Stability of parametrically forced linear systems

Rochester Institute of Technology

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Theses

Thesis/Dissertation Collections

1994

Stability of parametrically forced linear systems

Andrew J. Leccese

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Recommended Citation

Approved by:

Stability of Parametrically

Forced Linear Systems

Andrew

J.

Leccese

A Thesis Submitted

in

Partial Fulfillment

of the Requirements for the Degree of

MASTER OF SCIENCE

in

Mechanical Engineering

Prof.

J.

s.

Torok (Thesis Advisor)

Prof. H. Ghoneim

Prof.

C.

W. Haines (Department Head)

Department of Mechanical Engineering

College of Engineering

Rochester Institute of Technology

Pennission Grant

1,

Andrew J. Leccese, hereby grant pennission to the Wallace Memorial Library of the

Rochester Institute of Technology to reproduce this thesis in whole or in part. This

pennission is extended as long as no reproduction will be used for commercial use or

profit.

Abstract

The stabilityanalysis of constant-coefficientlinearsystems isextendedtosystems

with _{periodically-varying}coefficients. Althoughthis

theory

is_{mathematically}

well-understood,little work_{has been done regarding its} applicationtophysical problems. All

previous results arebasedon asymptotic analysis. Areview ofthe

theory

of

parametrically-forced linearsystemswillbe presented, followed

by

adetailed stability

Acknowledgments

Iwouldliketo thankeveryonewhomadethis thesis possible,especially:

Dr. Joszef

Torok,

for is guidance, insightandassistancein

finishing

this thesison

timeinsucha shorttime.

Thecommitteemembers, Dr.

Torok,

Dr.GhoneimandDr. Haines for_reviewing this thesisina shorterthannormal_{time, allowing my}todefend beforegraduation.

Dr.Charles HainesandtheentireMechanical

Engineering

faculty

whohavegiven metheexcellent education Ineedforasuccessful career.

My

parentswho'spatience and support made_mysuccess possible.

My

friendswhohadto dealwithmenot_onlywhenI wasat_{my best but}alsowhen Iwas at_myworst.

The wonderfulpeopleatPepsiCo for sellingtheirfineproduct,MountainDew,

ListofSymbolsUsed

A,B,...

-capitalletters generally denote matrices

-squarebrackets areplacedaroundthe

letterwhenneeded for clarity

ay

by,...

-elementin thematrix_using the same letterofthealphabet

x'

,y '

-vectors,denoted with_arrow, superscriptused where _{necessary for clarity}

Xj.yi,...

-generallyelements ofvectors, denotedwithsubscript

X

-eigenvalue, also called a characteristic factoror characteristic multiplier

v

-generallyaneigenvector_{(modal vector)}

r,

Table

of

Contents

SECTION PAGE NUMBER

Abstract i

Acknowledgments ii

List ofSymbols iii

CHAPTER 1

Introduction to

Stability Theory

1-1

CHAPTER 2

2.1 Fundamental Solutions ofLinearSystems 2-1

2.2 Fundamental Solutions 2-2

2.3 Examples ofFirst Order Linear Systems 2-4

CHAPTER 3

3.1 Constant Coefficient Systems 3-1

3.2 Examples ofPossibleCases 3-4

CHAPTER 4

4.1 Time

Varying

Systems 4-1

4.2 Floquet

Theory

forOne Dimensional Systems 4-1 4.3 ProofofFloquet's

Theory

for One Dimensional Systems 4-3

4.4One-Dimensional Examples 4-6

4.5 Van der Pol Equation 4-8

4.6 Floquet

Theory

for Systems 4-12

4.7 Floquet's Theorem forn-Dimensions 4-14

4.8 Hill's Equation 4-17

4.9 TheMathieu Equation FormofHill's Equation 4-20

CHAPTER 5

5.1 Higher Order

Study

5-1

5.2 Illustrative Examples 5-3

5.3 General Cases 5-13

5.4 NumericalExamples 5-17

5.5 Effects of

Damping

5-17

5.6

Summary

5-29

CHAPTER6

REFERENCES R-l

APPENDIX

Integration Example From Chapter 2 A-l

Explanationof eAt

B-l

Sample Plots ofSolutions to Mathieu Equation C-l

CHAPTER 1

Introduction:

Stability Theory

Anacceptable andphysically-realizabledynamic systemmust_satisfythe threebasiccriteria ofstability,

accuracy, and a_satisfactorytransientresponse. Ofthese criteria,stability isthemost important specification ofa system. Ifa physical systemisunstable, other propertiessuchastransientresponse

and steady-state errors are_onlysecondary,if_theyare relevantat all. Aspecifictransientresponseor

steady-stateerrorrequirement can notbepredictedforan unstable system. Thereare_{many definitions}

ofstability,

depending

uponthekindofsystemorthepointof view. Inthisinvestigation,stabilitywas

taken as aboundedresponse astimeapproachesinfinity.

