PSI - Issue 36

Ihor Javorskyj et al. / Procedia Structural Integrity 36 (2022) 122–129 Ihor Javorskyj et al. / Structural Integrity Procedia 00 (2021) 000 – 000

127

6

(a) (c) Fig. 4 The amplitude spectra of the deterministic oscillations for three stages of gear damage (b)

The first harmonics of the deterministic part spectra can be interpreted as the order harmonics of the shaft rotation frequency; the twenty fifth harmonic corresponds to the first harmonic of the mesh frequency. The frequencies of the higher harmonics are linear combinations of the mesh and rotation frequencies. In the first stage, the amplitude of the mesh frequency harmonic is the largest. The further analysis of the gearbox conditions was performed on the basis of the vibration residues obtained by means of the extracting of the estimator of the PCRP’s mean function from raw signal, i.e. ( ) ( ) ( ) ˆ   = − t t m t . The graphs of the covariance function and the spectral density estimators of the vibration residuals are given in Fig. 5. The covariance function estimators have the form of slowly damped groups following one after another over the rotation period. These groups become clearly observable for the second (Fig. 5b) and the third (Fig. 5c) stages. As the lag increases, the estimators decay to low-power fluctuations, so we conclude that the deterministic oscillations have been fully extracted from the vibration signal. The comb-like forms of the spectral densities estimators indicate narrow-band modulation of the PCRP carrier harmonics of both the low- and high-frequency range. This means that the modulating processes can be represented in the form of the sum of the low- and high-frequency narrow-band components. These components can be explained in a respective Rice representation. It should be mentioned, that conclusions about the correlations or lack of correlations between these components within the low- and high-frequency domains can be established only on the basis of the results of PCRP analysis.

(a) (c) Fig. 5 The estimators of the covariance function of the stochastic part of vibration signals For the detection of the hidden periodicities of the second order, a formula similar to the one described above has to be applied. The LS functional of the covariance function has the form: (b)

1

=− +  K n K

ˆ

(

)

(

)

2 R nh jh , ,

,

F jh



=



,

(10)

2

2 1 K

where