# Comparison of Crack Detection Performance of Low and High Contact Ratio Spur Gears
## 2 Gear Design and Determination of Crack Paths  
Although there are different gear profiles that satisfy the law of gearing, the involute profile is the most used profile in gear geometry today due to its advantages such as ease of manufacturing, not requiring precision of the distance between the axes for its operation, and the ability to manufacture different gears using the same tool. Gears with an involute profile are basically formed by the combination of two different curves, the side surfaces of which are involute and the root parts of which are trochoid. There are many studies in the literature on the design of gears with involute profiles ([PERSON] [25], [PERSON] et al. [26]). In this study, mathematical modeling of the gear geometries was undertaken using the rack-type cutting tool, and 2D CAD geometries were created. The cutting tool reference profile used in creating the models is shown in Figure 2.  
The cutting tool geometry consists of the vertical axis and two linear regions inclined by the pressure angle \"\(\alpha\)\" and the radius region at the head of the cutting tool. The linear regions form the involute curve of the tooth geometry, and the radius region in the cutting tool tip area forms the trochoid region of the tooth geometry. The \"\(\alpha\)\" angle with the vertical axis determines the pressure angle of the gear, ha determines the tooth head height (addendum), and h\({}_{\ell}\) determines the base height (dedendum). The (ce) and (df) regions of the cutting tool constitute the tooth root region. Tooth designs with different geometric features can be made by taking different values of all these parameters. In this study, LCR and HCR gear designs with different features were realized by changing the ha and hf parameters. The design parameters of LCR and HCR gears are given in Table 1. Details about the gear design using the rack-type cutting tool can be found in reference [27]. In order to determine the crack initiation point in the crack propagation analysis, static structural analysis was performed, and the coordinates of the point where the maximum stress occurred at the root of the tooth were obtained. Crack propagation analyses were carried out after the crack initiation point was determined as a result of static structural analysis. Crack propagation analyses were performed using the ANSYS 19.1 Workbench SMART Crack Growth module. This module needs an artificial crack to perform crack propagation analysis. Therefore, an initial crack was opened artificially at the determined crack starting point. The artificial crack initiation points were positioned at a 45\({}^{\circ}\) angle horizontally in the crack propagation analysis. Since the change in the crack initiation angle directly affects the direction of crack propagation, the initiation angle was determined to be 45\({}^{\circ}\) neutral concerning horizontal and vertical angles. Crack length and width were determined as 0.3 mm and 0.2 mm, considering the finite element mesh size used in the analysis (Figure 3). Figure 2: Rack-type cutting tool basic dimensions. The mechanisms of crack propagation resulting from repeated loading were analyzed using Paris' law. A standard fatigue crack propagation law describes the increase in crack length as a function of the stress intensity factor and the stress ratio [13]. The general form of Paris' law is presented in Equation (1). \[\frac{\text{da}}{\text{dN}}=\text{f(K,R)} \tag{1}\]  
In this context, a represents the crack elongation, N denotes the total number of loading cycles, and da/dN indicates the fatigue crack propagation rate per cycle. K is the Mode I stress intensity factor at a specific moment, while R, set to 0 for this study, refers to the load ratio. \[\text{R}=\frac{\text{K}_{\text{min}}}{\text{K}_{\text{max}}} \tag{2}\]  
Fatigue crack propagation is typically categorized into three distinct regions, as illustrated in Figure 4a. In the first region, Kth indicates that the threshold value has been surpassed, although crack growth remains very limited.