According to the different definitions, the 1.5-dimensional spectrum of the complex harmonic signal can detect the components involved in coupling and the components generated by coupling respectively.
The analysis and verification steps of the simulation signal are given as follows:
Set the simulation signal as follows:
Where: φ I is an independent random variable uniformly distributed on [0, 2 π).
Among them: f1-9 Hz; f2-30 Hz; f3-39 Hz; f4-40 Hz; f5-51 Hz; f6-100 Hz; f7-33 Hz; F3 = F1 + F2, φ 3 ≠φ 1 + φ 2, and the harmonic component F3 is produced by F1 and F2 through secondary frequency coupling, and their phase does not meet the coupling relationship; F6 = F1 + F4 + F5, φ 6 ≠ φ 1 + φ 4 + φ 5, harmonic component F6 is generated by F1, F4, F5 through third frequency coupling; F7 does not participate in any form of coupling.
The spectrum of the simulation signal is shown in the figure.
In figure (a), the spectrum shows all the frequency components in the simulation signal, so the fault characteristic frequency can not be distinguished, which increases the difficulty of fault diagnosis.
As can be seen from figure (b), only the three frequency components of 9 Hz, 30 Hz and 39 Hz with secondary frequency coupling are shown in the 1.5-dimensional spectrum, and do not need to meet the phase coupling, only need to meet the frequency coupling; while the frequency components with three coupling and without any coupling will be suppressed and will not be shown in the 1.5-dimensional spectrum. Therefore, the 1.5-dimensional spectrum can detect the frequency components which only satisfy the secondary frequency coupling.
In figure (C, d), more detailed frequency components can be seen. The frequency components involved in coupling are shown in figure (c) and the frequency components generated by coupling are shown in figure (d). This situation is conducive to extracting the required wear fault features.