Computer-aided analysis of rigid and flexible …hosting.umons.ac.be/html/mecara/grasmech/Intro.pdf1...

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GraSMech – Multibody 1

Computer-aided analysis of rigid and flexible multibody systems

GraSMech course 2005-2006


Organization and Teaching team

Part I (2005-2006): Modelling approaches and numerical aspects

Speakers :Dr. Olivier Brüls, OB, ULgDr. Gaëtan Kerschen, GK, ULgProf. Paul Fisette, PF, UCLDr. Joris Peeters (replacing Prof. Wim Desmet), JP, KULProf. Jean-Claude Samin, JCS, UCLProf. Olivier Verlinden, OV, FPMs

Part II (2006-2007): Flexible systems and special topics


Schedule

Lessons on wednesday from 14.00 to 18.00

Lesson 1 (February 15) :

Part one : Introduction (JCS) + applications (PF,OB,GK,OV)

Part two : approach based on Minimal coordinates (OV)

Lesson 2 (February 22) : approach based on Cartesian coordinates

(OV,JP)

Lesson 3 (March 1) : approach based on Finite elements (OB,GK)

Lesson 4 (March 8) : approach based on Relative coordinates (PF,JCS)

Lesson 5 (March 15) : numerical problems (integration,…)

After Easter (April 26 ?) : presentation of projects by students

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Introduction

Prof. J. C. SaminUniversité catholique de Louvain

[email protected]


Computer-aided analysis of rigidand flexible multibody systems


Introduction : multibody applications

Set of articulated bodies

MechanismsRailway vehicles

Robot manipulatorsSpace applications

t

y


What is a multibody system ?

Multibody System (MBS):

Bodies (rigid or flexible)

Joints

force elements

3


Principle

System defined from technical

elements

Equations of motion built (and eventually solved)

automatically


Purpose of simulation

To assess the behaviour of a system before its construction

so as to dimension its mechanical elements

so as to optimize its performances

so as to design a controller

…

Computer-aided design tool

To identify possible problems on an existing system


MBS versus FEM

Multibody approach

- Rigid bodies

- Large motions

MBS with flexible bodieslimit : efficiency versus universality

Why do we need to develop a new theory whereas structural dynamics and finite element theories are well established ?

Finite elements

- Flexible bodies

- Structural and modalanalysis (small motions)

p(t)KqqCqM =++ &&&

0)(

=+=+

g(q,t)λJQ,t)qc(q,qqM T&&&

4


Historical origins

Spacecraft : 1960 Mechanisms : 1970

Robotics : 1980 Flexible bodies : 1990


Historical origins

The American Explorer I flew in January 1958. Explorer I waslong 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.


Some unstable satellites

5


Historical origins

First publications :

Hooker and Margulies : « The Dynamical Attitude Equations for a n-Body 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.


Different approaches of MBS formalisms

Choice of coordinates

Mechanical principle to generate the

equations

Computer implementation



Absolute coordinates(a bad example…)

6 bodies:=> 36 differential eq. of motion6 revolute joints:=> 30 algebraic equations

TOTAL : 66 equations (DAE)for a 6 d.o.f. system

6



Absolute coordinates(a good example…)

Main body : 6 absolute coordinates Each bogie frame : 6 absolute coordinates Each wheelset : 6 absolute coordinates

All bodies “elastically” inter-connected6n equations (DAE)

for a 6n d.o.f. system



6 d.o.f.

6 relative coordinates

=> 6 ODE

Relative coordinates(a good example…)



Absolute or Relative coordinates ?(a medium example…= reality)

either :

3 relative coordinates

2 constraints

=> 5 DAE

5 absolute coordinates

4 constraints

=> 9 DAE

or :

1 d.o.f.

