# Parallel shaft pure rolling internal meshing line gear

Due to the complex tooth profile of traditional industrial gears, it is very difficult to miniaturize and micro nano application. In 2007, Chen Yangzhi put forward the meshing theory of spatial curve, through which the “simple entity” constructed by spatial curve instead of the traditional “complex entity” constructed by spatial complex surface, the space occupied by gear was minimized.

As shown in Fig. 1, the early linear gear composed of a cantilever beam structure with a hook bar and a cylindrical base, which was used in the transmission device under light load conditions. In recent years, many scholars have carried out research on gear design based on spatial curve meshing. Chen Yangzhi et al. Derived the basic design formula of skew axis and staggered axis gear, and constructed the line gear sample. In the follow-up work, the stability and continuity of transmission were verified through experiments. Chen Bingkui et al. Established the meshing equation along the direction of any contact angle, and proposed the method of equidistant envelope to construct the tubular tooth surface. Yao Li and others proposed a design method of line gear to meet the conventional power transmission, and studied the processing technology of line gear based on four axis machining center. Chen Zhen et al

The conjugate curve is designed according to the meshing line function, and the geometric design, meshing performance and mechanical performance of cylindrical gear, intersecting shaft gear and rack and pinion are introduced. The above research focuses on the field of external meshing transmission. At present, the research on internal meshing linear gear is less, which also leads to the limitation of the application of linear gear mechanism in gear train. Therefore, this paper mainly studies the parallel shaft internal meshing line gear mechanism.

(1) Based on the conjugate meshing theory of space curve, the construction method of a pure rolling internal meshing line gear pair is studied, and the mathematical models of convex convex contact, convex concave contact and convex flat contact are established. The relative curvature radius of convex and concave tooth surface is the largest, and the contact stress of tooth surface is the smallest under the same load condition.

(2) Taking the cylindrical helix as an example, a pair of conjugate contact lines are solved, and the pure rolling meshing conditions are derived: the contact lines of pinion and ring gear have the same direction of rotation, the same helix angle, and the equation E = m (i12-1).

(3) According to the tooth surface equation, the linear gear pair model is constructed and the kinematics simulation is carried out. The experimental results show that the parallel shaft pure rolling internal meshing linear gear mechanism constructed by this method can realize smooth and continuous transmission and meet the requirements of design transmission ratio.

When the tooth profile is meshed, due to the different speed and direction of the contact point, there will be relative sliding between the tooth profiles, which will cause adverse factors such as tooth wear, node impact and noise. In order to weaken the negative effect of relative sliding, as early as in the early 20th century, grant proposed to make the tooth top height close to zero, only through the axial coincidence to ensure the continuity of transmission, so that the two gears only mesh at the node to achieve pure rolling mesh. However, because the transmission power is very little and the center distance is small, it will be out of engagement, so it has not been applied in practice. Huang Xikai’s team proposed that a pair of pure hobbing profiles should only contact at the node and not at other rotating positions of the gear, and the pure rolling profile should be located in the involute and other conjugate profile curves, and found a variety of simple and practical pure hobbing profile curves to complete the development of the system. Based on the characteristics of non rigid and flexible impact of quintic polynomial like curve, Yi Yali et al. Designed the shock wave gear profile to realize the constant speed conjugate transmission of pure rolling movable teeth with arbitrary tooth difference. Tan Rulong et al. Studied the pure rolling contact Cycloid Bevel gear, and carried out experiments on the samples processed by the existing surface hobbing technology. The results show that the gearbox with pure rolling contact Cycloid Bevel gear can achieve higher transmission efficiency.

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