外文翻譯--部分頻譜與齒輪缺陷發(fā)現(xiàn)相互關(guān)系的實際應(yīng)用

附錄一 英文參考文獻 Application of slice spectral correlation density to gear defect detection G Bi, J Chen, F C Zhou, and J He The State Key Laboratory of Vibration, Sound, and Noise,Shanghai Jiaotong University, Shanghai,Peoples 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 13. 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 (af). 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 autocorrelation5 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 average The autocorrelation can also be written in the Fourier series because of its periodicity Where Combining 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 t The SCD exhibits the characteristics of the signal in af 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, either individually or in combination,cause the presence of sidebands within the spectrum of the signal. Band-pass filtering around one of the harmonics of the tooth meshing frequency is the classical signal processing for the detailed observation of the sidebands. The filtered gear vibration signal can be expressed as follows where fh denotes one of the harmonics of the tooth meshing frequency. The subscript m is ignored for simplification in this equation and in the following discussion. The study emphasis of this paper is the filtered gear vibration signal model in equation (7),and its carrier is a single cosine waveform and modulated parts are period functions. 4 CS ANALYSIS OF THE GEAR MODEL According to the analysis mentioned earlier, the gear vibration signal can be simplified as a periodic signal modulated in amplitude and phase. The modulation condition reflects the severity extent of potential defect in gear. In this section, AM and PM cases are studied individually, and the CS analysis of the gear model is developed on the basis of their results. 4.1 AM case The model of AM signal is derived from equation (7) The analytic form of x(t) in equation (8) can be written as Substitution of x(t) into equation (4) can deduce the CA of x(t) Where is the envelope of is equal to as a provider of modulation information.It is the Fourier transform of according to equation (11). In addition, the Fourier transform of with respect to the time lag is the corresponding SCD .thus can be computed using twice Fourier transform of with respect to time t and time lag t,respectively According to integral transform, becomes where H(v) is the Fourier transform of a(t) After substituting H(v) into equation (13) and uncoupling f and a using the properties of d function, the final expression of an be obtained has a totally symmetrical structure in four quadrants. Equation (15) is just a part of it in the first quadrant, and others are ignored for simplification. According to equation (15), is composed of some discrete peaks. In addition, these peaks regularly distribute on the a f plane. Despite the comparatively complex expression, the geometrical description of is simple. These peaks nicely superpose the intersections of the cluster of lines . Then, these lines can also be considered as the character lines of . 4.2 PM case PM signal derived from equation (7) is The CA of its analytic form can be represented as The CA in the PM case also has the envelopecarrier form, as in the AM case. Therefore, the envelope of the CA is used to extract modulation information from the signal. Its corresponding SCD is also denoted as .The PM part, b(t), comprises finite Fourier series.The CS analysis of the PM case starts with the sinusoidal waveform .Bessel formula is employed in the computation. The final result of this simple case can be expressed as The geometrical expression of equation (18) is also related to lines ,and is nonzero only at their intersections. The number of the lines does not depend on the number of harmonics in the modulation part, but is infinite in theory even for a single