The modal multifield approach in multibody dynamics [Elektronische Ressource] / von Andreas Heckmann

Andreas Heckmann

Informations

• Publié par: gottfried_wilhelm_leibniz_universitat_hannover
• Publié le: 01 janvier 2005
• Nombre de lectures: 91
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

The Modal Multi eld Approach
in
Multibody Dynamics
Von der Fakult t f r Maschinenbau
der Universit t Hannover
zur Erlangung des akademischen Grades
Doktor-Ingenieur
genehmigte
Dissertation
von
Dipl.-Ing. Andreas Heckmann
geb. am 7. Juni 1962 in Worms
20051. Referent: Prof. Dr.-Ing. habil. B. Heimann
2. Referent: Prof. Dr. rer. nat. habil. M. Arnold
Tag der Promotion: 13.04.2005iii
Preface
This thesis is the result of my scienti c work at the German Aerospace Center, DLR, in Oberpfaf-
fenhofen, which I began as a member of the Vehicle System Dynamics Department at the Institute
of Aeroelasticity in 2000. At the beginning of the year 2004 the department was reintegrated into
the Institute of Robotics and Mechatronics.
First of all, I would like to thank Professor Dr. Bodo Heimann from the University of Hannover
for his kind support and interest in my work. I would also like to thank Professor Dr. Martin
Arnold from the Martin-Luther-University in Halle-Wittenberg. Professor Arnold initiated the
subject of this thesis when he was a senior member of the Vehicle System Dynamics Department.
Although he accepted a professorship at the Institute of Numerical Mathematics in Halle in 2002,
he substantially supported the accomplishment of this doctoral thesis over the whole period.
Unfortunately, the head of the Department of Vehicle System Dynamics and co-initiator, Professor
Willi Kort m, PhD, did not live to see the end of this work, but nevertheless I am deeply indebted
to him.
Furthermore, I very much appreciate the kindness of Professor Dr. Hirzinger, head of the Institute
of Robotics and Mechatronics and Dr. Bals, head of the Control Design Engineering Department,
who gave me the opportunity to nish this thesis.
Moreover, I am grateful to Dr. Wolfgang Rulka, Dr. Stefan Dietz and Dr. Lutz Mauer from INTEC
GmbH in Oberpfaffenhofen who gave helpful advice and support whenever it was asked for.
Last but not least, many thanks to my colleagues from the Vehicle System Dynamics group. In
particular Dr. Ondrej Vacul nhad an open mind for every discussion. Dr. Wolf Kr ger, Dr. Klaus
Schott and Mrs. Christine Traurig in addition to Dr. Vacul ngave a great deal of useful hints on the
writing of the manuscript.
Oberpfaffenhofen, 15th April 2005
Andreas Heckmanngewidmet meinen beiden Kindern
Helena und Hannelorev
Contents
Preface iii
Notation vii
Kurzfassung xi
Abstract xiii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Overview on Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Objectives and Structure of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Theoretical Framework 11
2.1 Analytical Multibody Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.1 The Floating Frame of Reference . . . . . . . . . . . . . . . . . . . . . . 13
2.1.2 The Modal Approach for Displacements, Strains and Stresses . . . . . . . 14
2.1.3 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.4 The Equations of Motion of an Elastic Body . . . . . . . . . . . . . . . . . 19
2.1.5 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Material Constitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.1 The Electric Gibbs Potential . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.2 Alternative Material Constants . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.3 Physical Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Augmented Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.1 Generalised Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . 37
2.3.2 Strong Thermal and Electrostatic Field Equations . . . . . . . . . . . . . . 38
2.3.3 Modal Multi eld Approach . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.4 The Equations of Motion of a Piezo-Thermoelastic Body . . . . . . . . . . 42
2.3.5 The Electrostatic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3.6 The Thermal Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43vi Contents
2.3.7 Topological Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Basic Modelling Concepts and Processes 49
3.1 of Electromechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.1 An Analytical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.2 Piezo-Patches on Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1.3atches on Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.1.4 Control of a Metal Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.1.5 Discussion of the Approach . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2 Modelling of Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.1 The Effects of Coupling and Inertia . . . . . . . . . . . . . . . . . . . . . 67
3.2.2 Thermal Response Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2.3 Veri cation Example 1: Disc with Thermal Loads . . . . . . . . . . . . . 76
3.2.4 V 2: Hot Spot . . . . . . . . . . . . . . . . . . . . . . 80
4 Tools and Applications 91
4.1 Software Components and Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 Active Damping of Railway Car Body Vibrations . . . . . . . . . . . . . . . . . . 95
4.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2.2 Development Process and Environment . . . . . . . . . . . . . . . . . . . 95
4.2.3 Simpli ed Beam Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.2.4 Detailed Car Body Model . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.3 A Machine Tool with Thermoelastic Deformations . . . . . . . . . . . . . . . . . 103
4.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.3.2 Simulation Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.3.3 Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.3.4 Multibody Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5 Summary and Outlook 113
A Speci cation of Input Data 117
A.1 Control of a Metal Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.2 Veri cation Example 1: Disc with Thermal Loads . . . . . . . . . . . . . . . . . . 121
A.3 V 2: Hot Spot . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Bibliography 131vii
Notation
Speci c terms and their corresponding symbols are explained when they appear for the rst time.
Scalar Quantities
c speci c heat capacity
h lm coef cient, specifying boundary heat ux due to convectionf
j imaginary unit
m mass
t time
A cross section area of a beam
B boundary area
E Young’s modulus
G electric Gibbs potential
I geometrical moment of inertia of a beam
Q applied electric charge’
S heat source density
V volume
thermal expansion coef cient
entropy density
’ electric potential
# temperature increment w.r.t. reference temperature 0
% density
Poisson’s constant
thermal conductivity
absolute temperatureviii Notation
Vectorial Quantities
a translatory acceleration
b generalised vector of a exible body
c Lagrange co-ordinate
d electric displacement vector
e electric eld strength
f external force
h load term of discretised differential equations
n outer unit normal vectorB
q heat ux
r position vector
u displacement vector
v translatory velocity
y vector constituted by a minimum set of generalised co-ordinates
z generalised co-ordinates
angular acceleration
" strain tensor in vector format
vector of Lagrange multiplier
! angular velocity
stress tensor in vector format
u input vector in state space description
x state vector in state space
y output vector in state space description
Matrices
A rotation matrix, which transforms vector quantities de ned w.r.t. the referenceIR
frame (R) into the equivalent description w.r.t. the inertial co-ordinate system (I)
B matrix of the partial derivatives of the modal shape functions
D damping, heat capacity or coupling matrix, to be multiplied by
rst time derivatives of generalised co-ordinates
H material coef cient matrix based on the electric Gibbs potential
H elasticity tensor at constant electric eld and temperaturec
H piezoelectric tensor at constant temperaturee
H thermal moduli at constant electric eld
H permittivity tensor at constant strain and temperatureNotation ix
H pyroelectric tensor at constant strainp
H heat capacity coef cient at constant strain and electric elda
i;iI identity matrix with dimension i, I 2 Ri i
J Jacobian matrix
K stiffness, conductivity, electric capacity or coupling matrix, to be
multiplied by generalised co-ordinates
M mass matrix
thermal conductivity matrix
matrix of the modal shape functions
A system matrix in state space description
B input matrix in state space
C output matrix in state space description
D feed-through matrix in state space description
Generally Used Indices
( ) assigns a q