## 4 Mechanical Analysis  
The vibration performances of SRM depend on two features--the radial force and the SRM structural characteristics. The frequency of a natural vibration in an elastic body is called the natural frequency. There are many reasons to calculate the natural frequencies and mode shapes of a structure prototype SRM. One reason is to avoid the resonance problem. In the SRM, it is found that resonance occurs if the phase frequency or the harmonics coincide with the SRM natural frequency. The phase frequency is given as:  
\[F_{p}=\frac{\omega_{m}P_{r}}{2\pi}, \tag{22}\]  
where \(\omega_{m}\) is the speed [rad/s] and \(P_{r}\) is the number of rotor poles. Vibration is the maximum when any frequency of the harmonics \(F_{n}=nF_{p}\) is coincident with the SRM natural frequency. In References [1; 15], the first mode frequency \(\omega_{fm}^{2}\) of the SRM is given by:  
\[\omega_{fm}^{2}=\frac{Eb_{sy}^{2}}{6\rho(1-\gamma^{2})(\frac{\pi^{2}}{4}-2)( \frac{D_{p}-b_{sy}}{2})4}, \tag{23}\]  
Figure 13: The 3D models temperature profile of WJ configurations. (**a**) 17 channels with spoke shaft; (**b**) 17 channels with solid shaft; (**c**) rotor with spoke shaft; (**d**) rotor with a solid shaft. where \(D_{o}\) and \(b_{sy}\) are the outer diameter of the stator lamination stack and the stator yoke thickness, \(\gamma\) is the Poisson ratio, \(E\) is the modulus of elasticity and \(\rho\) is the mass density of the lamination material. The radial and tangential forces are the components of the air-gap electromagnetic force. The useful forces are the tangential ones; they are producing the torque. The radial force caused by the interaction of doubly salient teeth, during phase winding is excited, will lead to vibration of the motor [17; 18; 41]. The tangential force \(F_{t}\) and the major force of mechanical deformation in the motor which is called the radial force \(F_{r}\), they are given as:  
\[F_{t}=\iint\frac{1}{\mu_{o}}(\vec{B}\cdot\hat{n})\vec{B}dS=\frac{1}{\mu_{o}} \iint(\vec{B}_{r}\vec{B}_{t})dS, \tag{24}\]  
\[F_{r}=\iint\frac{1}{2\mu_{o}}\
abla B^{2}\hat{n}dS=\frac{1}{2\mu_{o}}\iint(B_{ r}^{2}-B_{t}^{2})dS, \tag{25}\]  
where \(B\) is flux density, \(\hat{n}\) is the unit normal vector, \(B_{r}\) and \(B_{t}\) are the radius and tangential component of the flux density in the air gap respectively. A 2D finite element models for the SRM has been developed, to obtain the waveforms and the Fast Fourier Transform (FFT) of the radial force, as shown in Figure 14. The developed SRM is in 72/48 combination. Thus, the fundamental component of the phase current is 84 Hz at 105 rpm. The SRM was simulated for a 174 A DC current at 105 rpm, producing torque of 7.28 kNm, such that the high magnitude of the radial force was 1.46 kN, around the fundamental operating frequencies. The numerical model of 3-phase 72/48 SRM was analyzed using the 3D FEA model to obtain the natural frequencies of the motor with two different structures of the shaft. The comparison between the results of these two different structures is presented in Table 8. It must be mentioned here that these natural frequencies vary depending on the boundary condition and the structure of the housing. Besides, the spoke shaft is much more weight-efficient than a solid shaft. This explains why the natural frequencies of a spoke shaft are greater than that of a solid shaft, as can be seen in Table 8. Figure 15 shows the total deformation of the 72/48 SRM with two different structures of the shaft at different frequencies. It can be observed that the minimum deformation is at the support points of the housing, and it increases gradually to reach a maximum at the top side of the housing. Figure 15 also shows that the total deformation of the SRM with a spoke shaft is larger than the deformation of SRM with a solid shaft. The minimum\(\backslash\)maximum total deformations for the two shaft structures at various natural frequencies are clearly given in Table 9.  
Figure 14: The radial force of the 72/48 SRM.