# Research on Nonlinear Vibration Characteristics of Internal Beveloid Gear Transmission System
## 3 Numerical Solution of Internal Beveloid Gear with Parallel Axes
### Analysis of Nonlinear Vibration Characteristics  
When solving by the Runge-Kutta method, it is necessary to first reduce the original 6 s\(-\)order ordinary differential equations to 12 first\(-\)order ordinary differential equations. This section investigates the impact of factors on the nonlinear vibration characteristics of the transmission system under the action of multi\(-\)excitation coupling and discusses the effects of torque fluctuation and bearing clearance on the vibration characteristics. Take the rotational speed \(n=1000\) r\(/\)min, excitation \(T_{at}=10\) N \(\cdot\) m, torque fluctuation \(T_{f}=0.05\)\(T_{at}\)N \(\cdot\) m, transmission error \(E=20\) um, tooth backlash \(2\)\(b=10\) um, and bearing backlash \(b_{b}=20\) um. The time domain and frequency domain responses of radial and axial vibration displacement of a gear pair are derived. Under the above parameters, the nonlinear vibration characteristics of the internal beveloid gear pair are analyzed. Taking into account the non\(-\)steady state response in the early phase of the vibration response, this paper solves the system response in the period 0\(-\)3 s, and uses the vibration\(-\)displacement response curve of the steady\(-\)state response period 2.95 s\(-\)3.00 s to solve the radial and axial vibration\(-\)displacement response curve of each gear and frequency domain response curve. As shown in Figure 2 below. On the basis of the vibration\(-\)displacement response curves of Figure 2a\(-\)d, it is evident that the relative displacement responses of the radial and axial directions of the gears correspond to each other, therefore the ensuing analysis only concerns the external gear. The radial vibration displacement of the external gear has an amplitude of 30.975 um, whereas the axial vibration displacement has an amplitude of 6.8650 um. It can be shown that, under these conditions, the axial vibration displacement has minimal effect on the transmission system's vibration amplitude, and its radial vibration is the dominating one. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Meshing Damping**} & \multicolumn{2}{c}{**Backlash**} \\ \cline{2-3}  & **Tooth Backlash** & **Bearing Backlash** \\ \hline \multirow{2}{*}{\(2595.6\) N \(\cdot\) s/m} & \multirow{2}{*}{\(f(x)=\begin{cases}x-b&(x>b)\\ 0&(-b\leq x\leq b)\\ x+b&(x<-b)\\ \end{cases}\)} & \multirow{2}{*}{\(\overline{f}_{b}(\overline{x})=\begin{cases}\overline{x}-b_{b}&(\overline{x}>b_{b}) \\ 0&(-b_{b}\leq x\leq b_{b})\\ \overline{x}+b_{b}&(x<-b)\\ \end{cases}\)} \\ \multicolumn{3}{c}{Error stimulation} & \multirow{2}{*}{\(\overline{e}(t)=e_{r}\sin\omega_{n}t\)} & \multirow{2}{*}{\(\overline{T}(t)=T_{at}+T_{f}\sin(\omega t)\)} \\ \multicolumn{3}{c}{Note: In order to facilitate the calculation, the seventh\(-\)order fitting can be used, and the fitting function is \(f(x,b)=\frac{7}{\sum\limits_{n=1}^{2}a_{2n-1}x^{2n-1}}=a_{1}x+a_{2}x^{2}+a_{3} x^{3}+a_{4}x^{4}+a_{5}x^{5}+a_{6}x^{6}+a_{7}x^{7}\). \(e_{r}\)—Gear error amplitude. In this paper, 20 μm is taken. \(\omega_{n}\) —Gear pair meshing angular frequency. \(T_{f}\) —First order amplitude of torque fluctuation (N \(\cdot\) m), valued at 5% of theoretical torque. \(\omega\) —First order harmonic angular frequency. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Additional gear transmission system parameters. At the same time, it is discovered that the radial and axial vibration responses oscillate based on the new steady\(-\)state equilibrium point after a certain offset. Although its axial vibration amplitude is tiny, its offset from the equilibrium point is not significantly different from the radial vibration offset. The subsequent study will reveal that the offset value of the equilibrium point of the vibration response has a significant link with the bearing backlash.