**6.1 Cage Frequency (***FC***)**

The rotational frequency of the cage can be expressed in terms of the pitch circle diameter (*Dp*), the diameter of the rolling element (*Db*) and the contact angle (*α*) as:

$$F\_{\mathbb{C}} = \frac{F\_s}{2} \left( 1 - \frac{D\_b}{D\_p} \cos \alpha \right) \tag{A-1}$$

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

350 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

 Firstly, the impulse wavelet is used as a de-noising technique to extract the fault pulses buried in the noisy signal for fault detection, and by evaluating the periodicity of these pulses through the calculation of the autocorrelation function the location of the fault

Bearing fault detection through the evaluation of the wavelet envelope power

Automatic bearing fault detection and diagnosis through the extraction of the input

a. The use of the wavelet analysis provides more information related to the bearing fault detection compared with the FFT frequency spectrum which can be used only for a stationary signal. Also the use of a shifted and scaled wavelet window over the analyzed signal produces better detection capabilities than that of the fixed size

b. The use of a wavelet base function with more similarity with the fault feature leads to enhance the wavelet analysis and generates wavelet coefficients with more information related to the bearing fault and as a result the efficiency of the fault diagnosis process can be increased. In this project the optimized *Impulse wavelet* and *Laplace wavelet* are

c. The use of more informative features as input vectors to the NN classifier can speed up the classification process and increase its accuracy by reducing the size of the NN

d. The bearing vibration signals obtained from the bearing simulation model that take into account the effects of the amplitude modulation and the slippage effects which are the main causes of non-stationary bearing signals, can be used to evaluate the performance of the proposed detection techniques with different simulated working conditions.

In general, the bearing inner race is attached to a shaft and therefore has the same rotational frequency as the shaft (*Fs*) while the outer race can be assumed stationary, since it is generally locked in place by an external casing (i.e. it has a constant rotational frequency of

The rotational frequency of the cage can be expressed in terms of the pitch circle diameter

zero). The bearing rotational frequencies can be obtained as follows (Figure 1-7):

(*Dp*), the diameter of the rolling element (*Db*) and the contact angle (*α*) as:

Compared with the previously conducted researches that used the normal time and/or frequency domain features as NN input vectors, the use of the wavelet analysis for feature extraction produces a most efficient classifier of the bearing faults with less input features. Furthermore, the use of the optimized wavelet and the most dominant wavelet coefficients in the feature extraction process leads to increase the accuracy and

used as new wavelet functions for fault detection and feature extraction.

through decrease of the input vectors and the hidden layers and nodes.

Secondly, the implementations of the complex Laplace wavelet for

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

feature vectors to the NN classifier.

can be identified.

spectrum.

window used in the STFT.

the success rate of the NN classifier.

**6.1 Cage Frequency (***FC***)** 

**6. Appendix (A): Bearing rotational frequencies** 

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

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

$$F\_{BPI} = \frac{F\_s}{2} \left( 1 + \frac{D\_b}{D\_p} \cos \alpha \right) N\_b \tag{A-2}$$

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

$$F\_{\rm BPO} = \frac{F\_s}{2} \cdot \left(1 - \frac{D\_b}{D\_p} \cos a\right) \cdot N\_b \tag{A-3}$$

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

The ball (or roller) spin frequency is the frequency at which a point on the rolling element contacts with a given race (inner or outer race), and given by:

$$F\_B = \frac{F\_s}{2} \quad \frac{D\_p}{D\_b} \quad \left(1 - \text{(}\frac{D\_b}{D\_p}\cos\alpha\text{)}^2\right) \tag{A-4}$$

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

**15** 

*Italy* 

Simone Delvecchio

*Engineering Department in Ferrara* 

**On the Use of Wavelet Transform** 

**for Practical Condition Monitoring Issues** 

Condition monitoring is used for extracting information from the vibro-acoustic signature of a machine to detect faults or to define its state of health. A change in the vibration signature not only indicates a change in machine conditions but also points directly to the source of

Fault diagnosis, condition monitoring and fault detection are different terms which are sometimes used improperly. Condition monitoring and fault detection refer to the evaluation of the state of a machine and the detection of an anomaly. Fault diagnosis could be set apart from other diagnoses since it is more rigorous and requires the type, size,

Due to their non-intrusive behaviour and use in diagnosing a wide range of mechanical faults, vibration monitoring techniques are commonly employed by machine manufacturers. Moreover, increases in computing power have helped the development and application of

Firstly, the monitoring procedure involves vibration signals to be acquired by means of accelerometers. Due to the selection of acquisition parameters being critical, the data acquisition step is not of minor importance. Sometimes, several steps, such as the correct separation of time histories, averaging and digital filtering is required in order to split the useful part of the signal from noise (electrical and mechanical), which is often present in

Secondly, signal processing techniques have to be implemented by taking into account the characteristics of the signal and the type of machine from which the signal is being measured (i.e. rotating or alternative machine with simple or complex mechanisms). In the final analysis, several features have to be extracted in order to assess the physical state of the

When the nature of the signal varies over time, repeating the Fourier analysis for consequent time segments could describe the temporal variation of the signal spectrum. This well known technique is called Short Time Fourier Transform (STFT). The principal limitations of

machine or to detect any incipient defects and determine their causes.

location and time of the detected faults to be determined.

**1. Introduction** 

the signal alteration.

signal processing techniques.

industrial environments.

this approach are:

#### **7. References**

