SOUND AND VIBRATION SIGNAL ANALYSIS USING IMPROVED SHORT-TIME FOURIER REPRESENTATION. June-Yule Lee

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1 Inernaional Journal o Auomoive and Mechanical Engineering IJAME ISSN: Prin; ISSN: Online; Volume 7 pp January-June 03 Universii Malaysia Pahang DOI: hp://d.doi.org/0.58/ijame SOUND AND VIBRATION SIGNAL ANALYSIS USING IMPROVED SHORT-TIME FOURIER REPRESENTATION June-Yule Lee Deparmen o Marine Engineering Naional Kaohsiung Marine Universiy 48 Jhongjhou 3 rd Road Cijin Disric Kaohsiung Taiwan juneyule@ms0.hine.ne ABSTRACT Time-requency imaging provides a sraighorward means o undersanding machinery condiions. The mehods o shor-ime Fourier ransorm STFT Wigner-Ville disribuion and smooh-windowed Wigner-Ville disribuion SW are applied o he condiion monioring o roaing machines. The sound and vibraion signals o a roaing an are esed and ime-requency images are illusraed in erms o STFT and SW. The resuls show ha he resoluion o STFT is low and he resoluion o is high bu wih inererence. To overcome he inererence in he image a variable smooh-windowed weighing uncion is applied. The smoohing window uncion resuled in inererence aenuaion bu also in reducing he concenraion. The SW is a compromise beween STFT and. The SW ehibis beer resoluion han STFT and has less inererence han. Keywords: Time-requency analysis; smooh window weighing; sound and vibraion; condiion monioring. INTRODUCTION The sound and vibraion signals o roaing sysems have become more and more comple as wind eciaion run up or shu down roor cracking and chaering can lead o requency modulaion and ampliude modulaion. For he condiion monioring o hese problems he mos imporan and undamenal variables in signal processing are ime and requency. The undamenal ideal o ime-requency analysis is o undersand and describe hose condiions where he requency conen o a signal changes in ime. The shor-ime Fourier ransorm STFT is he mos widely used ool or he display o saionary and non-saionary signals in ime-requency analysis. For beer resoluion he quadraic ime-requency signal represenaion o he Wigner-Ville disribuion is considered. Unorunaely he presens cross-erms inererences in mulicomponen signals. Thus he is a general radeo beween good ime-requency resoluion and small inererence erms. Signals analysis simulaneously considered in ime and requency domain is well reviewed Cohen 989; Hlawasch and Boudreau-Barels 99; Nuawi e al. 0; Hammond and Whie 996; Sejdic e al In recen years ime-requency mehods have been widely used in gearbo aul deecion Saszewski e al. 997 he diagnosis and surveillance o diesel engines based on vibraion and acousics signals Delvecchio e al. 00 and he vibraion monioring o roaing machines Lee e al. 00; Al-Badour e al. 0; Climene-Alarcon e al. 0; Peng e al. 0. In his paper a parameer-conrolled smooh-windowed mehod SW is presened. The SW allows he smoohing o he ime spread and he smoohing o he requency spread. In pracical applicaion he sound and 8

2 Sound and vibraion signal analysis using improved shor-ime Fourier represenaion vibraion signals o a roaing an are considered and ime-requency images are illusraed in erms o STFT and SW. SHORT-TIME FOURIER TRANSFORM The basic ideal o he shor-ime Fourier ransorm is ha i one wans o know wha requencies eis a a paricular ime hen ake a small par o he signal around ha ime and analyse i neglecing he res o signal. Since he ime inerval is shor compared o he whole signal his process is called a shor-ime Fourier ransorm. For a given signal he Fourier ransorm is FT e - -j d where is requency in Hz. Now we muliply he signal by a window uncion w o obain a weighed signal. Considering his signal as a uncion o and aking he Fourier ransorm he STFT is * j STFT w e d where * is a comple conjugae. In Eq. he smoohing window is applied by a parameer conrolled uncion Lee 00 ep or 0 / : w 3 ep or 0 / Γ is a gamma uncion and where 0 / 3/ /. The parameer conrols he symmeric window shapes on he le side <0 and righ side >0 in Eq. 3. The window shape o represens he Laplace disribuion and represens he Normal disribuion. When ends o zero he uncion ends o an impulse disribuion and when ends o ininiy he uncion ends o a uniorm disribuion. Figure a shows he window shapes o 0.5 and 0 and he corresponding specrum plos are presened in Figure b. For eample considering a mulicomponen signal consising o he sum o wo chirp waveorms s sin [ 4 ] sin [ 4 ] 4 where 5 Hz and 45 Hz. Figure a shows he ime series o Eq. 4 a a sampling rae khz. The signals include wo non-saionary chirp waveorms. As ime advances he irs chirp signals sar a 5 Hz wih increasing requency and he second chirp signals sar a 45 Hz wih decreasing requency. The specrum is shown in Figure b where inormaion on he signal s requency change over ime is no provided. Using he STFT mehod and Figure daa he ime-requency conour plo is shown in 8

