Areal Evaluation of Involute Gear Flanks with 3D Surface Data

With the evolution of size, weight, surface structure, and tolerance requirements in gear production and application, gear metrology is challenged at the same time to fulfill the cycle time requirements and to support the varieties in gear dimensions and geometry. The standard gear flank inspection is based on measuring two-dimensional line features (classically one profile and one helix per tooth) on selected gear teeth (usually four teeth selected), and graphically evaluating the lines for deviation parameters. When inspecting structured modifications on gear flanks, multiple lines are sampled to examine the whole surface, which significantly increases the measurement time. Optical methods start to gain attention for gear measurement with the benefits of fast sampling and high data density. However, the evaluations are still carried out by extracting and evaluating a limited number of lines from the large data cloud. This paper focuses on the evaluation of involute gear flanks with areal, three-dimensional surface data, providing holistic information of the gears. The three-dimensional gear model and the plumb line distance model enable the calculation of deviations in the surface normal direction of a gear flank. This paper presents the benefits of an area-oriented inspection of gear flanks, comprising mathematical approaches for areal descriptions of involute surfaces, deviations, and modifications, as well as the characterization of areal data with “3D gear deviation parameters.” Approximation and orthogonal polynomial decomposition methods are applied for surface reconstruction and parameter calculation. Measured gear data is analyzed and comparisons with conventional evaluation results are presented.

1: Introduction

Gears are decisive components in transmission systems. The quality of the gears decides the performance of the transmission system they construct, including the power transmission capacity, transmission accuracy, noise and vibration, reliability, and life time. The geometry of the gears is controlled by quality inspection processes to ensure their conformance with the design and tolerances. Until today, tactile measurements followed by a line-oriented evaluation procedure have been the dominant method to assess the deviations of gears from their nominal geometry.

During a tactile measurement, the probe physically approaches and contacts the preassigned nominal points or scans along the prescribed lines in the profile and helix direction. Due to the tactile sensing technology as well as the mechanical alignments and the motion control needed to ensure the correct path of probing, the measurement speed of tactile systems is limited. Therefore, only two lines on both flank surfaces of four teeth are measured for flank geometry inspection, as a convention in industry (at least three teeth required in ISO 1328-1 [1]). The relative positions of the flanks are measured by a single point on each flank, captured at the pitch measurement diameter. It takes about three minutes to inspect an automotive cylindrical gear following this standard procedure, while the same set of measurement tasks could take up to an hour on a large gear, used, for example, in an energy system.

Modern transmission systems impose increasing requirements on the performance of gears, leading to tighter tolerances on the flank geometry and more complex modifications of the flank surface. For example, sinusoidally shaped modifications have been investigated to reduce the noise levels of ground gears [2]. The modifications are designed for the entire flank surface, but the manufactured gears are checked along a very limited number of lines on the sampled flanks. Especially for modifications applied neither along the profile nor along the helix direction, the conventional two-line measurement will have difficulties capturing the complete feature. Topography measurements could reveal the surface condition over a broader range of evaluation, but a quantitative assessment by deviation parameters is still based on line evaluation, which is usually not sufficient to represent the entire surface. In addition, topography measurements of multiple teeth increase the measurement time significantly, which is a critical disadvantage in production. Therefore, faster and area-based inspection techniques are required to fulfill the needs of modern gear metrology.

Optical sensing technology offers a high measuring speed and generates a large amount of measured data, which has been investigated for gear measurement. Gear measuring instruments (GMI) and coordinate measuring machines (CMM) have been equipped with optical sensors, and integrated commercial devices were developed [3-6]. Tens of thousands to hundreds of thousands of measuring points could be captured on each gear flank within a cycle time comparable to the standard tactile measurement. These points cover the entire flank (areal measurements) instead of two lines and can be captured on every flank surface of the gear, instead of four teeth only. This provides a solid base for the evaluation of the entire gear.

Even though optical sensors provide high density data, the standard profile and helix line analysis method can only offer evaluations based on two-dimensional line features. The evaluation of areal data requires a new mathematical approach that processes three-dimensional surface points, characterizes them by areal features, and correlates these features to attributes of the gear flanks with proper parameters. This paper describes an areal evaluation method, developed to fulfill these demands. Holistic parameters are introduced to characterize an entire gear.

