Eigenvalue method with symmetry and vibration analysis of cyclic structures

(1)

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Eigenvalue method with symmetry and vibration

analysis of cyclic structures

Aurélien Grolet, Philippe Malbos, Fabrice Thouverez

To cite this version:

(2)

of y li stru ture

AurelienGrolet

1

,PhilippeMalbos

2

,andFabri eThouverez

1

LTDS,É ole entraledeLyon,

36avenueGuydeCollongue,69134 ECULLY edex,Fran e.

2

UniversitédedeLyon,ICJCNRSUMR5208,UniversitéClaudeBernardLyon1,

43boulevarddu11novembre1918,69622VILLEURBANNE edex,Fran e.

Abstra t. Wepresentanappli ationoftheeigenvaluemethodwithsymmetryforsolving

polynomialsystemsarisinginthevibrationanalysisofme hani alstru turewith

symme-try.Thesear hforsolutionsis ondu tedbytheso alledmultipli ationmatrixmethod

inwhi h thesymmetry ofthe systemis takenintoa ount by introdu inga symmetry

group

G

andbyworkingwiththesetofinvariantpolynomialsunderthea tionofgroup

G

.Byusingthismethod,we omputetheperiodi solutionsofasimpledynami system

omingfromthemodelofa y li me hani alstru turesubje tedtononlinearities.

1 Introdu tion

Manyengineeringproblems an bemodeledorapproximatedsu hthat thedeterminationof a

solutiongoesthroughtheresolutionofapolynomialsystem.Inthispaper,weareinterestedin

omputingperiodi solutionsofnonlineardynami equations.It anbeshownthattheFourier

oe ients of the (approximated) periodi solutions an be obtained by solving multivariate

polynomial equations resulting from the appli ation of the Harmoni Balan e Method [1,2℄.

Moreover,inourappli ations,thedynami alsystemisofteninvariantunder some

transforma-tions( y li permutation, hangeofsign,...)duetothepresen eofsymmetryintheme hani al

stru ture. This implies that the polynomial system to be solved is also invariant under some

transformations,andsodoesitssolutions.

Mostofthetime,inme hani alengineering,polynomialsystemsaresolvedbynumeri

meth-odssu h asaNewton-likealgorithm,whi houtputsonlyonesolutionofthesystemdepending

onthestartingpointprovided.AlthoughtheNewtonmethodisane ientalgorithm(quadrati

onvergen e),thesear hforallsolutionsofapolynomialsystem annotbe ondu tedin a

rea-sonabletime usingonlythis method. Inthe ontinuationmethodsframework[3℄, thestudyof

bifur ationsallowsto follownewbran hesofsolution,but doesnotwarrantythat allsolutions

are omputed(egdis onne ted solutions).

Homotopy methods [2,4℄ are an alternative to the Newton algorithm when sear hing for

all solutions of a multivariate polynomial system. Basi ally, homotopy methods relie on the

ontinuationofthe(known)rootsofastartingpolynomial

Q

(easytosolve) tothe(unknown)

rootsof atargetpolynomial

P

. The hoi eof thestartingpolynomialis akeypoint onwit h

depends the e ien y of the method. Indeed, if the starting polynomial has to many roots

ompared to

P

, most of the ontinuationswill lead to divergentsolutions, thus wasting time

(3)

appli ation of the method rather di ult. Moreover,it is not lear how to takeinto a ount

symmetrypropertiesinthepolyhedralhomotopy.

In this ontext, where numeri al methods are not entirely satisfa tory, omputer algebra

appearsasanattra tivealternative,sin ethereexistane ientmethodspe iallydevelopedfor

solvingsymmetri systemof polynomialequations. Themethod, relativelyre ent,isproposed

by Gatermann in [6℄ and is alled "eigenvalue method with symmetry". It is based upon the

multipli ationmatrixmethod [7,8℄,where solutionsofthepolynomialsystemare obtainedby

solvinganeigenvalueproblem. Moreover,ittakesinto a ountthesymmetryof thesystemby

working only onasubspa e ofthe quotient algebra.Themethod is verye ient sin etaking

into a ountsymmetry allowforredu ing thesize ofthe multipli ationmatrixsu hthat only

onerepresentativeofea horbitofsolution anbe omputed.

In this paper, we propose a new appli ation of the eigenvalue method with symmetry for

omputing periodi solutions of nonlinear dynami systems solved by the harmoni balan e

method.It onstitutesanattempttoevaluatesthe apabilitiesof omputeralgebramethodsin

theeldofme hani alengineering,inwhi hnumeri almethodsareoftenthenorm.

Thepaperisorganizedasfollow:se tion2presentsthetypeofsystemstudiedin thiswork.

Themotion'sequationsarepresentedalongwithabriefre alloftheHarmoni Balan eMethod,

and we also derive the polynomial equations solved in this study. Se tion 3 on entrates on

polynomialsystemssolving.Were allsomefa tsaboutthemultipli ation matrixmethodand

wedes ribehowtotakeintoa ountthesymmetryofthesystem.Wealsopresentourresolution

algorithminthisse tion.Se tion4isdedi atedtonumeri alexamplesandthepaperendswith

some on ludingremarks.

2 Dynami system and periodi solutions

2.1 Systemofinterest

Weaimatndingperiodi solutionsof(polynomial)nonlinearme hani alstru tureswithspe ial

symmetry. For exemple, bladed disks subje ted to geometri nonlinearities represent su h a

stru ture [1℄. Here,only asimple y li system (whi h an be seenas aredu ed order model

of a bladed disk, where all blades have been redu ed on their rst mode of vibration) will

be onsidered.The model onsists in

N

dung os illators linearly oupled, governed by the

followingmotionequation:

m¨

u

i

+ c ˙u

i

+ (k + 2k

c

)u

i

− k

c

u

i−1

− k

c

u

i+1

+ k

nl

u

3 i

= f

i

(t), i = 1, . . . , N

(1)

where

u

i

(t)

representsthe temporal evolutionof degree of freedom (dof) number

i

, and

f

i

(t)

representsthe temporalevolutionofthe ex itationfor e a tingondofnumber

i

.If thereisno

for e, note that this dynami system is invariantunder the a tion of the dihedral group

D

N

(symmetryof aregularpolygonwith

N

verti es).