sinusoidal PM signal. In fact, Bessel coefficients limit discrete peaks in a range centring around the zero point of af. The amplitude of other theoretical character peaks out of the range is close to zero with the distance far away from the zero point.When the PM function comprises several sinusoidal waveforms as shown in equation (16), components of it can be expressed as bi(t), where i is Application of SSCD to gear defect detection 1387 from 0 to I. The envelope of CA can be written as Where equals unity. According to the two-dimensional convolution principle, the corresponding SCD of can be represented by where the sign means the two-dimensional convolution on the bi-frequency plane. The expression of is shown in equation (18) with fs replaced byifs and B by Bi and b by bi. Despite more complex expression of the SCD in the multiple sinusoidal modulation case, the result of the two-dimensional convolution between has the same geometrical distribution, as it does in the single sinusoidal modulation case. The distance between the character lines of along the general frequency axis is the fundamental frequency fs. Therefore, convolution does not create new character peaks, but changes their amplitude. Equation (18) also represents the SCD of the signal in equation (16), although the coefficients Cln are changed by the two-dimensional convolution. 4.3 CS analysis of the gear vibration signal The second-order CS analysis of the general gear model in equation (7) is developed on the basis of the AM and PM cases. The CA of the analytic signal also has the envelopecarrier form, and the envelope of the CA is expressed as follows Two parts in the sign . in equation (21) are relatedto AM and PM functions, respectively. Therefore, the corresponding SCD of has the form of two dimensional convolution of two components issued from AM and PM functions The expressions of and are given in equations (15) and (18). The two-dimensional convolution between and just causes the superposition of the character peaks in and , as it does in the PM case. Owing to the same geometrical characters, the convolution can not change the distribution, but involves change in the number and amplitude of the effective character peaks (whose amplitude is larger than zero). Therefore,the CS characters of the gear model are also represented by lines , as it does in the AM and PM cases. 4.4 SSCD analysis of the gear vibration signal and its realization Three modulation cases have a uniform CS character, according to the above analysis. Lines f = on the bi-frequency plane are their common character lines.Figure 1 shows its distribution.Only the part in the first quadrant is displayed because of the identical symmetry of in four quadrants. The number of these discrete points and the amplitude of the spectrum peaks reflect the modulation extent of the signal.The SCD provides redundant information for gear modulation information identification. In fact, some slices of it are sufficient for the purpose. For the AM case, the slice of , where CF is (in the first quadrant), can be derived from equation (15) The slice contains equidistant character frequencies,and the distance between them is fs. The PM case and the combination modulation case have the similar result, which can also be expressed by equation (23), whereas the coefficients Cl have different expressions. Therefore, , where is composed of discrete peaks All these character spectrum peaks correspond toodd multiples of the half shaft rotation frequency.The number and amplitude of the peaks reflect the modulation extent, thereby reflecting the severity extent of the potential defect in the gear.Similar situations will be encountered when analysing other Fig. 1 Diagram of CS character distribution slices of the SCD where CF equals the integer multiples of the shaft rotation frequency.The information redundancy of the SCD function always