2: Areal Deviation Map of a Cylindrical Gear Flank

To assess the features of a measured gear flank in three-dimensional space, its deviations from the designed (reference) flank geometry are obtained first. In the line-based measurement of a cylindrical gear, the reference geometries are two-dimensional lines. For example, the reference geometry of a profile is an involute curve in the transverse plane. The deviations are determined as the distances between the measured curve and the reference curve, measured perpendicularly to the reference curve. It is the direction of the line of action and follows the generation principle of involutes. A deviation chart is formed by plotting the deviations versus the position of the measured point along the profile (usually the roll length). In the evaluation procedure, a least-square mean profile is approximated, and deviation parameters are calculated to quantitatively characterize the deviation plot. Modifications are also identified and included in the mean profile, described by modification parameters. Modifications are intended alterations of the gear flanks, whereas deviations are undesired errors of the actual gear.

Several error sources introduce uncertainties in the evaluation results. Firstly, the imperfections in motion control introduce uncertainties, since the measured path might deviate from the target transverse plane. Secondly, preassigned nominal points are usually used for deviation calculations, while the probed points might not be along the normal direction from these nominal points.

In the three-dimensional model, the reference geometry of a cylindrical gear flank is an involute surface [7] with designed modifications [8–9]. The following assumes that the gear is aligned mechanically or numerically. Given the coordinates of the sampled points measured on a flank surface, the deviations are determined as the distances from each measured point to the designed surface in the surface normal direction. The plumb line distance method [9] offers a direct and analytical calculation of such a distance in the surface normal direction (as illustrated in Figure 2), based on the coordinates of the measured points and geometric parameters of the designed gear.

The plumb line distance from an arbitrary measured point (indexing j) to an involute surface (on tooth number i) is given by

Λ_j,i depends on the z coordinate of the measured point and is related to the relative positions of the left and right flank as well as the index of the measured tooth. The subscripts “lot” in d_lot,um,j,i denotes the plumb line distance. “um” clarifies that this plumb line distance equation is calculated with respect to the “pure” involute surface as reference geometry. This means that it does not take into account any modification, referred to as “unmodified.”

With the plumb distance model, the preassigned nominal points are not necessary for the calculation of distances. Therefore, prescribed motion paths are not required, and the uncertainties caused by improper nominal points are eliminated. Figure 2 illustrates the plumb line distances for a series of measured points on a gear flank. The red dots represent the measured points, the arrows represent the distances in surface normal direction, and the dark green dots on the ideal surface are the corresponding correct nominal points.

To be consistent with the generation principle and the representation of deviations in conventional evaluation methods, a UVD coordinate system is constructed to present the deviations. Coordinate u is the generation along the profile direction (roll length, which is the same for the conventional profile evaluation); coordinate v is the generation along the helix direction (z coordinate of the corresponding nominal point, which is the same for the conventional helix evaluation); coordinate d is the amount of plumb line distance. The calculated plumb line distances collectively form a distance map of the measured flank in the UVD coordinate system. This map contains only the distances between the actual and nominal gear flank, so a pure involute surface in XYZ coordinate system is equivalent to a flat with zero d everywhere in the UVD coordinate system. Therefore, for an unmodified gear, whose reference geometry is pure involute surface, the distance map contains undesired deviations only, which is analyzed for the deviation parameters.

For a gear with modifications, the reference geometry is not zero in the distance map since the designed flanks are different from involute surfaces. Modifications are three-dimensional surface-based features, so their definitions should be extended from two-dimensional features to areal features as well. In addition, the modification features should be specified in the surface normal direction for three advantages: firstly, it agrees with the direction of force transmission; secondly, adding modifications does not change the surface normal directions; thirdly, different modifications superimpose in the surface normal direction, so that they can be linearly combined to construct a designed surface. As a result, each modification is a specific areal feature added to the zero plane in the UVD coordinate system, which can be described by
a continuous function of (u,v) coordinates. Since the deviations are defined in the surface normal direction as well, deviations and modifications are linearly added to form the actual surface. In other words, the modifications can be subtracted directly from the distance map to determine the undesired deviations of a gear with modifications.

