OPERATIONAL MODAL ANALYSIS SCHEMES USING CORRELATION TECHNIQUE

Acta Mechanica Solida Sinica, Vol. 18, No. 1, March, 2005ISSN 0894-9166Published by AMSS Press, Wuhan, China. DOI: 10.1007/s10338-005-0513-4OPERATIONAL MODAL ANALYSIS SCHEMES USINGCORRELATION TECHNIQUE*Zheng Min'Shen Fan2Chen Huaibai2(I College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China)(2 Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics,Nanjing 210016, China)Received 3 March 2004; revision received 1 November 2004ABSTRACT, For some large -scale engineering structures in operating conditions, modal param-eters estimation must base itself on response-only data. This problem has received a considerableamount of attention in the past few years. It is well known that the cross- correlation functionbetween the measured responses is a sum of complex exponential functions of the same form as theimpulse response function of the original system. So this paper presents a time domain operatingmodal identifcation global scheme and a frequency-domain scheme from output-only by cou-pling the cross- correlation function with conventional modal parameter estimation. The outlinedtechniques are applied to an airplane model to estimate modal parameters from response-onlydata.KEY WORDS modal analysis, operating mode, global identifcation, cross correlationI. INTRODUCTIONRecently, the problem about operating modal analysis (OMA) has gained more and more attention.For example, the EUROKE project SINOPSYS is designed to investigate in-operation modal identifi-cation for some structures such as offshore structures, bridges, aeroengines excited by wind, waves, ortraffic. Early in the development of approaches to estimating modal parameters from output-only data,there were peak picking from power spectral density (PSD) functions1l autoregressive moving average(ARMA) models[2, ITD/ random decrement processing[3), maximum entropy method (MEM)|4] andOperational Delection Shapes (ODS)5. In 1995, James and Carne working at Sandia National Labo-ratory of America put forward the concept of Natural Excitation Technique (NExT)6, which meansidentifying the modal parameters of large -scale structures in operating conditions with the unknowninputs. They applied the technique to a running wind turbine and presented some time-domain modalidentification schemes using cross-correlation functions instead of impulse response functions. The sub-space techniques71 have been used to identify modal parame中国煤化工e data in recentyears. Two operating modal identification global schemes (=d in this paper.One is done in the time domain and the other in frequencyfY;CNMHG* Project supported by the National Natural Science Foundation of China (No. 50205012), Aeronautics Foundation(No. 01152059) and Civil Aviation Foundation (No.1007-272001).Vol. 18, No. 1 Zheng Min et al: Operational Model Analysis Schemes Using Correlation TechniqueII. THEORETICAL ASPECTS2.1. The Cross-Correlation Function and the Impulse Response Function[8]The cross-correlation function Rnp(T) between the output n and the output p due to a white noiseinput can be expressed as2NRmp(T)= >vhrQpre>rTr=1where T- a time separation; N- number of modes; ψnr- -nth component of the rth mode shape; λr-rth complex eigenvalue; Qpr- new constant. It shows that the cross-correlation function in Eq.(1) is asum of complex exponential functions of the same form as the impulse response function of the originalsystem as follows:hni(t)= > ynrWire^rt(2)where hn(t)- -impulse response at point n due to the input force at point l; Wir- -modal participationfactor. Consequently, the classical modal parameter techniques using impulse response functions areappropriate to extract the modal parameters of in- operating structures from response -onlyl6I.2.2. Time-Domain Operating Modal Identification Global SchemeWe obtain the impulse response function matrix h(k)h(k) =④e4k^w3)where k is the number of sample points, 0 the time internal,亚the mode shape matrix, A the complexeigenvalue matrix and w the modal participation factor matrix.The impulse response function matrix in the conventional time -domain modal identifcation globalscbemel9] satisfiesh(k)- Alh(k-1)-...