## 3 Condition of Backlash Distribution of Dual-Input Counter-Rotating Gearbox
### Three-Dimensional Tolerance Analysis of Output Gear Backlash of Gearbox  
Three-dimensional tolerance analysis mainly has an extreme value method and a statistical method. The backlash calculated with the extreme value method is large. In this research, we used the Monte Carlo method in the statistical method for the tolerance analysis of the backlash.  
Monte Carlo simulation is a numerical method for solving approximate solutions to mathematical, physical, and engineering technology problems through stochastic simulations and statistical experiments. The method is computationally accurate and consistent with the actual generation, and it is suitable for the solution of assembly functions with nonlinear expressions.  
The dimensional and shape tolerances of the gearbox follow the principle of independence, and the tolerance zones of their dimensional and shape tolerances are symmetrically distributed and obey a normal distribution. The symmetrically distributed tolerances, regardless of whether they are defined as increasing or decreasing rings, have the same effect on the closed ring. Therefore, in the process of tolerance analysis, the tolerance of each component ring is expressed as \(T_{i}\pm\frac{t}{2}\) (\(t\) is the tolerance of the component ring).  
The closed-loop samples \(FR_{1}\) and \(FR_{2}\) conformed to a normal distribution, for which the tolerance can be calculated as follows:  
\[FR_{w}=\mu_{FR}\pm Z\delta_{FR} \tag{16}\]  
where \(\mu_{FR}\) and \(\delta_{FR}\) are the mean and the variance of \(FR\), respectively. The confidence level at 99.73% probability was used, and \(Z\) was taken as 3.  
The backlash \(jn\) value was calculated as follows:  
\[jn=(|+FR_{1w}|\cdot\cos(|+FR_{1\delta}|)+|-FR_{2w}|\cdot\cos(|+FR_{2\delta}|)) \times\sin\theta \tag{17}\]  
where \(FR_{1w}\) is the upper deviation of the \(Z\)-directional translation of closed-dimensional chain 1, \(FR_{2w}\) is the lower deviation of the \(Z\)-directional translation of closed-dimensional chain 2, \(FR_{1\delta}\) is the upper deviation of the \(Z\)-directional rotation of closed-dimensional chain 1, \(FR_{2\delta}\) is the lower deviation of the \(Z\)-directional rotation of closed-dimensional chain 2, and \(\theta\) is the angle of engagement of \(\mathrm{G_{d1}}\) and \(\mathrm{G_{d2}}\) with \(\mathrm{G_{o1}}\).  
The results of the \(jn\) calculation are shown in Table 2.  
The theoretical value of 0.2443 mm was used as the lower limit of the backlash, and the value calculated with the Monte Carlo method, 0.4150 mm, was used as the upper limit of the backlash. The actual measurement of the backlash should be distributed in this range.  
The contribution of each dimensional tolerance to the backlash closure ring is shown in Figure 9.  
As can be seen from Figure 9, the tolerance \(T_{3}\) has the highest contribution to the backlash at 38.65%, followed by the tolerances \(T_{8}\) and \(T_{1}\) at 17.24% and 16.41%, respectively. Tolerance \(T_{3}\) is the center distance error of the first stage gear, tolerance \(T_{8}\) is the phase angle error of the first and second stage gear. The error value of the center distance and phase angle is consistent with the direction of the backlash, and at the same time, due to the large value of the center distance error, it has the greatest impact on the backlash, followed by the phase angle error.