Numerical calculation algorithm for spherical tooth profile of noncircular bevel gear

It is an important issue for a wide-narrow distance transplanting mechanism with planetary gear trains to obtain the spatial planting trajectory that meets the wide-narrow distance transplanting. A noncircular bevel gear transmission, one kind of spatial non-uniform velocity transmission mechanism could form such a trajectory. What is more, the pitch curve of the non-circular bevel gear is general, and a transplanting mechanism with this kind of bevel gears can achieve more potential spatial planting trajectories for wide-narrow distance planting than those with specific bevel gears such as an elliptical bevel gear or an eccentric-noncircular bevel gear. In order to meet diverse agronomic requirements, a variety of wide-narrow distance transplanting mechanisms and noncircular bevel gears with different parameters and pitch curves are needed. However, due to the lack of a uniform tooth profile calculation method, the designer has to establish different tooth profile calculation models for different pitch curves. A uniform method which could be applied to calculate the tooth profile of the non-circular bevel gear is put forward in this paper. Because of the spherical tooth profile of the non-circular bevel gear and the standard parameters of the big end, the uniform expression of big end pitch curve and the numerical model of big end tooth profile are essential to designing the noncircular bevel gear. In this paper, a cubic Nurbs curve was used to fit the spherical pitch curve of a bevel gear, which can ensure second order continuity of the points on the pitch curve. A smooth, continuous and closed spherical pitch curve could be obtained from several data points of the spherical surface by the proposed method. According to the included angle of two adjacent tangent vectors of the points on the pitch curve, the concavity and convexity of the pitch curve could be determined, and the radius of the big end pitch circle of a maximum bevel gear for enveloping concave pitch curve gear could be calculated. Then, the allowable maximum gear modulus of the concave pitch curve could be determined by calculating the minimum undercutting number of the circular bevel gear. By using the tooth profile normal line method and spherical triangle property, the numerical calculation model of a spherical tooth profile and dedendum transition curve were established. Furthermore, the undercutting phenomenon could be judged by analyzing the concavity and convexity of the tooth profile curve. Finally, the numerical calculation program for the tooth profile was compiled in Matlab, and several noncircular bevel gears for the transplanting mechanism of a walking-type wide-narrow distance transplanter were designed. The transmission ratio obtained by simulation in ADAMS software was highly identical with the one obtained by theoretical calculation, which verified the feasibility of this method.