Parametric Transition as a Spiral Curve and Its Application in Spur Gear Tooth with FEA

Authors: S. H. Yahaya, J. M. Ali, T.A. Abdullah

Abstract:

The exploration of this paper will focus on the Cshaped transition curve. This curve is designed by using the concept of circle to circle where one circle lies inside other. The degree of smoothness employed is curvature continuity. The function used in designing the C-curve is Bézier-like cubic function. This function has a low degree, flexible for the interactive design of curves and surfaces and has a shape parameter. The shape parameter is used to control the C-shape curve. Once the C-shaped curve design is completed, this curve will be applied to design spur gear tooth. After the tooth design procedure is finished, the design will be analyzed by using Finite Element Analysis (FEA). This analysis is used to find out the applicability of the tooth design and the gear material that chosen. In this research, Cast Iron 4.5 % Carbon, ASTM A-48 is selected as a gear material.

Keywords: Bézier-like cubic function, Curvature continuity, Cshapedtransition curve, Spur gear tooth.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1057623

References:

[1] K. G. Baass, "The use of clothoid templates in highway design," Transportation Forum 1, 1984, pp. 47-52.
[2] D. J. Walton, and D. S. Meek, "Planar G2 transition between two circles with a fair cubic Bézier curve," J. Computer Aided Design, 31, 1999, pp. 857-866.
[3] D. J. Walton, and D. S. Meek, "The use of cornu spirals in drawing planar curves of controlled curvature," J. Computational and Applied Mathematics, 25, 1989, pp. 69-78.
[4] D. J. Walton, D. S. Meek, and J. M. Ali, "Planar G2 transition curves composed of cubic Bézier spiral segments," J. Computational and Applied Mathematics, 157, 2003, pp. 453-476.
[5] D. J. Walton, and D. S. Meek, "Curvature extrema of planar parametric cubic curves," J. Computational and Applied Mathematics, 134, 2001, pp. 69-83.
[6] D. J. Walton, and D. S. Meek, "A planar cubic Bézier spiral," J. Computational and Applied Mathematics, 72, 1996, pp. 85-100.
[7] Z. Habib, and M. Sakai, "G2 planar cubic transition between two circles with shape control," J. Computational and Applied Mathematics, 223, 2009, pp. 133-144.
[8] Z. Habib, and M. Sakai, "Circle to circle transition with a single cubic spiral," Proc. Int. Conf. on Visualization, Imaging, and Image Processing, ACTA Press, 2005, pp. 691-696.
[9] Z. Habib, and M. Sakai, "G2 planar cubic transition between two circles," Int. J. Computer Mathematics, 80, 2003, pp. 959-967.
[10] I. Juhász, "Cubic parametric curves of given tangent and curvature," J. Computer Aided Design, 25, 1998, pp. 1-9.
[11] J. M. Ali, H. B. Said, and A. A. Majid, "Shape control of parametric cubic curves," Proc. Fourth Int. Conf. CAD/CG, 1995, pp. 161-166.
[12] B. W. Bair, "Computer aided design of elliptical gears," J. Mechanical Design, 124, 2002, pp. 787-793.
[13] G. I. Sheveleva, A. E. Volkov, and V. I. Medvedev, "Algorithm for analysis of meshing and contact of spiral bevel gears," J. Mechanism and Machine Theory, 42, 2006, pp. 198-215.
[14] A. Belsak, and J. Flasker, "Method for detecting fatigue crack in gears," J. Theoretical and Applied Fracture Mechanics, 46, 2006, pp. 105-113.