Flexible multibody systems Relative coordinates...

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GraSMech – Multibody Computer-aided analysis of multibody dynamics (part 2) Flexible multibody systems - Relative coordinates approach Paul Fisette ([email protected]) GraSMech – Multibody Introduction In terms of modeling, multibody scientists must develop a « critical mind » => What is the « real » problem to solve …. and thus the optimal model to build From a « customer problem » statics basic dynamics advanced dynamics flexible bodies advanced dynamics « by hand », Matlab Matlab rigid MBS code flexible MBS code

Transcript of Flexible multibody systems Relative coordinates...

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GraSMech – Multibody

Computer-aided analysis ofmultibody dynamics (part 2)

Flexible multibody systems-

Relative coordinates approach

Paul Fisette([email protected])

GraSMech – Multibody

Introduction

In terms of modeling, multibody scientists must develop a « critical mind »

=> What is the « real » problem to solve …. and thus the optimal model to build

From a « customer

problem »

statics

basic dynamics

advanceddynamics

flexible bodies advanced dynamics

« by hand », Matlab

Matlab

rigid MBS code

flexible MBS code

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Introduction

It is particularily true when dealing with « flexibility »

=> What is the « real » problem to solve and thus the « minimal » model to build

Examples :Car suspension deformation :

Antenna deployment :

Flexible 2D mechanism :

Chassis torsion :

MBS + Finite segment / 2D beammodel

MBS + FEM (super-elements…)

MBS + static FEM (in post-process)

Full FEM or Lumped torsion (MBS-rigid) ?

GraSMech – Multibody

Introduction

FEM community MBS community

> 1990 : « handshake »

MBS community « credos » :• Simulation time efficiency …towards real time• Lumped and/or « macro » models are still very promising• High dynamics problems• Flexibility : small deformation – few modes• High interest in system control and optimization

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GraSMech – Multibody

Contents

Relative coordinates approach : review

MBS with flexible beams : finite segment approach

MBS with flexible beams : assumed mode approach =>chap. 9 of

Beam Model

Symbolic implementation

MBS with telescopic beams Kluwer Academic Publishers, 2003

GraSMech – Multibody

Contents

Relative coordinates approach : review

MBS with flexible beams : finite segment approach

MBS with flexible beams : assumed mode approach =>chap. 9 of

Beam Model

Symbolic implementation

MBS with telescopic beams Kluwer Academic Publishers, 2003

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GraSMech – Multibody

Concepts

joint

• Relative joint coordinates

Leaf body

base

• Tree-like structuresloop

• Closed structures

• Rigid bodies (+flexible beams)

body

1

6

5

43

2

0

8

7

• Topology (tree-like structure) :the « inbody » vector : inbody = [ 0 1 2 2 4 4 1 7]

Multibody structure

GraSMech – Multibody

Joints

Concepts - Definitions

Classical …

« Wheel on rail » joint (5 dof)« Knee » joint (1 … 6 dof)

More « Exotic »« Cam/Follower » joint

6 dof !

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GraSMech – Multibody

.. .

body i

body j

body k

Fi

Li

Fj

Lj

Fk

Lk

ωj

ωi

kin.

dyn.

Tree-like MBS: Recursive Newton-Euler

Origin : Robotics – inverse dynamics(O(Nbody) operations)

Forward kinematics :

ω j = ω i + Ω ij

Backward dynamics :

F i = F j + F k + m i x i …

L i = L j + L k + I i ω i + ...Joint projection :

...

GraSMech – Multibody

Tree-like MBS: Recursive Newton-Euler

Origin : Robotics – inverse dynamics (Luh, Walker, Paul, 1980)(O(Nbody) operations)

Objective : Recursive formulation for direct dynamics(O(Nbody

2) operations)

… to keep the advantage of the recursive formulation

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for unconstrained system

Closed-loop MBS

Kinematic loops

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Closed-loop MBS

Coordinate partitioning :

u

v

v

Velocity :

Acceleration :

Exact resolution of the constraints

Position :

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GraSMech – Multibody

Contents

Relative coordinates approach : review

MBS with flexible beams : finite segment approach

MBS with flexible beams : assumed mode approach =>chap. 9 of

Beam Model

Symbolic implementation

MBS with telescopic beams Kluwer Academic Publishers, 2003

GraSMech – Multibody

Finite segment formulation

A very simple idea

i jk ij

li lj

. . .. . .