In conclusion, three different maps are defined to separate the deviations from the designed features (intended modifications) and to evaluate them as three-dimensional areal features. Figure 3 offers an illustration of this procedure for an arbitrary sample:

The plumb line distance map consisting of distances from measured points to the unmodified pure involute surface (Figure 3a).

The modification map, generated from the designed areal modification parameters as assigned in the drawing, (Figure 3b).

The areal deviation map consisting of the distances from measured points to designed modified flank surface (Figure 3c). The areal deviation map results from a direct subtraction of the first two maps.

3: Areal Deviation and Modification Parameters for a Cylindrical Gear Flank

To characterize a gear flank with areal features, area-based deviation and modification parameters are proposed to describe these features quantitively. In this paper, they are defined in consistency with the meaning of the corresponding parameters in conventional gear evaluation. For example, in ISO 1328-1, the profile slope deviation is defined as illustrated in the deviation plot of Figure 4. A mean profile line is constructed, and the intersections with the profile control diameter and tip diameter define the profile slope deviation f_Ha .

Extended to an areal parameter in the deviation map, the mean feature is a tilted plane instead of a line with slope (see Figure 5). The boundaries indicating the evaluation range are two planes as well. The areal profile slope deviation can be defined as the distance between two facsimiles of the designed surface, which are intersected by the mean surface at the profile control diameter and the tip diameter. The illustrations in Figure 5 are based on a deviation map with profile deviations only. The deviation map is a three-dimensional surface. The black colored plane is the mean surface and is extrapolated to the plane, indicating tip diameter, datum face, and non-datum face. The facsimiles of the designed surface intersecting the mean surface are displayed. The areal profile slope deviation is marked in Figure 5b) as  f_Hα^A with a superscript ‘A’ denoting the areal parameters.

The similar definition can be applied to areal helix slope deviation that it is also a plane but tilted in the helix direction. Since all deviation features are defined in the surface normal direction and can be linearly superimposed and decomposed, a measured surface with both profile and helix slope deviations can be represented as the superposition of two tilted planes. Other than first order features, higher order and more complex features contained in the measured surface could be described independently in similar manners and then combined to depict the measured data.

Since in the conventional evaluation, the profile and helix lines are analyzed separately, there are cases where different line-based parameters essentially represent the same feature when they are referred to the flank surface. For example, in conventional line evaluation, twist is defined by two separate parameters in profile and helix direction. In the profile direction, it is obtained by measuring one profile near the datum face and another one near the non-datum face. The difference between the slopes of these two profiles is defined as the twist in the profile direction. A similar procedure is applied for the twist in helix direction. Due to the arbitrary choice of measuring positions, it is common to see an unequal twist measured in profile and helix direction. However, one parameter could represent this twist feature in the areal evaluation, since it describes a single second order surface, as illustrated in Figure 6. Face I is the datum face and II is the non-datum face.

In the scope of this paper, six deviation parameters are used to characterize the deviation maps, including up to second order components in the map. The second order components are defined as crowning and twist in the current ISO standard. Unfortunately, ISO does not clearly distinguish between deviations and modifications for these components. They are included as deviation parameters in this paper as well. The same symbols as in ISO 1328-1 are used with an added superscript “A” to indicate the areal parameters. Although serving as deviation parameters, the symbol “C” is used for crowning deviations. The six parameters are, respectively, the cumulative pitch deviation F_pi^A, the profile slope deviation, f_Hα^A, the helix slope deviation, f_Hβ^A, the profile crowning deviation, C_α^A, the helix crowning deviation, Cβ^A, and the twist deviation, S^A. Thus, each single flank surface is quantitively presented with a series of deviation parameters.

4: Holistic Evaluation of an Entire Gear

The various deviations on gear flanks result from a diversity of error sources in the manufacturing processes. Two types of deviations could be categorized in this procedure each having specific correlations with the gear manufacturing process and performance properties.