1 “minimal” coordinate

0 constraints

=> 1 ODE

or :

7



user friendliness…

Wheel carrier

Front of vehicle

Data input :of the reference configuration

Absolute/relative coordinates(user point of view …)

Absolute :• for each center of mass :

• for each body frame : 321 ,, ϑϑϑ

zyx ,,

Relative :• , but in a local body frame

• relative orientation of frames

zyx ,,



Conclusions

number of equations <=> computer efficiency

+/- user- friendly

kinematical and dynamic equations +/- complicated

+/- easy for MBS computer implementation

= leads to DAE equations

influence on :



An alternative approach :“natural” coordinates

Rigid body

Set of equivalent punctual masses

used by :Garcia de JalonNikravesh

No need of rotational coordinates !

8





equations



Which formalism ?

Newton – Euler (basic)

Virtual work or power principle

Lagrange equations

Recursive Newton-Euler algorithm

Recursive “ order N ” algorithm


Description of a multibody system (basic elements)

Which formalism ?

Topology

Example : « tree-like »

9



Bodies, defined by :

mass

position of the center of mass

body-fixed frame

inertia tensor (with respect to the body-fixed frame)

auxilliary body-fixed frames (attachment points)

Which formalism ?

Ground = fixed reference body (except for space applications)



Joints, defined between frames attached to bodies

Which formalism ?

revolute

1 relative d.o.f.(rot)

prismatic

1 relative d.o.f.(trans)




Which formalism ?

spherical

3 relative d.o.f.(3 rot)

universal

2 relative d.o.f.(2 rot)

cylindrical

2 relative d.o.f.(1 rot + 1 trans)

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Which formalism ?

Contact joint

5 relative d.o.f.

or 3 relative d.o.f. (rolling without slip)

Planar

3 relative d.o.f.(1 rot + 2 trans)

Helicoidal (screw)

1 relative d.o.f.(1 rot/trans)



Force elements, defined between bodies

Which formalism ?

Example : spring and damper

suspension elements


Which formalism ?


For each individual body, write :

Newton (vector) equation : xmF &&=∑Euler (vector) equation : HL &=∑

Collect all equations

… easy if absolute coordinates are used …

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Which formalism ?

Collect all equations

L

M

M

M

M

&&

&&

&&

&&

&&

&&&&

&&

&&

OLLL

LLL

LLL

+

2

2

2

1

1

1

1

1

1

2

2

2

133

132

131

123

122

121

113

112

111

1

1

1

zyx

zyx

mm

mIIIIIIIII

mm

m

ψϕϑ

F=

rely on sparse-oriented numerical methods



Which formalism ?

Collect all equations in your computer program

L&& +qM vecontributiF= constraintF+

1 contributive torque component

(spring, friction, actuator,…)

5 constraint force/torque components



Which formalism ?

eliminate (numerically) all constraint force/torque component

=+L&&qM vecontributiF constraintF+J J

complete the dynamical set of equations in your

computer program

with the kinematical joint constraint equations

… lead to a huge amount of equationsbut relatively simple to obtain …


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Which formalism ?

Newton – Euler (modified)

all internal force/torque of the appending subsystem disappear

for each joint, sum analytically all equations of the « descendant » bodies

I1

ˆ

2I

3I

i

h k

j

Li

Fi

Oi

I1

ˆ

2I

3I

i

h k

j

Li

Fi

Oi


Which formalism ?


all internal force/torque of the appending subsystem disappear

Translational (vector) joint equation : ),,,( extii FqqqfF &&&=

Rotational (vector) joint equation : ),,,,( extextii LFqqqhL &&&=

project analytically these vector equations onto the joint axis

the joint constraint force/torque components also disappear


Which formalism ?


implement these analytical results in your computer program

either in implicit form (inverse dynamical model)

),,,,( extext LFqqqQ &&&Φ=

),,,()( extext LFqqcqqMQ &&& +=

or in semi-explicit form (direct dynamical model)

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Which formalism ?