A lineartime-invariant systemisstableifthenatural response approaches zeroas time approaches

infinity. This definition isalsoknownas asymptotic stability. Sincethe totalresponseisthesumofthe

forcedand naturalresponses, the definitionof_stabilityimpliesthat_onlytheforced responseremainsas

thenatural response approacheszero. Alinearsystem with_time-varyingsystem parametersissaidto

bestableifthe response,forallinitialconditions, remainsboundedastime approachesinfinity.

A linearsystemissaidtobe unstable,iftheresponse growswithoutboundastime approachesinfinity.

Ifthesystemresponseneitherdecaysnor_grows,butremainsconstant oroscillates,thenthelinear

systemisreferredto as_marginallystable.

Physically,

an unstable systemcan causedamageto the system, to adjacentproperty,or eventohuman life. Thusthecriterion of_{stability is}evenmoreimportantthan _{its quality}of performance.

In

fact,

someunstablesystemsare not evenphysically-realizable,suchas_tryingto balancea pencil on

itstip.

Inorderto_effectivelyanalyzethe_stabilityofdynamicsystems, it becomes necessarytoformulate

precise definitionsofthenotion ofstability. Since dynamicsystems are_{mathematically}modeled_using differentialequations, _{stability is}characterized

by

thenature oftheassociated solutions.

Ideally,

onewouldlike to_explicitlycompute all solutionstoadifferentialequation orasystem of

differentialequations. However,thereare_{actually very few}equations

(

beyond linearequationswith

constantcoefficients

)

thatallowexplicit solutionintermsofanalyticfunctions. Inthis

investigation,

we_studysomeofthequalitative aspects ofthesolutions ofdifferentialequations. Theobjectivewillbe to analyzetheproperties ofthesolutions without_{explicitlysolving for}them. These ideaswerefirst

advanced

by

theindependent workoftwo mathematicians,AM. LyapunovandHenri Poincare atthe turnofthecentury. Theirideasremain _veryapplicabletothisday.

Numericalanalysis allowscalculationofaspecific solutiontoadifferentialequation. The

computations_onlygiveresults_{corresponding} toa specific set ofinitialconditions. Thisis fineifwe

desireinformationon_onlyonespecific solution.

However,

in manyproblems,forexample,inthe

designof complexsystems, automaticcontrols,and so

forth,

we wanttoextractinformationof a

qualitativenature about allthepossible solutionstoa setofdifferentialequations.

Moreover,

weoften wanttoknow whetheracertain_propertyofthesesolutionsremainsunchangedif

thesystemissubjected tovarioustypesofchanges

(usually

calledperturbations). Forsuch purposes,

thecomputerandthecalculationofafewspecific solutionsdonotprovide a_satisfactoryanswer.

Thesequalitative studies are also important fromthepracticalpoint ofview, because inmost problems

(including

simple spring-massorpendulum_problems) thedifferentialequationsandthemeasurement ofinitialvalues and various otherdatainvolveapproximations.

Indeed,

inalmost_everymathematical model of a physical problem a number of effectshave beenneglected. Itistherefore importantto _study howsensitivetheparticular modelisto small perturbationsorchangesin initialconditions and of

various parameters. Another drawback intheuseofnumerical approximationisthatoften itisof

interestto showthatasolutionof adifferentialequationtends to zero ast-*. While anumerical approximation method_maysuggestthat thisistrue,itcannotbeused toproveit.

Onequalitativephenomenon ofinterestofgreatimportance isthenotion of_stabilityofacertain

solutionof adifferentialequation. _{This investigation is devoted primarily}to the_studyofthis_property

andconditionsunder whicha solutionisstable. Thisconceptwillbemotivated and_{precisely defined in}

thenext section.

Theobjectivewillbe toconcentrateondynamicsystemswith_{periodically-varying}coefficients. Afull

discussionof systems with variablecoefficients is beyondthescopeofthiswork.

Besides,

_many

interesting

and complicateddynamicsystemsknownas parametrically-excitedsystems are modeled

by

differentialequationswith

periodically-varyingcoefficients. Oneexampleisastandardpendulumwith

a_{periodically-moving base. Such}systems are_{mathematically}well-understood,but _veryfewengineers are exposedtosuchanalysis.

The discussion beginswithareview oflinearsystemstheory. Forcompleteness, the_stabilityof

constantcoefficient systems willalsobereviewed. _{The stability}ofsystemswithperiodic coefficients willbetreatedastwoseparatecases. Theone-dimensional_theorywillbe discussed in detailtoprovide

insightto thehigher-dimensionaltheory. _{For clarity}ofexposition, thetwo-dimensional casewillbe

representativeofthe_theoryin higher dimensions.