Firstly, a repeated pattern could be recognized on all teeth that is caused by errors imposing the same influence on every tooth during manufacturing. For example, an error in the tool’s pressure angle introduces the same profile slope deviation on each tooth, since this tool cuts every tooth during machining. Positioning errors of the tool and systematic errors in the tool motion control leave the same deviations on each tooth, since they are repeated when cutting the individual teeth. These common patterns have direct correlation with the systematic errors in the manufacturing process, and hence should be identified and extracted.

Secondly, except for the repeated pattern, each tooth shows individual deviations that differ from tooth to tooth. They are caused by varying machining conditions in manufacturing such as temperature gradients within the tool and the machined gear, vibrations, and inhomogeneous material. When combined with a mating gear, individual deviations result in changing contact conditions and should be taken into consideration, especially when analyzing the performances of a gear pair.

Therefore, holistic evaluation procedures and parameters are needed to identify and quantify the repeated patterns observed on all the teeth and to establish global parameters that represent the common condition of the entire gear. In the conventional evaluation, where 16 lines on four teeth are measured, the extraction of a repeated pattern observed on all the teeth has never been reported. The current gear standards do not separate global gear parameters and individual tooth parameters either. In the context of this paper, five global parameters are calculated and presented for the measured gear sample in Section 6. They are identified with “g” subscriptions, which are the global profile slope deviation, f_Hα_g^A, the global helix slope deviation, f_Hβ_g^A, the global profile crowning deviation, Cα_g^A, the global helix crowning deviation, Cβ_g^A, and the global twist deviation, S_g^A.

5: Mathematical Approaches for Areal and Holistic Evaluation

With the deviation map of each tooth obtained as explained in Section 2, the evaluation procedure characterizes the surface, offering the areal parameters as explained in Sections 3 and 4. Different surface analyzing approaches could be applied to quantify the surface features within the deviation map, which focus on the construction of the “mean surface,” also referred to as a “reconstruction” of the deviation map in this paper.

5.1: Approximation of the Deviation Map

To obtain a “mean surface” from the deviation map, a two-dimensional approximation could be carried out taking the deviation map as a polynomial function of u and v coordinates. For the five parameters in this paper, the highest order polynomial terms included are second order terms, describing the profile crowning, the helix crowning, and the twist. Therefore, the approximated surface takes the format of a second order polynomial of u and v. Different objective functions could be used, where the L2-norm of the deviations at all the measured points is one of the most widely used objective functions. It is consistent with the conventional line evaluation method and is relatively well understood for uncertainty estimation.

With the “mean surface” obtained, the deviation parameters could be calculated from the coefficients of the approximated polynomial directly. The quantitative values of the deviation parameters are then derived from the geometric relations in the parameter definitions (for example, the geometric relations in Figure 5).

5.2: Surface Components Decomposition with Orthogonal Polynomials

An alternative approach is based on a series of orthogonal polynomials, which describes an arbitrary surface as a linear combination of the individual terms. In other words, the surface can be decomposed into a series of orthogonal polynomials, where each term reveals a specific feature in this surface. Two-dimensional Chebyshev polynomials are one example of such polynomial series, which are orthogonal on the domain of [-1,1]x[-1,1]. Equations 2 to 7 offer the mathematical representations of the first six terms as functions of two variables x and y. the subscripts are the commonly used indices of the two-dimensional Chebyshev terms.

One of the major advantages of using Chebyshev polynomials in analysis of gear flank surfaces is that they describe equivalent features as represented by the gear flank deviation parameters [11]. Direct correlations between the coefficients of Chebyshev terms and the gear deviation parameters can then be found due to this similarity. Figure 7 offers a comparison of Chebyshev polynomials (up to the second order) and the corresponding gear flank features. The “2D Chebyshev Polynomials” columns offer the diagrams of the Chebyshev polynomials and the “Gear Flank Deviation Parameters” columns offer both the illustrations of the areal parameters on a gear flank in 3D space and the corresponding deviation map in the UVD coordinate system. Comparing the Chebyshev polynomial maps with the deviation maps in UVD coordinate system, the gear flank deviation parameters could be expressed as functions of the coefficients of the Chebyshev terms. The comparisons here are qualitative, thus the scales are not specified. Extrapolation rules are not included in Figure 7.