Equation(1) anbewritteninthefollowingmatrixform:

M

_u

_¨

_{+ C ˙}

_u

_{+ Ku + F}

_nl

_{(u) = F}

_ex

_(t),

(2)

were

u(t)

is theve torof dof of size

N

,

M

= mI

is the massmatrix,

C

= cI

isthe damping

matrix,

K

= (k + 2k

c

)I − k

c

I

L

_{− k}

c

I

U

isthestinessmatrix,and

F

nl

(u) = k

nl

u

3

(4)

assumed to be periodi , with period

T =

2π

ω

, and we will sear h for periodi solutions

u(t)

,

usingtheharmoni balan emethod des ribedhereafter.

2.2 Harmoni Balan e Method

The harmoni balan e method (HBM), is a widely used method in nding approximation to

periodi solutions of nonlinear dierential equations su h as (2) [1,9℄. The solutions

u(t)

is

approximatedundertheformofatrun atedFourierseries,andasystemofalgebrai equations

isderivedbyapplyingGalerkinproje tions.Letusre allthemainstepsofthemethod.

Atrst,ea h omponent

u

i

(t)

oftheperiodi solution

u(t)

isapproximatedby

u

b

i

(t)

under

thefollowingform:

b

u

i

(t) = x

(0)

+

H

X

k=1

x

(k)

i

cos(kωt) + y

(k)

i

sin(kωt), i = 1, . . . , N.

(3)

Wesubstitute(3) in(2)andweproje ttheresultingequationsonthetrun atedFourierbasis:

2 T

R

T

0 R(

u) × 1 dt = 0,

b

2 T

R

T

0 R(

u) × cos(kωt) dt = 0, k = 1, . . . , H,

b

2 T

R

T

0 R(

u) × sin(kωt) dt = 0, k = 1, . . . , H.

b

(4) with

T = 2π/ω

and

R(

u) = M ¨

b

u

b

+ C ˙

u

b

+ K

u

b

+ F

nl

(

u) − F

b

ex

(t).

Equations(4) orrespondsto asetof

N (2H + 1)

algebrai equationswithunknowns

x

and

y

.

2.3 Equationsto be solved

Inourappli ation,

F

nl

(u) = k

nl

u

3

ispolynomialand(4) orrespondstoasystemofpolynomial

equations.Inordertosimplifythepresentationandredu ethenumberofvariables,wewillonly

onsiderasingleharmoni approximationoftheperiodi solution,i.e.,

H = 1

in(3).Moreover,

asthenonlinearityisodd,no ontinuous omponentwillberetained,i.e.,

x

(0)

_{= 0}

in(3).Under

thesehypothesis,(4) orrespondstoasystemof

2N

polynomialequationswhi h anbewritten

inthefollowingform(droppingtheharmoni index

(k)

):

α(ω)x

i

+ δ(ω)y

i

− βx

i−1

− βx

i+1

+ γx

i

(x

2 i

+ y

i

2 ) = f

c

i

, i = 1, . . . , N,

α(ω)y

i

− δ(ω)x

i

− βy

i−1

− βy

i+1

+ γy

i

(x

2 i

+ y

2 i

) = f

s

i

, i = 1, . . . , N,

(5) where

f

c

i

(resp.

f

s

i

)denotestheamplitudeoftheex itationfor esrelativetothe

cos(ωt)

(resp.

sin(ωt)

)term,andwiththefollowingexpressionforthedierent oe ients:

α(ω) = k + 2k

c

− ω

2 m,

β = k

c

,

γ =

3

(5)

For edsolutions. Inthefor ed ase(

f

c

_{6= 0}

or

f

s

_{6= 0}

),theangularfrequen y

ω

issetbythe

ex itationfor esand(5)willbesolvedfor

x

and

y

.Dependingonthesymmetryoftheex itation

for es,system(5)maypresentsomeinvarian eproperties.Wewill hoose

f

c

i

= 1

,

f

s

i

= 0

forall

i = 1, . . . , N

sothat system(5)will beinvariantunderthea tionofthedihedral group

D

N

.

Freesolution. Inthefree ase,weaimsatndingsolutionsofanunfor ed,undampedversion

ofsystem(2),also alledNonlinearNormalModes(NNM)[1012℄.Inorderto simplifywewill

onlysear h forsolutionswhereall dofvibrate "in-phase"(monophase NNM[13℄)by imposing

y

i

= 0

forall

i = 1, . . . , N

,thusresultinginthefollowingpolynomialsystemwith

N

equations:

α(ω)x

i

− βx

i−1

− βx

i+1

+ γx

3 i

= 0, i = 1, . . . , N.

(6)

Theangularfrequen y

ω

willbeset toanarbitraryvalue andsystem(6)will be solvedfor

x

.

Again(6) isinvariantunderthea tionofthedihedralgroup

D

N

anditisalsoinvariantunder

hangeofsign, hara terizedbythegroupwith2elements

Z

2 = {e, b | b

2 _{= e}}

with

b(x) = −x

.

3 Solving multivariatepolynomial systems

Inthisse tionwepresentthemethodusedtosolvesymmetri systemofpolynomialequations.

First,theeigenvaluemethodisdes ribed.Thenweshowhowtoin ludesymmetryofthesystem

inordertoredu ethenumberofsolutionasproposedin [6℄,leadingtotheso alledeigenvalue

methodwithsymmetry.Finallyweproposeanalgorithmtosummarizethepro ess.

3.1 Gröbner Basis

We will denote by

C

[x]

the ringof multivariate polynomials with omplex oe ients in the

variables

x

= (x

1 , . . . , x

n

)

. A polynomialin

C

[x]

has the form

f (x) =

P

α∈S

c(α)x

α

, where

S ⊂ N

n

isthethesupportof

f

,

x

α

= x

α

1

1 · · · x

α

n

isamonomialoftotaldegree

|α| =

P

i

α

i

,and

c(α) ∈ C

isthe oe ientofmonomial

x

α

.Wexamonomialorderon

C

[x]

.Intheappli ation,

wewill onsiderthegradedreverselexi ographi order

≤

grevlex denedfor

α, β

in

N

n

by:

α

≤

grevlex

β

≡ [|α| ≤ |β|]

or

[α

j

≥ β

j

and

α

i

= β

i

for

1 ≤ j ≤ i]

We will denote by lm

(f )

and l

(f )

the leading monomial and the leading oe ient of a

polynomial

f

,wewilldenotebylt

(f ) =

l

(f )

lm

(f )

itsleadingterm.

