Modal Parameter Extraction Based on Hilbert Transform of - Asian ...

Asian Transactions on Engineering (ATE ISSN: 2221-4267) Volume 02 Issue 01<strong>Modal</strong> <strong>Parameter</strong> <strong>Extraction</strong> <strong>Based</strong> on Hilbert Transform of<strong>Modal</strong> ResponsesZheng Min*, Shen Fan***College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, China**College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, China*zhengminsf@163.com, **shen0039@ntu.edu.sgAbstract—Two modal identification methods based onHilbert Transform of modal responses are used to extractmodal parameters of a structure in this paper. First, theacceleration response time history polluted by noises isprocessed using Empirical Mode Decomposition withSingular Value Decomposition. Then, the frequencies anddamping ratios of the system can be estimated byperforming Hilbert Transform for obtained modalresponses. The mode shapes can be achieved when theresponses at all degrees of freedom are measured. Theproposed methods are illustrated using a simulated airplanemodel with dense modes built in ADAMS (AutomaticDynamic Analysis of Mechanical Systems) software. Thesimulation results demonstrate that the outlined twomethods based on Hilbert Transform can identify modalparameters well with the help of EMD and SVD as signalpreprocessor even for the structure with the closely spacemodes, low-energy components.Index Terms—modal analysis, Hilbert transform, EmpiricalMode Decomposition, signal processingI. INTRODUCTION<strong>Modal</strong> analysis approach is an important tool forexperimental identification of a structural dynamicsmodels. Identified parameters can be used for design ofactive and passive control of structures and structuralhealth monitoring [1].There are numerous approaches that can be applied toextract modal parameters of structure. Traditionally,modal parameters are extracted by performing thereducing and curve-fitting procedures either on series ofmeasured frequency response function or on timeresponse, such as impulse response function [2] andfree-decay response [3]. Recently modal analysis basedon output-only, which has gained more and moreattention, can be done by utilizing the correlationfunctions and power spectra of the operating data [4].Some time–frequency analysis tools, such asWigner–Ville distribution, Wavelet Transform (WT) andHilbert–Huang Transform (HHT) [5] building onEmpirical Mode Decomposition (EMD) and Hilbertspectral analysis (HSA) also have been used for systemidentification [6], damage detection [7]-[10] and etc [11].For example, the vibration responses are processed usingHHT for fault diagnosis [12-13], the damage featuresdetection of composite aircraft wing-box structures [14]and health monitoring of civil structural models [15-16].HHT technique is also proposed to identifymulti-degree-of-freedom (MDOF) linear systems withintermittency criteria [17] or band-pass filtering on themeasured free vibration time histories [18]. In this paper,two modal identifications based on Hilbert Transformwith EMD and SVD as preprocessor are applied toextract modal parameters of a simulated airplane modelwith closely spaced modes built in ADAMS software forvalidating the modal identification capabilities.II. MODAL PARAMTER IDENTIFICATIONA. <strong>Modal</strong> ResponseConsidering a multi-freedom system, we can describethe behaviours of the system by the equations of motionasM ( t)X ( t) C(t)X( t) K(t)X ( t) F(t)(1)where M(t) is the mass matrix, C(t) is the dampingmatrix, K(t) is the stiffness, F(t) is the vector of randomMar 2012 ATE-70209018©Asian-Transactions 25

