Diagnosis of partial rotor stator rubbing using Variational Mode Decomposition

Transcription

1 The 4th IFToMM World Congress, Taipei, Taiwan, October 5-30, 05 DOI Number: /IFToMM.4TH.WC.OS4.07 Diagnosis of partial rotor stator rubbing using Variational Mode Decomposition S. Braut R. Zigulic G. Stimac-Roncevic A. Soblar University of Rijea Rijea, Croatia Abstract Rotor stator rubbing is occasional problem faced by rotating machines especially during startups and shut downs when passing through their critical speeds. The rubbing often occurs in rotating machinery at position with small clearances and can sometimes cause catastrophic breadown of machine. So it is important to develop reliable tools for rub diagnosis. This paper investigates partial rub occurrence during constant and slightly variable speed operation and try to define vibration diagnosis patterns for its detection with Variational Mode Decomposition (VMD). The VMD is relatively new method with promising results potentially interesting for machinery failures diagnosis.. Diagnostic tool is tested on laboratory test rig at two different rotor operating conditions i.e. without rotor stator rubbing and with partial rotor stator rub. Measurements were performed with non-contact eddy current displacement sensors. Results are presented in the shape of VMD spectrograms and are compared to Empirical Mode Decomposition (EMD) spectrograms Rotor orbits are also presented to additionally verify rubbing condition. Keywords: Rotor-stator partial rubbing, fault diagnosis, Variational Mode Decomposition, Empirical Mode Decomposition I. Introduction Light partial rubbing between rotor and stator may occasionally occur during the operation of the rotating machine especially during starting up and shutting down while passing its critical speed. If it happen during normal operation it could be indicator of some failure which could eventually lead to more serious full annular bacward whirl. The requirement of higher machine efficiency has reduced clearances between rotor and stator and thus increased the possibility of rubbing occurrences. Depending on rotor and stator structure configuration rotor-stator contact phenomena can be classified as a rigid rotor disc-rigid stator con-tact [,,3,4], bladed disc-stator contacts [5] and rotor-stator contact in retainer bearings of the active magnetic bearings [6,7]. Blade loss scenario presents also interesting rubbing related topic for designers from industry [8] while they trying to answer questions regarding structural integrity of machines. Some authors also considered shaft bow effect on rubbing phenomena [9]. sbraut@riteh.hr zigulic@riteh.hr gstimac@riteh.hr asoblar@riteh.hr The vibration generated by rotor stator contact always contains nonlinear and non-stationary signals. Recently, a number of new methods have been proposed to analyze such signals. One of the promising methods is the Hilbert Huang Transform (HHT) [0,,]. Although its original name was Empirical mode de-composition (EMD) it can be considered as the first step of the method while the second one is and Hilbert Transform (HT) [3]. EMD start first and it is in charge for the sifting process. It decomposes the signal into a set of Intrinsic Mode Functions (IMF s), not assuming linearity, stationarity, or any a priori bases for decomposition. HT is employed then to form instantaneous amplitude and frequency from the corresponding IMFs. Nowadays, HHT method has become very popular in various areas including vibrations of rotating machinery [4,5,6]. Although popular, IMFs of original version of HHT (EMD) suffer from the lac of mono-component property for the real signals containing noise. Bearing this in mind Dragomiretsiy and Zosso have recently proposed a new approach called Variational Mode Decomposition (VMD) [7]. This paper considers light rotor stator rub occurrence and test the capability of VMD method to detect it as well as its sensitivity of method parameters setup. Lateral rotor displacements are measured near and at critical speed i.e. without rotor stator contact and with noticed light partial rub. Results of VMD method diagnostic capability are compared to EMD method and it can be seen that both methods have promising results. II. Variational Mode Decomposition As the Authors [7] of Variational Mode Decomposition (VMD) explain their motivation to establish a better way to decompose a complex signal into principal modes, than it is done with Empirical Mode Decomposition (EMD), we will start with brief explanation what is EMD all about. A. Empirical Mode Decomposition The EMD is an empirically based adaptive data analysis method that was first proposed by Huang et al. [0] and has been utilized for nonlinear and non-stationary data analysis in various applications. The EMD is also nown as Hilbert Hualng Transform (HHT). The essence of the method is to identify the intrinsic oscillatory modes by their characteristic time scales in the signal empirically, and then decompose the signal. The method utilizes an iterative sifting process which successively subtracts the local mean from a signal. The sifting process consists of following steps:

