## 2 The Analytical Model of Mesh Stiffness of Helical Gear
### The Stiffness of Each Sliced Gear
#### 2.2.1. Tooth Stiffness  
There is deformation of the tooth with the effect of axial force. The tooth can be considered as a cantilever beam whose root is fixed, as shown in Figure 3.  
According to beam theory in elastic mechanics, the bending, shear, and torsional energy with the axial mesh force can be calculated by  
\[U_{\text{atb}}=\int_{0}^{d}\frac{M_{\text{atb}}^{2}}{2 EI_{\text{atb}}}dy,U_{ \text{atb}}=\int_{0}^{d}\frac{1.2F_{\text{a}}^{2}}{2 GA_{2}^{2}}dy,U_{\text{attt} }=\int_{0}^{d}\frac{M_{\text{att}}^{2}}{2 GI_{\text{atp}}}dy \tag{10}\]  
According to [PERSON]'s law, the potential energy can be written as follows:  
\[U_{\text{atb}}=\frac{F_{\text{i}}^{2}}{2k_{\text{atb}}},U_{\text{atb}}=\frac{F _{\text{i}}^{2}}{2k_{\text{atb}}},U_{\text{att}}=\frac{F_{\text{i}}^{2}}{2k_{ \text{att}}} \tag{11}\]  
As a result, the bending, shear, and torsional stiffness of the tooth with axial mesh force can be obtained as  
\[\frac{1}{k_{\text{atb}}}=\int_{0}^{d}\frac{\sin^{2}\beta_{\text{b}}\cdot(d-y)^ {2}}{EI_{\text{atb}}}dy,\frac{1}{k_{\text{atb}}}=\int_{0}^{d}\frac{1.2\cdot \sin^{2}\beta_{\text{b}}}{GA_{2}}dy,\frac{1}{k_{\text{att}}}=\int_{0}^{d} \frac{\sin^{2}\beta_{\text{b}}\cdot h^{2}}{GI_{\text{atp}}}dy \tag{12}\]  
where \(G\) and \(E\) represent shear and Young's moduli, respectively. \(I_{\text{atb}}\) denotes the cross-sectional moment of inertia to the neutral axis. \(I_{\text{atp}}\) denotes the cross-sectional polar moment of inertia.  
Moreover, there also exists deformation of the tooth with the effect of transverse force. The tooth also can be considered as a cantilever beam whose root is fixed. The sketch of the tooth with the transverse force is shown in Figure 4.  
Figure 3. The sketch of the tooth with the axial mesh force.  
Figure 2. The decomposition sketch of mesh force of a helical gear.  
According to beam theory in elastic mechanics, the bending, shear, and radial compression energy with the transverse mesh force can be calculated as follows:  
\[U_{\text{ttb}}=\int_{0}^{d}\frac{M_{\text{ttb}}^{2}}{2 EI_{\text{ttb}}}dy,U_{ \text{tts}}=\int_{0}^{d}\frac{1.2F_{\text{tb}}^{2}}{2 GA_{2}}dy,U_{\text{ttr}}= \int_{0}^{d}\frac{F_{\text{ta}}^{2}}{2 EA_{2}}dy \tag{13}\]  
According to [PERSON]'s law, the potential energy can be written as follows:  
\[U_{\text{ttb}}=\frac{F^{2}}{2k_{\text{ttb}}},U_{\text{tts}}=\frac{F^{2}}{2k_{ \text{tts}}},U_{\text{ttr}}=\frac{F^{2}}{2k_{\text{ttr}}} \tag{14}\]  
As a result, the bending, shear, and radial compression stiffness of the tooth with transverse mesh force can be obtained as  
\[\begin{array}{l}\frac{1}{k_{\text{ttb}}}=\int_{0}^{d}\frac{\cos^{2}\beta_{ \text{b}}\cdot[\cos a_{\text{tc}}\cdot(d-y)-h\cdot\sin a_{\text{tc}}]^{2}}{EI_{ \text{ttb}}}dy\\ \frac{1}{k_{\text{tts}}}=\int_{0}^{d}\frac{1.2\cdot\cos^{2}\beta_{\text{b}} \cdot\cos^{2}a_{\text{tc}}}{GA_{2}}dy\\ \frac{1}{k_{\text{ttr}}}=\int_{0}^{d}\frac{\cos^{2}\beta_{\text{b}}\cdot\sin^{ 2}a_{\text{tc}}}{EA_{2}}dy\end{array} \tag{15}\]  
where \(I_{\text{ttx}}\) denotes the cross-sectional moment of inertia to the neutral axis. \(A_{2}\) denotes the area of a tooth section perpendicular to the transverse face.  
The total energy stored in a tooth with total mesh force can be written as  
\[U_{\text{t}}=U_{\text{atb}}+U_{\text{ats}}+U_{\text{attt}}+U_{\text{ttb}}+U_{ \text{trs}}+U_{\text{trr}} \tag{16}\]  
As a result, the comprehensive tooth stiffness of a sliced gear can be computed as  
\[\frac{1}{k_{\text{t}}}=\frac{1}{k_{\text{atb}}}+\frac{1}{k_{\text{ats}}}+\frac {1}{k_{\text{attt}}}+\frac{1}{k_{\text{ttb}}}+\frac{1}{k_{\text{ttb}}}+\frac{ 1}{k_{\text{ttr}}} \tag{17}\]