# A Low-Cost, Small-Size, and Bluetooth-Connected Module to Detect Faults in Rolling Bearings
## 2 Evaluation of the Frequency Spectrum  
### Fundamental Frequencies of Rolling Bearings  
Rolling bearing faults are typically due to misalignment, lack of lubrication, or excessive load. A crack may origin at subsurface or at surface level, and gradually increases up to become a spall, i.e., a groove removal from the surface of the inner or the outer race. In such a case, the rolling bearing is definitely damaged and it has to be substituted. When the defect is localized to the inner or outer ring, the rolling elements, i.e., the rollers or the balls, impact on it at a given frequency, which depends on the bearing geometry and on the rotation frequency. The ball pass frequency of the outer race (\(BPFO\)) and the ball-pass frequency of the inner race (\(BPFI\)) and can be evaluated as [23]  
\[BPFO=\frac{Zf_{r}}{2}\left(1-\frac{d}{D}cos\alpha\right) \tag{1}\]  
\[BPFI=\frac{Zf_{r}}{2}\left(1+\frac{d}{D}cos\alpha\right), \tag{2}\]  
where \(Z\) is the number of rolling elements, \(f_{r}\) is the inner rotation frequency, \(d\) is the rolling element diameter, \(D\) is the pitch diameter, and \(\alpha\) is the contact angle. Similarly, if the defect is located on the cage or on a rolling-element, it will occur every Fundamental Train Frequency (\(FTF\)) or Ball Spin Frequency (\(BSF\)), respectively. Similarly, such frequencies can be evaluated as  
\[FTF=\frac{f_{r}}{2}\left(1-\frac{d}{D}cos\alpha\right), \tag{3}\]  
\[BSF=\frac{D}{2d}\left(1-\left(\frac{d}{D}cos\alpha\right)^{2}\right). \tag{4}\]  
The impact of a ball or a roller on a defect generates an impulsive transient, which will be reflected on both the vibrations and on the acoustic signal delivered by the bearing. Such an impulsive event excites the resonant frequencies of the structure, and, if the operating conditions are unchanged, it is cyclically repeated every \(BPFI\), \(BPFO\), \(FTF\), or \(BSF\), depending on the location of the defect. Thus, the more convenient way to check whether a defect is present or not, is to verify the amplitude of the frequency spectrum of the acquired signal at the defect fundamental frequencies. It can be noticed that such defect fundamental frequencies are valid both for ball and roller bearings [24]. Indeed, when a defect arises, the amplitude of the spectrum at the corresponding frequency will increase. On the other hand, it is possible to band-pass filter the signal, evaluate its envelope signal and the corresponding frequency spectrum, and finally to check the amplitudes at the same fundamental frequencies. Indeed, the frequency spectrum evaluation is the most expensive computational step, thus it should be carefully optimize referring to the target application. As such data processing must be performed by the module itself, and that dedicated accelerating hardware, such as FPGA or Application Specific Integrated Circuit (ASIC), cannot be exploited to keep the cost of the developed module low, a microcontroller should be used. A discussion on how to reduce the computational effort of the evaluation of the frequency spectrum is presented in the remainder of this section.