Flexible multi-body system

In technical mechanics , a branch of physics , a flexible multi-body system is a multi-body system in which one or more bodies are deformable.

definition

A deformable body is e.g. B .:

a longitudinally deformable rod,
a bending beam (deformation: bending and possibly longitudinal expansion),
a disk (plane deformation of a general body),
a plate,
a bowl,
a generally deformable spatial body (e.g. discretized by finite elements .)

The deformation can be both elastic and inelastic (e.g. elasto-plastic, viscous, ...). This is why one speaks of flexible and not elastic multi-body systems.

The difference to multi-body systems with rigid bodies is that additional degrees of freedom for describing the deformation are included in the formulation and must be calculated.

Many methods for describing flexible multibody systems have established themselves in the literature, most of them coming from the field of finite elements . One differentiates essentially

small deformations,
large deformations with small distortions,
large deformations with large distortions.

Some formulations for flexible multi-body systems

A basic distinction is made between large and small deformations in the formulations, and whether rotations are used for the discretization, or displacement quantities.

The following list tries to give some German translations for common English terms:

Floating frame of reference formulation
Incremental formulation
"Large rotation vector formulation"
Natural coordinates
Total Lagrange finite element formulation (total Lagrange)
Absolute nodal coordinate formulation

Moved reference configuration

To describe small deformations, the method of the moving reference configuration (floating frame of reference formulation) has proven itself. This method is used to describe the deformation of bodies that are subject to large rotations and was used even before the introduction of multi-body systems.

In this formulation, the position of a point in the body is composed of rigid body translation , rigid body rotation, expressed by a rotation matrix , and a (mostly small) deformation , ${\ displaystyle \ mathbf {r}}$ ${\ displaystyle \ mathbf {X}}$ ${\ displaystyle j}$ ${\ displaystyle \ mathbf {r} _ {0}}$ ${\ displaystyle \ mathbf {R}}$ ${\ displaystyle \ mathbf {u} ^ {(f)}}$

${\ displaystyle \ mathbf {r} = \ mathbf {r} _ {0} + \ mathbf {R} (\ mathbf {X} - \ mathbf {I}) + \ mathbf {R} \ mathbf {u} ^ { (f)}}$

The deformation is like in the finite elements z. B. discretized in place with the help of a Ritz approach:

${\ displaystyle \ mathbf {u} _ {f} = \ sum \ limits _ {i = 1} ^ {n} \ mathbf {N} _ {i} (\ mathbf {X}) \ mathbf {q} _ { i} ^ {(f)} (t)}$

Thereby make the shape functions are in place and are the generalized coordinates, which have to be additionally calculated using the equations of motion. ${\ displaystyle \ mathbf {N} _ {i} (\ mathbf {X})}$ ${\ displaystyle \ mathbf {q} _ {i} ^ {(f)} (t)}$ ${\ displaystyle n}$

Incremental formulation

In finite elements, the incremental formulation is mainly used for structural elements to model large deformations, whereby elements (e.g. beams, shells, ...) are described by means of position and rotation parameters. In the individual calculation steps, the increment of the rotations is used, which is derived from the linearized Rodriguez formula . This simplification leads to errors and possibly instabilities, but this formulation is one of the most common in the field of finite elements.

Vector large rotations

This formulation is used to model large deformations and was developed in the circle of the well-known researcher Juan C. Simo (1986). Rotations are interpolated and no approximations are made, which is also referred to as a geometrically exact formulation. When solving equations based on this formulation, errors occur in the approximation of the rotations, which have a significant effect on the conservation of energy and angular momentum. For this reason, time integration methods that have been specially developed for such systems are used, which also receive energy and angular momentum in the temporal approximation.

Natural coordinates

see J. Garcia de Jalon and E. Bayo in Kinematic and Dynamic Simulation of Multibody Systems .

Absolute node coordinate formulation

One of the latest formulations in the field of flexible multi-body systems, developed by AA Shabana , is based on so-called absolute node coordinates, which differs from the conventional definition in finite elements. The absolute nodal coordinate formulation uses both displacements (usually expressed in terms of position) and the directional derivatives of the position of several points on a body and is used to model large deformations. The node coordinates serve as degrees of freedom for describing the deformation of beams (usually 2 nodes) or shells (usually 4 nodes). Since no rotational degrees of freedom are introduced, the shape functions must be chosen in such a way that any rigid body translations and rotations as well as deformations can be represented.

This description prevents the usual problems of degrees of freedom of rotation (see above) no longer occurring, but there is an additional disadvantage that these elements contain extremely high rigidity (transverse expansion). The directional derivatives, which represent the gradient at the junction in the spatial formulation, can be used to connect two bars rigidly or freely rotatable with one another without any constraint.

Individual evidence

^ J. Garcia de Jalon, E. Bayo: Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge. Springer New York, 1994, ISBN 978-1-4612-2600-0 .

[1] J. Garcia de Jalon, E. Bayo: Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge. Springer New York, 1994, ISBN 978-1-4612-2600-0 .