# Mesh Stiffness and Dynamic Modeling and Analysis of Modified Straight Bevel Gears
## 4 Dynamic Model of Straight Bevel Gears
### Gear Mesh Model  
As shown in Figure 14, the input gear shaft and output gear shaft are coupled through a gear pair. The TVMS and time-invariant meshing damping (\(c_{m}\)) along the normal direction at the meshing point are connected between the gears. These are in series with geometric transmission error (\(e_{s}\)) [39]. As detailed in Section 2.1 via TCA, the \(r_{li}\) are separately  
Figure 13: Element model of gear rotor. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Pinion 1** & **Gear 1** \\ \hline Teeth number Z & 37 & 74 \\ Module m (mm) & 2 & \\ Pressure angle \(\alpha\) (\({}^{\circ}\)) & 20 & 20 \\ Face width B (mm) & 20 & 20 \\ Young’s modulus E (Gpa) & 206 & 206 \\ Poisson’s ratio & 0.3 & 0.3 \\ Addendum coefficient ha & 1 & 1 \\ Tip clearance coefficient c & 0.2 & 0.2 \\ Enter Torque load \(T\) (N\(\cdot\)m) & 10 \\ Density (kg/m\({}^{3}\)) & 7860 & \\ Diameter moment of inertia \(I_{dnt}\) (Kg\(\cdot\)m\({}^{2}\)) & \(1.53\times 10^{-4}\) & 0.003 \\ Polar moment of inertia \(I_{dnt}\) (Kg\(\cdot\)m\({}^{2}\)) & \(2.78\times 10^{-4}\) & 0.006 \\ Backlash \(bl\) (\(\mu\)m) & 20 & \\ Meshing damping \(c_{m}\) (Ns/m) & 1500 & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Parameters of straight bevel gears and bearing. calculated in the local coordinate systems of each gear during the meshing cycle, along with the time-varying normal vectors \(n_{li}\). Subsequently, as shown in equation (30), the \(\lambda_{li}\) values are computed for the pinion and gear, respectively [23]. Construct vector \(\mathbf{V}_{l}\) and the deflection of gear pair of action \(\mathbf{q}_{m}\) for the gear pair:  
\[\mathbf{V}_{l}=\left\{n_{lpx},\,n_{lpy},\,n_{lpx},\,\lambda_{lpx},\,\lambda_{ lpy},\,\lambda_{lpz},\,n_{lgx},\,n_{lgy},\,n_{lgz},\,\lambda_{lgx},\,\lambda_{ lgy},\,\lambda_{lgz}\right\} \tag{35}\]  
\[\mathbf{q}_{m}=\left\{x_{p},\,y_{p},\,z_{p},\,\theta_{px},\,\theta_{py},\, \theta_{pz},\,x_{g},\,y_{g},\,z_{g},\,\theta_{gx},\,\theta_{gy},\,\theta_{gz}\right\} \tag{36}\]  
In the equation, \(x_{i}\) represents the vibration displacement of the gear along the three directions x, y, and z, and \(\theta_{ix},\,\theta_{iy},\,\theta_{iz}\) represents the angular displacement around the x, y, and z axes. The gear overlap in straight level gears typically does not exceed 3, so the dynamic displacement of the gear pair along the line of action(DTE) [2] for the gear pair can be expressed as:  
\[\delta_{m}=ldr_{1}\mathbf{V}_{1}^{\mathrm{T}}\mathbf{q}_{m}+ldr_{2}\mathbf{V} _{2}^{\mathrm{T}}\mathbf{q}_{m}-e_{s}(t) \tag{37}\]  
where \(ldr_{l}\) denote the load distribution ratio on the gear meshing pair, it can be expressed as follows:  
\[lrd_{l}=K_{el}/K \tag{38}\]  
The dynamic meshing force along the meshing line direction between the straight level gears can be expressed as [2]:  
\[F_{m}=K(\gamma_{m0}\delta_{m}+\gamma_{m1}bl)+c_{m}\gamma_{m0}\delta_{m} \tag{39}\]  
where \(bl\) is half of the total gear backlash, and the index of the backlash function is:  
\[\gamma_{m1}= \begin{cases}-1&\delta_{m0}>bl\\ 0&|\delta_{m0}|\leq bl\,\,,\,\gamma_{m1}=\begin{cases}1&|\delta_{m0}|>-bl\\ 0&|\delta_{m0}|\leq bl\end{cases}\\ 1&\delta_{m0}<-bl\end{cases}  \tag{40}\]  
Based on the above analysis, the motion equations for the meshing pair of straight level gears in matrix form can be expressed as:  
\[ \begin{array}{l}\mathbf{M}^{\mathrm{m}}\tilde{\mathbf{q}}_{m}+(r_{1}^{2} \mathbf{C}_{11}^{\mathrm{m}}+r_{1}r_{2}\mathbf{C}_{22}^{\mathrm{m}}+r_{2}r_{1 }\mathbf{C}_{22}^{\mathrm{m}}+r_{2}\mathbf{C}_{22}^{\mathrm{m}})\gamma_{m0} \tilde{\mathbf{q}}_{m}+(r_{1}^{2}\mathbf{K}_{11}^{\mathrm{m}}+r_{1}r_{2} \mathbf{K}_{12}^{\mathrm{m}}+r_{2}r_{1}\mathbf{K}_{21}^{\mathrm{m}}+r_{2}^{2} \mathbf{K}_{22}^{\mathrm{m}})\gamma_{m0}\mathbf{q}_{m}\\ =(k_{m}\gamma_{m0}e(t)+c_{m}\gamma_{m0}e(t)-k_{m}\gamma_{m1}b)(r_{1}\mathbf{V} _{1}^{\mathrm{T}}+r_{2}\mathbf{V}_{2}^{\mathrm{T}})\end{array}  \tag{41}\]  
Figure 14: Straight level gear pair.