Use of Full Spectrum Cascade for Rotor Rub Identification

Transcription

1 Use of Full Spectrum Cascade for Rotor Rub Identification T. H. Patel 1, A. K. Darpe 2 Department of Mechanical Engineering, Indian Institute of Technology, Delhi , India. 1 Research scholar, 2 Assistant Professor tejas_er@yahoo.com; akdarpe@mech.iitd.ernet.in Abstract Rotor-stator rub is a commonly observed phenomenon in rotating machinery. The rotor vibration response resulting due to rotor-stator rubs exhibit characteristics typical of any nonlinear systems. As there are other faults such as rotor crack which exhibits nonlinearities, it is essential to understand exact nature of nonlinear vibration response of rotor with rub from fault diagnosis point of view and to understand the unique nature of response due to rub. In this paper, steady state vibration analysis is carried out for the rotor stator contact phenomenon at different rotation speeds. Study includes investigations related to different rotor rub regimes. The cascade full spectra have been effectively used to extract the distinctive directional features of rotor rub. The results indicate rotor stiffening and resonance shifting phenomena for the rub impacting rotors. Strong backward whirling 1X vibration motion is detected before the resonance. Superharmonics and subharmonics are found in the spectrum due to nonlinear rotor stator interactions. The vibration response near critical speed is found to be highly directional. The response at the certain supercritical speed range has been found to have stronger subharmonic frequency component (1/2 X) that dominates the frequency spectrum. Response features such as pseudo resonance, stronger backward whirling 1X frequency component, subharmonics and their whirl nature distinctly observed from the cascade spectra can be conveniently used for diagnostic purposes. Key Words: rotor-stator rub, fault diagnosis, full spectrum analysis 1

2 1. Introduction The present day rotors are more flexible and operate under tight clearances and harsh environment. Under these circumstances rotating elements are likely to make contact with stationary elements. Rotor to stator rubbing is considered as a secondary phenomenon resulting from primary cause, which perturbs the machine during normal operation. These primary causes could be rotor vibrations and/or displacements of rotor centreline due to rotor misalignment, gravity force, fluid forces etc. The rotor-stator rubbing is one of the serious malfunctions in rotating machines and there have been extensive research on it. Beatty [1] used the elastic impact-contact model with Fourier series expansion for the mathematically generated rotor response with rubbing condition. Ehrich [3] observed subcritical superharmonic response characterized by appearance of chaotic behaviour in the transition zone between successive orders of superharmonic response. Edwards et al. [4] highlights the importance of torsion in rotor stator contact problems. Chotic response was observed experimentally in the rotor s orbit by several researchers [2,7]. In most of the studies on rotor to stator rub, researchers focus on the steady state response in form of either super and sub harmonics or the typical nonlinear behaviour i.e. chaos and quasi-periodic motion. However, rotor to stator rub interactions is a complex problem and this involves phenomena like impacting, friction, stiffening etc. Depending upon the rotor configuration and system parameters one or other of the above mechanisms can govern the rotor response. In this paper, an attempt is made to demonstrate the different rotor-stator rub regimes using full spectrum analysis. 2

3 Fault identification in rotating machinery using vibration analysis is a constantly expanding field. Though different signal processing techniques were evolved over a period of time, spectral analysis using FFT algorithm is still widely used technique for vibration analysis of the rotors. However, basic drawback with the FFT is that, it treats vibration signal as real quantities so frequency spectrum looses important orbital information such as directivity i.e. forward or backward. Full spectrum overcomes this limitation by retaining the relative phase information between two measured vibration signals [5,6]. This attribute makes full spectrum one of the important diagnostic tools. In present study, the cascade full spectrum analysis is employed to extract the features of rub impacting rotors. The equations of motion of the rotor system are presented. The vibration response of the rotor is simulated in rotor-stator rub for sub critical and post critical speed ranges. Based on the study different rub related phenomena are revealed. 2 Full Spectrum analysis of vibration signal The full spectrum is based on the rotor vibration data from two YZ probes. At a glance, the full spectrum plot allows us to determine whether the rotor orbit frequency components are forward or backward in relation to the direction of rotor rotation. Conventional spectrums (also known as half spectrums) are independently calculated from each waveform. During this calculation, a part of the information contained in the waveform, in particular, the relative phase correlation between Y spectrum and Z spectrum components is not displayed. Full spectrum overcomes this limitation by retaining the directional information of each frequency components. The 3

