Basic mathematical principles for internal structure of new CAPP software application.

Monka, Peter ; Monkova, Katarina

1. INTRODUCTION

The objects in machine engineering as are the parts, machines, equipments and other is possible to model on the various stages with various goals. Every of these objects can be considered as system, which consists of other features. If it is possible to express the object property as multiple the basic units, we can speak about numeric properties. The basis of numeric formulation of numerical properties is an agreement by means of that we determine the basic unit. Some properties we don't know express by the mode listed above, and then we talk about non-numerical properties. On the other hand according to special agreement we can formulate non-numerical properties by numbers, too. It can be said that properties were formulated by numeric code. Apart from numeric code it can be used alphabetical or alphabetic-numeric codes. (Paholok, 2008)

As smallest unit of model is considered feature indicated by small cursive letter (with or without index), for example ". Features with same characteristics can be grouped into set. Set is defined, if about all its features can be decided whether they have specific property that is whether they belong in set. The sets are indicated by capital letters "[A.sub.1]". There are relations between sets. Some of them are shown on Fig.1. (Cerna, 1987)

[FIGURE 1 OMITTED]

We can meet with numerical expression in the machine engineering (e.g. at the part dimensioning, where conventional unit is millimetre), but we meet with non-numerical expression, too (for example at the coding of part in group technology).

2. INTERNAL MATHEMATICAL STRUCTURE

Data obtained during manufacturing process preparation are arranged to process plan by technologist what would ensure most profitable manufacturing manner of part according to choices criteria in specific conditions and at the same time they have to be suitable for technical requirements from the view of correct part function. (Lukovics & Sykorova, 2008)

It is necessary to localize and to separate a lot of data in the processing, therefore it is useful the manufacturing system consider to be set, which is unification of subsets marked as subsystems. (Fig. 2) (Bekes & Andonov, 1986) The created system can be mathematically expressed by the relation:

MS = S [union] E [union] O (1)

MS--manufacturing system,

S--Segment,

O--Operation,

E--Equipment.

For example set Equipment can be mathematically described:

E = MS [union] EP [union] ME [union] MTE [union] TE [union] SE (2)

E--Equipment,

MS--Manufacturing Spaces (halls, workshops ...),

EP--Equipment for Power production and distribution,

ME--Manufacturing Equipment,

MTE--Measuring and Testing Equipment,

TE--Transport Equipment,

SE--Storage Equipment and other.

All of these sets are union of subsets. For example the set ME can be written down as:

ME = M [union] T [union] F [union] AME [union] EMO [union] AE (3)

M--Machines,

T--Tools,

F--Fixtures,

AME--Auxiliary Manufacturing Equipments,

EMO--Equipments for Manual Operation shops,

AE--Assembly plants Equipments.

[FIGURE 2 OMITTED]

The set Machines can be devided according to technology type as follows:

M = MM [union] CM [union] WM [union] AM [union] MoM (4)

M--set of all plant Machines

MM--Machining Machines

CM--Casting Machines

WM--Welding Machines

AM--Assembly Machines

MoM--Moulding Machines

In the relation to the Machining machines:

MM = [M.sup.O1] [union] [M.sup.O2] [union] [M.sup.O3] [union] [M.sup.O4] [union] [M.sup.O5] (5)

[MM.sup.01]--machining machines

[M.sup.01]--turning machines,

[M.sup.02]--milling machines,

[M.sup.03]--drilling and boring machines

[M.sup.04]--grinding machines and other

Production plant disposes by the certain number of turning machines that are the features of the set [M.sub.01]:

[M.sup.01] = {[m.sub.1], [m.sub.2], [m.sub.3], [m.sub.4] ... [m.sub.i] ... [m.sub.n]} (6)

[m.sub.i]--is i-th turning machine and

n--is whole number of turning machines in the plant.

