The research of noncircular bevel gears' undercut condition

Noncircular bevel gear (NBG) is a kind of gear which can transmit spatial variable transmission ratio motion directly. It has the advantages of compactness, smooth motion, and accurate transmission. There, however, exists an urgent problem that the NBGs' exact undercut condition is unknown. The planar noncircular gears' undercut condition has been researched in different ways, yet the undercut condition research of NBGs is very limited [1-3]. This paper mainly aims at analyzing the undercut phenomenon with tooth profile curve evolute and finding an easy method to judge undercut. The research of this problem is meaningful because it reveals the essence of undercut in an analytical and intuitionistic way and consequently does help to judge the occurrence of the undercut of any teeth. Besides, the proper module chosen according to the undercut judging method will improve the contact ratio of NBG pairs which is good for the meshing property and bearing capacity. This paper is mainly based on analytical and computational methodology. Firstly, for a pair of NBGs, the included angle of pitch curves' mutual tangent arc and teeth' normal arc is researched. It can be proved that the included angle is a constant one and is equal to the tool's tooth profile angle. It should be emphasized that the tool is limited to rack and bevel gear shaper cutter only. Then, the evolute of the tooth profile curve is researched. By analyzing the geometric and kinematic relationship between tooth profile curve, pitch curve and pitch curve's evolute, the tooth's every center of curvature can be calculated with the sine and cosine formula of spherical triangle. In other words, the evolute equation is derived. Also, the critical point of undercut can be calculated since the tooth profile curve's evolute is the generative curve of the tooth profile, which means the intersection of the two curves is the critical point. Rack and bevel gear shaper cutter are discussed separately. The critical point is analyzed in spherical triangle to calculate the critical module value so that the occurrence of the undercut can be judged. Lastly, evolutes of high order spherical ellipses with specific condition are calculated with Matlab and their figures are demonstrated which can reveal the property of evolute directly. The critical module value of a pair of NBGs is calculated and their 3D models are generated by UG. It turns out that no undercut happens. The kinematical simulation is implemented in ADAMS. The actual angular velocity curve of the driven gear is very close to the ideal one. These results justify the correctness of undercut judgement method.