Asian Transactions on Engineering (ATE ISSN: 2221-4267) Volume 02 Issue 01forcing function, X(t) is the vector of randomdisplacements.Using a modal transformationNx( t) q ( t)(2)j1and the orthogonality of mode shapes, a set of equationsin the modal co-ordinates can be given as follows2( ) 2 ( ) ( ) Tjj jj jj( ) /jjq t q t q t F t m (3)where q j (t) is the jth generalized modal co-ordinate, ξ jand ω j are, respectively, the jth modal damping ratio andmodal frequency,modal mass.j is the jth modal vector, mjj is the jthWith the assumption of the existence of normal modes,due to a single impulse input force f l (t) at the lth DOF,i.e. f l (t)=F 0 δ(t), for all j≠l, (3) can be written asand2q ( t) 2 q( t) q ( t) 0 (4)jjF q t e tj0( ) lj j j j tj cos( )2djjm 12jj1 2 2where j tan 2 j1 j 12j natural frequency,factor. , ω dj is thejj is the modal participationljThe impulse acceleration responseDOF can be derived asNxpp pj j pjj1 j1 x ( t) q ( t) x ( t)From (5) we can obtainwhereN(5)t at the pth (6)j j t,sin , x t C e t (7)pj pl j dj pl jCF 0 pj ljpl,j (8)mjdj (9)pl, j j pl,jand is the phase difference between the pth elementpl,jand the lth element in the jth mode shape.B. Identifications of Natural Frequencies and DampingRatiosEmpirical Mode Decomposition (EMD) method [8]can extract the properties of signals even with nonlinearand non-stationary characteristics by decomposing anycomplicated data set via EMD into a finite, often small,number of Intrinsic Mode Functions (IMFs) that admit awell-behaved Hilbert transform. In this paper, EMD isapplied to decompose themeasured accelerationresponse excited by a multi-component loading with theimpulse, harmonic and noise properties to obtain modalresponses. Considering a system with an input signalapplied to the lth DOF, the measured accelerationresponsefollows:wheret z pat the pth DOF can be processed asN G H (10) z t x t x t x t r tp pj ph po pj1 g1 h1pjcomponents, x t are IMFs for impulse responsecomponents. The IMFsxpht are IMFs for harmonicpj x t can be processedthrough the Hilbert transform to determine modalparameters of the system andxph t can be identifiedas arising from harmonic excitation with zero damping.The analytical signalwhere ()pjYpjY pjt of x pjt is given~( t) x( t) ix ( t)(11)pjx t is the Hilbert Transform of x tEquation (11) also can be expressed aspj pj . pi, ji pj t (12)Y t A t e pjwhere A pj is the instantaneous amplitude and θ pj is theinstantaneous phase angle. Instantaneous modalMar 2012 ATE-70209018©Asian-Transactions 26

acc.(m/s 2 )acc.(m/s 2 )AmpAsian Transactions on Engineering (ATE ISSN: 2221-4267) Volume 02 Issue 0125III. SIMULATED EXPERIMENTAL CASE STUDYThe outlined two methods based on Hilbert Transformare applied to analyze the response data acquired from asimulated test conducted in ADAMS. The modalparameters are extracted and compared with those fromADAMS software.A. Description of Test Structure and TestsAn airplane model with closely spaced modes built inADAMS is considered. It is made of iron and fivemillimeters thick. The model was fixed at the coordinateorigin. The acceleration responses were measured only inthe vertical direction at 27 points for vertical excitation attwo symmetrical excitation points on the wingtip. Fig.1shows the test model and the distribution of the 27measurement points. The location numbers werearranged counterclockwise with the beginning ofairplane head.20151050-50 1 2 3 4 5 6 7 8 9 10time(s)Fig. 2. Time history of the excitation signalA total of 5000 response samples corresponding tosignal duration of 10s with a sampling rate of 500 Hz atthe 27 locations were acquired simultaneously, whichresults in a bandwidth of 0–250 Hz. Fig.3 shows thetemporal waveform of the 9th location and its FFTspectrum, respectively.3000200010000-1000-2000-30000 1 2 3 4 5 6 7 8 9 10time(s)450400350300250Fig. 1. Airplane model built in ADAMS with indication of thedistribution of measurement pointsAn input signal with the multi-component properties,which is composed of a sinusoidal harmonic frequency at80 Hz and an impulse at the moment of 0.1 second alongwith the stationary white noise, was applied to the 2excitation points. Fig.2 shows the time history of theexcitation signal.2001501005000 50 100 150 200 250Frequence(Hz)Fig. 3. Time history and FFT spectrum of measured accelerationresponse at the 9th location of the airplane modelB. Signal ProcessingIn practical applications, the closely space modesusually occur on some engineering structures. It is quitechallenging to extract characteristic parameters for suchstructures. SVD filtering is used to help EMD separateMar 2012 ATE-70209018©Asian-Transactions 28