2 Determine the local extrema (maxima, minim of the signal x(t). Connect the maxima with appropriate interpolation function, creating an upper envelope about the signal. Connect the minima with appropriate interpolation function, creating a lower envelope about the signal. Calculate the local mean m as half the difference between the upper and lower envelopes. Subtract the local mean from the signal, h = x(t) m and inspect whether the number of extrema and the number of zero crossings are equal or differ at most by one. Also, inspect whether all the local maxima are positive and all the local minima are negative. If signal meets definition of IMF (after iteration), designate it as a c = h First IMF can be now subtracted from original signal r = x(t) c () The reminder r should be treated as a new data set/signal and repeat the sifting process to obtain c. Described process should be repeated until the original signal is decomposed in terms of IMFs, that is x n () t = ci () t + rn () t i= () where c i (t) is the i-th IMF of the signal x(t) and r n is the final residue. Procedure ends with Hilbert transform of every IMF component i.e. mode. B. Description of VMD In contrast to EMD with its ad-hoc nature lacing mathematical theory and recursive sifting, VMD represents a method that determines the relevant bands adaptively and estimates the corresponding modes concurrently, thus properly balancing errors between them [7]. VMD has a goal to decompose an input signal into a discrete number of sub-signals (modes) that have specific sparsity property while reproducing the input signal. It is required that each mode is mostly compact around a center oscillation ω, which is to be determined along with decomposition. In EMD method it was observed that actual IMF in real signal application, often isn t a desired monocomponent function. To overcome this problem Dragomiretsiy and Zosso proposed another scheme: ) for each mode, compute the associated analytic signal by means of the Hilbert transform in order to obtain a unilateral frequency spectrum. ) For each mode, shift the mode s frequency spectrum to baseband, by mixing with an exponential tuned to the respective estimated center frequency. 3) The bandwidth is now estimated through the H Gaussian smoothness of the demodulated signal, i.e. the squared L norm of the gradient. The resulting constrained variational problem is solved by separate minimizations with respect to {u } (all modes) and with respect to {ω } (corresponding center frequencies). Complete optimization of VMD algorithm can be summarized as follows: At the beginning chose how many modes K are significant, Initialize { ˆ }, { }, ˆ u ω λ, n = 0 Repeat following loop n = n + for = : K Update û for all ω 0: ˆn ˆ n+ n λ ( ) ( ) ( ) ( ω) f ω uˆ i ω uˆ i ω + n+ ˆ i< i> u ( ω) = n + α ( ω ω ) (3) Update ω : ω n + = n+ ω uˆ 0 n+ uˆ 0 end for Dual ascent for all ω 0 ( ω) dω ( ω) dω (4) ˆ n + n n λ ( ω) = ˆ λ ( ω) + τ fˆ ( ω) u + ˆ ( ω) (5) Until convergence condition is met n n n uˆ uˆ uˆ < ε. (6) + III. Test Rig Experimental verification of the proposed algorithms for light rubbing detection is performed on a laboratory test rig shown in Fig.. The test rig, originally built for rotor stator contact dynamics investigation, consists of a rotor supported by two self-aligning roller bearings and connected via elastic coupling to induction motor with speed controller. Shaft has total length of m and diameter of 7 mm. Three discs are mounted very closely to each other at the shaft midspan. The biggest disc has diameter of 0 mm and is designed to establish contact with the stator ring while the other two smaller discs of diameter 80 mm are intended for measurements of lateral displacements with non-contacting probes and for phase. The total rotor mass is approximately 5 g. The stator is made of annular plate elastically suspended on four circular beams. The radial clearance between rotor and stator is 0.4 mm. According to modal testing first two bending normal modes have frequencies at 7.8 and 45 Hz while first torsional mode have frequency 34 Hz. Stator first lateral (bending concerning the support beams) is 90 Hz. The measurement system applied for this purpose is based on National Instruments PCI card NI 447 with 3 noncontacting displacement probes IN 085 made by Schenc ( for rotor radial displacement, one for stator radial displacement in horizontal direction and optical sensors P-84 also made by Schenc for phase measurement. Measurements were performed at 5 Hz sampling frequency.

