Optimization of the steady state parameters for machine tool gear trains

Abstract

The subject of this article is optimization of the construction of multishafted machine tool gear trains, consisting of double shaft gear transmissions with geometric progression of the output shaft speeds. The complexity of the problem and the large number of significant variables concerned, has led to the application of logic programmes, as this is the only known way to formulate economically based methods, providing that Digital Computers are employed. The following formulation of the logic principle has been developed. Should an optimal structures set be determined, the scope of guessing for the optimality constructional solution is to be limited to considerations of structural schemes within the determined set of structures and corresponding to the said structures. Hence, instead of a complete problem of statical optimization, the two following stages are considered separately—optimization of structures and investigations of structural schemes; optimization of the constructional solution; and, in the second stage of investigation, only a set of solutions obtained from the first stage of considerations are to be investigated. The mathematical model, the method of solution for optimization of the structures and considerations of structural schemes of the gear trains of single and multispeed ratios of the output shaft has been presented in the paper. In addition a logic method suitable for obtaining a solution of the mathematical model has been developed, taking into account the fact, that the algorithmic method is of unprofitable use in the digital computer in this case. As an optimality criterion for structure optimization, the minimal number of gears and couplings for creating direct kinematic ties between two shafts has been assumed so as to ensure a given set of speeds of the gear train output shafts. As a result of the first stage investigations the structure or set of optimal structures is obtainable as well as sets of structural schemes corresponding to the said structures. Results so obtained are the subject of consideration in the second stage, i.e. the stage of optimization of the constructional solution, in other words the stage of determination of optimal values for the principal constructional data of the gear train (moduli, number of teeth, materials, breadth of gear rims, materials and diameters of the shafts). The solution of the problem in the second stage of investigation is based on the analytic method of the optimization problem, i.e. the method which breaks down a given problem so that by investigation of partial problems the solution of the primary problem is obtained. Division for sub-problems has been made as required by the structure of the gear train, in other words, the partial problems are the problems of optimization of double shaft assemblies. Also the overall problem has been developed so that by coordination of the particular problem of optimization of the double shaft assemblies an optimal solution of the gear train as a whole is obtained. The mathematical model is presented in the article as well as the method of solution of the overall problem. Also an example of the mathematical model and the method of partial solution for the optimization of a double shaft gear shift assembly is presented. This approach to the statical optimality problem of machine tool gear train construction enables development of effective programmes for digital computing such that the range of optimality problems concerned cover requirements in practice. The programmes are recorded in FORTRAN IV language and checked in DC System 4–70 of ICL production. The programmes permit considerable reduction in designing time for the optimal gear trains, and in more complex cases the designing time may be reduced to several dozen hours including 1–3 hr of operation of DC, System 4–70 of ICL production. Should display arrangements be used, the DC operational time may furthermore be shortened and the designing process may be carried out on the basis of talk between designer and the DC.