becomes an obstacle to its practical application in the gear defect detection. The sampling frequency must be high enough to satisfy the sampling theorem. Simultaneously, identifying modulation character relies on the fine frequency resolution.Long data series are needed because of these two factors.Therefore, huge matrix operations bring heavy burden to the computation.Moreover, sometimes it is hard to find a clear representation for the redundant information in the three-dimensional space.Therefore, the SSCD, as shown in the above analysis,is presented as a competent substitute for the SCD in detecting gear defects. In this article, the SSCD is specialized to the slice of the SCD where CF equals a certain character frequency. The SSCD can be acquired directly from the time-varying autocorrelation without computing the CA matrix and other subsequent matrix operations. Its realization is detailed as follows: (a) use the Hilbert transform to get the analytic signal x(t)； (b) compute the time-varying autocorrelation of the analytic signal as described in equation (2)； (c) select the CF a0, which equals a certain prescient character frequency, and then compute the slice of the CA (a0 equals fs for gear defect detection)； (d) compute the envelope of the slice CA . It cannot be attained directly from the slice CA,therefore, a technique is involved for another form of Utilizing the equation , arrive at the squared modulus of ； (e) apply the Fourier transform of with respect to the time lag t and obtain the final result of the SSCD.The SSCD can be computed according to the steps listed above. Nevertheless, the manipulation of replacing the envelope slice CA by the squared modulus of it will change the spectrumstructure. Original half character frequencies are converted into integer form (lfs) together with the appearance of some inessential high frequency components.These changes do not impact the character identification capability of the SSCD, on the contrary,it gives more obvious representation. 5 SIMULATION Two modulated signals are used to identify the capability of the SCD and the SSCD in modulation character identification. All modulation functions of these signals are finite Fourier series. Figure 2 shows the AM case simulated according to equation(8). The AM function a(t) comprises three cosine waveforms, representing 10 Hz and its double and triple harmonics and amplitude of 1, 0.7, and 0.3 units, respectively. All initial phases in the model are randomly decided by the computer. The carrier frequency is 100 Hz, sampling frequency 2048 Hz,and the data length 16 384. Figure 3 shows the case of the combination of AM and PM simulated according to equation (7). The PM function b(t) comprises two sinusoidal waveforms with the frequency of 10 and 20 Hz and amplitude of 3 and 1 units,respectively. Other parameters are identical to the AM case.Figures 2(a) to (c) show the time waveform, the contour of its SCD analysis, and the SSCD where CF is equal to 10 Hz, respectively. Only the results of the SCD in the first quadrant are given because of its symmetry. All character points in the contour of the SCD are at the intersections of the lines f = . Their distribution is regular in the AM case. The Fig. 2 One simulated AM signal: (a) the time waveform, (b) the contour of its SCD, and (c) the SSCD at 10 Hz SSCD in Fig. 2(c) comprises Fig. 2 One simulated AM signal: (a) the time waveform, (b) the contour of its SCD, and (c)the SSCD at 10 Hzand its integer multiples and reflects themodulation condition in this signal as the SCD. Fig. 3 Another simulated modulated signal with modulation phenomena in amplitude and phase: (a) the time waveform, (b) the contour of its SCD, and (c) the SSCD at 10 Hz Figure 3 shows the case of the combination of AM and PM.All character points in the contour of the SCD are also at the intersections of the character lines 10 Hz. In