- Aph(k-p)=0(4)The auto and cross- correlation function matrix R(k) between N responses and P responses which serveas references can be written asR(k)= e4k^Q=业z*Q .5)where Z = eA0. The correlation functions matrix can be used instead of the impulse response functionsmatrix in Eq.(4) will result in:R(k)=-AiR(k-1)-...- ApR(k-p)(6)We knowR(k-i)=R(k-i)(i= 1,2,3..)(7)Unifying Eq.(6) and Eq.(7) givesR(k)= AR(k- 1)(8)R(k)-Ap 1R(k- 1) .where A =, R(k) =. By considering discrete time instantR(k-p+ 1)k,k+1 ,..k + p, a larger matrix equation can be constructed as follows:[原(k) R(k+1) R(k+2)..R(k+明=A[瓯(k-1) R(k) (R(+1)..R(k+p-1)](9)matrix A can be obtained, and then matrix A1, A2.中国煤化工Substituting Eq.(5) into Eq.(8), the following can be deriv= A亚MYHCNMHG(10)where重= [ψzk-1业z*-2 .z2-2.Solving an eigenvalue problem can yield matrices Z and业. From Z = eAo, modal frequencies anddampingratios can be obtained. Modal shape matrix业can be found from亚ACTA MECHANICA SOLIDA SINICA20052.3. The Cross-Power Spectral Density Function and the Frequency Response FunctionThe Fourier transform of the impulse response function hni(t) in Eq.(2) can be performed, then thefrequency response function Hnl(jw) can be expressed as follows:Hn(jw)='2NynrWr(11)The cross power spectral density function Gnp(jw) is the Fourier transform of the cross-correlationfunctionRnp(T) in Eq.(1), and can be expressed asVnrQprGnp(jw)=2jw- λ(12)It can be easily shown that the cross- power spectral density function in Eq.(12) is a sum of fractionfunctions that is of the same form as the frequency response function in Eq.(11). Consequently, thecross- power spectral density functions between responses can be used to estimate modal parametersfrom output-only instead of frequency response functions.2.4. The Frequency-Domain Operating Modal Identification Global SchemeThe traditional frequency. domain poly-reference modal identification using frequency responsefunctions[9] is a well-known technique for deriving global estimates of modal parameters. The presentfrequency- domain operating modal identification global scheme will be derived as follows:From Eq.(2), we obtain the impulse response function matrix h(t)h(t) =业e^tW(13)then the first derivatives of the above equation ash(t)= Ae^tW(14)The Laplace transform of Eq.(13) and Eq.(14) can be respectively derived asH(s)=ψ(s1- A)-1 w(15)anH(8)=A(sI- A)-1 w(16)where H(s)-transfer function matrix; I- unit matrix.Let us construct a new matrix equation with Eq.(15) and Eq.(16) as「业(sI- 1)-1 w = O(s)(17)[H(s)」= [4A]where重=[巫业]T, O(s)=(sI- A)-1w.The dynamics of the system are completely characterized by the eigenvalue matrix A and eigenvectormatrix业There exist a matrix X such that:X更+亞A=0(18)It can be rewritten as[X I]业]=[X I]更=0(19)]Postmultiplying the above equation by 0(s), according to中国煤化工MHCNMHG[X I]$O(a)=[X ] |H(s)]| ='U(20)Equation (20) also can be expressed asXH(8)+H(8)=0(21)Vol. 18, No. 1 Zheng Min et al: Operational Model Analysis Schemes Using Correlation Technique91.We knowH(s)=sH(8)- h(t)It=0(22)Substituting Eq.(22) into Eq.(21), then let 8= jw, we can getXH(jw) +jwH(jw)-h(t)|t=o=0(23)where H(jw)- frequency response function matrix.By using the cross power spectral density function matrix G(jw) instead of the frequency responsefunction matrix H(jw) and the auto-and cross correlation function matrix R(T) instead of the impulseresponse function matrix h(t) in Eq.(23), the following resultsXG(jw) + jwG(jw)- R(T)|r=o=0(24)where G(jw)- power spectral density function matrix of measured responses. According to Eq.(1), atT = 0, the following resultsR(T)|r=0=Q(25)Substituting Eq.(25) into Eq.(24), we can getXG(jw) + jwG(jw)- Q=0For all discrete frequencies W1, W2, .. wK in the frequency range of measurement, a set of scalarequations can be written asx -Q||τ 1...D= DNwhere D = [G(jw1) G(jw2)...G(jwk)], s = -diag[jw1I jw2I...jwKI]. From Eq.