To improve the bearing capacity and provide the mathematical model basis for vibration and noise reduction of the variable hyperbolic circular-arc-tooth-trace (VH-CATT) cylindrical gear, the tooth surface modification design method based on the parabolic forming blade and cutter inclination was proposed. The geometric design of inclined cutter was completed. The modified tooth surface equation was deduced, and the influence of modification parameters on tooth surface curvature characteristics was analysed further. The tooth contact analysis (TCA) was established, and the influence of modification parameters on the elliptical contact area was analysed. Research shows that the parabolic coefficient

Tooth surface modification technology can improve the pressure distribution of tooth surface, reduce the meshing-in and meshing-out impact, and improve the stability of system. It is an important technical means to reduce vibration and noise (Guo et al., 2021; Peng et al., 2020). For example, Nie et al. (2019) proposed a kind of Ease-off topology modification method for hypoid gears to improve meshing performance, the load tooth contact analysis (TCA) results indicate that the modified tooth flank contact stress distribution is improved. Jia et al. (2021) proposed a tooth modification design method of vibration reduction for involute helical gears considering the measured load spectrum, and the results show that the amplitude of loaded transmission error and the meshing-in impact force are effectively reduced after optimum tooth modification. Jiang et al. (2021) proposed a design and analysis method of tooth surface dynamic anti-wear modification to improve the contact performance.

The variable hyperbolic circular-arc-tooth-trace (VH-CATT) cylindrical gear is a new gear transmission. The tooth direction line of the VH-CATT cylindrical gear is an arc line, the tooth profile of the middle cross section is involute, and the other sections are hyperbolic (Ma et al., 2019). When the tooth surfaces are loaded, the contact area generally appears as an ellipse (Ma et al., 2021a). At present, the VH-CATT cylindrical gear has rich achievements about the meshing principle, 3D modelling and contact performances, processing and manufacturing, product application, etc. For example, Tseng and Tsay (2005), Zhao et al. (2016), and Song et al. (2010) deduced the tooth surface equation based on the gear-meshing principle, respectively. Li Hou realised the machining simulation of the VH-CATT cylindrical gears based on UG (Unigraphics NX), but there are machining marks on the tooth surface (Wang et al., 2012), and the 3D accurate model of the VH-CATT cylindrical gears was established according to the mathematical model (Zhao et al., 2016). Ma et al. (2018, 2021b) proposed the meshing contact impact hypothesis of the VH-CATT cylindrical gear based on the contact dynamics theory and gear transmission physical model; the gear mesh contact impact model was set up further to obtain the VH-CATT cylindrical gear's meshing contact impact properties. Wei et al. (2020) established the contact stress calculation model of the VH-CATT cylindrical gear based on the Hertz contact theory, which provided a reference for the contact stress calculation of the VH-CATT cylindrical gear. In addition, Fuentes et al. (2014), Fuentes-Aznar et al. (2017), Zhang and Xie (2016), and Chen and Lo (2015) established a 3D model of the VH-CATT cylindrical gear to study the contact performances by the finite element method compared with the spur gear and helical gear.

However, there are still some restricting factors in the industrial application of the VH-CATT cylindrical gear. The contact area of the VH-CATT cylindrical gear is the elliptical area on load. However, the ratio of the elliptical long axis to the gear width is small, and the tooth width is not fully used for the gear contact load. At the same time, due to the manufacturing errors, installation errors, and bearing contact deformation, the meshing-in and meshing-out impact of the unmodified VH-CATT cylindrical gear is larger. Because of the lack of tooth surface modification design technology, the optimisation design of tooth surface, system vibration, and noise reduction, improvement of bearing capacity cannot be completed.

Therefore, the tooth surface modification design method of the VH-CATT cylindrical gear was proposed. The tooth surface equation of the modified VH-CATT cylindrical gear was deduced based on the gear-machining development principle and the influence law of modification parameters on the main curvature of the tooth surface contact point was analysed. The TCA model of the VH-CATT cylindrical gear was established and the influence of modification parameters on elliptical contact area was analysed. Considering design efficiency and machining economy, only the tooth surface of the driven gear was modified in TCA. The research content provides a technical basis for improving the bearing capacity of the VH-CATT cylindrical gear and optimising the design.