addition, the SSCD also comprises 10 Hz and its several integer multiples.When PM is involved, the results from the PM part interact with those from the AM part by the two dimensional convolution. The number of the character peaks manifestly increases when compared with the original AM case in the contour of the SCD. The number of character peaks in the SSCD also augments.Therefore, according to the SCD or the SSCD, the same conclusion can be drawn: the second simulated signal is strongly modulated when compared with the first.Simulation results indicate that either the SCD or the SSCD has the capability of identifying the present and the extent of the modulation, disregarding its existence in amplitude or phase. The SSCD possesses the virtues of less computation and clear representation.These two factors seem to be indifferent for simulated signals, but are valuable when encountering very long data series in practice. 6 EXPERIMENTAL RESULTS Three experimental vibration signals employed in this section came from 37/41 helical gears. They represented healthy, slight wear (wear on addendum of one tooth of 41 teeth gear), and moderate wear status (wear on addendum of one tooth profile of 41 teeth gear and two successive tooth profiles of 37 teeth gear), respectively. The shaft rotation frequency of the 37 teeth gear minutely fluctuates 16.6 Hz. Signals were sampled at 15 400 Hz under the same load. The data length was 37 888. Before the SSCD analysis, all experimental signals were band-pass filtered around four-fold harmonics of the tooth meshin frequency in order to identify the change in themodulation sidebands in different defect status.These filtered signals are analysed by a conventional envelope technique and the SSCD. The comparison between their results dedicates the effect of theSSCD.Figure 4 shows the case of the healthy status.Figures 4(a) to (c) are the time waveform of the experimental signal, its envelope spectrum, and its SSCD analysis at the shaft rotation frequency of the 37 teeth gear, respectively. The envelope spectrum and the SSCD have the similar spectrum structure Fig. 3 Another simulated modulated signal with modulation phenomena in amplitude and phase: (a) the time waveform, (b) the contourof its SCD, and (c) the SSCD at 10 H Fig. 4 First experimental gear signal: (a) the time waveform, (b) the envelope spectrum, and (c)the SSCD comprising the rotation frequency and several negligible harmonics. Demodulated sidebands in these two spectra are few and low because there are some modulation phenomena during the gears normal operation. The fluctuation in the load, the minute rotational variation, and the circular pitch error in the machining process are the possible sources of the slight modulation. There is no comparability between numeric values of the envelope spectrum and the SSCD because of different computing procedures. The slight wear case is shown in Fig. 5. Wear on one tooth profile of one of the helical meshing gears does not result in significant deviation from its normal running. Therefore, there is a little increment in amplitude in the time waveform plot. In the envelope spectrum, compared with the normal case, the amplitude of these demodulated sidebands augments a little, and the extent seems to enlarge. The increment in number and amplitude of the sidebands is attributed to the modulation condition of the signal. However, the alteration is too slight to provide enough proof for the existence of some defect in the gear. In fact, a slight defect evidently always modulates the phase of the gear vibration signal and produces little change in the amplitude.Therefore, the envelope spectrum is not sensitiveto a slight gear defect due to its fail to the PMphenomena.Figure 5(c) shows the SSCD analysis of the slightlywearing gear. More sidebands are demodulated by the SSCD when compared with the normal case in Fig. 