# 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 the Bearing Clearance  
In rotating equipment, backlash, such as gear backlash and bearing clearance, easily leads to complex dynamic behaviors of the system [28,29]. Therefore, it is vital to analyze the effect of bearing clearance on motor systems. As the control parameter, three bearing clearances, b = 3 \(\times\) 10\({}^{-5}\) m, 8 \(\times\) 10\({}^{-5}\) m and 12 \(\times\) 10\({}^{-5}\) m, are chosen. The other system parameters are as follows: rotational speed \(\Omega\) = 100 rps, and eccentric distance of the rolling rotor \(\rho\) = 1 \(\times\) 10\({}^{-3}\) m. The corresponding dynamic responses are shown in Figures 7-9. Meanwhile, Figure 2 presents the vibration response when b = 5 \(\times\) 10\({}^{-5}\) m.  
Figure 6: Dynamic responses of the rotor (blue curves) and stator (red curves) at \(\rho\) = 5 \(\times\) 10\({}^{-3}\) m: (**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.  
Figure 8: Dynamic responses of the rotor (blue curves) and stator (red curves) at b = 8 \(\times\) 10\({}^{-5}\) m: (**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.  
Figure 7: Dynamic responses of the rotor (blue curves) and stator (red curves) at b = 3 \(\times\) 10\({}^{-5}\) m: (**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.  
Figure 9: Dynamic responses of the rotor (blue curves) and stator (red curves) at b = 12 \(\times\) 10\({}^{-5}\) m: (**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.  
When the bearing clearance increases from 3 \(\times\) 10\({}^{-5}\) m to 12 \(\times\) 10\({}^{-5}\) m, the spectral components of the rotor and stator are unchanged in the frequency spectrums of Figures 2b,f, 7b,f, 8b,f and 9b,f. For example, the multiplication frequency (\(3f_{r}\)) in the frequency plot of the rotor is the main response, and the rotational frequency (\(f_{r}\)) of the stator is the dominant response. The numerical results indicate that varying bearing clearance has no effect on the eccentric force and electromagnetic force. However, when the bearing clearance increases, the amplitudes of the vibration response of the rotor decrease gradually, and those of the stator increase. Specifically, the corresponding response amplitudes of the rotor are 0.126 mm, 0.122 mm, 0.119 mm and 0.113 mm, and those of the stator are 0.076 mm, 0.078 mm, 0.085 mm and 0.095 mm, respectively. Similarly, as the bearing clearance increases, the axis trajectory diagrams of the rotor become smaller, whereas the trajectory ranges of the stator gradually increase. According to the above analysis, to keep the better running performance of the rotor and the stator simultaneously, the bearing clearance should be selected properly.  
In addition, with different bearing clearances, the relative 3D dynamic response curves are shown in Appendix B.