Refer to the rotor specifications and design experience, the level and range of the design variables are shown in Table 1。 Eighty nine experimental design points are constructed though CCF method。文献综述

4。 MULTI OBJECTIVE OPTIMIZATION BASED ON

GENETIC ALGORITHM

Normally, there are two traditional methods to solve multi-objective optimization problems。 Transform multiobjective optimization into a single objective optimization problem by weighting method or retain only one goal and change the other objectives into constraints。 But these methods have many limitations, the optimal solution much depends on the designer's preference [12]。 At present, genetic algorithm is often used to solve the multi-objective optimization problem。 There are a large number of solving methods for the multi-objective optimization problem based on genetic algorithm。 Non-dominated Sorting Genetic

Algorithm- II(NSGA-Ⅱ) method based on Pareto is used in this work。 This method can quick non - dominated sorting solutions and maintain the elitist and the population's persity [13, 14]。 Initial population of genetic algorithm has a great effect upon NSGA-Ⅱconvergence。 In order to avoid early convergence and ensure global optimum Shifted Hamersley sampling technique and weighing function are adopted to produces initial population [15]。

Hamersley sampling technique is a kind of quasi random-sampling technique based on Hamersley algorithm。 Evenly distributed sample points can be produces in the n-dimensional feasible solution space through this technique [11]。 In this work Shifted Hamersley sampling technique is adopted to overcome the shortcomings that Hamersley sample points concentrate in the region of starting point in K-dimension cube。 Hamersley sample points are offset =N/2, the sample points are more even, smaller low-biased, it ensure that the multi-objective genetic algorithm quickly converge to the global optimal solution。 In this work, 500 sample points are evenly extracted in the feasible solution region using Shifted Hamersley technique。

Sort 500 samples by weighing function as shown in Eq。 (5)。

where n is the total number of objective function and constraint。 Mi is defined by Eq。 (6), where ymax is the maximum value of yi(X), ymin is yi(X) minimum, yt is the ideal solution of objective function yi(X), y is the current value of objective function yi(X)。 The smaller is weight function value, the better sample point is。 Select the first 300 Shifted Hamersley sample points as the initial population of genetic algorithm。

Evaluate the objective function by genetic algorithm in AWB DX。 The number of each iteration inpidual is 100, the maximum operating algebra is 100 generations, get the Pareto-optimal solutions, as shown in Fig。 (5)。 The abscissa and ordinate respectively represent an objective function in Fig。 (5)。 According to the design requirements, select 5 groups solutions from the Pareto optimal solution, as shown in Table 3。 Considering the design goal, the rotor first order natural frequency and the rotor radius is the most important, so select the fourth group of solutions。

In order to modify the optimal results and make the optimization results suitable for engineering applications, the change in rotor structural response with respect to design variables is analyzed through sensitivity analysis。 The sensitivity analysis results are shown in Fig。 (6)。 According to Fig。 (6), the rotor unbalance response is proportional to l1,l4, d1, and d1 is the biggest effect factor; the rotor mass unbalance response is inversely proportional to d3, h1, h2, and d3, h1 is the biggest effect factor, by contrast, h2 has little effect; the first-order natural frequency of the rotor is proportional to l1, l4, d1, d3, h1, and l2, h1 have little effect。 So it can be considered to increase the h1 properly to reduce the rotor mass unbalance response。 The increase of l1, l4, d1 can improve the first-order natural frequency of the rotor, but the increase of l1, l4 and d1 will cause the rotor mass unbalance response increase, so that the increase of l1, l4 and d1 is Limited。 The increase of d3 can enlarge the first-order natural frequency of the rotor body, at the same time, the increase of d3 can reduce mass unbalance response of the rotor, but the increase of d3 will make the mass of the rotor body increase, so that the increase of d3 is restricted; The first-order natural frequency is inversely proportional to h2, and it strongly affected by h2, consider increasing the first order natural frequency of the rotor by reducing h2。 Optimized design variables of the rotor are corrected according to the results of sensitivity analysis as shown in Table 4。