4(c). Moreover, the amplitude isapproximately ten Fig. 5 Second experimental gear signal: (a) the time waveform, (b) the envelope spectrum, and (c)the SSCD times that of the normal case. Changes between the status of these two operations in the SSCD are so remarkable that a conclusion of the existence of a certain gear defect can be affirmed. Different from the neglect of envelope spectrum to PM, the SSCD treats AM and PM equally. It picks up AM and PM characters simultaneously, that is to say,the SSCD is a whole embodiment of all modulation phenomena in the system. Therefore, this is an effective and reliable method for slight gear defects.The moderate wear case is shown in Fig. 6. Wear on one tooth profile of one of the meshing gears and two neighbouring tooth profiles of the other impact the running of the meshing gears. According to the time waveform, the vibration is more violent than the two cases mentioned eaerlier. In the Fig. 5 Second experimental gear signal: (a) the time waveform, (b) the envelope spectrum, and (c)the SSCD Fig. 6 Third experimental gear signal: (a) the time waveform, (b) the envelope spectrum, and (c)the SSCD Application of SSCD to gear defect detection 1391 envelope spectrum, the amplitude and the number of the sidebands continue to increase. The obvious changes, compared with the normal case, indicate the abnormality of the system. The sidebands demodulated by the SSCD also increase in amplitude and number. The SSCD is indicative of more serious defects, whereas AM phenomena are the major reflection of the moderate wear. Therefore, the envelope spectrum and the SSCD both reflect the severity extent of the modulation in the signal. Both fit to the detection of moderate gear defects. 7 CONCLUSION Gear vibration signal is a typical modulated signal.The changes of the modulation condition indicate the existence and the development of defects. The SSCD is introduced in this article as a valuable method to detect gear defects. It is verified to be a whole reflection of the modulation phenomena in gear vibration and is able to pick up AM and PM information simultaneously. Experimental results show the defect detection capability of the method not only for moderate gear defects, but also for slight defects. Therefore, the SSCD method has a bright future in identifying the presence of gear defects and monitoring their evolution. ACKNOWLEDGEMENTS This research was supported by the National Natural Science Foundation of China (no. 50175068) and the Key Project of the National Natural Science Foundation of China (no. 50335030). Experimental data came from the Department of Applied Mechanics of University Libre de Bruxelles. REFERENCES 1 Dalpiaz,G. and Rivola, A. Effectiveness and sensitivity of vibration processing techniques for local fault detection in gears. Mech. Syst. Signal Process., 2000, 14(3), 387412. 2 Capdessus, C. and Sidahmed, M. Cyclostationary processes:application in gear faults early diagnosis. Mech.Syst. Signal Process., 2000, 14(3), 371385. 3 Antoni, J. and Daniere, J. Cyclostationary modeling of rotating machine vibration signals. Mech. Syst. Signal Process., 2004, 11(18), 12851314. 4 Gardner, W. A. Exploitation of spectral redundancy in cyclostationary signals. IEEE Signal Process. Mag., 1991,8, 1426. 5 Gardner, W. A. Introduction to random processing with applications to signals and systems, 1990 (McGraw-Hill, New York). 6 McFadden, P. D. and Smith, J. D. A signal processing technique for detecting local defects in a gear from the signal average of the vibration. Proc. Instn Mech. Engrs,Part C: J. Mechanical Engineering Science, 1985, 199(C4), 287292. 7 Randall, R. B. A new method of modeling gear faults. J. Mech. Des., 1982, 104, 259267. 附錄二 英文文獻翻譯 部分頻譜與齒輪缺陷發(fā)現(xiàn)相互關(guān)系的實際應(yīng)用 G Bi, J Chen、 F C Zhou 和 J He 中華人民共和國，上海，上海交通大學，國家震動、聲音和噪音重點實驗室 原稿于 2005年 10月 16日完成，經(jīng)修改后于 2006年 5月 3日發(fā)表 DOI： 10.1243|0954406 JMES206 摘要：振幅和振動調(diào)制相位的變化能最直接的反映出齒輪的缺陷。