外文翻译--部分频谱与齿轮缺陷发现相互关系的实际应用

附录一 英文参考文献 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. 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Des., 1982, 104, 259267. 附录二 英文文献翻译 部分频谱与齿轮缺陷发现相互关系的实际应用 G Bi, J Chen、 F C Zhou 和 J He 中华人民共和国，上海，上海交通大学，国家震动、声音和噪音重点实验室 原稿于 2005年 10月 16日完成，经修改后于 2006年 5月 3日发表 DOI： 10.1243|0954406 JMES206 摘要：振幅和振动调制相位的变化能最直接的反映出齿轮的缺陷。部分频谱密度关系 （ SSCD）方法在本文中被用来从齿轮震动信号中提取调制数据；振幅和调制阶段数据能个别地或在组合中被提取。 这一个方法能用比较仪明显的发现调制相位的细微的缺陷和通过放大的调制振幅发现严重的缺陷。实验不同严苛的等级下当前齿轮缺陷的振动信号，证实了它的特性，鉴别能力，表明了 SSCD是一种有效的方法，特别是在发展早期发现缺陷。 关键字：部分频谱密度相互关系，齿轮，发现缺点，调制 1介绍 齿轮震动信号是一个典型的周期调制信号。调制现象随着齿轮缺陷的恶化更加严重。因此，调制在光谱中的边频带中的数量和振幅会增加。因此从边频带中提取调制信号是最直接发现齿轮缺陷的方法。传统的包络技术是实现这个意图方法中的一种。它对振 幅中的调制现象很灵敏，但是不是在相位中。微小的齿轮缺陷经常导致振动振幅的小的变化，但是它也经常同时伴随着明显的相位调制。开始的微小缺陷使用包络技术不能产生满意的结果。 近年来，循环统计理论被用在旋转机器振动信号，并且在控制条件和诊断结论 1 3上表现出了好的潜力。在本文中，波谱密度关系（ SCD）在二阶循环平稳度分析法中的功能对齿轮缺陷的发现被证实是一个多余的信息提供者。它在齿轮震动中同时显示出振幅和调制相位，对于检测微小的缺陷和监视它们的演变尤其有价值。 SCD系统将绘制的信号发送到循环频率（ CF）对一 般频率 ( -f)的二维功能中。考虑到其信息冗余 4 和巨大的计算， SCD部分在缺陷发现时循环频率即轴旋转频率是独立的计算，叫做部分频谱密度关系 （ SSCD）。 SSCD 结果显示，它和 SCD 号功能拥有相同的鉴定能力。它可以用很少的计算从时间自变量直接计算，并在同一时间，与 SCD的三维形式明确对比。 2二阶循环统计 一个随机过程一般有一个时间自变量 5 其中 E （ . ）是数学期望函数， T是时间差。如果自变量是以 T0 为周期的周期变量，那整体平均值约即时间平均值。 自变量因为它的定 期性也可以写成傅立叶函数。 结合等式 2，它的傅立叶系数给 5。 t 是平均运行时间， R (t)被称为循环变量（ CA），是循环频率。 SCD能够由循环变量里傅立叶积数的变换和时间差 t得到。 SCD 在一般频率双频率间展示出信号的特性。所有非零循环频率是以周期平稳信号（ CS）的特征为特征的。 3齿轮模型 变速箱振动中最重要的组成元件是齿啮合振动，这是因为偏离了理想的齿形。这种偏离的来源是齿加工过程中负载或原始配置错误下的变形。一般来说，调制现象发生在局部缺 陷通过啮合并产生周期性的变化使齿啮合在振幅和相位发生振动。正常齿轮，波动轴旋转频率和负载或微小差异的牙齿空间还允许轻微调幅（ AM）或相位调制（ PM） 。因此，一般齿轮模型可写为 6 ， 7 Fx是齿啮合频率和 Fs是轴旋转频率。 am （ t ）和 bm（ t）分别表示调幅和相位。 主要调制部分源于轴旋转频率及其谐波 ；其他分钟调制部分可以忽略不计。 AM 和 PM，无论是单独或合并，都会导致信号光谱边频带的出现。带通滤波周围之一谐波的齿啮合频率是传统的信号处理，用来提供对边频带的详细观察。齿轮振动的过滤信号 可表示如下： Fh是指齿啮合频率的谐波之一。