4 procedure for obtaining the full spectrum from the half spectrums of YZ probes is shown in the Fig. 1 [6]. In right half part, the full spectrum plot presents the amplitude of the forward whirling frequency components (also known as positive frequency components) of signal. The left half part shows backward whirling frequency components (also known as negative frequency components) of the signal. Fig. 1 Mathematical procedure for obtaining a full spectrum [5] 3 Equations of motion of the cracked rotor with rub-impact The equations of motion for the two degree of freedom Jeffcott rotor system in the fixed coordinate system can be written as, my cy ky F ( y, z) mu cos( t ) mg y mz cz kz F ( y, z) mu sin( t ) z 2 2 (i) Forces Fy ( y, z) and F ( y, z) are the non linear rubbing forces, which are z generated from the interactions between the rotor and stator. When rub occurs as shown 4

5 in Fig. 2, the radial impact force F N and tangential rub force F T can thus be expressed as, 0, for( e ) F and F F ( e ) ks, for( e ) N T N (ii) 2 2 Where, e y z, is radial displacement of the rotor, is the clearance between rotor and stator, is coefficient of friction, k s is stator stiffness. Fig. 2 Rotor to stator contact The forces Fy ( y, z ) and F ( y, z ) can be written as, z Fy ( y, z) F N y z f F T ; Fz ( y, z) F z y N f F T e e e e ; or F y ( e ) k 1 s f y ; for e, & F f 1 z z e F y 0 F 0 z Where, f is the function, to decide the direction of frictional forces, ; for e (iii) 1 for R vt 0 y z f 0 for R vt 0 and v t z y e e 1 for R vt 0 (iv) 5

6 4. Nonlinear response of the rub impacting rotors The nonlinear equations (i.e. Eq. i) are solved using Runge-Kutta numerical integration scheme. Values of the system parameters are as follow: mass of the disk, m = 4 kg; shaft stiffness, k 0 = 2.275E+05 N/m; unbalance eccentricity, u = 1E-05m; damping ratio, = 0.05; stator stiffness, k s = 60E+06 N/m; rotor stator clearance, = 1.735E-04 m and coefficient of friction, = Unbalance response of the rotor Unbalance response of the rotor is shown in the form of cascade full spectrum (Fig. 3). The cascade full spectrum is plotted for speed range from 0.1 cr to 2.5 cr (227 r/min to 5694 r/min) in the speed step of 0.1 cr, where, cr m/ k0 is the bending critical speed of the rotor. The full spectrum plot indicates that frequency components only at the rotational speeds are present. Frequency components along the negative frequency axis are totally absent in the spectra. It means vibration motion is synchronous with the rotation speed and in the direction of the rotor rotation through out the speed range considered. Maximum vibration amplitude is observed at the critical speed. Fig. 3 Full spectrum plot of unbalance response 6

7 4.2 Vibration response of the unbalanced rotor under rub-impact During rotor stator contact, the forces acting on the rotor are in the direction normal to the contacting surfaces and tangential to the surfaces. The tangential force is due to friction between the meeting surfaces and depends upon normal force and coefficient of friction at the interface. This friction force tires to accelerate the rotor centreline in reverse precession direction. For this reason, rub produces reverse components in the full spectrum plots. Fig. 4 is the cascade full spectrum plot of rotor stator rub, for the different rotor speeds. Response indicates the strong 1X motion, in presence of other spectral components. The frequency spectrum is rich in the super synchronous frequency components; however, magnitude of higher harmonic components is less compared to 1X component. Rotor systems with rub impact are nonlinear. The nonlinearity comes from the interaction between the stator and rotor during rotation. These nonlinearities are the main source of the higher harmonics in vibration response. Sub synchronous vibration motion is also observed for some speeds. Both forward and backward precession frequencies are present in the response. 1X vibrations are forward through out the speed range considered, except from 3188 r/min to 4099 r/min, before the resonance. Though the high amplitude of the response is observed in the speed range near the bending natural frequency (2277 r/min), resonance is not taken place at this speed, as was observed for the unbalanced rotor (Fig. 3). This is owing to the increased system stiffness in rotor-stator interactions. When rotor stator contact occurs, sudden increase in stiffness is observed. The stiffening of rotor shifts the resonance speed to a higher value and accordingly modifies the rotor response. 7

