# Virtual Reality Training Application for the Condition-Based Maintenance of Induction Motors
## 3 Theoretical Basis
### Fault-Related Frequency Components for Detecting Faults in Induction Motors and Gearboxes  
The analysis of fault-related frequency components has been considered the fundamental role of MCSA-based condition-monitoring approaches; in this way, it must be highlighted that the most common affectations that may damage an induction motor have characteristic patterns that are inherently induced by their occurrence. Consequently, when an induction motor operates under the influence of broken rotor bars, an affectation is produced that leads to an unbalance of the airgap magnetic flux because the current stops flowing through the broken rotor bar. Several theoretical studies have analyzed this effect and have established the fault-frequency components that facilitate the categorization of these characteristic frequencies according to Equation (1) [28]:  
\[f_{b}=(1\pm 2k\mathrm{s})f_{s},\;k=1,\;2,\;3,\;\ldots \tag{1}\]  
where \(f_{b}\) is the characteristic fault-related frequency component produced by the occurrence of broken rotor bars, \(f_{s}\) is the supply frequency, \(k\) is an integer value that can take values of \(k=1,\;2,\;3,\;\ldots\) and \(s\) is the unit slip that is estimated by Equations (2) and (3):  
\[s=\frac{n_{s}-n_{r}}{n_{s}} \tag{2}\]  
\[n_{s}=\frac{60f_{s}}{p} \tag{3}\]where \(n_{s}\) is the synchronous speed of the motor, \(n_{r}\) is the rotor speed and \(p\) is the number of pairs of poles. Thus, frequency harmonics will appear in an estimated current spectrum that can be appreciated as a side band harmonic around the supply frequency, \(f_{s}\): more precisely, first-order side-bands components may appear at (1) when \(k=1\) and these first-order components are the most significant with which to detect the fault occurrence associated with broken rotor bars. In addition, the two variables, rotational speed and load, affect the amplitude of this fault-related frequency, and the location of the sidebands can be slightly shifted outwards. On the other hand, the eccentricity problems occur due to an uneven airgap between stator and rotor. The problems associated with eccentricities have also been studied, and characteristic patterns that depict the fault-related frequency components have also been defined. In this regard, in the presence of eccentricity, low-frequency sidebands appear in the stator current spectrum given by Equation (4) [36]:  
\[f_{ccc}=f_{s}\bigg{[}1\pm k\bigg{(}\frac{1-s}{p}\bigg{)}\bigg{]},\ k=1,\ 2,\ 3,\ \ldots \tag{4}\]  
where \(f_{ccc}\) is the fault-related frequency component that depicts the occurrence of eccentricity problems, \(f_{s}\) is the supply frequency, \(s\) is the unit slip and \(k\) is an integer value that can take values of \(k=1,\ 2,\ 3,\ \ldots\) Thus, the dominant frequencies at low frequencies correspond to \(k=1\). The existence of these harmonics is due to the interaction between the two most common types of eccentricities: static and dynamic. Theoretically, the harmonics given by (4) will only occur when both eccentricities coexist. Similarly, the fault-related frequency components of the problems produced by misalignments where the induction motor shaft takes part of the coupling have been defined. Thus, the misalignment problem in induction motors may be detected through a spectral analysis following Equation (5) [37]:  
\[f_{mis}=f_{s}\bigg{[}1\pm k\bigg{(}\frac{1-s}{p}\bigg{)}\bigg{]},\ k=1,\ 2,\ 3,\ \ldots \tag{5}\]  
This can be rewritten as Equation (6):  
\[f_{mis}=f_{s}\pm f_{rm} \tag{6}\]  
where \(f_{mis}\) is the characteristic fault-related frequency component generated by misalignments, \(f_{s}\) is the supply frequency and \(f_{rm}\) is the fundamental rotational frequency of the misalignment shaft. Therefore, the misalignment condition may produce spectral modulations that are affected by \(f_{rm}\), and sidebands may appear. Moreover, over the past decade, MCSA has also been applied as a part of the condition assessment approaches to identify faults within rotating mechanical components such as gearboxes.