Ω = 150 rad/sec

Ressorthélicoïdal

θGlissière

« Computer methods in flexible multibody dynamics», Huston R.L., IJNME 1991 (also … Amirouche, 1986)

Flexible rod

Flexibility effects are modeled by spring (and possible dampers) between bodies=> Lumped flexibility formulation

ex. bending :

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GraSMech – Multibody

Finite segment formulation

Motivation

• to avoid the « difficult » mariage between rigid body dynamics and structural dynamics

• intuitive and direct method – perfect for a first « pre-study »

• can be implemented in a rigid multibody code (easy to implement !)

• incorporate the flexibilty effects into the global dynamic equations

• not intrinsically limited to elastic systems (=> viscoelastic, … nonlinear elastic …)

i jk ij

li lj

. . .. . .

Limitations • restricted to slender bodies (beams, tapered bodies, rods, …)

• no prove to satisfy the « Rayleigh » vibration criteria (in terms of eigenvalues approx.)

• deformation coupling : sequence dependent (but ok for small deformation)

• could be used « erroneously » : requires skills, insight and intuition

• computer efficiency : OK in 2D, heavy in 3D

GraSMech – Multibody

Finite segment formulation

Computation of the equivalent stiffness coefficients

Equivalent stiffness coefficient are computed from basic principle

of structual mechanics applied to bending, torsion and extension

Resulting Force –Torque in the MBS equations

Extension (example) :

m

µ dxContinuous :

Lumped :

=> Joint force (for extension) or torque (for torsion/bending)

l

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GraSMech – Multibody

Finite segment formulation

Proposed combinations (not exhaustive)

GraSMech – Multibody

Finite segment formulation

Proposed combinations (not exhaustive) COMBINED TAPERED SEGMENTS

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GraSMech – Multibody

Finite segment formulation

Example : a flexible slider-crank

Ω = 150 rad/sec

Ressorthélicoïdal

θGlissière

A

yA

Rod modal analysis around equilibrium (horizontal configuration)

GraSMech – Multibody

Finite segment formulation

Example : a flexible slider-crank

Ω = 150 rad/sec

Ressorthélicoïdal

θGlissière

FEM FSM

A

yA

yA yA

Lateral deflection of the mid-point A in frame Y

Y

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GraSMech – Multibody

Finite segment formulation

Example : a flexible slider-crank

Ω = 150 rad/sec

Ressorthélicoïdal

θGlissière

FEM FSM

B

xBShortening of the rod (point B) in frame Y

xB xB

Y

GraSMech – Multibody

Finite segment formulation

___ : FEM (+)…. : monomials shape function (+)-.-.-. : FSM (+)-----:modal shape function (-)

Kane’s benchmark : 2D rotating beam

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GraSMech – Multibody

Contents

Relative coordinates approach : review

MBS with flexible beams : finite segment approach

MBS with flexible beams : assumed mode approach =>chap. 9 of

Beam Model

Symbolic implementation

MBS with telescopic beams Kluwer Academic Publishers, 2003

GraSMech – Multibody

MBS with flexible beams

Flexible beam model

Shape functions

Beam Kinematics

Multibody kinematics (forward)

Joint dynamic equations( backward)

Deformation equations (beam per beam)

Symbolic computation

Applications

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GraSMech – Multibody

MBS with flexible beams

Flexible beam model

Shape functions

Beam Kinematics

Multibody kinematics (forward)

Joint dynamic equations( backward)

Deformation equations (beam per beam)