8 The response is similar to a nonlinear system with spring stiffness increases with displacement. Several researchers [3,4] have attempted to catch this phenomena by modelling the rotor stiffness by a piece-wise nonlinear spring. If vibration amplitude increases further, the dwell time of the rub is likely to increase, thereby increases the average stiffness and making the resonance even higher. The rotor, in effect, chases the resonance by increasing the average spring stiffness in the system as its speed increases. The phenomenon is clearly depicted in the full spectra of Fig. 4(a). In post critical zone, rotor behaves in a typical fashion. First, the rotor system chases the resonance till rotor speed reaches to a value 2961 r/min (Fig. 4(a) & Fig. 4(c)), level of vibrations is also increases along with. This increases the average system stiffness; which has been realized immediately with decrease in vibration amplitudes at next higher speed. Fig. 4(d) shows the full spectra for the speed range 3188 r/min to 4099 r/min. For the second time increase in vibration amplitudes with speed is observed in the full spectra. Rotor now remains in tight contact with stator and tends to roll over the stator. Rotational frequency component is whirling backward in this speed range. This also means that the bouncy motion of the rotor is minimised and replaced with the rolling of the rotor over stator. As forces in the direction of rotation becomes significant compared to backward whirling forces, with increase in rotation speed, +ve 1X component increases more compared to ve 1X. Response increases till the occurrence of resonance at 4327 r/min. It is important to notice that resonance appears just near twice (1.9 times) the bending natural frequency (i.e r/min) speed, known as pseudo critical speed of the rotor. Once the rotor crosses the resonance, vibration level decreases and the rotational frequency component remains forward directional. 8

9 Fig. 4 Cascade full spectrum plots of rotor stator impact at different speed ranges 9

10 Besides the other weak spectral components as well as their directional nature, presence of chaotic motion is clearly evident at some of the rotation speeds, e.g r/min, 2050 r/min, and 4783 r/min to 5466 r/min. Strong sub-harmonics are also observed in the speed range 4783 r/min to 5466 r/min, in the chaotic vibrations. It is important to notice that motion at 1/2X frequency is backward precession. Fig.5. Rotor rub response at ½ critical speed (1138 rpm) Figure 5 shows response of the rotor at ½ the bending critical speed. The vertical vibration response shows rotor bouncing thrice in each rotation that is reflected in the presence of higher harmonics (upto 6X) in the frequency spectrum. Sudden changes due 10

11 to rotor bouncing in vertical direction at three locations in the orbit plot are also observed. The full spectrum plot in Fig. 5 shows significant amount of backward whirl (-1X) rotational frequency component. The higher harmonic components also exhibit strong backward whirling motion. Fig. 6 shows the response of the rotor at the critical speed. Due to rubbing action against the stator the presence of higher harmonics is evident in the frequency domain signal of the vertical vibration. However, as the rotor does not hit against the stator horizontally, the horizontal vibration response does not exhibit higher harmonics as seen in Fig. 6. The vibration response is thus highly directional in nature. The orbit plot in Fig. 6 shows bouncing motion. The full spectrum of the rotor response shows strong backward whirling 1x frequency, much stronger than that observed at ½ the critical speed (Fig. 5). The even harmonics are found to have stronger backward whirling components (-2X, -4X, etc). It is important to note that the rotor generates stronger higher harmonic components even near critical speed. In case of other faults such as fatigue crack in a rotor generates higher harmonics due to nonlinearity of breathing. However, near critical speed the unbalance response at 1X dominates and higher harmonics are insignificant even for deeper crack case. In the rotor rub case, the higher harmonics are observed stronger in amplitudes. 11