Ampacc.(m/s 2 )acc.(m/s 2 )Ampacc.(m/s 2 )AmpAsian Transactions on Engineering (ATE ISSN: 2221-4267) Volume 02 Issue 01the dense modes in this paper.The SVD-decomposed subspaces are an orthogonalrepresentation of the original space, the relevant discretetime series y w (k) (w=1,2,…s) corresponding to thesubspace w should have strong correlation with theoriginal signal. With the aid of the correlationcoefficients, a screening process will be conducted andthe selected vital time series y w (k) will be remained.After reconstructing the signal y rec (k) by superposingthe retained time series y w (k) as followingrec sw1w y k y k(27)We process it through the band-pass filters each with afrequency band near the possible frequencies of thesystem. Each filtered time history is decomposed byEMD, and the resulting first IMFs will be processedthrough the Hilbert transform for the identification ofmodal parameters [18].The modal responses near the natural frequencies areshown in Fig.4. It can be seen that the 5 th selectedcomponent at 19 Hz that contains low-energy componentand the two closed space frequencies near the 3 th selectedcomponent at 52 Hz and the 2 nd selected component at54 Hz are clearly visible. It is noted that a segment of themodal response near t =0 is not a decaying functionmainly due to the phase shift. Such a segment should beremoved before the Hilbert transform is applied to thesemodal responses to extract modal parameters of thesystem.60504030201000 50 100 150 200 250Frequence(Hz)(a)Time history and FFT spectrum of the 1 st selected component250200150100500-50-100-150-200-2500 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1time(s)60504030201000 50 100 150 200 250Frequence(Hz)(b)Time history and FFT spectrum of the 2 nd selected component8006004002000150-200100-400-60050-8000 0.2 0.4 0.6 0.8 1 1.2time(s)0140-50120-10010080-1500 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5time(s)60402000 50 100 150 200 250Frequence(Hz)Mar 2012 ATE-70209018©Asian-Transactions 29

Ampacc.(m/s 2 )AmpAmpacc.(m/s 2 )acc.(m/s 2 )Asian Transactions on Engineering (ATE ISSN: 2221-4267) Volume 02 Issue 01(c)Time history and FFT spectrum of the 3 rd selected component304000203000200010100000-10-1000-2000-20-3000-300 0.5 1 1.5 2 2.5 3 3.5time(s)-40000 0.2 0.4 0.6 0.8 1 1.2time(s)8600750065400430032002110000 50 100 150 200 250Frequence(Hz)(d)Time history and FFT spectrum of the 4 th selected component1086420-2-4-6-8-100 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2time(s)21.81.61.41.200 50 100 150 200 250Frequence(Hz)(f)Time history and FFT spectrum of the 6 th selected componentFig. 4. Achieved modal responses at the 9 th location through theproposed signal processing techniqueC. <strong>Modal</strong> IdentificationIn order to avoid the bad influences of the phase shiftduring band-pass filtering and end effects of EMD onidentified results, after removing the segment from t =0to the maximum amplitude each modal response wereprocessed through the Hilbert transform to determine theinstantaneous amplitude and instantaneous phase. Theidentified results based on the response at 9th positionand the baseline models from ADAMS are listed inTable 1 for comparison purposes.10.80.60.40.200 50 100 150 200 250Frequence(Hz)(e) Time history and FFT spectrum of the 5 th selected componentMar 2012 ATE-70209018©Asian-Transactions 30