Consideramultivariatepolynomialsystemgivenby

P

(x) = [p

1 (x), . . . , p

n

(x)]

with

p

j

∈

C

_[x]

for

j = 1, . . . , n

.Wedenote by

I = hP i = hp

1 , . . . , p

n

i

theidealof

C

[x]

generatedbythe

polynomialsystem

P

. Theredu tionoperationmodulo

P

redu es apolynomial

f ∈ C[x]

into

aremainderofthedivisionof

f

byea helementof

P

,denedby:

f (x) =

n

X

i=1

µ

i

(x)p

i

(x) + r(x).

Su haremainderisgenerallynotuniqueanddependsonthedivisionorderandonthemonomial

order.However,theredu tionmoduloaGröbnerbasismakestheremainderuniqueduringthe

(6)

G

= [g

1 , . . . , g

m

] ⊂ I

with the property that for any nonzero polynomial

f

in

I

, lt

(f )

is

divisiblebylt

(g

i

)

forsome

i = 1, . . . , m

.Theremainderondivision of

f

byaGröbnerbasisis

uniquelydetermined,thusis allednormalformfor

f

anddenotednf

(f )

.Inpra ti e,aGröbner

basis an be omputeby the Bu hberger algorithm [14℄ and itsimprovements,e.g. [15℄.The

monomialordering hooseninuen esboththeformofthebasis

G

and omputationtime,and,

ingeneral, omputationwiththegrevlexorderingtendstobefasterthanwiththelexi ographi

ordering.Wedenoteby

A = C[x]/I

thealgebradened asthequotientof

C

[x]

bytheideal

I

.

Theset

G

beingaGröbnerbasis,themonomials

B

= {x

α

| x

α

/

∈ h

lt

(G)i}

form a basis of algebra

A

, as ave torspa e over

C

. If the polynomialsystem

P

(x) = 0

has

onlyanitenumberof solutions(say

D

solutions),the ideal

I

is zero-dimensional,and it an

beshown[7,16℄that,asaspa e,

A

isofnitedimension

D

.

3.2 Multipli ationMatri es Method

Given apolynomial

f ∈ C[x]

, we onsiderthe map

m

f

: A → A

, dened by

m

f

(h) = f h

,for

any

h

in

A

.Sin e

A

isanite-dimensionalalgebrathemap

m

f

anberepresentedbyamatrix

M

_f

relativetothebasis

B

.Thematrix

M

f

is alledmultipli ationmatrix andis hara terized

bythefollowingrelation(modulo

I

):

f B = M

f

B

mod

(I),

(7) orequivalently:

f B

i

=

D

X

j=1

M

i,j

f

B

j

mod

(I), i = 1, . . . , D.

The oe ientsof line

i

ofthe matrix

M

f

anbe obtainedby omputing thenormalform of

ea h produ t

f B

i

andbyexpressingtheresultsasalinear ombinationofelementsof

B

.

For parti ular hoi es of

f = x

p

,

p = 1, . . . , n

, it an be shown that the eigenvalues of

the multipli ation matri es

M

x

p

are related to the zeros of the polynomial system. Indeed,

substituting

f = x

p

into (7),forany

x

,wehave:

M

_x

p

− x

p

I

B(x) = 0

mod

(I).

(8)

It follows that the ve tor

M

x

p

− x

p

I

B(x)

an therefore be expressed asa ombination

of the polynomials in

P

. Now, let'ssuppose that

x

∗

is a root of

P

. Then

p

i

(x

∗

_{) = 0}

for all

i = 1, . . . , n

,and(8)showsthat

x

∗

p

isaneigenvalueof

M

x

p

asso iatedtotheeigenve tor

B(x

∗

₎

.

Notethattheeigenve torshouldbenormalizedsothatitsrst omponentequals1(inorderto

ma hwiththeasso iatedpolynomials

B

1 (x) = 1

).

Going further, it an be shown [7,16℄ that the omponents of the roots are given by the

eigenvaluesof

M

x

p

,

p = 1, . . . , n

, asso iatedwith ommoneigenve tors

B

k

.

Here, we follow themethod given in [4℄ (Chap.1.6.3.2), whi h onsists in onsidering only

onemultipli ationmatrixasso iatedwithalinear ombinationofthevariables

f =

P

n

i=1

c

i

x

i

,

where

c

i

arerationalnumbers hosensu hthatthevalueof

f (x

(k)

₎

isdierentforea hsolution

x

(k)

,

k = 1, . . . , D

. Generally, random hoi es for oe ients

c

i

are su ient to ensure this

propertiesalmost surely[4℄.Thesear hfortheroots ofsystem

P

isthensimply ondu tedby

solvingtheeigenvalueproblem

(M

f

−f I)B = 0

,andbyreadingthesolutionsintheeigenve tors

(7)

Invariantpolynomialsystems. Duetothesymmetryoftheme hani alstru ture( hangeof

oordinates,...),thepolynomialsystemstobesolvedinourappli ations(seese tion2.3)also

possessasymmetri stru ture.Herewewill onsiderthatthepolynomialsystemtobesolvedis

equivariantunderthea tionofagroup

G

, thatis

P

(g(x)) = g(P )(x), ∀g ∈ G

,where

g ∈ G

is

apermutationoperationdenedby

g(x) = [x

g(1)

, . . . , x

g(n)

]

.Theset ofinvariantpolynomial

under

G

is denoted

C

[x]

G

and dened by:

C

[x]

G

_{= {f ∈ C[x] | f (g(x)) = f (x), ∀g ∈ G}}

. We

denoteby

I

G

_{= I ∩ C[x]}

G

theidealinvariantunderthea tionofthegroup

G

.

