Vibration Signal Analysis of Journal Bearing Supported Rotor System by Cyclostationarity

Cyclostationarity has been widely used as a useful signal processing technique to extract the hidden periodicity of the energy flow of the mechanical vibration signature. However, the conventional cyclostationarity is restricted to analyzing the real-valued signal, which is incapable of processing the constructed complex-valued signal obtained from the journal bearing supported rotor system operating with oil film instability. In this work, the directional cyclostationary parameters, such as directional cyclic mean, directional cyclic autocorrelation, and directional spectral correlation density, are defined based on the principle of directional Wigner distribution. Practical experiment has demonstrated the effectiveness and superiority of the proposed method in the investigation of the instantaneous planar motion of the journal bearing supported rotor system.


Introduction
Journal bearings are one of the most widely used elements in high speed rotary machines and their performance is of great necessity to the normal operation of the rotor systems supported by oil film bearings.As such, plenty of research on journal bearing malfunction analysis has been conducted.
Oil whirl and oil whip are common failure modes of journal bearings induced by oil film instability when the rotating speed of the shaft exceeds the critical speed of the rotor system.Based on the mathematical model of a symmetric rotor supported by one rigid journal bearing and one oil film journal bearing, Muszynska [1,2] mathematically interpreted the dynamic phenomena related to synchronous vibration and self-excited vibration.Relying on a pair of orthogonally installed proximity sensors, the orbit of the shaft centerline is usually monitored to help in analyzing the system operating condition.In addition, Moosavian et al. [3] compared K-nearest neighbor (KNN) and artificial neural network (ANN) for the power spectral density technique based fault diagnosis of main journal bearings of internal combustion (IC) engine under different conditions.Ying et al. [4] studied the system dynamics characteristics of tilting pad journal bearing rotor system operating around natural frequency or on high-speed range by considering the influence of the pad moment of inertia.Dimond et al. [5] reviewed strengths and weaknesses of identification methods for fluid film journal bearing static and dynamic characteristics, particularly the bearing stiffness, damping, and mass coefficients based on measured data of different measurement systems.The developments and trends in improving bearing measurements were also documented.Moreover, Zanarini and Cavallini [6] illustrated detailed experimental results corresponding to oil whirl and oil whip mixed with misalignment, unbalance, and resonance.
In statistics on random processes, cyclostationary processes belonging to the classes of nonstationary processes are defined to represent the correlation features of periodic phenomenon.As such, it is necessary to analyze the cyclostationarity of the modulated vibration radiated by the journal bearing supported system with its components operating periodically.Gardner firstly defined the concept of cyclostationarity and foresaw the application of it.Then Gardner et al. [7] presented a concise review of the literature on cyclostationarity based on the investigation of an extensive bibliography.In addition, Serpedin et al. [8] attempted to provide a detailed classification group which represented a comprehensive list of references on cyclostationarity and its applications.Mccormick and Nandi [9] compared second order cyclostationarity with conventional spectral analysis and synchronous averaging in the condition monitoring of rolling element bearing.They found that the former can reveal significant fault features while the latter failed.Antoni [10] presented a tutorial on cyclostationarity, which introduced key concepts on actual mechanical signals and proved the superiority of cyclostationarity applied in machine diagnostics, identification of mechanical systems, and separation of mechanical sources.
Condition monitoring of journal bearing supported rotor systems usually relies on two orthogonally installed proximity sensors.The orbit signature of the center of the journal is then qualitatively analyzed to investigate system operating condition.Southwick [11,12] applied the full spectrum plot, as compared to the half-spectrum, to diagnose diverse rotating machinery malfunctions.They found that using the full spectrum the orbit ellipticity and vibration precession can be determined without being affected by probe orientation.In order to investigate the complex-valued signal of the instantaneous lateral vibration of the mechanical structure, Lee et al. [13] proposed the two-sided directional power spectra complex-valued vibration signals for the diagnosis of cylinder power faults of four-cylinder compression and spark ignition engines.Furthermore, Lee and Han [14,15] proposed the directional Wigner distribution to effectively determine the procession parameters of the planar motion both in timefrequency domain and in order-frequency domain.
Based on the previous introduction, in this work the concept of directional cyclostationarity of complex-valued signal is defined and used to investigate the vibration of a journal bearing supported rotor system which is influenced by oil film instability.The structure of the paper is organized as follows.In Section 2, the theory responding to the first and the second order cyclostationarity of real-valued signal is reviewed.Furthermore, the directional Wigner distribution is introduced.Subsequently, the proposed directional cyclostationarity is represented.In Section 3, the experiments conducted on a test bench are described and the results obtained from analysis compared and discussed using directional cyclostationarity, directional time frequency distribution, and conventional cyclostationarity, respectively.In Section 4, summary discussions and the conclusions are given.

