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Constraint analysis of pressure angle of involute elliptical gears
Shanming Luo* Anhua Chen†
Hunan University of Science and Technology, Xiangtan, P. R. China
Yue Wu‡
University of Exeter, Exeter, UK
Abstract—
A geometric relationship and mathematical model of pressure angle of elliptical gears are developed based on the theory of gearing, and some constraints of pressure angle are presented in order to prevent gear teeth from coming out of mesh. The results show that the pressure angle is a function of contact position and increases as the eccentricity of the pitch curve grows, which is very different from circular gears. For the sake of higher meshing efficiency, the pressure angle should not exceed an admissible value. The results of this work should be helpful in the design, manufacture and measurement of elliptical gears. Keywords: pressure angle, constraint, elliptical gear, pitch curve I. Introduction The elliptical gear is well known for its favorable characteristics such as variable gear ratio, compact size, accurate transmission and easy dynamic balance. For this reason, elliptical gears are useful in those mechanisms whose task is to force an output element to move according to a specific law of motion. The elliptical gear is the most commonly used noncircular gear in automatic machinery, flying shears, pumps, flow meters and other instruments. A vast amount of literature of elliptical gears is available focusing on kinematical analysis and computeraided design, and many significant advances in this area have been achieved in the recent decades. For instance, Litvin [1] proposed an enveloping method. This approach is based on the idea of using tools (rack and shaper cutters) similar to those usually employed in circular gear generation. According to this approach, conjugate tooth profiles are generated by performing a pure rolling of the cutter centrode along the given pitch curve. A set of necessary relations between cutter and gear motion was then derived for both rack and shaper cutter generation, in order to fulfill the condition of rolling without sliding. Chen and Tsai [2] described rack cutters with circular-arc profile teeth to generate elliptical gears which rotate about one of their foci. And a mathematical model for elliptical gears with circular-arc teeth was developed according togear theory. The effects of the circular-arc radius of the rack cutter on both the undercutting of teeth and on pointed teeth of the generated circular-arc elliptical gear were also studied. Chang, et al. [3] employed gear theory and the geometry of a straight-sided rack cutter to derive a mathematical model for elliptical gears, in which the number of teeth had to be odd, and also examined the undercutting conditions of the developed elliptical gears. A computer program was also developed to generate the tooth profile of elliptical gears. Bair [4, 5] proposed a mathematical model of profile-shifted elliptical gears used to avoid the tooth undercutting and pointed teeth. Moreover, a computer simulation program is developed to generate the tooth profiles of non-standard elliptical gears with few teeth and a compact size. Freudenstein and Chen [6] developed variable-ratio chain drives, and applied them to bicycles and variable motion transmission involving band drives, tape drives and time belts with a minimum slack. Emura and Arakawa [7] employed elliptical gears to analyze a steering mechanism able to turn a carrier with a small radius. Danieli and Mundo [8] proposed an algorithm to obtain the kinematic synthesis of a pair of variable radius gears, generating a prescribed law of motion. Then, the concept of constant pressure angle non-circular helical gears and a method to generate a CAD representation was introduced, which can greatly increase the contact ratio between the teeth of noncircular gears. Mundo [9] presented a new concept of epicyclical gear train able to generate a variable gear ratio law. The basic mechanical configuration consists of three noncircular gears in a typical planetary arrangement, whose pitch lines are all variable-radius curves. Lozzi [10] present a general way to generate base and involute contours associated with noncircular pitch outlines by graphic construction. The outlines used were ellipses of the first three orders. Cao, et al [11] analyzed the conjugate principle of variable ratio gears with constant center distance and contact angle by using normal mapping model. Tan and Yoshiyuk [12] pointed out that the pressure angle of noncircular gear is variable and increases with the eccentricity of ellipse. Chang and Tsai [13] presented the feasible design spaces of the pressure angle for spur gears by using polar angular functions, and investigated the possibility of cusp occurring in each design space of mating gears. Naji and Marshekpresented an analysis of determining the effect of pitch difference on roller chain articulation angles and pressure angles. Basic formulas and strategies were given for locating contact points. And results were compared with those of a graphical solution. In addition, Dooner [15], Tong [16], Simionescu [17], Lin [18] and Figliolini [19], et al. had also contributed to the design and generation of noncircular or elliptical gears from various angles. Summing up the aforementioned literature, we can obtain the following findings: (1) most of those studies about elliptical gears focused on the kinematical analysis and computer-aided design of tooth profiles; (2) almost all had been accomplished under the assumption of constant pressure angle; (3) few studies took the constraints of pressure angle of elliptical gears into account, perhaps because of its variability. Pressure angle is an important design parameter of elliptical gears and a function of contact position. It increases as the eccentricity of pitch curve of ellipse grows. The aim of this work is to investigate the change law of pressure angle of elliptical gears along with the contact position and arrive at useful constraints for the design of pitch curve. For this reason, a geometric relationship and mathematical model of pressure angle of elliptical gears are developed based on the theory of gearing. Some examples of numerical calculation are also illustrated in this paper.