# Dynamic Characteristic Analysis of Permanent Magnet Brushless DC Motor System with Rolling Rotor
## 3 Dynamic Response of the BLDCM System
### Analysis of the Effects of Rotational Speed  
For a motor system, rotational speed \(\Omega\) is one of the key physical parameters. Thus, this subsection mainly studies the effect of rotational speed on the dynamic response of the motor system, in which the vibration responses of the rotor and stator are taken for example. During normal operation conditions, the rotational speed of the motor system is in the range of 1200 r/min \(<\Omega<\) 9000 r/min, i.e., 20 rps \(<\Omega<\) 150 rps. Other system parameters remain unchanged and mainly include the bearing backlash \(b=5\times 10^{-5}\) m and the eccentric distance of the rolling rotor \(\rho\) = 1 \(\times\) 10\({}^{-3}\) m. Under three different rotational speeds, namely \(\Omega\) = 100 rps, 120 rps and 150 rps, the corresponding vibration responses along the \(y\) direction are shown in Figures 2-4, where subfigures a-d and subfigures e-h indicate the dynamic features of the rotor and stator, respectively. At \(\Omega\) = 100 rps, the rotor presents multi-period motion, which can be seen in Figure 2a,d. The frequency spectrum shows two main harmonic components in Figure 2b, i.e., the rotational frequency (\(f_{\mathrm{r}}\) = 100 Hz) and the multiplication frequency (\(3f_{\mathrm{r}}\) = 300 Hz). This is due to the eccentricity of the rolling rotor and the electromagnetic force. The response amplitude at \(3f_{\mathrm{r}}\) dominates, the amplitude at \(f_{\mathrm{r}}\) is second largest, and the amplitude at other frequencies is negligible. The axis trajectory diagram in Figure 2c shows a complicated closed non-circular \begin{table}
\begin{tabular}{l c} \hline
**Parameters** & **Value** \\ \hline Rated power (kw) & 1.5 \\ Mass of rotor \(m_{1}\) (kg) & 0.84 \\ Mass of rolling rotor \(m_{2}\) (kg) & 0.23 \\ Mass of left shaft neck \(m_{3f}\) (kg) & 0.05 \\ Mass of left sliding bearing \(m_{4f}\) (kg) & 0.19 \\ Mass of left support \(m_{3f}\) (kg) & 0.89 \\ Mass of shell \(m_{6}\) (kg) & 5.33 \\ Stiffness \(k_{1,2}\) (N/m) & \(3.07\times 10^{5}\) \\ Stiffness \(k_{4r,\ 4l}\) (N/m) & \(1.7\times 10^{5}\) \\ Stiffness \(k_{5r,\ 5l}\) (N/m) & \(3.6\times 10^{5}\) \\ Stiffness \(k_{6}\) (N/m) & \(2.8\times 10^{5}\) \\ Damping \(c_{1,2}\) (N (m-s)) & 800 \\ Damping \(c_{4r,4l}\) (N/(m-s)) & 660 \\ Damping \(c_{5r,\ 5l}\) (N/(m-s)) & 540 \\ Damping \(c_{6}\) (N/(m-s)) & 600 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of the BLDCM system. curve, which is caused by the combined action of the eccentric force and electromagnetic force. When \(\Omega\) increases to 120 rps, the vibration amplitude of the rotor reduces obviously in Figure 3a. However, the components of the frequency spectrum in Figure 3b are unchanged, where 3\(f_{\rm r}\) (360 Hz) is still dominant. Compared to the axis trajectory diagram in Figure 2c, the range of the axis trajectory diagram in Figure 3c decreases. Although the motion state in Figure 3d is similar to that in Figure 2d, the system under \(\Omega\) = 120 rps shows smaller vibration response. However, when \(\Omega\) further increases to 150 rps, the rotational frequency (\(f_{\rm r}\) = 150 Hz) in Figure 4b becomes the main response since the eccentric force increases. The corresponding axis trajectory diagram also shows a simpler curve in Figure 4c. This reveals that there is a better rotational speed to make the motor system more stable. Figure 3: Dynamic responses of the rotor (blue curves) and stator (red curves) at \(\Omega\) = 120 rps: (**a**) time history of the rotor; (**b**) frequency plot of the rotor; (**c**) rotor axis locus of the rotor; (**d**) phase portrait of the rotor; (**e**) time history of the stator; (**f**) frequency plot of the stator; (**g**) rotor axis locus of the stator; and (**h**) phase portrait of the stator.