The gears examined in this paper (Fig. 2, diameter approx. 39 mm) were produced by injection moulding using a polyoxymethylene (POM) of the type Hostaform C9021 by Celanese Services Germany GmbH, Kelsterbach. POM is often used in gear applications due to its self-lubricating properties and resulting good dry running capabilities and low friction and wear (Piorkowska and Rutledge, 2013). The semi-crystalline structure of the material leads to a process-dependent microstructure. In injection moulding processes, an optically amorphous edge layer is formed due to the high cooling rate in the areas with contact with the mould material. This leads to the formation of crystalline nanostructures with sizes below the wavelength of visible light. As a result, the edge layer appears optically amorphous (Menges et al., 1976). This semi-transparent edge layer leads to difficulties in optical measurements, especially with structured-light methods, due to the scattering of the projected lines into the material. Thus, the surface cannot clearly be determined. A further factor complicating optical measurements on this material is its gloss. To achieve useful measurements, the optical properties of the material have to be altered.

Using colourants, i.e. pigments with or without carrier materials, the optical properties of polymer materials can be altered easily and cost-efficiently. Pigments and other nano-sized particles act as scattering centres, reducing the translucency of the optically amorphous skin layers. They can also decrease the amount of directed reflection, reducing gloss (Raymond and Ronca, 2016). There are the three main methods of colourizing polymers with pigments, each with their own use case (Müller, 2012). Firstly, pre-coloured granulate, made by adding pigments to the base material, is used for large batches of material and frequently uses colours such as black and white (Müller, 2012). Secondly, for the individual colouration of smaller batches, master batches, highly pigment-filled granules, are used (Müller, 2012). Thirdly, compared to master batches, liquid colours allow for a more flexible mixing of pigments and a wider range of colours (Müller, 2012). The cost of colouration depends on the choice of pigment, way of colouration and dosage. Master batches and liquid colours are more expensive than pre-dyed material, but they are more flexible. Pigment cost can vary widely from inexpensive carbon to more expensive mineral- or metal-based effect pigments.

There are two main effects of colourants on part properties. On the one hand colourants can influence the crystallization either by acting as a nucleating agent or by hindering nucleation, depending on chemical interactions of the polymer and the colourant. On the other hand the colourant itself can effect changes, which can be subdivided into effects of the carrier material and effects due to the pigment. In general, these effects are superimposed and cannot be distinguished easily.

Nucleating effects of pigment-based colourants on the base material are observed frequently. This is due to the fact that both nucleating agents (Richaud, 2014) and colourants (Müller, 2012) are usually small in size (a few hundred nanometres) and require chemically compatible surfaces, good dispersion and insolubility to ensure proper functionality. As a result pigments can introduce additional crystallization centres leading to parts which exhibit a finer, more crystalline microstructure if compared to the base material (Müller, 2012). Lower cooling rates in the material can lead to a similar effect by allowing more time for the material itself to form nucleation centres (Richaud, 2014). In the case of POM two commonly used pigments, white titanium dioxide (Wacharawichanant et al., 2012) and carbon black (Richaud, 2014), show nucleating effects. One consequence of higher crystallinity is increased shrinkage and therefore deviations from desired geometric dimensions. Mechanical properties, like yield strength (Darras and Séguéla, 1993), break elongation, stiffness and impact strength (Piorkowska and Rutledge, 2013), are also often increased with a finer and highly crystalline microstructure. In pin-on-disk tests materials with a finer, highly crystalline morphology showed improved tribological properties, especially reduced wear rates (Künkel, 2005). However, even with titanium dioxide showing nucleation properties on POM (Wacharawichanant et al., 2012), a chemically related pigment, chrome antimony titanium rutile, showed de-nucleating effects and a related decrease in mechanical and tribological properties (Schubert et al., 2019). Thus, due to the complex interactions of pigment, carrier system and base polymer there is no general approach to predict whether a colourant has a nucleating, de-nucleating or no effect on the base polymer.

Due to the aforementioned complex interactions of pigments and base material, no general trend can be given for the effects of the materials itself. In regard to light and heat stability pigments can have widely varying impacts on POM. For example, carbon black has excellent properties as a stabilizer and UV absorber (Richaud, 2014) and titanium dioxide can selectively absorb harmful radiation and reduce material ageing (Wallner et al., 2017); in the presence of moisture, however, it can act as a photo-catalyst increasing degradation (Richaud, 2014). There are even pigments which almost exclusively have negative effects on light stability, especially red pigments (Klemchuk, 1983). Research on the influence of pigments and other nanoparticles, like carbon nanotubes (Yousef et al., 2016) or organoclay (Jose and Alagar, 2011), on the mechanical and tribological properties also show no general trends. Particle material, size and dosage highly influence whether nanoparticles have beneficial or disadvantageous effects (Wacharawichanant et al., 2012).

