PARADIGM CHANGE IN GEAR INSPECTION BASED ON A HOLISTIC. DESCRIPTION .... top view of the evaluation zone; (b) side view of the areal deviation ...
PARADIGM CHANGE IN GEAR INSPECTION BASED ON A HOLISTIC DESCRIPTION, MEASUREMENT AND EVALUATION OF GEAR FLANKS Gert Goch1, Kang Ni1, Yue Peng1 and Anke Guenther2 for Precision Metrology, UNC Charlotte, Charlotte, NC, USA 2Reishauer AG, Wallisellen, Switzerland
1Center
INTRODUCTION Gears have a decisive importance for the functionality of automotive, aerospace and machinery transmission systems [1]. The demanding geometric accuracy requirements for gears are indispensable to reduce the noise emission and increase the power density. Various manufacturing technologies from roughing [2] to finishing [3] have been significantly improved in the past decades to produce high quality gears with complex flank modifications. Gear inspection on a high level of accuracy and speed is indispensable for a modern gear production from both quality control and process improvement perspectives. CNC-controlled tactile Gear Measuring Instruments (GMIs) and Coordinate Measuring Machines (CMMs) have been dominating over the past decades. Line oriented measurement and evaluation of gear geometric features such as profile, helix and pitch deviations are well established in current gear tolerancing and evaluation standards [6], enabling a standardized inspection procedure for an automobile-size gear, in which several deficiencies have been identified [8]. Today, complex surface structures on gear flanks modifying “purely involute” gear flank geometry are well established at gear manufacturers that aim at improving the functional performance [6]. It is necessary to obtain topographical information on entire gear flanks to diagnose manufacturing process variations, to assess the gear quality, and even to predict the gear meshing conditions [4,5]. Therefore, a paradigm change from a line oriented gear inspection representing the stateof-the-art to an area oriented, holistic gear measurement and evaluation is needed requiring: • A complete 3D geometric description of gear flanks; • A non-contact measuring system that enables a reliable data acquisition of all flanks in reasonable time;
•
An effective evaluation method for calculating all gear deviation parameters from areal measurement data.
This paper offers a systematic approach for key technical aspects of future inspection technologies for the cylindrical gear. CONVENTIONAL LINE ORIENTED GEAR INSPECTION FIGURE 1 shows a typical CMM based measurement with multiple-styli configuration, highlighting the location of profile, helix (red lines) and pitch point (yellow dots) on a gear sample. Correspondingly, the evaluation charts and parameters for a measured profile, including total profile deviation (Fα), profile form deviation (ffα), and profile slope deviation (fHα) are illustrated in FIGURE 2, according to the latest ISO standard [6]. The marked Lα is the profile evaluation length and gα is the length of path of contact in profile direction.
FIGURE 1. Characteristic lines and points on gear flanks in conventional gear inspection.
plumb line distance is the shortest distance from the measured point to the 3D helical gear flank analytically expressed as: |𝑑⃗lot | = ηb + 𝑧m
𝑟b √1+tan2 βb
tan βb 𝑟b
}
{√
2 ρm
𝑟b2
2 ρm
− 1 − arctan (√
𝑟b2
− 1) − φm +
(2)
|𝑑⃗lot | is the plump line distance. The aggregated
FIGURE 2. Conventional gear evaluation charts for line oriented flank measurement [6]. HOLISTIC DESCRIPTION OF GEAR FLANKS WITH 3-D MODIFICATIONS For the 3-D flank of a helical gear (i.e. βb ≠ 0), the flank equation can be generated by “threading” the origin of the involute profile (yellow dot in FIGURE 3) along the base helix line (black line). This is mathematically realized by describing the initial angle Λ of the 2-D involute Equation (1) as a function of the third dimension 𝑧: 𝑥 = 𝑟𝑏 ∙ [cos(𝜉 + Λ(z)) + 𝜉 ∙ sin(𝜉 + Λ(z))] {𝑦 = 𝑟𝑏 ∙ [sin(𝜉 + Λ(z)) − 𝜉 ∙ cos(𝜉 + Λ(z))] , 𝑧=𝑧
(1)
Here, Λ(z) = ηb + 𝑧 ∙ tan βb /rb is a function of the helix angle on the base circle βb , the 𝑧 coordinate and the base space half angle 𝜂𝑏 . In Equation (1), the size of each tooth is determined by the range of the rolling angle 𝜉 and the position of each tooth is governed by a Λ angle, whose ηb angle is determined by the number of teeth of the gear [9].
