## 2 Theoretical Formulation
### Principle of Traveling Wave Resonance  
There are usually three modes of vibration for a wheel structure with a central axis and no rotation: node-circle (NC), ND and composite, as shown in Figure 2. Additionally, bevel gears are prone to ND vibrations. The ND vibration is superimposed by two cosine waves with the same shape. The one that rotates along the gear is called the forward wave, and the one that rotates against the gear is called BTW. During the propagation of ND vibration, the periodic fluctuation of each point on the wheel as the wave passes by is called TWR [3].  
After any point on the spoke plate of the gear rotates \(\theta\) Angle, the corresponding point phase angle on the mode expansion line of the joint diameter m rotates \(m\theta\). Then, the vibration displacement of the corresponding point on the expansion line perpendicular to the direction of the spoke plate can be expressed as follows [19]:  
\[y=B(r)\cos m\theta\cos\omega t \tag{5}\]  
where \(y\) is the transverse vibration displacement of the spoke plate of the gear, in mm; \(B(r)\) is a function of the transverse amplitude of the gear on the radius \(r\) of the gear indexing  
Figure 1: Hertz contact model.  
Figure 2: Vibration modes of disc structure: (**a**) 1 st NC; (**b**) 2 nd ND; (**c**) 1 st NC and 3 rd ND.  
circle, in mm. When the gear rotates relative to the coordinate system, it can be expanded in an exponential form and a triangular form, respectively, as in Equation (6) [19]:  
\[\left\{\begin{array}{l}y=\frac{1}{2}B(r)\big{[}\cos(m\theta)e^{j\omega t}+\cos( m\theta)e^{-j\omega t}\big{]}\\ y=\frac{1}{2}B(r)\big{[}\cos(m\theta-\omega t)+\cos(m\theta+\omega t)\big{]} \end{array}\right. \tag{6}\]  
\(t\) can be seen from the above formula that the nodal vibration displacement of the gear spoke plate can be decomposed into two vibrations with equal amplitude and opposite rotations. The resonant frequencies of the forward and BTWs correspond to their m-diameters, as follows [19]:  
\[\left\{\begin{array}{l}f_{F}=f_{d}+\frac{m\pi_{F}}{60}\\ f_{B}=f_{d}-\frac{m\pi_{B}}{60}\end{array}\right. \tag{7}\]  
where \(f_{F}\) is the FTW resonance frequency, in Hz, \(f_{B}\) is the BTW resonance frequency, in Hz, \(f_{D}\) is the dynamic frequency, in Hz, of the gear pitch diameter type resonance, \(n_{F}\) is the FTW resonant rotational speed, as r /min, and \(n_{B}\) is the BTW resonant rotational speed, as r /min.  
During the modeling of the FEM model, it is necessary to set the TWR rotational speed \(N\) of the driven level gear and the torque \(T\) of the gear shaft since the frequency of the excitation force is equal to the vibration frequency of the FTW and BTW of the gear. By transforming Equation (6), the expression of the rotational speed of the driving gear when the driven gear experiences TWR is as follows [19]:  
\[N_{2}=\frac{60f_{d}}{i\times(z_{2}\pm\ m)} \tag{8}\]  
where the FTW signal is \"\(-\)\"; The BTW signal is \"\(+\)\"; \(z_{2}\) is the number of teeth of driven level gears; \(m\) is the number of pitch diameters; \(N_{2}\) is driven level gear speed; \(i\) is the gear ratio; \(f_{d}\) is the dynamic frequency of the driven level gear. A static frequency is used instead of a dynamic frequency in the calculation.  
The formula for calculating the torque \(T\) of a driven level gear is as follows [19]:  
\[T_{1}=9.55\times 10^{6}\frac{P}{N_{1}} \tag{9}\]  
where \(P\) is the output power of the rotating shaft, in kW, which is 200 kW, and \(N_{1}\) is the speed of the driving gear, as r/min.