Virtual power (or work) principle

close to the modifed Newton-Euler approach(and historically inspired by it)

Translational vector equation

I1

ˆ

2I

3I

Gi

hk

jg

Li

Fi

O0i

z i°∆

Oi


Which formalism ?

Rotational vector equation

I1

ˆ

2I

3I

Gi

hk

jg

Li

Fi

O0i

z i°∆

Oi



Which formalism ?


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Which formalism ?

expand the r.h.s. of the equations in terms of the generalized coordinates and their derivatives

implement these analytical results in your computer program

),,,,()( gLFqqcqqMQ extext&&& +=



l.h.s.


Which formalism ?

)( jijoioj qAAA ⋅=

L& +Ω+= )( jjij qωω

L&& += ij ωω

ωωi

ω j

ωω h Xibody i

body j

body h

I1

2I

3I

Base : 0

Ωj

Recursive Newton-Euler algorithmespecially convenient for relative coordinates

first develop a « forward » kinematical recursion


Which formalism ?

Recursive Newton-Euler algorithmespecially convenient for relative coordinates

first develop a « forward » kinematical recursion

Oi

Oj

dij

body i

body j

body h

zi

I1

2I

3I

pi

pj

Base : 0

L++= )( iiij qzpp

L&& += ij pp

L&&&& += ij pp

15


Which formalism ?

second, develop a « backward » dynamical recursionin vector form


I

Gibody : , i m

hj

Oi

Fiext

I ii

L exti

-Fj

-Lj

Fi

Li

OjL&& iiji xmFF +=

L& +⋅+= iiji ILL ω


Which formalism ?


iiiii QLF =⋅+⋅ ϕψ

),,,,( jjiiii LFxQ L&&& ωΦ=

l.h.s.

r.h.s.

finally, collect all these dynamical equations :

implicit form (inverse dynamical model)

),,,,,( gLFqqqQ extext&&&Φ=


note : the semi - explicit form can also be obtained by isolating q&&


Which formalism ?

Recursive “ order N ” algorithm

a first « forward » kinematical recursion step

a second « backward » dynamical recursion step

a third « forward » recursion step

[ ]),,,()(1extext LFqqcQqMq &&& −= −

explicit form of the dynamical model

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Which formalism ?

comparison of computational costs(for simulation purpose : explicit form)

... ... ... ... ... ... ... ...

1

2 3

4 5 6 7

8 9 10 11 12 13 14 15

R1

T2 R3

T1 R2 T3 R1

T2 R3 T1 R2 T3 R1 T2 R3

I 3˜

^

Example : a binary tree multibody system


Which formalism ?

..Minimum number of operations to compute : q

Lagrange andvirtual principles Newton/Euler

recursive

Order(N)

# (+ - x :)

# dof





equations


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Numerical approach

generate numericallyall system equations

according to the generic MBD formalism

solve the system numerically

Numerical implementation

Initial conditionsiterate

MBS description

set of numerical data

including :the particular topologyconstant valuesinitial values qq &,



Symbolic approach

)( and)( qJqh

),( and )( qqcqM &

Generate all system equations :

can also be done by implementingthe MBD formalism in a symbolic

computer program



generate symbolicallyall system equations according to the generic MBD formalism

Specific model equations,given as a sub-routine

Symbolic implementationSymbolic approach

Solveiteratively

Numerical implementation

set of data

MBS description

including :the particular topologysymbols and zeros

18


Principal simulation software

MSC / ADAMS (1973, USA, reference in road vehicles)

LMS / DADS (Belgium, originally USA)

SIMPACK (DLR, 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 (FPMs, free library (GPL) developed for teaching)


Application examples


Computer-aided analysis of rigidand flexible multibody systems


ApplicationsFlexible multibody dynamics

Prof. O. BrülsUniversité de Liè[email protected]


Finite Element method [Géradin & Cardona ’01]

Samcef-MECANO software

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Deployable structures

Large space structuresCompact stowed configurationElastic effects:

lightweight designpre-stressingsmall damping



Large antennaNo axial symmetry1 kinematic dof !Slow deployment (15 min)1-g test rig ⇒ 0-g behaviour?