To exclude influences of carrier materials, two pigment-based colourants were chosen for this research. The pigments were added by means of direct compounding in order to achieve a non-translucent, low-gloss surface with homogenous pigment distribution to improve the optical properties of the gear specimens for further optical measurements. The following pigments were used: a white titanium dioxide (TiO₂) pigment of the type Dei^®Pow – Weiß (Fig. 2, right), provided by Deifel GmbH & Co. KG, Schweinfurt, and a pigment mixture in the ratio of 1:1 of the (Ti, Cr, Sb) O₂ pigment with a carbon black (C) pigment of the type Dei^®Pow – Spezialschwarz (Fig. 2, left), also provided by Deifel GmbH & Co. KG. An injection moulding machine of the type Arburg 370 U-700-30-30 with a plasticizing unit with an 18 mm screw diameter by Arburg GmbH & Co. KG, Loßburg, Germany, was used to produce the gear specimens. Table 1 shows the processing parameters used for manufacturing.

Figure 2Coloured gear wheels: ID_DG (identifier dark green, left) and ID_W (identifier white, right).

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Table 1Overview of processing parameters.

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The goal of this paper was to evaluate the locally distributed single point uncertainties for gear wheel measurements using a structured-light scanner. The used system was GOM ATOS CORE 200 5M (GOM GmbH, 2019) in combination with a rotatory stage (Fig. 3). The sensor was controlled by the GOM ATOS Professional 2018 software (GOM GmbH, 2018). The internal settings for the sensor were set to the least restrictive options possible, thus making sure that as many surface points as possible were recorded during the measurement. The illumination settings were determined by user-controlled selection of a target of interest on the live image once before each measurement series. During the measurements, all external light sources in the laboratory were dimmed to a minimum in order to ensure the exclusive illumination by the measuring device. The measurement object (gear wheel) was mounted onto the rotatory stage through its accessible rotatory axis. The used bracket was equipped with reference markers, which enabled the GOM ATOS Professional 2018 software to perform the referencing of each image with the angular position of the rotatory stage. The sensor is characterized by the following key data: measuring range 200×150 mm², lateral resolution 80 µm, measuring distance 250 mm and native camera resolution 5 million pixels. Before measurement, the sensor was calibrated using the CP40 170 40760 calibration plate consisting of 3657 calibration points. A single measurement consisted of 200 single images (projections), which were equally distributed over a full rotation (360^∘) of the rotatory stage, with two different sensor positions each: one position captured the gear wheel sloping from above the gear wheel and the other from below, with both angles roughly equalling 45^∘. As soon as all images were recorded from that position, the scanner was manually moved to the second position and the second set of 200 projections was recorded. The mounting of the measurement object (including the reference markers) was only done once and therefore not altered during the recording of all measurements. The reference markers were visible to the scanner from both senor positions, which made the data fusion of several image stacks possible. Thus, the GOM software merged all of the 400 images (200 for each sensor position) together into one surface representation of the gear wheel in a triangulated mesh format (STL). Each measurement was repeated 20 times to ensure a statistically valid result for the following single point uncertainty calculations. During the polygonization routine, which is required to transform the (point cloud) measurement data into the desired measurement result (surface data in the form of a triangle mesh), the GOM software offers five different polygonization settings, which differ in the number of surface points and thus (implicit) filtering of the measurement data: no post-processing, more details, standard, less details and smallest data volume. A reduction of the surface point density is problematic if additional measurement errors are introduced because of the reduction of the information density. But on the other hand, an unnecessary high point density can lead to increased computation requirements without adding to the quality of the measurement data. A perfect polygonization routine would implicitly (locally) filter out all spatial frequencies, which are higher than the (local) structural resolution of the measurement system. Both measurement series were processed with each of the five quality settings in order to be able to examine the effect of the polygonization routine on the observed measurement uncertainty. The measurements were performed in a temperature-controlled environment with temperatures set to 20 ^∘C ±1 K. Overall, two different gear wheels were examined, and thus 40 single measurements (each with two sensor positions) were performed. Both required reference measurements were captured using the Zeiss Metrotom 1500 industrial X-ray computed tomography system (Carl Zeiss Industrielle Messtechnik, 2019). The gear wheel was positioned on the rotatory stage of the CT system using the bracket already presented in Fig. 3. The upper part of the bracket consisted of polymer, which was easy to penetrate by X-rays and did not lead to CT-specific sources of errors due to a possibly high material density. CT systems are in general very well suited for measuring polymer parts with high accuracy. To ensure a high-quality measurement, lengthy device settings were chosen in order to guarantee a very low measurement uncertainty of the reference measurement system. Because of the fact that the polymer colouration had no measurable impact on the absorption properties of the measurement object, these following settings were used for both reference measurements: tube voltage 150 kV, tube current 400 µA, resulting nominal focal spot size 60 µm, X-ray filter 0.5 mm Cu, geometrical magnification 9.0, resulting voxel size 22.3 µm, integration time 1000 ms, number of projections averaged 10, number of projections per measurement 2050. The required surface determination was performed using VGStudio Max 3.2 (VGS) (Volume Graphics GmbH, 2019) using the settings automatic grey value threshold, search distance 16 voxels, remove all particles and voids, and iterative surface determination. The measured surface was then exported as a triangle mesh in the STL format using high-quality settings. The measurement was subsequently registered against the nominal geometry (CAD) using VGS. Finally, the gear sprocket was extracted from the full surface using our own MATLAB program by removing all triangles with an edge point distance to the rotatory axis of smaller than 16 mm. The authors are aware of the fact that an industrial CT system might not necessarily show improved measurement uncertainty characteristics for all applications compared to a structured-light scanner. Nonetheless, these measurement devices were chosen to demonstrate the principle approach for the determination and evaluation of the single point uncertainty framework as a tool in dimensional metrology.

