK ABSTRACT The new higher order spectral techniques, the noralized cross-covariances of coplex spectral coponents, are proposed for onitoring structure and achinery non-linearity and signal non- Gaussianity and estiating the haronic phase coupling of signals fro structures and achines. Noralization of the proposed techniques is also developed. It is shown by siulation that the proposed techniques provide effectiveness gain for detection of non-linearity in coparison with the HOS.. Introduction For onitoring of a structure and a achinery non-linearity and non-gaussianity of signals fro a structure and a achinery due to daage and estiating the haronic phase coupling of signals fro a structure and achinery, an input excitation (e. g. vibration excitation, acoustical excitation, etc.) excites the resonance oscillations of a structure or a achinery in question and resonance oscillations are processed by the higher order spectra (HOS) []. This approach has been widely investigated for stationary and non-stationary signals [-3]. For diagnosing structure/achinery nonlinearity, we propose to use the noralised crosscovariances of order n between n coplex spectral coponents. The physical sense of this proposition is that if coplex spectral coponents have appeared due to nonlinearity (daage), these coponents have non-zero cross covariances. It can be easily shown that the widely used noralized HOS: i.e. the bicoherence and the skewness for the HOS of order 3 and the kurtosis and the tricoherence for the HOS of order 4, do not present the exact noralized cross-covariances between three and four coplex spectral coponents respectively. It also can be shown that in the general case, the noralised HOS of order n also do not present the exact noralized cross-covariances between n coplex spectral coponents. Therefore, the noralised cross- covariances between three, four and n coplex spectral coponents cannot be estiated by these traditional noralized HOS. Thus, the ain novelty of this paper is the proposed new higher order spectral techniques that are the noralized cross-covariances between n coplex spectral coponents. The purposes of this paper are to: 525

2 propose the new higher order spectral techniques for structure and achinery health onitoring: the noralized cross-covariances of n coplex spectral coponents copare by siulation the proposed techniques with the noralized HOS 2. The new higher order spectral cross-covariances The novel condition onitoring techniques are proposed here: the higher order spectral techniques, the noralized cross-covariance of n coplex spectral coponents. For estiating the proposed techniques, the following steps should be undertaken: the tie doain signal should be divided into overlapping segents by the internal tie window, =...M, M defines the total nuber of overlapping segents in the signals. The generic expression of the proposed cross-covariances of order n based on the Fourier transfor is as follows: = () (, 2,, ) = = [ ( Σ ) ( Σ )] where X f ) is the Fourier transfor at frequency f j at segent duration ( j t of a signal, j =, n, f nσ is the accuulated frequency, f nσ = f j, is a sybol of the coplex = conjugate, is the ean value of variable. The proposed cross-covariances () are coplex valued, estiated by the Fourier transfors of a signal at n frequencies and depend on (n-) frequencies. Only in the particular case of the zeroean coplex spectral coponents, the functions () for order n, 3 and 4 are the classical unnoralized HOS of order n, the classical bispectru and the classical trispectru respectively. In the general case of the non-zero-ean coplex spectral coponents, the proposed functions of order 3 and 4 are not the bispctru and the trispectru. The physical significance of the proposed functions is that they provide a easure of cross-covariances between n coplex spectral coponents. It is known fro the classical statistical analysis that the cross-covariances should be noralised in order to avoid the isleading interpretation. The standard noralization of the cross-covariances is eployed here as follows for order n, 3 and 4 respectively: (, 2 ) = (, 2, 3 ) = (, 2,, ) = where var is the sybol of the variance. n j (, 2,, ) = ( Σ ) = { ( ) ( )} { ( 2 ) ( 2 )} { ( + 2 ) ( + 2 )} (3) [ ( )] [ ( 2 )] [ ( + 2 )] = { ( ) ( )} { ( 2 ) ( 2 )} { ( 3 ) ( 3 )} { ( ) ( )} [ ( )] [ ( 2 )] [ ( 3 )] [ ( )] In the general case of order n, the proposed noralised cross-covariances (2) differ fro the traditional noralised HOS of order n. It can be also seen fro expressions (3-4), that for orders 3 and 4, the noralised triple and fourth covariances differs fro the traditional noralized HOS, the bicoherence and skewness and the tricoherence and kurtosis respectively. This difference reains even for the zero-ean spectral coponents. (2) (4) 526

