# Comparison of Crack Detection Performance of Low and High Contact Ratio Spur Gears
## 4 Dynamic Analysis of LCR and HCR Gears with Tooth Cracks  
In this study, a dynamic model of a single-stage gearbox with six degrees of freedom was created for both LCR and HCR gears. The schematic image of the created six degrees of freedom model is shown in Figure 9. The effects of the motor and load inertias, the torsion of the shafts connected to the gears, the interactions of the gears with each other, and the bearings in which the gears are supported are examined. The change in dynamic force in LCR and HCR gears with different crack levels was examined as output by solving the dynamic model. Figure 8: TVMS results for LCR and HCR gears: (**A**) \(\mathrm{h_{a}}=1\)\(\mathrm{h_{f}}=1.25\) (LCR); (**B**) \(\mathrm{h_{a}}=1.5\)\(\mathrm{h_{f}}=1.75\) (HCR). The equation of motions of the motor, pinion, gear, load, and the bearings can be written as the following equations in (30)-(35). \[J_{M}\ddot{\theta}_{M}+C_{s1}\left(\dot{\theta}_{M}-\dot{\theta}_{P}\right)+K_{s1 }\left(\theta_{M}-\theta_{P}\right)=T_{in}. \tag{30}\]  
\[J_{P}\ddot{\theta}_{P}+C_{s1}\left(\dot{\theta}_{P}-\dot{\theta}_{M}\right)+K_{ s1}\left(\theta_{P}-\theta_{M}\right)=-DF_{b2}. \tag{31}\]  
\[J_{g}\ddot{\theta}_{g}+C_{s2}\left(\dot{\theta}_{g}-\dot{\theta}_{L}\right)+K_{ s2}\left(\theta_{g}-\theta_{L}\right)=DF_{b3}. \tag{32}\]  
\[J_{L}\ddot{\theta}_{L}+C_{s2}\left(\dot{\theta}_{L}-\dot{\theta}_{g}\right)+K_ {s2}\left(\theta_{L}-\theta_{g}\right)=-T_{out} \tag{33}\]  
\[m_{P}\ddot{Y}_{P}+C_{2}\left(\dot{Y}_{P}\right)+K_{2}\left(Y_{P}\right)=DF. \tag{34}\]  
\[m_{g}\ddot{Y}_{g}+C_{3}\left(\dot{Y}_{g}\right)+K_{3}\left(Y_{g}\right)=DF. \tag{35}\]  
where the dynamic force (DF) between teeth is:  
\[DF=K_{mesh}\Big{(}\theta_{P}r_{bp}-\theta_{g}r_{bg}+Y_{g}-Y_{P}\Big{)}+C_{ mesh}\Big{(}\dot{\theta}_{P}r_{bp}-\dot{\theta}_{g}r_{bg}+\dot{Y}_{g}-\dot{Y}_{P} \Big{)}. \tag{36}\]  
The dynamic transmission error can be defined as:  
\[DTE=\Big{(}\theta_{P}r_{bp}-\theta_{g}r_{bg}+Y_{g}-Y_{P}\Big{)}. \tag{37}\]  
The damping between the teeth is calculated by the following expression [31]:  
Figure 8:0  
where \(\tilde{\delta}_{t}\) is the damping ratio of the meshing gears and the value of \(\tilde{\delta}_{t}\) should be between 0.03 and 0.17 according to [32]. We selected a \(\tilde{\delta}_{t}\) of 0.1 in this study. The mass moments of inertias of motor, pinion, gear, and load are defined as:  
Figure 8:1  
Figure 8:2  
Figure 8:3  
Figure 8:4 6-DOF dynamic model of the one-stage gearbox. The torsional stiffness of the input and output shafts are defined as:  
Figure 8:5  
Figure 8:6  
where I\({}_{\text{s1}}\) is the polar moment of inertia of the shaft, G is the shear modulus of the material, and L is the length of the shafts. Damping of shafts is calculated with the following expression [32]:  
Figure 8:7  
Figure 8:8  
where \(\lx@overaccentset{{3}}{\mathcal{\tilde{2}}}_{\text{1}}\) and \(\lx@overaccentset{{3}}{\mathcal{\tilde{2}}}_{\text{2}}\) are the critical damping coefficients of shafts (defined between 0.005 and 0.075 [32]. The calculated constant values used in the dynamic model can be seen in Figure 8:9  
The MATLAB R2022b package program was used to solve the equations of motion. Six second-order differential equations were transformed into twelve first-order differential equations for the numerical solution. The equations of motion, written as twelve first-order differential equations, were solved by the fourth-order Runge-Kutta method. The change of dynamic force obtained from the dynamic model depending on the pin-ion rotation angle is given in \[J_{M}\ddot{\theta}_{M}+C_{s1}\left(\dot{\theta}_{M}-\dot{\theta}_{P}\right)+K_{s1 }\left(\theta_{M}-\theta_{P}\right)=T_{in}. \tag{30}\]0 The dynamic forces for the LCR gears (Case I) are seen in the left column and the dynamic force for the HCR gears (Case V) is seen in the right column. Dynamic analysis results obtained for other cases are shown in Figures A5-A7 in Appendix A.