Determination of upset ingot dimensions during computer-aided forging design under fuzzy goals

The paper describes an algorithm for determining reasonable dimensions of an upset ingot during computer-aided design of shaft press-forging under fuzzy goals. Fuzzy goals related to product metal elaboration quality and the complexity of its manufacture are formulated. Objective values of a design parameter for each goal are determined. Membership functions for possible solutions corresponding to these goals and the rule of choosing the most reasonable solution are constructed. Two examples of solving the problem depending on the selected goals are given. The proposed algorithm allows one to choose the most reasonable solution from a set of feasible ones and reduces human participation in the computer-aided design process. The developed approach can be applied to solving other problems of press forging, such as choosing an ingot, determining the necessity of the billet upset operation, determining the cogged billet dimensions, etc.

The computer-aided design of press-forging technology is a rather complicated process owing to the poorly formalized subject area. As a result, different technologies can offer different solutions for the same forged part depending on stated goals, even within the same enterprise. Such goals include, for example, reasonable metal saving, low labour consumption for forged part production, etc. In this case, the words "reasonable", "low", understandable to man, are fuzzy in terms of algorithms and programs, and they require special methods for their formalization. Therefore, the computer-aided design of a technological process is performed under conditions of fuzzy goals and limitations.
Due to the lack of subject area formalization, developers of computer-aided process planning (CAPP) systems, on the one hand, have to pay much attention to the interactive communication of users with the system [1][2][3]. On the other hand, the natural desire of the developers is to reduce the share of human participation in the computer-aided design process. Therefore, while creating CAPP systems, much attention has always been paid to improving their intellectual level by using various methods of artificial intelligence [4,5]. The theory of decision making under fuzzy conditions, according to the Bellman-Zadeh scheme [6], is promising for these purposes.
Fuzzy logic is used to solve a variety of problems arising during a design process. The study [7] described how the fuzzy modelling theory helps to resolve contradictions within a design process. The advantage of the fuzzy approach to computer-aided design of a forged part was demonstrated in [8]. Expert computer systems of technology design based on fuzzy logic, including those for cold stamping, were described in [9,10]. Those studies emphasized that design uncertainty is associated with the subjectivity of decision making and that it is advisable to use fuzzy logic to obtain final decisions.
In general, the technological process of press forging includes the following main stages. Ingot selection. A minimum-weight ingot is selected. This allows one to produce a required forged part taking into account inevitable technological losses of metal.
Preforging. Preforging includes rounding of the cast ingot (the resulting product is termed a billet), billet upsetting (upset billet) and cogging into a blank with a circular cross section.
Final forging. Sequential forming of the circular blank into the final forged part. Figure 1 exemplifies a fragment of a process sheet for shaft forging.
Preforging Billet  Thus, it is necessary to be guided by certain principles of rationality in terms of the total design process problem for the determination of a specific value of

Upset billet Cogged round billet
. At the same time, the selection of reasonable values for the process parameters from the intervals of their feasible values depends on the stated goals.
A concept of choosing a reasonable decision during computer-aided design of press forging under conditions of uncertainty and numerous pursued goals was described in [12]. This concept is based on the theory of decision making according to the Bellman-Zadeh scheme [6]. In this paper, the concept is concretized for solving the problem of the determination of reasonable upset billet dimensions during preforging (see figure 1).
While forging on presses, the main indicator of metal elaboration quality is the amount of strain in the final forged part, which is generally evaluated by the expression mm. The analysis of the existing generally accepted recommendations, enterprise instructions, and forging process sheets allow one to formalize requirements imposed on the upset billet dimensions and to write them in the form of the inequality system It should be noticed that, although the first restriction in the system of inequalities (1) corresponds to the generally accepted recommendations, the analysis of forging process sheets for real shafts, especially for critical forgings, shows that forging technology developers generally tend to set the amount of strain in a final forged part with some reserve, for example, 4 . They consider that, under this condition, the product metal will be elaborated with guaranteed quality. It is easy to see that the desire to increase the amount of strain in the upset billet, as well as in the final forged part, i.e. the desire to increase the value of D , is limited by the third inequality in system (1). This condition is interpreted as follows. An increase in D improves the quality of metal elaboration, but it degrades the manufacturability of ingot upsetting and increases the complexity of subsequent ingot processing, since a larger volume of metal will have to be moved during subsequent forging operations. The latter may lead to microcracking in the metal.
The problem of determining reasonable upset billet dimensions is solved in steps.
Step 1. ( n is the number of goals) for the design of the upset billet and the determination of the value of its diameter for each goal. When the upset billet diameter reaches this value, the corresponding i-th goal is fully achieved.
Let us define three design goals 1 C , 2 C , and 3 C ( 3 = n ). Goal 1 C . The complexity of subsequent forging of an upset billet should be as minimal as possible. The smaller the upset billet diameter, the smaller the metal volume should be moved during subsequent forging, hence 970 C . The quality of metal elaboration in an upset billet should be as high as possible. It follows from the third inequality in system (1) that 1070 C . The quality of metal elaboration in a final forged part should be guaranteed to be high. It follows from the inequality 4 Step 3. |The determination of the membership functions 1 µ , 2 µ , 3 µ for the upset billet diameter D corresponding, respectively, to the stated goals 1 C , 2 C , and 3 C . The concept of membership functions was introduced in the theory of fuzzy sets [6]. For a fuzzy set, unlike classical sets, each element can belong to a set only partially. The degree of the element membership in a fuzzy set is characterized by the membership function, the values of which lie within the interval [0, 1]. When this value is equal to 1, it means that the element fully belongs to the set. When it is equal to 0, it means that the element is absent from the set.
According to [12], the membership function i µ of the parameter Y corresponding to the stated goal is taken as 2 min max After substitution of the values min Y , max Y , and C i Y obtained in steps 1 and 2 into (2), we arrive at three membership functions, where [ ] , when the goal 2 C is 4 times as important as the goal 3 C , and the goal 1 C is 5 times as important as the goal 3 C , is shown in figure 3. For unequally important goals, the desired value of the upset billet diameter changes to become equal to 1017 mm.

Conclusion
The algorithm for solving the problem of determining the reasonable upset billet diameter during computer-aided design of shaft press forging in terms of fuzzy goals has been proposed. Fuzzy goals have been related to the quality of product metal elaboration and the complexity of forging production. The objective values of the design parameter for each goal have been determined. Membership functions for possible solutions corresponding to these goals and the rule of choosing the most reasonable solution have been constructed. Two examples of solving the problem depending on the selected goals have been demonstrated. The proposed algorithm allows one to choose the most reasonable solution from a set of feasible ones and reduces human participation in the computer-aided design process. The developed approach can be applied to solving other problems of press forging,