**5. Conclusions**

The novelty of this chapter is concerned with the applications of the wavelet analysis in two different new approaches:

Wavelet Analysis and Neural Networks for Bearing Fault Diagnosis 351

*s b*

2

*F D <sup>F</sup>*

*C*

(*Nb*), the theoretical balls (or rolling element) pass frequencies are:

contacts with a given race (inner or outer race), and given by:

*B*

Fig. A1. Basic frequencies and faults in a rolling element bearing.

*F*

2

2

**6.2 Ball Pass Frequencies (F***BPI,* **F***BPO***)** 

The inner race ball passes frequency (*FBPI*),

And the outer race ball passes frequency (*FBPO*),

**6.3 Ball Spins Frequency (***FB***)** 

1 cos

*D*

The rolling element (ball or roller) pass frequencies are the rate at which rolling elements pass a point on the track of the inner or outer race. Given the number of rolling elements

1 cos

 

1 cos

*s b BPO b p*

*<sup>F</sup> <sup>D</sup> F N D*

The ball (or roller) spin frequency is the frequency at which a point on the rolling element

<sup>2</sup> 1 ( cos ) <sup>2</sup> *p s b*

*b p*

*D D*

*F D D*

*s b BPI b p <sup>F</sup> <sup>D</sup> F N D*

*p*

(A-1)

(A-2)

(A-3)

(A-4)

	- Bearing fault detection through the evaluation of the wavelet envelope power spectrum.
	- Automatic bearing fault detection and diagnosis through the extraction of the input feature vectors to the NN classifier.

From the above wavelet applications the following points can be concluded:

