Research Article Comparison of the Time Domain Windows ...downloads.hindawi.com/archive/2016/3837971.pdfResearch Article Comparison of the Time Domain Windows Specified in the ISO

Research ArticleComparison of the Time Domain WindowsSpecified in the ISO 18431 Standards Used to EstimateModal Parameters in Steel Plates

Jhonatan Camacho-Navarro1 R Guzmaacuten-Loacutepez2 Sergio Goacutemez2 and Marco Floacuterez1

1Escuela de Ingenierıa Electronica Universitaria de Investigacion (UDI) Grupo de Investigacion en Procesamiento de Senales (GPS)Bucaramanga Colombia2Universidad Pontificia Bolivariana Bucaramanga Colombia

Correspondence should be addressed to Jhonatan Camacho-Navarro camachonavarrojhonatangmailcom

Received 29 April 2016 Revised 14 July 2016 Accepted 3 August 2016

Academic Editor Massimo Viscardi

Copyright copy 2016 Jhonatan Camacho-Navarro et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The procedures used to estimate structural modal parameters as natural frequency damping ratios and mode shapes are generallybased on frequency methods However methods of time-frequency analysis are highly sensible to the parameters used to calculatethe discrete Fourier transform windowing resolution and preprocessing Thus the uncertainty of the modal parameters isincreased if a proper parameter selection is not considered In this work the influence of three different time domain windowsfunctions (Hanning flat-top and rectangular) used to estimate modal parameters are discussed in the framework of ISO 18431standard Experimental results are conducted over an AISI 1020 steel plate which is excited by means of a hammer elementVibration response is acquired by using acceleration records according to the ISO 7626-5 reference guidesThe results are comparedwith a theoretical method and it is obtained that the flat-top window is the best function for experimental modal analysis

1 Introduction

Physical behavior of complex engineering systems can bestudied through prediction and simulation analysis by meansof specialized software [1 2] Thus it is possible to checkabnormal performance with the help of monitoring methodsbased on simulation tools In particular the analysis ofstructural dynamics can be addressed by determining modalparameters defined by mode shapes natural frequencies anddamping ratios In this sense bymeans of OperationalModalAnalysis it is possible to conduct nondestructive testingfatigue analysis and issues concerned with field in structuralanalysis [3] These tasks also involve updated finite elementmodels used to predict the dynamic behavior of the structurereliability [4] For instance an application of modal analysisis the assessment of highway-bridge by dynamic testing andfinite-element model updating [5] Also damage detectionhas been carried out in reinforced concrete beams by usingmodal flexibility residuals [6]

Because of the importance of modal analysis in thefield of structural analysis well established procedures toobtain proper estimations of modal parameters are requiredAlthough there exists a huge documentation about methodsused for modal analysis [7 8] their practical implementationis still difficult because there are many parameters involvedin the procedure which must be decided by engineers of dif-ferent areas and knowledge In this sense the 7626 standardISO specifies a guideline of methodological steps to conductexperiments in order to obtain the frequency responsemeasurement and the 18431 ISO describes the proceduresfor time-frequency analysis of vibrational records Thus byconsidering the recommendations of the ISO standards thefurther estimation of modal model parameters by means ofwell-known modal analysis methods (as eg peak-pickingor least squares among others) is facilitated

In this paper a practical implementation of the above-mentioned ISO standardswith a special emphasis on comput-ing the natural frequency values is demonstrated Thus the

Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2016 Article ID 3837971 7 pageshttpdxdoiorg10115520163837971

2 Advances in Acoustics and Vibration

influence of using three domain window functions (Hanningflat-top and rectangular) to estimate natural frequencies isdiscussed Experimental results are conducted over an AISI1020 steel plate which is excited by means of a hammerelement

2 Theoretical Framework

The procedure used in this paper to estimate the naturalfrequencies of a modal model is based on the analysis ofmeasurements fromFrequency response functions (FRFs) Inthis sense the extraction of relevant frequency informationis performed by applying spectrum estimation techniques tostructural vibrational records Thus FRFs are approximatedby the cross-power spectral density (PSD) between thevibrational responses

In this section the conceptual issues involved in theestimation of structural natural frequencies bymeans of FRFsare detailed Also fundamentals of methods used to estimatethe frequency decomposition are presented focusing ongiven details about the parameters with great influence for itsimplementation

21 Frequency Response Functions (FRFs) Dynamical modelof structures is constructed on physical knowledge andfundamental laws of motion according to [9]