12 Fig.6. Rotor rub response at critical speed (2277 rpm) 12

13 Fig. 7. Rotor rub response at 1.3 times critical speed (2961 rpm) Fig. 7 shows rotor response at 1.3 times the critical speed (supercritical speed range). The response due to severe rubbing, exhibits strong -1x frequency response (almost equal to the +1X frequency component). The response also shows backward whirling response at the even harmonics. The response at 1.6 times the critical speed as shown in Fig. 8 shows backward whirl response at 1X. At still higher speed (1.9 times the critical speed, 4327 rpm), the rotor exhibits peak response and vibration levels are maximum. This is referred to as pseudo resonance which is observed at close to twice the 13

14 bending critical speed. All the harmonics have equally strong forward and backward whirl components as shown in Fig. 9. The response shows dominant 1X frequency response and other higher harmonics are not as strong as at other speed. Fig. 8. Rotor rub response at 1.6 times critical speed (3644 rpm) 14

15 Fig. 9. Rotor rub response at 1.9 times critical speed (4327 rpm) Fig. 10 shows the cascade full spectra of the rotor response in the supercritical speed range of 2.1 to 2.4 times the critical speed. The response is aperiodic and exhibits strong subharmonics and a strong ½ X response that is backward whirling in nature. The subharmonics dominate the spectra. Response at one such speed (2.3 times the bending critical speed) is shown in Fig

16 -1/2X is more in comparison with +1/2X Fig. 10. Rotor rub response at speed range of times the critical speed Fig. 11. Rotor rub response at 2.3 times the critical speed 16

17 As mentioned earlier the response is aperiodic, with frequency spectrum dominated by harmonics of 1/6 times the rotational speed. The full spectrum shows strong ½ X frequency component that is again backward whirling. The orbit plot exhibits a complex picture. 5. Conclusions Relative phase information between the vibration signals in two directions has been seen to have great significance in the diagnosis of the rotor rubbing malfunctions. The Full spectrum analysis is a tool which utilizes this directivity information of the rotor orbit along with vibration amplitude in the form of frequency spectrum. Rotor stiffening and resonance shifting are the common phenomena for the rubbing rotors. Pseudo critical speed is observed at rotation speed nearly twice the bending critical of the unbalanced rotor. It has been shown for the first time that while approaching the bending critical speed, rotor system exhibits strong backward whirling 1X vibration motion before the resonance. Spectrum is rich in super spectral and sub spectral lines, due to nonlinear rotor-stator interactions. Backward whirling motion of the harmonics and sub harmonics can be utilised for rub identification, particularly when the rotor starts chasing the resonance. The backward whirling frequencies are excited as strong as forward whirling frequencies at most of the speed range. The subsynchronous frequency components are observed but not at all the speeds. This is an important finding of the extensive analysis carried out in the study that the subharmonics are observed at certain speed ranges and not an essential response feature of rotor rub as is usually stated in the literature. Beyond the pseudo resonance aperiodic response is observed. It is therefore suggested that the 17

18 rotor response during rotor coast up be represented in the form of full spectrum cascade as it gives much more useful information for rub diagnostic view point and the backward whirling 1X frequency component is proposed to be used for rub diagnosis. References [1] Beatty, R. F., 1885, Differentiating rotor response due to radial rubbing, J. Vibration Acoustic, Stress & Reliability in Design 107, pp [2] Chu, F., Lu, W., 2005, Experimental observation of nonlinear vibrations in a rubimpact rotor system, J. Sound and Vibration, 283, pp [3] Ehrich, F., 1992, Observations of subcritical superharmonics and chaotic response in rotordynamics, J. Vibration and Acoustics, 110, pp [4] Edwards, S., Lees, A.W., Friswell, M.I., 1999, The influence of torsion on rotor/stator contact in rotating machinery, J. Sound and Vibration, 225, pp [5] Goldman, P., Muszynska, A., 1999, Application of full spectrum to rotating machinery diagnostics, Orbit First Quarter, pp [6] Lee, C.W., Han, Y.S., Lee, Y.S., 1997, Use of directional spectra of vibration signal for diagnosis of misalignment in rotating machinery, 5 th Int. Congress on Sound and Vibration, Adelaide, Australia, pp [7] Li., G.X., Paidoussis, M.P., 1994, Impact phenomena of rotor-casing dynamical systems, Nonlinear Dynamics, 5, pp