3 Lee /Inernaional Journal o Auomoive and Mechanical Engineering Figure 3. The STTF image shows he complee inormaion in he ime and requency domains. Two direcions o moving requency are illusraed where he irs and he second chirp signals cross a.5 seconds. The colour bar indicaes he ampliude o he specrum in inensiy. Figure. Smoohed window N=8 or parameers =.0.0 and 0; a ime series b specrum. Figure. Simulaion signals o he sum o wo chirp waves: a ime series b specrum. WIGNER-VILLE DISTRIBUTION The STFT suers ime-requency resoluion limiaions: a good ime resoluion requires a shor window weighing uncion and a good requency resoluion requires a long 83

4 Sound and vibraion signal analysis using improved shor-ime Fourier represenaion 84 window weigh uncion. Thus a undamenal resoluion radeo eiss: i is impossible o simulaneously achieve boh good ime and requency resoluion. To improve he ime-requency resoluion a quadraic ime-requency represenaion o he Winger-Ville disribuion is considered Hlawasch and Boudreau-Barels 99. For a signal he is deined as. * d e j 5 Figure 3. Time-requency conour plo using STFT mehod and Figure daa. The STFT in Eq. is a linear ime-requency represenaion o saisy he superposiion principle ha i is a linear combinaion o some signal componens hen he STFT o is he same linear combinaion. However he quadraic represenaion o in Eq. 5 is violaed in erms o is lineariy srucure. Considering mulicomponen signals hen is given by 6 where * d e j 7. * d e j 8

5 Lee /Inernaional Journal o Auomoive and Mechanical Engineering The which saisy are auo-erms and are cross-erms *. 9 Thus cross-erms is a real and Eq. 6 is wrien as Re[ ]. 0 Equaion 0 indicaes ha he o he sum o he wo signals is no only he sum o wo auo-erms bu also includes he cross-erms. These cross-erms cause inererence in he disribuion over boh ime and requency. For eample using he mehod and Figure daa he ime-requency conour plo is shown in Figure 4. As epeced he image shows a high concenraion when compared o he STFT in Figure 3. Bu he cross-erms inererence occurs midway beween he wo chirp signals. Figure 4. Time-requency conour plo using mehod and Figure daa. SMOOTH-WINDOWED WIGNER-VILLE DISTRIBUTION In pracical applicaions he inererence erms o are oen a problem or condiion monioring. In order o suppress he cross-erms a window weighed mehod is applied. Two smoohed windows are processed or in boh he ime and requency direcion. 85

6 Sound and vibraion signal analysis using improved shor-ime Fourier represenaion This is a smooh-windowed Wigner-Ville disribuion and is deined as SW * j h g s s s dse d where h and g are smoohing windows or ime and requency respecively. In he ollowing invesigaion he window uncion in Eq. 3 is applied or h and g. The smoohing window process resuled in inererence aenuaion bu also in reducing he ime-requency concenraion. Thus he SW is a compromise o STFT and. For eample using he SW mehod and Figure daa he ime-requency conour plo is shown in Figure 5. As epeced he inererence erms in Figure 4 are suppressed in boh he ime and requency direcions. The colour bar o Figure 5 show ha he inensiy is reduced when compared o Figure 4. However he resoluion is ar beer han in Figure 3. Thus he SW shows a compromise resul beween STFT and. The SW has beer resoluion han STFT and has less inererence han. Figure 5. Time-requency conour plo using SW 0 mehod and Figure daa. SOUND AND VIBRATION SINGALS ANALYSIS For applicaion in condiion monioring he above mehods are applied o real roaing an signals. Figure 6 shows he measured vibraion acceleraion and sound pressure signals a a sampling rae o 5 khz and he corresponding specra are shown in Figure 7. The vibraion signal shows wo resonance peaks locaed a 50 and 700 Hz; and he main peak o he acousic noise is locaed a 80 Hz. Using he vibraion signals in Figure 6a 86

