Research on tooth surface design and side milling machining method of ruled line face gear

To solve the problem of generalized design and batch processing of face gear, a method of tooth surface design and side milling of ruled line face gears is proposed in this paper. First of all, the discrete points of the pitch cone surface of the face gear are calculated, and the straight-line cluster and section curve of the tooth surface is constructed to analyze the deviation of the ruled line face gear. Secondly, the tooth surface model of the ruled line face gear is constructed, and the curvature and contact trace are calculated. Thirdly, the curvature and tangent of the ruled line face gear tooth surface are analyzed, the feed plane of the equidistant surface and the guideline are obtained, the torsion angle of the tooth surface and the distribution of the tangent vector along the generatrix are calculated, and the tool axis position based on the two-point offset method is studied. Finally, the side milling processing test of the face gear is carried out, and the tooth surface of the face gear after side milling is measured. The measurement results verify the correctness of the tooth surface design and the side milling processing method of the ruled line face gear.


Introduction
The face gear drive has the advantages of small size, lightweight, strong bearing capacity, and good interchangeability, which has broad application prospects in the fields of aerospace, vehicles, and ships. The tooth surface of the face gear is a complex curved surface in space, the configuration process of the tooth surface is complex, and the processing of the tooth surface is difficult. To meet the needs of the engineering application of face gears, many scholars have carried out a lot of research on the design and processing of face gears.
Lin et al. presented a machining method for orthogonal variable ratio face gears using a five-axis CNC machine tool to solve manufacturing problems of the face gear [1]. Yang et al. studied a rough machining method for the face gear through plunge milling and planning and carried out theoretical research and numerical control simulation machining, respectively [2,3]. Shen et al. developed a method of shaving processing for spiroid face gear, and the kinematics of the shaving processing was analyzed. And they further developed a computer-integrated shaving processing of the gear on a five-axis computer numerical control machine [4,5]. Shih et al. proposed a disk tool cutting method for a five-axis machine whose five-axis coordinates for gear production are derived based on the target tooth surface [6]. Zhou et al. derive an advanced method, based on the geometry characteristic of the tooth surface, to calculate the points as an even distribution on the tooth surface [7]. They present an accurate measurement model of the face gear tooth surface, wherein the digital tooth contact analysis is implemented with a robust algorithm [8].
Peng et al. studied a method where the predesigned contact path and transmission errors are applied to the synthesis of the motion rule and presented the application of the modern Phoenix bevel gear manufacturing machine tool in the face gear generation by the plane-cutter [9]. Xiang et al. propose a method to analyze and compensate for geometric errors of sixaxis CNC grinding machines for spiral bevel gears [10]. Mo et al. studied the changing laws of load sharing coefficients influenced by flexible support when the sun gear is floating and the sun gear is normally supported [11]. They also studied the multi-power face gear split flow system [12]. Guo et al. presented the procedure of computerized generation and the results of simulation of meshing and stress analysis of face gear drive manufactured by circular cutters without considering the feed motion of the circular cutters and proposed a kind of approximate definition method of grinding worm surface with variable meshing angle for grinding face gears [13,14].
Dong et al. analyzed the characteristics of dynamic load sharing in a concentric face gear split-torque transmission system and introduced a concentric face gear torque split system used in the helicopter main transmission to transfer more power and reduce the structure weight [15,16]. Deng et al. presented a machining method referred to as power skiving applied to the machining of face gears [17]. Wang et al. have done a lot of work on the design and precision machining of face gear. They proposed a precision milling method for face gear by disk cutter, a precision generating hobbing method for face gear with the assembly spherical hob and a finishing method for the continuous generation of spur face gears with shaving cutters [17][18][19][20].
Many scholars have carried out a lot of research on the design and processing of face gears, but less research has been done on the generalized design and rapid machining of face gear tooth surfaces. In the face gear tooth surface design, there is much research on the modification optimization design based on the theoretical tooth surface, and the simplified design method of the face gear tooth surface is lacking. Affected by the complex topography of the face gear tooth surface, the universality of the face gear tooth surface processing method is low, special processing tools are required, and the processing cost and processing efficiency are difficult to reduce. Affected by the complex topography of the face gear tooth surface, special machining tools are required for the machining of the tooth surface, and it is difficult to reduce the machining cost and machining efficiency. The universal tooth surface design of face gears and efficient machining based on simple tools have become difficulties that need to be overcome urgently.
The theoretical tooth surface of the face gear can be obtained by the gear shaping method, indicating that the tooth surface of the face gear can be regarded as a ruled surface composed of a family of spatial straight lines. Side milling is a high-efficiency machining method applied to ruled surfaces. Compared with spot milling, side milling has a larger single-cutting amount, less tool wear, and higher surface machining accuracy and quality.
Based on the above analysis, this paper proposes a tooth surface design and side milling processing method of ruled line surface gear to solve the complex design and processing problems of the face gear tooth surface. Firstly, the discrete points of the conical surface of the face gear transmission pitch are studied, and the straight line of the face gear tooth surface is constructed. Secondly, the space curvature of the ruled gear is studied, and the tooth surface characteristics of the ruled gear are analyzed. Thirdly, the tool path of ruled line surface gear machining is studied, and the tool pose planning method is established. Finally, the correctness of the proposed method is verified by the side milling test of the ruled gear.
The remainder of part I of this paper is organized as follows. Section 2 provides the tooth surface equation of ruled line face gear. Section 3 describes the tooth surface characteristics of ruled line face gear. Section 4 describes tool path generation and tool pose planning for ruled line face gear. Section 5 conducts the side milling machining test of ruled line face gear. Finally, a conclusion of the theoretical results is presented in Section 6.