FE mechanical modelFlexibility (struts + panels)Local stiffnesses & frictionInertial forces = negligible1400 equations

Kinematic analysisImposed angle of the central bodyDriving torque?Forces in the struts?Hinge angles?

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a) 1-g model ⇒ experimental validationb) 0-g model ⇒ accurate prediction




Landing gears

Different phasesDeployment / retractionGround impactRollingBreakingTaxiing

Multidisciplinary approachDeformable mechanismTyreHydraulics (shock-absorber, brake, steering)Active / semi-active control

Concorde – nose undercarriage

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Landing gears

Motivations for modelling:

Optimal design configurationmass

Pre-prototyping checks motionloads, stressesstability (shimmy)complementarity with experimental tests


Landing gears

Deployment Wheel/ground impact


Landing gears

Landing of the A380

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Applications

Prof. P. FisetteUniversité catholique de Louvain

[email protected]


Hybrid modeling

Dynamical identification


Bogie actuation : hybrid modeling

« Multibody/Electrical » coupled modelSymbolic generation of the electro-multibody model

Dynamical model of the electrical actuators

Equational-level coupling : same « frequency range »

Application (collab. Bombardier-Transportation, Crespin, France)

Articulated bogie (High speed train)

3-phase induction motor

Effect of the electrical frequency on fatigue problems (chassis)


Bogie actuation : hybrid modeling

Problem: Torque oscillations lead to fatigue stress

in the bogie frame during acceleration

Force in themotor to chassis - joint

« Electro-multibody » model

Rr1

Rr2

Lr1

Lr2

Lr3

ROTOR

Rr3

Rs1

Rs2

Rs3

Ls1

Ls2 L

s3

us1

us2

us3

STATOR

Msr

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Semi-active suspension modeling

« Multibody/Hydraulic/Electrical » coupled modelSymbolic generation of the multibody model

Dynamical model of the hydraulic system

Fast simulation (« real time » in Simulink)

Application (collab. Tenneco-Automotive, Saint-Trond)

Audi A6 (more than 80 relative coordinates)

« Kinetic » system : cylinders, pipes, accumulators, …

Behavior and sensitivity analysis


Semi-active suspension modeling

Problem: Behaviour prediction of this new

suspension system on modern car

« Hydro-multibody » model

Carbody rollWrap excitation

HydraulicDynamics


Dynamical identification of Tree-like MBS

« Multibody symbolic dynamical model« Inverse dynamics » model

« Reaction dynamics » model

Parameter combination : morphoplogy-dependent

Application (collab. KULeuven)

KUKA Robot identification

( Human body identification )

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Serial manipulator

Human Body• 20 - 30 dof

« recursive »dynamic model

• … 6 dof

« in extenso »dynamic model

Find ? : Minimal set of dynamical parameters : p



l

jk

d

b

i

i

id

d

i

i

j

lzi

Gi

Bi

O0i

mi =X

mj

mibi =X

mj dij

Ki = Ii −X

mj dij

. dij

j: i j≤

j: i j≤

j: i j≤

“Augmented” body i

Barycentric mass :

Barycentric vector :

Barycentric inertia tensor :


Barycentric parameters


Recursive formulation (origin : Renaud-88 for linear chains)=> Extension to tree-like systems (recursive form)

body i

body j

body k

Fi

Li

Fj

Lj

ωi

kin.

dyn.