119872X + 119862X + 119870X = 119865 (1)

where 119870 119862 and 119872 are the stiffness damping and massmatrices respectively Likewise 119865(119905) and 119883(119905) denote theforcing vector and displacement Assuming that 119865(119905) is adelta-correlated exciting force and using properties demon-strated in the Natural Excitation Technique (NExT) it ispossible to write the law motion of (1) in the form of [9]

119872RX X119894 (120591) + 119862R

X X119894 (120591) + 119870RX X119894 (120591) = 0 (2)

where RX X119894 is a vector of correlation functions for all positive

lags 120591 gt 0 between the displacement vector and a referencesignal which must be uncorrelated with respect to excitationsignalThus (2) establishes that correlation function betweenacceleration records and a reference signal can be treatedas free vibration data which allows determining modalcharacteristics of the structures

Moreover to law motion and NExT equations used inmethodologies for modal parameter identification the FRFrelationship in structures must be specified The FRF for onen-degree of freedom system represented by (1) or (2) can bewritten in the form of partial fractions as is expressed in (3)(classical poleresidue) [10 11]

[119866119910119910

(120596)]

=

119898

sum

119896=1

[119860119896

]

119895120596 minus 120582119896

+[119860119896

]lowast

119895120596 minus 120582119896

lowast

+[119861119896

]

minus119895120596 minus 120582119896

+[119861119896

]lowast

minus119895120596 minus 120582119896

lowast

(3)

where119866119910119910

(120596) is the output PSDmatrix119898 is the total numberof modes of interest 120582

119896

= minus120590119896

+ 119895120596119889119896

is the pole of

1

|FT(120596

) |

1205961 1205962 120596i

(120596)

Figure 1 Typical FRF of an 119899-degree of freedom system

the 119896th mode 120590119896

is the modal damping (constant decay)120596119889119896

is the damped natural frequency and the lowast superscriptdenotes complex conjugate The typical PSD curve for theFRF determined by (3) is depicted in Figure 1

According to Figure 1 the modes (120596119896

) located at the peakcan be identified from spectral density through commonsignal processing as discrete Fourier transform

22 Frequency Domain Decomposition (FDD) Classicalapproach used to estimate modal parameters is often referredto as Peak-Peaking (PP) which is a nonparametric methodessentially developed in frequency domainThe main advan-tages of this method are its user-friendliness simple use andfast results obtaining In this method average normalizedpower spectral densities and frequency response functionsbetween all the measurement points of the structure areevaluated [12] As a result first the natural frequencies aresimply taken from the observation of the peaks on thegraphs of the magnitude of the response (Figure 1) Nextdamping ratios (120590

119896

) are calculated from the sharpness ofthe peaks obtained by half power band method Then themode shapes are calculated from the ratios of the peakamplitudes at various points in the structure [13] Finallymodal participation factors are computed to scaled modeshapes by using measures of force exciting In order to applythe PP method its procedure can be summarized as follows

(i) Determine the natural frequencies by means of thePSD computed from acceleration records by identi-fying all frequencies present at peaks (120596

119903

)(ii) Estimate damping ratios (120585) by means of the loss

factor (119899119903

) which depends of the half power bandfrequencies (120596

119886

120596119887

) as is illustrated in Figure 2

According to Figure 2 the half power band comprises thefrequencies where the PSD amplitude decays by 3 dB withrespect to its maximum value At once the half power bandis determined the loss and damping factors are computed byusing

119899119903

=120596119887

minus 120596119886

120596119903

120585 =120596119887

minus 120596119886

2120596119903

=119899119903

2

(4)

Advances in Acoustics and Vibration 3

20lg

xF

Δf

3 dB

120596a 120596r 120596b

Figure 2 Half power band frequencies

In this paper an enhanced method known as frequencydomain decomposition (FDD) is used instead which is anextension of PP method It consists of three main steps [14]

(i) Estimate the power spectral density matrix at discretefrequencies

(ii) Do a singular value decomposition of the powerspectral density

(iii) For an n-degree of freedom system then pick the 119899

dominating peaks in the power spectral densityThesepeaks correspond to the mode shapes

FDD technique removes disadvantages associated with clas-sical PP method as for example the difficulty to detectclose modes and bias estimation Furthermore FDD reducesthe uncertainty or hardness (even impossible) of dampingestimation Also it keeps the user-friendliness giving a feelingof the data it is dealing with

23 Spectrum Estimation Frequency response functionsbetween input and output are approximated as cross-powerspectral densities between responses while the impulseresponse functions are approximated by cross-correlationsbetween responses Cross-PSDs are obtained using Welchmethod (FFT based method) [15] Welchrsquos periodogramaveraging with overlapping is a method which introducesimproved properties based on basic periodogram methods

