Line gear is a newly developed gear mechanism with point contact meshing according to space curve meshing theory. This paper proposes a new form of line gear with a couple of concave convex arc tooth profiles. It has four characteristics. First, contact curve of the driving line gear is a cylindrical spiral curve. Second, two axes of a pair of line gears are located in the same plane with an arbitrary angle. Third, at the mesh point, normal tooth profiles of a line gear pair are a couple of inscribed circles. Namely, they form a couple of concave convex tooth profiles. Fourth, the tooth profile of a driving line gear is a convex, that of a driven line gear is a concave, and they are interchangeable. If only consider that the arcs of teeth at meshing point are tangent, the actual tooth surfaces may interfere outside of the meshing point. In this paper, the geometric condition of the tooth surface for a concave convex arc line gear mechanism is derived, and the optimal formulae of the tooth profile parameters are derived on basis of interference proof conditions. Finally, the 3-D modeling and kinematic simulation of line gear pairs show that the proposed line gear pairs can perform transmission normally. The proposed method will extend the application of line gear in the conventional power drive.

The concave convex arc line gear (Chen and Yao, 2015) is the line gear with arc tooth profile. The arc gear has advantages in large loading capacity. For instance, Sun et al. (2016) investigated the contact strength of the circular-arc-tooth-trace cylindrical gear (C-gear). The results reveal that the C-gear is superior to spur gear and helical gear in the contact strength. The curvilinear tooth gear proposed by Zhang et al. (2016) is superior to spur gear in contact and bending stress. The Double Circular-Arc Helical Gear developed by Wang (2012) can be used for heavily-loaded planetary gear reducer. The circular-arc gear designed by Zhou et al. (2016) can be used for a high-pressure and high-speed gear pump in the aerospace application. Because of the advantages of arc tooth, scholars have used the arc tooth in many different types of gear transmission. They are hyperboloidal-type normal circular-arc gears (Chen et al., 2016), helical gear with triple circular-arc teeth (Xie and Yang, 2014), quadruple-arc Profile Bevel Gears (Ren, 2014), a new kind of gear transmission with circular arc tooth profiles (Chen et al., 2014), etc. All of these belong to cylindrical gear. Arc tooth are also used in bevel gears (Dong and Wang, 2014; Zhang et al., 2011, 2012) and worm (Zhao and Zhang, 2011).

A pair of concave convex arc line gears at the meshing point.

The concave convex arc line gear is a new form of line gear. The early line gear is also called the space curve meshing wheel. It contains a driving wheel, a driven wheel, driving tines and driven tines (Chen et al., 2013b). A cantilever structure is formed by the combination of the tines and the wheel body. Compared with the previous proposed line gear, the concave convex arc line gear has the advantages of arc gear. It is better than the early line gears in loading capacity, bears much lower contact and bending stress. It can be applied to conventional power transmission with larger power. It can be processed by the metal cutting machine tool. Compared with other arc gear, it has the advantages of line gear. It has a large transmission ratio and compact structure. The gear shafts of the gear pair can be crossed at any angle. It means that the conventional gear pair must be a driven wheel corresponding to a driving wheel. The concave convex arc line gear pair can be designed as a driving gear and a corresponding plurality of driven gears. The minimum tooth number of line gear can reach 1 (Chen, 2014). But it can bear lower contact stress than other conventional forms of arc gear above due to point contact meshing.

The main objective of the work is to improve the original form of line teeth profile, to be capable of applying in conventional power transmission. If the line gear is only the combination of the characteristics of the space curve meshing wheel and the characteristics of the arc gear, it cannot be processed by the metal cutting machine tool. It also shows the importance of the gear profile and the parameter quantization. Moreover, the tooth profile parameters and gear parameters are quantified to provide the basis for the analysis of gear strength. A new design method of the concave convex arc line gear is presented in this paper, aiming to solve the problem of tooth surface geometry interference and the optimization of the tooth profile parameters. In order to verify the feasibility of the parametric design and tooth profile parameters optimization of concave convex arc line gears, this paper presents the specific example and the 3-D modeling and kinematic simulation of the line gear.