individual plumb line distances form a vector representation of the entire distance map, denoted by dlot. Setting dlot to 0 is equivalent to a surface function 𝐹(ρnom , φnom , znom ; 𝑟𝑏 , 𝛼𝑡 , 𝛽, z) = 0 for the nominal gear geometry. This means that a function dlot = F(ρm, φm, zm) = a denotes an equidistant surface with a constant deviation of a to the nominal helical gear flank. This interpretation of the dlot-formula offers another major advantage for gear metrology. As for other geometries inspected with CMM technology, where the nominal surface can be described by a function F(x, y, z) = 0 (like cylinders, spheres, circles, etc.), neither nominal point coordinates nor surface normal directions are required to evaluate the deviations between actual and nominal gear geometry. To be consistent with the generation principle and the representation of deviations in conventional evaluation methods, a UVD coordinate system is constructed as shown in FIGURE 4.
FIGURE 4. Generation of areal deviation and distance map (a), illustration of measured points and ideal involute surface; (b) representation of areal distance map in (u,v,d) coordinate system.
FIGURE 3. One 3D helical flank surface generated by basic geometry parameters. Applying the aforementioned “threading” model, a unique distance called “plumb line distance” is obtained for any measured point (subscripted with “m”) given in cylindrical or Cartesian coordinates (xm, ym, zm) = (ρm, φm, zm) [9]. The
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 value of plumb line distance. As shown in FIGURE 4 (b), 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 plane with d = 0 everywhere. Therefore, for an unmodified gear, whose reference geometry is a “ pure involute” surface, the distance map contains undesired deviations only, which is analyzed for the deviation parameters.
(a), location of multiple profiles measured on a left flank; (b) raw point clouds of measured flanks.
AREAL MEASUREMENT OF CYLINDRICAL GEAR FLANKS Two instruments were used to capture areal point clouds of a ground gear sample. As shown in FIGURE 5 (a), the gear was measured by a Leitz PMMF302016 CMM equipped with a “star” probe configuration of eight styli. Each flank was scanned in profile direction with 51 distinct lines along transverse planes from the bottom to the top of the gear. The collected point clouds of all flanks represent an areal dataset as shown in FIGURE 5 (b), obtained in about 2.5 hours. An optical CMM (Nikon HN3030) was also used to collect a second area oriented dataset from the same gear sample, in significantly shorter time (several minutes). as shown in FIGURE 6 (a) and (b). The optical instrument captured about 480,000 raw points per flank including sections of the bottom and top lands. A quarter of those points lie within the evaluation range and were used in areal evaluation. FIGURE 6. Areal flank measurement of the same gear sample as in FIGURE 5, using an optical instrument: (a), setup during optical scanning; (b), raw point clouds of measured flanks. AREAL EVALUATION OF GEAR FLANKS DEFINITIONS OF AREAL CHARACTERISTIC PARAMETERS To characterize the captured areal point clouds covering an entire flank, a set of areal parameters is preferred. According to [12], the areal parameters could be obtained by approximation algorithms based on least-squares method over the evaluation zone, which is bounded by the profile and helix evaluation ranges. Alternatively, a new set of areal parameters (denoted with a superscript “A”) is proposed, as an extension of the standardized line oriented parameters [11]. TABLE 1. Definition and symbolic of a selected list of line oriented [6] vs. areal gear parameters. (“dev.” abbreviated for deviation) FIGURE 5. Areal flank measurement using a tactile instrument (CMM without rotary table):
Definition Profile slope modification Profile slope dev. Profile crowning modification Profile crowning dev. Profile form dev. Helix form dev. Total profile dev.