Quotient de omposition. It anbeshownthat

C

[x]

anbede omposed into adire t sum

of isotypi omponents [6,17℄, su h that

C

[x] = V

1 ⊕ V

2 ⊕ . . . ⊕ V

K

, where the

V

i

's arethe

isotypi omponents(related to the

K

irredu ible representationsof group

G

[6℄), and where

therst omponentistheinvariantringitself:

V

1 = C[x]

G

.Bydening

I

i

= I ∩ V

i

,thealgebra

A = C[x]/I

anbede omposedintoadire tsumasfollows[6℄:

A = C[x]

G

_/I

G

_{⊕ V}

2 /I

2 ⊕ . . . ⊕ V

K

/I

K

(9)

Thespa e

C

[x]

G

anbede omposedintothefollowingdire tsum(Hironakade omposition)[6℄:

C

_[x]

G

_{= ⊕}

_i

_S

_i

C

_{[π] = C[π] ⊕ S}

₂

C

_{[π] ⊕ S}

₃

C

_{[π] ⊕ · · · ⊕ S}

_p

C

_[π]

where

π

= [π

1 , . . . , π

n

]

isthesetofprimarypolynomialinvariantsrelatedto

G

,and

S

2 , . . . , S

n

orrespondtothese ondarypolynomialinvariantsrelatedto

G

.Theprimarypolynomial

invari-ants

π

anbefoundbyusingtheReynoldproje tionoperatordenedfor

f ∈ C[x]

by[18℄:

Re

f

(x) =

1 |G|

X

g∈G

f (g(x)).

(10)

ApplyingtheReynoldsproje tor toanypolynomial

f ∈ C[x]

leadsto aninvariantpolynomial

Re

f

∈ C[x]

G

.Theprimaryinvariants anbe omputedbyapplyingtheReynoldproje tortoea h

monomials

x

α

with

|α| ≤ |G|

.In ertain ases,somemonomialswillleadtothesameinvariant,

or some invariants an be obtained as a ombination of the others. In those ases, we need

to eliminatetheredundan iesby omputingGröbner basis[18℄.Inthis work,we omputethe

primaryinvariantsusingtheinvariant_ring ommandofSingular.These ondaryinvariants

orresponds to a module basis of

C

[x]

G

as a

C

[π]

-module. It an also be omputed by the

invariant_ring ommand.

Usingtheprimary polynomialinvariants. Inthefollowing,theprimaryinvariantswillbe

usedtondthesolutionofaninvariantsystem.Let'ssupposethatwe anndthevaluesofthe

primaryinvariant

π

(k)

_{= π(x}

(k)

₎

forea hsolution

x

(k)

,thenbysolvingthefollowingsystems:

π(x) = π

(k)

,

k = 1, . . . , D

G

,

for

x

byaNewton-likemethod,one an omputeanuniqueo urren eofsolution

x

(k)

andthe

other anbegeneratedbyapplyingthegroup'sa tionson

x

(k)

,i.e.,

g(x

(k)

_{), ∀g ∈ G}

.

We will omputethe valuesof the primary invariants

π

(k)

forea h solution

x

(k)

(8)

suited basisof

A

, it is even shown that themultipli ation matri es asso iated to theprimary

invariantsare blo k diagonal[6, Thm.3℄, withea h blo k ontainingthesame eigenvalues[6,

Prop.8℄.Thus,onlytherstdiagonalblo k(relatedtothesubspa e

C

[x]

G

_/I

G

)isofinterestto

omputethevaluesoftheprimaryinvariants.

All thatis leftto dohere,isto ndabasis

B

′

of

A

thatmakesthemultipli ationmatri es

blo k diagonal.Morepre isely, itis su ientto ndabasis

B

G

= [B

′

1 , . . . , B

D

′

G

]

of

C

[x]

G

_/I

G

inagreementwiththedire tsumde ompositionin (9).

Constru tionofanadaptedbasis. Thegoalistondabasis

B

G

of

C

[x]

G

_/I

G

(with

#B

G

=

D

G

)inagreementwiththedire tsumde ompositionin(9),inorderto onstru ttherstblo k

ofamultipli ationmatrix.Asinthepreviousse tion,themultipli ationmatrixwillberelated

toapolynomial

f =

P

n

i=1

c

i

π

i

,where

c

i

arerational oe ients hosenrandomly.

Thebasis

B

G

shouldonly ontainsinvariantpolynomials,andtheirnormalformsshouldbe

su ienttoexpressallremainders

r

inthedivisionof

f B

G

i

by

I

(i.e.,

r =

P

D

G

j=1

M

G

i,j

nf

(B

G

j

)

).

WesupposethataGröbnerbasis

G

of

I

isknown.Letnfthenormalformoperatorfor

G

.

Atstart,weset

B

G

1 = 1

.

The onstru tion of thebasis then goesas follows. For

B

G

i

in

B

G

we ompute thenormal

form

r =

nf

(f B

G

i

)

.Then,untiltheremainder

r

equalszero,wesear hifthereexists

B

G

j

in

B

G

su h thatlm

(

nf

(B

G

j

)) =

lm

(r)

, thatislt

(r) = q

lt

(

nf

(B

G

j

))

, with

q ∈ C

if su h a

B

G

j

exists, then we divide

r

by nf

(B

G

j

):

r = M

G

i,j

nf

(B

G

j

) + h

and we save the

(numeri )matrix oe ient

M

G

i,j

. Finally, weae t

r = h

, and sear hfor anewdivisorof

lt

(r)

.

if not, we will reate a newbasis term

B

G

k

whose leadingmonomial equalslm

(r)

by

on-sidering theReynold proje tionof lm

(r)

, ie:

B

G

k

=

Re

lm

(r)

. However,it mayhappen that

lm

(

nf

(

Re

lm

(r)

)) 6=

lm

(r)

. In that ase, we modify the Reynold proje tion by subtra ting

thehighordertermuntillm

(

nf

(

Re

lm

(r)

)) =

lm

(r)

.Thisisdonebysear hingintothebasis

an element

B

G

j

0

su h that lm

(

nf

(B

G

j

0 )) =

lt

(

nf

(

Re lm

(r)

))

and by modifying the Reynold

proje tion:Re lm

(r)

=

Re lm

(r)

− c

j

0 B

G

j

0

.On etheinvariantis omputed,wedivide

r

bythe

newelement:

r = M

G

i,k

B

G

k

+ h

, andwe ansavethe(numeri )matrix oe ient.Finally,

weae t

r = h

,andsear hforanewdivisoroflt

(r)

.

This pro ess is repeated until allprodu ts

f B

i

, i = 1, . . . , D

G

, have been omputed.The

basis onstru tionissummarizedin Algorithm1.

