Based on the space curve meshing equation, a novel helical gear mechanism with improved geometry is presented. First, equations of the theoretical contact curves were derived based on the space curve meshing theory. Then, tooth surfaces with a concave–convex meshing form were constructed, depending on the contact curves. The tooth profiles were improved as the theoretical contact curves were corrected by predestining the designed transmission errors. The effect of the center distance error on the transmission errors was studied, and the effects of gear modifications on transmission errors and maximum contact stresses were also investigated. The results show that the transmission error curves of the improved gear drive become much smoother. Maximum contact stresses of the improved gear drive are decreased synchronously.

A line gear (Chen, 2014), which is also named space curve meshing wheels in the published papers (Chen et al., 2007), is applicable to micro-mechanical systems due to its small size, light weight and large transmission ratio. Line gear pairs can be applied to transmissions with perpendicular shafts (Chen et al., 2009), intersecting shafts (Zhen et al., 2013) or skew shafts (Chen et al., 2013). Based on the line gear theory, non-generated double circular arc helical gear drives and pure rolling cylindrical helical gear drives have been proposed (Long et al., 2022; Zhen et al., 2020b). Different kinds of gear profiles can be constructed based on the line gear theory.

There are many studies detailing the geometry-improved methods of different gears with the aim to improve transmission performance. Since the beveloid gear has been proposed (Beam, 1954), concave and convex modification analysis for skewed beveloid gears considering misalignment has been studied (Siyuan et al., 2019). The meshing theory and the simulation of non-involute beveloid gears has been studied (Li et al., 2004). Besides, Liu et al. (2018) have discussed effects of tooth modifications on the mesh characteristics of a crossed beveloid gear pair with a small shaft angle. Litvin (1992) derived that the transmission error (TE) is the main source of vibration and noise. Lots of research on gear modification has been studied. Litvin et al. (2003, 2006) found that predestining a parabolic type of TE means that it is able to absorb most of the linear function of TEs caused by misalignment, and predestining the TE has proved to be a method to improve the transmission precision and to reduce noise. Ni et al. (2017) have studied the transmission performance of a crossed beveloid gear transmission with a parabolic modification. Fuentes et al. (2010) have investigated the computerized design method of conical involute gears to improve the bearing contact and reduce noise. A simulation of the meshing and contact of hyperbolical-type normal circular-arc gears has also been studied (Chen et al., 2016).

The aforementioned research covers numerous efforts in improving the transmission performance of different gears through profile modification. The geometric design, meshing simulation and stress analysis of pure rolling cylindrical helical gear drives and pure rolling rack and pinion mechanisms are studied (Chen et al., 2020a, b).

Based on the line gear theory, a novel helical gear mechanism (NHGM) with a concave–convex meshing form is presented for parallel shaft transmission. The geometry modification of the NHGM is researched to improve the transmission precision. The effect of the center distance error on the TEs is analyzed. The effect of the geometry modification on the maximum contact stresses are studied. The principle is expounded through theoretical research, and the feasibility is verified by a numerical method.

As shown in Fig. 1, coordinate systems

Coordinate systems of the gear drive.

Schematic illustration of the tooth profile section and force analysis.

Space curve

Contact curve equations of the driving gear surface are expressed as Eq. (

According to the above formulas, the tooth surfaces equation of the driving and driven gears can be obtained, and then the induced normal curvature

According to the above formulas, a pair of NHGMs is designed. The parameters
are listed in Table 1. The 3D models of the NHGM were established according to Eqs. (

Design parameters of the NHGM sample.

As shown in Fig. 3, they were manufactured with a 3D printer. The 3D printer model is Lite450HD. The resolution is 0.001 mm. The material is a gray high-temperature, photosensitive resin (YGH-5001).

A pair of gear drive samples produced by a 3D printer.

