3.1. Determination of Wavelet Basis Function
Because wavelet transform can map the signal to time–frequency domain, and a series of wavelet basis functions are used to represent the signal, the hidden features in the signal can be effectively displayed. The assembly failure of spindle box studied in this paper is a part of rotating machinery failure. In order to better extract the characteristics of the collected samples, the wavelet analysis method is also used to process the fault signal.
The basic method of wavelet transform is to take the inner product of the basic wavelet function with the signal
to be analyzed at different scales
a. The basic expression of continuous wavelet transform is shown as Equation (
1):
where
is Wavelet coefficients,
a is scale that represents the frequency parameter, and
b is time or space position parameters.
For continuous wavelet transform, scale
a, time
t and offset
b are continuous. When using the computer to realize the calculation of wavelet transform, they need to be discrete processing, which is the discrete wavelet transform. Binary discretization of scale
a and offset
m is usually carried out, and the corresponding binary wavelet is shown in Equation (
2):
In the calculation procedure of binary discrete wavelet transform, only the approximate part (low-frequency sub-band) of the signal is divided by exponential interval from scale 2, and the detailed part (high-frequency sub-band) is not processed. Therefore, the resolution of the high frequency part is poor. In order to process and analyze the high frequency part, Coifman, Meyer and Wickerhauser put forward the concept of wavelet packet decomposition. Wavelet packet decomposition divides the frequency band into several levels, so as to further decompose the high frequency part and realize more detailed analysis of the signal. The decomposition structure obtained by decomposing signal
S into three-layer wavelet packet transform is shown in
Figure 7.
It can be seen from the analysis structure diagram that wavelet packet analysis can decompose the original signal into each frequency band, so that the frequency band matching the concerned frequency can be selected for analysis, and the time-frequency resolution of signal analysis can be improved.
The fault characteristic frequency studied in this paper is not limited to the rotation frequency or meshing base frequency of a rotating part. In order to realize fault diagnosis, their high harmonic frequencies are also analyzed. Therefore, the wavelet packet transform method can be used as a filter. A node only reflects the information related to the frequency range of the node but excludes the interference of other frequency components. In contrast, discrete wavelet transform usually ignores some high frequency information. In this paper, the advantages and disadvantages of wavelet transform and wavelet packet transform in signal representation will be compared quantitatively.
The evaluation criteria of wavelet transform method are mainly as follows:
(a) The energy content of the signal
The energy content of a signal is a parameter directly related to the characteristics of the signal, so it is usually used to describe the signal. The energy contained in the signal
can be calculated by the wavelet coefficient, as expressed by Equation (
3) [
16]:
If a major frequency component of a signal corresponds to a specific scale, the wavelet coefficients in that scale will have a relatively large amplitude when the major frequency occurs. This allows the energy associated with a particular frequency component to be extracted from the signal. Therefore, the energy content can be used as a criterion to evaluate the applicability of wavelet basis. For the same signal, the larger the calculated value of the energy content, the more effective and appropriate processing method is considered to show the characteristics of the signal, which is the maximum energy content standard.
(b) Shannon entropy of the signal
The wavelet coefficients are obtained by wavelet transform. The coefficients reflect the similarity between wavelet and signal. If the coefficient matrix obtained by wavelet transform is regarded as a distribution, the sparser the coefficient matrix is, the higher the similarity between signal and wavelet will be. Shannon entropy can be used as an indicator to evaluate sparsity. Shannon entropy is defined as Equation (
4):
where
is the energy probability distribution of the wavelet coefficients.
The probability distribution uniformity can be evaluated by calculating the Shannon entropy of the distribution. For example, the distribution is the most uncertain when it has equal probability property. This distribution has the highest Shannon entropy and the lowest sparsity. This evaluation method is applied to the calculation and evaluation of wavelet coefficient matrix. The smaller the Shannon entropy is, the more the corresponding wavelet analysis method matches with the signal. This is also the minimum Shannon entropy standard to evaluate the applicability of wavelet basis function and wavelet analysis method.
