## 1 Introduction
(2020) developed a combined analytic-numerical contact model to simulate gear dynamics, but they did not specifically investigate gear body induced tooth deflections. However, the analytical equations derived for \(L\), \(M\), \(P\), and \(Q\) in Ref. [1] are very complicated and hard to use. To address this problem, [PERSON] and [PERSON] [1] numerically calculated the values of \(L\), \(M\), \(P\) and \(Q\) based on the plane strain assumption over a realistic range of values (\(h\) between 1.4 and 7, \(\theta_{f}\) between 0.01 and 0.12) and fitted these values using polynomial functions of the form as shown in Equation (2)  
\[X_{i}(h,\theta_{f})=A_{i}/\theta_{f}^{2}+B_{i}h^{2}+C_{i}h/\theta_{f}+D_{i}/ \theta_{f}+E_{i}h+F_{i} \tag{2}\]  
where the constants \(A_{i}\), \(B_{i}\), \(C_{i}\), \(D_{i}\), \(E_{i}\), and \(F_{i}\) are given in Table 2.  
For a pair of spur gears with a contact ratio between one and two, single-tooth-pair meshing and double-tooth-pair meshing take place alternatively as shown in Figure 2. The formula reported in Ref. [1] can only be used in the single-tooth-pair meshing duration but not in the double-tooth-pair meshing duration as their model only considered a single loading force on a gear. To address this shortcoming, [PERSON] et al. (2020) introduced a correction factor to consider the load dependence in the double-tooth-pair meshing duration and to account for the errors of using Equation (1) in the double-tooth-pair meshing duration. Meanwhile, [PERSON] et al. (2020) derived new formulas for gear body induced deflections considering cubic and parabolic stress distribution around the tooth root and also load dependence between teeth. [PERSON] et al. (2020) extended the formula reported in Ref. [1] by considering the effect of tooth root crack on gear body-induced tooth deflections. \begin{table}
\begin{tabular}{c c c c c c} \hline  & \(\mathbf{L}\) & \(\mathbf{M}\) & \(\mathbf{P}\) & \(\mathbf{Q}\) \\ \hline [PERSON] (1999) & 5.2 & 1 & 1.4 & 0.294 \\ [PERSON] (1999) & & & & 0.32 \\ [PERSON] (1999) & 5.306 & 1.4 (plane stress) & & 1.534 & 0.32 \\ [PERSON] and Velex (1999) & \(L(h,\theta_{f})\) & \(M(h,\theta_{f})\) & \(P(h,\theta_{f})\) & \(Q(h,\theta_{f})\) \\ \hline \end{tabular}
\end{table} 
Table 1: \(L\), \(M\), \(P\) and \(Q\) values used in four different papers. Figure 2: Gear meshing scenarios. Based on our literature review, the formula reported in Ref. [1] is most widely used by researchers to calculate gear body-induced deflections. The formula reported in Ref. [24] is an improved version of the formula reported in Ref. [1]. However, these methods still generate a large error in calculating gear body-induced deflections. There are mainly two reasons for the errors. First, the formulas were derived considering the gear body as an elastic ring without considering the effect of teeth on body deflections. Second, the stress distribution assumed in deriving these formulas is not proper. Linear and constant stress variation around the tooth root circle was assumed in Ref. [1] while cubic and parabolic stress distributions were assumed in Ref. [24]. Based on our finite element analysis, these assumptions are far from real situations. Detailed descriptions will be given in this study. In this paper, we will discuss how to develop an accurate finite element model, and analyze the effect of different parameters such as gear number of teeth, location of loads, and gear geometry on gear body-induced deflections. Then, an improved analytical method will be proposed to deliver a more accurate estimation of gear body-induced deflections. The remaining part of this paper is organized as follows. Section 2 describes our finite element model and investigates the effect of gear parameters on gear body-induced deflections. Section 3 presents the proposed method to derive an improved formula to evaluate gear body-induced tooth deflections. Section 4 gives case studies to validate the accuracy of the proposed method. Section 5 draws a conclusion.