Finally,

adetailedanalysis of a_{parametrically forced} pendulum willbe documented.

CHAPTER

2

2. 1 Fundamental SolutionsofLinear Systems

Asa background forthe discussion to

follow,

wewillstartwiththesystem

x =A x

where x isan n-dimensional vector with variable components which arerealnumbers,

whichcan beexpressed_explicitlyas:

x=i x.(t)

x2(t) '

and x= -x2(t)

A(t).

*_(tr.

Also,

A is ann

by

n matrixinwhicheachcomponent_ay

(l<i,j<n)

_{may be}either constant

or a continuousfunctionoftime,that

is,

a,j=a,j(t) with

(l<i,j<n)

Sincethis isalinear homogeneoussystem, twosolutions x' and x 2

which_satisfythe

relations x1 = Ax1 and x2 = Ax2

canbecombinedto formthesolution ccx1

+fix2 which

willalso _satisfytherelation

(ax1

+[_x2

)

=

A(ax'

+(3x2

)

.

2.2 Fundamental Solutions

Acollection ofvectors

v',v2,...,v"

are called

linearly

independent if

c,v'+cA+...+cv =0

onlywhen

ci = _c2 = ... =cn=0

In other_{words, any}one vectorcan notbewritten as alinearcombinationofthe others.

Since wehavea n-dimensional system, therewill ben

linearly

independentsolutions.

Now let x\x2,...,xn

bethen

linearly

_independentsolutions ofthesystem x=_Ax.

Therefore x1 =

Ax1

foreachiand _{x'(t),x2(t),...,x"(t)} are

linearly

independent for all

time. Then jx1,^2,...,^"

j

is afundamentalset ofsolutions to x =Ax. When _arranged

ascolumns,

x(o=[x1(t)

;

x2(t)

;

;

ru)]

iscalled afundamental matrix of x =_{Ax (orthefundamentalsolutionifthesystem isone} dimensional). Since x',x2xn

are

linearly

independent,

det(X(t))^0 forallt.

Therefore

X(t)~

willexistforall t.

Suppose that:

X(0)=Iaxn^

1

which meansthat:

x'(0)=e'=i

0

When thisis true then

X(t)

isnot_onlya fundamental matrix, it is also referred to as the

principalfundamentalmatrix.

In general,if

X(t)

isa fundamental matrix, then thematrix defined

by

X.t)-X(O)"1 will bea

principal fundamental matrix.

Additionally,

_anymatrixdefined

by

X(t)Cwill also be a

fundamental matrix providedthatC is a non-singular matrix (det(C)*0).

Also it shouldbe noted that thefundamentalmatrix

X(t)

is commonlywritten as _{<_>(t)}with

columnsdenoted as _<j)'(t) insteadof x'

(t)

. Forcompleteness <_> can be writtenas

*(t)

=[$1

<DJ

i

?']

For anysystemwithconstant coefficients oftheform:

x=Ax

theprincipalfundamental matrixis definedas:

X(t)=eAt

The meaning oferaisedto thepowerof a matrixisexplainedintheAppendix.

Fortime_varyingcoefficients, thefundamentalmatrix canbe determined

by

the

following

procedure. First ageneralsolution withnconstants mustbefound:

x(t)=

i

cc,f,(t)

a,f,(t)

.

o,f_(t)J

thenchoose theconstants _ctksuchthat x'

(0) = e1

orin other words:

x(0H=i 1

1

Now

X(t),

the principalfundamental matrix,is knownso_any solution

(depending

on

arbitrary initial conditions _x(0))isgivenby:

__(_)

2.3 ExamplesofFirstOrderLinear Systems

1

)

Aone dimensionalsystem withaconstant, insteadofatime dependent,coefficient wouldbe x=ax.

The fundamentalmatrix (orthefundamental solutioninthiscase)would be:

X(t)=e^dtc=eA.

Now pick c suchthatX(0)=1.

Substituting

t=0 results inc=l to _satisfythiscondition. Therefore x(t)=X(t)x(0)=eatx(0). Thisresults inthefundamental solution

X(t)=ea .

2)

Now lookatthe system x =_{Ax where}A is _{an n}

by

_{n constant matrix.} If

X(t)

is the principalfundamentalmatrix, then:

X(0)=el

1

Al

it isalso true thatX(t)=e =exp[At].

3)

Nowconsiderthe

following

system:

[1

1]

X =

lj

=_{A3_}

x.

The twoequations ofinterestare:

Thegeneral solutions are:

Xi=Xi+X2

orforeach ofthen

linearly

_independentsolutions(2 inthiscase):

x,

|

I

c,e3t -i-c.e

'

x.