3 The noralisations of the proposed cross-covariances allow avoidance of the isleading interpretation of the proposed techniques () due to variations of the power spectral density of a signal. The proposed techniques can be also used for non-stationary signals by eploying the appropriate tie-frequency transfors (e. g. the chirp-wigner transfor [4], the short tie chirp-fourier transfor [5] etc.). This can be done by substituting the appropriate tiefrequency transfors for the Fourier transfor in equations (-4). To deonstrate that the proposed techniques can effectively detect structure non-linearity due to daage and to copare the with the traditional HOS, a siulation test with linear and nonlinear (bilinear) structures was perfored. An input rando cosine excitation with constant aplitude, rando initial phase and linearly changed instantaneous frequency in tie (i. e. the chirp signal) has been passed via the following nonlinear (bilinear) syste: ( ) x + hx + x = A Ω t x x + hx + x = A Ω ( t) x < 2 2 ωs cos, 0, 2 2 ωc cos, 0, (5) where X x =, X is the displaceent, c are the ass and daping coefficient respectively, k S and c ks kc h =, h is daping; ωs =, ωc =, and 2 k C are the stiffness for positive A displaceent and stiffness for negative displaceent respectively, A =, A is the constant aplitude of the input signal, Ω( t) ω ( t) dt, ( t) linearly changed angular frequency. = Ω is the instantaneous phase, ω () t is the The initial phase of the each siulated signal has been taken randoly and is uniforly distributed in the range [0; 2π ]. The output signal of the bilinear structure is transient, with variable instantaneous frequency; therefore, the noralised spectral cross-covariance of order 3 based on the chirp-fourier transfor [5] is eployed for non-linearity detection. The non-stationary rando cosine vibration excitation excited the resonance oscillations of a structure. 300 signals fro the linear structure and 300 signals fro the bilinear structure were tested for non-linearity detection. The resonance frequencies of the linear and bilinear structures are 4. Hz and 3.9 Hz respectively, the chirp rate is 0.5Hz/s, the stiffness ratio is The proposed noralised cross-covariances of order 3 and the bicoherence based on the chirp- Fourier transfor of the structure resonance oscillations have been eployed for detecting additional level of structure nonlinearity. The cross-covariances and the bicoherence at the fundaental and second haronics have been eployed. The following paraeters have been used for estiating the cross-covariances and the bicoherence: the frequency resolution is 3.7Hz (i.e. segent size is 0.27s), duration of signals is 5s, segent overlapping is 60%, the internal tie doain window is the Haing window, the sapling frequency is 3600Hz. The Fisher criteria [6] for detection effectiveness are 373 and 260 for the proposed technique and the bicoherence. It is known [6] that features with higher values of the Fisher criterion provide better detection effectiveness. 527

4 Thus, the proposed technique provides effectiveness gain.43 ties in coparison with the HOS and, therefore, is ore effective for non-linearity detection. 3. Conclusions. The new higher order spectral techniques, the noralized cross-covariances of n coplex spectral coponents, are proposed for onitoring structure and achinery non-linearity and signal non-gaussianity and estiating the haronic phase coupling of signals fro structures and achines. Noralization of the proposed techniques is also developed. 2. The proposed techniques differ fro the classical higher order spectral techniques. The proposed un-noralised spectral cross-covariances coincide with the classical un-noralised higher order spectra only for the particular case of the zero ean spectral coponents. The noralised spectral cross-covariances differ fro the classical higher order spectral techniques even for the particular case of the zero ean spectral coponents. 3. It is shown by siulation that the proposed techniques provide effectiveness gain.43 ties for detection of non-linearity in coparison with the HOS. 4. The proposed techniques could be extended for onitoring of structure and achinery nonlinearity and signal non-gaussianity due to daage and estiating the haronic phase coupling of signals fro structures and achines for non-stationary signals by eploying the appropriate tie-frequency transfors in equations (-4). The proposed techniques could be used in echanical and electrical engineering, telecounication, underwater acoustics, etc. 4. Acknowledgeent The author is very thankful to MSc student A. Stepien (Cranfield University) for carrying out a siulation and processing of the siulated data. 5. References. Fackrell J. W, White P. R, Haond J. K, Pinnington R. J. The interpretation of the bispectra of vibration signals. Mechanical Systes and Signal Processing 995; 9(3): Ki Y. C, Powers E. J. Digital bispectral analysis and its applications to non-linear wave interactions. IEEE Transactions on Plasa Science 979; 7 (2): Collis W. B, White P. R, Haond J. K. High order spectra: the bispectru and trispectru. Mechanical Systes and Signal Processing 998; 2 (3): Schreier P. J, Scharf L. L. Higher-order spectral analysis of coplex signals. Signal Processing 2006; 86 (): Mendel J. M. Tutorial on higher-order statistics (spectra) in signal processing and syste theory: theoretical results and soe applications. Proc. of the IEEE 99; 79 (3): Nikias C. L, Mendel J. M. Signal processing with higher-order spectra. IEEE Signal Processing Magazine 993; 0 (3): McCorick A. C, Nandi A. K. Bispectral and trispectral features for achine condition diagnosis. IEE Proc. of the Vision, Iage and Signal Processing 999; 46 (5): Gelan L, Petrunin I. The new ultidiensional tie/ulti-frequency transfor for higher order spectral analysis. Multidiensional Systes and Signal Processing 2007; 8 (4):

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