The paper presents possibilities of applying linear decimation procedure in frequency analysis of non-stationary signals. It shows the results of analytical experiments conducted with vibration signals registered during examination of car suspension which was activated to vibration on harmonic stand research. Working cycles identification algorithm was helpful to make LDP of working cycles parts of a signal. Signals spectrum results confirm amplitude selectivity for typical frequency ofresearching dynamic system

Janusz GARDULSKI

Rafał BURDZIK

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TRANSPORT PROBLEMS                                                                                                             2007 PROBLEMY TRANSPORTU                                                                                        Tom 2 Zeszyt 1     Rafał BURDZIK*, Janusz GARDULSKI Silesian University of Technology, Faculty of Transport,  Department of Automotive Vehicle Construction Krasińskiego St. 8, 40-019 Katowice, Poland *Corresponding author. E-mail: rafal.burdzik@polsl.pl     FREQUENCY ANALYSIS DECIMATION VIBRATION SIGNALS OF PASSENGER CAR’S SUSPENSIONS   Summary. The paper presents possibilities of applying linear decimation procedure in frequency  analysis  of  non-stationary  signals.  It  shows  the  results  of  analytical experiments  conducted  with  vibration  signals  registered  during  examination  of  car suspension which was activated to vibration on harmonic stand research. Working cycles identification algorithm was helpful to make LDP of working cycles parts of a signal. Signals  spectrum  results  confirm  amplitude  selectivity  for  typical  frequency  of researching dynamic system.    CZĘSTOTLIWOŚCIOWA ANALIZA DECYMOWANYCH SYGNAŁÓW DRGANIOWYCH ZAWIESZEŃ SAMOCHODÓW OSOBOWYCH  Streszczenie. W artykule zaprezentowano moŜliwości zastosowania procedury liniowej decymacji  w  częstotliwościowej  analizie  sygnałów  niestacjonarnych.  Przedstawiono wyniki  eksperymentów  analitycznych  przeprowadzonych  na  sygnałach  drganiowych zarejestrowanych  w  trakcie  badań  zawieszenia  pojazdu  pobudzanego  do  drgań  na stanowisku harmonicznym. Algorytm umoŜliwiający identyfikację kolejnych cykli pracy umoŜliwił przeprowadzenie PLD sygnału podzielonego wg cykli pracy. Uzyskane widma sygnałów  potwierdzają  selektywność  amplitud  dla  charakterystycznych  częstotliwości badanego układu dynamicznego.   1. ANALYSIS METHODS OF VIBRO-ACOUSTICS SIGNALS  Acoustics  or  vibrations  signal  as  the  results  of  changes  which  occur  in  technical  system  or associated processes is a medium of the vibro-acoustic information. As the information we understand everything  what  is  used  to  get  more  efficient  selection  of  the  leading  operation  to  the  specified objective [3]. Information is related indissolubly with the signal because this signal is the medium of it. It informs  us  about  conditions,  changes  or  process  of  physical  or  technical  system  taking  under consideration. Vibro-acoustics signals have the biggest information’s capacity which enables us to observe changes on a wide frequency band. That is why vibro-acoustics research methods are use in wide application in technical objects diagnostics. Vibro-acoustics signals analysis is very difficult. There are many methods of signals analysis. The main problem in vibro-acoustics research is the difficulty of useful signal components separation from the rest of signal without any important information. 24                                                                                                                                      R. Burdzik, J. Gardulski   It is possible to consider many of measurement problems on general signal level so it is considered as the total signal in observation time. To define this kind of signals it can be used with such quantities as: a)  amplitude domain, for example: rms value, mean value, peak value etc.; b)  time domain, for example: autocorrelation function, probability density function etc.; c)  frequency domain, for example: Fourier transform. Due to the complexity of this problem this paper focuses only on frequency domain analysis of vibration signals.    2. FREQUENCY ANALYSIS OF SIGNALS  Spectrum  of  signal  is  signal  energy  distribution  in  frequency  domain.  It  shows  dominant frequency components and dynamic changes in research system. Fourier transform signal conversion enables to obtain signal spectrum as shows this equation:  ( ) ∫− ⋅ =Tt j dt e t fTF0) (1 ω ω   (1)  In  digital  signal  transformations  Fast  Fourier  Transform  (FFT)  is  used.  Using  FFT  is  well-grounded only in case of stationary signals. In  analysis  of  non-stationary  signals  random  processes  have  fundamental  meaning,  in  which interesting random effects can be functions of frequency and time domain. For this kind of processes the most often used analysis methods are: −  short time Fourier transform (STFT), −  continuous wavelet transform (CWT), −  Wigner-Ville distribution (WVD). The  results  of  these  transformations  are  signal  distribution  in  time-frequency  domains. Multidimensional analysis methods of non-stationary signals enable to observe with good quality of distributed signal but they are time-consuming. Different approach in analysis of non-stationary signals is Linear Decimation Procedure (LDP) application [1, 5, 6].    