On the dynamic analysis of an elastic multi-bodies system.

Vlase, Sorin ; Purcarea, Ramona ; Munteanu, Mihaela Violeta 等

1. INTRODUCTION

The first step is to establish the motion equations for an elastic finite element with a general three-dimensional motion together with an element of the system. The type of the shape function is determined by the type of the finite element. For this reason we will present the motion equations in three different situations: for a three-dimensional finite element with a general three-dimensional motion, for a two-dimensional finite element with a plane motion and for an one-dimensional element with a general three-dimensional motion. We will consider that the small deformations will not affect the general, rigid motion of the system. The major difficulty using FEM is the non-linearity of the motion equations. The coefficients that appears in equations are time-position dependent and, in some practical application (mechanisms with a periodical motion) they can be periodical. To solve this problem the motion must be considered "frozen" for a very short interval of time. In this case the obtained equations can be considered linear. Writing the principle of minimum energy is possible to obtain the motion equations for a finite element with a three-dimensional rigid motion. These equations have some important particularities: in the equations exists Coriolis terms (conservative) and the rigidity is modified by the some terms determined by the "rigid motion" of element. They depend on element distribution mass and on the field of velocities and accelerations. More, the force term of equations is modified by the effect of inertia forces and momentum due to the relative motion. Generally, the multi-bodies systems have a great complexity and a strong non-linearity.

2. MOTION EQUATIONS FOR ONE SINGLE ELEMENT

The equations describing the time dependent motion of an elastic mechanical system is obtained via Lagrange equations. We consider that, for the all elements of the system, we know the field of the velocities and of the accelerations. We refer the finite element to the local coordinate system Oxyz, mobile, and having a general motion with the part of system considered.

We note with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the velocity and with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the acceleration of the origin of the local coordinate system. The motion of the whole system is refer to the general coordinate system O'XYZ. By [R] is denoted the rotation matrix. The velocity of point M' will be:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

The kinetic energy of the finite element considered is:

[E.sub.c] = 1/2 [[integral].sub.V][rho][v.sup.2]dV 1/2 [[integral].sub.V][rho][{[v.sub.M],}.sup.T]{[v.sub.M]}dV (2)

where [rho] is the mass density and the deformation energy is:

[E.sub.p] = 1/2 [[integral].sub.V][{[[delta].sub.e]}.sup.T] [[k.sub.e]{[[delta].sub.e]}dV (3)

where [[k.sub.e]] is the rigidity matrix for the e element. If we not with {p}={p(x, y, z)} the distributed forces vector, the external work of these is:

W = [[integral].sub.V][{p}.sup.T] {f}dv = ([[integral].sub.V] [{p}.sup.T][N]dV){[[delta].sub.e]} (4)

and the nodal forces {[q.sub.e]} produce an external work:

[W.sup.c] = [{[q.sub.e]}.sup.T] {[[delta].sub.e]} (5)

The Lagrangean for the considered element is obtained with the relation:

L = [E.sub.c] - [E.sub.p] + W + [W.sup.c] (6)

If we apply the Lagrange's equations:

d/dt {[partial derivative]L/[partial derivative] [[??].sub.e]} - {[partial derivative]L/[partial derivative][[delta].sub.e] = 0

with the notations explains in (Mocanu et al., 1990), (Vlase, 1985) and (Vlase, 1987), it result the motion equations for the finite element analyzed in a compact form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [??] represent the angular velocity and E the angular acceleration with the components in the local coordinate system.

The notation:

{[partial derivative]L/[partial derivative][[delta].sub.e]

we understand:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For an one dimensional element with a three dimensional motion the motion equation are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3. ASSEMBLING PROCEDURES

The unknowns in the elasto-dynamic analysis of a mechanical system with liaisons are the nodal displacements and the liaison forces. By assembling the motion equations written for each finite element we try to eliminate the liaisons forces and the motion equations will contain only nodal displacements as unknowns. The liaisons between finite elements are realized by the nodes where the displacements can be equal or can be other type of functional relations between these. When two finite elements belong to two different elements (bodies) the liaison realized by node can imply relations more complicated between nodal displacement and their derivatives.

The motion equations for the whole structure, referred to the global coordinate system, are (Vlase, 1992), (Vlase, 1994), (Vlase, 1997):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

4. THE INFLUENCE OF THE CORIOLIS TERMS

The matrix [c] is skew-symmetric. If we want to obtain the energy balance by integration, we obtain that the variation of energy due to the term skew-symmetric is null. Consequently, the Coriolis term only transfer the energy between the independent coordinates of the system and had no role in the dissipation of the energy.

If we consider now a motion mode on the form:

{q} = {A}sin(([omega]t + [phi])

and we introduce in the motion equations, where the forces are considered null, we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If we pre-multiply with [{A}.sup.T] we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We have considered that:

([omega][{A}.sup.T][c]{A} = 0

and

[{A}.sup.T][k.sub.[epsilon]]{A} = 0

because [c] and [[k.sub.[epsilon]] are skew-symmetric. It results:

[[omega].sup.2] = [{A}.sup.T][k]{A}+[{A}.sup.T] [k[[omega].sup.2]]{A}/[{A}.sup.T][m]{A}

This relation can not express, in a direct way, the influence of the matrix [c] in the eigen-values calculus, but this influence is present by the eigenvectors {A}. The term [c] has an influence on the values of the eigen-values. Some of the eigen-values increase and the other decrease. This variation is presented, extended, in the paper. Between these values there exist some interesting relations.

5. CONCLUSIONS

The problems involved by use of finite element analysis of an elastic system are the followings:

* the strong geometric non-linearity of the motions equations and the additional term that appear in these;

* the motion equations are valid only for the "frozen" system, for a very short interval of time.

5. REFERENCES

Mocanu, D.; Goia, I.; Vlase, S. and Vasu, O. (1990). Experimental Checking's in the Elasto-dynamic analysis of mechanism, by using finite elements. International Congress On Experimental Mechanics, Lyngby, Denmark, 1053-1058

Vlase, S. (1985). Elastodynamische Analyse der Mechanischen Systeme durch die Methode der Finiten Elemente. Bul. Univ. Brasov, pp. 1-6, Brasov

Vlase, S. (1987). A Method of Eliminating Lagrangean Multipliers from the Equations of Motion of Interconnected Mechanical Systems. Journal of Applied Mechanics, ASME trans., vol.54, nr.1

Vlase, S. (1992). Finite Element Analysis of the Planar Mechanisms: Numerical Aspects. Applied Mechanics--4. Elsevier, pp. 90-100

Vlase, S. (1994). Modeling of Multibody Systems with Elastic Elements. Zwischenbericht. ZB-86, Technische Universitat, Sttutgart

Vlase, S. (1997). Elimination of Lagrangian Multipliers. Mechanics Research Communications, Vol. 14, pp. 1722