## 2 Meshing Stiffness and Vibration Response for Cracked Gears in Different Shape Parameters
### Meshing Stiffness Calculation under a Universal Equation of Gear Profile
#### 2.1.3 Meshing Stiffness of a Gear Pair  
The meshing stiffness of a gear pair can be expressed by considering the meshing condition:  
\[\frac{1}{k_{\mathrm{sr}}}=\frac{1}{k_{h}}+\frac{1}{k_{bx1}}+\frac{1}{k_{xx1}} +\frac{1}{k_{xx1}}+\frac{1}{k_{f1}}+\frac{1}{k_{bx2}}+\frac{1}{k_{xx2}}+\frac{ 1}{k_{x2}} \tag{13}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **The Main Meaning** \\ \hline \(F\) & the meshing force on the tooth curve \\ \(F_{b}\) & the y-direction component of meshing \(F\) \\ \(F_{a}\) & the x-direction component of meshing \(F\) \\ \(l_{sg}\) & rotational inertia of gear on the excessive part curve \\ \(d_{C}\) & distance between the ending point on the excessive curve and tooth root \\ \(h_{G}\) & distance between the end point of excessive curve and tooth midline \\ \(h_{xg}\) & distance between the tooth midline and the contact point on the \\ \(x_{g}\) & the distance between the tooth root and the points on the excessive \\ \(x_{g}\) & curve in x-direction \\ \(\varphi_{f}\) & rolling angle at the tooth root \\ \(\varphi_{C}\) & the rolling angle corresponding to the excessive curve ending point \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters for the transition part of the curve.  
where  
1 gear of the gear pair, 2 is the pinion of the gear pair, \(k_{h}\) is the Hertz stiffness [11], and \(k_{f\!I}/k_{f\!2}\) is the fillet-foundation stiffness of the gear/pinion [13].  
To verify the accuracy of the universal equation of gear profile, the calculated meshing stiffness was compared with the FEM method, as shown in Figures 3 and 4. Both figures show the change in time-varying mesh stiffness (TVMS) with the change in the rotation angle. Figure 3 is the meshing stiffness calculated by FEM, and Figure 4 is the meshing stiffness by the universal equation of the gear profile. The max error of the universal equation of the gear profile was 2.15% when compared to the FEM, and the veracity of the universal equation of the gear profile has been proven.  
Differential equations are established in simulation modeling. The \(xy\)-direction motion equations are shown as follows:  
\[M\ddot{x}=F_{d} \tag{14}\]  
\[\Rightarrow\left[\begin{array}{cccc}M_{1}&&M_{2}&&\\ &&M_{2}&&\\ &&M_{1}&\\ &&&M_{2}\end{array}\right]\left[\begin{array}{c}\ddot{x}_{1}\\ \ddot{x}_{2}\\ \ddot{y}_{1}\\ \ddot{y}_{2}\end{array}\right]=\left[\begin{array}{c}-K_{x1}x_{1}-C_{x1} \dot{x}_{1}+f_{1}\\ -K_{x2}x_{2}-C_{x2}\dot{x}_{2}+f_{2}\\ -K_{y1}y_{1}-C_{y1}\dot{y}_{1}+N\\ -K_{y2}y_{2}-C_{y2}\dot{y}_{2}+N\end{array}\right]. \tag{15}\]  
The rotary motion equations are shown as follows:  
\[I\ddot{\theta}=T_{d} \tag{16}\]  
\[\Rightarrow\left[\begin{array}{cc}I_{1}&&\\ &I_{2}\end{array}\right]\left[\begin{array}{c}\ddot{\theta}_{1}\\ \dot{\theta}_{2}\end{array}\right]=\left[\begin{array}{c}R_{b1}N+T_{p}+M_{1} \\ M_{2}-R_{b2}N-T_{g}\end{array}\right] \tag{17}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Value** \\ \hline Meshing frequency & & \(f_{m}\)= 1000 Hz \\ Torsional stiffness of bearings & \(K_{x1}=K_{y1}=K_{x2}=K_{y2}=6.56\times 10^{8}\) Nm/rad \\ Damping coefficient of bearings & \(C_{x1}=C_{y1}=C_{x2}=C_{y2}=1.8\times 10^{3}\) Nm/rad \\ Damping coefficient of meshing teeth & & \(C_{t}\)= 67 ns/m \\ Coefficient of friction & & \(u\)= 0.06 \\ \hline \end{tabular}
\end{table}
Table 4: Main parameters of the transmission system.  
Figure 5: Dynamic model for a gearbox.  
where  
\[N=K_{t}(R_{b1}\theta_{1}-R_{b2}\theta_{2}-y_{1}+y_{2})+C_{t}\Big{(}R_{b1}\dot{ \theta}_{1}-R_{b2}\dot{\theta}_{2}-\dot{y}_{1}+\dot{y}_{2}\Big{)}. \tag{18}\]  
The \(f_{1}\) is the friction force of the pinion. It can be expressed as:  
\[f_{1}=uN_{1} \tag{19}\]  
The \(f_{2}\) is the friction force of the gear. It can be expressed as:  
\[f_{2}=-uN_{2} \tag{20}\]  
where  
\(N_{1}\) is the meshing force of the first tooth pair, and \(N_{2}\) is the meshing force of the second tooth pair.