Symbolic computation

Applications

GraSMech – Multibody

Flexible beam model : hypotheses

Geometry : prismatic beams - rectilinear centroidal axis

Material : homogeneous and isotropic - conforms to linear elasticity

Deformation model : Timoshenko 3D, conservation of plane cross sections, shear deformation and rotary inertia included

Kinematics : angular and curvature of the beam must remain small(rotation matrix linearized) - but still compatible to « capture »geometic stiffening effect

Topology : a beam has the same status as a rigid body in the MBS :

Rigid bodyJoint

Flexible beam

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GraSMech – Multibody

Flexible beam model : notations

Beam local rotation

Linearized rotation matrix

GraSMech – Multibody

Flexible beam model : notations

Centroidal axis deformation

Current = undeformed + deformed :

Displacement field v :

Vector position (section S) :

C

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GraSMech – Multibody

Flexible beam : shape functions

For the x, y, z, linear deformation of the centroidal axis C

θ Ez^

Ex^ v

For the x, y, z, angular deformation of the cross sections S

with :

with :

generalized coordinates (=amplitude of shape functions)

generalized coordinates (=amplitude of shape functions)

GraSMech – Multibody

Flexible beam : shape functions

Which kind of shape functions ?

θ Ez^

Ex^ v

• From a previous (FEM) modal analysis => assumed modes

• From a purely mathematical set of functions:

• cubic splines

• Legendre polynomials

• Monomials

• …

« global » shape functions

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GraSMech – Multibody

• They can « theoretically » approximate any plausible deformation

(think at a Taylor series which combines them)

• Being an invariable set of functions, there is no need for a prior modal analysis*• They are perfectly suitable for symbolic computation of integrals, ex.:

• But … they do not form a set of orthogonal functions (=> numerical unstabilities)

Flexible beam : monomials

Why monomials ?

*Monomials being not eigenmodes, the beam configuration (q,qd,qdd) may« move away » from the equilibrium state of a prior modal analysis

= !

GraSMech – Multibody

MBS with flexible beams

Flexible beam model

Shape functions

Beam Kinematics

Multibody kinematics (forward)

Joint dynamic equations( backward)

Deformation equations (beam per beam)

Symbolic computation

Applications

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GraSMech – Multibody

Flexible beam : kinematics

Angular velocity and acceleration

« Relative » kinematics

Linear velocity and acceleration

GraSMech – Multibody

MBS with flexible beams

Flexible beam model

Shape functions

Beam Kinematics

Multibody kinematics (forward)

Joint dynamic equations( backward)

Deformation equations (beam per beam)

Symbolic computation

Applications

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GraSMech – Multibody

MBS with flexible beam : kinematics

Forward kinematic recursion

etc. for accelerations…

GraSMech – Multibody

MBS with flexible beam : joint dynamics

Virtual velocity field :

MBS Virtual power principle:

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GraSMech – Multibody

MBS with flexible beam : joint dynamicsBackward dynamic recursion

Fi

Li

GraSMech – Multibody

MBS with flexible beams

Flexible beam model

Shape functions

Beam Kinematics

Multibody kinematics (forward)

Joint dynamic equations( backward)

Deformation equations (beam per beam)

Symbolic computation

Applications

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GraSMech – Multibody

Flexible beam : deformation dynamics

Beam deformation :

Displacement gradient :

Strain vector of the centroïdal axis :

Beam curvature vector :

with

where

GraSMech – Multibody

Flexible beam : deformation dynamics

… requires the relative derivatives of Γ and K :

MBS Virtual Power Principle:(see next slides)

Real :

Virtual :

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GraSMech – Multibody

Flexible beam : deformation dynamics

Virtual velocity field :

MBS Virtual power principle:

GraSMech – Multibody

Flexible beam : deformation dynamics

Local equations of motion (of the beam portion ds) :

Constitutive equations (linear elasticity) :

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GraSMech – Multibody

Flexible beam : deformation dynamics

By integrating by part the terms :

Final form :

… for « each » beam i

GraSMech – Multibody

Flexible beam : deformation dynamics

Example of computation :

… for « each » beam i

recall … :

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GraSMech – Multibody

Flexible beam : deformation dynamics

Example of computation :

Using monomials

Analytical integrals

GraSMech – Multibody

Flexible beam : deformation dynamics

Example of computation :

Interpretation :

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GraSMech – Multibody

Flexible beam : deformation dynamics

Example of computation :

!!! Second order terms required in Γ to be consistent with first order kinematics in the VPP) !!!

