## 1 Introduction  
The magnitude and variation of tooth pair compliance affect tooth loading and gear dynamics significantly [1]. For an elastic gear, when a force \(F\) is applied on a gear tooth, a tooth deflection \(\delta\) is generated along the line of force as illustrated in Figure 1. The tooth deflection can be calculated using the finite element method (FEM) or analytical method [2; 3]. With the development of computer science technology, the FEM can give very accurate results if the boundary conditions and the degree of discretization are properly defined [4]. Meanwhile, many researchers have been developing analytical formulas to achieve the fast and accurate calculation of gear tooth deflections. Analytical formulas are easy to use and time-saving in computation [5]. In Ref. [1], gear teeth were assumed to be rigid in evaluating gear-body induced tooth deflections in their finite element model. This assumption will generate some errors. We will remove this assumption in our analysis and give a detailed explanation of how to develop a finite element model to evaluate gear-body induced tooth deflections. In the existing analytical methods, tooth deflection is considered from two parts, the deflection from gear tooth (assuming rigid gear body) and the deflection caused by gear body. One popular analytical method to estimate the deflection from the first part is the potential energy method [6; 7], in which the gear tooth is modeled as a cantilever beam of variable cross-section and the energy stored in the beam is considered as the summation of Hertzian contact energy, bending energy, shear energy, and axial compressive energy. Many researchers have applied this method to evaluate the time-varying meshing stiffness of gears [8; 9; 10; 11]. The deflection induced by a gear body is also called tooth fillet/foundation deflection [1]. [PERSON] [12], [PERSON] [13], [PERSON] [14], and [PERSON] and [PERSON] [1] developed analytical formulas to calculate the gear body-induced tooth deflections. Their analytical formulas all have the same format as Equation (1) but with different \(L\), \(M\), \(P\), and \(Q\) values. \[\delta=\frac{1}{E}\frac{F}{b}\cos^{2}\alpha\left[L\left(\frac{u}{S_{f}}\right) ^{2}+M\left(\frac{u}{S_{f}}\right)+P\times\left(1+Q\tan^{2}\alpha\right)\right] \tag{1}\]  
where \(\delta\) is tooth deflection under a force \(F\) as shown in Figure 1, \(E\) represents [PERSON]'s modulus of gear body material, \(\alpha\) denotes the gear pressure angle, \(b\) is the tooth face width, \(u\) represents the distance between the root circle and the intersection of the tooth center-line and the gear meshing action line (see Figure 1), \(h\) denotes the ratio of gear root circle radius (\(R_{d}\)) and gear bore radius (\(R_{a}\)), \(\theta_{f}\) represents the angle between tooth center-line and the junction with the root circle, and \(S_{f}\) is the arc length corresponding to the angle \(\theta_{f}\). Table 1 gives a summary of the \(L\), \(M\), \(P\), and \(Q\) values used in the four papers [1; 12; 13; 14]. Authors in Refs. [12; 13; 14] used constant \(L\), \(M\), \(P\), and \(Q\) values in their formulas while [PERSON] and [PERSON] [1] considered \(L\), \(M\), \(P\), and \(Q\) as functions of \(h\) and \(\theta_{f}\). The formulas reported in Ref. [1] were developed based on the theory of [PERSON] [15] applied to circular elastic rings with the assumption of linear and constant stress variation at the root circle. It has been widely used in the gear meshing stiffness evaluation even for gears with a tooth crack [16; 17; 18; 19]. [PERSON] et al. [20] studied non-linear dynamics of spur gears using a multi-body model. In their model, the gear body induced tooth deflection was calculated using the model proposed by [PERSON] et al. [1]. [PERSON] et al. [21] gave one analytical formula to evaluate gear body induced tooth deflections. But, they did not give details how the formula was  
Figure 1: An elastic tooth subjected to a force \(F\). derived. [PERSON] et al.