Theoretical Study On The Unified Fuzzy Reliability Model And Reliability Assessment Modeling Of Blades To Avoid Resonance

There are a lot of uncertainties in engineering, including random uncertainties and fuzzyuncertainties. Conventional random reliability theory and methods were founded on theprobability theory and mathematical statistics, which can only deal with the problems withrandomness. Those reliability problems including fuzzy factors should be resolved by thefuzzy reliability theory and methods. In this paper, two generalized fuzzy reliability models,which were universally applicable in engineering, were proposed based on the cut-set methodand the equivalent transformation method respectively. Further, a unified generalized fuzzyreliability model which could deal with all the Posfust, Posbist, Profust and the randomproblems was proposed by exploring its convergence. For the resonance problems ofaero-engine blades, considering the fuzziness of the resonant criterion and the inherentfrequency, the fuzzy reliability models of blades to avoid resonance were proposed based onthe generalized models above. The details are shown as follows:(1) In practical applications, taking into account the limitations of the existed methods onthe construction of the membership functions, a new method to construct the membershipfunction which is suitable for the continuous fuzzy variables was proposed. The specific stepsby using experimental data were given, which is applicable to the construction of the commonL-R type membership functions in engineering, including point type and flat type. It is shownin the numerical simulation that the method is feasible in practical applications and has goodstability.(2) Considering that the existed equivalent transformation methods may lead to theuncertainties of the fuzzy variable are not equal before and after the transformation, and canbe just applied to handle the point symmetric fuzzy variables, two new equivalenttransformation methods were proposed. First, for the point symmetric fuzzy variables, entropyconservation methods were proposed based on the principle of uncertainty unchanged aftertransformation, including fuzzy entropy (ambiguity) conservation method and probabilityentropy conservation method. Then, for the general fuzzy variables, a direct normalizationmethod was proposed, and the numerical sampling method for the atypical random variablesobtained after transformation was given. The examples show that, as a new equivalenttransformation method, the entropy conservation method is an effective improvement on otherexisted methods, and the direct normalization method is feasible in dealing with the generalfuzzy variables.(3) Based on the general assumptions on the fuzzy reliability problems, two generalized fuzzy reliability models, which are universally applicable in engineering, were proposed byusing the cut-set method and the equivalent transformation method respectively. Meanwhile,the essential relationship between these two models was analyzed and discussed, and it isindicated that the model based on the equivalent transformation method is a cut-set reliabilitymodel under a special confidence level and cut-set distribution, and thus is a special andcoarse model based on the cut-set method. Therefore, the model based on cut-set method ismore general, and that based on the equivalence transformation method is more convenient incalculation.(4) The convergence of the generalized fuzzy reliability model based on cut-set method wasdiscussed, and a convergence theorem of the model with commonly-used cut-set distributionswas proposed and proved. The theorem points out that when the fuzzy variables graduallyconverge to the random variables in the real number field, the reliability assessment of thefuzzy reliability model with commonly-used cut-set distributions converges to a fixed value,which is just determined by the form of the safety criterions and has nothing to do with thespecific issues. This proposed theorem indicates the shortcomings and deficiencies of thecommonly-used cut-set distributions, that is, when the fuzzy variables converge to the randomvariables, the reliability assessment of the Posfust model does not converge to the result of thecorresponding Profust model or random model, but converges to a fixed value, which isobviously unreasonable. The correctness of the proposed theorem was verified by somenumerical examples.(5) In order to avoid the deficiencies of the commonly-used cut-set distributions, a familyof new cut-set distributions named reflecting-characteristic cut-set distributions was proposed,and three specific distributions named modified truncated normal distribution, truncatedlognormal distribution and truncated Weibull distribution were given respectively. Then theconvergences of the models with reflecting-characteristic cut-set distributions were discussed,and the results showed that when the fuzzy variables gradually converge to the randomvariables in the real number field, the Posfust model with reflecting-characteristic cut-setdistribution will converge to the corresponding Profust model or the random model, and thuscould deal with all the Posfust, Posbist, Profust and the random problems, which ensures theunity and generalize of the model. So the model with reflecting-characteristic cut-setdistribution could be named as unified generalized fuzzy reliability model. Numericalexamples verified the above theoretical analysis, and indicated that the proposedreflecting-characteristic cut-set distributions have obvious advantages over thecommonly-used ones. (6) For the fuzziness in the blades resonance problems, a normal bathtub function was usedas the membership function of the fuzzy event "blade resonance does not occur", andaccording to the concavity and convexity of the function in the neighborhood of the median,the fuzzy inherent frequency was described through trapezoidal, flat-normal and flat-parabolicmembership functions respectively. Then, based on the two generalized models, thecorresponding fuzzy reliability models of blades to avoid resonance and their calculationmethods were proposed. Finally, the proposed models were applied to some simulationexamples, and the influence of the fuzzy degree on the reliability of blades to avoid resonancewas studied. The results showed that the reliability to avoid resonance is inverselyproportional to the fuzzy degree of the resonance criterion, namely,"the fuzzier the resonancecriterion is, the smaller the reliability will be". In addition, when the membership function offuzzy inherent frequency is symmetric, the reliability decreases with increasing of the fuzzydegree of inherent frequency. While when the membership function is asymmetrical, thebigger the fuzzy degree on the side close to the mean of excitation frequency is, the smallerthe reliability is, and vice versa; but the bigger the fuzzy degree on the side far away the meanof excitation frequency is, the bigger the reliability is.