Directional Cyclostationarity
2.1.Cyclostationarity of Real-Valued Signals.Since cyclostationarity can indicate itself in statistics of any orders determined by the degree of nonlinear operation, there are various statistic parameters illustrated in the time, the frequency, and the cyclic frequency domains [10].The first and second order of cyclostationarity are firstly reviewed for simplicity but without losing accuracy.
The mean value of a cyclostationary signal (cyclic mean, CM) is defined as where  is the cyclic frequency of the signal and the set Ã contains all cyclic frequencies.[⋅] = lim  → ∞ ∫  (⋅) / is the mean operator.
Then the cyclic autocorrelation (CR) representing the energy of the signal can be defined in as By taking the Fourier transform of CR with respect to time lag , the spectral correlation density (SCD) is given by Finally by transforming the cyclic frequency  back to the time domain, the Wigner-Ville spectrum (WV) can be obtained in where   () and   () denote the forward and backward, respectively, procession of the rotor centerline.
The   () and   () are given by where p() is the Hilbert transform of ().
Then the auto-DWD is defined in where    (, ) and    (, ) are, respectively, the forward and the backward terms and  is the circular rotating frequency.
Figure 3: The lateral vibration of the shaft and its power spectrum at 5500 rpm.Cyclic frequency (Hz) The cross-DWD is given by In addition, the full spectrum (FS), another approach for shaft orbit signal analysis, is developed by Southwick [11,12], defined in

Proposed Directional Cyclostationarity of Complex-Valued
Signals.Originated from the cyclostationarity for real-valued signal, in this work the directional cyclostationarity for complex-valued vibration signal is developed according to the principle of the previously introduced directional Wigner distribution.The proposed method can be applied not only to characterize the hidden periodicities of the vibration radiated by the journal bearing supported system but also to determine the procession directivity of the planar motion in time-frequency domain by analyzing the combined complexvalued vibration signal.The directional cyclic mean (DCM) can be defined in (10) according to (1) and ( 7): Frequency (Hz) Frequency (Hz)  where DCM  (, ) and DCM  (, ) are, respectively, the forward and backward terms.DCM is able to identify the periodic components induced by the mean behavior of the signal.
The autodirectional cyclic autocorrelation (DCR) is defined in (11) according to (2) and ( 7): where DCR     * (, ) and DCR     * (, ) are, respectively, the forward and backward terms.The cross-directional cyclic autocorrelation is given in (12) according to ( 2) and ( 8): Both the auto-and the cross-directional cyclic autocorrelation coefficients can indicate the energy of the signal.
Subsequently, taking the Fourier transform of the autodirectional cyclic autocorrelation with respect to the lag  produces the auto directional spectral correlation density function (DSCD), given by (13) according to (3) and (7): In the particular case when the cyclic frequency  = 0, (13) becomes the power spectrum of the signal ().
The cross-directional spectral correlation density function is given by (14) according to (3) and ( 8): Both the auto-and the cross-directional spectral correlation density functions actually describe the density of the correlation of two spectral components spaced apart by  around the central frequency .As the frequency  in the conventional Wigner-Ville spectrum does not represent the periodicity of the vibration response as the cyclic frequency  does, the negative component of the frequency  will be of no meaning.Therefore, by transforming the cyclic frequency variable  back to the time domain, no variable with physical meaning can be obtained, which means that the directional Wigner-Ville spectrum does not exist.
It is noted that all the significant symbols in the equations are listed in the Nomenclature section.