3. LINEAR DECIMATION PROCEDURE  In  signals  digital  processing  for  maximal  spectral  resolution  diagnostics  signal  should  be synchronism sampling with sampling frequency so the sample number are integral multiple of N samples number for FFT analysis. To get the expected effect of series length we can make decimation of signal [2]. Signal decimation is the process of reduction of effective samples number of signal which is sampling frequency reduction.  Linear Decimation Procedure is coefficient-variable dynamic decimation. The method involves dynamic  sampling  signal  in  form  of  linear  increase  decimation  coefficient.  It  assumes  linear approximation of trend cycle in the observation window. The procedure eliminates linear cycle trend preserving its fluctuations in the transformation range. The sample number per cycle must be constant in the newly created vector, it is the condition of stationary characteristics.  Process  of  obtaining  secondary  signal  vector  after  LDP  from  primary  vector  is  presented  on figure 1. Frequency analysis decimation…                                                                                             25   Fig. 1. Linear Decimation Procedure [5, 7] Θ  –  cycle  time,  t(Θ)  –  observation  time,  N(n)  –  primary  ample-cluster  in  the  observation  window,             -  L(v)  –  secondary  ample-cluster  in  the  observation  window  after  LDP,  -  Dcp  –  initial  decimation coefficient, - Dck – final decimation coefficient. Rys. 1. Procedura liniowej decymacji [5, 7]  For the sake of FFT correctness the number of samples N (power of digit 2) is specified by the relation: C krfNs k ∈ ≥ = , 2   (2) where:  s f  - sampling frequency, which must satisfy a condition:  Dc f fs max 2 >   (3) where:  max f  - upper frequency of analyzed signal band limited by low-pass filter with frequency  f< max f . Decimation coefficient, with which the signal is resampled, agrees with the increase or decrease of rotational  speed,  that  is  cycle  change  (run  up,  braking).  For  stationary  characteristics  the  sample number per cycle must be constant:  const L L L p k j = = = Θ Θ Θ   (4) where:  j LΘ  - number of samples in the jth cycle of observation window for newly created vector,  k LΘ  - number of samples in the last cycle of observation window for newly created vector,  p LΘ  - number of samples in the first cycle of observation window for newly created vector.  Assuming, based on number of samples in the last period of signal, final decimation coefficient Dck we can determine in succession decimation coefficient Dcn finishing on nDk sample.  ( − + =ΘΘΘΘ ) 1kpDk kpk n NNnnNNDc Dc   (5) where: NΘp – number of samples in the first cycle of observation window of primary signal, NΘk – number of samples in the last cycle of observation window of primary signal, n – number of sample, nDk – number of sample finishing LDP: 26                                                                                                                                      R. Burdzik, J. Gardulski   k k Dk Dc n n − =   (6)   where: nk – number of last sample of observation window of primary signal.    4. RESULTS OF TRANSFORMATIONS  Within the framework some analytic experiments were conducted on vibration signals registered during research of automotive vehicle suspension at the station regarding harmonic enforcement, were performed such as: 1.  run up the stand – constant increase of rotational speed of the forced system (10 s), 2.  work with constant frequency of forced system (20 s), 3.  coasting – station shut-down and frequency decrease of the forced system with constant step (10 s). This method allows for vibration acceleration signals recording of springing and non-springing vehicle masses. It enables us to determine relative acceleration vibration signals of shock absorber. These are non-stationary signals so FFT is useless for diagnostics information to obtain. It shows the spectrum of exemplary analyzed signal on figure 3. There is only unequivocally selected amplitude of near 21 Hz which is the frequency of constant work of forced system the rest of spectrum picture is illegible. time [s]vibr a t i on ac c e l e r a t ions [m / s]2 Fig. 2. Relative vibration acceleration signal Rys. 2. Sygnał względnych przyspieszeń drgań   Fig. 3. Spectrum of relative vibration acceleration signal Rys. 3.Widmo sygnału względnych przyspieszeń drgań  For  signal  dividing  on  next  cycles  of  forced  machine  working  were  next  phases  of  vibration inductor working identification measure formulated. The markers of next cycles of forced machine working  measures  based  on  STFT  transformation  were  used.  The  main  reason  of  choosing  this transformation was short realization time. Example of STFT distribution is presented on 4 figure.  