The second advantage of Chebyshev polynomials is the orthogonality property. As explained in Section 2, the deviations are all defined in the surface normal direction of the flank. In addition, they are independent and linearly combined to form the deviation map. Therefore, the Chebyshev polynomials could serve as an efficient tool to decompose the measured deviation map, offering the coefficients of the six terms listed in Figure 7. The decomposition could be expressed mathematically by Equation 8.

5.3: Statistical Analysis of Individual Parameters for Holistic Evaluation

Each individual tooth can be evaluated with either one of the mentioned algorithms, resulting in a list of individual parameters for each tooth. As explained in Section 4, the repeated pattern is to be recognized from all teeth to represent the condition of the entire gear in its totality. The holistic evaluation aims at finding a proper and reasonable summary of the information valid for all teeth. Statistical analysis on individual parameters could offer a reasonable assessment for the entire gear. The average of the profile slope deviations on all left or right flanks of all teeth, for example, offers an estimation of the common profile slope deviation of the entire gear. In this case, it will be reasonable to consider the left and right flank parameter separately, since there might be different deviations on the left and right cutting edge of the tool. When trends or a certain distribution of the individual parameters are discovered, changing machining conditions or errors that propagate from tooth to tooth will be revealed.

5.4: Aligning and Combining Flank Surfaces for Holistic Evaluation

The relative positions of the teeth are described by single pitch deviations or individual cumulative pitch deviations. The individual cumulative pitch deviations of all teeth are their positioning errors with respect to the same reference flank. It is an individual tooth parameter in the areal evaluation, which is different and unique for each flank. In the measured flank data, it is contained as a uniform offset from the reference geometry across the entire flank in the surface normal direction. The corresponding feature in the deviation map is a plane, related to the 0th term of the two-dimensional Chebyshev term (refer to Figure 7) if evaluated with the algorithm explained in Section 5.2.

The pitch deviations do not contribute to the repeated pattern for all teeth. If they are quantified and removed from the deviation maps, the flank positions will be corrected so that every flank will be present at its nominal position relative to the reference flank. Since the nominal position of a flank is known, it can be rotated by a certain angle to be duplicated at the position of the reference flank. By aligning all teeth to the reference position, a combined surface consisting of all measured points on all flanks is constructed. This surface contains all the deviation information of the gear (except the cumulative pitch deviations), such that the set of global deviation parameters (see Section 4) characterizing this combined surface is a fully sufficient representation of the entire gear.

The prerequisite to ensure the effectiveness of the global parameters in this method is to evaluate and remove the pitch deviations correctly. Some difficulties might be caused by the large amount of input data during the numerical calculations. For example, the computation time might be increased significantly.

6: Evaluation of Both Optical and Tactile Measurement Data and Comparisons

A series of measurements and evaluations was carried out to implement the areal and holistic evaluation method. The same gear sample was measured using both tactile measuring devices and optical instruments. Some of the tactile inspections were based on conventional line-oriented measurements and evaluations. Areal measurements were realized (i) by scanning multiple lines across the surface of all teeth using tactile instruments and (ii) with an optical instrument performing high density laser scanning. The gear sample was a ground automotive gear, which is not a calibrated artefact. Therefore, the results presented here are not compared to access the performances of the measuring instrument, but to show the outcomes of the areal and holistic evaluations. Algorithms introduced in 5.2 and 5.3 are used to achieve the results listed in this section.

A Nikon HN-C3030 is used to obtain optical areal data. Approximately 480,000 points were measured on each flank including parts of the top and bottom lands and the datum and non-datum faces. About 105,000 points were identified as valid evaluation points within the evaluation range on each flank. The device and the obtained point data cloud are shown in Figure 8.

An areal measurement on all teeth was conducted on a CMM to compare the results with the optical measurements; 51 profile lines across the flank surface were measured on each flank of all teeth.