Algorithm1.Computation ofabasis

B

G

oftheinvariantspa e

C

[x]

G

_/I

,and onstru tionof

themultipli ationmatrixoftheinvariantvariable

f =

P

c

j

π

j

#Preliminaries

omputeaGröbnerbasis

G

of

P

withthegrevlexorder

initialize

f =

P

j

c

j

π

j

,

B

G

1 = 1

,

n = 1

#Basis Computation

j = 0

while

j < n

do

(9)

omputethenormalform

r =

nf

(f B

G

j

)

while

r 6= 0

do for

k = 1, . . . , n

do if lm

(

nf

(b

k

)) =

lm

(r)

then redu e

r

:

r = qB

G

k

+ h

save

M

j,k

= q

andupdate:

r = h

endif

endfor

if lm

(r) /

∈ B

G

then

omputetheReynoldproje tion

Re(x) = Re

lm

(r)

(x)

if lm(nf(Re))=lm(r)then ae t

B

G

n+1

= Re

else

whilelm

(

nf

(Re)) 6=

lm

(r)

, redu etheReynoldproje tion:

Re = Re − c

k

B

G

k

ae t

B

G

n+1

= Re

endif

redu ethenormalform

r

:

r = qRe + h

save

M

j,n+1

= q

andupdate:

n = n + 1

,

r = h

endif

endwhile

returnthemultipli ationmatrix

M

f

andthebasis

B

G

4 Numeri al appli ations

In this se tion, we apply the eigenvalue method with symmetry to the system given in

Se -tion 2.3. The numeri al appli ation will be ondu ted for system with

N = 2, 4

degrees of

freedom. In the two ases, free and for ed analysis are ondu ted. Solutions for a parti ular

frequen yare omputedwiththemultipli ationmatri emethod,andwegiveanoverviewofthe

systemdynami sbyapplying ontinuationmethods[3℄.Finally,anNNManalysisis arriedfor

2 ≤ N ≤ 6

in ordertoshowthede reaseinthenumberofsolutions.

4.1 Simpleexample with

2

degreesoffreedom

Asarstappli ation,westudyasystemwith

N = 2

degreeoffreedom.Inthis ase,(2)redu es

tothefollowingdynami system:

m¨

u

1 + c ˙u

1 + (k + k

c

)u

1 − k

c

u

2 + k

nl

u

3

1 = f

1 (t),

m¨

u

2 + c ˙u

2 + (k + k

c

)u

2 − k

c

u

1 + k

nl

u

3

2 = f

2 (t).

(11)

Theappli ationoftheHBMwith onlyoneharmoni (

u

i

= x

i

cos(ωt) + y

i

sin(ωt)

)leadstothe

followingsystemofpolynomialequations:

αx

1 − βx

2 + δy

1 + γx

1 (x

2

1 + y

2

1 ) = f

c

,

αy

1 − βy

2 − δx

1 + γy

1 (x

2

1 + y

1

2 ) = f

s

,

αx

2 − βx

1 + δy

2 + γx

2 (x

2

2 + y

2

2 ) = f

c

,

αy

2 − βy

1 − δx

2 + γy

2 (x

2

2 + y

2

2 ) = f

s

,

(10)

with

α = k + k

c

− ω

2 _m

,

β = k

c

,

γ =

3

4 k

nl

and

δ = ωc

.Thefrequen yparameterwill beset to

ω =

25

10

(howeverthesear hformultiplesolution anbe ondu ted foranyvalueof

ω

),leading

tothefollowingnumeri alvalues:

α =

−17

4 , β = 1, γ =

3

4 , δ =

1

10 , f

c

= 1, f

s

= 0.

(13)

Monophase NNM analysis. Wesear hfor monophaseNNM solutionsof(12) (undamped,

unfor ed).Inthis ase,thesystem(6)redu estothefollowing:

αx

1 − βx

2 + γx

3

1 = 0,

αx

2 − βx

1 + γx

3

2 = 0.

(14)

We onsider the order grevlexwith

x

1 > x

2

. Sin e the leading termof ea h equation are

o-prime,thepolynomialsystem

P

isalreadyinaGröbnerbasisform.We omputedanormalset

andweshowthealgebra

A = C[x]/ hP i

isofdimension

9

(i.e.,thesystemhas

9

solutions).

Thesystem(14)is invariantunder permutationof variableand under hangeof sign.This

invarian eproperty orrespondstothegroup

G = C

2 × Z

2

,where

C

2 = {e, a | a

2 _{= e }}

,where

a[(x

1 , x

2 )] = (x

2 , x

1 )

and

Z

2 = {e, b | b

2 _{= e }}

, where

b[(x

1 , x

2 )] = (−x

1 , −x

2 )

. All element

g ∈ G

anberepresentedbyamatrix

M

g

= A

i

g

B

i

g

where

A

and

B

aregivenbythefollowing:

A

₌

0 1

1 0

, B =

−1 0

0 −1

.

UsingSingular,weknowthattheprimaryinvariantof

G

are

π

1 = x

1 x

2

and

π

2 =

1

2 (x

2

1 +x

2

2 )

.

Weset

f = π

1 +

2

3 π

2

,andwe onstru tthemultipli ationmatrixof

f

in ansymmetryadapted

basisof

A

G

usingAlgorithm1.Thebasis

B

G

of

A

G

andthemultipli ationmatrix

M

f

aregiven

by

B

G

= [1,

1

2 (x

2

1 + x

2

2 ), x

1 x

1 , x

2

1 x

2

2 ], M

f

=







0

4 ₃

1

0

46

39

59

9

2

3

0

19

9

68

9

1

0

471

27 1187

27

68

9 



 .

The omputation of eigenvalues

λ

= f (x

∗

₎

and eigenve tors

B

G

(x

∗

₎

of

M

f

gives (after

normalizationoftherst omponent):

λ

=







0 16.3333

2.4444

1.4444





 ,

B

G

(x

∗

) =







1.00 1.00 1.00

1.00

0 7.00 2.83

4.33

0 7.00 −1.33 −4.33

0 49.00 1.77 18.77





 .

