Abstract
A method for reconstructing the digital real tooth surfaces of hypoid gears can be a significant foundation for a variety of dynamic performance and lifetime prediction. This study demonstrates a new digital real tooth surfaces modeling approach for hypoid gears based on non-geometric-feature segmentation and interpolation algorithm. In this method, the discrete data points, which are obtained by using acoordinate measure machine (CMM), are segmented in the form of Delaunay triangular meshes. In order to identify irregular local micro-geometry features, the segmentation method starts with a feature detection based on normal vectors of Delaunay triangular meshes, identifying wear regions around each discrete data point, and is followed by region growing steps to divide tooth surface. In addition, a revised interpolation algorithm is applied to describe local micro-geometry features on wear regions via weighted factors to locally qualify the triangular vertexes. And the revised fairing algorithm minimizes the effect of noisy points. Experimental results from reconstruction of real tooth surface after wear test demonstrate that our method can improve the computation precision of wear region on actual tooth surfaces.
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Abbreviations
- B :
-
the recursion formula of basis function
- E :
-
pinion position error along the shaft offset direction
- ΔE m :
-
blank offset
- ΔE i :
-
deviation between the control vertex and interpolation curve
- Δe :
-
absolute error of position vector norm
- G :
-
axial displacement of the gear
- J i :
-
moment of inertia
- M ji , L ji :
-
matrix of coordinate transformation from system Sito system S j
- n :
-
number of contact lines
- N g :
-
number of the gear teeth
- n i,j :
-
normal vector of V i,j
- n p , n g :
-
unit normal vectors in system S t to profile of pinion and gear, respectively
- P(u i ):
-
interpolation curve
- q :
-
cradle rotation angle
- R a :
-
ratio of roll
- R d :
-
contact ratio of reference tooth surface
- ΔR p , ΔR g :
-
deviations between the actual tooth surface and the reference tooth surface
- r B :
-
cutter radius
- r G :
-
cutter point radius
- r p , r g :
-
position vectors in system S t to profile of pinion and gear, respectively
- S a , S b :
-
coordinate systems for assisting the installment of the work piecen
- S c :
-
machine cradle coordinate system
- S i :
-
theareaof Tri(i)
- S m :
-
cutting machine frame coordinate system
- S p , S g :
-
coordinate systems are attached to the pinion and the gear, respectively
- S t :
-
the gear head-cutter coordinate system
- s r :
-
radial setting
- Tri(i):
-
Delaunay triangle
- t :
-
time
- t i :
-
position vector of control vertex
- u :
-
profile direction
- u p , u g :
-
surface parameters of the head-cutter to the pinion and the gear, respectively
- V ij :
-
control points
- v :
-
tooth trace direction
- v gt :
-
relative velocity vector of head-cutter to the gear
- v p , v g :
-
velocity vector of the pinion and the gear, respectively
- ΔX B :
-
sliding base
- ΔX D :
-
machine center to back
- α g :
-
blade angle of the head-cutter
- γ m :
-
machine root angle
- κ :
-
curve curvature
- l i (j):
-
measure points
- θ p , θ g :
-
surface parameters of the head-cutter
- δ i :
-
angle between normal vectors of Delaunay triangles
- Γ i :
-
tangential vector
- ω i :
-
weighted factor for control point
- ω p , ω g :
-
angular velocity of the pinion and the gear, respectively
- ω c :
-
angular velocity of the cradle
- Σ p , Σ g :
-
generating surface of pinion and gear, respectively
- ψ p , ψ g :
-
current rotation angles of the pinion and the gear, respectively
- ψ c :
-
current rotation angle of the cradle
References
Chen, B., Liang, D., and Li, Z., “A Study on Geometry Design of Spiral Bevel Gears based on Conjugate Curves,” Int. J. Precis. Eng. Manuf., Vol. 15, No. 3, pp. 477–482, 2014.
Park, D. and Kahraman, A., “A Surface Wear Model for Hypoid Gear Pairs,” Wear, Vol. 267, No. 9, pp. 1595–1604, 2009.
Park, D., Kolivand, M., and Kahraman, A., “An Approximate Method to Predict Surface Wear of Hypoid Gears using Surface Interpolation,” Mechanism and Machine Theory, Vol. 71, pp. 64–78, 2014.
Zhang, Q., Kang, J., Dong, W., and Lyu, S., “A Study on Tooth Modification and Radiation Noise of a Manual Transaxle,” Int. J. Precis. Eng. Manuf., Vol. 13, No. 6, pp. 1013–1020, 2012.
Litvin, F. L., Fuentes, A., Fan, Q., and Handschuh, R. F., “Computerized Design, Simulation of Meshing, and Contact and Stress Analysis of Face-Milled Formate Generated Spiral Bevel Gears,” Mechanism and Machine Theory, Vol. 37, No. 5, pp. 441–459, 2002.
Tsai, Y.-C. and Hsu, W.-Y., “The Study on the Design of Spiral Bevel Gear Sets with Circular-Arc Contact Paths and Tooth Profiles,” Mechanism and Machine Theory, Vol. 43, No. 9, pp. 1158–1174, 2008.
Litvin, F. L., Fuentes, A., and Hayasaka, K., “Design, Manufacture, Stress Analysis, and Experimental Tests of Low-Noise High Endurance Spiral Bevel Gears,” Mechanism and Machine Theory, Vol. 41, No. 1, pp. 83–118, 2006.
