Computeraided analysis of rigid and flexible multibody...

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1
GraSMech – Multibody 1
Computeraided analysis
of rigid and flexible
multibody systems
GraSMech course 20092010
GraSMech – Multibody 2
Organization and Teaching team
Part I (20092010): Modelling approaches and numerical aspects
Speakers
Dr Olivier Brüls, OB, ULg
Prof. Paul Fisette, PF, UCL
Prof. Olivier Verlinden, OV, UMONS
Part II (20102011): Flexible systems and special topics
Web site: http://mecara.fpms.ac.be/grasmech (hopefully copy of the
slides before the lesson)
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GraSMech – Multibody 3
Schedule
Lessons on wednesday from 14.00 to 17.20
Lesson 1 (February 17):
Part one : Introduction +applications (OB, OV, PF)
Part two : approach based on Minimal coordinates (OV)
Lesson 2 (February 24): approach based on Cartesian coordinates (OV)
Lesson 3 (March 3): approach based on Finite elements (OB)
Lesson 4 (March 10): approach based on Relative coordinates (PF)
Lesson 5 (March 17): numerical problems (integration,…)
After Easter : presentation of projects by students
GraSMech – Multibody 4
What is a multibody system ?
Multibody System (MBS):
Bodies (rigid or flexible)
Joints
force elements
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GraSMech – Multibody 5
Introduction : multibody applications
Set of articulated
bodies
Mechanisms
Railway vehicles
Robot manipulators
Space applications
t
y
GraSMech – Multibody 6
Principle
System defined
from technical
elements
Equations of motion built
(and hopefully solved)
automatically
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GraSMech – Multibody 7
Purpose of simulation
To assess the behaviour of a system before its construction
(virtual prototyping)
so as to dimension its mechanical elements
so as to optimize its performances
so as to design a controller
…
To identify possible problems on an existing system
Computeraided design tool
GraSMech – Multibody 8
MBS versus FEM
Multibody approach
 Rigid bodies
 Large motions
Finite elements
 Flexible bodies
 Structural and modal
analysis (small motions)
MBS with flexible bodies
limit : efficiency versus universality
Why do we need to develop a new theory whereas structural
dynamics and finite element theories are well established ?
p(t)KqqCqM =++ &&&
0=
+=+
g(q,t)
λJQ,t)qc(q,qM T&&&
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GraSMech – Multibody 9
Historical origins
Spacecraft : 1960 Mechanisms : 1970
Robotics : 1980 Flexible bodies : 1990
GraSMech – Multibody 10
Historical origins
The American Explorer I flew in January 1958. Explorer I was
long and narrow like a pencil. It was supposed to rotate around its
own centerline, like a pencil spinning about its lead. It was
definitely not supposed to rotate end over end, like an airplane
propeller or a windmill blade.
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GraSMech – Multibody 11
Some unstable satellites
GraSMech – Multibody 12
Historical origins
First publications :
Hooker and Margulies : « The Dynamical Attitude Equations for
a nBody Satellite », J. Atronautical Sciences, 1965.
Roberson and Wittenburg : « A Dynamical Formalism for an
Arbitrary Number of Interconnected Rigid Bodies, with reference to
the Satellite Attitude Control », proc. of the 3rd. Int. Congress of
Automatic Control, Butterworth and Co., Ltd, London 1967.
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GraSMech – Multibody 13
Multibody Systems
Database
GraSMech course 20092010
GraSMech – Multibody 14
MBS database  Basic elements
The basic elements to describe a MBS are
Bodies
Joints
Force elements
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GraSMech – Multibody 15
MBS Database  Bodies
Bodies, defined by
mass mi
position of center of mass Gi (rGi/i)
inertia tensor (ΦGi)
Body fixed frame (local reference)
Auxiliary body fixed frames or attachment points
(coordinate systems with fixed position and
orientation with respect to body)
Ground=fixed reference body (except space applic.)
GraSMech – Multibody 16
MBS Database  Bodies
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GraSMech – Multibody 17
MBS Database  Joints
In MBS, a joint is defined between two body fixed
frames and imposes their relative motion
A joint is characterized by
a number of n of degrees of freedom (dof)= number
of independent elementary motions (rotation,
translation) alllowed by the joint;
a number l of prevented elementary motions
(constraints on the relative motion) + l joint forces
(non contributive forces > no power)
Generally speaking, in 3D
l+n=6
GraSMech – Multibody 18
MBS Database  Principal joints
prismaticrevolute cylindrical
1 degree of
freedom (rot)
parameter: θ
1 degree of
freedom (trans)
parameter: d
2 degrees of
freedom
(1 rot+1 trans)
parameters: d, θ
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GraSMech – Multibody 19
MBS Database  Principal joints
planaruniversalspherical
3 degrees of
freedom (3 rot)
parameters: α, β, γ
2 degrees of
freedom (2 rot)
parameters: α, β
3 degrees of
freedom
(1 rot+2 trans)
parameters: dx, dy, θ
GraSMech – Multibody 20
MBS Database  Principal joints
Helicoidal
(screw)
1 degree of freedom (1 rot and
1 trans coupled: d=θ p/2π)
p=pitch
parameter: d or θ
Contact joint
Without slip: 3 degrees of
freedom (nonholonomic
constraints)
With slip: 5 degrees of
freedom
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GraSMech – Multibody 21
MBS Database – Force elements
A force element
is defined between two body fixed reference frames
Is function of the relative motion (position and
velocity) of the frames and/or time
Ex: spring, damper, tyre, elastic contact, motor torque
Example : spring and damper
suspension elements
GraSMech – Multibody 22
MBS Database  Force elements
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GraSMech – Multibody 23
Example: double pendulum
2 bodies
2 revolute joints (01.1 and 1.22.1)
2 forces (m1 g, m2 g)
GraSMech – Multibody 24
Example: four bar mechanism
3 bodies
4 revolute joints (01.1, 1.22.1, 2.23.1 and 3.20.1)
3 gravity forces (m1 g , m2 g and m3 g)
1 moment C z0 on 1.1 (reaction on 0)
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GraSMech – Multibody 25
Example: suspension
2 bodies
2 revolute joints (0.11.1 and 1.22)
2 gravity forces (m1 g, m2 g)
1 springdamper between 0 and 1
eventually a tireroad contact force
GraSMech – Multibody 26
Different approaches of MBS formalisms
Choice of coordinates
Mechanical principle to generate the
equations
Computer implementation
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GraSMech – Multibody 27
Choice of coordinates
Purpose of coordinates (configuration parameters q): to
express the motion (kinematics) of the bodies
Different approaches exist !
