Advances in Computational Multibody Systems

371 Pages·2005·8.112 MB·English

by  J. de García Jalón, E. Álvarez (auth.), Jorge A.C. Ambrósio (eds.)

ADVANCES IN COMPUTATIONALMULTIBODYSYSTEMS
Computational Methods in Applied Sciences
Volume 2
Series Editor
E. Oñate
Advances in Computational
Multibody Systems
Edited by
JORGE A.C. AMBRÓSIO
IDMEC, Instituto Superior Técnico,
Lisbon, Portugal
AC.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-10 1-4020-3392-3 (HB)
ISBN-10 1-4020-3393-1 (e-book)
ISBN-13 978-1-4020-3392- 6 (HB)
ISBN-13 978-1-4020-3393-3 (e-book)
Published by Springer,
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© 2005 Springer
No part of this work may be reproduced, stored in a retrieval system, or transmitted
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Printed in the Netherlands.
Table of Contents
Preface . . ..................................................................................................................vii
J. GARCÍA DE JALÓN, et al.
A Fast and Simple Semi-Recursive Formulation for Multi-Rigid-Body
Systems.....................................................................................................................1
O. ÖTTGEN and M. HILLER
Hardware-in-the-Loop for Quality Assurance of an Active Automotive
Safety System.........................................................................................................25
R. S. SHARP, S. EVANGELOU and D. J. N. LIMEBEER
Multibody Aspects of Motorcycle Modelling with Special Reference
to Autosim..............................................................................................................45
G. M. HULBERT, Z.-D. MA and J. WANG
Gluing for Dynamic Simulation of Distributed Mechanical Systems...................69
W. SCHIEHLEN, B. HU and R. SEIFRIED
Multiscale Methods for Multibody Systems with Impacts....................................95
J. MCPHEE
Unified Modelling Theories for the Dynamics of Multidisciplinary
Multibody Systems...............................................................................................125
J. A.C. AMBRÓSIO and M. P.T. SILVA
A Biomechanical Multibody Model with a Detailed Locomotion
Muscle Apparatus.................................................................................................155
P. E. NIKRAVESH
Understanding Mean-Axis Conditions as Floating Reference Frames................185
J.C. GARCÍA ORDEN and J.M. GOICOLEA
Robust analysis of flexible multibody systems and joint clearances
in an energy conserving framework.....................................................................205
J. M. MARTINS, J. SÁ da COSTA and M. AYALA BOTTO
Modelling, Control and Validation of Flexible Robot Manipulators...................239
M. BULLINGER, K. FUNK and F. PFEIFFER
An Elastic Simulation Model of a Metal Pushing V-Belt CVT...........................269
vi
J. P. DIAS and M. S. PEREIRA
Multicriteria Optimization of Train Structures for Crashworthiness...................295
L. KÜBLER, C. HENNINGER and P. EBERHARD
Multi-Criteria Optimization of a Hexapod Machine............................................319
E. PENNESTRÌ and L. VITA
Multibody Dynamics in Advanced Education.....................................................345
Preface
The area of Multibody Dynamics is a part of the Computational Mechanics scien-
tific field associated to solid mechanics. It can be argued that among all the areas
in solid mechanics the methodologies and applications associated to multibody
dynamics are those that provide a better framework to aggregate different disci-
plines. This idea is clearly reflected in the multidisciplinary applications in biome-
chanics that use multibody dynamics to described the motion of the biological en-
tities, in finite elements where multibody dynamics provide powerful tools to
describe large motion and kinematic restrictions between system components, in
system control where the methodologies used in multibody dynamics are the
prime form of describing the systems under analysis, or even in many applications
that involve fluid-structures interaction or aeroelasticity.
The ECCOMAS thematic conference Multibody Dynamics 2003 that took
place in Lisbon, Portugal was organized in turn of special sessions dedicated to
multibody dynamics in Biomechanics, Vehicle Dynamics, Contact and Impact,
Optimization and Design Sensitivity, Flexible Multibody Systems, Education of
Computational Kinematics, Dynamics and Multibody Systems, Multidisciplinary
Applications and Real-Time Applications. These sessions were organized by rec-
ognized experts in each of these areas and gathered together 127 participants from
22 countries, including Japan, Korea, India, USA, Mexico, Canada and many of
the European nations, who delivered 90 communications during the 4 days of the
conference.
This book contains the contributions of the special session organizers, or of par-
ticipants selected by the organizers, that reflect the State-of-Art in the application
of Multibody Dynamics to different areas of engineering. The chapters of this
book are enlarged and revised versions of the communications, delivered at the
conference, which were enhanced in terms of self-containment and tutorial quality
by the authors. The result is a comprehensive text that constitutes a valuable refer-
ence for researchers and design engineers which helps to appraise the potential for
the application of multibody dynamics methodologies to a wide range of areas of
scientific and engineering relevance.
