## 3 Vibration Analysis and Experiment at Operating Speed Range
### Electromagnetic Force Induced Vibration Analysis for the Operating Speed Range  
In this section, the electromagnetic force induced vibration characteristics of the variable-speed motor are analyzed using the 3-D entire FE model validated previously. The FE analysis results are validated in comparison with the experiment results. Then, the FE analysis results are evaluated over a range of 3500 Hz, at which the validation is performed by modal analysis. The conditions of the vibration experiment are determined in consideration of the measurement time and numerical analysis precision, as listed in Table 4.  
The electromagnetic-structural vibration coupled analysis method is used to analyze the vibration induced by the electromagnetic force of the motor. A transient analysis is performed by using the modal superposition method defined as Equations (5)-(7) [17].  
\[[M]\big{\{}\ddot{x}(t)\big{\}}+[K]\{x(t)\}=\{F(t)\}, \tag{5}\]  
\[\{x(t)\}=\sum_{j=1}^{k}\Phi_{j}y_{j}(t), \tag{6}\]  
\[[\Phi]^{T}[M][\Phi]\{\ddot{y}(t)\}+[\Phi]^{T}[K][\Phi]\{y(t)\}=[\Phi]^{T}\{F(t )\}, \tag{7}\]  
where, \([M]\) is a mass matrix, \([K]\) is a stiffness matrix, \(\{x\}\) is a displacement vector, \(\{F(t)\}\) is a force vector, \([\Phi]\) is a mode shape matrix, and \(\{y\}\) is a modal displacement.  
\begin{table}
\begin{tabular}{c c} \hline
**Item** & **Quantity** \\ \hline Frequency resolution & 5 Hz \\ The number of spectral lines & 1024 \\ Bandwidth & 5120 Hz \\ \hline \end{tabular}
\end{table}
Table 4: Vibration experiment conditions.  
Figure 11: Waterfall graph of electromagnetic force.  
Figure 10: 2-D FFT results of magnetic flux density at 2000 rpm.  
Also, the 3-D entire FE model validated in Section 2 and the electromagnetic force in the range of 200 to 2000 rpm calculated in Section 3.1 are used for the transient analysis. The vibration characteristics of the BLDC motor are evaluated over the range of operating speeds in the frequency domain by applying FFT to the transient analysis results.  
Waterfall graphs of the FE analysis and the experiment results are shown in Figures 12 and 13, respectively. As shown in Figure 12, a general vibration occurred around natural frequencies and electromagnetic force components of the 4\(k\)-order frequencies (\(k\) is a positive integer). In particular, at high operating speeds (1600 to 2000 rpm), the 32 nd-order electromagnetic force component overlapped with the region of the 1 st-order natural frequency, resulting in a large vibration. In addition, the 44 th-, 52 nd-, and 60 th-order electromagnetic force components overlapped with the region of the 1 st-order natural frequency at middle operating speeds (800 to 1600 rpm) and with the region of the 2 nd-order natural frequency at high operating speeds, resulting in a large vibration. These vibration characteristics of the BLDC motor, according to the operating speed, can also be found in the experiment results. It is confirmed that the vibration characteristics occurring in the actual motor can be predicted very accurately by using the 3-D entire FE model.  
On the other hand, the vibration characteristic calculated by using the electromagnetic-structural vibration coupled analysis on the operating speed range with the simplified FE model is shown in Figure 14. The simplified FE model consists of the motor housing and stator only [23; 27]. It is found that there was not any vibration response in the first natural frequency around 1000 Hz at which the first natural frequency of the motor is in bending mode caused by the axial mode of the rotor structure. Therefore, the analysis  
Figure 12: Waterfall graph of electromagnetic-structural vibration coupled analysis with 3-D entire FE model in the operating speed range.  
Figure 13: Waterfall graph of vibration experiment in the operating speed range.  
results using the simplified FE model cannot calculate the vibration characteristics caused by the interaction between the first natural frequency and the 44 th, 52 nd, and 60 th order of electromagnetic forces in the middle speed area.