modred 2010 challenges and experiences in model reduction for mechanical systems ... · elastic...

Institute of Engineering and Computational Mechanics

University of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter Eberhard

Challenges and Experiences in Model Reduction for Mechanical Systems Illustrated for the Reduction of

a Crankshaft

Christine Geschwinder, Jörg Fehr und Peter Eberhard

MODRED 2010



robotics

power engineering

Application EMBS

EMBSsaves experiments and prototypes due to simulationlightweight constructions requires consideration of elastic effects

automotive and drive engineering

researchmodel reduction

models gets moredetailed and largerno efficient simulationpossible without MORfor use of modal reduction it is necessary to have a lot of know-how and no simple error-control is possible



Principle of Elastic Multibody System

multibody system

elastic bodydiscretisation

finite element,finite difference,...

continuum

elastic multibody system

rigid body

bearings and coupling elements

p bodiesf degrees of freedomq reaction force

C

reduction of the elastic degrees of freedom

elastic multibody systems are an important tool for calculation of complex mechanical systemsmodels are getting larger and more detailed

FE-models has to be reduced cause of a large number of degrees of freedom



combustion engineimportant component: bearing of a crankshaft

calculation of hydrodynamic bearingforces for discrete crankshaft positions

deformation of the crankshaft has great influence

for realistic results:combination of elasto-hydrodynamicbearings and elastic multibody system

Technical and Industrial Application

rod bearingmain bearing

solving expenseelasto-hydrodynamic equationsequation of motion of the elastic bodies

rigid crankshaft

elastic crankshaft



Overview

motivation

model reduction of elastic bodiesshort overview of industrial state of art and current state of developmentsmodel reduction based on the frequency weighted Gramian matrices with POD

reduction of the crankshaft

conclusions and outlook



Model Reduction ofElastic Bodies

basic equation

linear model reductionapproachprojection of the elastic coordinates to a subspace with

to get a solution with vanishing residuum

reduced system

qVq euBqVKqVDqVM eeee

eWuBWqVKWqVDWqVMW Te

Te

Te

Te

T

uBqKqDqM eeee

)()( dimdim qq

qDqK

0uB

hqa

MMWVMM

eee

r

eerT

Terr

qCy e

qDqKuB

hqa

MMMM

eee

r

eer

Terr



Overview of Model Reduction Techniques

industrial state of art

traditional modal reduction (selection of modes by user)Craig-Bampton/ Component Mode Synthesis (CMS)

current state of developments

reduction with moment matching and Krylov-subspacesmethods based on Gramian matricescombinated reduction techniques

model reduction techniques used for elastic bodies



position and velocityGramian matricesGramian matrices play an important rolein balanced truncation model reductionfor second order systems:

Gramian matrices of the equivalent first order system

partitioning the Gramian matrices

alternative characterization of Gramian matricesconnection with -Norm

alternative observability Gramian matricesfor second order systems

with structural propertieswith same in- and outputs

Balanced Truncation of 2nd Order Systems

,po

opTv

Tp

v

p

PP

PPSS

SSP

po

opTv

Tp

v

pQQQQ

RRRRQ

TSSP TRRQ

2H Te1

evTee

T2H trtr2

BMQMBBQBH

TepeT trtr CPCCPC

1ev

Tepv :

MQMQ

Tee BC

pvp QP



Frequency Weighted 2nd Order Gramian Matrix

motivationin engineering problems, oftenknowledge about the interestingfrequency range is available

frequency weighted Gramian matrixfrequency range are emphasized by applying an suitable frequency filter

with

band pass frequency weighted Gramian matrixfor mechanical systems the filter matrices are often not available directly but the interesting frequency range is knownband pass Gramian matrix for an arbitrary frequency range maxmin ,

)i( W

djjjj21 HT

eH

ep GBWWBGP 1

eee2 )j(j KDMG

min

max

max

min

djj21djj

21, HT

eeHT

eemaxminp GBBGGBBGP

advantagesa priori error boundaryweighting of a specialfrequency range

disadvantagescalculation of the matrix integralonly efficiently possible for small models



max

min

djjjj21, TT

eeHT

eemaxminp GBBGGBBGP

Calculation of the 2nd Order Gramian Matrix

approximationcalculation of the integral by numerical integral at discrete frequency sampling point

node displacement snapshots

snapshots matrix

numerical integration

summarize of all snapshot matrices to one snapshot matrix

i

eiii jjˆ BGQU

iii ˆImˆRe~ UUU

l

1i

Tiiminmaxmaxminpmaxminp

~~1l

,~, UUPP

l

pl221l21 ~ˆˆ uuuUUUU

T

c

minmaxmaxminp 1l

,~ UUP



Proper Orthogonal Decomposition (POD)

basic ideareducing large number ofdependent variables to a small numberof uncorrelated variables

snapshotsfrequency response of a harmonic excitationsnapshot matrix with

approximation approachsearch of an orthogonal basis , which minimizes least squares difference between snapshots and the approximated subspace

with Langrange-function for constraints → eigenvalue problem

)j( ii Xu

s21 uuuU

2s

1j

m

1iii

jjmin

uuΦ

s

1j

m

1i

2i

jmax uΦ

m21 Φ

iiiT λ

s1

R

UU i

i

dominant eigensubspace of POD problem = dominant eigensubspace of Gramian matrix

Proper Orthogonal Value (POV)

Proper Orthogonal Mode (POM)

POD - kernelR

pl2s



POD Snapshot Method with Greedy Search

snapshots methodbased on data vector and theProper Orthogonal Modes span thesame linear space

POD eigenvalue problem

every snapshot affectson the POMs with

Greedy Searchadaptive search of the snapshotsproceeding

searching for the frequency at the maximum error of an error messure calculated for the interesting frequency rangethis frequency is used for the next snapshot

kiu

s

1k

kiki a u

s

1k

kiki

s

1j

Tjs

1k

ik

kj aλas1 uuuu i

ki

s

1k

ik

kj aλas1

uu

snapshots method reduce the POD-problem to an s dimensional eigenvalue problem

weight factor of snapshot

ika

maxmin ,



Overview

motivation






The Crankshaftbasic information

number of nodes 15 631number of elements 60 279number of indipendent DoF 46 860number of in- and outputs 35

modeling of inputs and outputs

reducing the number of inputs and outputs by interface nodes (RBE3) wich causeconstraint equations

interface node



Frequency Response

4036, 572,65822

265,794778,28609

5980,07101133,495010

828282

40nred

0,500,699

0,590,3715

nsnap

[10-10] fmax=200Hz

[10-10] fmax=2000Hz

7 3789000,00 210765,83

5 0,39 0,76

error in H∞-norm

H

H

reduction to nred = 82



Time Simulationrigid crankshaft

elastic crankshaft

reference



Overview

motivation






Conclusions and Outlook

an industrial application where model order reduction is imported is introducedcalculation of hydrodynamic bearings with the help of an elasticcrankshaft

overview of industrial and current state of art of model reduction techniques is givenalternative model reduction based on frequency weighted Gramian matrices with the help of POD is explained

results of reduction of the crankshaft is shown for the frequency range and the time domain

application of POD for large-scale industrial applicationsapplication and improvement of error estimator for POD reductions

summary

outlook

thank you for your attention

modred 2010 challenges and experiences in model reduction for mechanical systems ... · elastic...

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