(27), we canget X. Solving an eigenvalue problem of Eq.(18) will lead to the complex eigenvalue matrix A andeigenvector matrix业Then the modal parameters of the original vibrating system can be obtained.III. APPLICATIONThe experimental model is visualized in Fig.1. Thefuselage of the airplane model is 1000 mm long and the口suspending pointwingspan 1100 mm wide. It was suspended on three flexi-。measuring pointble threads at the fuselage. The responses were measuredby using acceleration transducers only in the vertical di-rection at 24 points for vertical excitation and at twoexcitationpoints on the wing.point下pointFirst, the modal parameters were obtained using theaee00ee.52multi-point pure mode excitation technique, yielding thebaseline model for comparison purposes. Then two pro-posed schemes in this paper were applied to the experi-Fig.1 The distribution of measured locations.mental structure. The white noise signals were generated by HP3562 and the MVMAS 3 multi-vibrationmeasurement and an analyzing system was employed to acquire the response data at all locations witha sampling rate of 512 Hz.After obtaining the experimental data, modal parameters estimation can be made. The procedureis divided into the following parts:(1) The time -domain scheme: The auto- and CrOSS- correlation functions of the responses at allmeasured locations were calculated and then modal parameters of the structure were identifed fromthe cross correlation difference model. The physical modes were separated from the computed modesusing the stability waterfall plot of the modal frequencies and中国煤化工rith the modelorder.(2) The frequency-domain scheme: The auto and cross-poMHC N M H Gasponses at allmeasured locations were calculated and then modal parameters of the structure were estimated. Whenthe total number of measured responses is greater than the number of the modes in the frequency rangeof analysis, the singular value truncation technique can be adopted to reduce the model orderl9. Thenthe order- reduced cross- power spectral density matrix was fed to the technique presented to extract the92ACTA MECHANICA SOLIDA SINICA2005更2020100t0-10pMryN个已-10-20-30- synthesized PSD-synthesized PSD---- measured PSDs -50020406080100120140160180200frequency (Hz)(a) auto-power spectral density of point 1(b) auto power spectral density of point2Fig. 2. Comparison between measured (dashed line) and synthesized (solid line) power spectral density.modal parameters from output-only. The auto- and cross- power spectral densities between responseswere ftted by using identified poles in the frequency band 0- 200 Hz. Figure 2 shows a comparison betweenmeasured and synthesized auto-and cross power spectral densities data. The dashed lines correspondto the measured power spectral densities, the solid lines show the synthesized values. Clearly, a goodft is obtained. The plot ilustrates how the synthesis of data can be used to validate the modal model.(3) The mode shapes for the first two modes from three techniques are shown in Fig3. The histogramsof the modal frequencies and damping ratios were plotted in Figs.4 and 5 to compare these results better.The MAC-values between the mode shapes extracted by the multi-point pure mode excitation methodand those by two technique described above are given in Fig.6.The results show that the errors of modal frequencies from two described techniques are very small.However, some damping ratios have large errors such as those at the second and third mode from thetime-domain method and at the second mode from the frequency- domain method. The all MAC-valuesare above 0.8, which ilustrates that there is a good agreement between the resultant modal shapes. Itcan be seen that the modal shapes from frequency-domain scheme are more accurate than those fromthe time-domain one.(a) the first mode shapes from three diferent methods中国煤化工EIYHCNMHG(b) the second mode shapes from three dfferent methodsleft: multi-point pure mode excitation methodmiddle: time-domain schemeright: frequency-domain schemeFig. 