Example : pure bending (=> Γ1 = 0 (by definition))

Second order :

pure bending =>

where :

First order :

pure bending => = 0 !!

GraSMech – Multibody

MBS + flexible beams : eq. of motion

Joint equations of body i :

Deformation equation of beam i :

Implicit equations of motion of the MBS

Rigid : Flexible (beam) :

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GraSMech – Multibody

MBS with flexible beams

Flexible beam model

Shape functions

Beam Kinematics

Multibody kinematics (forward)

Joint dynamic equations( backward)

Deformation equations (beam per beam)

Symbolic computation

Applications

GraSMech – Multibody

MBS + flexible beams : eq. of motion

with

Symbolic computation of the tangent matrices M, G, K with ROBOTRAN => recursive (efficient) derivation !!!!

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GraSMech – Multibody

MBS + flexible beams : eq. of motion

Global computation scheme (Robotran) :

GraSMech – Multibody

MBS + flexible beams : symbolic generation

Input file (for flexible beams)

Input file (for rigid bodies)(standard file) ROBOTRAN Symbolic

model…

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GraSMech – Multibody

MBS + flexible beams : symbolic generation

Symbolic model (example - Matlab)

Implicit form :

GraSMech – Multibody

MBS with flexible beams

Flexible beam model

Shape functions

Beam Kinematics

Multibody kinematics (forward)

Joint dynamic equations( backward)

Deformation equations (beam per beam)

Symbolic computation

Applications

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GraSMech – Multibody

MBS + flexible beams : examples

A cantilevered L-shaped structure : modal analysis

For each beam :• FEM : 10 beam elements• FSM : 10 interconnected bodies• Monomials : 5 shape functions in x, y, θ

GraSMech – Multibody

MBS + flexible beams : examples

Kane’s benchmark : 2D rotating beam

CPU time reduction factor :14

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GraSMech – Multibody

MBS + flexible beams : examples

Fisette et al. benchmark : 3D rotating beam

CPU time reduction factor :38

GraSMech – Multibody

MBS + flexible beams : examples

Jahnke, Popp’s benchmark : flexible slider-crank

FEM

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GraSMech – Multibody

Contents

Relative coordinates approach : review

MBS with flexible beams : finite segment approach

MBS with flexible beams : assumed mode approach

Beam Model

Symbolic implementation

MBS with telescopic beams

GraSMech – Multibody

MBS + telescopic beam : principle

Sliding section S, frame t

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GraSMech – Multibody

MBS + telescopic beam : principle

Position constraints :

Orientation constraints :

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Loop closure : pseudo-rotation constraints

… recall…(part 1)

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Let’s choose a subset of 3 independent constraints(general 3D rotation):

Loop closure : pseudo-rotation constraints

GraSMech – Multibody64

if all the constraints are satisfied :

because : = E

Loop closure : pseudo-rotation constraints

=> Coordinate partitioning method can be used to reduce the system => ODE

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Conclusion 1 :

with :

Loop closure : pseudo-rotation constraints

Pseudo-gradientPseudo-rotationconstraints

GraSMech – Multibody

MBS + telescopic beam : example

A telescopicflexible slider-crank

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GraSMech – Multibody

Conclusions

Flexible multibody systems in relative coordinates

Finite segment method : a « pragmatic » technique

MBS with flexible beams

Floating frame approach – set of « problem-independent » shape functions

Beam Model : Timoshenko (Euler-Bernouilli would certainly be a bit more efficient)

Symbolic implementation : OK and « fully » in case of monomial shape functions

CPU time : very powerfull (but limitation in terms of flexibility modeling)

MBS with telescopic beams : closed loop approach – symbolic implementation