(i) Simple Periodogram Quotient of the squared magni-tude of the Fourier transform of the signal and lengthof the signal

(ii) Modified Periodogram Certain window other thanrectangularwindow that is applied to the signal beforetaking the Fourier transform Windows solve theleakage problem

(iii) Bartlett Periodogram Averaging Averaging of thedifferent blocks of the signal which decrease varianceof the signal at the expense of resolution

Thus the cross-power spectra between 119883(119899) and some119884(119899) discrete signal computed through Welchrsquos method are

expressed by (5)Therefore the cross-PSD 119878119883119884

can be definedas the Fourier Transform of the cross-correlation function119877119883119884

(120591)

119878119883119884

=1

119870

119870

sum

119896=1

119878119883119884

(119896)

where 119878119883119884

(119896) =1

119873

119871minus1

sum

119899=0

10038161003816100381610038161003816119908 (119899) 119877

119883119884

(120591)(119896)

119890minus119895120596119899

10038161003816100381610038161003816

2

(5)

According to (5) the Welch method divides the signal into 119896

blocks and then increases the averaging by taking overlapsof the blocks Thus the 119871 samples of 119877

119883119884

are divided into119870 overlapping sections with 119873 samples in each block eachof which is then windowed by the WINDOW 119908(119899) Finallyit averages the periodograms of the overlapping sections toform 119878

119883119884

the power spectral density estimation of 119877119883119884

24 Windowing Effect In estimating power spectral density(PSD) of a signal there are two tradeoffs One is frequencyresolution and the other is noise in the signal To obtaina good estimation of PSD we should have large length ofthe signal but during measurements we have finite length ofsignal If we take small block size bad frequency resolutioncould introduce leakage in the spectrum To reduce leakagesignals are windowed that is multiplied with a windowin time domain Many windows are available each onehaving specific application in signal processing [16 17] Fig-ure 3 illustrates the spectral characteristics of time windowsrecommended by the ISO standard Hanning flat-top andrectangular

According to Figure 3 the frequency estimation formodal parameter identification is limited by the frequencyresolution of the spectral density Resolution refers to theability of distinguishing narrowband spectral componentsIn this sense the spectrum of the ideal window with highfrequency resolution and good noise suppression is narrowerthan the main lobe width and lesser than the side lobelevel Consequently in order to select the suitable windowfunction a compromise between noise suppression and res-olution is required In this sense the flat-top and rectangularwindows produce the worstbest leakage factors (9679ndash914) respectively which corresponds to low and highrelative side lobe attenuations Also it is noted that Hanningwindow offers balanced characteristics

25 The ISO Standards The 7626-5 and 18431 ISO standardsgive recommendations about recording protocol and selec-tion of window function in order to estimate modal parame-ters [18] Specifically in the document entitled ldquoVibration andShock Experimental Determination of Mechanical MobilityPart 5rdquo it is found that the guidelines formeasurements usingimpact excitation with an exciter which are not attached tothe structure Thus the ISO 7626-51994 specifies proceduresto obtain acceleration records using impact excitation Alsosignals analysis based on the discrete Fourier transform iscovered where a recommendation about Hanning window


10 20 30 40 50 60minus02

0

02

04

06

08

1

12

Samples

Am

plitu

deTime domain

0 02 04 06 08minus150

minus100

minus50

0

50

Mag

nitu

de (d

B)

Frequency domain

Flat-topHanningRectangular


Normalized frequency (times120587 radsample)

Figure 3 Spectral characteristics of recommended ISO-time windows

can be justified Similarly the document ldquoMechanical Vibra-tion and Shock Signal Processing Part 2 Time DomainWindows for Fourier Transform Analysisrdquo which containsthe ISO 18431-22004 standard describes three time domainwindows consisting ofHanning flat-top and rectangular thatare suggested to be used for preprocessing samples vibration

Considering the selected set of time windows specifiedin the ISO standards in this paper the results of applyingHanning flat-top and rectangular windows for modal modesestimation are discussed

3 Results and Analysis

The next sections show the results obtained by applyingexperimental and theoretical procedures For experimentalanalysis the three time windows specified in the ISO standardare considered Hanning flat-top and rectangular functionswhile the theoretical approach is based on finite elementmethod

31 Experimental Setup A steel plate was used in order toanalyze the influence in the estimation of natural frequencieswhen different window functions are considered In Figure 4a scheme of the structure is shown

