A preliminary study on the curved cylindrical gears Zhao Dengfeng Li Shuangyue Chen Yunbiao (Mianyang Economic and Technical College, Mianyang 621000, Sichuan, China) The meshing characteristics of the interference, induced curvature, sliding coefficient, coincidence coefficient, etc., and the working angle of the tool during cutting.
Curved bevel gears (usually including circular bevel gears and cycloidal bevel gears) are widely used for their excellent transmission performance and efficient machining methods. The curved cylindrical gear and the curved bevel gear discussed here are similar, and the principle of the tooth profile is as shown.
In the figure, a plurality of teeth are evenly distributed on the circumference of the cutter head, and the cutter head rotates around the shaft to form a main motion of the cutter teeth on the cutting of the tooth blank. The movement of the tooth blank is a development motion, and it is also divided into a cycloid and an arc: when the circular arc gear is cut, the tooth blank moves in the horizontal direction while rotating, which corresponds to the rolling of the tooth blank on the indexing plane. At the end of one tooth machining, the tooth blank needs to be rotated and the other tooth is machined.
When cutting the cycloidal gear, the tooth blank is superimposed on the basis of the above motion, and the rotating part is superimposed with a rotation proportional to the main moving speed of the cutter head (the ratio is related to the number of cutter teeth and the number of teeth of the cutter). The tooth flanks of the teeth, thus eliminating the need for additional tooth movement. The other details of the tool are similar to the bevel gear.
1 Analysis of coordinate system and motion is the coordinate system used in the analysis of curved cylindrical gears. Where (o-xyz) is a spatial coordinate system fixed to the space. (Factory xdydzd) The tool coordinate system aXayaza) and (bX/by/, Zb) which are fixed to the cutter head are the gear coordinate systems respectively fixed to the gear A gear B. Gear A and gear B are a pair of gears that are conjugate engaged. They are cut by blades of the same shape but complementary to each other.
The angle between the Xd axis and the x-axis is the rotation angle of the tool, and the distance 1 from o to d is the translation distance of the tool. It is assumed that the cutting edge is a straight line placed on the (dzdyd) plane, and the distance from the intersection of the yd axis to d is the indexing radius of the cutter head. The angle between the Rd and the zd axis direction is the tooth angle angle of the blade, and the straight line equation of the blade is sorted into In the two-gear coordinate system, the angle (or 0/) between the ya (or y/) axis and the y-axis direction is the angle of rotation of the gear A (or B). The distance Ra (or R/) from a (or /) to o is the pitch circle radius of the gear A (or B).
The motion relationship that should be satisfied when analyzing the circular arc gear is: the motion relationship that should be satisfied when analyzing the cycloidal gear is: where Rp is a constant, and the adjacent tooth should be processed to adjacent tooth surface, 2 toothed surface tooth shape The surface equation: coordinate transformation relation and (2.3) obtain the transformation relationship between the tool coordinate system and the gear A coordinate system: Md and Ma are the transformation matrix of the tool coordinate system and the gear A coordinate system to the space coordinate system. Equations (3.1) and (3.2) can be regarded as transformations from a parameter space composed of 0d, 0a, s, and 1 to a tool coordinate system. The tooth profile surface of the analysis gear A and all properties thereon are based on this two formula.
2) Finding the first-order differential to obtain its differential transformation, and sorting it into a matrix form is: where U is a 4×4 matrix, dZ=dsdl) It is known from the geometric knowledge that on the envelope surface of the tooth profile, the differential transformation is outside the gear Width, lateral position, and tool radius all have a large effect on the coincidence factor.
The qualitative situation of the influence of the tip circle diameter, the index circle diameter and the tooth angle on the coincidence coefficient is the same as that of the ordinary gear. The effect of the gear width is similar to that of a conventional helical gear, and the width increase coincidence coefficient also increases. As the tool radius decreases, the degree of tooth profile increases and the coincidence coefficient increases accordingly.
When x is 0, the meshing condition is the same as that of the spur gear, and it is obvious that the overlap coefficient of the curved gear is always larger than that of the spur gear.
In the two examples analyzed here, when the tooth radius of the two gear teeth is 210 and 160 respectively and the width is 120, the coincidence coefficient of the corresponding ordinary spur gear is 1.796, and the circular gear is 2.069. 2.109 has been significantly improved. A large overlap factor is advantageous for gear transmission force, motion stability, and lubrication.
4 The working angle of the tool is the same, but the movement of the tool during the actual cutting is more complicated, so that the working angle is seriously deviated from the marked angle. In severe cases, the working angle is less than 0 and the cutting process cannot be performed.
The relative cutting speed, labeling angle, working angle, etc. at the specified point of the blade are indicated. Ux, Uy, and Uz are the components of the actual cutting speed on the tool axis, which are the components in the blade direction and the vertical direction of the blade. Y, a is the front angle of the main section of the tool, which is the correction amount of the rake angle, the back angle and the rake angle. According to our setting of the cutting edge position, the blade inclination angle k is 0. The calculation result of the slow feed motion is as follows: (1) When machining the circular gear, the correction amount is 0 at the s = 0 deviation from s=0. The amount of correction is also small. The geometry of the tool can be determined by a common end mill.
Normally, the angle of the tool is marked with the angle of the end mill. The tool working angle is shown in Table 5. The working angle correction value is further analyzed by the Rp and a, which is mainly affected by Rp and Rd. The inverse sine of the ratio. Other factors have little effect.
Special emphasis must be placed on ensuring that the working clearance angle is greater than 0 in the most unfavorable situation. It should be noted that the working angle of the solid part of the tool on different sides of the cutting edge, the angle of the marking and the angle of the angle correction are different.
5 Conclusion According to the above analysis of the curved cylindrical gear and its transmission, it can be concluded that the curved cylindrical gear has the following main advantages: it has high production efficiency compared with the conventional gear processing method.
Since the tooth shape is bow-shaped and has a large coincidence coefficient, the curved cylindrical gear should have a higher bending strength.
Due to the small induced curvature and the long contact line, the curved cylindrical gear should have a high contact strength.
Curved cylindrical gears should have better lubrication, wear resistance and transmission stability.
However, the curved cylindrical gear also has the following deficiencies: the separability of the center distance of the common involute gear transmission is lost.
It is difficult to machine internal gears and multiple gears.
Like the bevel gear, the cycloidal gear is difficult to grind. However, the curved cylindrical gear is a new type of transmission gear worthy of further study.

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