Figure 9 shows the evaluation results of the tactile areal data. The dots with color gradients show the original deviation maps of the first seven teeth. An obvious common pattern of crowning in the helix direction could be recognized. The red surfaces show the “mean surface,” which are the reconstructed repeated pattern determined with Chebyshev method. It could also be noticed that the repeated pattern has different offsets from the original deviation map except the first tooth. This misalignment is caused by the cumulative pitch deviation of each tooth as explained in Section 5.4, which is not included in the repeated pattern since they are individual deviations of each flank. The areal deviation parameters calculated with the holistic evaluation procedure described in Sections 5.2 and 5.3 are also listed in Table 1.

Conventional line-oriented measurements and evaluations were also carried out using both GMIs and CMMs to compare the results with areal evaluations. The deviation parameters from one of the standard measurements are listed in Table 1. Four teeth were measured with a profile line and a helix line on each flank for the slope and crowning parameters. Two profile lines and two helix lines were measured on each flank of the first tooth to calculate the twist. The average of four profile slope deviations of the right flanks are taken as the global profile slope deviation to compare with those calculated from the areal data. Only right flanks are used here, since the areal evaluations are conducted on all right flanks for the results displayed in Table 1. The same procedure was applied for the other three global parameters. The average value of the two twist parameters calculated in profile and helix direction is taken as the global twist value.

Table 1 consists of results of three different measurement and evaluation processes. One of them represents the result of a conventional line-oriented inspection procedure. The other two are based on areal measurements and evaluations, one of them based on tactilely probed data and the other on optically measured data. It could be discovered that the three different measurements and evaluation methods result in variations of the deviation parameters. For the experiments reported here, the five investigated parameters vary within 2 µm. The differences might result from:

Different sampling conditions: the line evaluation is based on samples in one profile or helix line on the surface, whereas the areal evaluation is based on a larger number of points covering the entire area of each flank, all teeth involved.

Different definitions of area-based and line-based deviations: For example, the conventional profile deviations are defined in the transverse plane perpendicular to the involute curve, whereas the areal deviations are defined on the entire surface in the surface normal direction.

Different approximation methods: the line-oriented parameters are evaluated based on least- square profile or helix lines, whereas the areal evaluation is based on Chebyshev orthogonal polynomials.

Different measurement conditions: the gear was measured on different instruments by different operators under different lab conditions.

Conclusions and Future Work

Modern gear production requires advancements in inspection techniques: a faster and more comprehensive measurement as well as an improved evaluation. The shift of perspective to three-dimensional design and modeling, areal measurement, and holistic evaluation are some of the most significant improvements in gear metrology. They form the basis for innovative solutions of data interpretation, including improved manufacturing process assessment, and functional performance investigation. This paper presented:

Definitions of deviations in a three-dimensional gear model and the construction of areal deviation maps.

Suggestions to define extended deviation and modification parameters, covering also areal measurements and evaluations, and their meaning in terms of surface features.

Holistic evaluations of entire gears as an integration of information obtained from all teeth.

Effective algorithms to conduct areal and holistic evaluations.

Comparisons of different evaluation processes on a gear sample to show the capability of the current algorithms.

The scope of future research work will comprise:

Measurements and evaluations of a calibrated artefact using different instruments leading to traceable assessments of the measuring devices and algorithms.

Additional surface analyses, approximation, and reconstruction algorithms.

Modeling of complex modifications, correlation of parameters to manufacturing processes, and interaction between tooth flanks at meshing based on areal gear flank data. 

References

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9. Goch, G.; Günther, A., 2006, “Areal gear flank description as a requirement for optical gear metrology,”. Towards Synthesis of Micro /Nano-Systems, The 11th International Conference on Precision Engineering (ICPE) August 16–18, 2006, Tokyo, Japan, pp. 47–52.
10. ISO, 2007, “Gears – Cylindrical involute gears and gear pairs – Concepts and geometry,” ISO 21771.
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^{Printed with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, Suite 500, Alexandria, Virginia 22314. Statements presented in this paper are those of the authors and may not represent the position or opinion of the American Gear Manufacturers Association. (AGMA) This paper was presented October 2017 at the AGMA Fall Technical Meeting in Columbus, Ohio. 17FTM08}