Proceedings of IDETC/CIE 2005 ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference September 24-28, 2005, Long Beach, California, USA DETC2005-85218 EXPERIMENT OF OIL-FILM WHIRL IN ROTOR SYSTEM AND WAVELET FRACTAL ANALYSES Qing Kai Han +86-24-81541130, qingkai_han@sohu.com Zhi Wei Zhang +86-24-83680545, ttwwooblueyes@126.com Hong Liang Yao +86-24-83671429, yao_hongliang@sina.com Bang Chun Wen +86-24-83671429, bcwen1930@sina.com ABSTRACT An oil-film whirl experiment is carried out in a rotor test rig and some nonlinear analyses are achieved in the paper. Firstly, the experiment schemes for measuring oil-whirl of a rotor system are introduced. The shaft vibrations are measured both in stable rotating process and running up and coast down. And then, the detailed signals of level 3, reconstructed by wavelet transform, representing the oil-film whirl motions are analyzed. At last, a wavelet fractal method is proposed, based on wavelet package transform and fractal geometry theory, and the correlative dimensions of signals in every frequency bands are calculated. This method describes the chaotic properties of the oil-film whirl vibration well, and wavelet fractals prove to be more effective than any other nonlinear parameters of the corresponding original shaft vibration signals. KEYWORDS: rotor system, oil-film whirl, wavelet transform, wavelet fractal analysis INTRODUCTION Oil whirl is a typical self-excited rotor vibration instability mechanism and has brought about several catastrophic failures in rotating machinery. After discovered and presented by B. L. Newkirk and H. D. Taylor firstly [1], many studies have been conducted on it. Besides the pioneer works using linear dynamics, recently works made many efforts to study the nonlinear mechanisms [2-4]. The authors completed some numerical analyses in the bifurcation characteristics of oil whirl phenomenon based on a nonlinear oil-film force model in [5,6]. But until now, experimental and analytical researches on oil-film whirl are still not enough. The reasons mainly are that the oil-film dynamics is too complex, as well as the oil whirl is difficult to be explained even with non-linear mathematical models. It is known that, when oil whirl occurs, the rotor system will experience unstable nonlinear motion, such as bifurcation, quasi-periodic, and even chaos. In that case, wavelet transform (WT) is regarded as a better choice to study its time-frequency characteristics instead of traditional FFT or short-time FFT. WT indicates both the frequency contents and the time of occurrence of the transient vibration, for its varying window size and optimal time-frequency resolutions in all frequency ranges [7]. The frequency intervals, considered as octave bands, determine the frequency resolution of signals. In addition, fractional geometry can be used to quantitative study the nonlinearity of oil whirl rotor system. Fractal dimension is a quantity that is related to static property and describes the geometry of a strange attractor. The higher the fractal scalar dimension, the higher the degree of chaotic content in the motion. Wavelet transform and fractal geometry can be combined to study the nonlinearity of rotor system, which can be called wavelet-fractal process. In wavelet-fractal process, the nonlinear complex characteristics and irregularity of vibrating signals are described with the fractal dimensions in different frequency bands obtained by wavelet package decomposition. The calculated correlate dimensions [8], referring with positive maximum Lyaponuv exponents [9, 10], 1 Copyright 2005 by ASME
determine that the nonlinear oil-film whirl may behave as chaos. In the present paper, firstly, some experiment results of oil-film whirl in a rotor test rig are introduced. The vibration signals of the rotor shaft during oil-film whirl, both at a fixed rotating speed and during the process of running up and down, are recorded. The developing progresses of oil-film whirl in different operating cases are illustrated in time and frequency domain. Secondly, wavelet transform is used to study non-stationary signals of rotor oil-film whirl. Thirdly, the wavelet-fractal analyses are carried out for both the stable running and the running up and down process where oil-film whirls existing certainly. EXPERIMENT EQUIPMENTS AND TEST SCHEME The test rig is shown as Fig. 1. The rotor length is 275mm. The diameter and thickness of the disc are 75mm and 25mm, respectively. The equivalent eccentricity of the disc is 0.036mm. The rotor shaft diameter