Figure 3Figurative representation of the measurement setup, consisting of a structured-light scanner (here GOM ATOS Compact Scan 2M/300) in combination with a rotatory stage.

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Figure 4Examination of the geometry-altering mesh operations using a nominal–actual comparison and a Gaussian fit.

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As mentioned above, the sampling strategy normal vector requires trustworthy sampling vectors in order to compute the metrologically correct distances. CT measurements are often subject to high-frequency noise with location wavelengths considerably higher than the structural resolution of the measurement system. In order to remove very high spatial frequencies and to improve the triangle mesh properties regarding triangle edge size and homogeneity, the generated surface of the reference measurement was undertaken with a post-processing routine using GOM ATOS Professional 2018. First, the mesh was smoothed using the Mesh → Smooth… operation with settings very large filter radius, very low detail sharpness and surface tolerance 10 µm. After that the homogeneity of the surface point distribution was improved using the Mesh → other → Regularize… operation with settings surface tolerance 10 µm and maximum edge length 50 µm. Of course these kinds of operations need to be used with special care, because any filtration of the reference measurement results in altered uncertainty values for the examined structured-light measurement device. Because of that, the effect of the described mesh operations was investigated further by comparing the filtered mesh against the unfiltered mesh using a conventional nominal–actual comparison (VGS). Figure 4 shows that the deviations between the filtered and unfiltered geometry nearly form a perfectly shaped Gaussian distribution with the expected value μ=0. The comparison against the regression result of the deviations using a Gaussian model confirms the underlying normal distribution. Consequently, we can state, following the central limit theorem (Hazewinkel et al., 1988), that the removed part of the geometry consists of noise. Consequently, it is part of the random measurement error. Because of that, and because of the fact that the removed parts of the geometry due to the filtration are clearly below the voxel size (22.3 µm) and the X-ray focal spot (60 µm) of the CT system, it is stated that no relevant measurement information of the geometry form was removed by the mesh post-processing operations. After the post-processing of the reference geometry, each of the 20 measurement repetitions was registered against that reference measurement using VGS. Following a first rough alignment, the rotatory symmetry was dissolved by arrangement of the drilling in one of the tooth roots. The geometrical registration was finalized by a fine registration using the quality level 50 and consider current transformation settings. Then the gear wheel sprocket was extracted by removing all triangles with an edge point distance to the rotatory axis of smaller than 16 mm (same procedure as for the reference measurement before). This step reduces the overall number of processed triangles during the calculation of the single point uncertainty and is justified because the gear wheel sprocket was targeted for the dimensional investigations. Figure 5 shows the alignment of each single measurement with the reference geometry. It is clearly observable that the drillings in each mesh lie on top of each other (top region in the figure), thus confirming a valid registration result. After that, the geometric deviations of the measurement object were calculated by determining the distances from each sampling point SP_k of the reference geometry to the nominal geometry using the shortest distance sampling strategy. (These results are not used further in the subsequent evaluations, because they are not part of the measurement uncertainty. Nonetheless, these findings can be exploited in different usage scenarios.) Then, the single point uncertainty was calculated by determination of the distances from each of the same sampling points SP_k on the reference geometry to each of the measurements of the measurement series using the normal vector sampling strategy (as described in the first chapter). Finally, each sampling point SP_k was associated with a value describing the local geometrical deviations of the measurement object t and a (mathematical) set d^k characterizing the local measurement uncertainty.

Figure 5Alignment of the sprocket of the reference geometry (grey) and the 20 measurement repetitions (various colours) after the registration routine.

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Figure 6Visualization of the expanded measurement uncertainty (k=2) of gear wheel ID_W.

