computer-aided analysis of rigid and flexible multibody...

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GraSMech – Multibody 1

Computer-aided analysis

of rigid and flexible

multibody systems

GraSMech course 2009-2010


Organization and Teaching team

Part I (2009-2010): Modelling approaches and numerical aspects

Speakers

Dr Olivier Brüls, OB, ULg

Prof. Paul Fisette, PF, UCL

Prof. Olivier Verlinden, OV, UMONS

Part II (2010-2011): 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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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


What is a multibody system ?

Multibody System (MBS):

Bodies (rigid or flexible)

Joints

force elements

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Introduction : multibody applications

Set of articulated

bodies

Mechanisms

Railway vehicles

Robot manipulators

Space applications

t

y


Principle

System defined

from technical

elements

Equations of motion built

(and hopefully solved)

automatically

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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

Computer-aided design tool


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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Historical origins

Spacecraft : 1960 Mechanisms : 1970

Robotics : 1980 Flexible bodies : 1990


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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Some unstable satellites


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.

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Multibody Systems

Database



MBS database - Basic elements

The basic elements to describe a MBS are

Bodies

Joints

Force elements

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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.)


MBS Database - Bodies

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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


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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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



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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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


MBS Database - Force elements

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Example: double pendulum

2 bodies

2 revolute joints (0-1.1 and 1.2-2.1)

2 forces (m1 g, m2 g)


Example: four bar mechanism

3 bodies

4 revolute joints (0-1.1, 1.2-2.1, 2.2-3.1 and 3.2-0.1)

3 gravity forces (m1 g , m2 g and m3 g)

1 moment C z0 on 1.1 (reaction on 0)

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Example: suspension

2 bodies

2 revolute joints (0.1-1.1 and 1.2-2)

2 gravity forces (m1 g, m2 g)

1 spring-damper between 0 and 1

eventually a tire-road contact force


Different approaches of MBS formalisms

Choice of coordinates

Mechanical principle to generate the

equations

Computer implementation

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Purpose of coordinates (configuration parameters q): to

express the motion (kinematics) of the bodies

Different approaches exist !


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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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.



Main body : 6 absolute coordinates

Each bogie frame : 6 absolute coordinates

Each wheelset : 6 absolute coordinates

All bodies “elastically” inter-connected

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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Relative or absolute ?

Wheel carrier

Front of vehicle

That’s the question !

In this case, minimal coordinates could be particularly tedious …


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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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



choice of coordinates

mechanical principle to generate the

equations

computer implementation

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Which mechanical principle ?

Newton – Euler (basic)

Virtual work or power principle (d’Alembert)

Lagrange’s equations (Hamilton)

Recursive Newton-Euler algorithm

Recursive “ order N ” algorithm


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 Newton-Euler or

Lagrange

Relative coordinates: Recursive Newton-Euler,

Recursive O(N)

Finite elements: Lagrange

+ Choice of numerical methods !!

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Recursive Newton-Euler

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


Recursive Newton-Euler

Step 2

« backward

» dynamical

recursion

I

Gibody : , i m

hj

Oi

Fiext

Iii

Lexti

-Fj

-Lj

Fi

Li

Oj

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Form of the equations of motion

Second-order differential equations

And eventually constraint equations

=> set of differential-algebraic equations (DAE)

Without

constraints (ODE)




Mechanical principle to generate the

equations


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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 &,



set of data

generate symbolically

all system equations according

to the generic MBD formallism

Specific model equations,

given as a sub-routine

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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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)


Application examples (PF)

Future challenges (OB)


computer-aided analysis of rigid and flexible multibody...

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