Line CH
Areal
C
A Hα
fH
A f Hα
C
CαA
N.A.
f CαA
ff ff F
FfA total form
Total helix dev.
F
dev.
Single pitch dev.
fpi
f piA
Cumulative pitch dev.
Fpi
FpiA
f fA form dev.
The 3D illustration of the areal profile slope A deviation (𝑓Hα ) is shown in FIGURE 7.
FIGURE 8. Graphical illustration of the first eight terms (𝑖 =0,1, 2,…,7) of 2D Chebyshev polynomials 𝐶𝑖 . Based on their geometric similarity, each term of the low order 2D Chebyshev polynomial can be assigned to one areal component (deviation and/or modification). In addition, due to the orthogonality property of 2D Chebyshev polynomials, the amount of each areal component (characterized by the corresponding areal parameters) can be uniquely determined by the value of the corresponding coefficients for the polynomial base. Equation (3) gives the areal distance map, 𝐷(𝑥, 𝑦), which is approximated by a linear combination of 2D Chebyshev polynomials. 𝐷(𝑥, 𝑦) ≈ 𝐷′ (𝑥, 𝑦) = ∑𝑁 𝑖 𝐴𝑖 𝐶𝑖 (𝑥, 𝑦),
FIGURE 7. Illustration of the areal gear 𝐴 parameters 𝑓𝐻𝛼 obtained in a deviation map: (a), top view of the evaluation zone; (b) side view of the areal deviation map. CALCULATION OF AREAL PARAMETERS BY 2D CHEBYSHEV METHOD As explained in [10], the information of areal deviations and modifications are intrinsically contained in the areal distance map such as the one in FIGURE 7 (b). Since they are linearly and independently superimposed in the surface normal direction, a decomposition approach was applied to separate the various components into a set of orthogonal base functions, composed of low order 2D Chebyshev polynomials. They are graphically illustrated in FIGURE 8.
(3)
where, 𝑁 is the number of 2D Chebyshev terms used for reconstructing 𝐷′ (𝑥, 𝑦) as an approximation of 𝐷(𝑥, 𝑦) . The 2D Chebyshev coefficients 𝐴𝑖 are given by the double integral: 1
1
1 𝐷(𝑥,𝑦)𝐶𝑖 (𝑥,𝑦)𝑑𝑥𝑑𝑦
𝐴𝑖 = ∫−1 ∫−1 𝐾
√1−𝑥 2 √1−𝑦 2
,
(4)
where, 𝐾 is the normalization factor for 2D Chebyshev polynomials. VERIFICATION ON GEAR ARTIFACT Besides the proven effectiveness of the 2D Chebyshev method via numerical simulation [10], it is necessary to verify the method with experimental data. A modification artifact (FRENCO® Gear Artifact M, see FIGURE 9) was measured with the Leitz CMM. This artifact was specially manufactured to provide different types and amounts of gear modifications on its 36 left flanks. Four of the flanks with their corresponding modifications (profile slope, helix slope, profile crowning and helix crowning of left flanks indicated by “#10”, “#19”, “#20” and “#30”, respectively) were used to test the 2D Chebyshev algorithm as shown in the bottom left corner of FIGURE 9. For areal
evaluation, a total of 99 profile lines were scanned on all flanks without using a rotary table.
