Abstract :
Di erent nite-element-based strategies used to represent the components’ exibility in multibody systems
lead to various sets of co-ordinates. For systems in which the bodies only experience small elastic
deformations it is common to use mode component synthesis to reduce the number of generalized elastic
co-ordinates and, consequently, the equations of motion are written in terms of modal co-ordinates.
However, when the system components experience non-linear deformations the use of reduction methods
is not possible, in general, and the nite element nodal co-ordinates are the generalized co-ordinates
used. Furthermore, depending on the type of nite elements used to represent each exible body, the
nodal co-ordinates may include all node rotations and translations or only some of each. Regardless of
the type of generalized co-ordinates adopted it is required that kinematic joints are de ned. The complete
set of joints available in a general-purpose multibody code must include, for each particular type
of joint, restrictions involving only rigid bodies, or only exible bodies, or exible and rigid bodies.
Therefore, the e ort put into the development and implementation of any joint is at least three times
as much as the initial work done in the implementation of joints with rigid bodies only. The concept
of virtual bodies provides a general framework to develop general kinematic joints for exible multibody
systems with minimal e ort, regardless of the exible co-ordinates used. Initially, only a rigid
constraint between the exible and a massless rigid body is developed. Then, any kinematic joint that
involves a exible body is set with the massless rigid body instead, using the regular joint library of
the multibody code. The major drawback is that for each kinematic joint involving a exible body it is
required to use six more co-ordinates per virtual body and six more kinematic constraints. It is shown
in this work that for small elastic deformations, for which the mode component synthesis is applied,
the use of sparse matrix solvers can compensate for the computational overhead of involving more
co-ordinates and kinematic constraints in the system, due to the use of virtual bodies. For non-linear
deformations, where the generalized co-ordinates are the global positions of the nite-element nodes,
the use of the virtual body concept does not require an increase in the number of system co-ordinates
or kinematic constraints. By introducing the rigid joint between the exible body nodal co-ordinates
and the virtual body, with the use of Lagrange multipliers, and then solving the equations explicitly forthese multipliers the resulting equations of motion for the subsystem have the same degrees of freedom
as the original exible body alone. The di erence is that degrees of freedom associated to the virtual
body are used as co-ordinates of the subsystem instead of the nodal co-ordinates of the nodes of the
exible body attached to the virtual body