Measurements from acceleration response were con-ducted over 4 points of the structure (P6 P7 P8 andP12) Vibration data were recorded under impulse hammerexcitation type with sample time 119879

119904

= 20 120583s Then 25000samples for each accelerometer were processed which cor-respond to 05 seconds of data Figure 5 shows the recordedsignals

Figure 4 Bench structure

32 Numerical Analysis In order to calculate modal fre-quency of different deformation modes finite element sim-ulation through a numerical model was performed by usingANSYS software The model includes the plate detailedgeometry and it is characterized by density = 773375 Kgm3119864 = 200000Mpa and Poison = 029 parameters The usedfinite elements mesh is constituted by total number of 40052reduced integration hexahedral elements of eight integrationnodes C3D8R (see Figure 6)

The modal shapes studied by means of the simulationsoftware are depicted in Figure 7

33 Experimental Results In order to evaluate the influenceof windowing effect experimental measurements of acceler-ation records were processed considering windows describedin previous sections As a first result the 2 Khz range ofinterest for cross-power spectral density of the acceleration


0 01 02 03 04 05 06 07 0802468

10121416

Time (s) Time (s)

Time (s) Time (s)

minus40minus30minus20minus10010203040

Hammer inputAccel response P6

01 02 03 04 05 06 070

10

20

0 08minus10

0

10


0 005 01 015 02 025 03 035 04 045 05minus10

minus5

0

5

10

15


0 005 01 015 02 025 03 035 04 045 05minus40

minus30

minus20

minus10

0

10

20

30


Figure 5 Recorded vibration signals

Figure 6 Structural finite element model

measurements is depicted in Figure 8 The signals weredivided into eight sections with 50 overlap each sectionwas windowed with a Hamming window and eight modifiedperiodograms were computed and averaged

Also the frequency response function computed bymeans of frequency domain decomposition after processingthe CPSDof data in Figure 8 is depicted in Figure 9 where theamplitude in db corresponds to the first singular values of thePSD matrix estimated by using Hanning window function

A comparison of the modes estimated according to ISO7626-5 by using the three selected time window functions ispresented in Table 1

Table 1 Natural frequencies [Hz] estimated by using three differentwindows

Vibrationalmode

Numericalmodel Hanning Flat TOP Rectangular

First bending 636 6409 6348 6363First torsion 671 6714 6714 6683Secondtorsion 1472 1465 1469 1468

Secondbending 1640 1697 1691 1694

Table 2 Percentage error () of mode estimation with respect tonumerical model

Vibrational mode Hanning Flat-top RectangularFirst bending 07704 01887 00472First torsion 00596 00596 04024Second torsion 04755 02038 02717Second bending 34756 31098 32927

Finally the percentage errors for each natural frequencywith respect to theoretical numerical model are summarizedin Table 2

According to results inTable 2 the high error correspondsto the second bendingmode while the othermodesmaintain


(a) First bending 636Hz (b) First torsion 671Hz

(c) Second torsion 1472Hz (d) Second bending 1640Hz

Figure 7 Modal frequency obtained by numerical analysis

Cross-power spectral density P7ndashP12

0 02 04 06 08 1 12 14 16 18 2minus80

minus60

minus40

minus20

Frequency (kHz)

Pow

erfr

eque

ncy

(dB

Hz)

0 05 1 15 2 25 3 35 4 45 5Frequency (kHz)



Pow

erfr

eque

ncy

(dB

Hz)

0 05 1 15 2

minus80

minus60

minus40

minus20

Frequency (kHz)

Cross-power spectral density P6-P7

Pow

erfr

eque

ncy

(dB

Hz)

02 04 06 08 1 12 14 16 18 2

minus60

minus40

minus20

Frequency (kHz)

Cross-power spectral density P6ndashP8Po

wer

freq

uenc

y(d

BH

z)

0 02 04 06 08 1 12 14 16 18 2minus80

minus60

minus40

minus20

Frequency (kHz)


Pow

erfr

eque

ncy

(dB

Hz)

02 04 06 08 1 12 14 16 18 2Frequency (kHz)

minus80

minus60

minus40

minus20Cross-power spectral density P8ndashP12

Pow

erfr

eque

ncy

(dB

Hz)

