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A new type of non-orthogonal arc tooth face-gear is designed. The arc tooth cylindrical gear is used as an imaginary tool to envelop the working tooth surface and transition surface of the face gear. The mathematical model of the face-gear is obtained in Matlab. The full tooth surface tooth surface equations of the non-orthogonal arc tooth face-gear is derived，and the realization of tooth surface visualization through Matlab programming which laid the theoretical foundation for the subsequent research on non-orthogonal arc tooth face-gears.

Considering the advantages of the circular arc gear and the face gear,the gear drive with circular arc tooth profile is put forward. In order to advance the machining application of the face gear,the face gear blank size must be determined. By using the envelope forming theory,the tooth surface equation of the circular arc tooth profile is deduced,and the contour visualization is realized in MATLAB,and it is concluded that the tooth tip has two sharpening in the radial direction. The factors that affect the radius of the tooth tip sharpening are analyzed and calculated according to the numerical calculation. A special case with a helix angle of 0° is proposed and the approximate equation of inner diameter and outer diameter of tooth tip sharpening is obtained by least squares method. Finally,the feasibility of the theoretical calculation and the approximate equation is proved by the simulation experiment.

In view of the shortcomings of involute gear drive,the advantages of synthetic face gear drive are summarized. The circular arc profile is applied to the face gear,and the gear drive with circular tooth profile is put forward. By means of theoretical analysis,numerical calculation and physical experiment,the design of the tooth surface of the circular tooth profile is carried out. Based on gear meshing theory and differential geometry theory,the tooth surface equation of the conjugate tooth profile of the basic tooth profile of the circular rack is modeled firstly,and then use the envelope forming theory to derive the equation of gear face machining arc face gear,and the cutter and circular arc tooth profile gear for each envelope forming work on tooth surface and the derivation of the face gear transition surface equation. Finally,the MATLAB software program is used to solve the set of tooth points and imported into the CATIA software. The EXCEL table with macro program runs the macro program and generates the contour curve in the CATIA software. And then the 3D physical tooth surface of the arc gear profile is obtained by surface fitting method. Compared with the simulation model of gear cutting envelope,the accuracy is verified.

Aiming at the phenomenon of undercut in the process of circular arc tooth profile face gear,by using the research method combining theoretical analysis,numerical calculation and simulation experiment,the minimum inner radius of the face gear without undercut is studied. Firstly,the tooth surface equation of the tool gear is established,and the working tooth surface and the transition surface equation of the gear are deduced from the combination of the tool and the circular arc tooth profile. And then according to the circular arc tooth profile face gear does not produce undercut conditions theory,through the MATLAB software to write the program for numerical calculation. The calculated results are compared with the results obtained by the simulation experiments,and the correctness of the minimum radius values of the circular arc tooth profile face gear without rooting is verified.

In order to explicitly solve the tooth surface model of forming face gear,and exactly solve the minimum inner radius of face gear with top-cut. The orthogonal face gear envelope coordinate system is established and the tooth surface equation of generating gear is given. The face gear working tooth surface equation,transition surface equation and top-cut surface equation are derived. According to the tooth surface equation,the simulation analysis of face gear tooth surface is carried out by using the MATLAB 7. 5 and CATIA V5. According to the simulation comparison of tooth surface and the analysis of top-cut boundary line,the top-cut minimum inner radius key point is got,the key point is the intersection of the face gear tip surface and top-cut boundary line. Through the explicit solving the key points,the minimum inner radius of the face gear without top-cut can be got. Comparing with the existing literature results of the implicitly solving inner radius,and the difference on numerical is small between the two. Finally,through the simulation cutting tooth way,it shows the accuracy of this method is superior to the method of literature.