Study of coast-up vibration response for rub detection
Introduction
The increasing need for higher power and efficiency makes rotating machines to operate under severe mechanical stresses and tight clearances. There are number of applications in which rotors are designed to be light in weight and operate at very high speeds, usually beyond first few critical speeds. Under these circumstances, contact between rotating and stationary part is likely to occur, especially while coasting past the bending critical speeds. Rotor–stator contact vanishes once rotor radial excursion drops below the clearance between rotor and stator. The contact in such cases is for short period and the rotor mostly experiences partial rubbing. However, it had been reported [1] that the rotor once perturbed to interact with stator, might develop full rubbing from the partial rubbing. Continuous rubbing between rotor and stator accelerates wearing of these parts and increases the clearances between then, which in turn results in loss of efficiency and also economy. Rubbing might results in broken machine parts, for example, rubbing between a blade and stationary part could result in broken blade. Ignoring the occurrence of rubbing may lead catastrophic breakdown of the rotating machines.
The rotor to stator rubbing is considered as a secondary phenomenon resulting from a primary cause, which perturbs the machine during normal operation. These primary causes could be rotor vibrations (due to unbalance or other sources) and/or displacements of rotor centerline, due to rotor misalignment, gravity force, fluid forces, etc. Muszynska [1] gave a detailed literature review on rotor rub related phenomena and vibration response. Rotor stator contact is highly nonlinear phenomena and depending upon the rubbing conditions, it can generate variety of vibration response [2], [3], i.e. chaos, quasi-period, sub-harmonics and super harmonics. Beatty [4] proposed a simple mathematical model for rotor–stator contact problems and highlighted the destructive instability of rotors due to rubbing. Smalley [5] presented a study on rub-induced thermal bow vibrations during acceleration or deceleration of a steam turbine rotor. Al-Bedoor [6] studied lateral and torsional coast-up vibration signals of rotor–stator rub and observed rotor response anisotropy. Feng and Zhange [7] examined the effect of lateral perturbation, rotor–stator clearance and rotation speed on the occurrence and state of rotor–stator rub.
Many of the rotor–stator rub related investigations were focused on establishing the vibration symptoms of rotor rub from the conventional time-domain (i.e. vibration waveform, orbit plot, Poincare map, etc.) and frequency-domain (i.e. FFT) signal representation methods. Use of full spectrum plots to reveal the whirl character of the rotor is also gained popularity [8], [9] for the rub monitoring. However, many a times these methods are not sufficient and sometimes not appropriate to completely reveal the nonlinear nature of the non-stationary rub signal, hence, researchers have recently started using advanced signal processing techniques such as wavelet transform (WT), Hilbert–Huang transform (HHT), etc. These tools represent the vibration signal in the form of time–frequency–energy maps. This supposedly detects exact occurrence time of rub and associated excitation frequencies. Since last decade or so, wavelets are widely used [10] for extracting the time–frequency features of the non-stationary vibration signals. Peng et al. [11] demonstrated the application of wavelet analysis for numerical and experimental rub signal. Like other signal processing techniques wavelet transform also suffers from certain weaknesses, major of which is the leakage of energy in the neighbouring modes due to unequal time–frequency resolution defined by Heisenberg–Gabor inequality. Wavelet provides good time resolution and poor frequency resolution at higher frequency and poor time resolution and good frequency resolution at lower frequency. Once wavelet is selected, one will have to use it to analyze all the data, hence, its likely to overlook precise information of either low frequency components or high frequency components of the signal, particularly when signal carries wide frequency range such as coast-up or coast-down vibration signal.
Huang et al. [12] presented the empirical mode decomposition (EMD) technique to decompose any complicated vibration data into finite number of intrinsic mode functions (IMF). These IMFs are complete, adaptive and almost orthogonal representation of the complex vibration signal. Recently Hilbert transform based on IMFs (also known as Hilbert–Huang transform) derived using EMD is evolved as powerful tool for analysis of nonlinear and non-stationary signal [13], [14]. Yu et al. [13] presented local Hilbert transform and local Hilbert marginal transform based on EMD for the fault diagnosis of roller bearings. Qi et al. [14] demonstrated the use of EMD for rubbing fault diagnosis from steady state response. Though both Hilbert–Huang and wavelet transforms show time–frequency–energy distribution of the vibration signal, the former is adaptive and provides equal resolution at all frequencies and time instants, which makes use of Hilbert–Huang transform (HHT) more meaningful for transient vibration signals.
Most of the past research studies on rotor stator rub phenomena focused on the vibration response at constant rotational speeds and relatively less attention has been given to the detection of the rub from the coast-up or coast-down vibration signals. Since rotor operates with possible minimum clearance between rotor and stator, they are likely to make contact with stator while traversing through critical speed. Early detection of occurrence of the rub is important for avoiding any catastrophic damage it may bring to the rotating machines. In present study, rotor start-up lateral vibration signal is investigated with focus on detection of rub at its initiation stage. Coast-up vibration response is simulated for the Jeffcott rotor with two lateral degrees of freedom. Hilbert–Huang transform is applied for the first time to analyze the coast-up rub signal and results are compared with the wavelet transform.
Section snippets
Equations of motion
A schematic of a rotor–stator system is shown in Fig. 1. The rotor is a Jeffcott rotor supported on the simple rigid bearings. Central disk of mass M and radius R is suspended on the shaft of length L. Rotor has a static radial clearance of δ with respect to stator. The disk has an unbalance eccentricity ε (distance between centre of gravity of rotor, i.e. CG and geometric centre of the rotor, i.e. GC) at an initial angle of β with vertical direction. The clearance and unbalance eccentricity
Hilbert–Huang transform
Generally, the frequency is defined as the number of oscillations per unit time of a physical field parameter such as displacement, current or voltage. However, for non-stationary and nonlinear signals commonly encountered in machine vibration analysis applications, this definition becomes ambiguous and loses its effectiveness over the fact that the spectral characteristics of the signals vary with time [15]. In such situations, the notion of instantaneous frequency is viable. It has been
Analysis of a rotor–stator rub coast-up signal
The system parameters considered in present study are as mentioned below, mass of the disk, M = 6 kg; shaft diameter, d = 25 mm; shaft length, L = 0.7 m; external damping, C = 182.56 N s/m; stator stiffness, Ks = 140 × 10+6 N/m; coefficient of friction, μ = 0.2 and unbalance eccentricity, ε = 1.0597 × 10−5 m. Bending natural frequency of the rotor is ω0 = 304.3 rad/s (i.e. 48.42 Hz). Solution to the nonlinear nonautonomous equations of motion (Eq. (1)) is obtained using Runge–Kutta fourth order numerical integration
Conclusions
The present study examines the coast-up vibration response of the rotor–stator rub fault, while accelerating past the bending natural frequency. Two different rubbing conditions, i.e. heavy rub and light rub are simulated. Rotor–stator interactions are highly nonlinear and exhibit quite a complex vibration response. The rub in rotor–stator system is shown to reveal different vibration response in both the lateral direction, which suggests the directional nature of the rub fault. Rotor–stator
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