# Measurement of the Speed of Induction Motors Based on Vibration with a Smartphone
## 3 Results
### Motor Fed from a Inverter Coupled to a DC Generator  
In this configuration, a frequency converter was used to power the induction motor at frequencies from 30 Hz to 80 Hz. The motor shaft was coupled to a DC generator, feeding an adjustable resistance, acting as variable mechanical load for the motor. Figure 9a shows the test sketch, and Figure 9b shows six RMS spectra of the vibration signals, \(\mid\)F(\(j\)\(\cdot\)\(\omega\))\(\mid\), when the IM was fed at 400 V 45 Hz by the converter corresponding to the rotor speeds from 2369 r/min to 2673 r/min. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Resistance**} & \multicolumn{2}{c}{**Experimental**} & \multicolumn{2}{c}{**Algorithm**} & \multicolumn{2}{c}{**Relative Error**} \\ \cline{2-7}  & \(\Omega_{m}\) **(r/min)** & \(\omega_{m}\) **(Hz)** & \(\Omega\) **(r/min)** & \(\omega_{F}\) **(Hz)** & \(\varepsilon\) **(\%)** \\ \hline R7 & 2857 & 47.62 & 2859.4 & 47.66 & 0.0831 \\ R6 & 2872 & 47.87 & 2871.1 & 47.85 & 0.0316 \\ R5 & 2900 & 48.33 & 2900.4 & 48.34 & 0.0135 \\ R4 & 2920 & 48.67 & 2920.9 & 48.68 & 0.0308 \\ R3 & 2942 & 49.03 & 2941.4 & 49.02 & 0.0202 \\ R2 & 2962 & 49.37 & 2961.9 & 49.37 & 0.0029 \\ R1 & 2969 & 49.48 & 2970.7 & 49.51 & 0.0574 \\ R0 & 2990 & 49.83 & 2991.2 & 49.85 & 0.0405 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Motor fed from the mains at 400 V, 50 Hz, and coupled to a DC generator. Measured and calculated values of rotor speed and frequencies, as well as relative errors. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Resistance**} & \multirow{2}{*}{**Time Recording (s)**} & \multicolumn{3}{c}{**Relative Error, \(\varepsilon\) (\%)**} \\ \cline{3-5}  & & \multicolumn{3}{c}{**Smartphone Locations over the Motor**} \\ \cline{3-5}  & & _Pos1_ & _Pos2_ & _Pos3_ \\ \hline R7 & 5 & 0.1130 & 0.0831 & 0.0131 \\ R7 & 10 & 0.1130 & 0.0831 & 0.0131 \\  & 15 & 0.0106 & 0.0194 & 0.0131 \\ \hline R4 & 5 & 0.0696 & 0.0696 & 0.0626 \\ R4 & 10 & 0.1311 & 0.0308 & 0.0626 \\  & 15 & 0.0308 & 0.0308 & 0.0377 \\ \hline R1 & 5 & 0.0911 & 0.0574 & 0.0237 \\ R1 & 10 & 0.0911 & 0.0574 & 0.0237 \\  & 15 & 0.0911 & 0.0574 & 0.0237 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Motor fed from the mains at 400 V, 50 Hz, and a DC generator coupled to the motor shaft. Relative errors corresponding to different smartphone positions and different values of the resistance load and time recording length. Table 3 shows the results of the measured rotor speeds for frequencies of the power supply ranging from 30 Hz to 80 Hz for various mechanical loads of the motor. For each frequency, as the mechanical load decreased (R7\(\rightarrow\)R0), the rotor speed increased, as expected. Tables 4-6 summarise the results for 35 Hz, 60 Hz, and 70 Hz, respectively. Although a full set of vibration recordings was made in all the cases reported in Table 3 (including other frequencies, several positions of the smartphone, and various recording time durations), only the results corresponding to 35 Hz, 60 Hz, and 70 Hz were selected to be shown in this section to illustrate the quality of the experimental results. Each of these tables shows the measured rotor speed, \(\Omega_{m}\), the rotor shaft frequency, \(\omega_{m}\), the estimated rotor speed, \(\Omega\), the estimated rotor shaft frequency, \(\omega_{F}\), and finally, the relative errors.