## 3 An Improved Formula to Evaluate Gear-Body Induced Deflections
### Double-Tooth-Pair Meshing Duration  
In Section 2.1, we optimized the 24 coefficients used in Equation (1) that was derived by [PERSON] et al. [1]. Our coefficients give better performance in the evaluation of gear body-induced tooth deflections during the single-tooth-pair meshing period. However, this analytical formula is not proper to use directly for the double-tooth-pair meshing duration. This section aims to derive an improved formula for the double-tooth-pair meshing duration. In the double-tooth-pair meshing duration, there are two pairs of teeth meshing simultaneously. Figure 13 is a simple illustration of gear meshing forces and their resulting defections. In this illustration, we assume gear teeth do not have tooth deflection since our focus is only the gear body deflections. In the figure, \(F_{1}\) and \(F_{2}\) represent the gear meshing forces of the first and the second tooth pairs, respectively. And \(P_{1}\) and \(P_{2}\) denote the gear meshing points of the first and second gear tooth pairs, respectively. Imagine if the first tooth pair starts to mesh first and, correspondingly, a deflection \(\Delta_{11}\) is generated in the mesh point \(P_{1}\) and a deflection \(\Delta_{21}\) is generated in the meshing point \(P_{2}\). Later, when the second tooth pair starts to mesh (the first tooth pair is still in meshing), further deflections \(\Delta_{12}\) and \(\Delta_{22}\) are generated in the meshing points \(P_{1}\) and \(P_{2}\), respectively. Therefore, the total deflection is \(\Delta_{11}+\Delta_{12}\) (\(\delta_{1}=\Delta_{11}+\Delta_{12}\)) at the meshing point \(P_{1}\) and \(\Delta_{21}+\Delta_{22}\) (\(\delta_{2}=\Delta_{21}+\Delta_{22}\)) at the meshing point \(P_{2}\). The values of \(\Delta_{11}\) and \(\Delta_{22}\) can be obtained from the analytical equation derived for the single-tooth-pair meshing duration. Then, our remaining problem is to get the values of \(\Delta_{12}\) and \(\Delta_{21}\). As we described in the previous paragraph, if we apply the force \(F_{1}\) followed by \(F_{2}\), the energy stored in the system can be obtained as follows:  
\[W=\frac{1}{2}F_{1}\Delta_{11}+\frac{1}{2}F_{2}\Delta_{22}+F_{1}\Delta_{12} \tag{3}\]  
Similarly, if we apply the force \(F_{2}\) followed by \(F_{1}\), the energy stored in the system can be expressed as Equation (4) based on one assumption that the deflection is proportional to the load. This assumption is generally valid for elastic materials with small deflection. \[W=\frac{1}{2}F_{1}\Delta_{11}+\frac{1}{2}F_{2}\Delta_{22}+F_{2}\Delta_{21} \tag{4}\]  
From Equations (3) and (4), we can conclude:  
\[F_{1}\Delta_{12}=F_{2}\Delta_{21} \tag{5}\]  
Figure 13: Illustration of meshing forces and the resulting deflections. Further, we can get:  
\[\Delta_{21}=\frac{F_{1}}{F_{2}}\Delta_{12}=\frac{\lambda}{1-\lambda}\Delta_{12} \tag{6}\]  
where \(\lambda\) denotes the load sharing ratio of a pair of spur gears with the expression \(\lambda=\frac{F_{1}}{F_{1}+F_{2}}\). Equation (6) gives the relationship between \(\Delta_{12}\) and \(\Delta_{21}\) (see Figure 13 for details about these two deflections). For better illustration, \(\Delta_{21}\) can be expressed as follows  
\[\Delta_{21}=\frac{F_{1}}{F_{1}/\Delta_{21}}=\frac{F_{1}}{k_{21}^{A}} \tag{7}\]  
where \(k_{21}^{A}\) is the ratio of \(F_{1}\) and \(\Delta_{21}\). We call \(k_{21}^{A}\) affiliated body stiffness in this paper. Similarly, we can define the other affiliated body stiffness \(k_{12}^{A}\) with the expression \(k_{12}^{A}=\frac{F_{2}}{\Delta_{12}}\). From Equation (5), we can find that \(k_{21}^{A}\) is equal to \(k_{12}^{A}\). From Equation (7), we can see that the deflection \(\Delta_{21}\) can be obtained if the affiliated body stiffness is known. Then, from Equation (6), the deflection of \(\Delta_{12}\) is available if \(\Delta_{21}\) and the load sharing ratio \(\lambda\) are known.