real-time mbs formulations: towards virtual engineering

170 J. Cuadrado, M. Gonzalez, R. Gutierrez, M.A. Naya
dynamics of complex rigid-flexible multi-body systems; b) the experimental validation of the motions,
deformations and forces obtained through the application of the abovementioned formulations, so as to
verify that reality is being reasonably well imitated; c) the study of the aptitude of such formulations
for becoming part of a virtual reality environment, in connection with user-interface devices.
The paper has been organized as follows: Section 2 describes a real-time formulation developed by
the authors for the dynamics of rigid-flexible multi-body systems; Section 3 shows some examples of
rigid-flexible multi-body systems simulated with such formulation, paying attention to the achieved
efficiency, robustness and accuracy; Section 4 reports the results of the experimental validation of one
of the examples; Section 5 presents a simulator based on the proposed formulation; and, finally,
Section 6 summarizes the conclusions of the work.
2. THE REAL-TIME FORMULATION
The three main elements (modeling, dynamic equations and numerical integrator) of the proposed
approach to solve the dynamics of rigid-flexible multi-body systems are described in what follows.
For the modeling, fully-Cartesian coordinates, also known as natural coordinates, are used. Both rigid
and flexible links are modeled with this type of coordinates in a total compatible form. Further
explanation about these coordinates can be found in [1]. The dynamic equations are stated through an
improved version of the index-3 augmented Lagrangian formulation with mass-orthogonal projections
given in [2]. For the integration scheme, the unconditionally-stable implicit single-step trapezoidal rule
has been chosen [3].
2.1 Modeling
The modeling of rigid bodies follows the general rules given in [1], and won’t be described here,
since it can be considered as a well established technique.
The modeling of flexible bodies is carried out by a floating frame of reference approach with
component mode synthesis for small elastic displacements. The global motion of each flexible body is
described as a superposition of the large-amplitude motion of a moving frame, rigidly attached to a
certain point of the body, and the small elastic displacements of the body with respect to an
undeformed configuration, taken as reference. Any deformed configuration of the body is expressed as
a linear combination of static and dynamic modes, in the sense of the mode synthesis approach with
fixed boundaries. The static modes depend on the natural coordinates defined at the joints of the body,
while the number of internal, dynamic modes should be decided by the analyst. A detailed description
of the kinematics of this method can be found in [4].
However, the approach proposed in the abovementioned reference for the dynamics has not been
adopted, as long as it showed to be too involved, particularly in what refers to the form of the inertia
terms. Therefore, the co-rotational approximation suggested in [5] has been considered in the way
proposed by [6] when natural coordinates are used for the modeling. This modified approach provides
the same level of accuracy while notably simplifying the formulation of the mass matrix and velocity
dependent forces vector. In what follows, the approach is
briefly described.
Figure 1 shows a general flexible body. Point r o
and
orthogonal unit vectors a, b and c serve to define the body
local reference frame, and are always needed, so that they
must compulsorily be considered as problem variables.
Additional points and unit vectors (further problem
variables) may be defined at the joints of the body (they
will produce the static modes) in the typical way featured
by the natural coordinates. The position of any point of the
Figure 1. General flexible body.
body can then be expressed as