Effects of cutter modification on the meshing characteristics of intersected beveloid gear and involute cylindrical gear

The parabolic cutter is introduced into the design of intersected beveloid gear and involute cylindrical gear pair, and the numerical model of the normal section tooth profile of the parabolic cutter is established. Based on the hobbing theory, the equation of the beveloid gear face of tooth machining with a modification cutter is derived. The mesh characteristics analysis model of gear pair is established, and the impacts of modification coefficients on meshing characteristics that contact pattern, transmission error, meshing stiffness, contact force, and tooth root stress are studied.


Introduction
Beveloid gears belong to the general form of involute gears.Because its meshing performance in the case of small shaft angles is better than other traditional transmission forms, it is widely used in marine transmissions and automobile transmissions.The key problems such as the narrow internal space of the gearbox and bad lubrication condition of rotating parts caused by inclined installation of power transmission are improved, and the transmission efficiency is improved.In the selection of gear pair combination, the gear pair composed of beveloid gear and cylindrical gear can effectively avoid the phenomenon of tooth side clearance interference and jamming caused by excessive axial momentum as shown in Figure 1.However, the point contact mode between the beveloid gear and involute cylindrical gear will cause adverse effects on the gear pair, which restricts its application in industry.
Gear modification can effectively reduce the negative effects [1] .Therefore, it is of great significance to study gear modification to improve the meshing characteristics of gear pairs.At present, a lot of research on tooth modifications and meshing characteristics has been done.The concave modification of the beveloid gear tooth surface was proposed by Liu et al. [2] .Zhou et al. derived the equations of the gear face of the tooth with modifications [3] .A shaving model of beveloid gear with installation errors was proposed by Cao et al. [4] .Simon optimized the machine tool parameters to improve the meshing tooth surfaces [5] .Ni et al. optimized the meshing characteristics of the gear pair by modifying the rack-cutter [6] .Vu et al. presented a gear-modified method of closed-loop topology for gear honing [7] .
Therefore, this study introduces a modification method for the design of intersected beveloid gear and involute cylindrical gear pair to improve the meshing characteristics.The paper results have a certain reference value for the application of this method in gear optimization design.

Equations of the gear tooth surface with modification
According to the theory of Merritt and Mitome, when hob machining beveloid gear, the generating process of the gear tooth surface is equivalent to the enveloping process of an imaginary cutter with the same end face, as shown in Figure 2. Therefore, the original face of tooth equation is as follows: where 1, 2 i  represents beveloid gear and cylindrical gear, i  is the gear meshing rotation angle, i x ,  Therefore, the modified design of the gear can be achieved by changing the cutter shape when machining the beveloid gear.The tooth profile is divided into three sections that root M M and the working tooth surface M M .The working tooth surface is modified in Figure 3. Therefore, after the modification work, points on the tooth surface where the superscript represents the right face of the tooth, and the following table represents the left.Then, by coordinate transformation: Therefore, in n S , it is represented as: Then, according to the transformation of coordinates in Figure 4, the face of the tooth of the cutter in c S can be expressed as: The unit normal vector can be expressed as: Then, by substituting the modified cutter face of tooth i c R and the unit normal vector i c n into Equation ( 1), the modified gear tooth surface equation can be obtained [5] .According to the derived equations, the MATLAB program was written to obtain the tooth surface set.In Figure 5, the black dot set is the involute face of the tooth without modification, the blue dot set is the face of the tooth processed by the positive modified cutter, and the red dot set is the face of the tooth processed by the negative modified cutter. indicates the amount of face of tooth modification.

Mesh characteristics analysis model of gear pair
According to the geometric design parameters of intersected beveloid gear and involute cylindrical gear transmission [2] , the geometric design parameters are obtained in Table 1.The appropriate modification coefficients are selected by loaded tooth contact analysis, as shown in Figure 6.Data from Table 2 are selected for the meshing characteristics research.According to the above equations and data in Table 1, the entity model of the gear pair is established in commercial software.In Figure 7, 17CrNiMo6 was selected for the material of the model.The elastic modulus is 208 Gpa and Poisson's ratio is 0.295.It is just need to know the contact status of one gear face of a tooth from the process of engaging-in to engaging-out, so five pairs of meshing teeth are specifically analyzed, and the rest teeth are not involved in the analysis and are not specifically meshed.P 1 is the grid density of the meshing surface (70 (longitudinal direction) × 40 (profile direction)), and P 2 is the contact situation of the meshing surface.Under the load torque of 400 N m  , the corresponding gear contact pattern is obtained by simulation calculation and analysis.The instability of gear transmission is described by transmission error, which has a significant impact on the noise and vibration of the gear transmission system.Therefore, transmission error can be studied as an important parameter of meshing characteristics.The calculation equation of transmission error is as follows: where 10  and 20  are the initial angles when the beveloid gear and involute cylindrical gear enter the engagement respectively; 1  and 2  are the rotation angles; 1 N and 2 N are the tooth number.Therefore, the equivalent meshing stiffness [8] is: where tatal F is the total engagement force, and z M is the contact torque in the direction of the axis.Finite element mesh of beveloid gear and involute cylindrical gear pair.

Numerical example
Case 1: beveloid gear modified with a positive coefficient and cylindrical gear modified with a negative coefficient.The selection of modification coefficients is shown in Table 3. Table 3. Modification coefficients.

Modification coefficient (𝑚𝑚 ) A B C
0 0.0005 0.001  0 -0.0005 -0.001The CPRESS value at each moment of a cycle of beveloid gear and cylindrical gear were extracted to form the tooth surface impressions in Figure 8.It can be found that the contact area of the tooth surface does not change much.The peak-to-peak value of transmission error decreases with the modification coefficient increase, while the average of meshing stiffness increases in Figures 9-10.The contact force does not change much, but the tooth root stress of the gear decreases obviously in Figure 11.Case 2: beveloid gear modified with a negative coefficient and cylindrical gear modified with a positive coefficient.The selection of modification coefficients is shown in Table 4.  12, the contact area of the tooth surface does not change much.In Figures 13-14, the peak-to-peak value of transmission error decreases with the modification coefficient increase, while the average of meshing stiffness increases.The contact force does not change much in Figure 15, but the tooth root stress of the gear decreases obviously after modification.

Conclusion
The modified cutter is introduced in the hob machining of gear, and different degrees of modification of the beveloid gear is realized by changing the modification coefficient.The peak-to-peak value of transmission error and tooth root stress of the gear pair decreased obviously after parabolic modification, and the average value of time-varying meshing stiffness increased.In a certain range, the increase in the amount of modification can enhance the meshing performance and drive smoothness of the gear pair.

Figure 1 .
Figure 1.Different combinations of gear pairs transmission.

iy
, and i z are the position vectors on the gear face of the tooth, coordinates on the imaginary cutter, and pi r is the reference pitch radius.

Figure 3 .
Figure 3. Normal section tooth profile of the cutter.

Figure 5 .
Figure 5. Effects of the sign of modification coefficients on gear tooth morphology.

5
According to the point coordinates of the modification surface in Chapter 2 and Figure5, the modification of the face of the tooth is not uniform.The maximum amount of modification is the existing research, the modification amount is generally selected in the range of 20 m  -40 m  . is taken as the criterion for judging the value of the modification coefficient.

Figure 6 .
Figure 6.Design flow of load tooth contact analysis.