is 10mm. The length and diameter of the shaft inside Bearing 2 are 30mm and 25mm, respectively. The inner diameter and working length of Bearing 2 is 25.18mm and 7.1mm, respectively, which means that the average thickness of oil-film is 0.09mm. The lubrication oil is 30#. The measured oil film parameters of Bearing 2, for example, are k xx =1.029e8N/m, k xy =-7.108e7N/m, k yx =9.694e7N/m and k yy =-6.772e7N/m respectively, in stable running with 5340 r/min. The first critical speed of the rotor system is 80Hz, i.e. 4800 r/min. Two proximity probes are settled nearby Bearing in order to measure vibration displacements of the rotor shaft in horizontal direction x and vertical direction y, as shown in Fig. 1. The sampling frequency is 1000Hz. direction and y direction are drawn in Fig. 2. The amplitude spectrums of the two direction vibrations are remarkable with sub-harmonic components, 70Hz, which are nearly about 0.44 times of the rotating frequency. The amplitudes of the N/2 sub-harmonic components are even larger than that of the corresponding synchronous components. In addition, the trajectory orbit of the shaft center in Fig. 2(c) is obviously irregular and limited. (a) Displacement and its amplitude spectrum in x direction Disc Motor Bearing 1 y x Bearing 2 Transducers Figure 1: Schematic of test rig for oil-film whirl experiment Oil whirl appears as a sub-synchronous vibration, and it will happen when the journal bearings act as negative dampers to the lowest frequency forward-whirl rotor-bearing vibration mode. In experiment, the vibrations of the rotor shaft in this test rig were measured both in the processes of stable state and run-up and coast-down. In the steady running process, the rotating speed went up from 0 r/min to 9600 r/min, and then was kept on stable running. In running up and down, the rotating speed went up from 0 r/min to 13940 r/min gradually, and then coasted down to 0 r/min. (b) Displacement and its amplitude spectrum in y direction OIL-FILM WHIRL VIBRATIONS IN STABLE RUNNING AND COAST DOWN The oil-film whip occurs when the rotating speed is fixed at 9600 r/min, i.e. 160Hz. The displacement time histories in x 2 Copyright 2005 by ASME
(c) Orbit of shaft center Figure 2: Rotor shaft vibration at 9600 r/min WAVELET TRANSFORM OF OIL-FILM WHIRL VIBRATION IN STEADY AND RUNNING UP-DOWN Two segments of sampling displacements of rotor in x and y directions are decomposed into 4 levels with the mother wavelet db44 in wavelet transform. Four detailed signals and four approximate signals are reconstructed and compared with each other in time and frequency domain. As the sampling frequency is 1000 Hz, the detailed signal of level 3 (D3) covers the frequency range from 62.5 to 125Hz, which is just corresponded to the oil-film whirl. Figure 4(a) demonstrates the detailed signal of level 3 (displacement y) and its amplitude spectrum. The shaft center orbits with two detailed signals of level 3, obtained from the displacement signals in x and y direction, are shown as Fig. 4(b). Figure 3 shows the time history of displacement in y direction and its cascade plot during the rotor runs up and down. When the rotor speed increases to about 9600r/min, which is about twice times of the first critical speed of rotor system, the sympathetic resonance vibration disappears, and the oil-film whirl comes to appear. Moreover, the oil-film whirl keeps on even when the rotating speed goes on climbing. In fact, the whirl frequency remains as a constant one, which is about the first natural frequency of the system. The rotating speed begins to drop down from 13940 r/min on. The oil-film whirl disappears when the speed comes down to 6900 r/min. The resonant vibration occurs again when the speed decreases to the first critical speed. (a) Detailed signal of level 3 (a) Displacement time history when running up and down (b) Shaft center orbit with detailed signals of level 3 Figure 4: Rotor vibrations with wavelet transform (b) Cascade plot of displacement Figure 3: Rotor shaft vibration when running up and down The shaft vibrations in running up and down process are decomposed into 4 levels with mother wavelet db44, too. When oil-film whirl occurs, the rotating speed should be beyond twice times of the first critical speed, which is about 160Hz. So, the using of the detailed signal of level 3, which covers the frequency range from 62.5 to 125Hz, can also filter 3 Copyright 2005 by ASME
