# A Real-Time Novelty Recognition Framework Based on Machine Learning for Fault Detection
## 4 Proof of Concept--Digital Model
### Simulations  
Numerous simulations are performed to represent different fault cases. The chosen tests performed on the framework are the following:  
* Low Lubrication: a lubrication degradation condition is simulated inside bearings. This situation is represented by increasing the variance range of the impulse's force acting on the shaft. Two accelerometers are placed on the bearings and on the shaft to measure vibrations emitted and transmitted. Hence, in every simulation, the accelerometers measure an increase in general noise with a reduction of rotational viscous damping around z. The parameters change in each simulation in the same manner for the left and the right bearing. It can imagine that every simulation has different time instants in which a maintainer takes data from the machine to monitor; for this reason, each simulation is delayed by one day. This case is represented in Figure 12a,b in which the lubricated and not completely lubricated bearings are shown. The vibrations shown are measured on the right bearing:
* Inner Race Faults: as reported in Ref. [27], a general rolling bearing has four standard types of fault that can occur over time; these faults are \"translated\" as frequencies with the following names and shapes:
* Ballpass frequency outer race (BPFO), a fail occurs in the bearing's outer race:
* Ballpass frequency inner race (BPFI): a fail occurs in the bearing's inner race:
* Fundamental train frequency (FTF): a fail occurs in the bearing's cage:
* Ball spin frequency (BSF): a fail occurs in the bearing's rolling elements:
* \(BSF=\frac{D}{2d}(1-(\frac{d}{D}cos\phi)^{2})\) where \(f_{r}\) is the shaft speed, \(n\) is the number of spheres, \(\phi\) is the angle of the load with respect to the radial plane, \(d\) is the sphere diameter, and \(D\) is the bearing's radius between two rings. For the proof-of-concept of this paper and to test the framework in different situations, an Inner Race Fault is simulated, as shown in Figure 13a,b. Hence, this kind of simulation works similarly to the low lubrication one with the difference in the force application. In the low lubrication case, the force's impulse acts with a random amplitude at high frequency, while in this case, it acts with a period depending on the BPFI computed with the above formula. Similar to the low lubrication case, the force's amplitude increases in each simulation, reproducing a failure's worsening. The initial and final simulation gives the following results:
The framework is tested for both simulations to understand how it works. Figure 14a represents the framework application on the low lubrication data, while Figure 14b represents the application on the BPFI fault:  
Figure 14: (**a**) Prediction with low lubrication fault; (**b**) Prediction with BPFI bearing fault. Figure 12: (**a**) Lubricated bearing noise; (**b**) Low lubricated bearing noise. Figure 13: (**a**) Bearing without fault; (**b**) Bearing with Inner Race Fault. The blue dashed line represents the result obtained from the AIU, while the red lines represent the predictions computed by the MCU. Starting from Figure 13(a), the MCU is applied to the low lubrication case. An increase in the noise generated by the model corresponds to an increase in the error computed by the AIU, subjecting this to a substantial difference compared to the elements trained at the beginning. The parameters set for this evaluation are the following (Table 3):  
The average value gives the \"speed\" of the prediction: an MCU with a small average creates predictions that are able to better follow the real system's volatility; a high number of the average corresponds to a slower prediction response, compensating for the volatility and outliers of the real measures. The configuration used for the digital model allows for fast predictions that are able to follow the error trend. As reported in the previous paragraph, each predictor monitors a window computed after the preprocessing stage.