Flodin, A. and Andersson, S., “Simulation of Mild Wear in Helical Gears,” Wear, Vol. 241, No. 2, pp. 123–128, 2000.
Flodin, A. and Andersson, S., “A Simplified Model for Wear Prediction in Helical Gears,” Wear, Vol. 249, No. 3, pp. 285–292, 2001.
Bajpai, P., Kahraman, A., and Anderson, N., “A Surface Wear Prediction Methodology for Parallel-Axis Gear Pairs,” Journal of Tribology, Vol. 126, No. 3, pp. 597–605, 2004.
Kahraman, A., Bajpai, P., and Anderson, N., “Influence of Tooth Profile Deviations on Helical Gear Wear,” Journal of Mechanical Design, Vol. 127, No. 4, pp. 656–663, 2005.
Park, D., Kolivand, M., and Kahraman, A., “Prediction of Surface Wear of Hypoid Gears using a Semi-Analytical Contact Model,” Mechanism and Machine Theory, Vol. 52, pp. 180–194, 2012.
Kolivand, M. and Kahraman, A., “An Ease-Off based Method for Loaded Tooth Contact Analysis of Hypoid Gears Having Local and Global Surface Deviations,” Journal of Mechanical Design, Vol. 132, No. 7, Paper No. 071004, 2010.
Kubo, A., Tarutani, I., Gosselin, C., Nonaka, T., Aoyama, N., and Wang, Z., “On Simulation Methods of Performance of Hypoid and Spiral Bevel Gears,” Transactions of the JSME, Series C, Vol. 62, No. 599, pp. 2833–2850, 1996.
Takeda, R., Komori, M., Nishino, T., Kimura, Y., Nishino, T., et al., “Performance Analysis of Generated Hypoid Gear based on Measured Tooth Flank Form Data,” Mechanism and Machine Theory, Vol. 72, No. pp. 1–16, 2014.
Jiang, Q., Gosselin, C., and Masseth, J., “Simulation of Hypoid Gear Lapping,” Journal of Mechanical Design, Vol. 130, No. 11, Paper No. 112601, 2008.
Jiang, Q., Gosselin, C., and Masseth, J., “Computer-Aided Machine Setting for Lapping Optimization,” Journal of Mechanical Design, Vol. 131, No. 3, Paper No. 031003, 2009.
Abbas, A. T. M., “Enhanced CNC Machines Capabilities by Adding Circular Patterns Cycle,” Int. J. Precis. Eng. Manuf., Vol. 13, No. 10, pp. 1753–1758, 2012.
Simon, V. V., “Influence of Tooth Modifications on Tooth Contact in Face-Hobbed Spiral Bevel Gears,” Mechanism and Machine Theory, Vol. 46, No. 12, pp. 1980–1998, 2011.
Yun, J.-H., Jeong, M.-S., Lee, S.-K., Jeon, J.-W., Park, J.-Y., and Kim, G. M., “Sustainable Production of Helical Pinion Gears: Environmental Effects and Product Quality,” Int. J. Precis. Eng. Manuf.-Green Tech., Vol. 1, No. 1, pp. 37–41, 2014.
Meek, D. S. and Walton, D. J., “An Arc Spline Approximation to a Clothoid,” Journal of Computational and Applied Mathematics, Vol. 170, No. 1, pp. 59–77, 2004.
Piegl, L. A. and Tiller, W., “Parametrization for Surface Fitting in Reverse Engineering,” Computer-Aided Design, Vol. 33, No. 8, pp. 593–603, 2001.
Cripps, R. J., Hussain, M. Z., and Zhu, S., “Smooth Polynomial Approximation of Spiral Arcs,” Journal of Computational and Applied Mathematics, Vol. 233, No. 9, pp. 2227–2234, 2010.
Sitnik, R. and Błaszczyk, P. M., “Segmentation of Unsorted Cloud of Points Data from Full Field Optical Measurement for Metrological Validation,” Computers in Industry, Vol. 63, No. 1, pp. 30–44, 2012.
Liu, Y. and Xiong, Y., “Automatic Segmentation of Unorganized Noisy Point Clouds based on the Gaussian Map,” Computer-Aided Design, Vol. 40, No. 5, pp. 576–594, 2008.
Litvin, F. L. and Fuentes, A., “Gear Geometry and Applied Theory,” Cambridge University Press, pp. 97–118, 2004.
Tang, K. and Pang, A., “Optimal Connection of Loops in Laminated Object Manufacturing,” Computer-Aided Design, Vol. 35, No. 11, pp. 1011–1022, 2003.
Woo, H., Kang, E., Wang, S., and Lee, K. H., “A New Segmentation Method for Point Cloud Data,” International Journal of Machine Tools and Manufacture, Vol. 42, No. 2, pp. 167–178, 2002.
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Li, G., Wang, Z. & Kubo, A. The modeling approach of digital real tooth surfaces of hypoid gears based on non-geometric-feature segmentation and interpolation algorithm. Int. J. Precis. Eng. Manuf. 17, 281–292 (2016). https://doi.org/10.1007/s12541-016-0036-6
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DOI: https://doi.org/10.1007/s12541-016-0036-6