__. Ml_| M

x, 2c,eJt

Forsuch asystem wewant

x'(0) = e'

which resultsintherelations:

X ₍₀₎

2c,

-2c. "

Ol

for x1

theresults are:

and:

For x"

theresults are:

and:

and

"=A,H.

c,=0.5

J0.5J

c. =

|0.5

,

0.5e3A0.5eM

X =

e*-e-Cl=0.25

J

0.25

j

c. =-0.25^1-0.25

Thereforethe

fundamental

matrix is:

X(t)=

0.5e3A0.5e_l 0.25e3t

e -e

0.5e3t+0.5e"

4)

Now consider a 1

by

1 system witha variablecoefficient, x= _{a(t)x. In a one}

dimensionalsystem, thegeneral solution isalways:

X(t)

=_eJa,t)dtc

where cis a constant

Since thesystemisone

dimensional,

n=l andthefundamentalmatrix_{is simply}the scalar:

X(t)

=_ejaU)dtc

Ifc ischosensuch that

X(0)=1,

then _x=X(t)x(0) isthesolution which will _satisfythe

given differentialequationandinitialconditions.

5) Let,

forexample, thesystem be x=cos(t)x. Again, as shown inthe lastexample, fora

one dimensionalsystem thefundamental solutionis:

X(t)=e^("c= _esm<t,c

Againwe wantX(0)=1 so_setting ec=l results inc=l. Thisresults in

X(t)=esmlt),

whichis

theprincipalfundamentalsolution. Therefore x(t)=X(t)x(0)= _esmlt,x(0).

6)

Next,

considerthe

following

system:

sin(t) 0

0 cos(t). x,

=A(t)x

The twoequations ofinterestare:

x,=sin(t)xi

Sinceeach equation isonedimensional the general solutionswillbe:

-cos( t) xi(t)=e

""ll'c,

x2(t)=esml,,c2

or foreach ofthen

linearly

independentsolutions (2inthis case):

fx, 1

e'^'A,

x. esm(,A

Forsuchasystem we want

x1

(0)

= e'

whichresultsin therelations:

For x theresultis:

"ArHii

and

c1=e

1|

c, =₀ ^

lo]

and:

f (-co_(0+l)

x' =

For x2theresultis:

c, =0

01

cAbJ

and:

-X '

|esm(t)

X(t)=

-cos(t)+l

0 esin(t)

7) Finally,

considerthe

following

system:

A.

cos(t) 0

sin(t) -l+_{cos(t). A.}

where theequationsofinterestare:

X[=COS(t)Xi

x,=

sin(t)x1+(-l+cos(t))x2

Thefirstequationisonedimensional resulting inthegeneral solution:

x1(t)=

esm,,)c1

substitutingthis back intothesecond equation results in:

x2 =c1esm(,A(-l+_cos(t))x2

which can besolved yielding:

x2=e-,+s,n(t)c,+ cie' [' eu-.in(u)esm,u) du ^0

a more detailed procedureto gettheabove resultis shownin AppendixA. Theresultis:

x2=e-t+sm<t,c2 + c. e,+sin(t)

[

eu du

x,=c,esm,,,+cx-l+sm|,0)

which isthegeneral solution ofthesecondequation. Therefore foreachofthen

linearly

independentsolutions (again,2 inthiscase) wehave:

sin(t)

I

c,e

le^HcAcA'),

Forsuchasystem we want x'(0) =

e;, whichresults intherelations:

x'(0) =

'x.(O)

1

and

A0)4X'(0)L(

c>

}_(01

lx2(0)J

jc1+c3r

if

For x1

theresultis:

:::Hu

and:

gSin(t)

*

~{e-W(i-e-')i

For x2

the resultis:

and:

C=0 ₀

c,+c3

=_l

(ij

-t+si_(t)

Therefore

the principal

fundamental

_matrixis:

esm(t)

o

f

asin'"

CHAPTER

3

3. 1 Constant

Coefficient

Systems

Considera system ofdifferentialequationsofthe form:

x= _Ax

A is a matrixofconstantcoefficientsbut x and x areboth functionsoftime, hence the

doton x

denoting

aderivative withrespect to time. Therefore thesystem couldbe

writtenas:

x(t)=

Ax(t)

Thetime dependentnature should be understood andwill

frequently

notbeexpressed

explicitly.

Theeigenvectors, or modalvectorsand thesolutionofthegiven system ofdifferential

equations are bothdependentoneach other.

