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1 Assembly conditions and its calculation method 11 Assembly conditions 2ZX type epicyclic gear train widely used in engineering, the planetary gears need to be evenly installed between the two center wheels on the planet carrier to offset the planetary gears in the transmission. The centrifugal force generated and the support reaction force and vibration of the planet carrier are reduced. For this reason, the number of teeth and the number of planet wheels Cs must satisfy a certain ratio of the number of teeth, that is, the assembly conditions can ensure the assembly and transmission performance. In general, the assembly conditions can be summarized as follows: 1) For a single-row 2ZX planetary transmission, regardless of the number of teeth of the planetary gear, the sum of the number of teeth of the two center wheels should be an integral multiple of the number of planetary wheels; 2) For the planetary gear Double-row 2ZX planetary transmission consisting of double gears c and d: NN-type transmission (as shown), WW-type transmission (as shown) and NW-type transmission (as shown), if the double planetary gear is Cut out on the same blank, the relative position of the two gears can not be directly adjusted during installation. When the number of teeth of the center wheel is Za and Zb is an integral multiple of Cs, the calculation of the tooth is simpler, and when the number of teeth of one or two center wheels is not a planet When the number of rounds is an integer multiple of Cs, the assembly condition formula given in reference [1] is as follows: WW type transmission: Za ZbCs 1 ZdZcEan-ZaCs=n(1) NW type transmission:
Za ZbCs 1-ZdZcEan-ZaCs=n(2)
Where, Za, Zb are the number of teeth of the two center wheels; Zc, Zd are the number of teeth of the double planetary gears c, d; Ea, n are integers, and n is selected from 0, 1, 2, and 3. When Za/Cs=integer, Ea=Za/Cs; when Za/Cs is an integer, Ea is an integer slightly larger than Za/Cs, which can be proved to be equally applicable to the NN type transmission (1) (the proof is omitted).
The introduction of the new matching tooth calculation method (1), (2) is complicated in form, and the calculation of the tooth is inconvenient. The two formulas can be simplified by the elementary number theory method, for which the formula (1) is transformed into:
ZbZc-ZaZdCsZc ZdZc(Ean)=Integer (3)
The maximum common divisor of Zc and Zd is represented by M, which can be expressed as M = (Zc, Zd) according to the theory of the greatest common divisor.
Order: Zc=ZcM, Zd=ZcM; S=ZbZ -ZaZdCsM; X=Ean.
Since Ea and n are integers, then X=0,1,2. Therefore, equation (3) can be expressed as: S ZdXZc=integer (4)
ZdX and Z
c is an integer. Obviously, for equation (4) to be true, S must be an integer, that is, S = integer is a necessary condition for equation (4) to be established.
When S, Z d is an integer, Zc is a positive integer, and Z c and Z d are prime numbers, that is, (Zc, Zd) = 1, then a congruence Z dX S0 (modZc) d0 (modZ
c) There is an integer solution, (where mod is the remainder operator, the same below), the mathematical meaning is that ZdX S can be divisible by Z c , and Z d cannot be divisible by Z c because of Z c and Z d . It can be seen that the S=integer is a sufficient condition for the equation (4) to be established, namely: ZbZc-ZaZdCsM=integer (5)
Therefore, in the assembly conditions for expressing the NN type and WW type planetary transmission, the equation (5) and the equations (1) and (3) are completely equivalent, but the calculation is greatly simplified. According to a similar procedure, equation (2) can also be simplified by the method of elementary number theory:
ZbZc-ZaZdCsM=integer (6)
In order to facilitate the calculation of the teeth during design, equations (5) and (6) can be further transformed. Taking the NN type transmission as an example, the formula is treated as follows: the difference in the number of teeth of the two center wheels is Za-Zb=u, and the difference in the number of teeth of the double planetary gears is Zc-Zd=v, which is constrained by concentric conditions, generally |uv| 2, by taking in (5) can be obtained: vZb-uZdCsM = integer (7)
Expressed in elementary number theory as: vZb-uZd0(modCsM)(8)
Equations (7) and (8) give the assembly conditions of the NN-type transmission under normal conditions in the form of the difference in the number of teeth. Once the difference in the number of teeth is determined, a combination of many possible numbers of teeth can be quickly determined, which is very convenient for calculation design.
2 The calculation example of the tooth is based on the number of planetary wheels Cs=6, indicating the matching method of the NN type transmission with multiple planetary gears and the corresponding combination of the number of teeth is obtained. For general power transmission planetary transmissions, a very precise transmission ratio is not required. In the case of the known required transmission ratio ibaX, the number of teeth can be selected as follows:
1) According to ibaX, look up the number of planet wheels Cs;
2) According to the design requirements (strength, stability, gear ratio conditions, assembly conditions, etc.), determine the number of teeth of other teeth such as the sun gear. In this paper, for the assembly condition formula (8) of the NN type transmission, through the two possible values ​​of the two tooth differencees u and v, the combination of the teeth required by the formula (7) or (8) is treated by the first congruence method. The final gear ratio can be calculated as follows: ibaX=ZcZaZcZa-ZbZd(9)
For example, when the difference in the number of teeth u=v=1 is obviously M=1, then there is Zb-Zd0 (mod6), whereby many possible combinations of the number of teeth can be obtained, as shown.
1 When the difference in the number of teeth u=v=1, the calculation of the matching teeth is when u=2, v=1, obviously M=1, then there is Zb-2Zd0(mod6)20(mod6) because (2,6)=2 , so there is no integer solution in the above formula. It can be seen that Zb should be an even number to meet the assembly conditions, and thus a large number of combinations of teeth can be obtained, as shown by 2.
2 The difference in the number of teeth u=2, the matching calculation of the teeth when v=1 is the difference in the number of teeth u=3, and the combination of the teeth when v=1 is as shown in 3.
3 The difference in the number of teeth u=3, the calculation of the combination of the teeth at the time of v=1 The conclusion of the combination 3 The initial number theory method can simplify the assembly condition formulas (1) and (2), and discuss the relationship between the tooth difference and the number of planet wheels. The calculation table for matching teeth is obtained, which provides a reference for how to correctly select the number of gear teeth in the design of the double row 2ZX planetary transmission.
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