## 3 Mathematical Modeling of Gears with Uncertainties
### Thermal Deformation Uncertainties  
In this study, we assume that either mechanical uncertainties or thermal deformation can influence geometric errors in gears. While mechanical uncertainties have been considered as an important factor affecting the performance of gear pairs (or systems), thermal deformation has been relatively less studied and ignored in most previous works [9; 10]. Relative significance of the two types of uncertainties depends on the material properties. For example, in steel gears, thermal deformation barely changes geometric errors, and the impact of mechanical uncertainties is dominant. On the other hand, thermal deformation can be more influential than mechanical uncertainties. Figure 6 shows the 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. We use the machining error described in Section 3.1 when \(\epsilon=4\). The implementation of thermal deformation is described in the remainder of this section. Here, we consider deterministic thermal deformation, i.e., \(\eta=0\); see Equation (17) below. Figure 6 exhibits 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. Figure 5: Coordinate system of rack cutter profile with error. Friction between the gear meshing surfaces makes a gear pair operate and continuously generates heat on the tooth surface. The frictional heat increases the temperature of the gear pair, resulting in thermal deformation of the gear tooth. To estimate the thermal deformation, we measure the change in displacement of a gear tooth at the temporal steady-state when heat transfer is complete. By performing FE analysis at the steady-state, we estimate the transition of displacement from the initial state. The displacement change depends on the thermal expansion coefficient and steady-state temperature. To clarify, we consider the frictional heat as the only heat source and assume that the operating conditions such as torque, speed, lubrication, and ambient temperature are given a priori following the design process of the entire system. The heat convection parameters used in thermal analysis are constant values that are determined according to the lubrication condition and operation environment. Thus, when the frictional heat is constant, material properties are the only factors that determine the steady-state temperature and the displacement change. In this study, we assume randomnesses in five essential material properties, i.e., Young's modulus, thermal expansion coefficient, density, conductivity, and specific heat capacity, to investigate the uncertain thermal deformation. All material properties are assumed to follow uniform distributions within the ranges  
\[ \begin{split}\mu_{i}\in[\mu_{i}^{*}(1-\eta),\mu_{i}^{*}(1+\eta) ],\\ (\mu_{1}^{*}:\text{\emph{Young}}^{\prime}\text{\emph{s}}\text{ \emph{modulus}},\mu_{2}^{*}:\text{\emph{thermal expansion coefficient}},\mu_{3}^{*}:\text{\emph{density}},\\ \mu_{4}^{*}:\text{\emph{conductivity}},\mu_{5}^{*}:\text{\emph{ specific heat capacity}}).\end{split}  \tag{17}\]  
Here, \(\mu_{i}\) represents each material property with uncertainty, and \(\eta\) denotes the size of the uncertainties of the material properties. Frictional heat is related to the contact stress and relative sliding speed, which depends on where the contact occurs. Each position of the gear tooth has a different temperature when the gear pair reaches a steady state. Thus, the gear has a different shape compared to the involute curve after thermal deformation. Rolling friction, friction caused by deformation, and sliding friction are the three main aspects of the frictional heat generated between the gear meshing surfaces [28]. Rolling friction and friction caused by deformation have a negligible effect on heat generation. Therefore, only sliding friction is considered to predict the frictional heat.