# Measurement of the Speed of Induction Motors Based on Vibration with a Smartphone
## 4 Discussion
### Extended Method  
The extended method only requires the analysis of the spectra of the vibration signals, \(F_{REF}(\omega)\), corresponding to each reference axis, \(REF=\{X,Y,Z\}\). The mechanical frequency of the rotor, \(\omega_{F}\), is now identified as the frequency, \(\omega_{F\_{REF}}\), corresponding to the maximum of the peaks of the amplitudes of the three axial vibration signal spectra  
\[\omega_{F}=\omega_{F-REF}:\ |F_{REF}(\omega_{F-REF})|=MAX|F_{REF}(\omega)|,\ REF=\{X,Y,Z\}. \tag{6}\]  
Finally, the rotor speed, \(\Omega\), was calculated by means of (4) as previously. Figure 10 summarises the flow chart of the extended method.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Motor Rated** & \multicolumn{3}{c}{**Measurement**} & **Estimate** & **Error** \\ \cline{2-6}
**Power** & **Load** & \(\Omega_{m}\) & \(\omega_{m}\) & \(\Omega\) & \(\varepsilon\) \\  & **Condition** & **(r/min)** & **(Hz)** & **(r/min)** & **(\%)** \\ \hline
15 kW & No load & 1502.3 & 25.01 & 1500.62 & 0.11 \\  & Full load & 1454.0 & 24.37 & 1461.97 & 0.55 \\  & No load & 1498.0 & 24.95 & 1496.98 & 0.07 \\  & Full load & 1463.0 & 24.41 & 1464.36 & 0.09 \\  & No load & 1491.0 & 24.82 & 1489.08 & 0.13 \\  & Full load & 1396.0 & 23.20 & 1392.20 & 0.27 \\ \hline \end{tabular}
\end{table}
Table 7: Motor speed estimation with the motor fed from the mains (380 V, 50 Hz). Vibration signal sampled at 10 kHz for 10 s. Results from [26].  
As an example of the application of the extended method, a recording of the axial components of the motor vibration was analysed step by step. The case corresponds to a motor speed of \(\Omega_{m}=1581\) r/min (\(\omega_{m}=26.35\) Hz).  
As can be seen in Figure 11, the RMS value of the vibration spectrum shows a peak value of \(\mid F_{MAX}(\omega)\mid\) = MAX\(\mid\)\(\mid\)\(F(\omega_{F\_RMS}=73.8770\) Hz) \(\mid\) = 93.2191 p.u., which is placed at a frequency of \(\omega_{F\_RMS}=73.8770\) Hz. This is one of the rare cases where, using (3), the frequency corresponding to the highest peak of the RMS value of the spectrum does not match the mechanical frequency of the rotor shaft, \(\omega_{F\_RMS}=73.8770\) Hz \(\
eq\omega_{m}=26.35\) Hz.  
On the other hand, Figure 12 and Table 8 show that the peak value of the amplitude of the \(X\) component of the vibration signal spectrum is \(\mid\)\(F_{X\_MAX}(\omega)\mid\) = MAX\(\mid\)\(\mid\)\(F_{X}(\omega)\mid\) = \(\mid\)\(F_{X}(\omega_{F\_X}=26.3184\) Hz) \(\mid\) = 84.3203 p.u. Figure 12 and Table 8 also show that the maximum peak value of the amplitudes of the three axial vibration signal spectra corresponds to the X axis since, \(\mid\)\(F_{X}(\omega_{F\_X}=26.3184\) Hz) \(\mid\) = 84.3203 p.u. = MAX\(\mid\)\(\mid\)\(F_{X\_MAX}(\omega)\mid\) = 84.3203 p.u., \(\mid\)\(F_{Y\_MAX}(\omega)\mid\) = 55.0815 p.u., and \(\mid\)\(F_{Z\_MAX}(\omega)\mid\) = 69.0521 p.u. = 84.3203 p.u. That maximum peak takes place at \(\omega_{F\_X}=26.3184\) Hz. As a result, and according to (6), the mechanical frequency is taken as \(\omega_{F\_X}=\omega_{F\_X}=26.3184\) Hz. Finally, using (4), the motor speed results in \(\Omega=1579.10\) r/min, which agrees with the tachometers (\(\varepsilon=0.1199\%\)).  
As can be seen, the extended method, using the same data source (the spectrum of the axial components of the motor vibration), is able to circumvent the difficulties in the identification of the mechanical frequency that occurs when using the RMS value of the vibration spectrum, leading to a proper calculation of the motor speed.  
Figure 11: Frequency spectrum of the RMS value of the motor vibration at a rotor speed of \(\Omega_{m}=1581\) r/min (\(\omega_{m}=26.35\) Hz).  
Figure 10: Flow chart of the extended method.