Lisbon, Portugal
Jorge Alberto Cadete Ambrósio
Chairman
A Fast and Simple Semi-Recursive Formulation for
Multi-Rigid-Body Systems
J. García de Jalón1, E. Álvarez1, F.A. de Ribera1, I. Rodríguez2 and F.J. Funes3
1 Escuela Técnica Sup. de Ingenieros Industriales, Univ. Poli. de Madrid, Spain
2 STT Engineering and Systems, S.L, San Sebastián, Spain
3 Telefónica de España, Madrid, Spain
This work describes a topological semi-recursive formulation for multibody dy-
namics that is very simple and efficient. This formulation is called “semi-
recursive” because it uses recursivity, but at the end it needs to solve a system of n
linear equations, with n the numbers of degrees of freedom. With relative coordi-
nates the formulation shall include the closure-of-the-loop constraint equations.
These constraint equations are more complicated to take into account with fully
recursive O(n) formulations, which are the fastest for long, open-chain systems.
For this reason, several semi-recursive formulations simpler and easier to imple-
ment have been developed in the last few years. In this paper, some semi-recursive
formulations are reviewed and a new variant, that is simpler and more general, is
described in detail. A simple way to introduce the topology of the spanning tree is
presented. Special attention is paid to closed-loop multibody systems with rods,
and the benefits of opening the loops by removing these rods •while keeping its
inertia forces exactly• are explained. Some examples and numerical results illus-
trate the aforesaid theoretical developments.
1 Introduction
Complex multibody systems arise in many areas of engineering: automobiles,
machinery, robotics, aerospace, biomechanics, etc. Although the motion differen-
tial equations that govern their dynamic behavior have been known since the times
of Newton, Euler and Lagrange, their practical application started 40 years ago
when space and robotics problems demanded more precise mathematical models,
and digital computers provided the means to numerically integrate these differen-
tial equations in an acceptable elapsed time.
The first practical systems studied (spacecraft and robots) were open-chain sys-
tems, so relative coordinates were more appropriate than Cartesian coordinates. In
addition to this, relative coordinates had fewer storage requirements. In the seven-
ties and eighties the main emphasis switched to automobile applications, which
are inherently closed-chain systems. So, software packages that used highly con-
strained Cartesian coordinates, such as ADAMS and DADS, were developed.
1
J.A.C. Ambrósio (ed.), Advances in Computational Multibody Systems, 1–23.
© 2005 Springer. Printed in the Netherlands.
2 J. García de Jalón, E. Álvarez, F.A. de Ribera, I. Rodríguez and F.J. Funes
These programs are based on global formulations, in the sense that they consider
all mechanisms –open-chain and closed-chain; loosely or severely constrained– in
exactly the same way. As a consequence, the efficiency was low. In contrast with
the global formulations are the topological formulations, which try to take advan-
tage of the system topology to improve the efficiency of the dynamic simulations.
The topological formulations tend to use relative coordinates or special sparse
matrix techniques.
In this paper, after a review of some of the improvements published in the last
few years for global formulations, a simple semi-recursive formulation based on a
double velocity transformation will be described. It is known that the presence of
rods (slender elements with two spherical joints) present some difficulties when
relative coordinates are used. So, a very interesting option is to open the closed
chains by eliminating these rods (and perhaps also some joints in the system). For
kinematics, a rod element can be replaced by a single constraint equation, in fact a
constant distance condition, but for dynamics its inertia forces should be kept in an
exact way. Later on, the procedure followed to take into account the rod inertia
will be explained in detail.
In the descriptor form, using Cartesian dependent coordinates, the motion dif-
ferential equations take the form,
M(cid:11)q(cid:12)q(cid:6)(cid:6)(cid:16)(cid:301)T(cid:540)(cid:32)Q(cid:11)q,q(cid:6)(cid:12) (1)
q
whereq is the vector of Cartesian coordinates that defines the system position, q(cid:6)
and q(cid:6)(cid:6) are its first and second order time derivatives, M is the inertia or mass ma-
trix, Q is a vector that includes the external and velocity dependent inertia forces,
(cid:41) is the Jacobian matrix of the kinematic constraint equations and (cid:79) the vector of
q
Lagrange multipliers. The position, velocity and acceleration vectors in Equation
(1) must satisfy the corresponding constraint equations,
(cid:301)(cid:11)q(cid:12)(cid:32)0 (2)
(cid:301)(cid:6) (cid:11)q(cid:12)(cid:32)(cid:301) q(cid:6)(cid:14)(cid:301) (cid:32)0 (3)
q t
(cid:301)(cid:6)(cid:6)(cid:11)q(cid:12)(cid:32)(cid:301) q(cid:6)(cid:6)(cid:14)(cid:301)(cid:6) q(cid:6)(cid:14)(cid:301)(cid:6) (cid:32)0 (4)
q q t
Equations (1) and (4) constitute a system of index 3 DAEs. If only Equations
(1) and (4) are considered, the following index 1 DAE system –equivalent to an
ODE system– is obtained:
(cid:170)M(cid:11)q(cid:12) (cid:301)T(cid:186)(cid:173)q(cid:6)(cid:6)(cid:189) (cid:173)(cid:176) Q(cid:11)q,q(cid:6)(cid:12) (cid:189)(cid:176)
(cid:171)(cid:171)(cid:172) (cid:301)q 0q(cid:187)(cid:187)(cid:188)(cid:175)(cid:174)(cid:540)(cid:191)(cid:190)(cid:32)(cid:174)(cid:176)(cid:175)(cid:16)(cid:301)(cid:6)qq(cid:6)(cid:16)(cid:301)(cid:6)t(cid:190)(cid:176)(cid:191) (5)
The matrix in this system of linear equations is known as the augmented matrix
[26] or a matrix with optimization structure [26,33]. The system of differential
equations (5) presents a constraint stabilization problem. As only the acceleration
constraint equations have been imposed, the positions and velocities provided by
the integrator suffer from the “drift” phenomenon. Two popular solutions to this
A Fast and Simple Semi-Recursive Formulation for Multi-Rigid-Body Systems 3
problem are the Baumgarte stabilization method [5,14] and the mass-orthogonal
projections of position and velocity vectors [25].
