# A Novel Measurement Standard for Surface Roughness on Involute Gears
## 3 Investigations of the Roughness of an Industrial Gear  
To provide a physical measurement standard, PMS, that is sufficiently representative of the gears used in the industry, a typical work piece has been chosen as the master work piece (see Figure 4). The master work piece represents a typical gear of a common industrial application. It is a spur gear as used in reverse gears in the automotive industry with a base radius of \(r_{\text{b}}=39.9\) mm and \(z=17\) teeth. The surfaces are finished by gear-grinding processes with rolling in accordance with the involutes. Even after super fine end finishing without rolling, the roughness features are visible and vary between root and tip of a tooth. Figure 5 shows a variation, the so-called _gradation_, of roughness parameters \(R_{\text{a}}\), \(R_{\text{q}}\) and \(R_{\text{k}}\) as functions of the position of the evaluation interval of the roughness profile \(z^{\text{R}}\) on the master work piece. To facilitate the examination of a tooth flank of the master work piece by the PTB home built profilometer HRTS, one of the teeth has been removed. Then, profiles have been scanned on the exposed tooth from the root to the tip. Scanning a tooth on the gear enables a comparison measurement on the P40 instrument. The scanning direction is referred to as \(x\)-direction. The profiles have been processed by form removal and waviness filtration. The measured profiles have a length of 8 mm. The roughness parameters in Figure 5 have been obtained, as explained in Section 2.5, by using an evaluation length of \(l_{\text{e}}=4.0\) mm. The roughness parameters as functions of the positions \(x_{\text{m}}\) of evaluation intervals \([x_{\text{m}}-\frac{1}{2}l_{\text{e}},x_{\text{m}}+\frac{1}{2}l_{\text{e}}]\), i.e., as \(R_{\text{a}}(x_{\text{m}})\), \(R_{\text{q}}(x_{\text{m}})\) and \(R_{\text{z}}(x_{\text{m}})\), clearly show a change between root and tip region of a tooth. This reveals that the relocalisation of the evaluation region is  
Figure 4: A typical spur gear used in industry with \(z=17\) teeth and base radius \(r_{\text{b}}=39.9\) mm taken as master piece for investigations to aid the process of designing a physical measurement standard. Figure 5: Roughness parameters as function of the position \(x_{\text{m}}\) of the evaluation interval; the master gear shows that the roughness is _graded_ along the profile. an important issue. For the design of a physical measurement standard, this issue includes two aspects:  
* One aspect is that the dependence of the roughness parameters on the position of the evaluation region should be negligible by preparing a surface texture that is homogeneous. * The other aspect is the macroscopic geometry which is the involute shape with root and tip of a tooth and its orientation and position relative to the base circle. The involute shape is a function of the base radius \(r_{\text{b}}\). In order to remove the form from the profile measured by the HRTS, the data points of the cartesian system of the profiler are related to the involute shape and orientation, as explained in Section 2.2. This optimization procedure has specifically been developed for these investigations and is not available in any kind of software of commercial form testers and profilers. Therefore, an alternative form operator that is more easily obtainable, such as a smoothing spline, has been employed to compare the influence of form operation. To compare the gear measuring system P40 with the profilometer HRTS, the same tooth of the master work piece has been scanned with both instruments. Furthermore, the HRTS measured profile has been processed by three different methods resulting in four types of primary profiles \(z^{\text{P}}\):  
1. P40 measurement system;
2. HRTS: form removal via the fitting algorithm delivering lateral coordinates on the arc of the involute and vertical perpendicular to the arc;
3. HRTS: form removing the form via a smoothing spline filter delivering lateral coordinates on a chord;
4.