# Optimum Slot and Pole Design for Vibration Reduction in Permanent Magnet Synchronous Motors
## 3 Results
### Analysis of the Influence of the Number of Slots and the Number of Poles on the Vibration Response  
Equations (2)-(4) determined that the spatial orders of the pressure waves are \(kp\) and \(kp\pm v\mathcal{Q}_{s}\) (see Section 2.1) and, in accordance with the model shown in Section 2.2, the mode excited is of the same spatial order (\(n\)) as the pressure wave (see Equation (16)). Figure 7 shows the modes of the lowest radial-tangential orders in the radial-tangential plane. In order to identify the modes that produce a higher vibration, the lowest natural frequencies obtained by the analytical model proposed in [29] are listed in Table 5.  
The above table shows that the modes of lower spatial orders have lower natural frequencies. Nevertheless, the frequencies do not increase with the order when orders 0 and 1 are considered. Hence, the first aspect to take into account is the lowest spatial order excited \(n_{min}\), which is obtained by the Greatest Common Divider (GCD) of the number of slots (\(Q_{s}\)) and the number of poles (\(2p\)) [13; 18; 42]. The Bezout's identity gives \(n_{min}=\gcd(Q_{s},2p)\) from \(n=ip\pm jQ_{s}\). Therefore, it is suggested that \(\gcd(Q_{s},2p)\) should be maximised in order to reduce vibrations and noise [13; 18]. However, not only does the mode order excited has to be taken into account. The amplitude of the harmonic exciting that mode must also be considered, which is of lower amplitude for higher harmonics. In addition, the radial forces of the 0 th spatial order are often significant, as is the vibration response that they may also induce [42]. Therefore, the main vibration peaks arise when the lowest vibration mode order (\(n_{min}\)) and the 0 th mode order (breathing mode) are excited. Thus, by taking into account the harmonics which excite those mode orders, it is possible to determine the harmonics which induce the highest vibrations. The harmonics of the magnetic pressure \(k\) and the slots \(v\) that induce the highest vibration peaks can be calculated by the equation \(n=|kp\pm v\mathcal{Q}_{s}|\):  
* The harmonics that excite the lowest order \(n=\gcd(Q_{s},2p)\) are obtained finding the integers \(k\) that fulfil the equation \(\gcd(Q_{s},2p)=|kp\pm v\mathcal{Q}_{s}|\), where \(k\) the harmonic of the pressure due to the magnetomotive force can only be even. * The harmonics that induce 0 th order force, where \(n=0\), are those that fulfill \(kp=v\mathcal{Q}_{s}\), which is calculated with the value of the LCM. Therefore, \(k\) and \(v\) can be obtained from \(\operatorname{lcm}(Q_{s},2p)=kp\) and \(\operatorname{lcm}(Q_{s},2p)=v\mathcal{Q}_{s}\). To determine the design guidelines, a number of motors of the same dimensions but different numbers of poles and slots were compared. The width of the teeth and the slots, and the angle of the poles were changed in accordance with the number of slots and poles, to maintain the same proportions. The machines were named by the number of slots and  
Figure 7: Lowest four mode orders. The deformed mode shape is indicated in red and the undeformed mode shape in blue. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Mode (n,m)** & **(2,1)** & **(1,1)** & **(3,1)** & **(4,1)** & **(5,1)** & **(6,1)** & **(0,1)** \\ \hline Frequency (Hz) & 675.7 & 700.3 & 1144.5 & 1614.1 & 1881.1 & 2169.4 & 2263.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Natural frequencies of the lowest radial mode orders of the assembly of the stator and windings. the number of pole pairs separated by \(p\). In Table 6 the GCD and LCM of each machine are set out, as well as the smallest harmonics of the pressure due to the magnetomotive force and slotting. Table 6 shows that the zero-order harmonic may dominate in the machines 24p8 and 48p8. The lowest mode orders excited by those machines are 8 and 16, respectively, whose natural frequencies are significantly higher than that of the mode of 0 spatial order. The amplitude of the pressure of 0 spatial order is linked to the LCM value, which in this case is also the same.