## 4 Statistical Analysis for Gear System with Uncertainties
### Monte Carlo (MC) Simulation  
Monte Carlo (MC) simulation was used to evaluate the effects of the respective uncertainties on the transmission error. The uncertainties in the pitch, machining, and material properties are characterized by the parameters \(E_{p}\), \(\epsilon\), and \(\eta\), respectively. The parameters indicate the range or magnitude of the respective uncertainties. To estimate the influence of these uncertainties on the spur gear pair, we generated gear model samples under a randomly given pair \((E_{a},\epsilon,\eta)\). Figure 8: Nonuniform cantilever beam gear model with uncertainties. Specifically, for the pitch uncertainty, we randomly choose \(N\) values of \(\epsilon_{p}^{i}\in[-E_{p},E_{p}]\), \((i=1,2,\cdots,N)\), for a given uncertainty parameter \(E_{p}>0\). From Equation (13), the movement of the rack cutter \(\phi_{g}\) is measured for each \(e_{p}^{i}\). Second, to obtain the rack cutter profile samples in which machining uncertainty is considered, we adopt the Latin hypercube sampling (LHS) method [33; 34]. It is known that LHS exhibits a faster rate of convergence than a simple random sampling method. For a given machining uncertainty parameter \(\epsilon>0\), the LHS operates by dividing the uncertainty ranges into non-overlapping subspaces of equal size and obtains a random point from each subspace. Figure 9 shows an example of the sampling for the machining uncertainty. Here, we suppose that \(\epsilon=1\) and divide the \(X,Y\) axes into \(m_{sub}(=8)\) subspaces of equal size. Out of \(m_{sub}^{2}\) subspaces in \(\mathbb{R}^{2}\), we choose \(m_{sub}\) subspaces such that their rows and columns are not overlapping with each other. In each subspace, a point is randomly selected. Then, as described in Section 3.1 and Figure 5, we obtain a rack cutter profile sample by interpolating these points. For a fixed \(\epsilon>0\), we can generate \(N\) different sets of points from the LHS subspaces, which leads to \(N\) rack cutter profile samples. In the numerical tests, we use a fixed number of subspaces for the LHS sampling \(m_{sub}=8\); see Algorithms 1 and 2 and Section 5. Finally, to calculate the geometric error of thermal deformation, uncertainties in the material properties are considered. Five material properties (\(\mu_{1}\) : Young's modulus, \(\mu_{2}\) : thermal expansion coefficient, \(\mu_{3}\) : density, \(\mu_{4}\) : conductivity, \(\mu_{5}\) : specific heat capacity) generated randomly within a given range \([-\mu_{i}(1-\eta),\ \mu_{i}(1+\eta)]\), \((i=1,2,\cdots,N)\) were used for the thermal analysis. We can obtain data on the geometric error generated by thermal deformation through thermal analysis using FEM. Two types of materials (steel and plastic) were considered to investigate the influence of thermal deformation on the transmission error. In summary, from a given pair \((E_{\epsilon},\epsilon,\eta)\), we have \(N\) samples of gear models in which all three types of uncertainties are considered. The pitch and machining uncertainties are related to the \(N\) rack cutter samples and their movements. Through the rack cutters, the corresponding \(N\) gear profiles are generated, where the displacement due to thermal deformation is affected by the material property. After obtaining the \(N\) gear model samples, we combine them with each other and generate \((N(N-1))/2\) gear systems with uncertainties. Then, LTCA is conducted to calculate the transmission error under the loaded condition. Depending on which gears are  
Figure 9: Latin hypercube sampling (LHS) method for the case of machining tool uncertainties: \(\epsilon=1\). By interpolating these points, we obtain a rack cutter profile sample; see Section 3.1 and Figure 5.  
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1:Inputs:
2:\(\{G^{i}:i=1,2,\cdots,N\}\): gear model samples. 3:\(\{d_{t}^{i}:i=1,2,\cdot,N\}\): Displacement due to thermal deformation of gear model sample.