# Study on machining of conjugate curve bevel gear

Based on the principle of conjugate curve meshing, the general equation of conjugate curve meshing theory is deduced and its basic properties are discussed. On this basis, a conjugate curve bevel gear pair based on conjugate curve meshing is proposed. Aiming at the three cases of single point pure rolling contact, single point non pure rolling contact and multi-point contact, the model of bevel gear pair is deduced, and their geometric, kinematic and mechanical characteristics are studied, including the condition of no local interference, the motion law of contact point, tooth strength and so on. The full text is summarized as follows:

① According to the conjugate curve, the tooth profile surface with bearing capacity can be generated, that is, the conjugate curve obtained from the curve meshing theorem can be extended into a local conjugate tooth surface. The contact trace of this pair of tooth surfaces in the meshing process is a conjugate curve, and the local conjugate tooth surface does not interfere in the local contact.

② A new method for constructing the tooth profile of bevel gear is proposed, that is, the tooth profile surface with bearing capacity is obtained from the conjugate curve equation. The contact trace of the tooth profile surface is a conjugate curve, and the normal section tooth profile can be any regular curve. Based on this method, the conjugate curve bevel gear with double arc section is designed. The concave arc can increase the thickness of the tooth root and reduce the bending stress; For large and small gears, the cross-section curve can be designed uniformly, which means that the large and small gears can be processed by a tool.

③ According to the conjugate curve meshing principle, the conjugate curve equation of pure rolling contact conjugate curve bevel gear is deduced. For this bevel gear, the conjugate curve is located on the pitch cone, that is, the contact point is always located on the pitch cone. At the same time, it can be obtained from the conjugate curve equation, Γ 1 and Γ 2 has the same expression, that is, they are the same type of conical curve.

④ The theoretical model of conjugate curve bevel gear with pure rolling contact is established, the models of equiangular spiral bevel gear, equidistant spiral bevel gear and long epicycloid bevel gear with pure rolling contact are deduced, and the solid model is established. The stress condition is analyzed. The results show that the pure rolling contact bevel gear model has the potential to be applied to power transmission.

⑤ According to the conjugate curve meshing principle, the conjugate curve equation of non pure rolling contact conjugate curve bevel gear is deduced and the force analysis is carried out. For non pure rolling conjugate curve bevel gear, since the contact point is no longer limited to the pitch line, the contact line can be taken arbitrarily, which has greater flexibility for the tooth surface design of bevel gear. The finite element analysis results show that the contact stress will not be greatly affected when there is a small displacement at the design contact point under load.

⑥ This paper discusses the general principle of multi-point contact conjugate curve bevel gear, mainly discusses its design principle, calculation criteria and configuration method, and designs the actual parameters according to the proposed design method of multi-point contact conjugate curve bevel gear. On this basis, the three-dimensional solid models of two-point contact conjugate curve bevel gear and five point contact conjugate curve bevel gear are constructed, The finite element analysis of these models is carried out. When the instantaneous contact point increases from single point to two points, the contact stress decreases by 15%, and when the instantaneous contact point increases from two points to five points, the contact stress decreases by 10.1%. Therefore, it can be seen that with the increase of instantaneous contact points, the load between bevel gear tooth surfaces is shared by multiple contact points, which can significantly reduce the contact stress.

⑦ The manufacturing and assembly of conjugate curve bevel gear transmission prototype are completed, and the test bench of sew gearbox and new conjugate curve bevel gear box is built. The test bench is used to test the gearbox. The test results show that controlling the contact point near the pitch line can reduce the friction loss of gear pair and improve the transmission efficiency.

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