# Research on Influence of Switching Angle on the Vibration of Switched Reluctance Motor
## 3 Influence of Switching Angle on Vibration
### Influence of Switching Angle on Vibration  
When the switched reluctance motor is running, the frequency of the excitation current and its harmonic components is determined by the rotational speed and the number of rotor poles, and the frequency of the \(n\)-th order current harmonic is:  
\[f_{e}=\frac{\omega}{2\pi}\cdot N_{r}\cdot k \tag{19}\]  
where, \(f_{e}\) is the excitation current harmonic frequency, \(\omega\) is the motor speed, \(N_{r}\) is the number of rotor poles and \(k\) is the current harmonic order. [PERSON] et al. showed that when the harmonic frequency of the current is consistent with the natural frequency of the motor, it will cause the motor to vibrate. The two-step commutation method is one of the control strategies to reduce the vibration of switched reluctance motors. In the two-step commutation method, the upper bridge switch of the power amplifier is first turned off to induce free damped oscillation, and the oscillation frequency is equal to the characteristic frequency of the main mode shape. Then, after half a cycle, the lower bridge switch is turned off, causing another oscillation with the same amplitude and a phase shift of 180\({}^{\circ}\). After the two oscillations are superimposed on each other, the vibration at the characteristic frequency can be significantly reduced. This method shows that a specific shift in the radial force harmonic spectrum can be produced by adjusting the turn-off angle. Therefore, the radial force spectrum can be shifted by adjusting the turn-off angle. When the switched reluctance motor is commutated, the current harmonic component that is the same as the natural frequency of the motor can be minimized, which is helpful for vibration suppression. In order for the \(k\)-th current harmonic to have a phase offset of 180\({}^{\circ}\), the interval between the two turn-off angles is equal to half the excitation period:  
\[t_{i}=\frac{1}{2f_{e}}=\frac{1}{2\cdot\frac{\omega}{60}N_{r}\cdot k} \tag{20}\]  
The spacing angle (electrical angle) between the two off angles is  
\[\theta_{i}=t_{i}\cdot\omega\cdot N_{r}\cdot 60 \tag{21}\]  
Substituting Equation (21) into Equation (20), we get  
\[T_{k}=2\theta_{i}=\frac{180}{k} \tag{22}\]  
\(T_{k}\) represents the period in which the current harmonic amplitude changes with the turn-off angle. When the turn-off angle moves by \(2\theta_{i}\), the \(k\)-th order harmonic value of the current also changes periodically. This conclusion is consistent with the conclusion drawn in previous section when the turn-on angle is constant and the turn-off angle changes. The vibration of a switched reluctance motor can be represented as a linear superposition of a finite number of single-degree-of-freedom systems in modal coordinates. In modal coordinates, the relationship between radial force and its induced vibration can be described in the transfer function  
\[Z_{\omega}^{n}=\frac{F_{r}(\omega)}{a(\omega)} \tag{23}\]  
\(Z_{\omega}^{n}\) is the transmission impedance of radial force at a specified frequency in mode n, which can be identified analytically or experimentally. For the \(n\)-th order radial force harmonic, the corresponding vibration in the \(m\)-\(th\) order mode is:  
\[v_{n}^{m}=Trans(f_{n})\cdot F_{rn}=F_{rn}/Z_{\omega}^{n} \tag{24}\]  
As shown in Equations (3)-(22), the frequency of the higher harmonics of the excitation current is an integer multiple of the fundamental frequency, and the integer \(k\) is the harmonic order. Therefore, the critical harmonic depends on the rotational speed,which causes differences in vibration characteristics when the speed changes. Vibration can be reduced by reducing the harmonic components of radial forces that coincide with the eigen frequency of the body. According to the vibration test results of the switched reluctance motor, the vibration of the low-order mode is more obvious, and the low-order characteristic frequencies of the motor used are 634 Hz and 1220 Hz, corresponding to the first-order and second-order vibration modes, respectively.