Figure 8 CPSD for acceleration records

comparable values to the theoretical ones with errors lowerthan 1

4 Conclusion

Although no meaning differences were found when thethree windows specified in the ISO standard were used toestimate natural frequencies a slight better result is obtainedfor flat-top function This implies that for modal parameter

estimation purposes the selection of time windowing func-tion has low influence with major errors for the highestmodes However the influence of the windowing prepro-cessing for the analysis of different modal parameters asshape mode and factor participation should be studiedAlso further analysis should be conducted with respect toadditional parameters involved in the spectrum estimationsuch as overlap FFT length and segmentation Moreover itis recommended to include uncertainty analysis to evaluatethe influence of using different time windows


10minus1

10minus2

10minus3

10minus4

10minus5

Am

pl (d

B)

200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

X 6409Y 00114

X 6714Y 03656

X 1465Y 00003244

X 1697Y 05765

Figure 9 Frequency response function

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work has been developed as part of a collaborativework between researches fromUniversitaria de Investigaciony Desarrollo (UDI) and Universidad Pontificia Bolivariana(UPB) Bucaramanga Colombia

References

[1] E Madenci and I Guven The Finite Element Method andApplications in Engineering Using ANSYS Springer BerlinGermany 2015

[2] H H Lee Finite Element Simulations with ANSYS Workbench16 SDC Publications 2015

[3] R Brincker and C Ventura Introduction to Operational ModalAnalysis John Wiley amp Sons New York NY USA 2015

[4] M ImregunW J Visser andD J Ewins ldquoFinite elementmodelupdating using frequency response function data ITheory andinitial investigationrdquoMechanical Systems and Signal Processingvol 9 no 2 pp 187ndash202 1995

[5] J MW Brownjohn P Moyo P Omenzetter and Y Lu ldquoAssess-ment of highway bridge upgrading by dynamic testing andfinite-element model updatingrdquo Journal of Bridge Engineeringvol 8 no 3 pp 162ndash172 2003

[6] B Jaishi and W-X Ren ldquoDamage detection by finite elementmodel updating using modal flexibility residualrdquo Journal ofSound and Vibration vol 290 no 1-2 pp 369ndash387 2006

[7] D J EwinsModal Testing Theory and Practice vol 6 ResearchStudies Press Letchworth UK 1995

[8] N M M Maia and J M M e Silva Eds Theoretical andExperimental Modal Analysis Research Studies Press TauntonUK 1997

[9] J M Caicedo Two structural health monitoring strategies basedon global acceleration responses development implementationand verification [MS thesis] Washington University 2001

[10] S Gade N B Moslashller H Herlufsen and H Konstantin-Hansen ldquoFrequency domain techniques for operational modalanalysisrdquo in Proceedings of the 1st International OperationalModal Analysis Conference pp 261ndash271 St Louis Mo USA2005

[11] R Brincker L Zhang and P Andersen ldquoModal identificationfrom ambient responses using frequency domain decomposi-tionrdquo in Proceedings of the 18th International Modal AnalysisConference (IMAC rsquo00) San Antonio Tex USA February 2000

[12] A J Felber Development of a hybrid bridge evaluation system[PhD thesis] University of British Columbia 1993

[13] N M M Maia Extraction of valid modal properties frommeasured data in structural vibrations [PhD thesis] ImperialCollege London University of London 1988

[14] R Brincker L Zhang and P Andersen ldquoModal identification ofoutput-only systems using frequency domain decompositionrdquoSmart Materials and Structures vol 10 no 3 pp 441ndash445 2001

[15] S M Kay Modern Spectral Estimation Prentice Hall UpperSaddle River NJ USA 1988

[16] A V Oppenheim R W Schafer and J R Buck Discrete-TimeSignal Processing vol 2 Prentice-Hall Englewood Cliffs NJUSA 1989

[17] C Michael and A F Audrey ldquoThe fundamentals of FFT-basedsignal analysis and measurementrdquo Strategies for ChoosingWindows National Instruments Application Notes 041

[18] S B Blaeser and P D Schomer ldquoAcoustical standards newsrdquoThe Journal of the Acoustical Society of America vol 136 no 1pp 439ndash448 2014

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influence of using three domain window functions (Hanningflat-top and rectangular) to estimate natural frequencies isdiscussed Experimental results are conducted over an AISI1020 steel plate which is excited by means of a hammerelement

2 Theoretical Framework

The procedure used in this paper to estimate the naturalfrequencies of a modal model is based on the analysis ofmeasurements fromFrequency response functions (FRFs) Inthis sense the extraction of relevant frequency informationis performed by applying spectrum estimation techniques tostructural vibrational records Thus FRFs are approximatedby the cross-power spectral density (PSD) between thevibrational responses

