# Comparison of Crack Detection Performance of Low and High Contact Ratio Spur Gears
## 3 Calculation of TVMS for LCR and HCR Spur Gears with Root Cracks
\begin{array}{lcl}\sum\limits_{\text{i}=\text {L}\text{II}}\text{K}_{\text{i}}&\text{in}&|\text{AB}|\\ \sum\limits_{\text{i}=\text{I}}\text{K}_{\text{i}}&\text{in}&|\text{BD}|\\ \sum\limits_{\text{i}=\text{I},\text{III}}\text{K}_{\text{i}}&\text{in}&|\text {DE}|\end{array} \right. \tag{20}\]  
The power is transferred by sequentially engaging three and two teeth pairs in HCR spur gears. When the mesh is in the \(|\text{AB}|\), \(|\text{CD}|\), and \(|\text{EF}|\), the power is transmitted through the three teeth pairs. In other regions, power is transferred through two gear pairs, as seen in Figure 7.  
The equations of critical points on the line of action in HCR gears are given in Equations (21)-(28). \[\text{r}_{\text{pB}}=\sqrt{\left(\sqrt{\text{r}_{\text{ap}}^{2}-\text{r}_{ \text{bp}}^{2}}-2\text{p}_{\text{b}}\right)^{2}+\text{r}_{\text{bp}}^{2}} \tag{21}\]  
\[\text{r}_{\text{gB}}=\sqrt{\left(\sqrt{\text{r}_{\text{ag}}^{2}-\text{r}_{ \text{bg}}^{2}}-(\varepsilon_{\text{ad}}-2)\text{p}_{\text{b}}\right)^{2}+ \text{r}_{\text{bg}}^{2}}. \tag{22}\]  
\[\text{r}_{\text{pC}}=\sqrt{\left(\sqrt{\text{r}_{\text{ap}}^{2}-\text{r}_{ \text{bp}}^{2}}-(\varepsilon_{\text{ad}}-1)\text{p}_{\text{b}}\right)^{2}+ \text{r}_{\text{bp}}^{2}}. \tag{23}\]  
\[\text{r}_{\text{gC}}=\sqrt{\left(\sqrt{\text{r}_{\text{ag}}^{2}-\text{r}_{ \text{bg}}^{2}}-\text{p}_{\text{b}}\right)^{2}+\text{r}_{\text{bg}}^{2}}. \tag{24}\]  
\[\text{r}_{\text{pD}}=\sqrt{\left(\sqrt{\text{r}_{\text{ap}}^{2}-\text{r}_{ \text{bp}}^{2}}-\text{p}_{\text{b}}\right)^{2}+\text{r}_{\text{bp}}^{2}}. \tag{25}\]  
\[\text{r}_{\text{gD}}=\sqrt{\left(\sqrt{\text{r}_{\text{ag}}^{2}-\text{r}_{ \text{bg}}^{2}}-(\varepsilon_{\text{ad}}-1)\text{p}_{\text{b}}\right)^{2}+ \text{r}_{\text{bg}}^{2}}. \tag{26}\]  
\[\text{r}_{\text{PE}}=\sqrt{\left(\sqrt{\text{r}_{\text{ap}}^{2}-\text{r}_{ \text{bp}}^{2}}-(\varepsilon_{\text{ad}}-2)\text{p}_{\text{b}}\right)^{2}+ \text{r}_{\text{bp}}^{2}}. \tag{27}\]  
Figure 7: Meshing of HCR gears: (**A**) line of action; (**B**) calculation of mesh stiffness. \[\mathrm{r_{gE}}=\sqrt{\left(\sqrt{\mathrm{r_{ag}^{2}-r_{bg}^{2}}}-2\mathrm{p_{b}} \right)^{2}+\mathrm{r_{bg}^{2}}}. \tag{28}\]  
The TVMS is calculated with the following equation for the HCR gears. \[\mathrm{K_{mesh}}=\left\{ \begin{array}{ll}\sum\limits_{\mathrm{i=I,II,III}} \mathrm{K_{i}}&\mathrm{in}&\mathrm{|AB|}\\ \sum\limits_{\mathrm{i=II,II}}\mathrm{K_{i}}&\mathrm{in}&\mathrm{|BC|}\\ \sum\limits_{\mathrm{i=I,IV,II}}\mathrm{K_{i}}&\mathrm{in}&\mathrm{|CD|}\\ \sum\limits_{\mathrm{i=IV,I}}\mathrm{K_{i}}&\mathrm{in}&\mathrm{|DE|}\\ \sum\limits_{\mathrm{i=V,IV,J}}\mathrm{K_{i}}&\mathrm{in}&\mathrm{|EF|}\\ \end{array} \right. \tag{29}\]  
TVMS for different crack sizes, which are 0, 1.5, 3, and 4.5 mm, is calculated numerically. The variation of TVMS for LCR (Case I) and HCR (Case V) gears, depending on the mesh position, is shown in Figure 8. The change of TVMS for other cases can be seen in Figure A4 in Appendix A. The HCR gears' TVMS values are higher than the LCR gears for healthy and all crack sizes, due to more gear pairs taking part in power transmission. When the contact ratio increases, addendum and total tooth height increase; thus, the TVMS is decreased. Moreover, the regions where power is transferred by the three tooth pairs in the HCR gears increase considerably. The mesh stiffness of both LCR and HCR gears decreases significantly with the formation of the tooth root cracks. When the cracks propagate along the tooth width, the TVMS decreases continuously. As the crack progresses, TVMS decreases less in the HCR gears compared to the LCR gears.