# Mesh Stiffness and Dynamic Modeling and Analysis of Modified Straight Bevel Gears
## 2 Time-Varying Meshing Stiffness Calculation Model
### Standard Straight Bevel Gear  
Based on elasticity, the total meshing stiffness of a single gear pair can be expressed as [6]:  
\[K_{e}=\frac{1}{\frac{1}{k_{h}}+\frac{1}{k_{f}^{2}}+\frac{1}{k_{b}^{2}}+\frac{1}{k _{a}^{2}}+\frac{1}{k_{s}^{2}}+\frac{1}{k_{f}^{3}}+\frac{1}{k_{b}^{4}}+\frac{1}{ k_{a}^{4}}+\frac{1}{k_{s}^{4}}} \tag{8}\]  
The bending stiffness, shear stiffness, axial compression stiffness, tooth blank base stiffness, and Hertz contact stiffness for tooth pair \(i=p,g\) are denoted as \(k_{b},k_{s},k_{a},k_{f}\), and \(k_{h}\), respectively. As the tooth shape of SBGs varies along the width direction, it is not possible to directly calculate the complete SBG bending stiffness, shear stiffness, axial compression stiffness, and tooth blank base stiffness. Therefore, as illustrated in Figure 4, base on reference [29], slicing is applied to SBGs. The slices are oriented perpendicular to the contact line, and the TCA outcomes are integrated into each cross-section to compute the meshing stiffness. The bending stiffness, shear stiffness, and axial compression stiffness for each slice \(j\) (\(j=1,2,\ldots\), n) are represented as \(\delta k_{b}\), \(\delta k_{s}\) and \(\delta k_{a}\), respectively. These values are cumulatively obtained by summing them according to the number of slices, resulting in \(k_{b},k_{s}\) and \(k_{a}\).  
According to the potential energy method, the bending energy \(\delta U_{b}\), shear energy \(\delta U_{s}\), and axial compressive energy \(\delta U_{a}\) for each slice are expressed as follows:  
\[\delta U_{b} =\frac{F^{2}}{2 dk_{b}}=\int_{0}^{d}\frac{3(F_{b}(t-x)-F_{a}h)^{2}} {4 Eh_{x}^{3}b}dx, \tag{9}\] \[\delta U_{s} =\frac{F^{2}}{2\delta k_{s}}=\int_{0}^{d}\frac{1.2F_{b}^{2}}{4 Gh_{ x}b}dx,\] \[\delta U_{a} =\frac{F^{2}}{2\delta k_{a}}=\int_{0}^{d}\frac{F_{a}^{2}}{4 Eh_{x}b }dx\]  
Bending stiffness \(\delta k_{b}\), shear stiffness \(\delta k_{s}\), axial compressive stiffness \(\delta k_{a}\) for each slice can be obtained [30].  
\[\delta k_{b} =\frac{F^{2}}{2\delta U_{b}}=\int_{0}^{d}\frac{2 Eh_{x}^{3}b}{3( \cos\alpha(t-x)-\sin\alpha h)^{2}}dx, \tag{10}\] \[\delta k_{s} =\frac{F^{2}}{2\delta U_{s}}=\int_{0}^{d}\frac{2 Gh_{x}b}{1.2\cos \alpha^{2}}dx,\] \[\delta k_{a} =\frac{F^{2}}{2\delta U_{a}}=\int_{0}^{d}\frac{2 Eh_{x}b}{\sin \alpha^{2}}dx\]  
Figure 4: Slicing the straight bevel gear.  
\(h\) and \(l\) denote the distances from the meshing point to the gear teeth's line of symmetry and to the base circle, respectively. At this meshing point, the interaction force \(F\) can be decomposed into a radial force \(F_{a}\) and a tangential force \(F_{b}\) along the direction of the meshing line. The variables for the elastic modulus and shear modulus are represented by \(E\) and \(G\), respectively.  
\[k_{b}=\sum_{j=1}^{n}\delta k_{b},k_{a}=\sum_{j=1}^{n}\delta k_{a},k_{s}=\sum_{j =1}^{n}\delta k_{s} \tag{11}\]  
The Hertz contact stiffness is obtained based on the applied load [6]:  
\[k_{h}=\frac{E^{0.9}B^{0.8}F^{0.1}}{1.275} \tag{12}\]  
The fillet-foundation stiffness is obtained based on the geometric shape of the tooth germ [31].  
\[U_{f}=\frac{1}{2G}\underset{lh}{\int}dx\underset{A}{\int}\left(\frac{T\rho}{ I_{p}}\right)^{2}dA \tag{13}\]  
\[K_{f}=\frac{F^{2}}{2 Ul_{f}} \tag{14}\]  
\(T\) is the torque, \(I_{p}\) is the polar moment of inertia of the section, \(\rho\) is the distance from the point on the cross-section to the center of the circle.  
Based on TCA and determining single or double-tooth contact regions through overlap assessment, the TVMS of SBGs is formulated as follows:  
\[K=\sum_{l=1}^{2}K_{el} \tag{15}\]  
where \(l\) denotes the number of meshing pairs.