## 4 Statistical Analysis for Gear System with Uncertainties
### LTCA for Gear System with Uncertainties  
To calculate the transmission error \(\Delta\psi^{U}\) under the unloaded condition, we considered the gear profile with uncertainties. The equation of meshing satisfied by the envelope of the family of rack cutter surfaces cannot be rearranged as an explicit function because of the uncertainties of the rack cutter surface. Therefore, we need to consider the equation of  
Figure 7: Thermal analysis results. (**a**): contour of displacement on \(\overline{x}\) (left) and displacement on \(\overline{x}\) along \(u_{p}\) (right). (**b**) : contour of displacement on \(\overline{y}\) (left) and displacement on \(\overline{y}\) along \(u_{p}\) (right). meshing as an additional condition along with the kinematic conditions (8) satisfied at the contact points, which results in three kinematic conditions:  
\[ \begin{split}\vec{S}_{e1}-\vec{S}_{e2}&=0,\\ \vec{N}_{e1}\times\vec{N}_{e2}&=0,\\ (\frac{\partial\vec{r}_{pi}^{*}}{\partial u_{p}}\times\frac{ \partial\vec{r}_{pi}^{*}}{\partial\theta_{p}})\cdot\frac{\partial\vec{r}_{pi}^ {*}}{\partial\phi_{p}}&=0\quad(i=1,2),\end{split}  \tag{21}\]  
where  
\[ \begin{split}\vec{S}_{e1}(u_{p1},\phi_{p1},\theta_{p1},\psi_{1} )=&\begin{bmatrix}\cos\psi_{1}&\sin\psi_{1}&0&0\\ -\sin\psi_{1}&\cos\psi_{1}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}\vec{r}_{T1},\\ \vec{S}_{e2}(u_{p2},\phi_{p2},\theta_{p2},\psi_{2})=& \begin{bmatrix}-\cos\psi_{2}&-\sin\psi_{2}&0&0\\ \sin\psi_{2}&-\cos\psi_{2}&0&d_{g}\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}\vec{r}_{T2},\\ \vec{N}_{ei}=&\frac{\partial\vec{S}_{ei}^{*}}{\partial u_{p}}\times\frac{ \partial\vec{S}_{ei}^{*}}{\partial\theta_{p}},\quad(i=1,2).\end{split}  \tag{22}\]  
Here, \(\vec{S}_{ei}\), \(\vec{N}_{ei}\), \(\vec{r}_{pi}\), and \(\vec{r}_{T1}\)\((i=1,2)\) are the position vector, normal vector of the gear surface with uncertainties in \(\mathbf{C}_{si}\), position vector of the gear surface in \(\mathbf{C}_{p}\), and position vector of the gear surface with thermal deformation in \(\mathbf{C}_{g}\), respectively. The subscripts \(i=1,2\) denote driving and driven gears, respectively. Further, \(\vec{S}_{e1}^{*}\), \(\vec{S}_{e2}^{*}\), and \(\vec{r}_{p}^{*}\) are vectors obtained by eliminating the last elements of \(\vec{S}_{e1}\), \(\vec{S}_{e2}\), and \(\vec{r}_{p}\), respectively. To simplify Equation (21), we assume that there is no variation on gear surface in the \(z\)-axis, the longitudianl direction of gear. Note that parameter \(\theta_{p}\), the \(z\)-coordinate of the position vector \(\vec{r}_{p}\), can be eliminated as following the 2D simplification. From Equation (21), five equations \(P_{ei}\)\((i=1,2,3,4,5)\) are derived:  
\[ \begin{split}\mathbf{P}_{e1}(u_{p1},\phi_{1},\psi_{1},u_{p2},\phi _{2},\psi_{2})&=(\vec{S}_{e1}-\vec{S}_{e2})\cdot\vec{i}=0,\\ \mathbf{P}_{e2}(u_{p1},\phi_{1},\psi_{1},u_{p2},\phi_{2},\psi_{2})& =(\vec{S}_{e1}-\vec{S}_{e2})\cdot\vec{j}=0,\\ \mathbf{P}_{e3}(u_{p1},\phi_{1},\psi_{1},u_{p2},\phi_{2},\psi_{2})& =(\vec{N}_{e1}\times\vec{N}_{e2})\cdot\vec{k}=0,\\ \mathbf{P}_{e4}(u_{p1},\phi_{1})&=(\frac{ \partial\vec{r}_{p1}^{*}}{\partial u_{p1}}\times\frac{\partial\vec{r}_{pi}^{*} }{\partial\phi_{p1}})\cdot\frac{\partial\vec{r}_{p1}^{*}}{\partial\phi_{p1}}=0,\\ \mathbf{P}_{e5}(u_{p2},\phi_{2})&=(\frac{ \partial\vec{r}_{p2}^{*}}{\partial u_{p2}}\times\frac{\partial\vec{r}_{p2}^{*} }{\partial\phi_{p2}})\cdot\frac{\partial\vec{r}_{p2}^{*}}{\partial\phi_{p2}}=0.\end{split}  \tag{23}\]  
Using the Newton-Raphson method, we can obtain the six unknown parameters \((u_{p1}\), \(\phi_{1}\), \(\psi_{1}\), \(u_{p2}\), \(\phi_{2}\), and \(\psi_{2})\) that satisfy Equation (23). The elastic deflection \(\delta^{E}\) is affected by the gear profile, material properties, and applied load. However, to focus on gear profile and material properties related to uncertainties, we exclude the load sharing effect in this study. The mesh stiffness changes according to the contact zone, the shape of the profile, and elastic modulus.