1、PDF外文：http:/ Computerized tooth profile generation of elliptical gears manufactured by shaper cutters Biing-Wen Bair Department of Mechanical Engineering, National Lien Ho Institute of Technology, Abstract This work simulates an elliptical gear drive, the axis of rotation of which
2、 is coincident with its geometric center, manufactured by shaper cutters. The mathematical model of an elliptical gear is developed based on the theory of gearing and gear generation mechanisms. In addition, the tooth undercutting of the gear is also investigated based on the developed mathematical
3、model of the elliptical gear, its unit normal vectors and a numerical method. A geometric relationship is developed and applied to prevent the occurrence of pointed teeth on elliptical gears. Further, this study also develops computer simulation programs to generate the tooth profile of ellipt
4、ical gears without tooth undercutting and pointed teeth. Comparison of the angular velocity variations of the elliptical gear drives is also made. The results show that the developed elliptical gear drive can be utilized as an oil pump with a larger pumping volume and less angular velocity variation
5、. 2002 Elsevier Science B.V. All rights reserved. Keywords: Elliptical gears； Undercutting； Pointed teeth 1. Introduction An elliptical gear drive, the rotation center of which coincides with one of its foci, is kinematically equivalent to the crossed link, and can be used to produce irr
6、egular rotations. In addition, it is well known for providing excellent characteristics such as accurate transmission, compact size, and ease of dynamic balance. Hence, elliptical gear drives have been applied successfully in various types of automatic machinery, quick-return mechanisms, packaging m
7、achines, and printing presses 1. This type of gearing can also be used to develop non-circular gears, which belong to high-order elliptical pitch curves. Second- order elliptical gear drives, the rotation center of which coincides with one of its foci, can find use in the design of instruments such
8、as pumps and flow meters 1. However, this type of gear set has two speed changes for each revolution, these two-cycle variations inducing a wave fluctuation that is so severe that the second-order elliptical gear set cannot be used as oil pumps for steady oil pumping. The design and manufactur
9、e of an elliptical gear are difficult because the pitch curve of the gear is an ellipse. Some studies 2 6 have focused on kinematic analysis and computer-aided design of elliptical pitch curves. Kuczewski 7 used a spur gear to approximate the profile of an elliptical ge
10、ar. Emura and Arakawa 8 used an elliptical gear to analyze a steering mechanism, where this steering mechan- ism can turn a carrier with a small radius. Also, Freudenstein and Chen 9 developed variable-ratio chain drives (e.g. elliptical gear drives), which were applied to bicycles and variable moti
11、on transmissions involving band drives, tape drives, and time belts with a minimum slack. Moreover, Litvin 10 adopted the concept of evolute curves to form the tooth profile, and also derived the tooth evolute of an ellipse. Chang and Tsay 11 used a shaper cutter and applied the inverse mechanism re
12、lationship and the equation of meshing to produce the mathematical model of elliptical gears, the rotation center of which turns around one of its foci. Also, Chang et al. 12 used a rack cutter and the same method to produce the mathematical model and undercut- ting conditions of the same type of el
13、liptical gears. However, when the elliptical gear surfaces are generated by shaper cutters, pointed teeth may appear and the tooth addendum is reduced. Pedrero et al. 13 proposed an approximation method for modifying the tooth addendum and contact ratio, and
14、 computer simulation results also show that the gear contact ratio depends on the tooth addendum. Additionally, Liou et al. 14 analyzed spur gears with low contact ratios (a contact ratio of less than 2) and high contact ratios (equal or larger than 2) when subjected to dynamic loads by applying the
15、 NASA gear dynamics software DANST. The DANST program determined the instantaneous contact teeth and contact ratio based on the gear average stiffness, commonly referred to as mesh stiffness. Recently, Bair and Tsay 15 proposed a tooth contact analysis (TCA) program to
16、 calculate the instantaneous contact teeth and average contact ratio of a dual-lead worm gear drive. The DANST and TCA methods confirm that reducing the gear addendum results in the decrease of the gear instantaneous contact teeth and the average contact ratio. Based on
17、the position of the rotation center, elliptical gear drives fall into two types: the elliptical gear which rotates about its geometric center (type 1)； the elliptical gear which rotates around one of its elliptical foci. An n- order driven non-circular gear, of which the rotation axis is one of its
18、foci, is one in which the driving elliptical gear performs n revolutions for one revolution of the driven non- circular gear. A second-order elliptical gear (type 2) is defined as n = 2. Under the same eccentricity and major axis, the size of the type 1 elliptical gears is smaller than that of
19、 the type 2. Further, if the type 1 elliptical gear set is applied to oil pumps, the wave fluctuation of the pumping oil is smaller and smoother than that of the other type. The elliptical gear tooth profile is usually produced by a hob- bing or shaping machine with a hob cutter or shaper cutter. Th
20、is study simulates the manufacture of elliptical gears via a shaper cutter on a shaping machine. It is known that if a spur gear is produced by a shaper, the profile of the shaper cutter should be the same as that of the mating spur gear. Therefore, the mathematical model of a shaper cutter is the s
21、ame as that of a spur gear, which can be derived from the generation mechanism with a rack cutter. According to the theory of gearing, the mathematical models of elliptical gear tooth profiles, which rotate about their geometric center (type 1), are develope
22、d based on the proposed generation mechanism with shaper cutters. Due to the complex characteristics of this type of elliptical gear, undercutting and pointed teeth may exist on its tooth profile. The tooth undercutting of this type of elliptical gear is affected by its pressure angle, number of tee
23、th, module, and major axis. The strength of the gear tooth root can be increased by applying a positive-shifted modification for the cutter during the gear generation. However, an over-positive-shifted modification may result in the appear- ance of pointed teeth. Pointed teeth are generated when the
24、 right- and left-side involute tooth profiles intersect on or below the addendum circle of the gear. Further, the pointed teeth are usually generated on the two major axis of an elliptical gear. If a profile index is defined to prevent pointed teeth generati
25、on on the two major axis of an elliptical gear, then no other pointed tooth will be generated for all elliptical gear profiles. Thus, the computer program developed here can calculate and provide proper design parameters for the designed elliptical gears to avoid tooth undercutting and pointed teeth
26、. 2. Mathematical model of the elliptical gear surfaces Shaper cutters are used to generate elliptical gears, and the profiles of shaper cutters are the same as those of spur gears. Hence, the mathematical model of the shaper cutter is the same as that of the spur gear, which is generated from rack cutters. A complete elliptical gear tooth profile consists of three surface regions, i.e. the working region, the fillet and the bottom land. Therefore, the profile parameters of a shaper cutter can be represented by the
