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1 The basic principle of the matching tracking principle matching tracking algorithm is:
Select the atom that best matches the signal in the atomic set of the dictionary, and extract the component n(n=0,1,,M) that best matches the selected atom from the original signal x(t); the remaining residual signal x1 (t) Decompose again by the above process, and so on, until the energy of the residual signal xm(t) (m=1, 2,) is less than the set threshold, and the iteration ends. Thus, the original signal can be gradually extracted from the dictionary atom, and its essence is the orthogonal projection process of the signal to ensure the uniqueness of the dictionary atomic decomposition.
Let D={n}n(=RR2) be a dictionary formed by a vector with a norm of 1. The matching pursuit algorithm first selects 0(t) from the set of dictionary atoms D to make it most similar to x(t). The expansion coefficient is a0=x(t), 0(t)=maxnD|x(t), n(t)|(1) After determining a0 and 0(t), the residual signal of the next step is calculated. The mth residual signal xm(t), the expansion coefficient am, and the m1th residual signal xm 1(t) are obtained:
Am=xm(t),m(t)=maxnD|xm(t),n(t)|(2)xm 1(t)=xm(t)-am(t)(3)
From equations (1) and (3), the mth approximate expansion of the original signal x(t) can be obtained x(t)=Mn=0an(t) xm 1(t)(4)
The residual signal is decomposed successively, and the energy of the residual signal decreases exponentially and shrinks to zero (after m 1 decomposition).
The final original signal x(t) can be approximated as x(t)Mn=0an(t)(5)
2 Gear simulation signal example The vibration condition of the gearbox is the result of the combined action of all the meshing gears. The vibration of each meshing frequency and its multiplication component is modulated by the axial frequency to form an amplitude modulation, frequency modulation or mixed modulation signal. Usually, the narrowband signal near the meshing frequency of the gear is used as the object, and the change of the gear modulation condition is studied to judge the running state of the gear. The narrowband signal near a certain meshing frequency is a mixed modulation signal with a single modulation frequency, amplitude modulation and frequency modulation. The model is x(t)=c(t)cos[2fct b(t)](6) where: fc is The meshing frequency; c(t) and b(t) are amplitude modulation and frequency modulation signals respectively, which are periodic signals composed of the frequency conversion of each rotating shaft in the gearbox and its harmonic frequency, and the expression is: c(t)=rqCr , qcos(2tqfo r,q)(7)
b(t)=rqBr,qcos(2tqfo r,q)(8)
Cr, q, Br, q are the modulation factors of the modulation components in the amplitude modulation and frequency modulation signals respectively, fo is the frequency conversion of any one axis in the gearbox, and the subscripts r and q(r, qN) are determined by the actual modulation conditions. , , is the initial phase.
In general, the axial frequency of a pair of meshing gears that have a fault in the gearbox is the largest in the modulation component.
A simulated signal is obtained according to the narrow band signal near the meshing frequency of the analog gear of equation (6). In order to clearly observe the analysis results, the modulation frequency is the harmonic component, and the simulation signal only includes the amplitude modulation. The sampling frequency is 1.024 kHz, the data length is 512, fc=200 Hz, fo=10 Hz. Since the Gabor atom has a good time-frequency Local characteristics, and its translation relative to the wavelet packet can be performed on a finer time-frequency grid, so the internal structure of the signal can be expressed more clearly, so the atom in this paper uses Gabor atom, and its expression is gl, s ( w)=g(w-ul)e2ivsw(9)
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It can be seen from the figure that due to the amplitude and frequency modulation in the time domain, the meshing frequency, the frequency conversion, the side frequency and other frequency components in the frequency domain are combined, and since the impact signal is a broadband signal, the frequency domain cannot be distinguished in the frequency domain. The time domain waveform and the power spectrum can not effectively extract the characteristic components of the gear signal; the local frequency local positioning ability is poor, the resolution is low, and the periodic frequency components such as the meshing frequency, frequency conversion, and sideband cannot be distinguished (see 1(a)); and based on the matching pursuit algorithm, since the best matching atom with the signal is adaptively selected from the dictionary in each decomposition process, the time-frequency distribution is the superposition of the Wigner distribution of the single atom, and the time-frequency The resolution can be close to the Wigner distribution, and the interference term is eliminated, and the best time-frequency resolution is obtained. Therefore, the periodic component and the impact signal can be well extracted, and the interference of the sideband component can be effectively suppressed and has good performance. Noise resistance (see 1 (b)).
3 Gear fault diagnosis experimental device based on matching tracking is shown in 2(a). The gear pair is 4137, the modulus is 5, the contact angle is 20, the wheelbase of the gear is 200mm, the meshing coefficient is 1.648, the tooth width is 41.944mm, and the material is low carbon steel (prone to surface defects). During the experiment, the acceleration sensor It is mounted on the gearbox housing and uses two phase-detection signals to ensure signal synchronization during data acquisition. A comparative analysis of normal conditions and gear fatigue wear faults, in which fatigue wear faults are formed by electrical discharge machining (see 2 (b)).
The data analysis length is 4096. The time-frequency diagram of normal gear operation is shown in 3(a). It can be seen from the figure that there is no impact signal for the normally operating gear base, only the periodic signal exists in the low frequency part, and there is no periodic shock signal; the time-frequency diagram of the fault gear operation is as shown in (b). It can be seen from the figure that the matching tracking method can extract the meshing frequency and its harmonic and impact signals. The shock signal on the time-frequency map is quasi-periodic, and the interval time is about 0.1s, which coincides with fo=10Hz, which indicates the gear. The teeth still have local damage. Therefore, the matching pursuit algorithm can detect gear faults efficiently and accurately.
4 Conclusion This paper studies the principle and algorithm of the matching pursuit method, and applies the method to the simulation signal and experimental signal of the gear. The decomposition result is compared with the short-time Fourier transform, and it is found that the adaptive time with high resolution is obtained. Frequency distribution. The results show that the method has high time-frequency resolution and can accurately determine the running state of the gear, which provides an effective means for identifying gear defects.