## 2 Analytical Calculation Procedure
### Calculation of the Vibration Response  
First of all, the magnetic pressure acting on the stator teeth is calculated using the Maxwell stress tensor. Usually, it is assumed that the vibrations of the stator are mainly produced due to the radial component of the pressure, and the tangential component is neglected due to its reduced effect [27; 30]. Thus, the radial pressure (\(q_{r}\)) is calculated, as shown in (15):  
\[q_{r}(t,\theta)=\frac{B_{\mathrm{s}}(t,\theta)^{2}}{2\mu_{0}} \tag{15}\]  
where \(\mu_{0}\) is the magnetic permeability of the air. The calculation procedure of the vibration response is explained in [26]. The pressure can be expressed as a sum of harmonic waves, as shown in Equation (16):  
\[q_{r}(t,\theta)=\sum\limits_{k=1}^{\infty}\sum\limits_{v=1}^{\infty}P_{kv} \cos(\omega_{k}t)\cos(\lambda_{v}\theta) \tag{16}\]  
where \(P_{kv}\) is the amplitude of the pressure, \(\omega_{k}\) is the rotational frequency of the pressure and \(\lambda_{v}\) is the wave number. The modal superposition method is used, which considers that the response of the system is the result of the sum of the contribution of its vibration modes. According to this method, the vibration response in the radial direction can be expressed as:  
\[u_{r}(x,t,\theta)=\sum\limits_{i=1}^{\infty}\eta_{i}(t)U_{ri}(x,\theta) \tag{17}\]  
where \(\eta_{i}\) is the modal participation factor and \(U_{ri}\) the radial mode shape of mode \(i\). As the excitation force is a sum of harmonic pressure waves, the steady-state response is also a sum of harmonic responses, with a phase lag (\(\phi\)) respective to the force. \[\eta_{i}(t)=\sum\limits_{k=1}^{\infty}\Lambda_{ki}\mathrm{e}^{\mathrm{i}(\omega_ {k}t-\phi_{i})} \tag{18}\]  
The amplitude of the response can be calculated by means of (19):  
\[\Lambda_{ki}=\frac{F_{ki}}{\sqrt{\left(\omega_{i}^{2}-\omega_{k}^{2}\right)^{2 }+4\xi_{i}^{2}\omega_{i}^{2}\omega_{k}^{2}}} \tag{19}\]  
where \(\xi_{i}\) and \(\omega_{i}\) are the damping ratio and the natural frequency of mode \(i\). The natural frequencies can be analytically estimated [31]. The damping is estimated according to the empirical expression proposed by [PERSON] [32]. However, its influence will be negligible in steady-state conditions if the excitation frequencies are not close enough to the natural frequencies. \(F_{ki}\) is the modal force corresponding to the \(k\)-th pressure wave and \(i\)-th mode, which depends on the pressure distribution and the mode shapes of the structure. Considering the stator as a cylinder, the general subscript \(i\) can be substituted by subscripts \(n\) and \(m\), representing the radial-tangential and axial mode shapes, respectively. Thus, the modal force can be represented by means of (20):  
\[F_{kmm}=\frac{1}{\rho h_{\mathrm{s}}C_{nm}}\int\limits_{0}^{L_{2}}\int\limits_ {0}^{2\pi}q_{r}U_{nmr}r_{\mathrm{s}}\mathrm{d}\theta\mathrm{d}x \tag{20}\]  
\[C_{nm}=\int\limits_{0}^{L_{2}}\int\limits_{0}^{2\pi}\left(U_{nmr}^{2}+U_{nm \theta}^{2}+U_{nmx}^{2}\right)r_{\mathrm{s}}\mathrm{d}\theta\mathrm{d}x \tag{21}\]  
where \(L_{\mathrm{s}}\), \(h_{\mathrm{s}}\), \(r_{\mathrm{s}}\) and \(\rho\) are the length, the thickness, the equivalent radius and the equivalent density of the stator. \(U_{nmr}\), \(U_{nm\theta}\) and \(U_{nmx}\) are the mode components of the flexural, torsional and longitudinal vibrations. The mode shapes are considered to be those of a cylindrical shell:  
\[U_{nmr}=\hat{U}_{r}\cos(n\theta)\sin(\frac{\pi mx}{L_{\mathrm{s}}}) \tag{22}\]  
\[U_{nm\theta}=\hat{U}_{\theta}\sin(n\theta)\sin(\frac{\pi mx}{L_{\mathrm{s}}}) \tag{23}\]  
\[U_{nmx}=\hat{U}_{x}\cos(n\theta)\cos(\frac{\pi mx}{L_{\mathrm{s}}}) \tag{24}\]  
Introducing the expressions of the mode shapes in Equation (21):  
\[q_{r}(t,\theta)=\sum\limits_{k=1}^{\infty}\sum\limits_{v=1}^{\infty}P_{kv} \cos(\omega_{k}t)\cos(\lambda_{v}\theta) \tag{16}\]0  
Thus, the amplitude of the modal force can be obtained for every radial-tangential and axial mode, and for every pressure component.