Shock and Vibration

Volume 2015 (2015), Article ID 542472, 12 pages

http://dx.doi.org/10.1155/2015/542472

## Cyclostationary Analysis for Gearbox and Bearing Fault Diagnosis

^{1}School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China^{2}Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China

Received 5 May 2015; Revised 24 July 2015; Accepted 27 July 2015

Academic Editor: Dong Wang

Copyright © 2015 Zhipeng Feng and Fulei Chu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Gearbox and rolling element bearing vibration signals feature modulation, thus being cyclostationary. Therefore, the cyclic correlation and cyclic spectrum are suited to analyze their modulation characteristics and thereby extract gearbox and bearing fault symptoms. In order to thoroughly understand the cyclostationarity of gearbox and bearing vibrations, the explicit expressions of cyclic correlation and cyclic spectrum for amplitude modulation and frequency modulation (AM-FM) signals are derived, and their properties are summarized. The theoretical derivations are illustrated and validated by gearbox and bearing experimental signal analyses. The modulation characteristics caused by gearbox and bearing faults are extracted. In faulty gearbox and bearing cases, more peaks appear in cyclic correlation slice of 0 lag and cyclic spectrum, than in healthy cases. The gear and bearing faults are detected by checking the presence or monitoring the magnitude change of peaks in cyclic correlation and cyclic spectrum and are located according to the peak cyclic frequency locations or sideband frequency spacing.

#### 1. Introduction

Gearboxes and rolling element bearings are critical mechanical components and widely used in many types of machinery [1–3]. Gear and bearing faults will result in deficiency of transmission or even shut-down of the entire machinery. Therefore, gearbox and bearing fault diagnosis play an important role.

The vibration signals of gearboxes and rolling element bearings are usually cyclostationary, since their statistics (in terms of ensemble average) change periodically with time due to their rotation. Hence, cyclostationary analysis is suitable to process gearbox and bearing vibration signals. Dalpiaz et al. [4] made a comparison study between various vibration signal analysis methods (including cepstrum, time-synchronous average, wavelet transform, and cyclostationary analysis) for gear localized fault detection, and they found spectral correlation density is effective in monitoring gear crack development. Sidahmed et al. [5, 6] showed that time-synchronous average can be considered as a first order cyclostationarity and spectral correlation as a second order cyclostationarity, found that gear vibration signals have second order cyclostationarity, and early detected gear tooth spalling using spectral correlation analysis. Zhu et al. [7] investigated the effectiveness of cyclostationarity from the first order to the third order, that is, spectrum of time-synchronous average, cyclic spectrum and cyclic bispectrum, in gearbox condition monitoring. Bi et al. [8] proposed to extract the amplitude modulation and phase modulation information from gear vibration signals using slice spectral correlation density, so as to detect gear defects. Li and Qu [9] deduced the cyclic correlation and cyclic spectrum of amplitude modulation signals and applied them to rolling element bearing fault diagnosis. Recently, Antoni et al. [10–15] conducted a series of researches on cyclostationary signal analysis and applied it to fault diagnosis of rotating machinery. To reduce the computational complexity of cyclic energy indicator based on cyclic spectral density, Wang and Shen [16] proposed an equivalent cyclic energy indicator for rolling element bearing degradation evaluation. These researches illustrate the effectiveness of cyclostationary analysis in gearbox and bearing fault diagnosis. However, the explicit relationship between the cyclostationary features of vibration signals and the gearbox and bearing dynamic nature still needs further investigation, in order for thoroughly understanding the vibration signal characteristics and thereby effectively diagnosing fault.

