Multi-objective Robust Optimization For Planetary Gear Reducer Of Earth-pressure-balance Shield Machine

The torque that drives the cutterhead of the Earth-pressure-balance shield machine(EPBSM) is first analyzed, and the load equilibrium and allocation of high-capacitythree-stages planetary gear reducer (PGR) are discussed to define the environment ofEPBSM for optimizing the PGR. For its4major elements, that is gear train, shaft,planetary carrier and gearbox, the gear train is a global commander so that the PGR canbe optimized by optimizing each constituent, their mathematical&mechanical modelsof multiple objectives are thus constructed according to the robust design demands. Inthe meanwhile, one or more very interesting algorithms that are selected from thecomputational intelligence (CI) are modified to become or be even combined to a newalgorithm that has better convergence than ever, and then they are used to solve themodels above.At last, the simulation test for the optimized PGR are done.The concretereason, methods, principles and effectiveness are successively as follows.To overcome the intrinsic defect of the grey theory in itself and the appearingproblem of particle swarm optimization (PSO) algorithm when it is used to slove theinteger or mixed one programming problems, the sigma is introduced into greyrelational degree, the notion of the reliably grey relational degree(RGRD) is put forward.For PSO algorithm, the demand of the integer solution is met by rounding their initialposition and speed of each flight in the course of answering the problems. The RGRD isacted as the alternation strategy to control PSO, and their mechanism and processes areillustrated to provide a practical and convenient method for multi-objective integer ormixed one programming problems subject to complex non-linear conditions. Themathematical models are constructed for three-stage PGR in EPBSM with4objectivefunctions, that is to make its volume smallest, its efficiency highest and the reliability ofits contacting and bending strength highest, which are subject to the conditions such asteeth, modification coefficient, interference, strength, equal strength, equal life etc,which are solved by MATLAB programming language that applied the reliably greyPSO algorithm. The research results show that1) the strength and lifetime in everystage are basically equal under the conditions of ensuring high reliability, and thevelocity of the proposed algorithm is faster than that of the penalty function method(PFM)(see P29-31),2) the generalization ability not only resolves the strength problemthat there is a little weak in the original3rdstage of the PGR, but decreases the gear train volume by11.55%and increases its efficiency by0.56%and its transmission ratio by3.67%.In view of the dissatisfactory convergence of the current immune genetic algorithm(IGA),6inherent drawbacks are found which existing in the current rationale of IGA.For these reasons,6new strategies which are corresponding to the drawbacks above areproposed to accelerate its convergence. The relationship between real-coded antibodiesand variables vector is illustrated, and the complicated algorithm mechanism isintuitively described by the figure, which make us understood easily and practicallyoperate on IGA. The stochastic perturbation method of the reliability is combined withits sensitivity analysis to deduce their function formulas on mechanical parts whoseprobability distributions of random parameters are arbitrary shape so that thetwo-objective mathematical models are created on minimizing the reliability sensitivityof the shafts of3-stage PGR in EPBSM with respect to design variables and theirvolume. A new model is proposed that can realize the dynamic balance betweenobjectives, by which the objectives are smoothly combined with the image set method.Finally, the shafts are optimized by means of the MATLAB programs languages thatapply the modified IGA above. The results show that the robustness of the improvedIGA not only decreases the total volume of shafts of the PGR by13.65%, but also itsconvergent velocity and accuracy are superior to that of the unimproved IGA (seeP50-51).In the practical engineering, to answer the small failure probabilities withhigh-dimensional correlated variables, the subset simulation (SS) is combined togetherwith the Monte Carlo simulation and importance sampling (IS) method. The samplesfrom the probability density functions (PDF) of the importance sampling are used toconstruct the intermediate failure events, by which the small failure probabilities areturned into a hybrid Markov chain (HMC), which is a continuous product made of aseries large failure probability or conditional failure probability (CFP) which is easilyanswered, on which the structural reliability sensitivity (RS) can be efficiently simulatedby directly obtaining the samples with correlated ones.Multi-objectives optimizationmodels are established about minimizing the RS of failure probability of the3planetarycarriers of the PGR with respect to the variable mean, variance (including the correlatedcoefficient between them) respectively and volume etc, and the collaborativeoptimization idea for multi-objectives is put forward, in the meantime, in view of theproblem that it is difficult to converge for multi-objectives to be collaboratively optimized because of the errors when the RS is used as an objective function, the ideaand method that utilize the errors are proposed. To increase the convergent velocity ofgenetic algorithm (GA) and PSO, the elite strategy that have elitist cloned and to takepart in evolution simultaneously and the idea that have an individual to mate theindividual who is most similar to it are put forward. And the excellent individuals fromthe modified GA are hybridized with those individuals from PSO to update thepopulation and to further enhance their convergence. Finally, the3planetary carriers areoptimized according to the algorithm above, the results show that1) the SS of the ISwith correlated variables can highly simulate failure probability and its sensitivity,2)the convergent velocity of the collaborative algorithm of hybrid GA-PSO is superior tothat of the GA and PSO (see P79-81), it can reduce the total volume of the planet carriersby7.06%when the correlated coefficient is equal to0.7,3) it is confirmed that theproposed idea and method that utilize the errors are feasible and correct when the RSacts as objective function.To solve the reliability and its sensitivity for structural system whose implicitnonlinear performance function (PF) are complicated, changeable and of non-normalvariables, the advantages of the saddlepoint approximation (SA) and line sampling (LS)are merged and the merits of dichotomy and the solution efficiency of the goldensection method are combined to propose the saddlepoint approximation-line samplingmethod (SA-LS) based on the dichotomy of the golden section point, namely, it is quickto find the zeropoint in PF corresponding to each sample along the important linesampling direction by the dichotomy above so that the structural failure probability canbe transformed into the mean of a series linear PF failure probability, and by whichreliability sensitivity can be solved, thus the multi-objectives are inferred about the RSof failure probability of the three-stage gearboxes of the PGR with respect to thevariables mean and variance and structural lightweight. To increase the convergence ofthe algorithm, after the PSO and shuffled frog-leaping algorithm (SFLA) are modifiedso that the former is changed into a self-adaptive PSO (SAPSO) algorithm, then it ishybridized with the improved SFLA to produce a new hybrid SAPSO-SFLA, and thehybrid algorithm is applied to answer the foregoing multi-objectives. Researches showthat1) the SA-LS method based on the dichotomy of the golden section point is of highprecision and fast velocity in solving the reliability and its sensitivity of thesophisticated nonlinear PF2) the convergence velocity of the proposed hybridSAPSO-SFLA is superior to that of the modified PSO and SFLA (see P106-108), and its robustness can decrease the volume of the gearboxes by8.42%.As can be drawn from the optimization of the4main components of the PGR inthe EPBSM,1) the generalization of the constructed4mathematical&mechanicalmodels and the proposed4computational intelligence not only resolves the strengthproblem that there is a little weak in the original3rdstage of the PGR, but can decreasethe volume of the whole PGR by21.58%, and increase its efficiency and transmissionratio by0.56%and3.67%,respectively,2) The practicable volume decrease of the wholePGR is5.12than that of the gear train when the volume decrease rate of the gear train isapplied to approximately predict that of the whole PGR.What’s more, the simulationexperiment also prove that the each index of the optimized PGR meets the presetteddemand, and that its reliability is good.