The Reliability Optimum Design Of The Gear Transmission Of Ball Mills Based On Kriging Model

Ball mill is one of the most universal machines widely used in metallurgy, cement, construction materials and chemical industries. Large scale is the overhead developing trend of ball mills, and generally they are called large scaled ball mills when their diameters are above 4 meters. Those ball mills whose powers are bellow 10000 kW. are commonly drived by bevel gears. Thus, how to improve the reliability of gear transmission is one of the key technologies in large scaled ball mills. As a lot of parameters are random variables in gear transmission design, reliability optimum design methods are used to improve the driving reliability and to reduce the total volume of gears at well.Mechanical reliability is calculated by the stress-strength theory. There are three ways to calculate the reliability: probabilistic approach, joint density function, and by Monte Carol random simulations.The stress and strength are usually functions of some random variables. For those random variables whose distribution is known, there are three ways to determine the mean and standard error: general function method, Taylor expansion method and variation coefficient method. Generally speaking, general function method is used when the functions are simple and the Tailor expansion method or variation coefficient method is used when the functions are complicated. The variation coefficient method is especially applicable when the stress or strength is functions in the form of product of many random variables.The tooth surface contact fatigue strength and the tooth root bending fatigue strength were mainly considered in the reliability of gear transmission. Improving the reliability of contact fatigue strength in order to avoid the occurrence of pittance and improving the reliability of bending fatigue strength in order to avoid the occurrence of breaking up. Both the stress and strength of gear are considered in lognormal distribution, and variation coefficient method is suited to determine the mean and standard error. The reliability can be calculated by joint density function method once the mean and standard error are known.At the same time, one can use Monte Carol random simulation method to verify the accuracy of the results. The basic thought of Monte Carol method is that establishing a probability model and then doing some digital experiments. The total sum is computed as the solution of the problem.After the gear transmission reliability calculation of theφ5.5m×8.5m ball mills, we found that the reliability of contact fatigue strength is too small to generate pittance. The bearing capacity is then falling frequently. This situation matches the practice. It's needed to improve the reliability and to minimize the total volume through reliability optimum design. We established the mathematical model of the reliability optimum design. The modulus, the tooth number of small gear, the wideness and the helical angle are the four design variables, and the object function is a combination of the total volume of the gear and the reliability.Considering the complexity of the object function, this thesis used Kriging model for approximation and the Bayesian analysis algorithm is used to find the global design optimization. The basic principle of Bayesian analysis algorithms is: (1) Obtain a set of initial samples of the true functions through Latin Hypercube Sampling and then fit a Kriging model to each function; (2) Numerically maximize a infill sampling criterion known as expected improvement function to determine where to sample next; (3) Sample the points of interest on the true functions and update the Kriging models. As iterations going, more and more sample points were obtained and the Kriging models are much similar to the true functions. At the same time, the optimal solution gradually converged to the global optimal solution.Other methods such as genetic algorithms or gradient-based algorithms require very little computational effort in determining where to evaluate the functions next. However, they require a large number of function evaluations to converge on a good solution. The benefit of the overhead of Bayesian analysis algorithms is that each iteration uses as much information as possible in determining where to evaluate the functions next, enabling them to locate good solutions with fewer iterations. This makes the Bayesian analysis algorithms best suited to situations where the functions are expensive, and the designer cannot afford to perform a large number of function evaluations.This thesis used Bayesian analysis algorithms based on Kriging model to solve the gear transmission reliability optimum design problems ofφ5.5m×8.5m ball mills. Good parameters of gear transmission were attained which could improve the reliability and minimize the volume at the same time.