Gearbox and rolling element bearing vibration signals usually feature amplitude modulation and frequency modulation (AM-FM), and the modulation characteristics contain their health status information [17–21]. Cyclic correlation and cyclic spectrum are effective in extracting modulation features from amplitude modulation (AM), frequency modulation (FM), and AM-FM signals. Feng and his collaborators [22–24] derived the expressions of cyclic correlation and cyclic spectrum for gear AM-FM vibration signals and proposed indicators based on cyclic correlation and cyclic spectrum for detection and assessment of gearbox fault. Nevertheless, the carrier frequency of rolling element bearing vibration signals (resonance frequency) is completely different from that of gear vibration signals (gear meshing frequency and its harmonics). Therefore, it is important to investigate the cyclic correlation and cyclic spectrum of bearing vibration signals in depth, considering both the AM and the FM effects due to bearing fault. Meanwhile, how to explain the cyclostationary features displayed by the cyclic correlation and cyclic spectrum and to map the modulation characteristics to gear and bearing fault are still important issues for application of cyclostationary analysis in gearbox and bearing fault diagnosis. In this paper, we derive the explicit expressions of cyclic correlation and cyclic spectrum for general AM-FM signals, summarize their properties, and further extend the theoretical derivations to modulation analysis of both gear and rolling element bearing vibration signals, thus enabling cyclostationary analysis to detect and locate both gearbox and bearing fault.

#### 2. Cyclic Correlation

##### 2.1. Definition

The statistics of cyclostationary signals have periodicity or multiperiodicity with respect to time evolution. Cyclic statistics are suitable to process such signals. Among those, second order cyclic statistics, that is, cyclic correlation and cyclic spectrum, are effective in extracting the modulation features of cyclostationary signals.

For a signal , the cyclic autocorrelation function is defined as [25]where is time lag and is cyclic frequency.

##### 2.2. Cyclic Correlation of AM-FM Signal

During the constant speed running of gearboxes and rolling element bearings, the existence of fault, machining defect, and assembling error often leads to periodical changes in vibration signals. For gearboxes, such periodical changes modulate both the amplitude envelope and instantaneous frequency of gear meshing vibration. For bearings, such repeated changes excite resonance periodically. The excited resonance vanishes rapidly due to damping before next resonance comes, resulting in AM feature. Meanwhile, in one repeating cycle, the resonance exists in early period and the instantaneous frequency equals approximately the resonance frequency, while in later period, the resonance vanishes due to damping and the instantaneous frequency becomes 0. That means the instantaneous frequency changes periodically, resulting in frequency modulation (FM). Hence the vibration signals of both gearboxes and rolling element bearings can be modeled as an AM-FM process [17, 21] plus a random noisewhereare the AM and FM functions, respectively, and are the magnitude of AM and FM respectively, is the carrier frequency (for gearboxes, it is the gear meshing frequency or its th harmonics; for rolling element bearings, it is the resonance frequency of bearing system), is the modulating frequency equal to the gear or bearing fault characteristic frequency, , , and are the initial phase of AM and FM, respectively, and is a white Gaussian noise due to random background interferences.

Without loss of generality, consider only the fundamental frequency of the AM and FM terms; then (2) becomes

According to the identity [26]where is Bessel function of the first kind with integer order and argument , the FM term in (5) can be expanded as a Bessel series, and then (5) becomes

For such a signal expressed as (7), the time-varying feature of its autocorrelation function is mainly determined by the AM-FM part, since the autocorrelation function of a white Gaussian noise (i.e., its Fourier transform has peak at 0 only, and therefore the noise does not affect identification of modulating frequency via cyclic correlation and cyclic spectrum analysis). In addition, it is the AM and FM effects on the carrier signal that leads to the cyclostationarity of bearing and gearbox vibration signals, and we rely on detection of the modulating frequency of such AM and FM effects to diagnose bearing and gearbox fault. Therefore, we neglect the noise and focus on the deterministic AM-FM part only in the following analysis. Then the cyclic autocorrelation function of (7) can be derived as [22]in lower cyclic frequency domain, andin higher cyclic frequency domain, where the intermediate functions

Set the time lag to 0, yielding the slice of cyclic autocorrelation functionin lower and higher cyclic frequency domains, respectively, where the intermediate functions

According to (10a) and (10b) and (11a) and (11b), the slice of cyclic autocorrelation function has two clusters of cyclic frequencies: one cluster concentrates around the cyclic frequency of 0 Hz separated by the modulating frequency , and the other cluster spreads around twice the carrier frequency with a spacing equal to the modulating frequency .

