# Analysis of gear failure by cepstrum analysis

There are two ways to express the amplitude of spectrum: one is in the form of amplitude, which is called amplitude spectrum; the other is in the form of energy, which is called power spectrum. Power spectrum is used to study the distribution of energy in frequency domain.

There are two ways to express the scale of the vertical coordinate of the spectrum chart: one is linear coordinate, the other is logarithmic coordinate. The advantage of linear coordinate is intuitionistic, but the disadvantage is that it can not display the components with large numerical difference at the same time; on the contrary, the logarithmic coordinate can display the frequency components with large numerical difference (1000 times, or even higher), but these components are not linearly proportional.

The vertical coordinate of amplitude spectrum is linear coordinate, and the vertical coordinate of power spectrum is logarithmic coordinate.

The logarithmic coordinates are expressed in decibels [DB], defined as

Ad = 20 LG (A / AR) or ad = 10 LG (A2 / ar2)

In the formula, ad ～ reference amplitude (or reference amplitude), usually ad = 1V, ad = 1 for dimensionless quantity

A ~ amplitude, the unit is the same as AD.

From the above formula, it can be seen that the decibel value is 6, and the ratio of amplitude is 2, which is commonly known as “6 decibels turn over”; the decibel value is 20, and the ratio of amplitude is 10; the decibel value is 60, and the ratio of amplitude is 1000.

The definition of cepstrum is the power spectrum of power spectrum logarithm. Its expression is

C(τ)={F-1[logG(f)]}2

Where, the power spectrum of G (f) – time signal FZ, i.e. g (f) = {f [FZ (T)]} 2

The time variable of τ – cepstrum is called cepstrum.

The vertical coordinate and the frequency spectrum of the cepstrum can adopt the same unit, while the horizontal coordinate is the cepstrum, and the unit is milliseconds [MS].

Cepstrum analysis plays a special role in gear fault diagnosis, especially in side band analysis. Because the frequency spectrum of the actual gearbox vibration signal is very complex, when there are several side bands crossing each other, it is not enough to rely on the frequency refinement analysis method only, and it is often difficult to see the structure of the side band. The cepstrum can show the periodic structure component edge frequency in the spectrum clearly.

If a complex spectrum is regarded as a time history signal, there are many sidebands around the meshing frequency and its harmonics. The interval frequency of the sideband is the fault frequency. The higher the harmonic order of the fault frequency is, the smaller the amplitude value is, the average envelope of the harmonic peak is close to a periodic wave. Another spectrum analysis (that is, inverse Fourier transform) is made for this power spectrum, which is converted to a new domain, so that the periodic frequency structure is clearly displayed, which is called cepstrum analysis method. Cepstrum is another concentration of the energy of the periodic frequency structure component on the original spectrum, and it gives higher weight to the low amplitude component in the logarithmic conversion of power, and lower weight to the high amplitude component, so the small signal period is highlighted. Therefore, the cepstrum can effectively identify the periodic components in the spectrum. This is the first advantage of cepstrum analysis.

The second advantage of cepstrum analysis is that it is little affected by the signal transmission path. The installation position of the sensor on the gearbox is different, the signal transmission way is different, and different transfer functions are formed. These transfer functions reflect different results on the output spectrum. Sometimes, the amplitude of some frequency components is very different, resulting in false grasping and missing grasping of fault feature information. However, after the simultaneous interpreting of the signals of different transmission paths, the two inverted spectrum images have the same high frequency components, which provides a very favorable condition for gearbox fault diagnosis.

In addition, due to the simultaneous existence of amplitude modulation and frequency modulation, as well as the phase changes of the two modulation, the side frequency is unstable, which often leads to the lack of symmetrical side band on the power spectrum, which makes it difficult to identify the side band. However, in the cepstrum, the amplitude representing the modulation level is not affected by the stability. After the two vibration signals are modulated, even though the amplitude of the side frequency at the corresponding position on the power spectrum is quite different, the effect of the phase difference on their frequency spectrum will not be reflected in the cepstrum at all. In the cepstrum, the cepstrum peaks of the two vibration signals are completely the same. 