(c) The ratio of energy content to Shannon entropy (E–S ratio)
Both maximum energy content standard and minimum Shannon entropy standard can be used to evaluate wavelet analysis. These methods are based on the description of the original signal information by wavelet coefficients. The evaluation standard derived from the combination of these two is called energy–Shannon entropy ratio (E–S ratio), and its definition is shown in Equation (
5). Corresponding to the maximum energy content standard and the minimum Shannon entropy standard, the wavelet transform method with the maximum E–S ratio has the strongest ability to characterize the signal. The above three evaluation criteria will be verified and compared in the analysis of actual signals later:
In different calculation methods of wavelet transform, discrete transform and wavelet packet transform can be implemented by the computer, and they can decompose signals into different frequency scales. Regarding which method is more suitable for the fault diagnosis of spindle box, the above-mentioned criteria (as shown in Equations (3)–(5)) are used for comparison. Using the same wavelet basis function, the methods of discrete wavelet decomposition and wavelet packet transform are compared. By analyzing the advantages and disadvantages of these two methods for decomposition of original signals, Shannon entropy, energy content and E–S ratio under different decomposition modes are calculated, respectively.
The wavelet basis function
in the Coiflet wavelet system is used, and the discrete wavelet transform is applied to carry out 3-layer decomposition of the original signal. For the wavelet packet, the decomposition of the original signal with the scale of 3 is also carried out. In addition, this comparative analysis was carried out for different samples. Data collected on each axis are analyzed and calculated, and the results are shown in
Table 1.
By comparing the data values of each row in the table, it can be seen that Shannon entropy value of signal through wavelet packet transformation is lower than that of discrete wavelet transform. The energy value is larger than the energy obtained by the discrete wavelet transform, and the E–S ratio is correspondingly larger than the discrete wavelet transform. It can be verified that wavelet packet decomposition can better show the change rule of the signal, characterizing the characteristics of signal and the information contained in it. In order to demonstrate the universality of the above verification, the wavelet basis function was replaced to process the same data, and the calculation results of the two analysis methods were still compared. Using wavelet basis function sym4, which is different from wavelet basis waveform coif1, the calculation result is similar to that of coif1 wavelet, which proves that wavelet packet analysis can reflect the characteristics of signal more precisely. From the perspective of principle, wavelet packet analysis not only decomposes the low-frequency signal, but also decomposes the high-frequency signal that has not been decomposed in the discrete wavelet transform, so the description of signal features is more appropriate. Therefore, better calculation results can be obtained when various indexes are introduced for evaluation. In addition, some of the characteristic frequencies studied in this research are at the higher harmonic frequency of the meshing frequency, so the high-frequency part of the signal also needs to be paid attention to. Therefore, wavelet packet decomposition can better meet the research needs.
Based on Shannon entropy, energy content and E–S ratio, the wavelet basis suitable for the analysis of spindle box vibration signal is evaluated and selected. Different wavelet basis functions are used for wavelet packet decomposition transformation, and the results are shown in
Table 2.
It can be seen from the calculation results that the values of each index obtained after the processing of different wavelet basis functions are different. Among these wavelet families, the calculated values of Bior and R-bior wavelet families are superior to other wavelet families. The results of wavelet functions of different orders in the same family of wavelets are different. The basic rule is that the higher the order, the better the result of each index. In terms of energy content, the maximum value of Bior3.5 was calculated in the analysis of different groups of sample data. In the calculation of Shannon entropy, Bior3.5, Coif4 and Rbio3.5 respectively obtained a primary minimum value. As for the E–S ratio, the maximum value of Bior3.5 is obtained. It can also be seen that the maximum E–S ratio standard can help us select the most suitable wavelet. Bior3.5 was selected for wavelet packet decomposition in this research.
3.2. Fault Feature Extraction
The signal collected from the spindle box of the machine tool is analyzed. The spindle box is known to have gear misalignment errors among the shafts. Sampling signal acquisition in shaft II bearing end cover. The tooth transmission relation between machine each axis: shaft I/II is 28/47, shaft II/III is 30/43. The motor speed measured during sampling was 999.6 rpm and the sampling frequency was 8192 Hz. After calculation, the rotation frequency of the test shaft is 9.93 Hz, and the engagement frequency of the gears installed on the shaft is 466.48 Hz and 297.75 Hz. The original time domain waveform of the acquired signal is shown in
Figure 8. The period of some faults cannot be directly identified from the information in the figure, and the waveform peak is disordered. Bior3.5 wavelet is used for five-layer wavelet packet decomposition, so the signal is decomposed into 32 frequency bands, and a five-layer binary tree structure is obtained. The last layer is [5 0], [5 1], …, [5 31], a total of 32 nodes, and the corresponding frequency band is 0–128 Hz, 128–256 Hz, …, 3968–4096 Hz. The characteristic frequency associated with the mismeshing fault of the gear is the first, second and third meshing frequency, and the corresponding frequency of conversion modulation, that is, the meshing frequency plus or minus 9.93 Hz.
