附錄一 英文參考文獻Application of slice spectral correlation density to gear defect detectionG Bi󰀀, J Chen, F C Zhou, and J He The State Key Laboratory of Vibration, Sound, and Noise,Shanghai Jiaotong University, Shanghai,People’s Republic of China The manuscript was received on 16 October 2005 and was accepted after revision for publication on 3 May 2006.DOI: 10.1243/0954406JMES206 Abstract: The most direct reflection of gear defect is the change in the amplitude and phase modulations of vibration. The slice spectral correlation density (SSCD)method presented in this paper can be used to extract modulation information from the gear vibration signal; amplitude and phase modulation information can be extracted either individually or in combination. This method can detect slight defects with comparatively evident phase modulation as well as serious defects with strong amplitude modulation. Experimental vibration signals presenting gear defects of different levels of severity verify to its character identification capability and indicate that the SSCD is an effective method, especially to detect defects at an early stage of development. Keywords: slice spectral correlation density, gear, defect detection, modulation 1 INTRODUCTION A gear vibration signal is a typical periodic modulation signal. Modulation phenomena are more serious with the deterioration of gear defects. Accordingly, the modulation sidebands in the spectrum get incremented in number and amplitude.Therefore, extracting modulation information from these sidebands is the direct way to detect gear defects. A conventional envelope technique is one of the methods for this purpose. It is sensitive to modulation phenomena in amplitude, but not in phase. A slight gear defect often produces little change in vibration amplitude, but it is always accompanied by evident phasemodulation. Employing the envelope technique for an incipient slight defect does not produce satisfactory results. In recent years, the theory of cyclic statistics has been used for rotating machine vibration signal and shows good potential for use in condition monitoring and diagnosis [1–3]. In this article, spectral correlation density (SCD) function in the second-order cyclostationarity is verified to be a redundant information provider for gear defect detection. It simultaneously exhibits amplitude and phase modulation during gear vibration, which is especially valuable for detecting slight defects and monitoring their evolution.The SCD function maps signals into a two-dimensional function in a cyclic frequency (CF) versus general frequency plane (a–f). Considering its information redundancy [4] and huge computation,the slice of the SCD where CF equals the shaft rotation frequency is individually computed for defect detection,which is named slice spectral correlation density (SSCD). The SSCD is demonstrated to possess the same identification capability as the SCD function. It can be computed directly from a time-varying autocorrelation with less computation and, at the same time, has clear representation when compared with a three-dimensional form of the SCD. 2 SECOND-ORDER CYCLIC STATISTICS A random process generally has a time-varying autocorrelation[5]Where is the mathematic expectation operator and t is the time lag. If the autocorrelation is periodic with a period T0, the ensemble average can be estimated with time averageThe autocorrelation can also be written in the Fourier series because of its periodicityWhereCombining with equation(2), its Fourier coefficients can be given as [5]Where is the time averaging operation, is referred to as the cyclic autocorrelation (CA),and a is the CF. SCD can be obtained by applying Fourier transform of the CA with respect to the time lag tThe SCD exhibits the characteristics of the signal in a–f bi-frequency plane. All non-zero CFs characterize the cyclostationary (CS) characters of the signal. 3 THE GEAR MODEL The most important component in gearbox vibration is the tooth meshing vibration, which is due to the deviations from the ideal tooth profile. Sources of such deviations are the tooth deformation under load or original profile errors made in the machining process. Generally, modulation phenomena occur when a local defect goes through the mesh and generates periodic alteration to the tooth meshing vibration in amplitude and phase. To a normal gear, the fluctuation in the shaft rotation frequency and the load or the tiny difference in the teeth space also permits slight amplitude modulation(AM) or phase modulation (PM). Therefore, the general gear model can be written as [6, 7]where fx is the tooth meshing frequency and fs is the shaft rotation frequency. am(t) and bm(t) denote AM and PM functions, respectively. The predominant component of the modulation stems from the shaft rotation frequency and its harmonics; other minute modulation components can be neglected.AM and PM, 。