To effectively reduce the meshing-in and meshing-out impact of the gear
system, the parabolic modification curve is adopted for the tooth profile
forming cutter blade, shown in Fig. 1. The second-, fourth- or higher-order parabola curve can be obtained by changing

Curve of tooth profile modification blade.

The parabolic blade profile in the coordinate system

To increase the length of the contact line along the tooth line direction, a
modification method of the VH-CATT cylindrical gear along the tooth line
direction by the cutter-inclined milling is proposed. Figure 2 is the
schematic diagram of the conventional milling cutter head for the VH-CATT
cylindrical gear. The cutter head rotates around the axis

Conventional large gear milling cutter.

Gear cutter-inclined large milling cutter.

To ensure that the pitch line of the cutter blade is tangent to the pitch
circle of the gear blank, when the inclined tool is installed as standard,
the cutter head is installed at an inclination angle, shown in Fig. 4. At
this time, the cutter head rotates around the axis

Installation position of a large cutter for inclined milling.

To derive the modified mathematical model of the VH-CATT cylindrical gear,
the coordinate transformation system of the modified tooth surface is
established, shown in Fig. 5. Figure 5b is the cutter head position and
geometric parameters. The dynamic coordinate system of the gear blank is

Machining coordinate system of double-edged milling with large cutter head.

According to Eq. (2) and the coordinate transformation relationship shown in
Fig. 5, the expression of the inclined cutter in

According to Litvin (2008), the normal vector and relative speed at the meshing point of the modified tooth surface are Eqs. (6) and (7), respectively:

According to Litvin (2008), the product of relative speed
and normal vector is zero:

Equation (9) is simplified by Eq. (8) and

The MATLAB numerical calculation shows that there are

Figure 6 is the modified tooth surface. The important parameters are as
follows: tooth number

Modified tooth surface.

The curvature of the tooth surface is an important parameter which reflects
the geometric characteristics of the tooth surface. The curvature characteristic analysis of the modified tooth surface plays an important role in the contact performance analysis. According to Ma et al. (2021a), the calculation expressions of the contact ellipse and principal
curvature are Eqs. (16) and (17), respectively, and the meanings of parameters in the equations can be found in Ma et al. (2021a):

Figure 7 shows the influence of the cutter inclination angle

Influence of cutter inclination angle

Influence of cutter inclination angle

According to Figs. 7a and 8a, it is obvious that the main curvature along the tooth line direction of the concave tooth surface gradually decreases from the tooth root to the tooth top, and the main curvature along the tooth line direction of the convex tooth surface gradually increases from the tooth root to the tooth top. The reason for the increase or decrease in the main curvature is the following: the rotary surface of the concave tooth surface forming blade is an inverted cone, and the rotary surface of the convex tooth surface forming blade is a positive cone. At the same time, it can be seen that the main curvature along the tooth line direction of the concave tooth surface increases with the increase in the cutter inclination angle, and the main curvature along the tooth line direction of the convex tooth surface decreases with the increase in the cutter inclination angle. It indicates that the degree of the crescent bay increases, which has an important influence on the long axis of the contact ellipse.

According to Figs. 7b and 8b, the main curvature along the tooth
profile direction gradually decreases from the tooth root to the tooth
top. Theoretically, the curvature at the base circle is infinite. It can also
be seen that the cutter inclination angle

Figure 9 shows the influence of the parabolic coefficient

Influence of parabolic coefficient

Influence of parabolic coefficient

According to Figs. 9a and 10a, the tooth line direction's main curvature of the concave and convex tooth surfaces increases with the increase in the parabolic coefficient from the tooth root to the pitch circle. Contrastingly, it decreases with the increase in the parabolic coefficient from the pitch circle to the tooth top, and the closer the tooth root or tooth top is, the greater the effect of parabolic coefficient on the main curvature is. The reason is the following: the parabolic shaping blade cuts off part of the material on the tooth surface, and the cutting amount at the tooth root and tooth top is the largest, so the tooth line direction's main curvature of tooth surface is changed, and the parabolic coefficient has the greatest influence on the tooth line direction's main curvature at the tooth root and tooth top.