3. Comparison of the mode shapes from three different methods.Vol. 18, No. 1 Zheng Min et al: Operational Model Analysis Schemes Using Correlation Technique93. .全603 504(0莒3(昌1026mode order0 MPPMEM■TDOMIGS 0 FDOMIGSFig. 4. Model frequencies from three diferent methods.6「”t0 MPPMEMI TDOMIGS 0 FDOMIGSFig. 5. Damping ratios from three different methods.105 (100区9;90858075dd045口TDOMICS -MPPMEM( I FDOMIGS -MPPMEMFig. 6. MAC-value between the mode shapes extracted by multi-point pure mode excitation method and those extractedby two presented techniques.IV. CONCLUSIONSThe operational modal analysis for structures is gaining m中国煤化工is ise, basedon the assumption about white noise input, two operational m3 are proposedby coupling correlation function with classical modal identi:YHc N M H Gn experimenton an airplane model shows that there is a good agreement between the two techniques described andthe baseline model.The identical characteristics of the two operational modal analysis techniques can be summarizedas follows:ACTA MECHANICA SOLIDA SINICA2005(1) Both of them assume the white noise excitation and are based on the fact that the correlationfunction of responses has the same form as the impulse response function.(2) Both of them can extract the modal parameters of the structures in operating conditions.(3) Both of them can yield the global modal parameters of the system using simultaneously themeasured data at all locations.The different characteristics:(1) The time-domain scheme uses the correlation function of the responses; the frequency-domainscheme uses the power spectral density function of the responses.(2) For the time-domain scheme, when the number of measured locations is great, the physical modesare separated from the computed modes by using the stability plot of the modal parameters increasingwith the model order; for the frequency-domain scheme, when the number of measured locations isgreater than the number of the modes in the frequency range of analysis, the singular value truncationtechnique can be adopted to reduce the model order.(3) The frequency-domain scheme is more convenient for checking the quality of the estimated modalparameters by overlaying the synthesized and the measured power spectral densities.References1] Luz,E. and Wallaschek,J, Experiment modal analysis using ambient vibration, The International Journalof Analytical and Experimental Modal Analysis, Vol.7, No.1, 1992, 29 39.[2] Hermans,L, Auweraer,H.V. and Mathieu,L. et al, Modal parameter extraction from in-operation data,In: Society of Experimental Mechanics, Proceedings of the 15th International Modal Analysis Conference,Bethel: SEM, 1997, 531 -539.[3] Mohanty,P. and Zixen,D.J, A Modified ibrahim time domain algorithm for operational modal analysisincluding harmonic excitation, Journal of Sound and Vibration, Vol.270, No.1, 2004, 93 109.[4] Parloo,E, Guillaume,P. and Cauberghe,B., Maximum likelihood identification of non-stationary opera-tional data, Jourmnal of Sound and Vibration, Vol.268, No.5, 2003, 971-991.[5] Kromulski,J. and Hojan,E, Application of two experimental modal analysis methods for the determinationof operational deflection shapes, Journal of Sound and Vibration, Vol.196, No.4, 41996, 26- 438.[6] James,G.H. and Carne,T.G., The natural excitation technique (NExT) for modal parameter extraction fromoperating structures, The International Jourmnal of Analytical and Experimental Modal Analysis, Vol.10,No.4, 1995, 260-277.7] Qin,Q., Li,H.B. and Qian,L.Z, Modal identifcation of Tsing Ma Bridge by using improved eigensystemrealization algorithm, Journal of Sound and Vibration, Vol.247, No.2, 2001, 325-341.8} Shen,F, Zheng,M. and Shi,D.F., et al. Using the cross-correlation technique to extract modal parameterson response -only data, Jourmal of Sound and Vibration, Vol.259, No.5, 2003, 1163 1179.[9] Fu, Z.F. and Hua,H.X., Theory and application of modal analysis, Shanghai: Shanghai Jiao Tong UniversityBooks, 2000 (in Chinese).中国煤化工MYHCNMHG