Therefore,

with this inmind we can lookat

theform ofthecoefficient matrix. Forthe_previouslygiven system:

x=_Ax

which, fora two-dimensional system,could alsobewrittenis the

following

form:

x^aiiX!-._a]2X2

X2=a2]X2+a22X2

Inordertodeterminethe_stability ofsystemssuch asthese one needstoinvestigatethe

eigenvalues and eigenvectors ofthesystem. Eigenvalues are_{the values,} denotedasX or

X,,

which _satisfytherelation:

Ax=A,x

forsome non-zero vector x.

A necessaryandsufficient conditionfortheexistenceof sucha solutionis therelation:

det(A->.I)=IA-Xll=0

As

long

as thisrelationis true,thesystem willhave a non-trivial solution.

To besurethat this isclear,hereis an exampleof

finding

theeigenvaluesfor an_arbitrary2

Let:

AM

4 -5

2 -3

[A-Xl]=

4-X -5

2 -3-X

|A-XI|

= (4-A.)(-

3->l)-(-5)(2)

=A.2->.-2 =₀

The roots ofthisequation are:

Xx=2

X2=-\

Eigenvectors arethenfound

by

substitutingone eigenvalue at atimeback into thesystem

[A->d]

x=0 and then_{solving for}avectorin termsof an unknown parameter. Hereis an example_continuingwiththissamesystem:

[A->J]x

=

4-X -5

2 -3-X x.

|o|

lol

For

Xl=2:

(4-2)x1-5x2=2x1-5x2=0

2x,+(-3-2)x2=0

Thisresults intwo identicalequations:

2xr5x2=0

Finding

arelation betweenx< _andx2it is foundthat_x2=0.4x! whichresultsinthe

eigenvector:

vi =

ii.ol

10.41

Any

multiple ofthiseigenvectoris alsoavalid eigenvector. Thisis a_relativelystandard

formsinceit ismaximumnormalized, thelargestterm is forcedtobepositive 1.

Thesame procedurecan beperformed_usingthesecond eigenvalue

X2=-\.

Thisresultsin

v,

This isa general procedurefor

finding

the eigenvalues and eigenvectors fora2

by

2

system of equations.

However,

we wantto transform theequations somewhat. Sincethe

eigenvectors_{corresponding} todifferenteigenvalues are all

linearly

independent,

wecan

investigate thedynamics ofthesystem:

x= _Ax

by

examining the dynamicsofthesolutions _{corresponding} todifferenteigenvalues and eigenvectors.

We wanttochangefrom thecoordinate systemx to thecoordinate system_ythrough the

lineartransformation:

x =

Py

Thismatrix Piscomposed oftheeigenvectorsarranged as columns.

Substituting

the

aboveequation intoour originalsetofequationsresultsin the

following

equation:

Py

=

APy

Since Pis themodalmatrix, iswillbenon-singular, therefore P"1

will exist. Now

pre-multiply bothsides

by

P"1

whichresultsin:

y=

P_1APy

By

renamingthematrix P"1

AP as anew matrix

B,

the transformedsystem can be

representedas thefollowing:

Y=

By

or

y(t) =

By(t)

Againthe timedependentnatureshouldbe understood andwill

frequendy

notbe

expressed explicitly. Thiswillbecome ournew system which will revealthedynamicsof

each eigenspace. ThematrixB willbeassimple aspossible(incanonical

form)

and willfit

intooneof six possible cases. Thesystem willhave initialconditionsdefined as:

y0 =

P"%

y=

y(t)= ce"=

y

e-Where thevalues

denoted

as_c, are constants

Knowing

theexpected formofthesolutions, we will take a closerlookatall ofthe six possibleformswhich these solutions cantake forsuch a system.

3.2Examplesof_{Possible Cases}

Asmentioned _{previously, using} such a2

by

2_system, the matrixB willfit into one ofsix

cases as follows.

Casei)

B=

X

o

0

X,

where

X2<X\<0

or

0<X2<X!

Here

Xi

and

X2

arethe twoeigenvalues. Theeigenvectors are:

<.=

Hand^=r,

Aphase plot_consistingof_yi versusy2 canbeconstructed to

help

demonstrate whetherthe

systemisstable ornot. As a sidenote, incases wherethe2

by

2system isa setof coupledfirstorderequationswhich was derived froma single second orderequation, y2 will_actually

by

thefirst derivativeofthevariableyi.

ForthecasewhereX2<^i<0thephase plotwillappearlike figure 3.1 below:

Forsuch a system thesolutions are oftheform:

y(t)

yi(t)l

=

Jyi(0)e

.y2(t)J

jy2(0)e^

Since^i

and

X2

arebothnegative inthisinstance thesolutions will

decay

_{exponentially}

and approach zero as t

. . Therefore this solutionis

inherently

stable.

When

0<A,2<Xi

the phase plotwill appearlikefigure 3.2shownhere:

(Fig.