Discrete points on the cone of face gear
The face gear transmission is a special transmission form by spur gear and bevel gear. In the face gear transmission, there is an instantaneous axis surface equivalent to the pitch cone in the bevel gear transmission. As shown in Fig. 1, the angle between the transmission axis of the involute pinion and the face gear is γ. The spur gear uses a cylinder with a radius of r as the pitch surface, and the pitch surface of the face gear can be regarded as a conical surface with a half cone apex angle γ 2 , and the face gear and pinion are pure rollings at the node P. Take any point P i (i = 1, 2, 3...n) on the pitch line, and the coordinate of P i is R 2x , R 2y , R 2z , then the coordinate of P i is expressed as: where r s is the radius of the indexing circle of the pinion and R 1 and R 2 represent the inner and outer radii of the face gear, respectively.
Discretize the above formula to obtain the set of nodal line points of the face gear, denoted asR i (i = 1, 2, 3...n) . As shown in Fig. 2, the pitch line is located in the middle part of the face gear tooth surface, and the distance from the tooth top and the tooth root is equal.

The straight-line structure of the face gear tooth surface
The face gear tooth surface is intercepted by plane y = y 0 parallel to the Y-axis, and the curve y = y 0 of the intersection of the face gear tooth surface and l 2 is obtained.
where u is the involute parameter and t is the pinion angle. According to the above formula, the section curve l 2 of the face gear can be obtained. To further analyze the bending degree of the curve l 2 , the curvature q of each point on the curve is calculated.
The modulus of the face gear is 4; the pressure angle is25 • ; the number of teeth of the gear shaper and the face gear is 37 and 51, respectively; the inner and outer radii are 97.3 mm and 112.2 mm, respectively, and the coordinates of the tooth top and tooth bottom are − 70 mm and − 79 mm, respectively.
Wheny 0 = 115 mm, the curvature of the discrete points of the cross-section curve is obtained according to Formulas (4) and (5). The calculation results are shown in Fig. 3. The maximum curvature at the tooth top is 0.0015, and the minimum curvature at the tooth bottom is2.43 × 10 −4 .
It can be seen from the above calculation that the curvature of the face gear section along the Y-axis is approximately 0, so the curve on the face gear section can be approximately regarded as a straight line, and the starting points of the straight-line segment are the tooth tip and the tooth root, respectively.
The start and end points of the straight-line segment are fixed at the tooth top and tooth root of the face gear,   The equation of the straight line passing through the two points of the tooth tip and the tooth root is as follows.
Thus, the family of straight-line equations Σ 1 is obtained. Figure 4 shows the relationship between the line family on the face gear and the face gear surface and nodes. The black points in the figure are the discrete points of the pitch line, and the straight line approximates the discrete points of the pitch line.

Face gear tooth surface line family and corresponding section curve
The linear family equations of face gear are constructed from the section curves of face gear, and each line is related to the  corresponding Y-axis section curves. To analyze the relationship between the section curve and the line, the points on the constructed line l need to be further discretized to calculate the distance between the section curve and the constructed line.
Take the points on the YOZ surface of the face gear; let y and z take discrete values along the direction of tooth height and tooth length, respectively; and obtain the corresponding point R 2 x 2 , y 2 , z 2 on the cross-section curve of the face gear and point R l x l , y l , z l on the straight line.
Thus, the distance d between the discrete points on the section curve and the corresponding points on the construction line is The distance from the straight line of the meshing area to the tooth surface is shown in Fig. 5. In the direction of tooth height, the distance from the ruled line to each point on the corresponding face gear first increases and then decreases and reaches the maximum in the pitch line area. Along the direction of tooth length, the distance from each straight line from the inside to the outside to the tooth surface gradually increases. The largest deviation occurs in the pitch line section, which is 0.015 mm.