FIGURE 9. Basic geometry of the artifact and the intended types of flank modifications on four left flanks at selected teeth. The measurement results of those four flanks are summarized in TABLE 2. All of the line oriented parameters are adopted from the calibration certificate by standard single line measurement, which have a claimed ± 3 μm extended measurement uncertainty. The areal parameters are evaluated by 2D Chebyshev method. The closeness between the certified line oriented measurement and the areal results shows the effectiveness of the 2D Chebyshev method. However, a detailed investigation of measurement and evaluation uncertainty of the areal parameters (i.e. measurands) is needed in future. TABLE 2. Comparison between standard line oriented evaluation and the areal evaluation results for four types of flank modifications. Type of flank Intended Line Areal modification [μm] [μm] [μm] Profile slope 10 10.2 9.5 Helix slope -20 -17.6 -16.5 Profile crowning 5 5.7 4.5 Helix crowning 10 9.5 9.6 HOLISTIC EVALUATION OF AREAL DATA OF AND GLOBAL PARAMETERS FOR A WHOLE GEAR Deviations on gear flanks can result from a diversity of error sources in gear manufacturing processes. In general, two types of deviations
could be categorized, each having specific origins in manufacturing processes. The first type is a repeated pattern of deviation, recognized on all teeth, that is caused by errors imposing the same influence on every tooth during manufacturing. For example, an error in a hobbing tool’s pressure angle introduces the same profile slope deviation on each tooth, since this tool cuts every tooth during machining. Repeated positioning errors during tooling or systematic kinematic errors in the tool motion control system leave the same deviations on each tooth during a cyclic cutting operation. The other type of deviation is observed on individual flanks, which might be caused by varying machining conditions in manufacturing such as, temperature gradients within the machine tool, tool wear, vibrations, or inhomogeneous material properties of gears etc. Therefore, a holistic evaluation procedure and corresponding sets of parameters are suggested to identify and quantify both repeated patterns observed on all teeth and the individual patterns varying from tooth to tooth. Both types of deviation parameters are necessary for a comprehensive assessment of to fully assess the variation of manufacturing processes and its impact on gear quality. The areal measurement data of the gear sample shown in FIGURE 5 and FIGURE 6 are used as example in the following discussion. This gear was finished by continuous generation grinding. All 20 right flanks were evaluated by 2D Chebyshev method, in which the first 6 2D Chebyshev coefficients were used to reconstruct the areal distance maps. The areal reconstruction results (tactile CMM) of the first 7 right flanks are shown FIGURE 10. The color-coded plots represent the reconstructed individual areal maps, among which a repeated pattern (slope and crowning) is obvious on every flank in the profile direction.
FIGURE 10. Areal distance maps of first 7 reconstructed individual flanks and the repeated pattern for all 20 teeth (measured by tactile CMM, areal evaluation).
The same procedure is used for displaying the optical measurement results, as shown in FIGURE 11.
FIGURE 11. Areal distance maps of first 7 reconstructed individual flanks and the repeated pattern for all 20 teeth (measured by optical CMM, areal evaluation). In FIGURE 10 and FIGURE 11, the repeated pattern (in red) is constructed by its corresponding areal global parameters listed in TABLE 3. The reconstructed individual areal maps show a variety of vertical shifts with respect to the repeated pattern, which is the individual cumulative pitch deviation. The discrepancy of the cumulative pitch deviation between tactile and optical measurement is caused by a different clamping, resulting in a different eccentricity that influences the cumulative pitch deviation. The latter is manifested in FIGURE 10 and FIGURE 11. In the conventional evaluation, only 16 lines on four teeth are measured in total. However, there is no publication found about an overall evaluation of all flanks in order to extract common shares of flank deviations. Also, the current gear evaluation standards do not distinguish between global gear parameters and individual tooth parameters. To address this issue, five global parameters are investigated for the gear sample measured by tactile and optical instruments. The global parameters are denoted with “g” subscriptions as listed in TABLE 3 for either line oriented or areal evaluation results. The three results coded by the columns A, B and C are obtained from different measurement strategies and evaluation methods. One is the result of a conventional line-oriented inspection procedure (using averaged parameters collected on four teeth only). 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.
