## 2 Bearing Diagnostics
### Fault Signal Given by Amplitude Modulation  
Whenever the rolling element in the bearing strikes a local fault in the inner or outer race or vice versa during operation, an impact signal is produced at the period of one of the bearing fault frequencies, such as Ball Pass Frequency Outer race (_BPFO_), Ball Pass Frequency Inner race (_BPFI_), Fundamental Train Frequency (_FTF_), and Ball Spin Frequency (_BSF_), which are defined as follows [15].  
\[BPFO = \frac{nf_{s}}{2}\left\{1+\frac{d}{D}cos\alpha\right\} \tag{1}\] \[BPFI = \frac{nf_{s}}{2}\left\{1-\frac{d}{D}cos\alpha\right\}\] (2) \[FTF = \frac{f_{s}}{2}\left\{1-\frac{d}{D}cos\alpha\right\}\] (3) \[BSF = \frac{Df_{s}}{d}\left\{1-\left(\frac{d}{D}cos\alpha\right)^{2}\right\} \tag{4}\]  
where \(D\), \(d\), \(\alpha\), \(n\) and \(f_{s}\) are the pitch diameter, ball diameter, contact angle between the ball and the cage, number of rolling elements, and the rotating speed of bearing (Hz), respectively. Usually, the impact signals at these fault frequencies are modulated by the much higher resonant frequencies of the bearings, which results in the signal like the one in Figure 1, the exponentially decaying ringing occurring periodically at the fault frequencies [16].  
This phenomenon is mathematically the same as the amplitude modulation (AM), of which the concept is illustrated by a simple example in Figure 2. The fault and the resonance signal of the bearings are assumed to be Figure 2a,b which are the modulating and carrier signal in the signal processing theory, respectively. Then, the AM signal becomes the carrier signal with variable amplitude given by the modulating signal as shown in Figure 2c. After taking the FFT, the frequency domains of modulating, carrier, and the AM signals are given in Figure 2e-g, respectively. The result shows that the AM spectrum has peaks at the resonance (carrier) frequency (\(f_{r}\)) with the sidebands spaced apart by the fault (modulating) frequency (\(f_{f}\)) [17]. The AM signal can be expressed by the following equation [18]:  
\[x(t)\ =\ [1+f(t)]\times r(t)\ =\ \left\{1+Msin\left(2\pi f_{r}t\right)\right\}Acos \left(2\pi f_{r}t\right) \tag{5}\]  
where \(f(t)\) is the fault signal (plus DC offset to ensure that the mean amplitude of AM signals is zero) with the amplitude \(M\) and frequency \(f_{f}\), \(r(t)\) is the resonance signal with the amplitude \(A\) and frequency \(f_{r}\), and \(x(t)\) is the AM signal. Equation (5) can be transformed to  
\[x(t)\ =Acos(2\pi f_{r}t)+\frac{A\cdot M}{2}\left[sin2\pi\left(f_{r}+f_{f} \right)t-sin2\pi\left(f_{r}-f_{f}\right)t\right] \tag{6}\]  
Figure 1: Bearing geometry and impact signal.  
which shows that the spectrum of the AM signals consists of three components \(f_{r}\), \(f_{r}\pm f_{f}\), with the amplitudes \(A\) and \(A\cdot M/2\), respectively, as shown in Figure 2f. Therefore, it can be concluded that, when a bearing has fault, it appears as the sideband around the resonance signal. Then, the remaining task is to extract or separate the original fault signal Figure 2a or its FFT (e) from the AM signal Figure 2c or its FFT (g). This is realized by the envelope analysis, which is an amplitude demodulation technique as will be addressed in the subsequent section.  
Figure 2: Amplitude modulation and envelope signal: (**a**) modulating signal, (**b**) carrier signal, (**c**) AM signal, (**d**) envelope signal, in the time domain; (**e**) modulating signal, (**f**) carrier signal, (**g**) AM signal, (**h**) envelope signal, in the frequency domain.