Optimization of module, shaft diameter and rolling bearing for spur gear through genetic algorithm
Introduction
Conventional methods have been widely used in various mechanical design problems. They are deterministic in nature and use only a few geometric design variables due to their complexity and convergence problems (Chakraborthy, Kumar, Nair, & Tiwari, 2003). The main disadvantages of them are slow convergence along with local minima (or maxima) problems. When the number of design parameters increase, the complexity increases drastically. Many practical optimum design problems are characterized by mixed continuous-discrete variables, and discontinuous and non-convex design spaces. If the optimization problem involves the objective function and constraints that are not stated as explicit functions of the design variables or which are too complicated to manipulate, it is hard to solve by classical optimization methods. Therefore, some optimization methods such as genetic algorithm (GA) have been developed to solve complex optimization problems recently (Rao & Tiwari, 2007).
GA, which is one of the stochastic methods of optimization, has been commonly used for the optimal design machine systems. The main advantages of genetic algorithms (GAs) are an assured convergence without use of derivatives and functions with discrete and non-derivable variables (Marcelin, 2005). GA one of the optimization methods for solving complex optimization problems has been applied to many area including machine design. Rao and Tiwari (2007) applied GA to rolling element bearings design problem. Adeli and Cheng, 1994, Hasancebi and Erbatur, 2000 applied GA in structural optimization. Marcelin (2001) applied GAs for optimum design of gears. Choi and Yoon (2001) used to optimize automotive wheel-bearing unit using GA. Their study has presented maximized system life of the wheel bearing. Periaux (2002) discussed in detail the application of GAs to aeronautics and turbomachinery. Chakraborthy et al. (2003) investigated design optimization problem for rolling element bearings with five design parameters using GAs. Marcelin (2004) proposed a different numerical approach based on genetic algorithms and some neural networks allowing optimization of gearbox.
In this study selection of optimum module, shaft diameter and rolling bearing for spur gear has been carried out using genetic algorithms.
Rest of this paper is organized as follows. Section 2 presents briefly description of GA. Section 3 presents formulation of problem. Section 4 presents results of optimization problem. In Section 5, conclusions of this paper are presented.
Section snippets
Genetic algorithm
GAs are global optimization methods based on the principles of natural selection and evolutionary theory (Goldberg, 1989, Holland, 1975). The algorithm is provided with a set of possible solutions (represented by chromosomes) termed a population. Solutions from one population are taken and used to form a new population. This is motivated by a hope that the new population will perform better than its predecessors. Solutions chosen to form new solutions (offsprings) are selected based on their
Model formulation
Optimization of module, shaft diameter and rolling bearing selection is carried out by using GA with help of a program developed on Matlab 7.0 platform. Optimum dimensions are obtained for the design of the gearbox. Factors (torque, material, width of tooth, tip speed and so on) having influence on gear module selection are given in Table 3.
Results for optimization of gear volume
In order to end algorithm, all chromosomes in the population were converged to the best chromosome. Optimum fitness function and gear variables are illustrated in Fig. 3, Fig. 4, respectively.
In Fig. 3, the best, average and poor curves are illustrated in the search space. The fitness function is optimized when the average curve is converged to the best curve after thirty-fifth generation. Optimum variables are obtained in Table 7.
Results for optimization of shaft volume
The optimum fitness function and variables of shaft diameter are
Analytical method
To facilitate the evaluation of genetic algorithm, a program was been developed on Borland Delphi 6.0 platform. Solution space was constructed by all combinations of variables in the program. Selection parameters belong to gear and the obtained results are illustrated in Fig. 9. Solution space scanning interval in this window was selected in a way that it was same with the genetic algorithm variable increasing intervals. The results obtained by the analytic method (AM) and GA for gear volume
Conclusions
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Minimization of gear, shaft and rolling bearing volumes was accomplished using GA. Gear volume was found to be equal to 1099 dm3 using AM. Gear volume was equal to 1083 dm3 using GA. These results showed that GA is a better method than AM to obtain minimum gear volume.
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Module was found to be equal to 4.5 mm and 5.1 mm through GA approach and AM, respectively. The number of teeth was found to be equal to 25 and 22 through GA approach and AM, respectively. Width ratio was found to be equal to 21.02
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