下标 m是为简化该方程和下面讨论时被忽略。 这项研究的重点是本文过滤齿轮振动信号模型的方程（ 7 ） ，它的传输是一个单一的余弦波形和调制部分是阶段功能。 4 齿轮模型的周期平稳信号（ CS）系统 根据之前的分析，齿轮振动信号能够在调整振幅和相位后被简化成一个周期信号。调制条件反映了齿轮潜在的严重的缺陷。在本节，调幅（ AM）或相位调制（ PM）将独立学习，齿轮模型的周期平稳信号（ CS）系统在它们结果的基础上发展。 4.1调幅（ AM） 该模 型的调幅信号来自方程式（ 7 ） 等式（ 8）中的 x(t)的解析式可以写为 将 替换进等式（ 4）能够推出循环变量（ CA） 是包络值 是 提供调制数据。它是通过等式（ 11） 的傅立叶变换，另外， 的傅立叶变换时间差与 SCD 的 相一致。于是 能够用两次傅立叶变换 的时间 t 和时间差计算得到。 通过整体变换 变成 H(v)是 a（ t）的傅立叶变换 在 H(v)替换进等式（ 13）后解开 f，用函数的值，最后表达式 就能得到 有四个完全对称的结构。等式（ 15）只是其中的一个 象限，其他的简化忽略不 计。通过等式（ 15）， 是一连串不连贯的波峰，另外，这些波峰规律的分配在 a f 平面。尽管表达比较复杂，但是 的几何描述是简单的。这些波峰重叠交叉成组和线 。这些线也可以被认为是 的线性特征。 4.2 相位调制（ PM） 相位调制信号由等式（ 7）得到 CA分析可以表示成 CA在 PM 中也有像 AM 中的包络传输。因此， CA包络被用来从信号中提取调制数据。它和 SCD一样也用 表示。 PM 部分， b(t)，由连续的傅立叶函数构成。 PM 中的 CS 分析从正弦曲线波形开始，贝塞尔公式 在计算中使用。最终结果可以表示为 等式（ 18）的几何表达也可以表达成 ， 只在它们的交叉处非零。这 些式子的值不只靠调制部分的谐波值，在无限理论上甚至为 PM 信号的一正弦曲线 。事实上，贝塞尔系数在 a f 的零点的周围限定了分离波峰。其他理论上的振幅波峰特征距离离零点远，超出了范围接近零。 PM功能包括像等式（ 16）中的一些正弦曲线波形，它的组成可以表达成，i是从 0到 I， CA包络可以写成 等式统一。通过二维卷积原则，相应 SCD 的 可表达为 符号 表示双频率平面的二维卷积。 的表达在等 式（ 18）中用 取代，用 B表示 Bi， b取代 bi。尽管 SCD在多重正弦曲线调制情形中表达更加复杂， 中的二维 卷积结果有相同的几何分配，正如它在单一正弦曲线的情形。 线性特征和一般频率轴之间的距离是基本的频率 Fs。因此，卷积不能得出新的 波峰特征，但是改变了它们的振幅，方程式（ 18）也在方程式（ 16）中表达了 SCD信号，尽管系数 被二维卷积改变。 4.3 齿轮振动信号的 CS分析 二位普通齿模型轮 CS分析在等式（ 7）中以 AM 和 PM为基础发展。 CA信号分析也有包络传输形式， CA包络的表达如下 符号 . 在等式（ 21）中的两个部分与 AM 和 PM功能相关。因此， SCD对应的 有两个 AM,PM功能组成分配的二维卷积。 和 的表达方法在等式（ 15）和（ 18）中给出。 和间的二 维卷积只能导致 和 间的特征波峰的重叠，就像它在 PM中。由于相同的几何特征，卷积不能改变分布，但是需要在数值和振幅的有效特征波峰上改变（振幅大于零）。因此，齿轮模型的 CS 特征也被表达为 ，与 AM,PM中一样。 4.4 齿轮振动信号的 SSCD分析和实现 通过上述分析，三种调制有相同的 CS特征。双频 率平面中 F=是它们的常见特征式。计算 1表示了它的分布。只有第一象限部分显示出来因为的四个象限是 统一整齐的，这些分离点和光谱波峰的振幅数值反映了信号的调制范围。 SCD提供了鉴别齿轮调制信息的多余数据，事实上，它的一部分足够达到目的，对于AM，部分， CF（在第一象限），能由方程（ 15）得到 包括等距特征频率，它们之间的距离是 fs。 PM 和联合调制有