Here

π

1

and

π

2

belong to the invariantbasis

B

G

(

π

1 = B

G

3

and

π

2 = B

G

2

), so that their

val-ues

π(x

∗

₎

an dire tlybe read into theeigenve tors

B

G

(x

∗

₎

(atline 3 and line 2), leadingto

the4followingsystemsof equations:

(π

1 (x), π

2 (x)) ∈ { (0, 0), (7, 7), (−1.33, 2.83), (−4.33, 4.33) }

(15)

(11)

solu-a tuallysolutionsof

P

(x) = 0

by omputingthevaluesof

kP (x

∗

_)k

inTable4.1.Toassessthe

quality ofthe realsolutions,we omparethem with renedsolutionsobtainedwith aNewton

algorithm applied on

P

with starting points

x

0 = x

∗

, see Table 4.1. It is seenthat solutions

from the eigenvalue method are indeed very lose to the a tual roots of

P

, as their relative

dieren esliebelow

0.5%

.Inany ases,afewNewtoniterationsshouldbeappliedtoover ome

thenumeri alerrorduetonumeri alroundingofrationalnumbersinthemultipli ationmatrix.

(x

1 , x

2 ) ∈ { (0, 0), (−2.65, −2.65), (2.31, −0.58), (2.08, −2.08) }

(16)

0

1

2

3 −4

−3

−2

−1

0

1

2

3

4 dof

amplitude

1

2 −4

−3

−2

−1

0

1

2

3

4 dof

amplitude

1

2 −4

−3

−2

−1

0

1

2

3

4 dof

amplitude

1

2 −4

−3

−2

−1

0

1

2

3

4 dof

amplitude

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6 −3

−2

−1

0

1

2

3

4 amplitude H 1

frequency [Hz]

solution 1

solution 2

solution 3

solution 4

sym. sol. 3

sym. sol. 4

sym. sol. 2

sym. sol. 3

sym. sol 3

continuation

sym. continuation

x

i

, i=1,..,2

Fig.1. Left:Form of the real solutions of system (14) found by the invariant multipli ation matrix

method. Right: Frequen y ontinuation of the solution obtained at

f =

1 2π

25

10

and their symmetri

relativetothegroupoperation

solution 1 2 3 4

value

kP (x

∗

_)k

0.040.110.000.04

relativedi.fromNR sol.(

%

) x 0.230.000.32

Table1.assessmentofthesolutionqualityof(14)at

ω =

25

10

The appli ation of the groupa tions generates 5other solutions. Atthe end the total set

ofsolutions ontains9elementsasindi atedbythedimensionofthequotientspa e. However,

theuseofsymmetryde reasedthesizeoftheeigenvalueproblemfrom9to4,leadingtoonly4

solutions(oneforea horbitofsolutions).

In order to give an overview of the system dynami s, we usethe four solutions in (16) as

starting pointsfor a ontinuation pro edure on theparameter

ω

. Theresults are depi tedon

Fig.1and orrespondto the monophasenonlinear normalmodes ofthe systems.Three types

ofsolution an beidentied,anin-phasesolution(sol.1), anout-of-phase solution(sol.4)and

alo alizedsolution(sol.3) whi h orrespondsto abifur ationoftheout-of-phasesolution.

For ed analysis. Wenowturn tothefor edanalysisof system(12).We omputeaGröbner

basis

G

with 12elementsrelativelyto thegrevlexorderwith

y

2 < y

1 < x

2 < x

1

.We ompute

(12)

G = C

2 = {e, a | a

2 = e }

with

a(x

1 , y

1 , x

2 , y

2 ) = (x

2 , y

2 , x

1 , y

1 )

. The representation of

G

is

hosensu hthat

a

isrepresentedby

M

a

=

0 I

₂

I

₂

0

.

Theprimary invariantof

G

aregivenby

π

1 =

1

2 (x

1 + x

2 )

,

π

2 =

1

2 (y

1 + y

2 )

,

π

3 = x

1 x

2

and

π

4 = y

1 y

2

;andthemultipli ationmatrixis omputedfor

f = π

1 + π

2 + π

3 + π

4

.

ByusingAlgorithm1we omputeabasis

B

G

of

A

G

with7elements.

All primary invariantsare in

B

G

ex ept for

π

3

. Thus, the normal form of

π

3

is omputed

andtheresultisexpressedintermsofelementsof

B

G

:

π

3 = c

T

_B

G

.Aftersolvingtheeigenvalue

problem,thevaluesof

π

3

atthesolutionspointaregivenby

π

3 (x

∗

_{) = c}

T

_B

G

(x

∗

₎

.

Thesolutionof

P

(x) = 0

arethenevaluatedbysolvingthe7nonlinearsystems

π

= B

G

(x

∗

₎

orresponding to ea h eigenve tor:7solutions(5 real and 2 omplex) arefound by aNewton

algorithm,andtheformoftherealsolutionsaredepi tedin Fig.2.

1

2

0

0.5

1

1.5

2 dof

amplitude

1

2 −2

−1

0

1

2 dof

amplitude

1

2 −1

0

1

2

3 dof

amplitude

1

2 −2

−1

0

1

2 dof

amplitude

1

2 −0.2

−0.1

0

0.1

0.2 dof

amplitude

0

0.5

1

0

0.2

0.4

0.6

0.8

1 dof

amplitude

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.5

1

1.5

2

2.5

3 frequency [Hz]

amplitude ddl 1

Solution 1

Solution 2

Solution 3

Solution 4

Solutiion 5

Sym. Sol 3

Sym. Sol 5

Continuation

Sym. continuation

x

i

, i=1,..,2

y

i

, i=1,..,2

Fig.2. left: Form of the real solutions of system (12) found by the invariant multipli ation matrix

method. Right: Frequen y ontinuation of the solution obtained at

f =

1 2π

25

10

and their symmetri

relativetothegroupoperation

Assessmentofthesolution'squalityisgiveninTable2.Notethat solutionsfromthe

eigen-valuemethod are losetothea tualrootsof

P

, astheirrelativedieren esliebelow

3%

.

solution 1 2 3 4 5

value

kP (x

∗

_)k

0.000.000.000.000.02

relativedi.from NRsol.(

%

)0.030.020.000.002.80

Table 2.Assessmentofthesolutionsqualityfor(12)at

ω =

25

10

Toobtainthefullsetofsolution,weapplythegroupa tionsandgenerate4moresolutions,

leadingto atotalof 11 solutions(7 realand 4 omplex) asindi ated bythe dimensionofthe

quotientspa e.