Experiment
3.1.Experimental Setup and Data Description.In this work, the experiment is conducted to investigate the dynamics of the asymmetric journal bearing supported rotor system on a test rig illustrated in Figure 1.The experiment setup consists of a rigid cylindrical shaft supported by two cylindrical journal bearings, with the supporting journal bearing on the right end near the driving motor and an oil film journal bearing at the left end for simulating oil film instability faults.Two discs are mounted on the shaft, with one at the midplane between the two bearings and the other near the left oil film bearing.Two pairs of accelerometers are used for measuring pedestal translational vibrations, with one pair mounted on the outboard of each of the two bearing pedestals.In addition, two proximity sensors are mounted just to the right-hand side of the center disc, to record the lateral vibrations of the rotor at that position.
In order to simulate the oil whirl and oil whip phenomenon, the rotor rotating in fluid lubricated cylindrical journal bearings is lightly loaded with an unbalanced mass.When the shaft rotates with a slow rotating speed, the stable synchronous vibration with low amplitude is observed.With the increasing rotation speed, the system reaches the first resonance with significant vibration amplitude at the first natural frequency.Then the vibration returns to normal after the resonance.At higher speed, the oil whirl appears along with higher vibration.In addition, the characteristic frequency of oil whirl (around half of the rotating frequency) can be observed.When the rotating speed approaches the double resonance speed, the oil whirl pattern becomes the oil whip with the characteristic frequency remaining to the first natural frequency of the system.Furthermore, the amplitude of the rotor under oil whip becomes higher than that of oil whirl.The vibration signals of journal bearing supported system during startup procedure are listed in Figure 2. Results obtained under different rotational speeds with various artificially simulated defects are listed in Table 1.

Cyclostationarity of the Real-Valued Signal.
For completeness, the statistical parameters characterizing cyclostationarity of real-valued signal of the journal bearing supported system are also illustrated from Figures 12,13,14,15,16,17,18,and 19.Similar conclusions to the directional cyclostationarity plots can be found except for information on the procession directivity of the planar motion.
Compared with conventional Wigner-Ville distribution, the Wigner-Ville spectrum reduces the influence of the interference terms on exhibiting how the signal energy flows in the time and frequency domain [10].In Figure 18, only the WV Figures 18(a) and 18(b) derived from the rotor vibration are able to clearly describe the periodicity induced by oil whirl fault.In Figure 19, both the WV plots of the rotor vibration and the pedestal vibration obviously show the periodic phenomenon responding to oil whip fault.

Discussions and Conclusions
In the work presented here, the characteristics of the vibration signals obtained both from the rotor and the pedestal of a rotor system supported by journal bearing were investigated.The experiments were conducted on a test bench to simulate the operation status of the rotor system suffering from oil whirl and oil whip.
The directional cyclostationary parameters, such as directional cyclic mean, directional cyclic autocorrelation, and directional spectral correlation density, were defined and utilized to analyze the complex-valued lateral vibration signal of the rotor system.The proposed method is able to identify the oil whirl or oil whip fault.In addition, the information on the procession directivity of the planar motion can also be obtained.Furthermore, it is found that actually the full spectrum can be interpreted as the first order directional cyclostationarity and the directional Wigner distribution as the second order directional cyclostationarity.Therefore the developed directional cyclostationarity integrates both the advantages of conventional cyclostationarity and the directional Wigner distribution.

Figure 1 :Figure 2 :
Figure 1: Test bench and its main components.

Figure 4 :
Figure4: The lateral vibration of the pedestal and its power spectrum at 5500 rpm.

Figure 5 : 9 FrequencyFigure 6 :
Figure 5: The lateral vibration of the shaft and its power spectrum at 6200 rpm.

Figure 7 :Figure 8 :Figure 9 :
Figure 7: Directional cyclic mean of the lateral vibration of the shaft and pedestal.

Figure 10 :
Figure 10: Directional Wigner distribution of the lateral vibration of the shaft and pedestal.

Figure 11 :
Figure 11: Full spectrum of the lateral vibration of the shaft and pedestal.

Figure 12 :Figure 13 :Figure 14 :Figure 15 :
Figure 12: The cyclic mean of the lateral vibration of the shaft and pedestal at 5500 rpm.

Table 1 :
System operating states at different rotating speeds.