In this section the conceptual issues involved in theestimation of structural natural frequencies bymeans of FRFsare detailed Also fundamentals of methods used to estimatethe frequency decomposition are presented focusing ongiven details about the parameters with great influence for itsimplementation

21 Frequency Response Functions (FRFs) Dynamical modelof structures is constructed on physical knowledge andfundamental laws of motion according to [9]

119872X + 119862X + 119870X = 119865 (1)

where 119870 119862 and 119872 are the stiffness damping and massmatrices respectively Likewise 119865(119905) and 119883(119905) denote theforcing vector and displacement Assuming that 119865(119905) is adelta-correlated exciting force and using properties demon-strated in the Natural Excitation Technique (NExT) it ispossible to write the law motion of (1) in the form of [9]

119872RX X119894 (120591) + 119862R

X X119894 (120591) + 119870RX X119894 (120591) = 0 (2)

where RX X119894 is a vector of correlation functions for all positive

lags 120591 gt 0 between the displacement vector and a referencesignal which must be uncorrelated with respect to excitationsignalThus (2) establishes that correlation function betweenacceleration records and a reference signal can be treatedas free vibration data which allows determining modalcharacteristics of the structures

Moreover to law motion and NExT equations used inmethodologies for modal parameter identification the FRFrelationship in structures must be specified The FRF for onen-degree of freedom system represented by (1) or (2) can bewritten in the form of partial fractions as is expressed in (3)(classical poleresidue) [10 11]

[119866119910119910

(120596)]

=

119898

sum

119896=1

[119860119896

]

119895120596 minus 120582119896

+[119860119896

]lowast

119895120596 minus 120582119896

lowast

+[119861119896

]

minus119895120596 minus 120582119896

+[119861119896

]lowast

minus119895120596 minus 120582119896

lowast

(3)

where119866119910119910

(120596) is the output PSDmatrix119898 is the total numberof modes of interest 120582

119896

= minus120590119896

+ 119895120596119889119896

is the pole of

1

|FT(120596

) |

1205961 1205962 120596i

(120596)

Figure 1 Typical FRF of an 119899-degree of freedom system

the 119896th mode 120590119896

is the modal damping (constant decay)120596119889119896

is the damped natural frequency and the lowast superscriptdenotes complex conjugate The typical PSD curve for theFRF determined by (3) is depicted in Figure 1

According to Figure 1 the modes (120596119896

) located at the peakcan be identified from spectral density through commonsignal processing as discrete Fourier transform

22 Frequency Domain Decomposition (FDD) Classicalapproach used to estimate modal parameters is often referredto as Peak-Peaking (PP) which is a nonparametric methodessentially developed in frequency domainThe main advan-tages of this method are its user-friendliness simple use andfast results obtaining In this method average normalizedpower spectral densities and frequency response functionsbetween all the measurement points of the structure areevaluated [12] As a result first the natural frequencies aresimply taken from the observation of the peaks on thegraphs of the magnitude of the response (Figure 1) Nextdamping ratios (120590

119896

) are calculated from the sharpness ofthe peaks obtained by half power band method Then themode shapes are calculated from the ratios of the peakamplitudes at various points in the structure [13] Finallymodal participation factors are computed to scaled modeshapes by using measures of force exciting In order to applythe PP method its procedure can be summarized as follows

(i) Determine the natural frequencies by means of thePSD computed from acceleration records by identi-fying all frequencies present at peaks (120596

119903

)(ii) Estimate damping ratios (120585) by means of the loss

factor (119899119903

) which depends of the half power bandfrequencies (120596

119886

120596119887

) as is illustrated in Figure 2

According to Figure 2 the half power band comprises thefrequencies where the PSD amplitude decays by 3 dB withrespect to its maximum value At once the half power bandis determined the loss and damping factors are computed byusing

119899119903

=120596119887

minus 120596119886

120596119903

120585 =120596119887

minus 120596119886

2120596119903

=119899119903

2

(4)


20lg

xF

Δf

3 dB

120596a 120596r 120596b















(120591)

119878119883119884

=1

119870

119870

sum

119896=1

119878119883119884

(119896)

where 119878119883119884

(119896) =1

119873

119871minus1

sum

119899=0

10038161003816100381610038161003816119908 (119899) 119877

119883119884

(120591)(119896)

119890minus119895120596119899

10038161003816100381610038161003816

2

(5)



119883119884


119883119884






10 20 30 40 50 60minus02

0

02

04

06

08

1

12

Samples

Am

plitu

deTime domain

0 02 04 06 08minus150

minus100

minus50

0

50

Mag

nitu

de (d

B)