3 more stable / fraction order vibration. Truly, it is more visible in Fig. 9 b for the vertical rotor vibration but still could be observed in Fig. 9. On VMD spectrograms shown in Fig. 0 one frequency component greatly oscillates around 3/ fraction order vibration what could be indication of partial rubbing. Fig.. Test Rig IV. Implementation and results To test light rub detection capabilities of VMD spectrograms, many tests are performed but due to limited space only two of them will be presented. The idea was to compare two similar measurements, first of them without rotor stator contact (at the speed 5,8 Hz i.e. 548 rpm) and the other one with noticed partial rotor stator rub (at the speed 7,6 Hz which is almost at critical speed). During tests rotor had slight unbalance (5e-5 g m) but also shaft bow equal to 0, mm. Measurements are analyzed via VMD spectrograms and in the shape of EMD spectrograms as well as in the form of classical FFT spectra. Fig., presents composite raw input signal obtained by non-contacting displacement probe. Fig. 3, Fig. 4 and Fig. 5, Fig 6, Fig 7a and Fig. 8a represents situation without rotor stator contact. Fig. 3 shows IMF component of a rotor lateral signal for a case without contact. In Fig. 4 EMD spectrograms are shown of two lateral rotor displacements in horizontal and vertical directions. Beside expected oscillating instantaneous frequency at about value of rotating speed (5,8 Hz) there are some frequency component greatly oscillating in time from 0,x up to 0,5x harmonic. VMD spectrograms on Fig 6 are clear from fractional sidebands. According to Muszynsa [8] and Peng et al. [9] if partial rotor stator rub is happening there must exist stable subharmonic vibration with frequencies equal to exact fraction of the rotating speed (most often this fraction is ½) harmonic. Muszynsa [8] further explains that during partial rub it should be also observed presence of the higher fraction harmonics e.g. 3/ and 5/. FFT of raw input signal in vertical lateral direction (Fig. 7 show no noticeable 0,5x harmonic whatsoever. Accordingly it can be concluded that there was no partial rubbing in the first measurement. The orbit shown in Fig. 8 confirms that claim. On contrary orbit in Fig. 8 suggest that some rubbing should happen. During this second test metal punching sound was heard. FFT spectrum of raw input signal for vertical lateral vibration signal of the second measurements (Fig. 7 now clearly shows noticeable /x harmonic as well as 3/x harmonic. Fractional harmonic 5/x is somewhat less visible but is also present in spectra. On EMD spectrograms shown in Fig. 9 now can be seen Fig.. Composite input signal, y direction No rubbing Fig. 3. IMFs from EMD, y direction no rubbing

4 Fig. 5. VMD u (t) continued Fig. 4. EMD component spectrogram no rubbing, x-direction, y- direction Fig. 6. VMD component spectrogram no rubbing, x-direction, y- direction Fig. 5. VMD u (t)

5 Fig. 7. FFT spectrums, y-direction, no rubbing, light partial rubbing Fig. 9. EMD component spectrogram light partial rubbing, x-direction, y-direction Fig. 8. Rotor orbits, no rubbing, light partial rubbing Fig. 0. VMD component spectrogram light partial rubbing, x-direction, y-direction

