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GraSMech – Multibody 1

Flexible multibody systemswith finite elements

O. Brüls, G. Kerschen

[email protected]

ULg – Department of Aerospace and Mechanical Engineering

GraSMech courseComputer-aided analysis of rigid and flexible multibody systems


Outline

Main approaches in flexible multibody dynamics

The elastic bar and the elastic beam

Superelement and corotational formulation

Energy conserving schemes

Link with other finite element fields

2


Outline

General description of the FE approach

FE assembly

Structure of the equations of motion

Time integration

Nonlinear description of flexible bodies

Corotational formulation

Superelement


Link with other FE fields


Finite element approach

Nodal coordinates:

Interpolation inside each element:

3


Inertia, internal and external forces can be computed at element level:

rigid bodybeam (2 nodes)hinge (2 nodes)…

Finite element assembly

Equations at mechanism level ?


I II III

x1 x2 x3 x4

Bar in extension:


4


FE assembly:


Loop overelements e

Similar principle for inertia, external and constraint forces mass, damping and stiffness matrices


Inertia forces ?rigid body: one node at the center of mass

Equations of motion

constant

not constant

5


Constraint forces ? 2D prismatic joint: one node in translation

Equations of motion

uv

r

x


Time integration algorithm: Newmark

Time integration

Correction (Newton):

Prediction:

6


Time integration

Correction

Linear (n+m) x (n+m) systemat each iteration of each time step !


Large FE model:many pairs of dofs are uncoupledsparse solver

qi

Time integration

qj

Population of a stiffness matrix

7


.NOEUDI 1 X 1. Y 0. Z 0.I 2 X 0. Y 0. Z 0.

.CLMFIX NOEUD 2 COMP 1 2 3 4 5 6FIX NOEUD 1 COMP 4 5 6

.MCEI 1 RIGID NOEUD 1I 2 SPRING NOEUDS 1 2

.MCCI 2 SPRING KR 1. C 0.1I 1 RIGID MASS 1.

.CLM CHARGE NOEUD 1 COMP 1 VALEUR 10.

.SUBANALYSIS I1MECA 2 IMPL 21 ALFA 0.05 TIME 0. A 5. PAS 0.01

Nodes and positions

Fixations

Elements topology

Elements properties

Loads

Time integration

Model definition (SAMCEF-MECANO)

N1N2


Kinematic eqn Constitutive eqnin material axis

x

y

XY

f

α

In material axis:

In spatial axis:

Configuration dependent

2D bar :

Flexible bodies

Rigid body components are filtered

8


component (1,1) of thematerial strain tensor

component (1) of thematerial force vector

Locally:

FE discretization ⇒ ⇒

Flexible bodies


Flexible bodies

Strain matrix B: L1

2

Nonlinear definition of ε(x)

B depends on qgeometric stiffness

α

x

y

9


Flexible bodies

Constitutive equations are expressed locally at point P

Spatial configuration

x(s)Material deformations

ε(s)Material forces

N(s)

Based on a local representation in material coordinates,⇒ no restriction on the amplitude of motion

FE discretization


Outline



Introduction

Corotational frame definition

Kinematics

Internal forces

Inertia forces

Superelement



10


Corotational approach: introduction

Felippa 2000 http://caswww.colorado.edu/courses.d/NFEM.d

A standard FE tooldefinition of a corotational frame relative displacements are small


Corotational approach: introduction

constitutive law at each material point:

linear force/displ law for the element:

?

How to extract local displacements ?purge the rigid body components !generic procedure (independent of the element type)

Local nodaldisplacements

NFE discretization

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Mean axisTisserand (relative kinetic energy is minimized)Buckens (deformation is minimized)Average position (~ center of mass):

Constant weights, such that

only makes sense for small rotations

Addition of large rotations does not make sense



Incremental approach:Rref


e.g. rotation at previous time step

12


Linear elasticity theory (standard FE procedure)

Internal forces

At the element level, local translation and rotation coordinates

?


Internal forces

Material position Spatial positionSmall deformation

13


rotation composition:

Nodal coordinates :

Internal forces

What about rotations ?


We need a relation:

Differentiate the translation kinematics

with

⇒ quite long expressions… (Géradin & Cardona, 2001)

Internal forces

14


Summary:1. Nodal coordinates:2. Corotational frame:

3. Corotational displacements

4. Corotational forces:

5. Generalized forces:

Internal forces


Internal forces

(Felippa, 2000)

15


Internal forces

Advantage: unique procedure for any type of element

Linear FE code (single body)

Corotational procedure

+

Flexible multibody software

Low implementation effort !