Frequency domain











119904







0 01 02 03 04 05 06 07 0802468

10121416

Time (s) Time (s)

Time (s) Time (s)



01 02 03 04 05 06 070

10

20

0 08minus10

0

10


0 005 01 015 02 025 03 035 04 045 05minus10

minus5

0

5

10

15


0 005 01 015 02 025 03 035 04 045 05minus40

minus30

minus20

minus10

0

10

20

30








Vibrationalmode



Secondbending 1640 1697 1691 1694










0 02 04 06 08 1 12 14 16 18 2minus80

minus60

minus40

minus20

Frequency (kHz)

Pow

erfr

eque

ncy

(dB

Hz)

0 05 1 15 2 25 3 35 4 45 5Frequency (kHz)



Pow

erfr

eque

ncy

(dB

Hz)

0 05 1 15 2

minus80

minus60

minus40

minus20

Frequency (kHz)


Pow

erfr

eque

ncy

(dB

Hz)

02 04 06 08 1 12 14 16 18 2

minus60

minus40

minus20

Frequency (kHz)


wer

freq

uenc

y(d

BH

z)

0 02 04 06 08 1 12 14 16 18 2minus80

minus60

minus40

minus20

Frequency (kHz)


Pow

erfr

eque

ncy

(dB

Hz)

02 04 06 08 1 12 14 16 18 2Frequency (kHz)

minus80

minus60

minus40


Pow

erfr

eque

ncy

(dB

Hz)



4 Conclusion




10minus1

10minus2

10minus3

10minus4

10minus5

Am

pl (d

B)

200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

X 6409Y 00114

X 6714Y 03656

X 1465Y 00003244

X 1697Y 05765


Competing Interests


Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










20lg

xF

Δf

3 dB

120596a 120596r 120596b















(120591)

119878119883119884

=1

119870

119870

sum

119896=1

119878119883119884

(119896)

where 119878119883119884

(119896) =1

119873

119871minus1

sum

119899=0

10038161003816100381610038161003816119908 (119899) 119877

119883119884

(120591)(119896)

119890minus119895120596119899

10038161003816100381610038161003816

2

(5)



119883119884


119883119884






10 20 30 40 50 60minus02

0

02

04

06

08

1

12

Samples

Am

plitu

deTime domain

0 02 04 06 08minus150

minus100

minus50

0

50

Mag

nitu

de (d

B)

Frequency domain











119904







0 01 02 03 04 05 06 07 0802468

10121416

Time (s) Time (s)

Time (s) Time (s)



01 02 03 04 05 06 070

10

20

0 08minus10

0

10


0 005 01 015 02 025 03 035 04 045 05minus10

minus5

0

5

10

15


0 005 01 015 02 025 03 035 04 045 05minus40

minus30

minus20

minus10

0

10

20

30








Vibrationalmode



Secondbending 1640 1697 1691 1694










0 02 04 06 08 1 12 14 16 18 2minus80

minus60

minus40

minus20

Frequency (kHz)

Pow

erfr

eque

ncy

(dB

Hz)

0 05 1 15 2 25 3 35 4 45 5Frequency (kHz)



Pow

erfr

eque

ncy

(dB

Hz)

0 05 1 15 2

minus80

minus60

minus40

minus20

Frequency (kHz)


Pow

erfr

eque

ncy

(dB

Hz)

02 04 06 08 1 12 14 16 18 2

minus60

minus40

minus20

Frequency (kHz)


wer

freq

uenc

y(d

BH

z)

0 02 04 06 08 1 12 14 16 18 2minus80

minus60

minus40

minus20

Frequency (kHz)


Pow

erfr

eque

ncy

(dB

Hz)

02 04 06 08 1 12 14 16 18 2Frequency (kHz)

minus80

minus60

minus40


Pow

erfr

eque

ncy

(dB

Hz)



4 Conclusion




10minus1

10minus2

10minus3

10minus4

10minus5

Am

pl (d

B)

200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

X 6409Y 00114

X 6714Y 03656

X 1465Y 00003244

X 1697Y 05765


Competing Interests


Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10 20 30 40 50 60minus02

0

02

04

06

08

1

12

Samples

Am

plitu

deTime domain

0 02 04 06 08minus150

minus100

minus50

0

50

Mag

nitu

de (d

B)

Frequency domain











119904







0 01 02 03 04 05 06 07 0802468

10121416

Time (s) Time (s)

Time (s) Time (s)