Deviation analysis of linear constructed face gear
The ruled lines constructed on the face gear are discretized, and the deviation of the two surfaces is analyzed by comparing the new surface with the theoretical face gear. The face gear tooth surface points are discretized according to (9 × 15) the direction of tooth width and tooth height, as shown in the following equations.
where R 1 and R 2 represent the inner and outer radii of the face gear, respectively. rm and ra respectively represent the distance between the bottom and top of the face gear in the tooth depth direction. The tooth surface deviation d f is expressed as the normal distance between discrete points P 1 and P 2 as: where α is the angle between the normal line of the face gear tooth surface point at point P 2 and vector �������� ⃗ P 1 P 2 . The fitting deviation results of the theoretical surface and the structural surface of the face gear are shown in Fig. 6. The maximum normal deviation between the constructed linear face gear and the theoretical face gear is 0.0204 mm, which appears in the pitch line local area.  Further, the tooth surface equation of the ruled line face gear can be obtained. It can be seen from the above formula that the face gear tooth surface is a ruled surface, the generatrix of the ruled surface is ⃗ l(y) , and the directrix is�� ⃗ r a (y) . The three-dimensional model of the ruled line face gear is thus obtained as shown in Fig. 8.

Spatial curvature comparison between ruled line surface and theoretical tooth surface
The face gear tooth surface can be expressed as the surfacer s , s , and then, the normal vector of any point P on the surface is expressed as: where ���� ⃗ r s is the partial derivative of the surface concerning s and ��� ⃗ r s is the partial derivative of the surface concerning s . The normal curvature of the tooth surface is as follows: where E, F, and G are the first-class fundamental quantities of the surface and L, M, and N are the second-class fundamental quantities of the surface.
According to Formulas (13) and (14), the calculation formula of the main curvature of the tooth surface can be obtained as follows: Thus, the main curvatures K1 and K2 of the ruled line face gear can be obtained. Further, the face gear tooth surface point coordinates are discretized, and the principal curvature of the tooth surface corresponding to each discrete point is calculated to obtain the face gear curvature distribution cloud diagram as shown in Fig. 9.

Representation of the contact trace of the ruled line face gear
The meshing coordinate system of the ruled line face gear is established as shown in Fig. 10. S p0 x p0 , y p0 , z p0 and S g0 x g0 , y g0 , z g0 are the fixed coordinate systems of pinion and ruled line face gear; S p x p , y p , z p and S g x g , y g , z g are the rotational coordinate systems of pinion and ruled line face gear. γ is the axis angle of the ruled line face gear and the pinion, = 90 • . 1 and 2 are the rotation angles of the pinion and the ruled line face gear, respectively.
Visualize the actual meshing process of the ruled line face gear and the pinion. As shown in Fig. 11, the meshing process of the ruled line face gear and the pinion belongs to  Fig. 13. It can be seen from the figure that the tangent vector of the directrix is evenly distributed and the tangent vector of the lower directrix changes more than that of the upper directrix.
Further calculate the curvature of the directrix on the ruled line face gear, as shown in Fig. 14. The curvature calculation result is between the interval(0.01, 0.06) , and the curvature change is stable and small.

Equidistant surface and feed plane of directrix
Given the maximum error ε, construct equidistant tooth surfaces s 1 (u1, t1) and s 2 (u2, t2) parallel to the tooth surface and with a distance ε along the normal direction of the ruled line face gear. The equidistant tooth surface equation is expressed as: where s(u, t) is the tooth surface of the ruled line face gear and n(u, t) is the normal vector.
Any point P k on the directrix L i is the theoretical tangent point, and the tangent l i of the curve L i through P k , the plane formed by the tangent l i and the normal vector n k (u, t) of the P k point is the feed plane ∑ of the tool, as shown in the following equation. The two equidistant surfaces of the ruled line tooth surface are determined by the maximum error, which limits the maximum range of the actual machining path movement of the tool. Within this motion range, the fewer contact points, the less the number of interpolation motions in the actual machining process, and the higher the machining efficiency. The feed plane represents the direction of the tool's interpolated movement.
The constant-bow height error variable-step interpolation is adopted, the maximum arc-height error is set to 0.005 mm, the alignment line l on the tooth surface of the ruled line face gear is selected, and the constant-bow height error method is used to discretize the curve. To meet the maximum bow height error, the discrete distance is selected as 0.129, and there are 124 discrete points in total, and the tooth surface alignment is discrete. The discrete results are shown in Fig. 15.
The result error calculation of isoparameter discreteness is shown in Fig. 16. Under the variable-step length and equal bow height error, the bow height error of each interpolation point satisfies the maximum allowable error.