Joint formulation(inter-body)

+


Inertia forces

x~x*

y~y*

Small amplitude motion

x

y

Large amplitude motion

x*y*

with

or

M* is available from the linear FE package

Inertia forces : contribution of 3 translational dofs

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Inertia forces

Corotational computation of the kinetic energy :

absolute velocity in the corotated frame(as if the frame were frozen)

M* = constantM = ??= M


Inertia forces

particle in translation:

⇒ same M in any frame

⇒

FE discretization:

Invariant under frame rotation M = M*

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Inertia forces

However, in general (e.g. Craig-Bampton description)

frame-dependent shape functions

whereas corotated shape functions ~ frame-invariant !

Objective: avoid the explicit computation of M(R0) and derivatives

compute the inertia forces directly from M*

M(R0) ≠ M*


Inertia forces

For rotational dofs, material angular velocities are used

and the material inertia tensor J is frame invariant

with

For the whole body:

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Inertia forces

Why is it difficult to compute the inertia forces ?

What can we say about: ?

Think about a rigid body…


Inertia forces

Hamilton: the actual motion satisfies

if

Integrationby part

19


Corotational velocities are non-integrable

since

Transfo into integrable quantities for integration by part :

Inertia forces

then develop


At the end of the day…

Inertia forces

20


Corotational approach

Summary:Small displacements in the corotated frameGeneric procedure for any elementSimple computation of internal forces

In many particular cases, M = M*simple computation of the inertia forces

In general, M ≠ M*corotational computation of the inertia forces


Outline



Superelement

Introduction

Linear reduction techniques

Craig-Bampton method

Deployable antenna

Landing gear



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n + m nonlinear equations at each time step…Can we reduce n ?

Superelement: introduction

Reduction techniques:transformation into modal coordinates


Component mode synthesis:modal coordinates at the body levelin a corotational frame

Linear structural dynamicsSubstructuring: Modularity and efficiency


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First step: single bodylinear analysis under small displacement assumption

Second step: multibody systemuse the reduced model in a corotational approach

Transfo into modal coordinates


(small q)


Eliminate the high-frequency modes, while keeping accuracy at low-frequency ?

FE model with thousands of dofs

High-frequency modes result from the discretization processdominated by numerical effects

Dynamic responseDominated by the low-frequency modesHigh-frequency modes filtered by the integrator


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Initial FE discretization:

Modal transformation:



Linear equations of motion:

=

+ =

+ =

Galerkin


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=

=

= Everything is in …



The superelement should behave as a usual element⇒ Interface dofs for assembly with the rest of the structure

Suitable description at the boundary dofsexact static response is requireddynamic response is approximated


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Selection of ?

Craig-Bampton : static boundary modes + internal vibration modes

McNeal / Rubin : free-free modes + residual flexibility correction

Balanced truncation :maximize controllability / observability

Krylov subspace:interpolate the FRF



boundary dofs static modes

internal dofs internal modes

(Géradin & Rixen, 1997)

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static boundary modes internal vibration mode

⇒ exact representation of the boundary static response




If no boundary dofs:

Fully populated submatrices:the sparse structure of M and K is lost…corotational inertia forces ≠

and

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The superelement looks like any other elementinterface nodes and dofs stiffness and mass matrices

Defined for small displacements q, but can be used in a corotational formulation !



Corotational formulation of the superelement:


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Body analysisLinear FE model of the bodyPartitioning: boundary / internal dofsMode shapes: static + modal analysisReduced stiffness and mass matrices

Multibody analysisConnect the boundary dofs to other bodiesEnjoy the corotational implementation to computeelastic and inertia forces…

Superelement: summary


Deployable structures

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

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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



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Landing gears

Different phasesDeployment / retractionGround impactRollingBreakingTaxiing

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

Concorde – nose undercarriage


Landing gears

Motivations for modelling:

Optimal design configurationmass

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

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Landing gears

Superelement for the elastic wings


Landing gears

Landing of the A380

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Superelement: conclusions

Linearity in a corotated frame

Reduced number of modal coordinates

Sparsity is lost

Systematic and modular approach

Cheap computation of elastic forces

Non trivial computation of inertia forces


Outline



Superelement


Motivation

Formulation: linear, nonlinear and constrained systems

Outlook


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Motivation

Two motivations for energy conserving schemes:

Without an accurate energy balance,no qualitatively acceptable solution

Stability of the solution in the nonlinear regime(without artificial numerical damping)


One-dof oscillator Euler explicit

-1 1

x

x

Scalar test equation

Stability region in thecomplex plane

Motivation

35


One-dof oscillator Euler implicit

-1 1

x

x

Scalar test equation

The stability regionincludes the left half plane

Motivation


FE model with thousands of dofsAccuracy is only required for the slow modesStability is also required for the fast modes

Newmark without numerical dampingAny high-frequency disturbance can oscillate for ever

Newmark with numerical dampingSecond-order accuracy is lost

Generalized-α with numerical dampingSecond-order accuracy is preservedLimited impact at low-frequency

Stability limit

frequency

Motivation

36


Importance of kinematic constraints

Undamped scheme Damped scheme

stability+

accuracyWeak instability

due to the constraints

Motivation

(Géradin & Cardona, 2001)


Motivation

Linear stability ⇒ nonlinear stabilityX

(Kuhl & Crisfield, 1999)

Conserving scheme

37


Motivation

Good reasons to look at energy conserving schemesQualitative aspect of the solutionSystem prone to instability

Low physical dampingStiff equations (FE discretization)DAE equations

High degree of nonlinearitynonlinear elasticitynonlinear constraints

Complex system: large numerical damping is required⇒ accuracy is reduced


Formulation: linear systems

Equation of motion

Momentum form

Energies

Mid-point balance

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Mid-point ruleSecond-order

accuracy !

Discrete equations

Energy conserving scheme(e.g. Simo and co-authors)



Discrete energy balance ?

use

Q.E.D.


39


Formulation: nonlinear systems

Discrete energy balance:

= ≠

Modify the algorithm s.t. the energy balance is verified

is replaced by

(discrete directional derivative)

X



Properties of the discrete directional derivative (Gonzalez, 1996)

2nd order approximation of the mid-point derivativedirectionality:

which ensures energy conservation

How to build a function which satisfies those conditions ?basics rules are availableformulation-dependent treatmentto be developed for each type of element !

40


Example: linear elasticity

Observing that

The directionality property

is verified by




Momentum equation for a rotating body

SummaryEquations in terms of momentumMid point rule ⇒ jump in Discrete directional derivative ⇒ jump in

41



Mid-point kinematics

very often,

translations

rotations

i.e.half rotation ?

Simpler using Euler parameters

Influence on the choice of parameterization


Formulation: constrained systems

Energy conserving scheme:

With the discrete directional derivative of the constraints s.t.

Discrete energy balance

X X

42


Formulation: constrained systems

Sustained numericaloscillations

Energy conservation

Numerical example (Lens, Cardona & Géradin, 2004)


Energy conserving schemes: Outlook

Stability ?Energy conservation is not sufficient for stiff DAEsEnergy decay is required at high frequencies

Energy decaying schemes (e.g. Bauchau & Théron, 1996)energy decay inequality for elastic bodiesalgorithmic work of the constraints vanishesnon-linear stability is ensured,

what about accuracy ?how to control the amount of dissipation ?

43



Structure preserving algorithmsFirst integrals: energy and momentumSymplecticity

Several approachesAlgorithms which structurally preserve the energy(described above)

First integrals as algebraic constraints(conservation laws explicitly imposed at each step)

Variational integrators



Variational integrators (e.g. Kane, Marsden, Ortiz & West, 2000)

conservation laws ⇔ variational principles

Variational principle

Equations of motion+ conservation laws

Variational principletime

discretization

Discretized equationsconservation laws

Discrete variational principle

Discretized equations+ conservation laws

time discretization

44


Discretization of the action integral (Hamilton principle)


=Discrete Lagrangian

Mid-point rule

Minimization of the discrete action ⇒ Numerical algorithm


Energy conserving schemes: Summary

qualitatively better solutionstability for nonlinear systemsmid-point rulediscrete directional derivative

not a generic integrator ⇒ multidisciplinary problems?

algorithmic work of the constraints vanishesStill a hot research topic…

energy decaying schemespreservation of momentum and symplecticity

45


Outline



Superelement



Introduction

Vehicle crash analysis (visco-plasto-elasticity)

Helicopter dynamics (aeroelasticity, composites,

piezoelectric actuation)