部分頻譜密度關(guān)系 （ SSCD）方法在本文中被用來從齒輪震動信號中提取調(diào)制數(shù)據(jù)；振幅和調(diào)制階段數(shù)據(jù)能個別地或在組合中被提取。 這一個方法能用比較儀明顯的發(fā)現(xiàn)調(diào)制相位的細微的缺陷和通過放大的調(diào)制振幅發(fā)現(xiàn)嚴重的缺陷。實驗不同嚴苛的等級下當前齒輪缺陷的振動信號，證實了它的特性，鑒別能力，表明了 SSCD是一種有效的方法，特別是在發(fā)展早期發(fā)現(xiàn)缺陷。 關(guān)鍵字：部分頻譜密度相互關(guān)系，齒輪，發(fā)現(xiàn)缺點，調(diào)制 1介紹 齒輪震動信號是一個典型的周期調(diào)制信號。調(diào)制現(xiàn)象隨著齒輪缺陷的惡化更加嚴重。因此，調(diào)制在光譜中的邊頻帶中的數(shù)量和振幅會增加。因此從邊頻帶中提取調(diào)制信號是最直接發(fā)現(xiàn)齒輪缺陷的方法。傳統(tǒng)的包絡(luò)技術(shù)是實現(xiàn)這個意圖方法中的一種。它對振 幅中的調(diào)制現(xiàn)象很靈敏，但是不是在相位中。微小的齒輪缺陷經(jīng)常導致振動振幅的小的變化，但是它也經(jīng)常同時伴隨著明顯的相位調(diào)制。開始的微小缺陷使用包絡(luò)技術(shù)不能產(chǎn)生滿意的結(jié)果。 近年來，循環(huán)統(tǒng)計理論被用在旋轉(zhuǎn)機器振動信號，并且在控制條件和診斷結(jié)論 1 3上表現(xiàn)出了好的潛力。在本文中，波譜密度關(guān)系（ SCD）在二階循環(huán)平穩(wěn)度分析法中的功能對齒輪缺陷的發(fā)現(xiàn)被證實是一個多余的信息提供者。它在齒輪震動中同時顯示出振幅和調(diào)制相位，對于檢測微小的缺陷和監(jiān)視它們的演變尤其有價值。 SCD系統(tǒng)將繪制的信號發(fā)送到循環(huán)頻率（ CF）對一 般頻率 ( -f)的二維功能中?？紤]到其信息冗余 4 和巨大的計算， SCD部分在缺陷發(fā)現(xiàn)時循環(huán)頻率即軸旋轉(zhuǎn)頻率是獨立的計算，叫做部分頻譜密度關(guān)系 （ SSCD）。 SSCD 結(jié)果顯示，它和 SCD 號功能擁有相同的鑒定能力。它可以用很少的計算從時間自變量直接計算，并在同一時間，與 SCD的三維形式明確對比。 2二階循環(huán)統(tǒng)計 一個隨機過程一般有一個時間自變量 5 其中 E （ . ）是數(shù)學期望函數(shù)， T是時間差。如果自變量是以 T0 為周期的周期變量，那整體平均值約即時間平均值。 自變量因為它的定 期性也可以寫成傅立葉函數(shù)。 結(jié)合等式 2，它的傅立葉系數(shù)給 5。 t 是平均運行時間， R (t)被稱為循環(huán)變量（ CA），是循環(huán)頻率。 SCD能夠由循環(huán)變量里傅立葉積數(shù)的變換和時間差 t得到。 SCD 在一般頻率雙頻率間展示出信號的特性。所有非零循環(huán)頻率是以周期平穩(wěn)信號（ CS）的特征為特征的。 3齒輪模型 變速箱振動中最重要的組成元件是齒嚙合振動，這是因為偏離了理想的齒形。這種偏離的來源是齒加工過程中負載或原始配置錯誤下的變形。一般來說，調(diào)制現(xiàn)象發(fā)生在局部缺 陷通過嚙合并產(chǎn)生周期性的變化使齒嚙合在振幅和相位發(fā)生振動。正常齒輪，波動軸旋轉(zhuǎn)頻率和負載或微小差異的牙齒空間還允許輕微調(diào)幅（ AM）或相位調(diào)制（ PM） 。因此，一般齒輪模型可寫為 6 ， 7 Fx是齒嚙合頻率和 Fs是軸旋轉(zhuǎn)頻率。 am （ t ）和 bm（ t）分別表示調(diào)幅和相位。 主要調(diào)制部分源于軸旋轉(zhuǎn)頻率及其諧波 ；其他分鐘調(diào)制部分可以忽略不計。 AM 和 PM，無論是單獨或合并，都會導致信號光譜邊頻帶的出現(xiàn)。帶通濾波周圍之一諧波的齒嚙合頻率是傳統(tǒng)的信號處理，用來提供對邊頻帶的詳細觀察。齒輪振動的過濾信號 可表示如下： Fh是指齒嚙合頻率的諧波之一。下標 m是為簡化該方程和下面討論時被忽略。 這項研究的重點是本文過濾齒輪振動信號模型的方程（ 7 ） ，它的傳輸是一個單一的余弦波形和調(diào)制部分是階段功能。 4 齒輪模型的周期平穩(wěn)信號（ CS）系統(tǒng) 根據(jù)之前的分析，齒輪振動信號能夠在調(diào)整振幅和相位后被簡化成一個周期信號。調(diào)制條件反映了齒輪潛在的嚴重的缺陷。在本節(jié)，調(diào)幅（ AM）或相位調(diào)制（ PM）將獨立學習，齒輪模型的周期平穩(wěn)信號（ CS）系統(tǒng)在它們結(jié)果的基礎(chǔ)上發(fā)展。 4.1調(diào)幅（ AM） 該模 型的調(diào)幅信號來自方程式（ 7 ） 等式（ 8）中的 x(t)的解析式可以寫為 將 替換進等式（ 4）能夠推出循環(huán)變量（ CA） 是包絡(luò)值 是 提供調(diào)制數(shù)據(jù)。它是通過等式（ 11） 的傅立葉變換，另外， 的傅立葉變換時間差與 SCD 的 相一致。于是 能夠用兩次傅立葉變換 的時間 t 和時間差計算得到。 通過整體變換 變成 H(v)是 a（ t）的傅立葉變換 在 H(v)替換進等式（ 13）后解開 f，用函數(shù)的值，最后表達式 就能得到 有四個完全對稱的結(jié)構(gòu)。等式（ 15）只是其中的一個 象限，其他的簡化忽略不 計。通過等式（ 15）， 是一連串不連貫的波峰，另外，這些波峰規(guī)律的分配在 a f 平面。盡管表達比較復(fù)雜，但是 的幾何描述是簡單的。這些波峰重疊交叉成組和線 。這些線也可以被認為是 的線性特征。 4.2 相位調(diào)制（ PM） 相位調(diào)制信號由等式（ 7）得到 CA分析可以表示成 CA在 PM 中也有像 AM 中的包絡(luò)傳輸。因此， CA包絡(luò)被用來從信號中提取調(diào)制數(shù)據(jù)。它和 SCD一樣也用 表示。 PM 部分， b(t)，由連續(xù)的傅立葉函數(shù)構(gòu)成。 PM 中的 CS 分析從正弦曲線波形開始，貝塞爾公式 在計算中使用。最終結(jié)果可以表示為 等式（ 18）的幾何表達也可以表達成 ， 只在它們的交叉處非零。這 些式子的值不只靠調(diào)制部分的諧波值，在無限理論上甚至為 PM 信號的一正弦曲線 。事實上，貝塞爾系數(shù)在 a f 的零點的周圍限定了分離波峰。其他理論上的振幅波峰特征距離離零點遠，超出了范圍接近零。 PM功能包括像等式（ 16）中的一些正弦曲線波形，它的組成可以表達成，i是從 0到 I， CA包絡(luò)可以寫成 等式統(tǒng)一。通過二維卷積原則，相應(yīng) SCD 的 可表達為 符號 表示雙頻率平面的二維卷積。 的表達在等 式（ 18）中用 取代，用 B表示 Bi， b取代 bi。盡管 SCD在多重正弦曲線調(diào)制情形中表達更加復(fù)雜， 中的二維 卷積結(jié)果有相同的幾何分配，正如它在單一正弦曲線的情形。 線性特征和一般頻率軸之間的距離是基本的頻率 Fs。因此，卷積不能得出新的 波峰特征，但是改變了它們的振幅，方程式（ 18）也在方程式（ 16）中表達了 SCD信號，盡管系數(shù) 被二維卷積改變。 4.3 齒輪振動信號的 CS分析 二位普通齒模型輪 CS分析在等式（ 7）中以 AM 和 PM為基礎(chǔ)發(fā)展。 CA信號分析也有包絡(luò)傳輸形式， CA包絡(luò)的表達如下 符號 . 在等式（ 21）中的兩個部分與 AM 和 PM功能相關(guān)。因此， SCD對應(yīng)的 有兩個 AM,PM功能組成分配的二維卷積。 和 的表達方法在等式（ 15）和（ 18）中給出。 和間的二 維卷積只能導致 和 間的特征波峰的重疊，就像它在 PM中。由于相同的幾何特征，卷積不能改變分布，但是需要在數(shù)值和振幅的有效特征波峰上改變（振幅大于零）。因此，齒輪模型的 CS 特征也被表達為 ，與 AM,PM中一樣。 4.4 齒輪振動信號的 SSCD分析和實現(xiàn) 通過上述分析，三種調(diào)制有相同的 CS特征。雙頻 率平面中 F=是它們的常見特征式。計算 1表示了它的分布。只有第一象限部分顯示出來因為的四個象限是 統(tǒng)一整齊的，這些分離點和光譜波峰的振幅數(shù)值反映了信號的調(diào)制范圍。 SCD提供了鑒別齒輪調(diào)制信息的多余數(shù)據(jù)，事實上，它的一部分足夠達到目的，對于AM，部分， CF（在第一象限），能由方程（ 15）得到 包括等距特征頻率，它們之間的距離是 fs。 PM 和聯(lián)合調(diào)制有