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1 Fuzzy reliability design of arc gear transmission The fuzzy reliability design considers the working stress of the gear and its influence parameters as random variables obeying a certain distribution law, and regards the allowable stress value as a fuzzy with some continuous membership function. Variables are constructed using fuzzy reliability theory to establish fuzzy reliability design criteria.
1.1 Stress distribution In arc gear transmission, the main failure modes are gear tooth bending and tooth surface pitting and wear. Therefore, in the design, the influence of the root bending stress σF causing the tooth fracture and the contact stress σH causing the tooth surface wear is mainly considered. It is known from the literature [1] that the working stress approximates the lognormal distribution and its probability density function. Let f(x)=1x2πSexp-(lnx-σ)2S2(1) where:
σ is the logarithmic mean of the stress variable x; S is the logarithmic standard deviation of the stress variable x.
Each allowable stress is regarded as a fuzzy variable, and its ambiguity is characterized by a membership function.
The membership function refers to the change law of membership degree in the intermediate transition zone that is completely allowed to be completely unusable.
In this paper, the fuzzy membership function of the allowable stress is the semi-trapezoidal distribution function [2], and its distribution image is 1, which is of the form μσ(x)=10a2(2) in the formula (2): the upper and lower bounds a2 of the transition interval A1 is the allowable stress value. This paper uses the amplification coefficient method to determine, a1=σ, a2=1.05σ, σ is the allowable stress value; μσ(x) is the membership function.
1.2 Calculation of fuzzy reliability The fuzzy reliability expression is R=∫ ∞-∞μσ(x)
f(x)dx(3) substitutes equations (1) and (2) into equation (3), where R = a2a2-a1 < (lna2-σS)-a1a2-a1<(lna1-σS)-1a2-a12πSexp (σ S2)ln(a2a1)-exp(σ)a2-a1<(lna2-σS-S)-<(lna1-σS-S)(4) where: distribution function of standard normal distribution, other symbolic meaning Cit.
With the fuzzy reliability R, the fuzzy reliability design criterion is RR ≥ 0, and R is the reliability allowable value.
2 Fuzzy reliability optimization design of arc gear transmission Fuzzy reliability optimization design is to introduce fuzzy reliability constraint function in conventional optimization design.
2.1 Design variables and objective function determination The main indicators of arc gear design are performance indicators, and the second is economic indicators. Under the premise of satisfying the given conditions and performance, the production cost is reduced as much as possible, and the economic indicators are improved.
When the gear volume is the smallest, the cost is generally low, so it is reasonable to use the gear volume as the optimization objective function.
If the minimum volume of a pair of circular gears is the optimization target (approximate by the sum of the cylindrical cylinders of the gear indexing), the objective function is F(x)=πφd4(1 i2)(mnz1cosβ)3 where: φd is the tooth The width factor, i is the gear ratio, mn is the normal modulus, z1 is the pinion number, and β is the helix angle.
The independent design parameters of the gear transmission are z1, mn, φd, β, so the design variables are taken as X=z1, mn, φd, βT=x1, x2, x3, x4T2.2 constraints. The fuzzy reliability constraint constraint expression of intensity is G1(X)=ROF1-RF1≤0G2(X)=ROF2-RF2≤0 where: RF2 and RF1 are the calculated fuzzy reliability of the tooth root bending strength of the large and small arc gears respectively. , ROF2, ROF1 are respectively the reliability of the gear root bending strength required for the design.
As long as the log mean σF1, σF2 and the logarithmic standard deviation SσF1, SσF2 of the stresses σF1 and σF2 are determined, RF1 and RF2 can be calculated by the equation (4).
The root bending stress is calculated as [3]σF1=(T1KAKVK1KH2με KΔε)0.79YE1YuYβYF1YEND1Z1mn2.37σF2=(T1KAKVK1KH2με KΔε)0.73YE2YuYβYF2YEND2Z1mn2.19(5)
According to the principle of mathematical statistics, the logarithmic mean of bending stress is σF1=(T1KAKVK1KF1με KΔε)0.79
YE1YuYβYF1YEND1Z1mn2.37σF2=(T1KAKVK1KF2με KΔε)0.73
In the YE2YuYβYF2YEND2Z1mn2.19 equation, the relevant coefficients and their value methods are described in the literature [3].
The logarithmic standard deviation is SσF=CσFσF where CσF is the coefficient of variation.
The fuzzy reliability constraint constraint expression of contact strength is G3(X)=ROH-RH≤0, where ROH is the reliability of the tooth surface contact strength required by the design, and RH is the calculated fuzzy reliability of the tooth surface strength.
The tooth surface contact stress is calculated as [3]σH=(T1KAKVK1KH2με KΔε)0. 7 ZEZuZβZαZ1mn2.1(6) Same as above, the logarithmic mean of contact stress is σH=(T1KAKVK1KH2με KΔε)0.7ZEZuZβZαZ1mn2.1 The logarithmic standard deviation is SσH = CσHσH, where CσH is the coefficient of variation.
Curved gear operation