the fundamental modes mostly. The rotating speed increases from 0 r/min to 13940 r/min, and then decreases to 0 r/min. The detailed signal of level 3 in y direction is shown in Fig. 5(a). The three-dimensional amplitude spectrum of it for running up and down is shown in Fig. 5(b). with the delay time of 1. Figure 6(b) shows the pseudo-attracter of the displacement y with oil-film whirl in the case of stable running. The obtained pseudo-attracter pattern is obviously irregular and limited. (a) The largest Lyapunov exponents of y displacement (a) Detailed signal of level 3 (b) Cascade plot of detailed signal of level 3 Figure 5: Rotor vibration with wavelet transform in case of running up and down WAVELET-FRACTAL ANALYSES OF OIL-FILM WHIRL VIBRATION OF ROTOR SYSTEM Just as stated in the above section, wavelet transform is effective in decomposing the rotor displacement signals into interesting frequency ranges for investigation. In this section, the displacement signals are firstly decomposed into continuous independent frequency bands with wavelet packet transform technique, and then, the fractal dimensions of reconstructed data in each frequency band are calculated respectively. The correlation dimensions in different frequency bands are used to describe the nonlinear characteristics of oil-film whirl, especially for its complexity, chaotic characteristics and/or irregularity. Before wavelet packet decomposing and fractal calculating, the maximum Lyapunov exponents of original vibration signals in steady rotating speed are computed firstly. The maximum Lyapunov exponents are useful indications of chaotic motion. The maximum Lyapunov exponents of shaft vibration in stationary rotating process are shown in Fig. 6(a), where the magnitudes of them are in the range of 0.3~0.4 within the time interval of 4 seconds or more than 600 rotating cycles. In addition, a pseudo-attractor, according to phase space reconstruction theory of nonlinear time series X(n), is depicted (b) Pseudo-attractor of y displacement time series Figure 6: Nonlinear description of rotor stable vibration with oil-film whirl The correlative dimensions are calculated from the original vibration signals measured for the stationary rotating and the running up and down process, respectively. The correlative dimension curves of the above two cases are drawn in Fig. 7(a) and (b). (a) The case of stable running (b) The case of running up and down Figure 7: Fractal dimensions of original rotor vibrations 4 Copyright 2005 by ASME
As shown in Fig. 7, the fractal dimension values of original signals in case of stable running are in the small range near 1.8. The fractal dimension values of original signals in case of running up and down are in the range from 2.4 to 1.1. In order to study the perfect fractal geometry of oil-film whirl motion, wavelet packet transform are achieved with Daubechies mother wavelet of db44. The decomposition level is set to 4 and there are 8 frequency bands with equally frequency intervals, 0 ~ 31.25, 31.25 ~ 62.5,, 218.75 ~ 250Hz, respectively. The curves of Fig. 8(a) show the fractal dimensions of each reconstructed signals in 1~8 frequency bands, from top to bottom, in the case of stationary rotating process. The curves of Fig. 8(b) are in the case of running up and coast down. As shown in Fig 8(a) and (b), the correlative dimensions in different frequency bands change in quite different ways. Some of them vary obviously even in the case of stable rotating process, such as in band 7 and 8. In contrast, some fractal dimensions do not change notably in some other frequency bands even in the case of speeding up and down, such as the band 1, 2 and 5. It means that, even in stable rotating process with oil-film whirl, there exists strong nonlinearity in some frequency bands. On the other hand, in unstable rotating process, the nonlinearities in some frequency bands are weak. Compared with the fractal analyses of the original oil-whirl vibration, in which the fractal dimensions may change slightly, as shown in Fig. 6, the results of wavelet fractal analysis exhibit different nonlinear characteristics of the oil-film whirl rotor system in many frequency bands. (a) In case of stable running CONCLUSIONS In this paper, experiments were carried out to investigate the instable vibration behaviors of oil-film whirl on a rotor test rig. The rotor shaft vibrations are measured and analyzed with wavelet transform technique and a wavelet-fractal method for both steady and up-down running. With the sub-harmonics of 0.44 times of per rev and three dimensional spectra of rotor shaft vibration displacements, the emerging and developing