3.2)

Againthesolutionhas thesameform:

y(t)=

'y,(t))=

y,(0)ev

j2(t)J

1y2(0)ex''

Howeversince

Xx

and

X2

arebothpositivein thisinstancethesolutionswill grow

exponentiallyand willbecome unstable.

Case

ii)

B=

'X

0

0 X

where

X>0

or

X<0

Here

X

isthe _repeatingeigenvalue. As beforetheeigenvectorsare:

4>i

=

When

X>0

thephase plot looks like Figure 3.3:

(Fig

3.3)

The solution would havetheform:

y.

y(t) =

y1(t)l

=

ly1(0)ex

72

(0

J

1y2(0)eA

Since

X

is positivethesolutionswillincreaseexponentially, therefore thesystem is

unstable. Since both _yi and _y2increase atthesameexponentialrate, the _resultingphase

plotshows alinear relationship betweenthese two variables.

WhenX<0thephase plotlooks like figure 3.4:

(Fig.

3.4)

Againthe solution wouldhavetheform:

?>

y(t) =

y.(0)el

Since

Xi

X2

are both negative, the solutionsconverge to zero ast> ascan beseenin the

phase plot. Therefore thissolution willbe stable. Againsincebothvariablesdecrease at

thesameexponentialrate thelinesonthe phaseplot show alinear_{relationship between}

them.

Case

iii)

B=

X

o

0

x2

where

Xi

and

X2

arethe eigenvaluesand againtheeigenvectors are:

AH

and

A

Inthiscasethe phase plot looks like figure 3.5

y

y>

(Fig.

3.5)

Thesolution willhavetheform:

y(t) =

'yi(0

y2(t)

Case

iv)

B=

'X

1 0

X

where

X>0

or

X<0

Again

X

represents the _repeatingeigenvalue (repeatstwice). However this time

eigenvectors appear as:

Herethe solutionhasthe form:

*i=^}and*2=r

y(t)

yi(0)e^+y2(0)te'

Negative valuesofXwill resultinstable solutions and resultsin thephase plot shownin Figure 3.6. Asyou mightexpect,positive values of

X

willresultin unstable solutions. For such a situation the arrowson thephase plot wouldbereversed.

Case v)

(Fig.

3.6)

B=

a v

-v a

where o,v*Q

and a>0 oro<0

Xi=G+vi

and

X2=c-vi

whichare not as _readilyapparent as in other cases. Alsonot so obvious are the

eigenvectors which are:

Here thesolution has theform:

y(t) = y_

(t)

1

_

Jy,

(0)em

cos(v _t)+_y2(0)em sin(v _t)

^2

(OJ

jy,

(0)em

sin(v _t)+_y2(0)emcos(v

t).

This casewillbeunstablewhen thereal part oftheeigenvalues

(o)

ispositive. This is the

situation shownin thephase plot shown in figure 3.7. Whenoisnegativethedirections

ofthe arrowsonthe phase plotwouldbereversed andthesolutions wouldconverge to

zero as t. .

(Fig.

3.7)

Case vi)

B=

0 v

-v 0

v*0

1

(J),

=

\P

_and 0- =

Thesolution ofthiscase has theform:

y(t)

Vi

(t)

1

f

yi

(0)

cos(v _t)+_y2

(0)

_sin(v _t)

y2(0 -y,

(0)

sin(v _t)+_y2

(0)

cos(v

t)J

Since die eigenvalues are_purely

imaginary

thesystem isstablefor any real value ofv.

The system doesnot

decay

down tozero asotherstable systems do but itsresponse

repeats over and overthrough the same cycle. Since it is bounded it isthen stable.

Inthiscasethe phase plotlooks like figure 3.8:

CHAPTER

4

4. 1 Time

Varying

Systems

Attention willnowbefocusedonlinearsystems whichhavetime_varyingcoefficients

insteadof constant coefficients. Thegeneral

theory

of systems withtime_varying

coefficientsis beyond thescope ofthis

investigation.

Forconcreteresults, we restrict

ourselvestosystems with

periodically-varying

_{coefficients. Suchsystems represent an}

importantclass of systems regarded asparametrically-forcedsystems. This definition

results fromthefactthatcertain

forcing

_mayresultin

time-varying

system parameters.

4.2Roquet

Theory

forOne Dimensional Systems

Startwith a systemofthe form:

x(t)=a(t)x(t)

whichcan also bewritten_simplyas:

x=a(t)x

(4-1)

The coefficienta(t)wtil beperiodic, witha periodof

T,

sothat a(t+T)=a(t).

The fundamentalsolutionis any solutionto

(4-1)

for whichx(0)=l.