Another way to solve the constraint stabilization problem is to use velocity
transformations, which map the dependent Cartesian velocities q(cid:6) on a minimal
set z(cid:6) of truly independent velocities. Let matrix R be the orthogonal complement
of the Jacobian matrix (cid:41) , that is an n×f matrix whose columns are a basis of the
q
nullspace of (cid:41) . The dependent velocities q(cid:6) can be expressed as a linear combi-
q
nation of the columns of matrix R. The coefficients of this linear combination are
the independent velocities z(cid:6) ,
q(cid:6) (cid:32)R z(cid:6) (cid:14)R z(cid:6) (cid:14)...(cid:14)R z(cid:6) (cid:32)Rz(cid:6) (6)
1 1 2 2 f f
MatrixR can be computed very easily by a coordinate partition of vector q(cid:6) on
dependent and independent velocities. The dependent velocities are those veloci-
ties related with the columns of the pivots in the Gauss factorization of matrix (cid:41) .
q
The independent velocities z(cid:6) can be expressed as the projections of the dependent
ones on the rows of a full rank (f×n) constant matrix B in the form,
z(cid:6) (cid:32)Bq(cid:6) (7)
The rows of matrix B shall be linearly independent of the rows of the Jacobian
matrix(cid:41) . Equations (3) and (7) can be expressed together in the form,
q
(cid:170)(cid:301) (cid:186) (cid:173)b(cid:189)
(cid:171)(cid:172) Bq(cid:187)(cid:188)q(cid:6) (cid:32)(cid:175)(cid:174)z(cid:6)(cid:191)(cid:190), b(cid:123)(cid:16)(cid:301)t (8)
Because of the conditions established for matrix B, the matrix in this system of
linear equations can be inverted. Consider this inverse matrix in partitioned form,
(cid:170)(cid:301) (cid:186)(cid:170)(cid:301) (cid:186)(cid:16)1 (cid:170)(cid:301) (cid:186) (cid:170) I 0 (cid:186)
(cid:171) q(cid:187)(cid:171) q(cid:187) (cid:32)(cid:171) q(cid:187)(cid:62)S R(cid:64)(cid:32)(cid:171) m m(cid:117)f (cid:187) (9)
(cid:172) B (cid:188)(cid:172) B (cid:188) (cid:172) B (cid:188) (cid:172)0f(cid:117)m If (cid:188)
This expression is used to define matrices S and R, which are part of the referred
inverse matrix. By introducing the result of Equation (9) in Equation (8), the fol-
lowing result is obtained for the velocity transformation,
(cid:170)(cid:301) (cid:186)(cid:16)1(cid:173)b(cid:189) (cid:173)b(cid:189)
q(cid:6) (cid:32)(cid:171)(cid:172) Bq(cid:187)(cid:188) (cid:175)(cid:174)z(cid:6)(cid:191)(cid:190)(cid:32)(cid:62)S R(cid:64)(cid:175)(cid:174)z(cid:6)(cid:191)(cid:190)(cid:32)Sb(cid:14)Rz(cid:6) (cid:11)b(cid:123)(cid:16)(cid:301)t(cid:12) (10)
MatrixS in Equations (9) and (10) is never computed explicitly: only the prod-
uct (Sb) need to be computed. According to Equation (10), this product is given
by the dependent velocities q(cid:6) computed with null independent velocities (z(cid:6) (cid:32)0).
The constant matrix B can be computed in several ways. It can be computed by
the QR factorization (or the SVD) as the orthogonal complement of the Jacobian
matrix in a previous position. The matrix B so computed is valid as far as the
inverse matrix in Equation (9) exists and is well conditioned. However, there is a