Contributions on final elaboration model of a feed kinematic chain (L.C.A.).

Enciu, George ; Nicolescu, Adrian ; Stanciu, Mihai 等

1. INTRODUCTION

Within the framework of a contract of the Machines and Production Systems (MSP) Department, from the Engineering and Management of the Technological Systems (IMST) Faculty, we had to determine the model of a feed kinematic chain for a machining centre for milling and reaming operations (Breaz et al., 2007; Ispas et al., 1999; Neugebauer et al., 2007). The experimental stand is presented in Figure 1 and Figure 2 and has the same mechanisms as the machining centre (Bausic & Diaconu, 2000; Zaeh & Baudisch, 2003).

2. IDENTIFICATION OF TRANSFER FUNCTION OF THE MECHANICAL STRUCTURE

The experimental researches were at different stages and complexity levels and, of course are impossible to be here presented. We present only some aspects about the identification of the transfer function.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

One of the reasons of this research was to collect the information necessary to apply an identification method.

For determination the transfer function of the frequency characteristics it was used the method of the smallest ponderous (balanced) squares

Assuming that the process it was described by its frequency characteristics:

([j.sub.wk]) = [Re.sub.(wk)] + [jIm.sub.(wk)]; k = 0, ... p (1)

it was trying the approximation of frequency spectrum that it was determinated with the model help

[M.sup.(s)] = [[summation].sup.a.sub.k=0] [A.sub.K] x [S.sup.k]/1 + [[summation].sup.b.sub.i=1] [B.sub.i] x [s.sup.i] = A(S)/B(S); S = Jw (2)

To minimize the error occurred between the real frequency characteristics and the model characteristics the matrix equation should be solved

V x X = W (3)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

In the relations above have been used the notations:

[S.sub.x] = [[summation].sup.p.sub.k=0] [w.sup.x.sub.k] (5)

[T.sub.x] = [[summation].sup.p.sub.k=0][[Re.sub.(wk)]] x [w.sup.x.sub.k] (6)

[U.sub.x] = [[sumamtion].sup.p.sub.k=0][[Re.sup.2.sub.wk]) + [Im.sup.2.sub.(wk)]] x [w.sup.sub.k] (7)

and the index value [r.sub.1], ... [r.sub.2] has been determined with the relations;

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

For application of the method of smallest balanced squares we need to know the grades a and b of the polynoms of the numerator and respectively the denominator of the transfer function. Because this is a rare situation, the respective method is applied for more combinations of grades a and b and we have to choose that combination that approximates the best the frequency spectrum determined experimentally.

[FIGURE 3 OMITTED]

For the aplications of smallest methods squares has been realised with a virtual instrument (.vi) in graphic programming Lab VIEW environment. The program required the existence data file contains the value R([[omega].sub.k]) and Im([[omega].sub.k]).

In the file of entrance each of two groups of date must be occupy one column, and the dates from same line must be sepparated with special character TAB, have been the ASCII code.

The previously mentioned programme was rolled successively for more combinations of grades a and b of the transfer function polynoms, the resulted values of A and B coefficients for the combinations used are presented in Table 2.1.

The comparation with the identified model (Fig.3.), through the Nyquist diagram has not led to the selection of any variant and that to be proceeded to extend the grade and we have obtained a series of values for the coefficients.

[H.sub.mt](s) = [U.sub.t](s)/[U.sub.m](s) = [c.sub.t] x [c.sub.m] x (s x [[xi].sub.12] + [k.sub.1]/[absolute value of [DELTA]]/s (9)

Where, [absolute value of [DELTA]] is the matrix determinative from (9) relation so the computation of [absolute value of [DELTA]] leeds at:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

If for easier calculations we note R + L=V we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Replace:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

It's obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

According of structure 1 .c.a. moulded from c.c. engine mechanical--stand structure, the total function will be:

[H.sub.t](S) = [H.sub.mt](s) x [H.sub.ids](s)/[H.sub.mt](s) x [H.sub.ids](s)+1 (14)

Made the replacement and adequate calculation have been obtained the final transfer function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

3. CONCLUSIONS

The conclusions of the experimental and research activity integrate in the procedures which have become well-known for the elaboration of the model and for the transfer function of a positioning LCA for machine-tools.

There have been used dynamic testing methods to determine the stability diagrams and through identification the elaboration of a real transfer function. It is to be mentioned that the transfer function obtained through identification has a total different degree and shape than the standard ones used in the theory of the automated systems theoretically allocated to LCA.

We also mention that the stability diagrams obtained from the final transfer function obtained through identification have a behaviour degree in comparison with the stability diagrams obtained in the intial phase, which are between 5-7%.

The developed researches will allow for our department the determination of the transfer functions for positioning LCA through identification method.

4. REFERENCES

Bausic, F. & Diaconu, C. (2000) Dinamica masinilor (Machines dynamics), Conspres Publishing, ISBN: 973-99571-3-7, Bucharest

Breaz, R.E., Bologa, O.C., Oleksik, V.S. & Racz, G.S. (2007). Computer Simulation for the Study of CNC Feed Drives Dynamic Behavior and Accuracy, EUROCON, 2007. The International Conference on "Computer as a Tool", pp. 2229-2233, ISBN: 978-1-4244-0813-9, Warsaw, September 2007.

Ispas, C., Mohora, C. & Caramhai, S., (1999) Simularea sistemelor integrate de fabricate (Simulation of integrated fabrication systems), Bren Publishing, ISBN: 973-9493-157, Bucharest

Neugebauer, R., Denkena, B. & Wegener, K. (2007). Mechatronic Systems for Machine Tools, CIRP--Annals Manufacturing Technology, Vol. 56, Issue 2, pp. 657-686, ISSN: 0007-8506, Imprint ELSEVIER.

Zaeh, M F & Baudisch, T., (2003). Simulation environment for designing the dynamic motion behaviour of the mechatronic system machine tool, Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, Vol. 217, No. 7, (2003), pp. 1031-1035, ISSN 0954-4054, Professional Engineerng Publising.

Tab 1.

 a=2,b=3 a=2,b=4 a=2,b=5 a=2,b=6

B2 1.39e-07 2.3e-07 2.3e-07 3.34e-07
B4 8.55e-15 8.55e-15 2.99e-07
B6 7.56e-22
B1 6.93e-05 4.04e-05 2.72e-05 8.06e-06
B3 9.46e-12 5.51e-12 2.49e-12 7.38e-13
B5 -1.21e-19 -3.57e-20
A0 -3.16e-06 -3.02e-06 -2.35e-06 -8.37e-06
A2 -1.02e-12 -1.32e-13 -1.00e-13 -4.56e-13
A1 1.79e-05 1.04e-05 1.05e-05 [3.10.sup.e]-06