According to Figs. 9b and 10b, the tooth profile direction's main curvature of the concave and convex tooth surfaces increases with the increase in the parabolic coefficient from the tooth root to the tooth top, and the parabolic coefficient has a great influence on the tooth profile direction's main curvature of the tooth root. The reason is the following: the parabolic shaping blade cuts off part of the material on the tooth surface, the arch degree along the tooth profile direction increases, so the tooth profile direction's main curvature decreases. Moreover, because the curvature radius near the tooth root is small, the main curvature of the tooth profile near the tooth root changes greatly.

Figure 11 shows the influence of the vertex position

Influence of parabolic vertex position

According to Figs. 11a and 12a, the main curvature along the tooth
line direction of the concave and convex tooth surfaces increases with the
increase in the parabolic vertex position

Influence of parabolic vertex position

Because the ratio of the elliptic long axis to the gear width is small,
the tooth width is not fully used for the gear contact load. To analyse the
influence of modification on the elliptical contact area, it is necessary to
accurately calculate the meshing points of the VH-CATT cylindrical gear
transmission pair first. Therefore, the meshing transmission coordinate
system of the VH-CATT cylindrical gear pair is established as shown in Fig. 13. The tooth surfaces of the driving gear are normal (unmodified) and the driven gear are modified. The tooth surface of the driving gear is

Equation (18) is the tooth contact analysis model, there are 6 unknown
variables such as

Meshing transmission coordinate system of the gear pair.

Figure 14 shows the influence of the cutter inclination angle

Influence of cutter inclination angle

Effect of cutter inclination angle

Modified gear contact pair (black surface is the driving wheel, and red surface is the driven wheel).

Limit value of the cutter inclination angle.

Figure 18 shows the influence of the parabolic coefficient

Influence of parabolic coefficient

Effect of parabolic coefficient

Figure 20 shows the influence of the parabolic vertex position

Influence of parabolic vertex position

Effect of parabolic vertex position

This paper discusses the surface modification and tooth contact analysis (TCA) of the VH-CATT cylindrical gear. Firstly, the tooth surface modification method was proposed. Next, the modified tooth surface equation was deduced, and the 3D reconstruction is realised. Thereafter, the influence of modification parameters on curvature characteristics was analysed. Finally, the TCA model was established to discuss the influence of
modification parameters on the elliptical contact area. The main conclusions
can be expressed as follows.

The main curvature along the tooth line direction of the concave tooth surface increases with the increase in

The elliptical contact area increases with the increase in

The modified tooth surface mathematical model of the VH-CATT cylindrical gear can provide a mathematical basis for vibration and noise reduction of the system, and the cutter inclination method can effectively improve the bearing capacity of the system.

All the code and data used in this paper can be obtained upon request to the corresponding author.

DQ and YL conceived the presented idea. DQ and YL established an overall paper research framework and the model. DQ conducted data calculation for the overall paper. All the authors discussed the results and contributed to the final paper.

The contact author has declared that neither of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors would like to thank anonymous reviewers for their valuable comments and suggestions that enabled us to revise the paper.

This work supported by the National Natural Science Foundation China (grant no. 51765032), the Guizhou Provincial Science and Technology Projects (Qiankehejichu – ZK [2021] yiban 273), the Zunyi Normal University Serving Local Industrial Revolution Project (Zunshihedifangchanyezi [2020] 01), and the Zunyi Normal University 2021 Rural Revitalization Project (Qiaojiaohe KY Zi [2012] 017-5).

This paper was edited by Daniel Condurache and reviewed by three anonymous referees.