Examplesof variousonedimensional linearsystems are asfollows:

i)

Forthesystem x=ax,

'a'

willbeaconstant. Asolutiontosucha system willbe

x(t)=eat. Sincethissatisfiestheconditionx(0)=l, this willbeafundamentalsolution.

ii)

Forthesystem x=cos

(t)x,

asolution willbe x(t)=esin(t). Againthissatisfies the

condition that x(0)=l,sothis solutionisalso afundamentalsolution.

iii)

Forthesystem x=(l-sin(t))x, asolution will be

+cos(t

. Onceagainthis does

satisfytheconditionthatx(0)=l,therefore thiswdl againbe afundamentalsolution.

Nowletx(t) be anysolution. Thesystem given

by

(4-1)

willhold forall_t,so:

butsince _a(t+T)=a(t) itfollows that:

dt

[x(t+T)]=a(t)x(t+T)

therefore x(t+

T)

isalso a solution of(4-1).

But anysolution of

(4-1)

is a multiple ofthe fundamentalsolution. Therefore:

x(t+T)=x(t)c

where cisconstant (and a scalar sincethisexampleisa onedimensionalsystem). This follows becausea(t)isperiodic.

Hence,

such systemshave a_{very important}multiplicative

property. Thisconcept canbeseenin thesketchbelow:

Ascan beseen inthis sketch, values separated

by

the period,

T,

differ

by

afactorofcand

values separated

by

afactorof2T differ

by

afactorofc2.

Stated

differently,

thesolutionin anytime intervaloflength T isctimes the solution atthe

correspondingpointin theprevious interval.

Therefore,

it is only necessary tosolve

x=a(t)xoverasingle period0<t<T.

Statedexplicidy,x(t)=x(F+nT)=x(t )-cD

for 0<F<Twhere t isthe_{location in any}given

periodasshown inthe

following

figure:

/.

t

t *

2T

.

where

Now ifx(t)is a principle fundamentalsolutionthenx(0)=l so x(T)=x(0)-c=c, x(2T)=c

, x(nT)=cn. Ifx(t) is any

(fundamental)

solution,butnotaprincipal fundamental x(T)

solution, thenc=

.

x(0)

The factorc is definedas thecharacteristic multiplier. Nowset . Therefore

rT=ln(c), andr=ln(c)/T which can becomplexwhen c<0. Thenumberrisreferred toas

thecharacteristic exponent.

4.3 ProofofRoquet's Theorem (One-Dimensional)

Let_{x(t) be any}solution of x=a(t)x,wherea(t) isa continuous periodicfunction with

period T. Then thereis aperiodic functionp(t) such that .

Proof:

-rt -rt Constructp(t)=x(t) . p=x -r-x(t) =x -rp

1)

is asolution of

(4-1)

asshown here:

/ /_. ru rt rt (p(t)e )= pe =rpe

dt

substituting p=

xert-r-p

^-(p(t)ert)=(xe'rt-r-p)ert+rx

dt

d rt rt

(p(t)e )=x-rp-e +rx dt

substituting x=a(t)xand

d (p(t)e )=a(t)x-rx+rx dt therefore (p(t)en)=a(t)x dt

2)

Nowshow that_p(t) is periodic.

p(t+T)=x(t+T)e-r(t+T)

p(t+T)=c-x(t)e"VrT

p(t+T)=c-x(t)eV

p(t+T)=x(t)e-rt

p(t+T)=p(t)

therefore p(t) isperiodic

Onic)

''

Again,

the _{Floquet decompositionis} given

by

x(t)=p(t)e"=p(t)exp t V i _J

Thus,

since _p(t) isperiodic and_{continuous, it is bounded.}

Therefore,

stabtiity is determined

by

thecharacteristic exponent r.

Recalling

that

ln(c)

r=

T

stabilityofthesystem wdlbe determinedasfollows: r<0 stable

r>0 unstable

r=0 _neutrallystable

(periodic/bounded)

Therefore,

reiterating Roquet

Theory,

thesolution _x(t) of a system such as x=a(t)x wdl

be equalto p(t)ert. Here p(t) is a periodic functionand

ert

willcause _x(t)to_{exponentially}

Whenmultiplied

by

thefunctione":

The product isthe modulatedfunction x(t):

4.4One-Dimensional Examples

As an example of a onedimensionalsystem, let

x= _a(t)x

where a(t) isperiodic, therefore:

a(t+T)=a(t).

