The cross axis structure of conical fan teeth is often used to realize the non-uniform speed transmission of the rack pair of the non-circular gear fan of the automobile steering gear. The middle teeth of the gear fan are used most frequently

and so that the broken teeth generally appear at the small end of the middle teeth. Therefore

it is of great engineering significance to study the influence of the contact line distribution and meshing trajectory on the wear of the gear surface. At present

most of the research on this type of transmission is carried out in the big end plane of the tooth fan

and the spatial contact state of the tooth surface cannot be accurately analyzed. Based on the principle of space meshing

this paper deduces the motion equation of the tooth surface of the non-circular fan

and establishes the three-dimensional model of the rack pair of the non-circular fan by using the envelope method. The contact line

meshing surface and meshing trajectory of the tooth surface are solved

and it is found that the meshing trajectory of the small end is longer than that of the big end. The influence of geometric parameters such as pressure angle α

tooth cone angle φ and tooth depth ap on the distribution of instantaneous contact lines is studied in detail. The theoretical method for improving the contact condition of the tooth surface and parametric design of the non-circular rack and pinion pair is provided.