## 2 Finite Element Analysis of Gear Body-Induced Tooth Deflections
### Comparisons between Rigid Teeth and Elastic Teeth Finite Element Models  
In [1; 24], finite element models were developed for gears with the assumption of the elastic body and rigid teeth to evaluate gear body-induced deflections. In these models, a contact force \(F\) was applied along the action line and the tooth deflection \(\delta\) was considered as the deflection caused by gear body along the action line. The rigid teeth in these finite element models generate large errors because these rigid teeth increase gear body stiffness. In our analysis, we will use elastic teeth and body to demonstrate the difference. Through our analysis, we find there are two ways to accurately obtain gear body-induced deflections through FEM. We will describe them one by one in the following two paragraphs. Method 1: Build an elastic gear (elastic teeth and body) using FEM and apply a contact force \(F\) on a tooth (see Figure 1). Extract acting forces generated by the force \(F\) on the tooth root (see the acting forces in Figure 3). Then, apply the same acting forces on the root of the gear rather than apply the force \(F\) on a rigid tooth, and record gear tooth deflection \(\delta_{\text{b}}\) along the action line. In this way, \(\delta_{\text{b}}\) is the gear body-induced tooth deflection (the deflection induced by the tooth is not included) corresponding to the contact force \(F\). Method 2: Build an elastic gear (elastic teeth and body) using FEM and apply a contact force \(F\) on a tooth (see Figure 1). Record the tooth deflection \(\delta\) along the action line that is the deflection induced by both gear teeth and gear body. Then, in the finite element model, change the gear body to be rigid. Apply the same contact force \(F\) on the same location of the tooth and record the tooth deflection \(\delta_{\text{t}}\) along the action line that is the deflection caused by the gear tooth. The gear body-induced tooth deflection corresponding to the contact force \(F\) can be calculated as \(\delta_{\text{b}}=\delta-\delta_{\text{t}}\). A finite element model reported in Ref. [11] is modified to test the performance of our two methods and the method used in Refs. [1,24]. The gears are modeled using the element type SOLID185 and the element shape of mapped hexahedral. Six teeth are refined when the middle one or two teeth are in meshing. In our model, meshing nodes of the pinion and gear are defined theoretically and then coupled. Gear parameters are given in Table 3 and Figure 3 shows the finite element model (a two-dimensional plane strain model) developed in Ansys. Figure 4 gives a comparison of three methods: the method reported in [1,24] and our methods 1 and 2. A constant force \(F\) (contact force per unit of tooth width \(F\)/\(b\) = 10 N/m) is applied on a single gear tooth from the tooth tip to the tooth root. The force \(F\) mimics the meshing force between a pair of gears with the transmission ration 1:1. The \(x\)-axis is the gear angular displacement from tooth tip to tooth root. The tooth tip corresponds to the angle 0 at the very left of the \(x\)-axis while the tooth root corresponds to the angle at the very right of the \(x\)-axis. The \(y\)-axis is gear-body induced deflection along the action line. From the comparison, we can see that the rigid teeth in the finite element model will lead to less tooth deflection. The rigid teeth make the gear more rigid, which generates errors. Our two methods give almost the same result and they are more accurate than the model reported in [1,24]. Figure 5 gives deformation and stress information (obtained from our FEM model) along a tooth root when a meshing force (10 N/m) is applied on the tooth tip. In Figure 5, the \(x\)-axis represents the range of tooth angle on the root circle with the angle 0 corresponding to the tooth central line. The very left and right angles in the \(x\)-axis of Figure 5 correspond to tooth fillet.