7 Lee /Inernaional Journal o Auomoive and Mechanical Engineering and he acousic noise in Figure 6b he ime-requency conour plos o he STFT and SW mehods are illusraed in Figures 8 9 and 0 respecively. As epeced he STFT image shows he requency conen in ime bu he ime-requency resoluion is poor. The image shows a high resoluion bu wih inererence in he ime and requency domains. Thus or mulicomponen signals analysis he STFT and mehods migh mislead he sysem condiions due o he low resoluion or inererence. However he SW image indicaes ha he inererence is signiicanly reduced and has beer resoluion han he STFT mehod. Figure 6. Time series o roaing an signals: a vibraion b sound pressure. Figure 7. Specrum o roaing an signals: a vibraion b sound pressure. 87

8 Sound and vibraion signal analysis using improved shor-ime Fourier represenaion Figure 8. Time-requency conour plos using STFT = mehod and Figure 6 daa: a vibraion b sound pressure. Figure 9. Time-requency conour plos using mehod and Figure 6 daa: a vibraion b sound pressure. Figure 0. Time-requency conour plos using SW =0 mehod and Figure 6 daa: a vibraion b sound pressure. 88

9 Lee /Inernaional Journal o Auomoive and Mechanical Engineering CONCLUSION The ime-requency represenaions o he STFT and SW mehods are presened and applied o he condiion monioring o a roaing sysem. The resuls demonsrae ha he ime-requency image is a sraighorward means o undersanding machinery condiions. To improve he resoluion o he SFFT and o overcome he inererence o he parameer conrol o he smooh-windowed SW mehod is demonsraed. The resuls show ha SW is a compromise version beween STFT and. The SW has beer resoluion han STFT and less inererence han. REFERENCES Al-Badour F. Sunar M. and Cheded L. 0. Vibraion analysis o roaing machinery using ime-requency analysis and wavele echniques. Mechanical Sysems and Signal Processing 5: Climene-Alarcon V. Anonino-Daviu J. Riera-Guasp M. Pons-Llinares J. Roger-Folch J. Jover-Rodriguez J. and Arkkio A. 0. Transien racking o low and high-order eccenriciy-relaed componens in inducion moors via TFD ools. Mechanical Sysems and Signal Processing 5: Cohen L Time-requency disribuions: A review. Proceedings o he IEEE 777: Delvecchio S. D Elia G. Mucchi E. and Dalpiaz G. 00. Advanced signal processing ools or he vibraory surveillance o assembly auls in diesel engine cold ess. ASME Journal o Vibraion and Acousics 30008: -0. Hammond J.K. and Whie P.R The analysis o non-saionary signals using ime-requency mehods. Journal o Sound and Vibraion 903: Hlawasch F. and Boudreau-Barels G.F. 99. Linear and Quadraic Time-requency Signal Represenaions. IEEE Signal Processing Magazine 9: -67. Lee J.Y. 00. Parameer esimaion o he eended generalized gaussian amily disribuions using maimum likelihood scheme. Inormaion Technology Journal 9: Lee S.U. Robb D. and Besan C. 00. The direcional Choi-williams disribuion or he analysis o roor-vibraion signals. Mechanical Sysems and Signal Processing 54: Nuawi M.Z. Ismail A.R. Nor M.J.M. and Rahman M.M. 0. Comparaive sudy o whole-body vibraion eposure beween rain and car passengers: a case sudy in Malaysia. Inernaional Journal o Auomoive and Mechanical Engineering 4: Peng Z.K. Zhang W.M. Lang Z.Q. Meng G. and Chu F.L. 0. Time-requency daa usion echnique wih applicaion o vibraion signal analysis. Mechanical Sysems and Signal Processing 9: Sejdic E. Djurovic I. and Jiang J Time-requency eaure represenaion using energy concenraion: an overview o recen advances. Digial Signal Processing 9: Saszewski W. Worden K. and Tomlinson G.R Time-requency analysis in gearbo aul deecion using he wigner-ville disribuion and paern recogniion. Mechanical Sysems and Signal Processing 5: