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Lorenzo
Trainelli’s research Teoria
delle rotazioni e del moto rigido Theory of rotation and rigid motion
In many engineering
applications one faces problems characterized by an independent rotation field
associated to the customary displacement field. Rotations can be
formalized as elements in a nonlinear space (the 3D special orthogonal
group), for which a suitable parametrization must be given. Research in this field
has led to the formulation of a very general family of parametrizations,
known as 'vectorial
parametrizations’, which encompasses several techniques adopted in
the literature (exponential, Cayley-Gibbs-Rodrigues, Wiener-Milenkovich,
Euler-Rodrigues, etc.). I contributed in this
work originated by prof. O.A. Bauchau at GaTech. For a detailed account on
this topic, see this paper:
Representation and parameterization of rigid motion The theory of rotation
finds a natural
extension in the theory of rigid motion, where rotation is coupled
with translation. All of the important
results obtained in the theory of rotation have an analogue in the theory of rigid
motion, which can be represented in many ways: for example, by vectors and
tensors in a 4 or 6-dimensional space, and by 3-D ‘dual’ vectors and tensors. This implies the
possibility of describing a general rigid motion by a ‘global’ parameterization, that
inherently couples translations and rotations. I contributed in this
work originated by prof. M. Borri at the Politecnico di Milano. For a complete
presentation, see this paper:
As a natural consequence,
the vectorial
parameterization concept developed for rotation (see above) can be
extended to rigid motion, as shown on detail in this report:
Subsequent work has extended
all the well-known non-vectorial parameterizations of rotation
(such as Euler-type angles, quaternions, etc.) to rigid motion. This is the subject of
the following works:
Geometric integration of structural
and multibody systems The approach described
above for the formulation of kinematics naturally couples with a
6-dimensional fixed-pole
formulation of the equations of motion. This consistes in taking a
fixed placement as the pole of reduction of torques, and in the subsequent
casting of the governing equations (force and torque balance eqs.) in compact
6-dimensional format. The framework established
in this way reveals convenient for geometric time and space integration, i.e., a numerical procedure that
preserves, at the discrete level, important qualitative properties of the
exact solution that are typically lost with conventional procedures. Among these properties we
find:
Properties (1) and (2)
imply a twofold
invariance: on the one hand, the numerical solution is indifferent
to the choice
of the absolute reference frame (both axes and origin); on the
other hand, it is indifferent with respect to the choice of the material reference entity
(a material point and a vector triad for a rigid body, an oriented line for a
beam, an oriented surface for a shell). Properties (3) and (4)
are detailed on the pages devoted to structural and multibody dynamics and to numerical integration
algorithms. The working group on this
topic comprises prof. M. Borri, prof. C.L. Bottasso
and myself. These papers focus on the
formulation for rigid bodies, beams, and complex flexible multibody systems:
This work specifically
addresses the objectivity issue:
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Last updated 27/09/2006. |
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