of oil-film whirl are demonstrated clearly. The rotor vibration signals with oil-film whirl are decomposed into 4 levels with wavelet transform technique, in which the mother wavelet is set as db44. The detailed signals in level 3 illustrate the oil-film whirl motions within the two processes including both stable running in 9600 r/min and speeding up and coast down from 0 to 13940 r/min and to 0 again. Wavelet-fractal analyses describe the specific nonlinearity and chaotic characteristics of oil-film whirl motions in octave frequency bands. Compared with fractal values of the whole original signals, the correlative dimensions of each different frequency regions reveal the complexity of the oil-film whirl motions in a detailing and comparative way. (b) In case of running up and down Figure 8: Fractal dimensions of every decomposed data in frequency bands 1~8 5 Copyright 2005 by ASME
ACKNOWLEDGMENTS This work is financed by National Natural Science Foundation of China. (No. 10402008 and 50275024) and the High Technology Developing Project of China (No. 2002AA118030). REFERENCES [1] Newkirk, B. L., Taylor, H. D., 1925, Shaft whipping due to oil action in journal bearing, General Electric Review, 28, pp. 559-568. [2] Adiletta, G., Guido, A. R., Rossi, C., 1996, Chaotic motions of a rotor in short journal bearings, Nonlinear Dynamics, 10, pp. 251-269. [3] Jing J. P., Meng G., Sun Y., Xia S. B., 2004, On the non-linear dynamic behavior of a rotor-bearing system, Journal of Sound and Vibration, 274, pp. 1031-1044 [4] Sundarajan, P., Noah, S.T., 1998, An algorithm for response and stability of large order nonlinear systems application to rotor system, Journal of Sound and Vibration 214, pp. 695 723. [5] Wen, B. C., Wu, X. H., Ding, Q., Han, Q. K., 2004, Dynamics and experiments of fault rotors, Science press, China, (in Chinese) [6] Han, Q. K., Yu, T., Yu, J. C., Wen, B. C., 2004, Nonlinear dynamic analysis of unbalanced single-span double-disc rotor bearing system, Journal of Mechanical Engineering, 40, pp. 16-20. (in Chinese) [7] Lamarque, C. H., and Pernot, S., 2001, An overview on a wavelet muti-scale approach to investigate vibrations and stability of dynamical systems, Nonlinear Analysis, 47, pp. 2271-2281 [8] Paciard, N. H., Grutchfield, J. P., Farmer, J. D., et al., 1980, Geometry from a time series, Phys Rev Lett, 45, pp. 712-716. [9] Kantz, H., 1994, A robust method to estimate the maximal Lyapunov exponent of a time series, Phys. Lett. A, 185, pp. 77-87. [10] Sano, M., and Sawada, Y., 1985, Measurement of the Lyapunov spectrum from a chaotic time series, Phys. Rev. Lett, 55, pp. 1082. 6 Copyright 2005 by ASME
Scitation Abstract View 页码,2/2 ASME Conf. Proc. / Year 2005 / Volume 6: 5th International Conference on Multibody Systems, Nonlinear Dynam 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control / Experiments With Nonlin Experiment of Oil-Film Whirl in Rotor System and Wavelet Fractal Analyses Paper no. DETC2005-85218 pp. 1149-1154 (6 pages) doi:10.1115/detc2005-85218 ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC/CIE2005) September 24 28, 2005, Long Beach, California, USA Sponsor: Design Engineering Division and Computers and Information in Engineering Division Volume 6: 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C ISBN: 0-7918-4743-8 ABSTRACT Author(s): Qing Kai Han Shenyang, P. R. China Hong Liang Yao Shenyang, P. R. China Zhi Wei Zhang Shenyang, P. R. China Bang Chun Wen N th t U i it Sh P R Chi An oil-film whirl experiment is carried out in a rotor test rig and some nonlinear analyses are achieved in the paper. Firstly, the experiment schemes for measuring oil-whirl of a rotor system are introduced. The shaft vibrations are measured both in stable rotating process and running up and coast down. And then, the detailed signals of level 3, reconstructed by wavelet transform, representing the oil-film whirl motions are analyzed. At last, a wavelet fractal method is proposed, based on wavelet package transform and fractal geometry theory, and the correlative dimensions of signals in every frequency bands are calculated. This method describes the chaotic properties of the oil-film whirl vibration well, and wavelet fractals prove to be more effective than any other nonlinear parameters of the corresponding original shaft vibration signals. 2005 ASME doi: http://dx.doi.org/10.1115/detc2005-85218 2010American Institute of Physics All rights reserved. Privacy Policy Terms of Use http://scitation.aip.org/vsearch/servlet/verityservlet?key=freesr&s... 2010-1-17