# 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  
The TVMS of the LCR and HCR gears are calculated by using numerical methods such as finite element analysis and MATLAB programming in this study. The single tooth stiffness (STS) of the LCR and HCR gears must be calculated for the definition of the TVMS. The STS of the LCR and HCR gears are calculated according to the following equation. More details are given for calculating the STS using the numerical methods used in this study in [11; 20; 23; 30]. \[\mathrm{k_{pl,gl}}=\frac{\mathrm{F_{n}}}{\mathrm{x_{pl,gi}}}. \tag{11}\]  
where k is the STS of any radius of the gear, p represents pinion, g represents the gear, \(\mathrm{F_{n}}\) is the applied force on the gear, and x is the displacement values obtained from the FE analysis. In spur gears, power transfer occurs in a straight line. This line is specifically called the line of action (Figure 6). Gear mesh begins at the tooth tip of the driven gear and at the point slightly above the base circle of the pinion gear. If this point is defined as point A, the radius of point A can be calculated with the following expression. \[\mathrm{r_{Ap}}=\left[\mathrm{r_{bp}^{2}}+\left(\left(\mathrm{r_{p}}+\mathrm{r_{ g}}\right)\mathrm{sin}\,\alpha-\left(\mathrm{r_{ag}^{2}}-\mathrm{r_{bg}^{2}} \right)^{0.5}\right)^{2}\right]^{0.5}. \tag{12}\]  
When the mesh reaches point B, a gear pair comes out of contact. This point is named the lowest point of single tooth contact (LPSTC) and the radius of the LPSTC is defined using the following equation. \[\mathrm{r_{Bp}}=\left[\mathrm{r_{bp}^{2}}+\left(\left(\mathrm{r_{ap}^{2}}- \mathrm{r_{bp}^{2}}\right)^{0.5}-\mathrm{\pi m_{n}cos}\,\alpha\right)^{2} \right]^{0.5}. \tag{13}\]  
The power is transferred between the \(\mid\)BD\(\mid\) via a single pair of teeth. When the mesh reaches point D, a new gear pair begins to engage, and again, the moment is transferred by two gear pairs. This point is named the highest point of single tooth contact (HPSTC) and the radius of the HPSTC is defined using the following equation. \[\mathrm{r_{Dp}}=\left[\mathrm{r_{bp}^{2}}+\left(\left(\mathrm{r_{bp}}+\mathrm{ r_{bg}}\right)\mathrm{tan}\,\alpha-\left(\mathrm{r_{ag}^{2}}-\mathrm{r_{bg}^{2}} \right)^{0.5}+\mathrm{\pi m_{n}cos}\,\alpha\right)\right]^{2}\right]^{0.5}. \tag{14}\]  
The meshing process ends when the pinion gear reaches the tooth head. This point is also described as point E and the addendum circle. The radius of the addendum circle can be calculated using Equation (15). \[\mathrm{r_{Ep}}=0.5\big{(}\mathrm{m_{n}}\mathrm{r_{p}}+2\mathrm{m_{n}}\big{)}. \tag{15}\]  
The total length of the contact can be written as:  
\[\mathrm{\left|AE\right|}=\sqrt{\mathrm{r_{ap}^{2}}-\mathrm{r_{bp}^{2}}}+\sqrt{ \mathrm{r_{ag}^{2}}-\mathrm{r_{bg}^{2}}}-\mathrm{a_{d}}\mathrm{sin}\,\alpha. \tag{16}\]  
The pitch of the gear is defined as:  
\[\mathrm{P_{b}}=\left|\mathrm{BE}\right|=\left|\mathrm{AD}\right|=\mathrm{\pi m _{n}cos}\,\alpha. \tag{17}\]  
The contact ratio of the spur gears can be calculated using the following equation:  
\[\varepsilon_{\alpha}=\frac{\left|\mathrm{AE}\right|}{\mathrm{\pi m_{n}cos}\, \alpha}. \tag{18}\]  
Figure 6: Meshing of LCR gears: (**A**) line of action; (**B**) calculation of mesh stiffness. The teeth in contact during the mesh are considered as springs connected in series with each other. A single pair in the mesh is calculated using the following equation with this approach. \[\text{K}_{\text{i}}=\frac{\text{k}_{\text{pi}}\text{k}_{\text{gi}}}{\text{k}_{ \text{pi}}+\text{k}_{\text{gi}}}. \tag{19}\]  
Gear pairs in synchronous meshing are considered as springs connected in parallel with each other. Thus, the TVMS is calculated using the following expression for the LCR gears. \[\text{K}_{\text{mesh}}=\left\{