Research on Multi-objective Optimization Design of Two-stage Planetary Reducer

Planetary gear reducers have the advantages of high transmission efficiency and compact structure, and have been widely used in various vehicles, construction machinery and other transmission systems. However, its design is a complex issue. Its volume, weight and load carrying capacity are mainly determined by the choice of transmission parameters. The existing optimized design scheme with the smallest volume as the target does not reflect the actual working conditions because it does not consider its vibration factor and single target, and it is easy to miss the real optimization scheme, and the satisfactory comprehensive effect is not obtained. In this paper, optimization theory and technology are used to establish a multi-objective optimization model with the smallest volume and the largest coincidence degree. For the constraint conditions, the feasibility enumeration method is used for effective judgment; for the multi-objective problem, the multiplication and division method is used for optimization, and finally the multi-objective optimal solution is obtained.
There are many factors affecting the smoothness of the output shaft of the planetary helical gear reducer, such as manufacturing precision and assembly process, surface roughness of the tooth surface, etc., but the improvement of manufacturing precision and the reduction of the surface roughness value of the tooth surface will greatly increase the production cost. . Optimizing the meshing parameters of the gear pair during design will achieve twice the result with half the effort. Taking the secondary planetary helical gear spur gear reducer as an example, the optimal gear meshing parameters of the smooth output of the reducer are found by the optimized method under the condition of key parameters (such as reduction ratio, etc.) and structure determination. In order to study the vibration caused by the change of the number of meshing teeth during gear transmission, the other factors are idealized, that is, the deformation of the shaft, the bearing and the casing are not considered, the manufacturing error and assembly error of the gear are neglected, and the lubrication is in a good state.
The gearbox uses a two-stage reduction drive, as shown. The three basic components of the first-stage reduction gear (the sun gear!, the planetary gears) and the inner ring gear can be moved; the second-stage reduction gear is actually a 2K-H (NG*) mechanism, and the transmission diagram is as shown. The shaft of the sun gear! is the input shaft of the gearbox and is connected to the motor.
The mechanism transmission sketch planet carrier is the output shaft of the first stage reduction gear, and is also the input shaft of the second stage reduction gear; the carrier % shaft is the input shaft of the gear box, and is connected with the traction sheave, and the rotation speed of the inner ring gear is equal to The speed of the planet carrier %. The reduction ratio of the whole machine is: under the premise that the strength and rigidity of the part are guaranteed, in order to minimize the vibration of the output shaft, the coincidence degree of the gear meshing is required to be the largest. Because it is a reduction transmission, the vibration of the previous stage is reduced by the reduction ratio and the output shaft has only 1/2 of the original. The formula of the coincidence degree of the planetary gear meshing is as follows: the design parameters such as the coefficient and the meshing angle cannot be reflected in the objective function. These values ​​are reflected in the constraint of tooth profile non-overlapping interference, pinion transition curve interference, coincidence degree, etc. In the solution of optimization mathematical model, these parameters are often set according to the experience of traditional design, sometimes resulting in unsatisfied Restrictions. Therefore, the sun gear and the planetary gear calculate the volume of the gear by the diameter of the tip circle. The internal gear calculates the volume by the square of the diameter of the top circle minus the square of the diameter of the root circle.
The second optimization goal is to obtain the smallest volume of the planetary gear transmission reducer. The main parameters affecting its volume are the sun gear a, the volume 'a, 'A, the planetary gear (, G volume 'g, '*, internal gear!, * Volume ', 'B. Then: dag, da * star wheel (, G tip diameter; G, - planetary wheel (, G tooth width; dab, daB - internal gear!, * tooth top) Round diameter; !, b - internal gear!, * tooth width; dfb, dffi - internal gear!, * root diameter of the tooth.
Design variables include tooth number, modulus, pressure angle, helix angle, tooth width, coincidence degree, total gear ratio, helical gear surface modulus, small helical gear teeth number, high speed gear ratio, small helical gear tooth width, helix angle, Planetary wheel displacement coefficient, planetary gear width, internal gear displacement coefficient, tip height coefficient, meshing angle and internal gear wall thickness.
According to the design requirements, the constraints are divided into the following categories: tooth constraints, strength constraints, motion performance constraints, structural constraints and other constraint constraints, each constraint contains its own specific constraints.
3p - the number of planet wheels; a + ac a, the center distance of the c gear meshing pair; 1c - the distance between the centers of two adjacent planet wheels.
Thus two constraints can be established.
Concentric conditions The so-called concentric condition is that the actual center distance of all the meshing gear pairs of the center wheel and the planet gear must be equal. Thus two constraints can also be established.
The contact fatigue strength and bending fatigue strength of the high-speed helical gear satisfy the strength requirement; the bending fatigue strength of the low-speed internal mesh gear meets the strength requirement.
(1) High-speed stage strength constraints The gear unit designed to ensure sufficient tooth surface contact strength tooth surface can obtain the following two constraints: R&D and manufacturing T1, T2 - torque transmitted by gears 1, 3 YFa - tooth profile coefficient; Sa - stress correction factor; allowable bending stress of F material.
(2) Low-speed strength constraints The gear unit designed to ensure sufficient root bending strength, the following constraints can be obtained: the kinematic performance constraints should first prevent the occurrence of tooth-tooth overlap interference, that is, require ma, planetary gear, Internal gear tip round pressure angle.
No transition curve interference requirements: external engagement: internal engagement: a'w - internal gear machining engagement angle; aa03 - tool tip pressure angle; aa3 - internal gear tip pressure angle.
Constraints under the constraint condition that the planetary gear and the internal gear do not have tooth-tooth interference are: the diameter of the planetary gear circle should ensure that the rolling bearing is installed in the planetary gear hole.
Fourth, the optimization method For the discrete problem, there is a very simple solution, that is, all possible combinations of design variables are listed, calculate the objective function value of each combination one by one, and then compare the optimal value, which is the piece Raise the law.
The basic idea of ​​the feasible enumeration method is: Before calculating a combined objective function value of a variable, it may be necessary to first determine whether the combination satisfies all the constraints. Once it finds that it does not satisfy any of the constraints, it immediately deletes this. combination. If it satisfies all the constraints, it should be further judged whether the value of the objective function calculated from this combination is better than the value of the objective function that has been obtained from other combinations. If there is no such possibility, then it is still deleted. Go to this combination. It is only for the combination of design variables that satisfy all the constraints and it is possible to obtain the optimal objective function value, and then calculate the corresponding objective function values.
Design a planetary gear reducer. Input power 0=11k, input speed)1=1500r/min, required output speed)1=The total gear ratio iT=31.Compared with the result of optimization design and conventional design, the volume is reduced by 14.73%, and the coincidence is also increased. Degree (increased by 9.32%), which improves the overall performance of the reducer.
By analyzing the mutual constraint relationship between the design parameters of multi-target planetary gear reducer mechanism, a mathematical model with maximum coincidence and minimum multi-objective optimization is proposed. This optimized mathematical model can fully reflect the mutual interaction between design parameters and global optimality. Relationships make the overall performance of the design better. The example shows that this optimized design has a significant improvement over the conventional design and is more in line with the needs of engineering design. This optimized mathematical model provides design and manufacturing value for planetary gear reducers.

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