The tooth contact analysis (TCA) is the main method to obtain TEs. The equation of TCA is expressed as Eq. (

TEs can be changed by applying preset transmission errors (PTEs; Litvin,
1992). After presetting the transmission errors, the rotation angle of the driven gear is expressed as Eq. (

While the tooth surface equations are obtained, the maximum contact stress of a loaded gear drive could also be obtained by the application of the Hertz contact theory.Shown in Fig. 2, force of the gear drive can be represented with Eq. (

Based on the Hertz contact theory, the maximum contact stress can be obtained by putting Eq. (

The following conditions of the gear drives were considered, and then the NHGMs were designed. Then, kinematic analysis was carried out with no center distance error

Rotation speed of the driven gear with the speed of driving gear at
600 (rad s

TCA was applied to obtain TEs which were caused by center distance error

In order to make the TEs controllable, an improved NHGM with preset
transmission errors is designed. Considering that trigonometric functions have stable oscillation characteristics, PTEs were designed as a trigonometric function. The function of the PTEs is expressed as Eq. (

TEs of the unimproved NHGM with different center distance errors.

TEs of the improved NHGM with different center distance errors.

Maximum contact stresses of the different tooth surfaces.

As shown in Fig. 7, the maximum contact stress of the unimproved NHGM shows a
linear change, while the maximum contact stress of the improved NHGM shows a
nonlinear change. When

The maximum contact stresses of meshing gears can be calculated by the finite
element method. For this case, maximum contact stresses that can be calculated by the finite element method are shown as Fig. 8, where

Maximum contact stress calculated by the finite element method, where

Compared with the results obtained by the finite element method shown in Fig. 8, the relative error in the maximum contact stress which is obtained by the theoretical calculation method shown in Fig. 7 is 1.32 % and 0.93 %, respectively.

As shown in Fig. 9, the values of the long and short axes of the contact ellipse of the unimproved gear drive decrease linearly. The values of the long and short axes of the contact ellipse of the improved gear drive are larger than the unimproved one, and they are not linearly changed. For the improved gear drive, the values of the long and short axes reach the minimum, where

Contact ellipses of the improved and unimproved NHGM.

The size of the contact ellipse of the improved NHGM decreases first and then
increases. While

Based on the above analysis, we can conclude that the improved NHGM has a stable transmission performance and stronger bearing capacity than the unimproved one.

For line gears, the driving and driven tines can both be regarded as cantilever beams which cannot absorb enough loading for conventional mechanical applications. Thus, line gears are suitable for micro-mechanical systems with low power. A novel pure rolling circular arc helical gear mechanism (CAHGM) with concave–convex meshing is presented for parallel transmission applications (Zhen et al.,2020). Due to the concave–convex contact form, it is able to carry more loads than the line gear. However, assembly errors have not be considered in constructing the gear surface. Litvin (2006) used the rack cutter with a parabolic profile to generate the profile-crowned helical gear and analyzed the contact characteristics. Based on the line gear theory, the NHGM with a concave–convex meshing form is presented for parallel shaft transmission. A geometry modification of the NHGM has been researched to improve the transmission precision. Transmission error analysis results show that the novel gear pair can realize a stable transmission process, and the contact stress analysis result shows that the novel gear pair has a stronger bearing capacity.

The previously presented discussions, computations and numerical examples
enable us to draw the following conclusions:

The NHGM can achieve a stable transmission ratio without a center distance error.

For the NHGM with improved geometry, linear transmission ratios which are caused by the center distance error become periodically changed.

For the NHGM with improved geometry, the maximum contact stress is symmetrically distributed, and it reaches its maximum value when

The software code of this study are available from the corresponding author, upon reasonable request.

The data that support the findings of this study are available from the corresponding author, upon reasonable request.

EH was responsible for the main research work of the article, and SY helped to complete the picture editing and part of the calculation work.

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.

This research has been supported by the National Natural Science Foundation of China (grant no. 51175180) and the Science Foundation of Hubei Provincial Department of Education (grant no. Z2018118).

This paper was edited by Guowu Wei and reviewed by two anonymous referees.