Individual turning machines are characterized by properties, which define the machine selection for turning of concrete part. In an exemplary way: for the machining too long shafts are suitable centre lathes, etc. The relationship between turning machines and its properties can be written down by functional dependency:

[M.sup.01] = f([p.sub.1], [p.sub.2], [p.sub.3], [p.sub.4], [p.sub.5], ... [p.sub.j]) (7)

[M.sup.01]--set of all plant turning machines

[P.sub.j]--j-th property of turning machine

[p.sub.1]--lathe type

[p.sub.2]--control system

[p.sub.3]--tool equipment

[p.sub.4]--dimensions

[p.sub.5]--energy and presission

If the examined object has more than one substantial property from the view of solved task goals, then feature set, which contains lathes suitable for turning operation according to selected property, can be obtained by partial derivation:

[partial derivative][M.sup.01]/[partial derivative][p.sub.1] = [M.sup.O1.sub.1] (8)

Simultaneously holds [M.sup.O1.sub.1] [subset] [M.sup.01], where

[M.sup.O1.sub.1] is set of turning machines with some features--for example {[m.sub.1], [m.sub.3], [m.sub.4], [m.sub.9] ...}, which satisfy request for machine selection according to specific property [p.sub.1] (e.g. fitness for very long part machining)

Similarly, other sets [M.sup.O1.sub.j] that originate by [M.sup.O1] differentiation with respect to j-th property can be obtained

[M.sup.O1.sub.j] = [partial derivative][M.sup.01]/[partial derivative][p.sub.j] (9)

Simultaneously for every [M.sup.O1.sub.j] holds [M.sup.O1.sub.j] [subset] [M.sup.O1].

Every of these subsets include various features, alternatively said individual requests [p.sub.j] satisfies always different turning machine combination.

Set of turning machines that conformed to all requests can be found as intersection of sets

[M.sup.O1.sub.1] [intersection] [M.sup.O1.sub.2] [intersection] [M.sup.O1.sub.3] [intersection] [M.sup.O1.sub.4] [intersection] [M.sup.O1.sub.5] ... = [M.sup.O1.sub.opt] (10)

Follow 3 cases can occur for [M.sup.O1.sub.opt] : (Monkova & Hloch, 2008)

1/ It is one-feature set, what means: if all conditions are fulfilled, concrete part is possible to turn only on one suitable machine.

2/ Set has several features, what means, that it is necessary to choose another criteria for the selection of machine (e.g. it is necessary to find out, which of the machines is free for the machining in this time).

3/ Set is empty. In this case it is needed to leave out one of the machine choice according to some of properties (nearly always it is the property that influences selection process at least) or the situation has to be solved by another manner (e.g. by producing of selected part in alternative plant; or by buying of new machine with needed parameters).

The principle of this simplified model is applicable not only for lathest or other machines but also for next subsets of Equipment subsystem, even for subsystems Segment and Operation, too.

3. CONCLUSION

Mathematical structure listed above was usesd at the suggestion of new software application at TU Kosice, Faculty of Manufacturing Technologies with the seat in Presov. It was generated on the strength of data flow analysis and synthesis in small and medium plants. New software application was built so as all problems could be solved logical by means of computer from the view of large volume data searching and processing. Mathematical basis enables to generate new coding system for objects of manufacturing system and consequently it enables to select parts with similar material and size characteristics, which will be process by affined manufacturing process plans. This approach has an impact on batch time and so on whole plant economy. It allows increasing of effectivity already at the beginning of its design and to improve the process of technological documentation creation without of the influence on its complexity. Generated application is built by modular manner to allow flexible adapt data structure to user specific conditions and to satisfy the specification of simple implementation into already existing information structure of the plant. The output data of the system is able to utilize not only for the generating of technological documentation but also to the processing of details for manufacturing, store, economic and wage records, thereinafter for the creating and archiving of NC programs and for the data registration, too. It is assumed the practical verification of the final product in real conditions of manufacturing plants.

4. REFERENCES

Bekes, J. & Andonov, I. (1986). Analysis and synthesis of machining objects and processes, 60-080-86, ALFA Bratislava, Slovakia

Cerna, R. (1987). Basis of numerical mathematics and programming, 04-003-87, SNTL Praha, Czech republik

Lukovics, I. & Sykorova, L. (2008). Materials machined by laser. In: IX. International conference NTMT, ISBN 978-80-553-0044-3, p.73-76, Presov, Slovakia

Monkova, K. & Hloch, S. (2008). Coding of manufacturing system objects within IAV software, Manufacturing Engineering, vol. 7, no. 1 p. 65-69, ISSN 1335-7972

Paholok I. (2008). Simulation as scientific method, Electronic journals for philosophy, Prague, 08/2008, p. 3-18, ISSN 1211-0442