In lower cyclic frequency domain, the cyclic frequency locations of present peaks are dependent on the difference of the two Bessel function orders . For any peak at a specific cyclic frequency, . This leads to constant complex exponentials in (11a). Furthermore, according to the addition theorem of Bessel functions [26]peaks appear in the slice of cyclic autocorrelation function in lower cyclic frequency domain, if and only if the orders of the two Bessel functions are equal to each other; that is, . Therefore, the cyclic autocorrelation slice in lower cyclic frequency domain can be further simplified as

According to (13), in lower cyclic frequency domain of the cyclic autocorrelation slice, peaks appear at the cyclic frequencies of 0 Hz, the modulating frequency , and its twice . If higher order harmonics of the modulating frequency are taken into account, then peaks also appear at the modulating frequency harmonics .

The addition theorem of Bessel functions does not apply to the slice of cyclic autocorrelation function in higher cyclic frequency domain. The cyclic frequency locations of present peaks are dependent on the sum of the two Bessel function orders . For any specific peak, . This does not mean . According to (10b) and (11b), in higher cyclic frequency domain of the cyclic autocorrelation slice, sidebands appear at both sides of twice the carrier frequency , with a spacing equal to the modulating frequency .

According to the above derivations, we can detect gearbox and bearing fault by monitoring the presence or magnitude change of sidebands around the cyclic frequency of 0 or twice the carrier frequency, that is, twice the meshing frequency for gearboxes and twice the resonance frequency for bearings, in the 0 lag slice of cyclic autocorrelation function. We can further locate the gearbox and bearing fault by matching the sideband spacing with the fault characteristic frequencies.

#### 3. Cyclic Spectrum

##### 3.1. Definition

For a signal , the cyclic spectral density is defined as the Fourier transform of the cyclic correlation function [25]where is frequency.

##### 3.2. Cyclic Spectrum of AM-FM Signal

Without loss of generality, we still consider the simplified gearbox and bearing vibration signal model, (5), by focusing on the fundamental frequency of the AM and FM terms. Its cyclic spectral density can be derived as [23, 24]in lower and higher cyclic frequency domains, respectively, where the intermediate function

According to (15a) and (15b), peaks appear at specific locations only on the cyclic frequency−frequency plane. Their cyclic frequency locations in lower cyclic frequency domain, as well as their frequency locations in higher cyclic frequency domain, are dependent on the difference of the two Bessel function orders . For any specific peaks, . Then, the addition theorem of Bessel functions [26], (12), applies to the cyclic spectral density, (15a) and (15b). Hence, on the cyclic frequency−frequency plane of the cyclic spectral density, peaks appear if and only if the orders of the two Bessel functions are equal to each other; that is, . Then cyclic spectral density can be simplified asin lower cyclic frequency domain andin higher cyclic frequency domain.

According to (17a) and (17b), for any peak on the cyclic frequency−frequency plane, its frequency location in lower cyclic frequency domain, as well as its cyclic frequency location in higher cyclic frequency domain, is dependent on the Bessel function order . Meanwhile, the peak magnitude only involves a few Bessel functions. Thus (17a) and (17b) can be further simplified asin lower and higher cyclic frequency domains, respectively.

According to (18a), in lower cyclic frequency domain, peaks appear at the cyclic frequencies of 0 Hz, modulating frequency , and its twice . If higher order harmonics of modulating frequency are taken into account, then peaks also appear at the cyclic frequencies of the modulating frequency harmonics . Along the frequency axis, these peaks center around the carrier frequency , with a spacing equal to the modulating frequency .

According to (18b), in higher cyclic frequency domain, peaks appear at the frequencies of 0 Hz, modulating frequency , and its half . If higher order harmonics of modulating frequency are taken into account, then peaks also appear at the modulating frequency harmonics and their half . Along the cyclic frequency axis, these peaks center around twice the carrier frequency , with a spacing equal to twice the modulating frequency .

Observing the peak locations in (18a) and (18b), the peaks in cyclic spectrum distribute along the linesThese lines form diamonds on the cyclic frequency−frequency plane, as illustrated by Figure 1, which is the cyclic spectrum of an AM-FM signal with a carrier frequency of 256 Hz and a modulating frequency of 16 Hz.