Gi =Xj∈ ı

Gj−Fiext+βi .biz+ mi γi

Fi = miα∗i+Gi

L i =Xj∈ ı

(L j +edijz .Gj)−L iext−ediiz .Fiext

+ K iz . ωi+ ωi .K i

z .ωi+ebiz .αi

Qqqq =δϕ ),,( &&&


25


Minimal set of body j parameters Redefinition of parent body i parametersmj (mi := mi)

bj1, bj2, b

j3 (bi := bi)

/ Ki :=Ki +Kj

Minimal set of body j parameters Redefinition of parent body i parameters/ (mi := mi)

bj1, bj2 bi := bi + bj3 ϕ

j

Kj12,K

j13,K

j23,K

jd,K

j33 Ki :=Ki − bj3(d

ij. φ

j+ φ

j. d

ij)−Kj

s φj. φ

j

with Kjd = Kj

11 −Kj22 with Kj

s = Kj11 + Kj

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Prismatic joint : combination and report rules

Revolute joint : combination and report rules

Set of parameters : p



1. Symbolic Computation of the theoretical combination rules

2. Symbolic Inverse or Reaction Dynamics

4. Detection of the « zero » columns in D

,,,,,,,,

,,,,,,,11*1*1*1*1133

3333333

dyyyzxzxyzxsy

dyyyzxzxyzx

KKKKKbbKb

KKKKKbbp =

correspond to parameters that must be removed from p

5. Results : => Minimal Parameters set δ ∗

=> Identification Matrix ϕ ∗ Qqqq =*),,(* δϕ &&&

=

xxxxxxxx

xxxxxxxxxxxx

D

0...00...0

.....................0...00...00...0

Partial derivative matrix D3. Symbolic Differentiation w.r. to p



Internal model inverse dynamicsExternal model reaction dynamics1 2 3 4 5 6

nr. parameter exact parameter σi σe σcvalue

P1 Kr111 32.0920 0.0916 0.0873 0.0611P2 Kd2 −12.3614 0.0906 0.0332 0.0304P3 K2

12 0.6174 0.1150 0.0351 0.0313P4 Kr213 1.5078 0.1231 0.0528 0.0467P5 K2

23 −0.2974 0.0990 0.0439 0.0423P6 b31 0.0580 0.0259 0.0080 0.0075P7 b32 2.5780 0.0392 0.0111 0.0107P8 Kd3 2.2511 0.0818 0.0322 0.0296P9 K3

12 0.5953 0.0909 0.0239 0.0227P10 K3

13 −0.2978 0.0562 0.0251 0.0237… … … … … …

Standard deviation

KUKA IR 361: Dynamic Parameter Estimation Combining Internal and External models


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2D volley-ball motion

Parameter estimation : mean error (%)

3. trunk

2. thigh

1. shank

5. arm

4. head 6. forearm

I1

I3

Reaction measurement at the foot

Reaction measurement at the trunk

error [%]

error [deg]

error [%]

error [deg]


ApplicationsRailway vehicles

Prof. O. VerlindenFaculté polytechnique de Mons

[email protected]



Building railway vehicle models

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Principal dedicated softwares

GENSYS (Sweden)MEDYNA (Germany, no longer distributed)NUCARS (USA, Association of American Railroads)VOCODYM (France, SNCF and INRETS)VAMPIRE (UK, AEA Technology)SIMPACK (Germany, DLR)MSC.ADAMS/Rail (USA, MEDYNA routines for contact)

Can be considered as reliableERRI benchmark, Manchester benchmarksComparison with measurements


Wheel-rail Contact - Geometric problem

Contact configuration (position, curvatures, ...) in terms of the lateral displacement


Difficulties of the geometric problem

Double contact

Discontinuities and contact jumps (several contact points)

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Normal contact force

Options for determination of normal contact force

contact=geometric constraint: N=Lagrange multiplier -> contact zone by Hertz theory

Advantages: ability to tabulate the contact, numerically more stableDrawbacks: not adapted to several contact points, contact area always assumed elliptical

contact=force element with wheel-rail penetration -> contact area and normal force by elastic forces.

