# Optimum Slot and Pole Design for Vibration Reduction in Permanent Magnet Synchronous Motors
## 4 Discussion  
Using the calculations and the measurement of the natural frequencies in Section 3.1, it can be concluded that the natural frequencies of the stator are roughly the same for significantly different numbers of teeth. Nevertheless, it should be noted that the diameters of the stators and the volume of teeth and windings are almost the same. It is therefore possible to change the number of slots without altering the structural behaviour, if the dimensions of the stator core and the volume of teeth and windings remain the same. The equations given in Section 3.2 are proven to be a reliable tool to determine the harmonics of the vibration response with highest amplitudes (see Figure 11). In order to reduce vibrations, in addition to the maximisation of \(\gcd(Q_{s},2p)\), it may also be advantageous to maximise the LCM of the number of slots and the number of poles (\(\operatorname{lcm}(Q_{s},2p)\)). This is because a higher LCM means that a higher harmonic of the pressure is generating that harmonic, thus, the amplitude of that harmonic will be lower. Nonetheless, maximising \(\gcd(Q_{s},2p)\) and \(\operatorname{lcm}(Q_{s},2p)\) is not possible, because their product equals the  
Figure 11: Comparison of the vibration response of the machines of 18 slots and 8 pole pairs, and 48 slots and 8 pole pairs. Figure 10: Comparison of the vibration response of the machine with 48 slots and 8 pole pairs. product of \(Q_{s}\) and \(2p\). For this reason, a simple design rule to choose the best numbers of slots and poles to minimise vibrations could not be determined. Considering the results obtained (see Figure 8), it can be concluded that low GCD values and high LCM values concentrate vibrations in the harmonics that excite the lowest modes. The amplitudes of these peaks are generally higher as they correspond to a mode with a low natural frequency. In the contrary case, if GCD is high, LCM is low, and mode 0 becomes significant. Thus, the lowest mode  obtained with the GCD should not be excessively high. In 24p8, the amplitudes of the harmonics were quite similar, with 8 being the order of the lowest mode excited. In the case of 48p8, mode 0  dominated, because the lowest mode was too big (16) and became insignificant. With nonexcessively big GCD values, the amplitudes are similar, therefore,  the maximum amplitude obtained is minimised. Taking this into account, it is concluded that 24p8 is the best among the machines compared. Nevertheless, depending on the application, a certain frequency range must be avoided, thus, the machine should be designed to have a few dominant peaks out of that frequency range. In this case, the GCD or the LCM should be maximised. Depending on the harmonic amplified in each of those cases and the frequency range where low vibrations are desired, the most appropriate combination can be chosen. For instance, 18p8 with a low GCD (2), which means a high amplitude of the second harmonic, can be an optimum design. If the predominant harmonic is out of the frequency range where the vibration should be low, the vibratory behaviour of that machine would be good due to the low amplitudes of most harmonics. Machine 48p8 with a low LCM also presents high amplitudes in a small number of harmonics. In this case, the sixth is the predominant harmonic, but a good performance may be possible by keeping that harmonic out of the frequency range to be avoided. In some cases, the problem is noise and, in general, low frequencies are not the main contributors. Thus, 18p8 may be a good design as the second harmonic is the predominant one. In other applications, the problem is the transmission of the vibrations to the surrounding elements and low-frequency vibrations are the most difficult to attenuate; in that case, 48p8 may be a good alternative. 18p8 and 48p8 have been measured (see Figure 11), hence, the design rules proposed are experimentally validated. These rules could prove crucial for the early design stage in which the number of slots and pole pairs are defined.