Fundamental features of technological systems rigidity.

Tonoiu, Sergiu ; Catana, Madalin

1. INTRODUCTION

Because the rigidity, characteristics of rigidity and some entities related to technological system rigidity are not unitary and generally defined in the literature (see Tonoiu, 1999, and Weck et al., 1989), several proposals concerning these topics were presented by Tonoiu & Doicin, 2002; Tonoiu et al., 1997a; Tonoiu et al., 1997b. This paper develops the previous researches of the authors.

2. PRELIMINARIES

Study method characteristics. Besides the essential phenomenological parameters, the definition and theoretical or/and experimental determination of an entity are influenced by the study method characteristics, such as: calculus /modeling hypotheses, measurement instrumentation accuracy, etc.

Technological system structure. A technological system, TS, is defined as a reunion of physical entities, [E.sub.i], constructive-functional interdependent, i.e.: TS = [union] [E.sub.i].

Let D(STR) represent the set of the defining characteristics of the technological system structure, i.e. characteristics referring to the system type, entity type, etc.

Technological system state. It is being considered that the state of a technological system or of its component entities is type of: rest; quasi-rest; passive functioning; active functioning.

Let D(STA) represent the set of the defining characteristics of the technological system state, i.e. characteristics referring to the state type, working parameters, etc.

References. It is being considered that the references associated to technological systems are of type of: physical or geometrical reference; absolute or relative reference. Physical references are of type of guide, fixed centre, etc. Geometrical references are commonly of type of Cartesian coordinate system.

Let RPR and RGR represent a relative physical reference and a relative geometrical reference associated to a given technological system (Fig. 1), so that: RPR [equivalent to] [E.sub.r], RGR [equivalent to] Oxyz.

Let D(REF) represent the set of defining characteristics of the references of a technological system, i.e. characteristics referring to the reference type, physical entity being reference, etc.

3. LOADINGS AND DEFORMATIONS

Loadings. Loading is a quantity of type of force, moment or pressure. The forces are of fixing, of machining, of excitation etc.; the moments are of type of bending moment or torsion moment; the pressure is acting because of a corresponding force etc. Let [bar.F] represent one loading or a resultant of more loadings.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The [bar.F] loading is applying on an element, [E.sub.f], [E.sub.f] [member of] TS (Fig. 1). Let [A.sub.f] represent a point, [A.sub.f] [member of] [E.sub.f], which is identical with the application point of a force or is participating to define the acting space of a moment or pressure; the position of [A.sub.f] is defined e.g. by its positional vector, [[bar.r].sub.f], (Fig. 1).

The direction of [bar.F] can be indicated in a pre-established manner. Let F represent magnitude of the [bar.F] loading, i.e. F = [absolute value of [bar.F]].

It is defined as being loading--time function the relationship between the loading magnitude, F, and the time, T, i.e.: F=F(T).

Dependent on the intensity of the F variation in time, the loading is static, [bar.F]st, or dynamic, [bar.F]dyn, if the F variation in time is "slow" (Fig. 2a) or "rapid" (Fig. 2b), respectively; a dynamic loading is occurring with a certain "frequency". Dependent on the monotony type of the F variation in time, a loading is of type of charging, constant or of discharging, if the F(T) function is increasing, constant or decreasing, respectively (see Fig. 2).

Let D([bar.F]) represent the set of the defining characteristics of the [bar.F] loading, i.e. characteristics referring to the application point, direction, etc.

Deformations. It is considered as being deformation a quantity of type of: elastic deformation, plastic deformation or displacement; linear deformation or angular deformation. Let [bar.U] represent one deformation or a resultant of more deformations. The [bar.U] deformation is considered to be of a point [A.sub.u], belonging to an entity, [E.sub.u], of the considered technological system. The position of [A.sub.u] is defined e.g. by its positional vector, [[bar.r].sub.u], (Fig.1). Thus: [E.sub.u] [member of] TS, [A.sub.u] [member of] [E.sub.u]. It is to be noticed that: [E.sub.u] [equivalent to] [E.sub.f] or [E.sub.u] [not equal to] [E.sub.f]; [A.sub.u] [equivalent to] [A.sub.f] or [A.sub.u] [not equal to] [A.sub.f].

The direction of deformation can be indicated in a pre-established manner.

Let U be the magnitude of the [bar.U] deformation, i.e.: U = [absolute value of [bar.U]].

[FIGURE 3 OMITTED]

Deformation-time function is the relationship between the deformation magnitude, U, and the time, T, i.e.: U=U(T).

Dependent on the intensity of the U variation in time, the deformation is static, [bar.U] st, or dynamic, [bar.U] dyn, if the U variation in time is "slow" (Fig. 2 a) or "rapid" (Fig. 2 b), respectively; a dynamic deformation is producing with a certain "frequency". Dependent on the monotony type of the U variation in time, a deformation is of type of charging, constant or discharging, if the U(T) function is increasing, constant or decreasing (see Fig. 2).

Let D([bar.U]) represent the set of defining characteristics of [bar.U].

4. LOADINGS-DEFORMATIONS RELATIONSHIPS

The useful relationships are of type of loading--deformation function/curve and deformation--loading function/curve, as follows.

The loading--deformation function is the dependence relationship between the loading magnitude, F, and the deformation magnitude, U, i.e.: F=F(U).

The deformation--loading function is the dependence relationship between the deformation magnitude, U, and the loading magnitude, F, i.e.: U=U(F).

