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Journal of Computational Design and Engineering   Volume 1, Number 1, January 2014, pages 27-36
Parametric surface and properties defined on parallelogrammic domain
Shuqian Fan, Jinsong Zou and Mingquan Shi

Abstract     [Full Text]
    Similar to the essential components of many mechanical systems, the geometrical properties of the teeth of spiral bevel gears greatly influence the kinematic and dynamic behaviors of mechanical systems. Logarithmic spiral bevel gears show a unique advantage in transmission due to their constant spiral angle property. However, a mathematical model suitable for accurate digital modeling, differen- tial geometrical characteristics, and related contact analysis methods for tooth surfaces have not been deeply investigated, since such gears are not convenient in traditional cutting manufacturing in the gear industry. Accurate mathematical modeling of the tooth surface geometry for logarithmic spiral bevel gears is developed in this study, based on the basic gearing kinematics and spherical involute ge- ometry along with the tangent planes geometry; actually, the tooth surface is a parametric surface defined on a parallelogrammic domain. Equivalence proof of the tooth surface geometry is then given in order to greatly simplify the mathematical model. As major factors af- fecting the lubrication, surface fatigue, contact stress, wear, and manufacturability of gear teeth, the differential geometrical characteris- tics of the tooth surface are summarized using classical fundamental forms. By using the geometrical properties mentioned, manufactura- bility (and its limitation in logarithmic spiral bevel gears) is analyzed using precision forging and multi-axis freeform milling, rather than classical cradle-type machine tool based milling or hobbing. Geometry and manufacturability analysis results show that logarithmic spiral gears have many application advantages, but many urgent issues such as contact tooth analysis for precision plastic forming and multi- axis freeform milling also need to be solved in a further study.
Key Words
    Spiral bevel gear; Mathematical modeling; Parametric surface; Geometrical characteristics; Manufacturability
Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing, China

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