Multiobjective Optimization of Steering Mechanism for Rotary Steering System Using Modified NSGA-II and Fuzzy Set Theory
Multiobjective Optimization of Steering Mechanism for Rotary Steering System Using Modified NSGA-II and Fuzzy Set Theory
Hongtao Li,1 Wentie Niu,1 Shengli Fu,2 and Dawei Zhang1
1Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, Tianjin 300072, China
2CNPC Bohai Drilling Engineering Company Limited, Tianjin 300457, China
Received 28 January 2015; Accepted 11 May 2015
Academic Editor: Mitsuhiro Okayasu
Copyright © 2015 Hongtao Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Due to the complicated design process of gear train, optimization is a significant approach to improve design efficiency. However, the design of gear train is a complex multiobjective optimization with mixed continuous-discrete variables under numerous nonlinear constraints, and conventional optimization algorithms are not suitable to deal with such optimization problems. In this paper, based on the established dynamic model of steering mechanism for rotary steering system, the key component of which is a planetary gear set with teeth number difference, the optimization problem of steering mechanism is formulated to achieve minimum dynamic responses and outer diameter by optimizing structural parameters under geometric, kinematic, and strength constraints. An optimization procedure based on modified NSGA-II by incorporating dynamic crowding distance strategies and fuzzy set theory is applied to the multiobjective optimization. For comparative purpose, NSGA-II is also employed to obtain Pareto optimal set, and dynamic responses of original and optimized designs are compared. The results show the optimized design has better dynamic responses with minimum outer diameter and the response decay decreases faster. The optimization procedure is feasible to the design of gear train, and this study can provide guidance for designer at the preliminary design phase of mechanical structures with gear train.
1. Introduction
Gear trains are widely used in mechanical engineering for advantages of compact structure, high reliability, and large power transmission. However, the design of gear trains is a complex process, and the traditional design process of gear trains depends on the designer’s intuition, experience, and skills, which is not satisfactory to the increasing demands for compactness, efficiency, and reliability in engineering application. Therefore, the optimization for gear trains has been a necessary process to solve the above problems at the preliminary design phase of gear trains, and many different optimization techniques have been reported in the literatures on gear trains.
The sequential quadratic programming (SQP) method was employed, respectively, by Bozca [1] and Huang et al. [2] to obtain a light-weight-gearbox structure by optimizing the geometric parameters of the gearbox. Chong et al. described a method for reduction of geometrical volume and meshing vibration of cylindrical gear pairs while satisfying strength and geometric constraints using a goal programming formulation [3]. Based on the random search method, Zarefar and Muthukrishnan investigated the optimization of helical gear design [4]. Ciavarella and Demelio investigated the optimization of stress concentration, specific sliding, and fatigue life of gears with numerical methods [5]. Huang et al. developed an interactive physical programming in order to optimize a three-stage spur gear reduction unit [6]. A Random-Simplex optimization algorithm was developed by Faggioni et al. for gear vibration reduction by means of profile modifications [7]. Based on min-max method combined with a direct search technique, Abuid and Ameen had done the optimization problem containing seven objective functions: gear volume, center distance, and five dynamic factors of shafts and gears [8]. Thompson et al. optimized minimum volume and surface fatigue of multistage spur gear reduction units by employing quasi-Newton method [9].
The mentioned optimization algorithms above are cataloged as conventional optimization techniques. Though they are efficient for some optimization problems in application, difficulties still exist in tackling some special problems with noncontinuous variables, complex constraints, and strongly nonlinear objectives. Therefore, some modern optimization methods such as genetic algorithm (GA) have been proposed to solve such problems in gear trains. By using the genetic algorithm, Mendi et al. investigated optimization of the modulus of spur gears, the diameters of shafts and rolling bearing [10]. Chong and Lee presented a design method to optimize the volume of two stage gear trains by using the genetic algorithm, which shows that the genetic algorithm is better than other conventional algorithms for solving the discrete, intege (...truncated)