On the topological description of the multibody systems.

Vlase, Sorin ; Munteanu, Mihaela Violeta ; Scutaru, Maria Luminita 等

1. INTRODUCTION

The graph theory was used in the mechanical system analysis in two different ways:

* in one way the constitutive elements of the system are considered nodes and the liaison between elements are considered edges (Thoma, 1975), (Voinea & Atanasiu, 1964). This treatment is equivalent with the Kirchhoff equations for the electrical systems. This analysis leads to the independent cycle method, developed by various authors in many variants. Some descriptions (Shai & Pennock, 2006), (Tsai et al., 1998) continue this case.

* another way is to consider the liaison between elements as nodes and the vector defined by the points of the liaison as edges. This description, less used in the literature (Vlase, 2007) permit to utilize the natural vector description of the system. In the paper is analysed this treatment using a topological approach. The planar case, presented in the paper, can be extended to the spatial systems.

2. TOPOLOGICAL DESCRIPTION OF THE

MECHANISM

2.1. Kinematical Equivalent Mechanism

In order to present the method we must to attach a graph to mechanism. The first step is to build a mechanical system made only by bars equivalent, from mechanical point of view, with the system done. We will analyze the planar mechanisms; the developed methods could be applied to other mechanical systems.

It is considered the mechanism presented in fig.1. The liaison between two elements it is accomplished by joints and slide bars.

We build a mechanism kinematical equivalent with the mechanism defined like:

[FIGURE 1 OMITTED]

I. We replace any mobile element type plate (fig.2a) with a rigid structure from fig. (2b), formed with triangles made by bars. It is introduced in this way the equations which describe the kinematical state of a fundamental cycles system (independent cycles) "false", but what they facilitate the equations representation.

[FIGURE 2 OMITTED]

II. It bonds the points that accomplish the bonding with the fixed element through bars type elements that close the polygon (fig.3) and then it is eliminated one of these elements (it results from vectors sum with changed sign of all other elements). It is introduced like this the "false" elements which can modify the length, but not the position, so always angular speed and angular acceleration of such element are null.

By these two constructions it obtains an equivalent kinematical mechanism with the initial mechanism, but made it only with bars (fig.3).

In the moment of establishing the kinematical behavior of the mechanism elements, we consider that knowing all the bars length from the equivalent kinematical mechanism and the angles made of them with Ox axis of a xOy system.

[FIGURE 3 OMITTED]

Putting to each bar element a vector with the initial point into a liaison and the end point into other liaison, we can build a graph which has, like nodes, the joints and slide of the mechanism, and like edges the mechanism bars, the sense of each line being determined by the direction of the vector attached to correspondent bar. (Fig.4).

[FIGURE 4 OMITTED]

2.2. Independent Cycles

To understand the presentation we summarise some basic notions in graph theory [1],[2]. Being r+1 the number of the nodes and s the number of the edges of the obtained graph. The complete incidence matrix is the matrix [[A.sub.a]] = [[a.sub.ij]] having the dimension (r+1) x s with:

* [a.sub.ij] = 1 if the edge j is incident to the node i and go out from node;

* [a.sub.ij] = -1 if the edge j is incident to the node i and go in to node;

* [a.sub.ij] = 0 if the edge j is not incident to the node i.

The matrix [A] obtained by eliminating a line in [[A.sub.a]] is the incidence matrix (or reduced incidence matrix ).

The selection of a tree into the graph allows a re ordination and a partition of the [A]:

[A] = [[A.sub.t] [??] [A.sub.l]] (1)

following the edge of the tree and the branches (if we attach the branches to tree we reconstruct the initial graph).

The complete cycle's matrix is the matrix: [B.sub.a] = [[b.sub.ij]] with:

* [b.sub.ij] = 1 if the edge j makes part from cycle i and its orientation coincide with the choice sense for passing the cycle;

* [b.sub.ij] = -1 if the edge j makes part from cycle i and its orientation is in contrary sense with the choice sense for passing the cycle;

* [b.sub.ij] = 0 if the edge j makes not part from the cycle i.

The rank of the matrix [B.sub.a] is s-r. The matrix [B.sub.f] with rank s-r obtained from [B.sub.a] by retaining of s-r independent lines is the cycles fundamental matrix. Between [B.sub.f] and A there is the following relation:

[B.sub.f] = [[[-[A.sup.-1.sub.t][A.sub.l]].sup.T] [??] I] (2)

3. KINEMATICAL DESCRIPTION

With some notations (Voinea & Atanasiu, 1964) the relations obtained for velocities and accelerations are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where index refer to the unknown kinematical quantities and index c refer to the independent quantities.

4. CONCLUSIONS

The presented method offers the kinematical unknowns in the case of the planar mechanism, using the topological properties of the mechanical systems with the associated graph. This method, using the natural description of a mechanical system with the vector algebra leads to an automatic description of such system. It is possible to obtain the velocities and the accelerations. At the positions level the problem is more complicated due to the multiple solutions and the nonlinearity of the equations.

5. REFERENCES

Berge, C. (1958). Theorie des graphes et ses applications. (Theory of graph) Collection Universitaire de Mathematiques, II Dunod, Paris,

Gross, J.L.; Yellen, J. (2003). Handbook of Graph Theory, CRC

Shai, O.; Pennock, G.R. (2006). Extension of Graph Theory to the Duality Between Static Systems and Mechanisms. Journal of Mechanical Design, Volume 128, Issue 1, pp. 179-191

Thoma, U.J., (1975). Introduction to Bond Graphs and Their Applications, Pergamon Press, Oxford

Tsai, L.-W.; Chen, D.-Z.; Lin. T.W. (1998). Dynamic analysis of geared robotic mechanisms using graph theory. Journal of mechanical design (J. mech. des.), vol. 120, No. 2, pp. 240-244

Vlase, S. (2007). Mecanica. Cinematica (Mechanics. Kinematics.) INFOMARKET Press, ISBN 978-973-820496-6, Brasov

Voinea, R.; Atanasiu, M. (1964). New analytical methods in the theory of mechanisms. Ed. Tehnica, Bucuresti