The frequency points concerned are 297.75 Hz, 595.5 Hz, 893.25 Hz, 466.48 Hz, 932.96 Hz, 1399.44 Hz, and the modulation frequencies around them. These frequency points are located in frequency bands corresponding to nodes [5 2], [5 3], [5 4], [5 6], [5 7], [5 9]. These nodes were reconstructed and Hilbert transformation was performed for spectral analysis, and the waveform obtained was shown in
Figure 9.
Figure 9a,c,e deal with pairs of nodes [5 2], [5 4] and [5 6]. Gear meshing frequency between shaft II and shaft III is 297.75 Hz, and the second and third order has a larger peak near the frequency doubling, and they all have a modulation of about 9 Hz. From the size of its peak value, it can be seen that the energy at the second and third order frequencies does not decay, and is even greater than the intensity of the meshing fundamental frequency, which is consistent with the fault characteristics.
Figure 9b,d,f processing the node [5 3], [5 7], [5 9] represent axial meshing frequency between shaft I and shaft II, the second order and third order meshing frequencies and the corresponding amplitude modulation frequencies. Through the characteristics shown in these processing results, it is proved that this processing method can be used to obtain significant changes related to specific faults in the frequency–power spectrum, and this ability to capture abnormal features can be used to extract features under different faults.
By analyzing the amplitude of signal at certain frequency in signal power spectrum, the fault type can be judged. Therefore, it is very meaningful to obtain the energy values at these characteristic frequencies for fault analysis. Because there are only certain kinds of assembly faults of the spindle box of the heavy horizontal lathe, and the detection of some faults needs to be based on the amplitude analysis of some rotating frequencies. These can be used as the basis of the characteristics of the frequency mainly includes the following: (1) shaft frequency and its harmonic frequency, shaft rotation frequency of 1/2 frequency, 1/3 frequency; (2) meshing frequency and doubling frequency of the gear mounted on the shaft; (3) the gear mesh frequency plus or minus the frequency of the shaft rotation frequency value. With the amplitudes at these frequency points, they can be arranged into a set of eigenvectors to represent the operation of the axis. The feature vectors extracted from known faults can be used to train the intelligent classifier, while the features extracted from unknown fault signals can be used to judge the fault type of the axis.
Considering the fault types to be diagnosed comprehensively, the feature frequency points that should be extracted are summarized in
Table 3. Considering that there are two gears meshing on some axes, the information of their frequency points should be extracted. For the side frequency modulation of the meshing frequency, the fault can be diagnosed only by knowing whether the unilateral modulation phenomenon exists. Therefore, a representative value of the two side frequencies can be extracted. For each signal, a 23-dimensional vector is extracted to represent its characteristics.
In order to extract the power spectrum value at the above characteristic points, the variables should be clearly defined, including the number of teeth of each relevant shaft of the machine tool spindle box, motor speed during testing, gear position during sampling, sampling frequency, and which axis the sensor is arranged in during sampling. After calculating the characteristic frequency, the wavelet packet decomposition node corresponding to the frequency band is reconstructed and spectral analyzed, and then the value of the characteristic frequency point is obtained. At this time, in the algorithm, the maximum value in the range of –2 Hz to +2 Hz can be extracted in order to cope with the rotation speed change in the above analysis. As for the meshing edge frequency band information, only one value needs to be extracted, so the value with a large side frequency is selected to extract it. Since the extracted values are likely to be large, they should be normalized to facilitate subsequent analysis and calculation. represents the amplitude of a certain frequency point, and it is the energy at a discrete point i representing the characteristic frequency after reconstruction of the node j. The total energy . For the extracted 23-dimensional eigenvector T= [, , …, ], each value corresponds to the value of a certain . After normalization, a new vector can be obtained, = [/E, , …, ], which will be used for the subsequent intelligent fault classification.