The torsion angle and generatrix tangent of the ruled line face gear
The angle between the tangent vector between the busbar and the intersection of the upper and lower directrix is shown in the following equation.
where � ⃗ a and � ⃗ b denote the direction vectors of the upper and lower directrix, respectively.
As shown in Fig. 17, the intersection points of the same generatrix parameter and the upper and lower directrix lines are P t1 and P t2 , respectively. The tangent vectors ��� ⃗ l 1t and ��� ⃗ l 2t of the directrix are made through P t1 and P t2 , and the ��� ⃗ l 1t is translated to P t2 . The angle between the two vectors is the torsion angle of the tooth surface.
The normal vector of the ruled line face gear can be expressed as: Calculate the normal vectors � ⃗ n of discrete point positions along a generatrix, and the calculation results are shown in Fig. 18.
It can be seen from the above figure that for a generatrix corresponding to the same parameter y, the normal vector of each discrete point along the generatrix is not in the same plane, and the resulting machining error is shown in Fig. 19.
The axis of the tool tooth surface is l, a point on the tool axis is P, and the tool is projected to the tooth surface along the normal vector � ⃗ n through point P. The intersection of the normal vector � ⃗ n and the tool axis is , the projection point is M, and the intersection of the projection vector and the tooth surface is H. The corresponding machining error is expressed as: where the positive direction represents the residual and the negative direction represents the overcut.

Tool axis pose planning based on the two-point offset method
The principle of the two-point offset method is shown in Fig. 20. Select the generatrix P 1t P 2t corresponding to a certain parameter on the ruled surface, P 1t and P 2t are respectively two points on the guideline, and two points Q t1 and Q t2 are arbitrarily selected on the generatrix. Offset the radius r along the normal vectors ���� ⃗ n t1 and���� ⃗ n t2 , respectively, and then the tool axis vector is The projection point Q t of the lower guideline point P 2t to � ⃗ c is the tool nose point, and the position vector of the tool axis can be determined according to Q t and the tool axis vector � ⃗ c . Take 0.25 and 0.75 of the directrix along the reference point in the direction of the directrix, and obtain the axis pose of each interpolation point according to the directrix interpolation method, as shown in Fig. 21. It can be seen from the figure that the double-point offset takes into account the projection of the normal vector of two points on the generatrix, so it has a good degree of coincidence with the discrete points of the generatrix along the normal vector offset point.

Side milling machining test of ruled line face gear
Under the guidance of the above research, the NC machining program was compiled, and the side milling of the face gear is completed on the J1VMC540W vertical machining center, and the machine tool structure is shown in Fig. 22. The machine tool movement includes two moving axes in the X and Y directions of the worktable, the main shaft rotates in the C direction around its axis and moves along the Y-axis, and the turntable rotates around the B-axis and the A-axis around its axis. The spindle speed was set to 3000r/min, the feed rate was 0.1 mm, and the processing method was down-milling. The side milling process is shown in Fig. 23. The three-coordinate measuring machine is used to test the tooth surface of the face gear after processing. The test results are shown in Fig. 24. It can be seen from the figure that the maximum deviation occurs at the undercut of the inner diameter tooth bottom, which is 0.02 mm, which meets the accuracy requirements.

Conclusion
To effectively improve the design and machining efficiency of face gears, a method of tooth surface design and side milling of ruled line face gears is proposed in this paper. Firstly, through the structure of discrete points and straight-line clusters on the face gear tooth surface, the tooth surface equation and deviation analysis method of the ruled line surface gear is established. Secondly, according to the mathematical model of the tooth surface of the ruled gear, the characteristics of the tooth surface of the face gear are studied, and the method of representing the contact trace of the ruled gear is established. Thirdly, the tool path of the ruled gear is established, and the tool path planning is completed. The simulation results show the correctness of the planning method. Finally, under the guidance of the theory and simulation results, the side milling processing test of the ruled line gear was carried out. The maximum deviation of the gear detection after processing was 0.017 mm, which verifies the correctness of the design and processing method.
Author contribution Xiaomeng Chu contributed to the conception of the study, wrote the manuscript, and funding acquisition; Hong Zeng contributed to the conception of the study and manuscript review; Yanzhong Wang contributed to analysis and manuscript review; and Yizhan Huang contributed to the experimental verification..

Declarations
Ethics approval Not applicable.