6 V. Conclusions This paper considers possibility of partial rub detection using VMD method. For this purpose two measurements are performed on the specially designed test rig. First measurement presents subcritical rotor operation without rotor stator contact. Corresponding VMD as well as EMD spectrograms clearly have x harmonic of the rotor speed but on the same time didn t have stable subharmonic /x vibration. FFT spectra of raw input signal confirmed absence of fractional sidebands. On contrary in EMD spectrogram of the second measurement (critical rotor operation with established partial rotor stator ru it is observed more stable /x fraction order vibration. In the same time VMD spectrogram showed presence of 3/x harmonic. Corresponding FFT spectra of raw input signal in vertical lateral direction now clearly shows noticeable /x, 3/x harmonic as well as 5/x harmonic. It can be concluded that both VMD and EMD spectrograms can detect partial rotor stator rub condition and have potential for improvement to become valuable diagnostic tools. Syst. Signal Process, , 007. [5] Ricci, R., Pennacchi, P.: Diagnostics of gear faults based on EMD and automatic selection of intrinsic mode functions, Mech. Syst. Signal Process. 5 (3), 8 838, 0. [6] Braut, S., Žigulić, R., Štimac, G., Soblar, A., Rotor-stator partial rub diagnosis using Hilbert Huang transform, In 9th IFToMM International Conference on Rotor Dynamics,. Milano, Springer, pp. -9, 04. [7] Dragomiretsiy, K. and Zosso, D. Variational Mode Decomposition, IEEE Trans. on Signal Processingm, 6(3):53 564, February 04. [8] Muszynsa, A.: Rotordynamics. CRC Press, Taylor and Francis Group, Boca Raton, USA, 005. [9] Peng, Z. K., Tse, P.W., Chu.F. L.: An improved Hilbert Huang transform and its application in vibration signal analysis. J. Sound Vibr. 86, 87-05, 005. References [] Choy, F. K., Padovan, J.: Non-linear transient analysis of rotor-casing rub events. J. Sound Vib., 3(3), , 987. [] Bartha, A. R.: Dry Friction Bacward Whirl of Rotors. PhD. Thesis, Swiss Federal Institute of Technology Zurich, 000. [3] Von Groll, G., Ewins, D. J. (00) A Mechanism of Low Subharmonic Responce in Rotor/Stator Contact Measurement and Simulation. J. Vib. Acoust. 4, , 00. [4] Torhany, M., May, L., Voinis, P. Light, medium and heavy partial rubs during speed transients of rotating machines: Numerical simulation and experimental observation, Mech. Syst. Signal Process 9, 45-66, 0. [5] Ahrens, J., Jiang, J., Ulbrich, H., Ahaus, G.: Experimentelle Untersuchungen zum Schaufelanstreifen. In: Schwingungen in rotierenden maschinen SIRM V Tagung. Wien, Austria, pp , 00. (in German) [6] Fumagalli, M., Varadi, P., Schweitzer, G.: Impact dynamics of high speed rotors in retainer bearings and measurement concepts. In:4th International Symposium on Magnetic Bearings, ETH Zurich Switzerland, pp , 994. [7] Orth, M., Nordmann, R.: ANEAS A modeling tool for nonlinear analysis of active magnetic bearing systems, In nd IFAC Conference on Mechatronic Systems, Berley, USA, pp , 00. [8] Braut, S., Žigulić, R., Soblar, A., Štimac, G., Butović, M., Joić, M., Dynamic Analysis of the Rotor-Stator Contact due to Blade Loss, In Twelfth World Congress in Mechanism and Machine Science, Besancon, pp. -6, 007. [9] Braut, S., Žigulić, R., Butović, M.: Numerical and experimental analysis of a shaft bow influence on a rotor to stator contact dynamics, Stroj. Vest. J. Mech. Eng. 54 (0); , 008. [0] Huang, N.E., Shen, Z., Long, S.R., et al., The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis. Proceedings of the Royal Society of London 454, , 998. [] Rilling, G., Flandrin, P., Gonçalves, P., Lilly, J. M.: Bivariate Empirical Mode Decomposition. IEEE Signal Process. Lett. 4(), , 007 [] Rilling, G., Flandrin, P.: One or Two Frequencies? The Empirical Mode Decomposition Answers. IEEE Trans. on Signal Process.56(), 85-95, 008. [3] Feldman, M. Hilbert transform in vibration analysis. Mech. Syst. Signal Process. 5 (3), , 0. [4] Qi, K., He, Z., Zi, Y., Cosine window-based boundary processing method for EMD and its application in rubbing fault diagnosis, Mech.