In the case of static loadings and deformations, [bar.F]st and [bar.U]st, respectively, the F(U) and U(F) functions could be nonlinear or linear (Fig. 3), in correspondence with the deformation type, as follows: the F(U) and U(F) functions which are nonlinear (Fig. 3, curves 2, 5, 6, 7, 10) or linear not constant (Fig. 3, curves 1, 3, 9) correspond to different deformation types--elastic, plastic, displacement (see [section] 3.2); a constant F(U) function and a variable U(F) function (Fig. 3, curves 4, 11) correspond to deformations of type of displacement caused by clearances from joints; a constant U(F) function and a variable F(U) function (Fig. 3, curves 8, 12) correspond to "not deformable" structure--in the considered conditions.

5. RIGIDITIES AND COMPLIANCES

In numerous theoretical and applicative developments, two important specific system characteristics are of major interest: rigidity and compliance.

Rigidity is the characteristic that expresses the intensity of loading--deformation relationship. Thus, quantitatively, the rigidity, K, is the ratio between the loading magnitude variation, [DELTA]F, and the deformation magnitude variation, [DELTA]U, i.e.: K=[DELTA]F/[DELTA]U, [DELTA]U[not equal to]0.

Compliance is the characteristic that expresses the intensity of deformation--loading relationship. Thus, quantitatively, the compliance, C, is the ratio between the deformation magnitude variation, [DELTA]U, and the loading magnitude variation, [DELTA]F, i.e.: C=[DELTA]U/[DELTA]F, [DELTA]F[not equal to]0.

In the case of loadings and deformations for which F(U) and U(F) are nonlinear, let ([F.sub.j];[U.sub.j]) and ([F.sub.1]; [U.sub.1]) represent two different couples/points associated to different states, j and l, j[not equal to]1, of the system.

There are being defined the relative rigidity, Kjl, and the relative compliance, [C.sub.j1], as:

[K.sub.jl] = [F.sub.1]-[F.sub.j]/[U.sub.1]-[U.sub.j], [C.sub.j1] = [U.sub.1] - [U.sun.j]/[F.sub.1] - [F.sub.j], [F.sub.1] [not equal to] [F.sub.j], [U.sub.1] [not equla to] [U.sub.j] (1)

In a point ([F.sub.1], [U.sub.1]) where F(U) and U(F) are differentiable, there may be defined the instantaneous rigidity, Kl, and the instantaneous compliance, [C.sub.1], as: [K.sub.1] = [(dF/dU).sub.1], [C.sub.1] = [(dU/dF).sub.1].

An important particular case is when, for a well defined [bar.F] [phi] loading--where [phi] is associated to the [bar.F] direction, the [bar.U] deformation is of unknown direction. In such of case, let [[bar.U].sub.[lambda][phi]] represent the components of [bar.U] parallel to the axes of Oxyz reference, [phi] and [lambda]= x,y, z. Thus, [K.sub.[lambda][phi]] and [C.sub.[lambda],[phi]] expressed on the basis of the definition relationships, are:

[K.sub.[lambda][phi]] = [DELTA][F.sub.[phi]]/[DELTA] [U.sub.[lambda][phi]], [C.sub.[lambda][phi]] = [DELTA][U.sub.[lambda][phi]]/ [DELTA][F.sub.[phi]], [phi] and [lambda]=x, y, z (2)

6. MATRICES OF RIGIDITIES AND COMPLIANCES

For each representative couple of points ([A.sub.f], [A.sub.u]), [[bar.F].sub.[phi]] loading and [[bar.U].sub.[lambda][phi]] deformation there may be defined the matrix of fundamental rigidities, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the matrix of fundamental compliances, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For example, the matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

7. CONCLUSION

Defining fundamental elements have been introduced referring to study method, structure, state, references, deformations, rigidity and compliance associated to technological system. The considered elements have been developed until a level that assures a high degree of generality.

The assembly of entities proposed by this paper represents an important theoretical basis for new interpretations and developments of standards, determination methodologies and of the other matters linked to the technological system rigidity or compliance.

8. REFERENCES

Tonoiu, S. & Doicin, C.V. (2002). Technological manufacturing systems' (TMS) rigidity, Proceedings of the 4th workshop "Human Factor and Environmentalist", Katalinic, B. (Ed.), pp. 101-102, ISBN 3-901509-37-2, Kosice-Slovakia, December 2002, DAAAM International, Vienna

Tonoiu, S. (1999). Contributions on the study of machining technological systems rigidity, Ph.D. Thesis, POLITEHNICA University of Bucharest, 1999, Romania (in Romanian)

Tonoiu, S.; Dulgheru, L.; Catana, M. & Purcarea, M. (1997a). Methods for experimental determination of static rigidity for machining technological systems, Proceedings of the 9th International Conference on Machine Tools, pp. 501-508, ISBN 973-31-1139-2, Bucharest-Romania, 1997, Ed. Tehnica, Bucharest (in Romanian)

Tonoiu, S.; Purcarea, M. & Catana, M. (1997b). Considerations on contact rigidity and dumping in machining technological systems, Proceedings of the 9th International Conference on Machine Tools, pp. 509-516, ISBN 973-31-1139-2, Bucharest-Romania, 1997, Ed. Tehnica, Bucharest (in Romanian)

Weck, M.; Eckstein, R & Schafer, W. (1989). Methods for determination of machine tool static rigidity, Mechanik, No. 4, 1989, pp. 125-129 (in Polish)