## 3 Establishment of the Dynamic Model for Spiral Bevel Gears
### Loaded Tooth Contact Analysis (LTCA)  
Referring to the work of [PERSON] et al. (2018), the LTCA method is employed to optimize the formula for solving transmission errors in this approach. The tooth surface gap values obtained from Equation (14) are further utilized for the gear LTCA. The following figure illustrates the contact analysis:  
The initial gap between the teeth of the wheel before deformation can be expressed as the following:  
\[\mathbf{w}_{k}=\mathbf{\delta}\prime_{\text{k}}+\mathbf{d}_{\text{k}}\quad(k=1,2) \tag{15}\]  
where the initial gap is a collection of initial gaps at each position, and the normal gap and intertooth gap are collections of gaps at each position:  
\[\left\{ \begin{array}{l}\mathbf{w}_{k}=\left[w_{1},\ldots,w_{i}\ldots,w_{n}\right] ^{\text{T}}\\ \mathbf{d}_{k}=\left[d_{1},\ldots,d_{i}\ldots,d_{n}\right]^{\text{T}}\\ \delta\prime_{\text{k}}=\delta\prime_{\text{k}}[1,\ldots,1,\ldots 1]^{T} \end{array} \right. \tag{16}\]  
Figure 4: Gear cutter disk. where \(n\) represents the number of discrete points taken, and \(d_{i}\) is the tooth surface normal gap at the \(i\)-th discrete point. As derived from Equation (14), \(\delta t_{i}\) represents the tooth gap before deformation. This can be obtained from Equation (15). It can be observed that under the action of discrete load P, the gear undergoes elastic deformation. Assuming the small gear is fixed, and the large gear experiences a normal displacement Z due to the load, the coordinated Equation (17) for the deformed displacement is as follows:  
\[\mathbf{F_{k}p_{k}+w_{k}=Z+s_{k}\;(k=1,2)} \tag{17}\]  
The main focus of dynamic analysis is on the transmission error obtained from LTCA. The definition of transmission error in a gearbox, based on the transmission model, is presented in this paper, taking into account surface contact. Transmission error is defined as the product of the difference between the actual measured output position and the measured input position, multiplied by the gear transmission ratio. The LTCA module is implemented through the MASTA software (2022a Edition), enabling the analysis of time-varying mesh stiffness and tooth surface forces for spiral level gears [32; 33]. Based on the gear surface Equation (12), the expression of the gear surface Equation (18) in their respective coordinate systems is obtained:  
\[r_{i}(u_{i},\theta_{i})\in C^{2}\frac{\partial r_{i}}{\partial u_{i}}\times \frac{\partial r_{i}}{\partial\theta_{i}}\
eq 0(u_{i},\theta_{i})\in E_{i}(i=1,2) \tag{18}\]  
Two distinct tooth surfaces are denoted by subscripts 1 and 2. The unit normal vector of the curved surface is represented as follows:  
\[\mathbf{n}_{i}=\frac{\frac{\partial r}{\partial u_{i}}\times\frac{\partial\mathbf{n}_ {i}}{\partial\theta_{i}}}{\left|\frac{\partial r}{\partial u_{i}}\times\frac{ \partial\mathbf{n}_{i}}{\partial\theta_{i}}\right|} \tag{19}\]  
Pinion 1 is presumed to rotate about a fixed axis in the rigid coordinate system \(S_{m}\), while gear 2 rotates about another fixed axis in the assumed coordinate system \(S_{q}\). The position and orientation of \(S_{q}\) relative to \(S_{m}\) simulate installation errors, consolidating all installation errors into gear 2. The representation of the two tooth surfaces in \(S_{m}\) is formulated as the following:  
\[r_{m}^{(1)}=M_{m1}r_{1}r_{m}^{(2)}=M_{mq}M_{q2}r_{2} \tag{20}\]  
The unit normal vector is represented in the \(S_{m}\) coordinate system as the following:  
\[\mathbf{n}_{m}^{(1)}=\mathbf{L}_{m1}\mathbf{n}_{1}\mathbf{r}_{m}^{(2)}=\mathbf{L}_{mq}\mathbf{L}_{q2} \mathbf{n}_{2} \tag{21}\]  
where \(\mathbf{M}\) is the position transformation matrix, and \(\mathbf{L}\) is the vector transformation matrix. By using the established gear surface Equation (12) in the rigidly fixed coordinate system \(S_{m}\) of the frame, an engagement contact analysis of the gear surface Equation (12) is performed.