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Figure 6 shows the expanded measurement uncertainty with a coverage factor of k=2 for the gear wheel measurement using a structured-light scanner as well as a white colour additive (ID_W) combined with the no postprocessing polygonization option. The expanded uncertainty is calculated by the quadratic sum of the systematic measurement error and the standard deviation of the random measurement errors and is then multiplied by a (positive integer) coverage factor k (Eq. 1) (Härtig and Krystek, 2008, Eq. 11).

A coverage factor of 2 results in having a level of confidence of approx. 95 %, assuming a normal distribution (, ). The findings in Fig. 6 show that the limited optical accessibility of the gear roots resulted in a substantial increase in the expanded measurement uncertainty over 100 µm. The gear wheel side also exhibits a larger measurement uncertainty because of the angled image capturing. The upper parts of the gear flanks and the tooth tips were measured with a comparably low expanded measurement uncertainty significantly below 50 µm. Figure 7 shows the expanded uncertainty (k=2) for the evaluated measurement series of gear wheel ID_DG. It can be seen that the measurement uncertainty is slightly lower (more dark blue regions) compared to gear wheel ID_W. The gear roots are also characterized by a lower measurement uncertainty. The visual examination of the distribution of the measurement uncertainty can be used subsequently to validate a given measurement strategy.

Figure 7Visualization of the expanded measurement uncertainty (k=2) of gear wheel ID_DG.

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Next, the effects of the tested polygonization settings during the measurement data evaluation are demonstrated. The numbers of unique vertices used to build the triangle meshes of the measurement repetitions are given for both measurement series ID_DG and ID_W in Tables 2 and 3, respectively. Even though the shape of the measured gear wheel geometry is not changed, the number of vertices and triangles is different for each polygonization setting. This possible impact was the target of the subsequent investigations.

Table 2Statistical overview of the number of vertices of the different triangle meshes for measurement series ID_DG. The number of vertices (sampling points SP_k) of the used reference geometry is 1 935 969.

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Table 3Statistical overview of the number of vertices of the different triangle meshes for measurement series ID_W. The number of vertices (sampling points SP_k) of the used reference geometry is 1 930 759.

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The following results give a statistical overview of the consequences of using different polygonization settings during the measurement data evaluation. The analysis of the systematic measurement error shows only marginal differences between the different polygonization settings (Fig. 8). Nonetheless it is important to note even though the observable differences are small, those errors are introduced purely based on the data processing pipeline. The comparison between both measurement series indicates that distributions of the calculated systematic measurement errors vary slightly. This was purely caused by the different colour additives because the manufacturing process parameters as well as the measurement setup and the data processing pipeline were chosen identically for both variants. Consequently, both the colourant and the polygonization settings do have an impact on the observed systematic measurement errors.

Figure 8Cumulative visualization of the systematic measurement errors for different polygonization settings for both gear wheels ID_DG and ID_W.

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Figure 9Cumulative visualization of the random measurement errors for different polygonization settings for both gear wheels ID_DG and ID_W.

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The evaluation of the random measurement error shows different characteristics for both measurement series: there is no clear correlation between the polygonization setting and the random measurement error observable for gear wheel ID_DG (Fig. 9). This is not the case for the second series, where a clear distinction between the different settings is visible. The more details option results in slightly lower error statistics than the no post-processing option, which indicates the successful filtration effect of the reduced polygon count. A further reduction of the vertex count using the standard, less details and smallest data volume settings leads to a continuous increase in the random measurement errors in that named order. Thus, there is a (non-trivial) optimum regarding the polygonization setting, which is different from the maximum possible number of polygons. The settings should be chosen taking into account the desired structural resolution.

In this paper, we present a methodology to determine the single point uncertainty of polymer gear wheels using structured-light scanning. The algorithms used for this purpose, the parameters for the generation of the measurement data and all data processing steps are described in detail in the paper. Additionally, publications cited in the paper describe the algorithms used to determine the single point uncertainty. Thus, all information needed for a reproduction of the presented results is available to the interested reader.

AMM contributed to conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, software, validation, visualization, writing the original draft, writing the review and editing. DS and DD contributed to conceptualization, methodology, funding acquisition, project administration, resources, supervision, validation, writing the original draft, writing the review and editing associated with Sect. 2 as well as the production of the examined gears. TH contributed to conceptualization, funding acquisition, project administration, supervision and writing the review and editing.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Sensors and Measurement Systems 2019”. It is a result of the “Sensoren und Messsysteme 2019, 20. ITG-/GMA-Fachtagung”, Nuremberg, Germany, 25–26 June 2019.

This research has been supported by the Deutsche Forschungsgemeinschaft (grant nos. HA 5915/9-2 and DR 421/21-2).

This paper was edited by Sascha Eichstädt and reviewed by two anonymous referees.