## 3 An Improved Formula to Evaluate Gear-Body Induced Deflections
### Single-Tooth-Pair Meshing Duration  
This sub-section aims to optimize the coefficients \(A_{i}\), \(B_{i}\), \(C_{i}\), \(D_{i}\), \(E_{i}\), and \(F_{i}\) of Equation (2) in order to get higher accuracy when we evaluate gear body induced tooth deflection in the single-tooth-pair meshing duration. The objective function of this optimization problem is to minimize the difference between the results generated by Equation (1) and a well-established finite element model. A finite element model reported in Ref. [11] is modified to evaluate the gear body-induced tooth deflections of 72 standard involute spur gears. 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 coupled. ANSYS is used to implement the finite element analysis. These gears (\(72=9\times 8\)) are a combination of of 9 teeth and 8  
Figure 10: Effect of gear bore size on tooth root stress distribution. different \(h\) values, as shown in Table 5. These gears cover a realistic range of \(\theta_{f}\) and \(h\) values: \(\theta_{f}\) between 0.03 and 0.15 (radian), and \(h\) between 2.1 and 7 (remember that \(L\), \(M\), \(P\) and \(Q\) are functions of \(\theta_{f}\) and \(h\)). All the gears have a module of 3.175 and a pressure angle of \(\alpha=20^{\circ}\). Gear body-induced tooth deflections of these 72 gears are evaluated using the finite element model to optimize the coefficients in Equation (2). Based on our finite element analysis, gear body-induced tooth deflections are independent of the module of gears. This conclusion can also be confirmed by the analytical formulas reported in Ref. [1]. We can see that the module of gears is not a variable in Equation (1). Therefore, our equations derived in this paper can be used for a spur gear with any modulus. However, since we only used the pressure angle of \(\alpha=20^{\circ}\) in our finite element analysis, the optimized coefficients may not give accurate results for other pressure angles of a spur gear. In the reported finite element models [1; 23], gear teeth were assumed as rigid in evaluating the gear body-induced tooth deflections. However, this assumption causes errors of about 10% in the gear body-induced tooth deflections based on our analysis. This assumption is released in our finite element analysis. When a force \(F\) is applied on the tooth profile (see Figure 3), we first extract acting forces generated by the force \(F\) on the tooth root. Then, we apply the same acting forces on the root of the gear and record gear tooth deflection \(\delta\) along the force action line. In this way, \(\delta\) is the gear body-induced tooth deflection and it is obtained from an elastic gear (elastic gear teeth and body). A two-dimensional (2-D) plane strain model is used in our finite element analysis with element type PLANE183 as it can give similar accuracy in meshing stiffness evaluation as a 3-D finite element model. We use the 2-D model because it requires much lower computational cost. To demonstrate that it is acceptable to use the plane strain conditions, we did tests on gears with the same parameters (number of teeth: 19, gear module: 3.175, bore radius: 17.5 mm, meshing force per unit of tooth width \(F/b=10\) N/m, transmission ratio 1:1, pressure angle \(\alpha=20^{\circ}\)) except for the tooth face width. Figure 11 shows the comparison results. The \(x\)-axis of Figure 11 is the gear angular displacement corresponding to the meshing force moving from the tooth tip to root. The \(y\)-axis is the gear body stiffness (some researchers call it body stiffness) per unit of tooth width which is expressed as \(k_{b}=\frac{F/b}{\delta}\), where \(F\) is the meshing force, \(b\) represents tooth width, and \(\delta\) denotes the gear body-induced tooth deflection. In the whole process, \(F\) is applied on a single gear tooth since we focus on the single-tooth-pair meshing duration in this sub-section.