On the analysis of a centrode type mechanismwith flexible and unextensible element.

Moldovan, Cristian Emil ; Perju, Dan ; Maniu, Inocentiu 等

1. INTRODUCTION

Mechanisms are designed for one of the following purposes: motion, path or function generation. With a classic four-bar linkage it is possible to impose a maximum of five precision points/positions. In comparison, with a centrode type mechanism there is a infinite number of precision points/positions which can be imposed. This statement is based on the theorem according with, a given curve can be reproduced by rolling whitout slipping of the other two curves, one of them being chosen conveniently so the imposed curve becomes a roulette curve. (Perju, 1971) The algorithm to determin the function between the input and the output element is presented.

A centrode type mechanism Fig.1 a). is composed of: 1- frame; 2-crank; 3- flexible and unextensible wire; 4-profiled wheel and it functions as follows: the wire 3 is winded and unwinded onto element 4, profiled wheel (Perju, 1987). The mechanism is equivalent to the instantaneous four-bar linkage OABD.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

A field where such mechanisms could be used is the liniarisation of a measuring scale where deviations are present. Fig. 2.

Because of difficulties in the analysis and manufacturing process, a circular and eccentric profile was accepted to replace the general profile, Fig.1b.

2. ANALYSIS OF THE FUNCTION GENERATING MECHANISM WITH ECCENTRIC CIRCULAR WHEEL AS OUTPUT ELEMENT

The first problem which appears during the analysis is that the length of the AB and BD elements are permanently changing.

For the mechanism in Fig.1b the analysis was made. OA,CB,CD,OD--are known.

To solve the problem three positions were taken into account:

--first, a extremum (minimum) position, designated 0, where the angle between the OA and AB elements is [pi] (Fig.3.)

--second, a extremum (maximum) position, where the angle between the OA and AB elements is 0, designated M (Fig.4.)

--third, a current position, where the angle between OA and AB is not known.(Fig.5.)

For the first position, illustrated in Fig. 3 O[C.sub.0] is determined in two ways, in [DELTA]O[C.sub.0]D and [DELTA]O[B.sub.0][C.sub.0] resulting the function

[([l.sub.2] + [l.sub.0]).sup.2] + [r.sup.2] = [l.sup.2.sub.1] + [l.sup.2.sub.4] + Z[l.sub.1][l.sub.4] cos[[psi].sub.0] (1)

Also [B.sub.0]D is determined in two ways, in [DELTA]O[B.sub.0]D and [DELTA][B.sub.0][C.sub.0]D, resulting a function.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

If [[psi].sub.0] is imposed [l.sub.0] and [[phi].sub.0] can be determined.

[l.sub.0] = [square root of [l.sup.2.sub.1] + [l.sup.2.sub.4] + 2[l.sub.1][l.sub.4] cos[[psi].sub.0] - [l.sub.2]] (3)

[(cos[[phi].sub.0).sub.1,2] = a(b sin[[psi].sub.0] - c)[+ or -][square root of [DELTA]/[b.sup.2] + [c.sup.2] - 2bcsin[[psi].sub.0] (4)

[FIGURE 3 OMITTED]

with:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

In the second position, illustrated in Fig. 4. O[C.sub.M] is determined in two ways, in [DELTA]O[C.sub.M]D and [DELTA]O[B.sub.M][C.sub.M]. resulting the function

[([l.sub.M] - [l.sub.2]).sup.2] = [l.sup.2.sub.1] + [l.sup.2.sub.4] - [r.sup.2.sub.4] + [2l.sub.1][l.sub.4]cos[[psi].sub.M] (6)

Also [B.sub.M]D is determined in two ways, in [DELTA]O[B.sub.M]D and [DELTA][B.sub.M][C.sub.M]D, resulting the function.

[r.sup.2.sub.4] + [l.sup.2.sub.4] - [2r.sub.4][l.sub.4]cos[[beta].sub.M] = [l.sup.2.sub.1] + [([l.sub.M] - [l.sub.2]).sup.2] + + [2l.sub.1]([l.sub.M] - [l.sub.2])cos[[phi].sub.M] (7)

Also it is needed to determin the length of the winded wire on the profile, this results from the difference between positions 1 and 2, resulting the function.