A pair of concave convex arc line gears is comprised a driving line gear and
a driven line gear. As shown in Fig. 1a, the axes of the driving line gear
and the driven line gear are arbitrary angle intersected axes in one plane.
The angle is

The contact curve of the driving line gear, namely a driving contact curve,
is set as a spatial cylinder spiral curve. And the parametric equation in

The contact curve of the driven line gear is conjugated with the driving
contact curve, and the equation in

A driving line gear is formed by driving line gear teeth radial attachment to a cylindrical wheel body. The gear teeth can be convex or concave, and the gear with convex teeth is taken as an example here. As shown in Fig. 2, a convex tooth of a driving line gear is formed by the motion of a convex normal profile along a driving contact curve and two tooth thickness auxiliary curves.

Schematic diagram of the driving line gear.

The two tooth thickness auxiliary curves have two functions. Firstly, they
lead the movement of the auxiliary line of the gear tooth profile. Secondly,
they provide the normal tooth thickness of the gear teeth. The auxiliary
curves of tooth thickness comprise a first auxiliary curve and a second
auxiliary curve. A first auxiliary curve 2 is arranged between a contact
curve 1 and a second auxiliary curve 3, and its equation in

The driving line gear is designed by determining the following parameters.
Parameters of a driving line gear include driving contact curve parameters,
tooth profile parameters and wheel body parameters. Driving contact curve
parameters cover

Schematic diagram of the normal tooth profile of the driving line gear.

A driven line gear is formed by the driven line gear teeth radial attachment to a conical or cylindrical wheel body. The driven line gear teeth can be concave or convex, and the gear with concave teeth is taken as an example here. As shown in Fig. 4, a concave tooth of a driven line gear is formed by the motion of a concave normal profile along a driven contact curve and two tooth thickness auxiliary curves.

For the driven line gear, two tooth thickness auxiliary curves have also two
functions. The auxiliary curves of driven line gear tooth thickness comprise
a first auxiliary curve and a second auxiliary curve. A first auxiliary curve 2
is arranged between a driven contact curve 1 and a second auxiliary curve 3, and
its equation in

Schematic diagram of the driven line gear.

Parameters of a driven line gear also include driven contact curve
parameters, tooth profile parameters and wheel body parameters. The driven
contact curve parameters cover

Rotary surface of the driven line gear.

Schematic diagram of the normal tooth profile of the driven line gear.

As shown in Fig. 7a, if only consider that the tooth profile arcs are
tangent at meshing point, the actual surfaces of the driving line gear tooth
and driven line gear tooth may interfere with each other outside of meshing
point, resulting in abnormal engagement. The tooth surface can be considered
as a set of tooth profiles on normal plane of each point at the contact curve
of the driving line gear. The relationship between the driving line gear
tooth profile and the driven line gear tooth profile on the normal plane of
meshing point is shown in Fig. 7b–d.
Coordinate system

The origin of the coordinate system

Diagram of the tooth profiles interference of the driving line gear and the driven line gear. (1 – the tooth profile arc of the driving line gear; 2 – the tooth profile arc of driven line gear on the normal plane of the contact curve of the driving line gear; 3 – an interference area.)

The tooth surface of the driven line gear is formed by a circular arc with a
radius of

Taking

From Fig. 7c, d and Eq. (9), the interference proof
condition of tooth surfaces between a driving line gear and a driven line
gear is described as Eqs. (14) and (15).

Considering of interference proof, the tooth profile parameters and the wheel
body parameters for the driving line gear and driven line gear can be
optimized by combining Eqs. (14) and (15). And,

If

If

In the design of line gears, the teeth clearance should also be considered.
In the meshing process of the convex and concave teeth, the height of the top
of the convex tooth is a priority. As shown in Fig. 3, the addendum of the
driving line gear can be estimated by as Eq. (16).