Link with other FE fields: Introduction

Geometric nonlinearities: finite displacements and deformations

(change in the geometric configuration)

Material nonlinearities:nonlinear stress-deformation relationsvisco-elasticity (creep)Plasticity

Coupling in the domain: thermoelasticity, piezoelasticity…

Coupling with an external domain:force / displacement relation (interaction with a fluid)intermittent contact

46


Kinematics Constitutive eqn Kinematics*

Imposeddisplacements

Applied loads

- Geometric nonlinearities- Material nonlinearities- Coupling in the domain- Coupling with an external domain

Link with other FE fields: Introduction


Crash analysis

Geometric nonlinearitiesVisco-elasticityPlasticity

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Crash analysis: viscoelasticity

dynamic stress / deformation lawstress / strain rates play important roles

Maxwell: Kelvin:

Constitutive law ⇒ differential equation !

= relaxation time (creep) = retardation time


Crash analysis: viscoelasticity

Differential equation: How can we get ?

differentiation of the strain / displacement couldlead to a rate / velocity relation…

memorize the successive values of and compute by finite difference

The computation of the constitutive equationinvolves a time integration procedure !

48


Crash analysis: plasticity

influence of the loading sequence(but not of the rate of loading)

The sequence should be memorized for computations !


Crash analysis

Detailed elasto-visco-plastic FE models are still challenging(in particular for crash analysis of a full vehicle)

complex formulations and algorithmshigh computational load (very small time steps)constitutive parameters are difficult to estimate(non-trivial experimental identification)

Such models are not appropriate if repeated simulations are required…

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Crash analysis

Plastic hinge approach: localized plastic deformations

If their location is a priori known,the chassis can be modelled asa set of articulated bodies…

Empirical moment /angle relation

(Ambrosio & Diaz, 2005)


Helicopter dynamics

Adaptation of the blade pitch during both rotation and manoeuverThe rotor support the weight of the helicopterHigh vibration level (cockpit isolation problem)

50


Challeging issuesNonlinear deformations of the blades(centrifugal stiffening)

Aeroelastic couplingNew materials (composites)Distributed actuation

Rotorcraft mechanism

Helicopter dynamics


Helicopter dynamics: aeroelasticity

Analysis of Lift and dragVibrations excitationFlutter instability

Simplified model:

Non-symmetric coupling…

Coupling through the boundary conditionsSolid: pressure is imposedFluid: velocity is imposed

51


Helicopter dynamics: aeroelasticity

Unsteady analysis of the fluid motionhuge Computational Fluid Dynamics software (CFD)usually, finite volume formulation (and not FE)

It is not realistic to develop a monolithic softwarefor both the fluid and the multibody system

Different software / solvers for the different subsystemsiteration from one subproblem to the otherinterfaces to exchange coupling variablesmanage non-conforming meshes

Manoeuver simulation: simplified aeroelastic force models


Anistropic properties for each layer:A longitudinal force can induceshear, bending and twisting

Helicopter dynamics: composites

Composite blade

homogenization

composite skin

Model of the micro-structure+ constitutive equations

Average constitutiveequation for the composite

Multiscale modelling

52


Plate element for compositesin a FE flexible multibody softwarecoupling anisotropy / geometric nonlinearities

(Das, Berut & Madenci 2004)

Helicopter dynamics: composites


(Masarati, 1999)

Helicopter dynamics: piezoelectricity

Active vibration controlReduce vibrations in therotor and in the cockpit

3 actuator techniques

DistributedPiezoelectric actuation

53


Electric displacement Electric field

compliance

Piezoelectric constitutive equation

The FE method can be used to solve the multiphysics problem



Bending at the mast

Multiphysics-multibody model (Masarati, 1999)

Warping of a blade section due to actuation

Coupling: anisotropy, piezoelectricity, geometric nonlinearity


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Outline



Superelement




Finite element modelling: Conclusions

The finite element approach allows to modelfinite motion of flexible and rigid bodiesjoints in articulated structuressmall and large deformations

Numerical approachsystematic and generaluseful for simulation / optimisation

The field is openMultidisciplinary, multiscale, multiphysics modelling…

Challenging large-scale applications

55


Finite element modelling: Conclusions

Classical finite elementsflexible bodiessmall displacements

Classical multibody dynamicsrigid bodieslarge motion

Flexible multibody dynamics

Nonlinear FE

Corotational FE …

Floating frame

Multiscale systems Multiphysics systems

Nonlinear mechanics

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