Fuzzy Reliability Optimization Design of Three-ring Planetary Reducer

In the mechanical industry design, the science and technology forum should consider the randomness and ambiguity of design parameters, and can carry out multi-parameter design, make the design scheme optimal, and predict the reliability of new products after design, which requires reliable An organic combination of sexuality and optimal design. K-H three-ring planetary reducer is one of the external eccentric shafts in the involute and small-tooth difference gear transmission. It is a new type of reducer. The utility model has the advantages of simple structure, small volume, large carrying capacity and long service life. In recent years, it has been widely used in transportation, metallurgy, cement, construction, shipbuilding and other industrial sectors. The basic structure of the three-ring planetary reducer is shown in Figure 1.
Then the fuzzy reliability of bending strength is: 1.3 Constraint condition 1.3.1 Fuzzy reliability constraint of bending strength Due to the randomness and ambiguity of stress and strength, in the strength check, the failure probability should be considered, that is, the fuzzy reliability constraint is established. If the reliability is R 0 , then R is the fuzzy reliability of the root bending strength of the outer and inner gears.
1.3.2 Deterministic Constraints In the internal meshing drive of the three-ring reducer, due to the small difference in the number of teeth between the inner and outer gears, interference is often caused during cutting and assembly, which is avoided. Therefore, these interferences should be avoided in the design. In addition, there are certain restrictions on some parameters.
1 tooth top thickness constraint inner and outer gear tooth tip can not be sharpened, the tooth top thickness should be satisfied, the outer gear tooth top thickness: internal gear tooth top thickness type: x - external gear displacement coefficient; - external gear Tip crown pressure angle; - internal gear tip round pressure angle.
2 The inner gear tooth top circle is larger than the base circle. Because there is no involute line in the base circle, the internal gear tooth top circle should be larger than the base circle, that is, the 3 coincidence limit is for the internal meshing transmission. In order to improve the stability of the transmission, The coincidence degree is increased as much as possible, and ε ≥ 1 is satisfied, that is, there are: 4 inner and outer gears do not produce transition curve interference limitation in the technical forum type: Z - the number of teeth of the pinion cutter; Pressure angle; - the angle of engagement when the shaper is machined.
5 cutting the inner ring gear does not produce the constraint of the top cutting 6 does not occur the constraint of the tooth overlap interference due to the characteristics of the internal meshing drive, especially when the number of teeth of the two gears is relatively close, the pair of teeth are not far from the end of the meshing It may interfere with each other, that is, tooth-tooth overlap interference occurs. Since the radius of curvature of the internal gear tooth tip is small, the radius of curvature of the external gear tooth tip is large, so the tooth profile overlapping interference often occurs at the top of the inner and outer gear teeth. The constraint that does not produce overlapping interference is: where: 7 rib bearing life constrained jib bearings should ensure safe operation within the required life, that is, in the constrained type: C - bearing dynamic load rating; P - bearing equivalent movement Load; the life limit of the required bearing is due to the fuzzy reliability optimization design of the above-mentioned available three-ring reducer: 2, the solution to the fuzzy reliability optimization model is to solve the fuzzy reliability optimization problem into ordinary Optimize the problem and solve it with ordinary optimization methods.
2.1 Processing of Fuzzy Boundaries In the fuzzy constraints of this model, their boundaries are not strictly defined. In the vicinity of the permissible value, it reflects the gradual process from the full permit to the complete disapproval, and the membership function is used to express the change law. Commonly used membership functions include trapezoidal distribution, normal distribution, and Cauchy distribution. Considering the convenience of solving, we use the trapezoidal membership function distribution diagram 2.
Figure 2: The membership function of the membership function reliability index R of the trapezoidal distribution obeys the semi-trapezoidal distribution diagram 2a, meaning that the reliability should not be lower than the reliability allowable lower limit. The augmentation coefficient method can be used to determine the upper and lower bounds of the permissible transition interval, and the specific calculation process is omitted. This converts the fuzzy constraints into non-fuzzy constraints.
2.2 Solution of Fuzzy Optimization Problem At present, the more mature fuzzy optimization model transformation method is the optimal horizontal intercept method, and the cut-off level λ is another basic concept of fuzzy mathematics. The magnitude of the numerical value indicates the degree of constraint. If the constraint condition can be satisfied to the maximum at a certain λ intercept level, the optimal solution of the design can be obtained. The intercept level is the optimal cut level λ*. Under the premise of ensuring the design is safe and reliable, considering the fuzzy factors such as design level, material quality and importance, and importance, the second-level fuzzy comprehensive evaluation method is used to comprehensively evaluate and determine an optimal cut-off level. A general optimization model can be obtained by calculating λ=0.668.
3, design examples and optimization results analysis 3.1 design examples and solution design a K-H type three-ring planetary reducer. The given gear ratio is 37.5, the input shaft speed is 1500r/min, the output shaft torque is 77, the external gear Z=75, the number of teeth of the gear teeth is Z=40, the modulus is 3nun, the tooth width is 35±0.35nun, the gear Material 40Cr carburizing and quenching requires a reliability of 0.999, making the structure the most compact.
An initial point X T is taken from the original design. Considering that the design parameters contain discrete design variables, the hybrid discrete variable optimization method is used to solve the problem. When the machine is calculated, the original design scheme is taken as the initial point, and the fuzzy reliability optimization design result is obtained, the objective function and the constraint conditions are satisfied. Compared to the original solution, the volume is reduced by 10.4%.
3.2 Results Analysis 1 Based on the reliability design and optimization design, combined with the theory of fuzzy mathematics, the mathematical model of fuzzy reliability optimization design of K-H three-ring planetary reducer is established, and the solution method is given.
2 The calculation results of the design example show that the fuzzy reliability optimization design not only further optimizes the design results, but also gives the specific reliability, and considers many practical factors in the design, making the design results more scientific and reasonable, more in line with objective reality. .
1 Zhang Guorui and so on. Planetary transmission technology. Shanghai: Shanghai Jiaotong University Press. 1989 2 Lu Yuming. Reliability design of mechanical parts. Beijing: Higher Education Press. 1989 3 He Zhongxiong. Fuzzy mathematics and its applications. Tianjin: Tianjin Science and Technology Press. 1982 4 Mechanical design fuzzy optimization principle and application. Beijing. Science published on page 40. The authentication service distributor sends the authentication information carried by the internal object to a specific authentication interface module, and the processing result is sent to the XML synthesis interface to form an XML authentication result available for POP3, and finally sent to the POP3 server by the XML authentication service interface and ends. Certification.
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