The appli ation of the ontinuation pro edure for the 5 real solutions from the invariant

system (Fig.2) shows that 3 solutions belong to the prin ipale resonan e urve, and that 2

solutionsbelong to losed urves orrespondingto alo alized motion. The appli ation of the

groupa tiongeneratesanother losed urvesolution orrespondingtothe hangeof oordinates

(13)

4.2 Simpleexample with

4

degreesoffreedom

For

N = 4

,theappli ationoftheHBMwithoneharmoni on(2)leadstothefollowingsystem:

αx

i

− βx

i+1

− βx

i−1

+ δy

i

+ γx

i

(x

2 i

+ y

2 i

) = f

c

i

,

i = 1, . . . , 4,

αy

i

− βy

i+1

− βy

i−1

− δx

i

+ γy

i

(x

2 i

+ y

i

2 ) = f

s

i

,

i = 1, . . . , 4,

(17) with

α = k + 2k

c

− ω

2 _m

,

β = k

c

,

γ =

3

4 k

nl

and

δ = ωc

. Inthe NNM analysis,the frequen y

parameterwill besetto

ω =

31

10

,leadingto thefollowingnumeri alvalues:

α =

−661

100 , β = 1, γ =

3

4 , δ =

1

10 , f

c

= 1, f

s

= 0.

Inthe for ed analysis,the angular frequen ywill be set by

ω =

25

10

, leading to the numeri al

valuesin (13)severalvaluesofthefrequen yparameterwillbe onsidered.

Monophase NNM analysis. Forthemonophaseanalysisthesystemisthefollowing:

αx

i

− βx

i+1

− βx

i−1

γx

3 i

= 0,

i = 1, . . . , 4.

(18)

Asinthepreviousexample,thesystemisalreadyin agröbnerbasisformforthegrevlexorder,

andthedimensionofthequotientpa eisgivenbydim

(A) = 81

(the systemhas81solutions).

Theinvarian egroupistakenas

G = C

4 × Z

2

,where

C

4

orrespondtothe y li groupwith4

elements( y li symmetry),and

Z

2

isthegrouprelativetothe hangeofsignasintheprevious

se tion.Theprimaryinvariantof

G

aregivenby:

π

1 = x

1 x

3 + x

2 x

4 , π

2 = x

1 x

2 + x

2 x

3 + x

3 x

4 + x

4 x

1 , π

3 = x

2

1 + x

2

2 + x

2

3 + x

2

4 , π

4 = x

1 x

2 x

3 x

4 .

The appli ation of Algorithm 1 leadsto the onstru tionof a basis

B

G

with 14 elements.

Followingmethodexposedinthepreviousse tion,14realsolutionsareobtainedbysolvingthe

invariantsystems,andtheirformsaredepi tedinFig.3.Theassessmentofthesolutionsquality

is givenin Table3, showingthat all solutionsofthe invariantsystemsare indeed solutionsof

thepolynomialsystem

P

.

solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14

residual

kP (x

∗

_)k

0.580.680.000.230.000.970.000.000.000.000.000.000.000.00

rel.di.from NRsol.(

%

) x 0.740.000.900.012.110.010.010.010.010.010.000.000.00

Table 3.Assessmentofthesolutionsqualityfor(18)at

ω =

31

10

solution 1234567891011121314total

o uren e128288488 8 4 4 8 8 81

Table 4.Appli ationofthegroupa tiontothesolutionof(18):numberofgeneratedsolutions

(14)

0

2

4 −4

−2

0

2

4 dof

amplitude

0

2

4 −4

−2

0

2

4 dof

amplitude

0

2

4 −4

−2

0

2

4 dof

amplitude

0

2

4 −4

−2

0

2

4 dof

amplitude

0

2

4 −4

−2

0

2

4 dof

amplitude

0

2

4 −4

−2

0

2

4 dof

amplitude

0

2

4 −4

−2

0

2

4 dof

amplitude

0

2

4 −4

−2

0

2

4 dof

amplitude

0

2

4 −4

−2

0

2

4 dof

amplitude

0

2

4 −4

−2

0

2

4 dof

amplitude

0

2

4 −4

−2

0

2

4 dof

amplitude

0

2

4 −4

−2

0

2

4 dof

amplitude

0

2

4 −4

−2

0

2

4 dof

amplitude

0

2

4 −4

−2

0

2

4 dof

amplitude

x

i

, i=1,..,4

Fig.3.Formoftherealsolutionsofsystem(18)foundbytheinvariantmultipli ationmatrixmethod

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0

1

2

3

4

5 frequency

amplitude ddl 1

0.3

0.35

0.4

0.45

0.5

1

1.5

2

2.5

3

3.5 frequency

amplitude ddl 1

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0

0.5

1

1.5

2

2.5

3 frequency

amplitude ddl 1

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

0.5

1

1.5

2

2.5

3

3.5

4 frequency

amplitude ddl 1

Fig.4.Frequen y ontinuationofthesolutionobtainedat

f =

1 2π

31

10

andtheirsymmetri relativetothe

groupoperation(onlypositiveamplitudesoftherstdofare depi ted).Fromtopleft tobotomright:

Mode1(solution2);Mode2(solutions7,11,12,13,14);Mode3(solutions4,5,6,9,10);Dis onne ted

solutions(solutions3,8)

For ed analysis. Wenowturn to thefor ed analysis of system(17). First,the angular

fre-quen yparameterissetto

ω =

25

10

.Inthis asethe omputationofaGröbnerbasisandanormal

set forthe grevlexorder tellsus that thequotient spa e

A

is of dimension

147

.The invariant

group

G

isthedihedralgroup

D

4

oforder 4representedin

R

8

bythefollowingmatri es:

M

_r

₌







0 I

2 0 0

0 0 I

2

0 0 0 0 I

2 I

2 0 0 0





 ,

M

s

=







I

2 0 0 0

0 0 0 I

2 0 0 I

2

0 0 I

2 0 0





 .