And, a design formula of the dedendum of the driven line gear is obtained as Eq. (17).

The size of the wheel is designed to satisfy the in-Eq. (18).

Take three pairs of line gears as design examples, the values of various
parameters are shown as Table 1. The values of

3-D solid model 1.

3-D solid model 2.

3-D solid model 3.

By the parameters in Table 1, three sets of line gear mechanisms are obtained, as shown in Figs. 8–10 respectively. From Table 1 and the figures, three pairs of line gear mechanisms have the same driving line gear. And the 3-D kinematic simulations on them were carried out smoothly, without interference. On one hand, it verifies the feasibility of the design formulae of parameters. On the other hand, compared with the traditional gear pair, the novel line gear mechanism has a better replacement.

Compared with other circular arc gears, the concave convex arc line gears can drive with two arbitrary intersecting axes. In the design of the concave convex arc line gear pair, the same driving wheel can be engaged with different driven wheels to drive with a variable crossed axes angle. Due to concave convex arc tooth profile, the concave convex arc line gear pair is better than the early form of line gear in the bearing contact stress. Due to point contact engagement, it bears lower contact stress than the other conventional forms of arc gear pair with line contact engagement. The tooth number of concave convex arc line gear can reach 1. The concave convex arc line gear pair has large transmission ratio and compact structure.

In this paper, a design method for a concave convex curve line gear mechanism is presented. And the design formulae of parameters of the tooth profile are deduced on the basis of interference proof condition proposed. Conclusions can be drawn as follows.

It is easy to mass production. The teeth of the driving line gear and driven line gear are attached to the wheel body radially. And this kind of concave convex curve line gear is easy to be processed by the numerical control machine tool, so that it is easy to mass production.

It has good replacement. For the transmission of a line gear pair with an arbitrary angle in the same plane, only one of driving line gear needs to be designed, which can match to many of driven line gear with different geometric parameters or arbitrary crossed angles. Therefore, compared with the traditional gear pairs and the other arc gear pairs, this kind of concave convex curve line gear mechanism has a good replacement.

Parameters of three line gear pairs.

It has power transmission capacity. Compared with previous proposed line gear mechanism, this kind of concave convex curve line gear can bear much greater power transmission capacity according to Hertz contact theory, but smaller than that of traditional concave convex surface gear pair due to its point contact meshing principle.

It has compact structure. The least number of driving line gear teeth of this kind of concave convex curve line gear mechanism can reach 1. Compared with the traditional gear pair, such as spur gear pair or helical gear pair, its structure is much more compact, that can greatly save installation space.

Compared with the traditional gear pairs and the other arc gears, the characteristic of concave convex arc line gears is the space curve meshing theory. It makes the line gear have good replacement and compact structure. Compared with the early form of line gears, the characteristics of the concave convex arc line gears are the tooth profile and the relationship between the wheel body and line gear tooth. On the one hand, it is strengthened in the tooth contact strength and the bending strength of the gear tooth. This makes it available for use in the field of conventional power transmission. And that allows the transmission with a larger power. On the other hand, it can be processed by the metal cutting machine tool. And that is more convenient for efficient processing of concave convex arc line gear and the expansion of its application. The tooth profile design and tooth formation are considered by the tool shape and the motion trajectory of the machine tool. The appearance of special NC machine tool for the concave convex arc line gear can greatly improve the machining efficiency.

The remainder of this work is to design and make a special-purpose numerical control machine tool for the line gear, to facilitate the efficient processing of gear, to expand the scope of application. We are doing these research works on contact pressure, sliding speeds and efficiency, conducted by newly test rig on transmission ability and dynamic performance experiments.

Funding supports from the National Natural Science Foundation of China (No. 51175180, No. 51575191) are gratefully acknowledged. Edited by: D. Pisla Reviewed by: three anonymous referees