## 2 Modal Analysis and Test of BLDC Motors Considering all Motor Components
### Normal Mode Analysis and Test  
To validate the 3-D entire FE model, a normal mode analysis and modal test are performed. As shown in Figure 4, a digital acquisition (DAQ) system, an impact hammer, 1-axis and 3-axis accelerometers, and a fast Fourier transformation (FFT) analyzer is used in the modal test. The specifications of the test instruments are listed in Table 2. Conservatively considering the bandwidths of all instruments, the bandwidth of measurement is set to 3500 Hz. To measure the mode shape and natural frequency, impulse signals are given at 26 points on the surface, and the acceleration is measured as shown in Figure 5a. Figure 5b  
Figure 3: 3-D entire finite element model of the BLDC motor.  
Figure 2: 3-D entire geometry of the BLDC motor including stator core, rotor, rotor housing, permanent magnets, stator housing, fin, coil, and other various components.  
shows the type of accelerometer attached to the surface. To validate the 3-D entire FE model, the isotropic material properties are modified to obtain a correlation with the test results.  
The results of the modal test and FE analysis are listed in Table 3. The relative error is less than 3% for four natural frequencies within the bandwidth, which indicates that the FE analysis results are very accurate. Figure 6 shows the test and FE analysis results of the mode shape occurring at natural frequencies. The mode shapes computed by FE analysis are very similar to the test results. The second mode shape is due to wave-shaped deflections known as the circumference mode of the stator. This type of mode shape is able to be calculated using 2-D FE analysis.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Specifications**} & \multicolumn{2}{c}{**Accelerometer**} & \multirow{2}{*}{**Impact Hammer**} \\ \cline{2-3}  & **1-Axis** & & **3-Axis** \\ \hline Bandwidth (Hz) & 5000 & 5000 & 3500 \\ \hline Maximum measurement deviation (\%) & 2.2 & 3.6 & - \\ \hline \end{tabular}
\end{table}
Table 2: Specifications of the equipment.  
Figure 4: Equipment for the impact hammer test: (**a**) digital acquisition (DAQ) system and impact hammer; (**b**) single-axial (left), and tri-axial (right) accelerometer.  
Figure 5: Modal test of the BLDC motor. (**a**) sensors attached to the BLDC motor; (**b**) sensor attachment points.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Mode**} & **FE Analysis Results (Hz)** & **Test Results (Hz)** & **Relative Error (\%)** \\ \hline
1 & 1010 & 1004 & 0.6 \\
2 & 1801 & 1810 & \(-\)0.5 \\
3 & 2121 & 2122 & 0 \\
4 & 3191 & 3284 & \(-\)2.9 \\ \hline \end{tabular}
\end{table}
Table 3: Comparison of the natural frequencies between the FE analysis and modal test.  
However, it can be seen that the first-order mode shape is the bending mode of the front housing generated by the rigid body mode of the rotor. In addition, the third-and fourth-order mode shapes are bending and torsion modes of the entire structure. These mode shapes can only be computed by using a 3-D entire FE model, not a 2-D or simplified FE model or an analytical model.