Frequency analysis decimation…                                                                                             27  frequency [Hz] part of freuqency bandtime [s] Fig. 4. STFT spectrum with marked part of the frequency band for further analysis Rys. 4. Widmo STFT z zaznaczonym pasmem wyciętym do dalszej analizy  There was 21-22 Hz frequency band cut out from STFT spectrum as presented on figure 4. Based on time function of cut off frequency band identifying algorithm of end of stand run up and start of stand coasting time coordinates was created. This is presents on figure 5.   stand run upcoordinatestand coastingcoordinatetime [s]STFT amplitude m/[s]2 Fig. 5. Time function of STFT coefficients for analysis frequency with end of rotational speed stand iput function increment and start decrease of rotational speed stand input function Rys. 5. Przebieg czasowy współczynników STFT dla analizowanej częstotliwości z zaznaczeniem końca rozbiegu i początku wybiegu   Elaborated algorithm is based on comparing the next value of analysed frequency band („analysis of edge”) around set parameters.  Locating of end of stand run up and start of stand coasting enables to divide signal on three time windows. First window for fragment of signal growing according to constant frequency increase of the 28                                                                                                                                      R. Burdzik, J. Gardulski   forced system. Second window for signal with constant frequency and the third one for coasting stand – decrease of signal amplitudes according to constant frequency decrease of the forced system. Signal  decimation  requires  earlier  use  of  lowpass  signal  filtration  to  avoid  aliasing  after decimation [7]. In the case of linear decimation procedure it is indispensable to filter with changeable stopband. It was suggested to filter with changeable stopband inversely proportional to changeable decimation coefficient. After the newly created vector summation we get the whole signal after the linear decimation procedure,  which  is  quasistationary  (fig.  6).  Spectrum  of  exemplary  signal  after  the  LDP  was presented on figure 7.  time [s]vibration acceleration [m/s]2 Fig. 6. Analysed signal after LDP Rys. 6. Analizowany sygnał po PLD frequency after LDP [Hz]amplitudeselected amplitudes  Fig. 7. Spectrum of analysed signal after LDP Rys. 7. Widmo analizowanego sygnału po PLD  As it is observed on the figure, LDP used for non-stationary vibration signals enables to select characteristic single amplitudes from the spectrum. Frequency scale signal after LDP is not exactly like frequency of primary signal. That is why domain is described as frequency after LDP on figure 7. However we can conclude that selected signal amplitudes correlate with resonance frequencies of springing and non-springing vehicle masses.    5. SUMMARY  The paper presents possibilities of technical condition identification of mechanical object based on analysis of vibro-acoustic signals recorded during work of the system. Basic methods and measures in signal processing and criteria of their applications were presented. Linear Decimation Procedure as the tool rarely used in non-stationary signals processing were widely discussed. Example results of the analytic experiments conducted on vibration signals registered during research of automotive vehicle suspension  were  presented.  Elaborated  algorithm  of  next  working  cycles  of  the  station  regarding harmonic enforcement identifying enabled to signal divided and LDP made for next working cycles. Primary and after LDP signals spectrum compare effectiveness of presented method confirm. These results enable to identify technical condition of the system based on spectral analysis of non-stationary vibration signals. Presented method of diagnostics information receiving as the results of vibration signals processing enables elaborate of automatic system of passenger cars shock absorbers diagnosis. However it is necessary to make more investigation in this direction.        Frequency analysis decimation…                                                                                             29  Literature  1.  Adamczyk  J.,  Krzyworzeka  P.,  Cioch  W.:  Wykorzystanie  decymacji  dynamicznej  w diagnozowaniu  maszyn.  XXVIII  Ogólnopolskie  Sympozjum  Diagnostyka  Maszyn,  Katowice 2001, s.7-18. 2.  Adamczyk  J.,  Krzyworzeka  P.,  Łopacz  H.:  Systemy  synchronicznego  przetwarzania  sygnałów diagnostycznych. Kraków, Collegium Columbianum, 1999. 3.  Basztura C.: Komputerowe systemy diagnostyki akustycznej. PWN, Warszawa 1997. 4.  Burdzik  R.,  Gardulski  J.,  Konieczny  Ł.:  Nowe  metody  diagnozowania  amortyzatorów. VI Konferencja Naukowa Telematyka i Bezpieczeństwo Transportu, Katowice 2006, s.86-95. 5.  Cioch  W.:  Sztuczne  sieci  neuronowe  w  diagnostyce  zagroŜeń  eksploatacyjnych  systemów technicznych. Praca doktorska, Kraków 2002. 6.  Krzyworzeka P.: Wspomaganie synchroniczne w diagnozowaniu maszyn. BPE, Radom 2004. 7.  Krzyworzeka P., Adamczyk J., Cioch W., Jamro E.: Monitoring of nonstationary states in rotating machinery. ITE, Kraków 2006. 8.  Lyons R.: Wprowadzenie do cyfrowego przetwarzania sygnałów. WKŁ, Warszawa 1999.    Received 23.11.2006, accepted in revised form 25.05.2007  

Frequency analysis decimation vibration signals of passenger car’s suspensions

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