Advantages: naturally adapted to several contact pointsDrawbacks: computational burden, stiff motion equations


Tangential contact forces


Assessment of performances

Major concernsStability, critical speedRide quality – comfortUIC 518 criteria (tests): stability, track loads, derailment

Main design variablesGlobal layout (distribution of bogies)Primary suspensions and contact geometry (stability and track loads)Secondary suspension (comfort)

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Stability – kinematic yaw


Linear analyses

Equations of motion can be linearized about a stationnary motion -> study of small perturbations


Root locus

Root locus: evolution of poles (roots) with velocity

Unstable(Re>0)

Stable(Re<0)

10% limit

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Root locus of a bogie

There exists a critical speed !


Critical speed vs concity


Tramway of Minneapolis

Rigid wheelsets

Independent wheels

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Pole leading to instability at low conicity



Pole leading to instability at high conicity


Nonlinear stability analysis

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Cityrunner

Short carbodies: adapted to tortuous networksMore carbodies than bogiesCentral carbody not connected to the railComfort of the central carbody driven by the global dynamics of the vehicle


Cityrunner


Derailment study

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Y/Q for Minneapolis


Conclusions

Reliable commercial softwares exist with many possiblitiesThey are classically used in industry

Attention to modelling mistakes !A prototype will always be built for final validation


Today Challenges


Multi – physics modeling and simulation

Real time computation

optimization, «hardware in the loop» simulations, …

Parameter identification

Education

…

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Perspectives: multiphysics

Mechatronics = science of motion controlMechanismActuators Sensors Control law Cars (suspension, ESP…)

Large manipulators

High-speedmachines

Magnetic bearings


Perspectives: multiphysics

Integrated analysis and designMultibody dynamicsElectronicsHydraulicsPneumaticsElectrical circuitsPiezoelectricityMagnetics…

Next step: micro-mechatronicsnew technologies / new physical effects

Radial comb motor

Bistable mechanism

Micropump


Multibody System Optimization

Optimization of « dynamical » performancesFast computation of the cost function (ex. symbolic model)

Optimization algorithms : deterministic or stochastic methods

Geometrical/morphological optimization of mechanismsRobust assembly method (loop closure)

Problem of local optima (related to mechanism reconfiguration)

Problem of computer time (=> parallel computation)

Illustration Car suspension

Planar mechanism

Acherman steering mechanism

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Suspension geometricaloptimization

for lateral dynamicstability

Test : lane changeat 65 km/h


Student final project



Optimal design of path-following mechanisms• Deformation of mechanism to circumvent assembling constraints

• objective : strain energy of the bars• parameters : natural lengths of the springs

Non-Linear Least Squares Optimization

Use of natural coordinates

Initial Optimal



• Different starting points different local optima

• Example : Six-bar steering linkage mechanismto fulfill the Ackermann condition

Other design criteria to choose best mechanim(e.g. robust design)

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Multibody Real time computation

For « greedy » simulations, such as:Sensitivity analysis

Identification process (many run of the model / many « exciting »trajectories

Optimzation of a dynamic transient behaviour

For real « Real time » requirementReal time control of robots (including an inverse dynamic model)

« Hardware in the loop » analysis

Others (Haptic systems, games, …)


Multibody Real time computation

Example : semi-active control of car

Non-Model-based controller …=> Model-based controller

KUL-PMA


Multibody Education via projects !

At the end of a MBS project, students should be able to:formulate consistent hypotheses (Engineering)manipulate 3D kinematic tools (Physics)use the Newton-Raphson method to find system equilibrium (Numerical methods)build a well-structured simulation program (Programming)make a 3D drawing of the vehicle and a 3D virtual animationanalyse their results (Physics)question their model (Engineering)

at the basis of the final examination

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Multibody Education via projects !

…numerical methods…a « real » system

… a group

…theoretical formalisms

….a CAD system

Given …

….programming tools

…system parameterization…system animation …system analysis

… Produce


Problem to solve : truck “jacknifing”


Typical Results

… …

Solved !

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Problem to solve : Sidecar side-skidding

Solved ?

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