## 3 Mathematical Modeling of Gears with Uncertainties
### Thermal Deformation Uncertainties
The sliding friction is determined by the contact stress, relative sliding  
Figure 6: Geometric error induced from the machining error (black) and from thermal deformation (red), respectively, when a plastic gear operates at 200 RPM under 20 Nm. The parameters for machining error and thermal uncertainties are set as \(\epsilon=4\) and \(\eta=0\), respectively. It shows that thermal deformation can lead to higher geometric error than machining error in the case of a plastic gear, which is unlikely to happen in steel gears. speed, and friction coefficient between the gear meshing surfaces. The frictional heat at each contact point is used for thermal analysis based on the FEM [29]. The thermal deformation in the steady state is measured by the nodal displacements of the tooth surface in the coordinate system \(C_{c}\). Here, \(U_{x}\) and \(U_{y}\) denote the \(x\)- and \(y\)-directional nodal displacements, respectively. The nodal displacements \(U_{x}\) and \(U_{y}\) can be expressed in terms of \(u_{p}\), the \(x\)-coordinate in \(C_{p}\), using the relationship between \(u_{c}\) and \(u_{p}\) in Equation (16). To construct continuous displacement curves of the thermal deformation, we interpolate the discrete nodal displacements \(U_{x}\) and \(U_{y}\). Note that the profile of the gear surface consists of \((n+1)\) nodes, whose nodal positions are denoted by \(u_{pi}\) in the coordinate system \(C_{p}\), \(i=0,1,\cdots,n\). We denote the nodal displacements \(U_{x}\) and \(U_{y}\) at \(u_{p}=u_{pi}\) by \(U_{xi}\) and \(U_{yi}\), respectively. Figure 7 (black diamonds) shows the discrete nodal displacements \(U_{xi}\) and \(U_{yi}\), \(i=0,1,\cdots,n\). By using the monomial polynomials \(u_{p}^{j}\), \(j=0,1,\cdots,n\), the displacement curves of thermal deformation in the \(x\)- and \(y\)-directions, denoted by \(H_{x}\) and \(H_{y}\), respectively, can be expressed as  
\[H_{x}(u_{p})=\sum_{j=0}^{n}a_{j}^{x}\,u_{p}^{j},\quad H_{y}(u_{p})=\sum_{j=0}^ {n}a_{j}^{y}\,u_{p}^{j}, \tag{18}\]  
where  
\[ \begin{bmatrix}1&u_{p0}&u_{p0}^{2}&\ldots&u_{p0}^{n}\\ 1&u_{p1}&u_{p1}^{2}&\ldots&u_{p1}^{n}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&u_{pn}&u_{pn}^{2}&\ldots&u_{pn}^{n}\\ \end{bmatrix}\begin{bmatrix}a_{1}^{x}\\ a_{1}^{x}\\ \vdots\\ a_{n}^{x}\end{bmatrix} = \begin{bmatrix}U_{x0}\\ U_{x1}\\ \vdots\\ 1&u_{pn}&u_{pn}^{2}&\ldots&u_{pn}^{n}\\ \end{bmatrix} ,\quad \begin{bmatrix}1&u_{p0}&u_{p0}^{2}&\ldots&u_{p0}^{n}\\ 1&u_{p1}&u_{p1}^{2}&\ldots&u_{p1}^{n}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&u_{pn}&u_{pn}^{2}&\ldots&u_{pn}^{n}\\ \end{bmatrix}\begin{bmatrix}a_{1}^{y}\\ a_{1}^{y}\\ \vdots\\ a_{n}^{y}\end{bmatrix} = \begin{bmatrix}U_{y0}\\ U_{y1}\\ \vdots\\ U_{ym}\end{bmatrix} . \tag{19}\]  
Figure 7 (red circles) shows the interpolation curves for the thermal deformations, i.e., \(H_{x}(u_{p})\) and \(H_{y}(u_{p})\). To represent the gear geometry with thermal deformation, the continuous curves \(H_{x}\) and \(H_{y}\) should be considered in the mathematical method for designing cylindrical gears. The position vector \(\vec{r}_{T}\), which expresses the gear geometry with thermal deformation, is represented as  
\[\vec{r}_{T}(u_{p},\phi_{p},\theta_{p})= \mathbf{M}_{T}\vec{r}_{g}(u_{p},\phi_{p},\theta_{p}),\quad \mathbf{M}_{T}= \begin{bmatrix}1&0&0&H_{x}(u_{p})\\ 0&1&0&H_{y}(u_{p})\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} , \tag{20}\]  
where \(\mathbf{M}_{T}\) is a transformation matrix used to consider the effect of thermal deformation.