01 02 03 04 05 06 070

10

20

0 08minus10

0

10


0 005 01 015 02 025 03 035 04 045 05minus10

minus5

0

5

10

15


0 005 01 015 02 025 03 035 04 045 05minus40

minus30

minus20

minus10

0

10

20

30








Vibrationalmode



Secondbending 1640 1697 1691 1694










0 02 04 06 08 1 12 14 16 18 2minus80

minus60

minus40

minus20

Frequency (kHz)

Pow

erfr

eque

ncy

(dB

Hz)

0 05 1 15 2 25 3 35 4 45 5Frequency (kHz)



Pow

erfr

eque

ncy

(dB

Hz)

0 05 1 15 2

minus80

minus60

minus40

minus20

Frequency (kHz)


Pow

erfr

eque

ncy

(dB

Hz)

02 04 06 08 1 12 14 16 18 2

minus60

minus40

minus20

Frequency (kHz)


wer

freq

uenc

y(d

BH

z)

0 02 04 06 08 1 12 14 16 18 2minus80

minus60

minus40

minus20

Frequency (kHz)


Pow

erfr

eque

ncy

(dB

Hz)

02 04 06 08 1 12 14 16 18 2Frequency (kHz)

minus80

minus60

minus40


Pow

erfr

eque

ncy

(dB

Hz)



4 Conclusion




10minus1

10minus2

10minus3

10minus4

10minus5

Am

pl (d

B)

200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

X 6409Y 00114

X 6714Y 03656

X 1465Y 00003244

X 1697Y 05765


Competing Interests


Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 01 02 03 04 05 06 07 0802468

10121416

Time (s) Time (s)

Time (s) Time (s)



01 02 03 04 05 06 070

10

20

0 08minus10

0

10


0 005 01 015 02 025 03 035 04 045 05minus10

minus5

0

5

10

15


0 005 01 015 02 025 03 035 04 045 05minus40

minus30

minus20

minus10

0

10

20

30








Vibrationalmode



Secondbending 1640 1697 1691 1694










0 02 04 06 08 1 12 14 16 18 2minus80

minus60

minus40

minus20

Frequency (kHz)

Pow

erfr

eque

ncy

(dB

Hz)

0 05 1 15 2 25 3 35 4 45 5Frequency (kHz)



Pow

erfr

eque

ncy

(dB

Hz)

0 05 1 15 2

minus80

minus60

minus40

minus20

Frequency (kHz)


Pow

erfr

eque

ncy

(dB

Hz)

02 04 06 08 1 12 14 16 18 2

minus60

minus40

minus20

Frequency (kHz)


wer

freq

uenc

y(d

BH

z)

0 02 04 06 08 1 12 14 16 18 2minus80

minus60

minus40

minus20

Frequency (kHz)


Pow

erfr

eque

ncy

(dB

Hz)

02 04 06 08 1 12 14 16 18 2Frequency (kHz)

minus80

minus60

minus40


Pow

erfr

eque

ncy

(dB

Hz)



4 Conclusion




10minus1

10minus2

10minus3

10minus4

10minus5

Am

pl (d

B)

200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

X 6409Y 00114

X 6714Y 03656

X 1465Y 00003244

X 1697Y 05765


Competing Interests


Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














0 02 04 06 08 1 12 14 16 18 2minus80

minus60

minus40

minus20

Frequency (kHz)

Pow

erfr

eque

ncy

(dB

Hz)

0 05 1 15 2 25 3 35 4 45 5Frequency (kHz)



Pow

erfr

eque

ncy

(dB

Hz)

0 05 1 15 2

minus80

minus60

minus40

minus20

Frequency (kHz)


Pow

erfr

eque

ncy

(dB

Hz)

02 04 06 08 1 12 14 16 18 2

minus60

minus40

minus20

Frequency (kHz)


wer

freq

uenc

y(d

BH

z)

0 02 04 06 08 1 12 14 16 18 2minus80

minus60

minus40

minus20

Frequency (kHz)


Pow

erfr

eque

ncy

(dB

Hz)

02 04 06 08 1 12 14 16 18 2Frequency (kHz)

minus80

minus60

minus40


Pow

erfr

eque

ncy

(dB

Hz)



4 Conclusion




10minus1

10minus2

10minus3

10minus4

10minus5

Am

pl (d

B)

200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

X 6409Y 00114

X 6714Y 03656

X 1465Y 00003244

X 1697Y 05765


Competing Interests


Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10minus1

10minus2

10minus3

10minus4

10minus5

Am

pl (d

B)

200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

X 6409Y 00114

X 6714Y 03656

X 1465Y 00003244

X 1697Y 05765


Competing Interests


Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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