Asian Transactions on Engineering (ATE ISSN: 2221-4267) Volume 02 Issue 01TABLE ITHE IDENTIFIED FREQUENCIES AND DAMPING RATIOSIn Table 1, we read that the identified harmonicfrequencies are 79.998 Hz and 80.053 Hz withcorresponding damping ratios of 0.02% and 0.01%Mode No.Frequencies(Hz)Damping Ratios(%)in method 1 and 2 respectively. Note that, althoughBase Method Method Base Method Methodthe damping should be zero in theory, it is verysmall in the results and therefore an analyst couldhardly be misled this pole as a true eigenpole of thesystem. The results also demonstrate that theproposed methods in this paper can identify themodal frequencies with great accuracy and thecorresponding damping ratios even when the densemodal frequencies exist near 52Hz. The modalparameters near the frequencies of 10Hz and 19Hzwith the low-energy, which are not clear in the FFTspectra of Fig.3, were extracted well. Generally, thedamping ratios from method 2 were less accuratethan those from method 1.123456Model10.26919.03643.66751.71753.79480110.27118.83343.72850.79752.85579.998210.28218.75943.84050.73652.93680.053Model0.590.621.031.491.16010.610.681.101.271.250.0220.700.811.151.211.330.01The results from the responses of other locations aloneare very close to those based on the 9th one. With theacceleration responses at all positions measured, themode shapes of the system can be extracted. Theachieved mode shapes are visualized in Fig.5 with thetheoretical mode shapes for comparison. It is observedthat the identified results have a good agreement with thebaseline model.25020015010050010006004005002000-200-4000 -600(a)The 1 st mode shapes25020050150100500010008006004002000-600-400-2000200400600-5010005000-600-400-2000200400600Mar 2012 ATE-70209018©Asian-Transactions 31

Asian Transactions on Engineering (ATE ISSN: 2221-4267) Volume 02 Issue 0150150105-50100005000-2000-400-600(b)The 2 nd mode shapes200400600-510005000-600 -400 -200 0 200 400 60086420-210005000-600 -400 -200 0 200 400 600(d)The 4 th mode shapes1050-5-10-1510005000-600 -400 -200 0 200 400 600251050-5-10-15100020151050-510005005000-600-400-20002004006000-600 -400 -200 0 200 400 600(c)The 3 rd mode shapes302520151050-510005000-600-400 -2000 200400 600(e)The 5 th mode shapesFig. 5. Identified mode shapes from the measured responsesMar 2012 ATE-70209018©Asian-Transactions 32

Asian Transactions on Engineering (ATE ISSN: 2221-4267) Volume 02 Issue 01(upper-ADAMS, middle-Method1, lower-Method2)IV. CONCLUSIONIn this study, two identification methods of structuraldynamic characteristics based on the Hilbert Transformof modal responses have been illustrated through anairplane model example built in ADAMS. The resultsdemonstrate that with the EMD and SVD aspreprocessors both of the outlined methods offereffective tools for yielding modal parameters even ifdense modes and low-energy component exist. When theharmonic excitations are present, the dynamic responsewill include the associated forced harmonic componentswhich can be seen as virtual non-damped modes. Thesevirtual modes can be achieved easily through theproposed identifications. Further investigation can bemade of modal identification for practical largestructures.ACKNOWLEDGMENTThis work has been funded by National NaturalScience Foundation of China (61179056).REFERENCES[1] R. Segawa, S. Yamamoto, A. 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Part 1:normal modes,” Earthquake Engineering and StructuralDynamics, vol.32, no.10, pp.1443–1467, Jul.2003.[19] J. Xiao, “Study on time-frequency analysis of modalidentification for time-varying system,” M.S. thesis, College ofCivil Aviation, Nanjing University of Aeronautics andAstronautics, Nanjing, China, Mar.2006.[20] M.J. Brenner, “Non-stationary dynamics data analysis withwavelet-SVD filtering,” Mechanical Systems and SignalProcessing, vol.17, no.4, pp.765–786, Jul.2003.Zheng Min joined Nanjing University of Aeronautics and Astronauticsin 2002 as an academic staff. She became a Member of ChineseSociety of Aeronautics and Astronautics in 2006. She went to NationalUniversity of Singapore as a research fellow in 2008. She achieved hisPhD degree at College of Aerospace Engineering in Nanjing Universityof Aeronautics and Astronautics in 2000. Her research interests includestructure dynamics, signal processing and energy efficiency.Mar 2012 ATE-70209018©Asian-Transactions 33