[l.sub.M] = [l.sub.0] + [r.sub.4][[theta].sub.M] (8)

[[theta].sub.M] = [[beta].sub.0] - [[beta].sub.M] (9)

where [[theta].sub.M] is the angle of the unwinded wire, [[beta].sub.0] is determined from the first position and [[beta].sub.M] is determined from the second position.

[[beta].sub.0] = [pi]/2 - ([[psi].sub.0] - [[psi].sub.0) (10)

From (8) (9) and (10), [[beta].sub.M] is determined

[[beta].sub.M] = 1/[r.sub.4] ([l.sub.0] + [r.sub.4][[beta].sub.0] - [l.sub.M]) (11)

With these relationships, [[psi].sub.M], [l.sub.M] and [[phi].sub.M] can be determined depending of the setup of the first position, namely 0.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

In the third position illustrated in Fig. 5 the mechanism's parameters are determined as being the instantaneous equivalent four-bar linkage OABD and calculating its kinematic parameters. To do so, the length of BD element from [DELTA]BCD is determined, resulting in the inslantaneous length [l.sub.4.sup.*]. The angle [[psi].sub.N] ia also determined:

[[psi].sub.N] = [psi] + [delta] (12)

where [delta] is CDB angle.

To determin [delta], the sine theorem is applied in [DELTA]BCD resulting

[delta] = arcsin ([r.sub.4]sin[beta]/[r.sup.2.sub.4] + [l.sup.2.sub.4] + [2r.sub.4] [l.sub.4] cos[beta]) (13)

the elements are projected on the Ox and Oy axes and the a parameter is eliminated resulting the following relationship:

[[psi].sub.N] = 2arctg -B + [square root of [A.sup.2] + [B.sup.2] - [C.sup.2]/A + C (15)

with

A = 2[l.sup.2.sub.4]([l.sub.1] - [l.sub.2]cos[phi]) (16)

B = [2l.sub.2][l.sup.*.sub.4]sin[phi]

C = [l.sup.2.sub.3] + [2l.sub.1][l.sub.2]cos[phi] - ([l.sup.2.sub.2] + [l.sup.*2.sub.4] + [l.sup.2.sub.1])

if [phi] is imposed, than [psi] can be determined from (12) resulting: [psi] = [[psi].sub.N] - [delta] (17)

3. CONCLUSION

All kinematic parameters are resulting in fuction of the startup position.

The research is important because, with this analysis method, and with existing synthesis methods(Lovasz, 2000), centrode type mechanisms can be designed and used where 5 precision points/positions are not sufficient in order to obtain an optimal solution.

Other field of intrest where this type of mechansims could be used is the force equilibration, for example the compensation of the archimedic effect.

4. FUTURE WORK

The development of a Synthesis--Analysis PC Algorythm in order to optimize the design of a mechanism of this type for any given application.An other point of intrest is the development of analysis algorithms for other configurations of centrode type mechanisms.

5. REFERENCES

Lovasz, E-C (1998). Function generating mechanism with applications in Precision Engineering (de: Synthese der Ubertragungsgetriebe mit Anwendungen in der Feinmechanik). PhD thesis, T.U. Dresden/U.P.T Timisoara

Perju, D. (1971). Contributions to the Pathgenerating Mechanisms Synthesis (ro: Sinteza mecanismelor plane pentru conducerea unui punct pe o curba data). PhD thesis, Politechnic Institute Bucharest

Perju, D.; Modler, K-H; Lovasz, E-C & Mesaros-Anghel, V (2001). A new type of function generating mechanism with variable link length. Proceedings of IFToMM International Symposium, Bucharest

Perju, D. (1987). Mechanisms with variable link length. Proceedings of IFToMM World Congress on TMM, Sevilla

Lovasz, E-C.; Perju, D. & Mesaros-Anghel, V (2000). On the mechanisms synthesis of centroidal type. Proceedings of IFToMM International Symposium, Liberec