TABLE 3. Line oriented vs. area evaluation results of the ground gear sample. Parameter Symbol A B C [μm] Profile A fHαg , fHαg 9.7 10.9 8.9 slope dev. Helix slope f A , f -3.6 -3.4 Hβg Hβg -4.1 dev. Profile A Cαg , Cαg 9.0 7.1 8.6 crowning Helix A Cβg , Cβg -0.5 0.0 0.0 crowning S gA , S g Flank twist 2.0 -0.5 -1.2 The automobile gear sample used in this comparison is not a calibrated artefact, therefore, the evaluation results presented in TABLE 3 do not aim to access the performance of certain measuring instruments or sensor principles. Instead, they aim at a comparison of the proposed evaluation methods alone. This result shows the capability of the 2D Chebyshev method, which offers evaluation results comparable to conventional line oriented methods. The discrepancy can be explained by the fact that the measured data were captured with different instruments (two tactile and one optical), that the evaluations were carried out based on different surface areas of the measuring object (entire flank versus two lines), and that the measuring conditions might have been different. CONCLUSIONS AND FUTURE WORK In this paper, a paradigm shift from line oriented measurements and evaluations to a holistic area oriented gear inspection is proposed, comprising areal gear measuring method and areal evaluation strategies. New areal gear parameters are introduced as extensions of the conventional line oriented gear parameters describing both modifications and deviations. The effectiveness of using an orthogonal decomposition algorithm based on 2D Chebyshev polynomials to calculated areal gear parameters meets the increasing demand for an improved holistic evaluation of whole gear flanks. This especially applies when a large amount of point clouds become available by means of optical gear measuring technologies. Future work includes two areas, firstly to investigate the measurement uncertainty of the presented areal parameters (measurands); secondly, to apply the developed areal gear
evaluation method to the correlation between the areal parameters, the manufacturing process variations and the functional performance of cylindrical gears. ACKNOWLEDGEMENT This work is funded by the Affiliates Program of the Center for Precision Metrology (CPM) at UNC Charlotte. The authors gratefully acknowledge the support from Hexagon-Leitz at WetzlarGermany, Siemens Energy at Charlotte-USA, Reishauer at Wallisellen-Switzerland, Nikon at Yokohama-Japan and Boston Gear at CharlotteUSA. REFERENCES [1] Goch G. Gear metrology. CIRP AnnalsManufacturing Technology 2003; 52, Vol. 2: 659-695. [2] Bauzakis K.-D., Lili E., Michailidis N., Friderkos., Manufacturing of cylindrical gears by generating cutting processes: A critical synthesis of analysis methods. CIRP Annals – Manufacturing Technology 2008; 57, Vol. 2: 676-696. [3] Karpuschewski B. Knoche H.-J., Hipke M., Gear finishing by abrasive processes. CIRP Annals – Manufacturing Technology 2008; 57, Vol. 2: 621-640 [4] Lotze, W., Härtig, F. 2001, “3D Gear measurement by CMM,” Fifth International Conference of Laser Metrology and Machine Performance (LAMDAMAP), WIT Press, pp.333-344. [5] Pfeifer T. Kurokawa S., Meyer S.,Derivation of parameters of global form deviations for 3-dimensional surfaces in actual manufacturing processes. Measurement 2001; 29: 179-200. [6] ISO, 2013, “Cylindrical gears - ISO system of flank tolerance classification - Part 1: Definitions and allowable values of deviations relevant to flanks of gear teeth,”, ISO 1328-1. [7] ISO, “Gears - Cylindrical involute gears and gear pairs - Concepts and geometry,” ISO 21771 (2007). [8] Goch G., Ni K., Peng Y., Guenther A. Future gear metrology based on areal measurements and improved holistic evaluations. CIRP Annals – Manufacturing Technology 2017; 66, Vol. 1: 469-474. [9] Guenther A., 1996, Flächenhafte Beschreibung und Ausrichtung von Zylinderrädern mit Evolventenprofil,
Diploma Thesis, University of Ulm, Germany. [10] Ni K., Peng Y., Characterization and evaluation of involute gear flank data using an areal model. Proceedings of the 31st ASPE Annual Meeting 2016; 184-189. [11] Peng Y. Ni K. Goch G. Areal evaluation of involute gear flanks with three-dimensional surface data, Proceedings of the 2017 AGMA Fall Technical Meeting. [12] Stoebener D., von Freyberg A., Fuhrmann M., Goch G., Areal parameters for the characterization of gear distortions. Mat.wiss.u.Werkstofftech. 2012; 43: 120 – 124.