Theprimary invariantof

G

aregivenby:

π

1 = y

1 + y

2 + y

3 + y

4 ,

π

2 = x

1 + x

2 + x

3 + x

4 ,

π

3 = y

1 y

3 + y

2 y

4 ,

π

4 = y

1 x

3 + y

3 x

1 + y

2 x

4 + y

4 x

2 ,

π

5 = x

1 x

3 + x

2 x

4 ,

π

6 = y

1 y

2 + y

2 y

3 + y

3 y

4 + y

4 y

1 ,

(15)

With Algorithm1we omputeabasis

B

G

with33elements,andthemultipli ation matrix

asso iated to the polynomial

f =

P

i

c

i

π

i

is also of size 33. Inthis ase all primary invariant

arein thebasisex eptfor

π

7

, forwhi h we omputeitsnormalform andexpressitin termof

elementsof

B

G

as

π

7 = c

T

_B

G

.Thesolutionoftheeigenvalueproblemthenleadsto33possible

values (5 real and 28 omplex) for the primary invariants. Finally the solution of the 5 real

invariantsystemsleadto5realsolutionsofthepolynomialsystem

P

(x) = 0

depi tedonFig.5.

1

2

3

4

0

0.5

1

1.5

2 dof

amplitude

1

2

3

4 −2

−1

0

1

2 dof

amplitude

1

2

3

4 −2

−1

0

1

2 dof

amplitude

1

2

3

4 −1

0

1

2

3 dof

amplitude

1

2

3

4 −0.2

−0.1

0

0.1

0.2 dof

amplitude

0

0.5

1

0

0.2

0.4

0.6

0.8

1 dof

amplitude

0.1

0.2

0.3

0.4

0.5

0.6

0.5

1

1.5

2

2.5

3 frequency [Hz]

amplitude ddl 1

sol. 1

sol. 2

sol. 3

sym. sol. 3

sol. 4

sym. sol. 4

sol. 5

sym. continuation

continuation

x

i

, i=1,..,4

y

i

, i=1,..,4

Fig.5. Left: Form of the real solutions of system (17) found by the invariant multipli ation matrix

method at

ω =

25

10

. Right: Frequen y ontinuation of the solution obtained at

f =

1 2π

25

10

and their

symmetri relative to the group operation. The ba kbone urve of NNM 1, NNM 2, NNM3 and a

bifur ationofNNM2arealsodepi ted

The appli ationof the group'sa tionson thereal solutionsgeneratesonly twoother

solu-tions(i.e., the symmetri of solution3and 4). Thefrequen y ontinuationof the solutionsis

depi tedonFig.5.Again,three solutionsbelongtotheprin ipalresonan e urve

( orrespond-ingto amotionshapeonthe rstNNM), andtwosolutionsbelong toa losed urvesolution

orrespondingto amotionshapeonabifur ation ofthe se ondNNM (i.e.,alo alized motion

ononlytwodof orrespondingtothemonophaseNNMsolution11inFig.3).

4.3 NNM analysisfor

3 ≤ N ≤ 6

Inthislastappli ation,we onsiderthemonophaseNNManalysisofsystem(2).Theappli ation

oftheharmoni balan emethod, leadsto thepolynomialsystem(6).Inorder toillustratethe

redu tioninthe numberof solution,Algorithm1isapplied for

N

from 3to 6.The invarian e

groupeis takenas

G = C

N

× Z

2

, where

Z

2

is related to thetransformation

x

→ −x

.Results

aresummarizedinTable5.

N dim(

C

[x]/I)

dim(

C

[x]

G

_/I

G

) redu tionratio 3 27 6 22.22

%

4 81 14 17.2

%

5 243 26 10.70

%

6 729 68 9.33

%

(16)

10%

ofthetotalnumberofsolution.Thisnumbershouldbeevensmalleriftakingintoa ount

invarian eby ree tion (i.e.,

G = C

N

× Z

2 × Z

2

). Inall ases, the resolution of the invariant

problemsleadsto amaximumnumberofrealsolutionsforthepolynomialsystem(6)(i.e.,the

systemhasdim

(C[x]

G

_/I

G

₎

realsolutions).

Thisappli ationalsoshowsthelimitationoftheproposedmethod.Indeed,the omputation

ofprimaryinvariantsforthedihedralgroup

D

N

isverytime onsumingwhen

N > 6

.However,

further investigations should be arried to see if there exist a way to dire tly ompute the

primaryinvariantofthedihedralgroupforlarge

N

.

5 Dis ussion, Con lusion

Thispaperpresenttheappli ationoftheso alledeigenvaluemethodwithsymmetryforsolving

polynomialsystems arisingin the vibrationsstudy of nonlinearme hani alstru tures by the

harmoni balan emethod.Thesystemunder onsideration orrespondto

N

dungos illators,

linearly oupled. The appli ation of theharmoni balan e method with oneharmoni on this

systemgeneratespolynomialequations,whi hareinvariantundersometransformations( y li

permutation, hangeofsign,...).

Theappli ationoftheeigenvaluemethodwithsymmetryforsolvingtheinvariantpolynomial

system shows that this method is well adapted for this kind of problem. Indeed, taking into

a ount symmetry an greatly de rease the size of the multipli ation matrix. Ea h obtained

solution is dierent and orresponds to aunique orbit of solutions that anbe generated by

applying the group's a tions. Moreover, the obtained solutions are very lose to the a tual

solutionsofthepolynomialsystem,eveninthepresen eofrounding-oerrors.

Thebestresultsareobtainedwhensear hingforfreesolutions(NNM)ofthedynami system.

Inthefor ed ase,themethodisonlyinterestingwhenthespa ialdistributionoftheex itation

alsopresentssymmetry properties. Intheworst ases enario(symmetrybreaking ex itation)

thesystemisnotlongerinvariant,andthemethod nolongerappli able.

Furtherappli ationstolargersystemsseemslimitedbyseveralfa tors.Therstdrawba kis

relatedtoGröbnerbasis omputation.Forlargenumberofvariables,it antakeagreatamount

of time even with thegrevlex ordering.Se ond, it is not lear how to e ientlynd primary

invariantsoflargegroupssu ha

D

N

or

D

N

× Z

2

forlarge

N

.However,the omputationofthe

invariantsis needed onlyon e perinvarian egroup asthey anbe reusedfor any subsequent

omputationonsystemhavingthesameinvarian eproperties.

Although thismethod haslimitations,wehaveto re allthat numeri almethods, su h has

homotopie, arealso subje tedto limitationsthat restri tthesize of thepolynomialsystemto

be solved. In this ontext, the fa t that the eigenvalue method with symmetry automati ally

sortsthesolutions(i.e., omputesonlyonerepresentativeofea h orbits)isanimprovementas

itsimpliestheanalysisof thesystem.

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(17)

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