New multi-curve gears
If the number of teeth of the small gear Z1 is small, the degree of coincidence is small, which increases the noise; the number of teeth is too large, and the volume of the transmission increases when the modulus is constant. The mid-point tooth height coefficient K mid-point tooth height factor ensures a proper meshing depth in the middle of the tooth. Increasing the midpoint tooth height factor can increase the midpoint working tooth height, increase the working contact area, reduce the tooth surface contact stress, and improve the tooth surface contact strength.
The tangential displacement coefficient xt is generally considered to be that the hypoid gear does not require tooth thickness correction. However, in the matching relationship between the size of the wheel and the wheel, the correction of the tooth thickness is beneficial to the life design of the wheel and the like. Increasing the tangential displacement coefficient can increase the small tooth thickness, and correspondingly reduce the large wheel tooth thickness.
Fuzzy constraints are applied to the fuzzy optimization design of hypoid gears. There are two constraints: performance constraints and geometric variable constraints. For performance constraints such as stress, the intermediate transition process from full permit to complete disallowment must be considered. For geometric variable constraints, The ambiguity of the actual existence of its boundaries must be considered. These constraints are fuzzy subsets of the design space, and the constraints are established as follows (see the literature [1] for the meaning of each symbol in the formula): (1) Contact stress constraints. According to the literature [1] there are: H = ZHZEZZZKAKVKHKHFmtdm1beHu2 1u-HP (2) (2) bending stress constraints. According to the literature [1]: F = FmtbeFmmnYFYsYYYKAKVKFKF-FP (3) (3) modulus constraints.
Me3-(4) Upper and lower limit of tooth width. 4-meb10-meb13Re (5) helix angle constraint. 35m50 (6) tooth number constraint. 40Z1 Z260 (7) midpoint tooth height coefficient constraint. 35K4-(8) tooth thickness coefficient constraint. The symbol ~ denoted by XtXtXt above constraints indicates ambiguity.
14 Fuzzy optimization mathematical model In summary, the specific fuzzy optimization model of this example can be expressed as: X=[me,b,m,Z1,K,xt]T=[x1,x2,x3,x4, X5,x6]TminF(x) performance constraint: i(x)-ip(i=H,F1,F2) geometric constraint: HjHj(x)Hj(j=1,2) Variable constraint: xKxKxK(K=1, 2,,6)(4)2 Fuzzy Constraint Membership Function The boundary of each fuzzy constraint above the transition from fully permitted to completely unusable, has a certain membership function, and should be designed according to the nature of each constraint. Statistical methods, experimental statistics, expert assessment methods, etc. are obtained, and it is common to use typical functional patterns. In this paper, linear membership functions are used for fuzzy subsets of various constraints.
Performance constraints: =10LU-U-LU0 (other cases) (5) For geometric and variable constraints: membership of performance constraints on allowable values; x membership of geometric constraints and variable constraints on permitted values; Each stress of the curved gear; X design variables.
The non-fuzzy optimization design model on the optimal horizontal cut-off set can use a series of values ​​[(0,1)] to intercept the fuzzy set according to the decomposition law of the fuzzy set, and obtain a series of horizontal cut sets under different fortification levels. The larger the more secure, the smaller the economy is. Therefore, there must be an optimal value that is both safe, reliable, and economical. The problem is to find the value corresponding to the horizontal cutoff at the optimal fortification level. According to the optimal horizontal cut-off method, equation (4) can be transformed into the non-fuzzy optimization model on the optimal horizontal cutoff: X:[x1,x2,x3,x4,x5,x6]TminF(x)i (x)Ui-(Ui-Li))(i=H,F1,F2)HLj (HUj-HLj)Hj(x)HUj-(HUj-HLj)(j=1,2)XLk (XUk-XLk) XkXUk-(XUk-XLk)(k=1,2,,6)(7) What is the optimal value of the horizontal value, which is affected by many fuzzy factors. When considering these fuzzy factors, the second-level fuzzy synthesis is adopted. The judging method is determined. Then the appropriate optimization method is used to solve the general optimization model, and the optimal solution of the original problem can be obtained.
The optimization results and the determination of the analysis and the determination of the upper and lower limits of the fuzzy constraint transition interval are evaluated by the second-level fuzzy comprehensive evaluation: the upper and lower limits of the 0.526 fuzzy transition interval are based on the experimental and empirical data of each constraint parameter. Make sure that the lower bound is about 0.800.95, and the upper bound is about 1.051.30.
The calculation results use different values ​​[(0,1)] to transform the fuzzy optimization mathematical model into the ordinary optimization mathematical model as the equation (7), 1 is the general optimization, and is the fuzzy optimization. The optimization method can be solved by the direct optimization method, the regular polyhedron method, or the indirect optimization of the internal and external point penalty function method. Take =1 and 0.526 respectively, take the parameters of the original design as the initial point X=[4,30.000,45.000,12,3.700,0], and use the complex method to find the best on the microcomputer. The result is as follows. The optimal solution of continuous variables is compared with the original design: volume reduction after ordinary optimization: F0-F1F0=19.2 Volume reduction after fuzzy optimization: F0-FF0=30.6 Fuzzy optimization is smaller than normal optimization volume: F1-FF1= 14.2 In the continuous optimal solution of fuzzy optimization, the number of teeth needs to be rounded (ie, discretized). It can be seen from the data in the table that after the partial data is rounded, the volume is still smaller than the original design.
The result analysis fuzzy optimization design takes into account the various fuzzy factors affecting the transmission. Therefore, the optimization model and the optimization scheme are closer to the objective facts than the ordinary optimization. The calculation results of this example show that the fuzzy optimization continuous variable optimal value is better than the ordinary continuous variable optimal value. The drop is 14.2, which is 30.6 lower than the conventional design and has unique advantages. After rounding the parameters, it can still be reduced by 30.1. It has significant economic benefits in industrial production.
Synthetic cryolite is a white crystalline powder or a sandy-size granularity, and a pinkish crystalline powder or a sandy-size granularity as well. Sp.gr.2.95-3.01g/cm3, melting point about 1000°C, specific heat 1.056j/g°C at 18-100°C. It`s slightly soluble in water, but insoluble in anhydrous hydrogen fluoride. The content of its crystal water will be decreased while the increasing of the molecular ratio, therefore its loss on ignition will be also decreased while the increasing of the molecular ratio. After the paste of synthetic cryolite with different molecular ratio dehydrates, the loss on ignition at 800°C will appear 10.34%, 6.22% and 2.56% when the molecular ratio reaches 1.74, 2.14 and 2.63 reactively.
Uses:
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