# Optimum Slot and Pole Design for Vibration Reduction in Permanent Magnet Synchronous Motors
## 2 Materials and Methods
### Analytical Calculation of the Vibration Response  
To analyse the effect of the design parameters, analytical calculations need to be considered. The analytical approach has proven to be a good alternative in terms of accuracy, as observed in [28; 29]. As shown in Section 2.1, the electromagnetic forces are a sum of harmonic pressure waves [30]. Thus, modal superposition is applied, which involves the summation of the responses of a significant number of modes of vibration. The amplitude of vibration for each mode is determined by the excitation frequencies and natural frequencies, and on the modal participation factor. This, in turn, is subjected to the mode shapes and the pressure distributions [29; 31; 32]. Consequently, the mode shapes and natural frequencies of the system are required, and can be calculated either by FE calculations or analytically. The vibration response for the radial direction is  
\[u_{r}(x,\theta,t)=\sum_{i=1}^{\infty}\eta_{i}(t)U_{ri}(x,\theta), \tag{6}\]  
where \(\eta_{i}(t)\) is the modal participation factor and \(U_{ri}(x,\theta)\) is the radial mode shape of the \(i\)th mode. As the excitation force is a sum of harmonic pressure waves (see Section 2.1), the steady-state response is also a sum of harmonic responses with a phase lag (\(\phi_{i}\)) with respect to the force  
\[\eta_{i}(t)=\sum_{k=1}^{\infty}\Lambda_{ki}e^{i(\omega_{k}t-\phi_{i})}, \tag{7}\]  
where the amplitude of the response is  
\[\Lambda_{ki}=\frac{F_{ki}}{\sqrt{(\omega_{i}^{2}-\omega_{k}^{2})^{2}+4\xi_{i}^ {2}\omega_{i}^{2}\omega_{k}^{2}}}, \tag{8}\]  
where \(\omega_{i}\) is the natural frequency, \(\omega_{k}\) is the frequency of the \(k\)th pressure wave, \(\xi_{i}\) is the modal damping of mode \(i\), and \(F_{ki}\) is the modal force of the \(k\)th pressure wave and \(i\)th mode. To account for axial and radial-tangential mode shapes, the subscripts \(i\) are changed to \(nm\), with \(n\) denoting the radial-tangential mode order and \(m\) the axial mode order. Considering the stator as a cylinder [29; 31; 32], for the radial pressure \(q_{kr}\) and the radial mode shape \(U_{nmr}\), the modal force is obtained as  
\[F_{knm}=\frac{1}{\rho hC_{nm}}\int_{0}^{l}\int_{0}^{2\pi}q_{kr}U_{nmr}a\mathrm{ d}\theta\mathrm{d}x, \tag{9}\]  
\[C_{nm}=\int_{0}^{l}\int_{0}^{2\pi}(U_{nmr}^{2}+U_{nm\theta}^{2}+U_{nmr}^{2})a \mathrm{d}\theta\mathrm{d}x, \tag{10}\]  
where \(U_{nmr}\), \(U_{nm\theta}\), and \(U_{nmr}\) are the mode components of the radial, tangential, and axial vibrations, and \(\rho\), \(h\), and \(a\) the equivalent density, thickness, and radius of the stator [29; 31; 32]. The modal shape functions can be assumed to be the same as for a cylindrical shell:  
\[U_{nmr}=R\cos(n\theta)\cos\Big{(}\frac{\pi mx}{l}\Big{)} \tag{11}\]  
\[U_{nm\theta}=S\sin(n\theta)\sin\Big{(}\frac{\pi mx}{l}\Big{)} \tag{12}\]  
\[U_{nmr}=T\cos(n\theta)\sin\Big{(}\frac{\pi mx}{l}\Big{)}. \tag{13}\]  
Entering the mode shapes for the fix ends case in Equation (10) [29]:\[C_{nm}=\int_{0}^{l}\int_{0}^{2\pi}\Bigg{[}\left[\mathcal{U}_{x} \cos(n\theta)\cos\left(\frac{\pi mx}{l}\right)\right]^{2}+\left[\mathcal{U}_{ \theta}\sin(n\theta)\sin\left(\frac{\pi mx}{l}\right)\right]^{2}+ \tag{14}\] \[+\left[\mathcal{U}_{r}\cos(n\theta)\sin\left(\frac{\pi mx}{l} \right)\right]^{2}\Bigg{]}a\mathrm{d}\theta\mathrm{d}x=\frac{\pi tal}{2}( \mathcal{U}_{x}^{2}+\mathcal{U}_{\theta}^{2}+\mathcal{U}_{r}^{2}),\]  
where \(\mathcal{U}_{x}\), \(\mathcal{U}_{\theta}\), and \(\mathcal{U}_{r}\) are the amplitudes of the mode shapes. The equation can be written in terms of one of them, so that the result depends on that arbitrary amplitude, which vanishes when substituted into Equations (6) and (9). If the boundary conditions were those of free ends or any other case different to that of the fixed ends, the constant term of the equation changes but a similar expression is obtained. The fixed ends condition is taken into consideration due to the assumption of stiff end-plates on both sides of the stator.