GraSMech – Multibody 28
Principal types of coordinates
3 basic choices
1 minimal coordinate
0 constraint equation
1 (complex) ODE
3 relative coordinates
2 constraint equations
5 DAE
9 cartesian coordinates
8 constraint equations
17 (simple) DAE
But also: natural coordinates,
finite element coordinates
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GraSMech – Multibody 29
Choice of coordinates
Example of a 6 DOF robot
With absolute coordinates
6 bodies:
=> 36 diff. eq. of motion
6 revolute joints:
=> 30 algebraic equation
TOTAL : 66 equations (DAE)
With relative (or minimal)
coordinates
6 relative coordinates
TOTAL : 6 equations (ODE)
Kinematic chain: relative coordinates come down to minimal coord.
GraSMech – Multibody 30
Choice of coordinates
Main body : 6 absolute coordinates
Each bogie frame : 6 absolute coordinates
Each wheelset : 6 absolute coordinates
All bodies “elastically” interconnected
More natural and efficient with cartesian
coordinates
6x12 equations (ODE)
With any approach
Example of a railway vehicle
Elastically connected bodies: abs. coord. come down to min. coord.
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GraSMech – Multibody 31
Choice of coordinates
Relative or absolute ?
Wheel carrier
Front of vehicle
That’s the question !
In this case, minimal coordinates could be particularly tedious …
GraSMech – Multibody 32
Natural coordinates: ~ point coordinates and unit
vectors (No rotational coordinates)
Finite element coordinates
(node coordinates + shape functions)
Other coordinates
Rigid bodyused by :
Garcia de Jalon
Nikravesh
used by : MECANO
(SAMCEF), University
of Milano, …
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GraSMech – Multibody 33
Choice of coordinates  Conclusions
Choice of coordinates has an impact on the form and
number of equations. The important is the computer
efficiency of the global problem
CoordinatesFormulation
Of eq. of motion
Numerical
Methods
Computer
implementation
GraSMech – Multibody 34
Different approaches of MBS formalisms
choice of coordinates
mechanical principle to generate the
equations
computer implementation
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GraSMech – Multibody 35
Which mechanical principle ?
Newton – Euler (basic)
Virtual work or power principle (d’Alembert)
Lagrange’s equations (Hamilton)
Recursive NewtonEuler algorithm
Recursive “ order N ” algorithm
GraSMech – Multibody 36
Choice of mechanical principle
The choice is closely linked to the choice of coordinates
Minimal coordinates: virtual work principle, eventually
Lagrange (less efficient)
Cartesian coordinates: basic NewtonEuler or
Lagrange
Relative coordinates: Recursive NewtonEuler,
Recursive O(N)
Finite elements: Lagrange
+ Choice of numerical methods !!
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GraSMech – Multibody 37
Recursive NewtonEuler
Works on kinematic chains => well adapted to relative
coordinates
Step 1
« forward »
kinematical
recursion
ωωi
ωj
ωωh Xi
body ibody j
body h
I1
2I
3I
Base : 0
Ωj
GraSMech – Multibody 38
Recursive NewtonEuler
Step 2
« backward
» dynamical
recursion
I
Gibody : , i m
hj
Oi
Fiext
Iii
Lexti
Fj
Lj
Fi
Li
Oj
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GraSMech – Multibody 39
Form of the equations of motion
Secondorder differential equations
And eventually constraint equations
=> set of differentialalgebraic equations (DAE)
Without
constraints (ODE)
GraSMech – Multibody 40
Different approaches of MBS formalisms
Choice of coordinates
Mechanical principle to generate the
equations
Computer implementation
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GraSMech – Multibody 41
Computer implementation
Numerical approach (one simulation program)
MBS description
set of numerical data
generate numerically
all system equations
according to the generic
MBD formallism
solve the system
numerically
Numerical implementation
Initial conditionsiterate
including :
the MBS database
initial values qq &,
GraSMech – Multibody 42
Different approaches of MBS formalisms
set of data
generate symbolically
all system equations according
to the generic MBD formallism
Specific model equations,
given as a subroutine
Symbolic implementation
Symbolic approach (one program per application)
Solve
iteratively
Numerical implementation (one per application)
MBS description
including :
the particular topology
symbols and zeros
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GraSMech – Multibody 43
Principal simulation softwares
MSC/ADAMS (USA, reference in road vehicles)
LMS/Virtual Motion (Belgium, originally DADS in USA)
SIMPACK (Germany, leader in railway dynamics)
In Belgium :
SAMCEF/MECANO (Samtech/ULg, based on finite elements)
ROBOTRAN (UCL, advanced symbolic generation of
equations of motion)
EasyDyn (UMONS, free library (GPL) developed for teaching)
GraSMech – Multibody 44
Application examples (PF)
Future challenges (OB)
GraSMech course 20092010