102
E. Chircan et al.
In the study, an analysis is made of the motion equations in the case of a beam
in rotation in plane, in order to determine the domain of instability without actually
calculating the eigenvalues or to integrate the obtained equations of motion. This
problem can be important in the engineering of the multibody with elastic elements.
2 One-Dimensional Finite Element
The problem of the study of a one-dimensional finite element in a centrifugal field
was made by many researchers, for a general three-dimensional motion and for a
plane motion [12, 14]. In the following, we will use the notation from [4, 5] in order to
obtain the motion equations for a single element. We need this approach to apply our
proposal concerning the study of such a system. Let’s consider a point M of beam and
its displacements [δ(u, v, w)] that can be expressed in terms of nodal displacements
at the ends as follows:
{δ} =
⎧
⎨
⎩
u
v
w
⎫
⎬
⎭
= [N ]
δ e,L
= [N ]
δ _1
δ _2
(1)
where we have the vector of nodal displacements {δ e }:
δ e,L
=
δ _1
δ _2
= {δ e }
(2)
where {δ e } is the displacement vector for e-th finite element in the local coordinate
system, δ _1 and δ _2 , are, respectively, the displacement vectors of the nodes one and
two.
Consider a finite element with a rotational motion around an axis. The nodal
coordinates: the displacements of the beam ends in the three directions x, y, and z,
the torsion angles at the end, the angles of rotation β and γ of the cross section at
ends around the two y and z axes, and the curvatures of the neutral axis in the two
xOz and xOy planes at both ends. If we consider the two ends, then the displacements
at the ends, the rotations, and the curves are [20]:
{ f 1 } =
⎧
⎨
⎩
u 1
v 1
w 1
⎫
⎬
⎭
; { f 2 } =
⎧
⎨
⎩
u 2
v 2
w 2
⎫
⎬
⎭
; {φ 1 } =
⎧
⎨
⎩
α 1
β 1
γ 1
⎫
⎬
⎭
; {φ 2 } =
⎧
⎨
⎩
α 2
β 2
γ 2
⎫
⎬
⎭
;
{m 1 } =
m x Oz1
m x Oy1
; {m 2 } =
m x Oz2
m x Oy2
(3)