Asshown inchapter2 the fundamentalsolution isgiven by:

X(t)=e'amdt

Now _x(t)can berewritten as:

Ja(t)dt

e

or

, , r fa(t)dt-bt bt

x(t)=[eJ ]ebt

Furthermore,

thecharacteristic multiplier can be writtenas:

x(T)=cJA)dt

x(0)

sothat thecharacteristic multipher

(eigenvalue)

is:

c=exp

ja(t)dt

The correspondingcharacteristic exponentis:

ln(c)

r=

T

or

1 rT

r=

fa(t)dt

jJo

1 rT a i f -

fa

(t dt J" JO

In summary, solutionsto x=a(t)x are:

stableif a <0

periodic if a=0

unstable if a >0

Asanother example_usingactual

functions,

consider the

following

system:

x= _cos2(t)x

withthesolution:

(AAi|

x(t)=el24 '

Since thisis a 1

by

1 system, the coefficient

"matrix"

A_{is simply}thescalarcos2(t) which

by

definition has a periodofT. Thisisthe requirementofthecoefficient (matrix). Also the fundamentalmatrix

X(t)

isthe scalar_x(t)as shown above. Therefore x(t+

T)

mustbe

equal toascalarmultiple ofx(t), or x(t+T)=cx(t).

Substituting

t+

T,

whereT isequal to k

forcos2(t), intothe abovesolution wdlresultin:

1 t Tt 1 t

cos(f)sin(.)++

cos(Osi_(0+T

e2 - '

=_c e~

wherecwill beequal to

sP1

or_{approximately 4.8105} whichisthecharacteristic multiplier. Therefore, thefact that x(t+T)isascalar multiple of_x(t) has been verified.

si_2t t

x(t)=e~-e2

Also,

as expected,x(t) has beenexpressedas

x(t)=p(t)ebt , where p(t+T)=p(t).

4.5 Van der Pol Equation

An

important

and

often-occurring

equation in_{non-linear mechanicsisthe Van der Pol} equation:

z+ (z2-l)z+co2z=₀

(4-2)

inwhich co ande are realparameters. Thisequation hasno analytical_{solution, but it} is

wellknown tohavealimitcycle,that

is,

an

isolated

_{periodic solution.}

Theanalysis proceeds

by

_writing

(4-2)

_{in state variablefo}

rm:

i x2

x2=-co2x1+e(l-Xl2)X2

(4"3)

Inorderto analyzethelimitcycle, weintroduce the_{polar variables}

xL=psin(co0) and x2=pcocos(co0) to obtain:

0=1-(l-p2sin2(co0))sin(2co0)

(4-4)

p= e(l-p2 sin2(co0))pcos2(co0)

(4-5)

Letp_,denote the

trajectory

ofthelimitcycle.

Then,

onthelimitcycle, thesolutionto

(4-4)

and

(4-5)

_{may be}expressed as:

P(t)=p-(t)

which isa periodicfunctionofperiodT=27r/a>.

Integrating

(4-4)

with respectto 0and

ignoring

thehigher harmonicsresultsin 6=1. Notethat this isequivalent to_averaging the

angular_velocity 0 overthelimitcycle.

Substituting

0=tinto

(4-5)

resultsin:

p =

e(l-p2sin2

(ox))-p-cos2

(ox)

(4-6)

Equation

(4-6)

isstill anon-linearequation, butwe proceedtoanalyze the_stabilityofthe

limitcycle.

(8p)' = e(8p-3p2 5psin(ox))cos2(cot) resulting in: (Sp)* =e(cos2

(cot)

-3p2 sin(cot) cos2

(cot))

8p

Here the variation

8p

represents a perturbationofthelimitcyclesolution p_

Now,

ifwelet

p=pc_+r(

andignore higherorder_terms, we obtaintheperturbation equation

t\ -e(cos2(cot)-3pj sin2

(ox)

cos2 (cot))-r\ that

is,

r|= ecos2(cot)U-3pM2sin2 (cot))-r|

We nowhave alinearequationin the perturbation,r|(t),withavariablecoefficient

a(t)=ecos2(ox)(1-3poS_n2(cot))

of period

2k/

'co.

Asshown in theprevioussection, thestabilityofthe solution ofthisequation isgoverned

by

the average value:

a=E

1 3(0

f--> . i

p" sin"

(cot)

cos"

(cot)dt 2 2k

a= 3e _p

"

sin"(2TCu)cos"(2Ttu)du

2 Jo

Integrating

overthelimitcycle

fp,,/

sin2

(2tiu)

cos2

(2rcu)du=4484

a=-.8452e

Thisanalysis verifies thatfore>0, the limitcycleis stable,and _nearbysolutions converge

to the limitcycle at a rate of

^at -8452a

e =_e

A nearbysolution forwhichx(0)=2.5 and x(0)=0 isplotted. Itcan_{be clearly}seen that the solutionconverges _veryquicklyto the limitcycle. Thisplot appearsbelow:

Using

x(0)=2.5 Van der Pol Equation

Quickly

Converges to Limit Cycle

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

Using

initialconditionsofx(0)=2 and x(0)=0the solutiona_very special casewhich

begins

veryclose to the limitcycle asseen inthe

following

plot:

Limit Cycle

for_{Van der Pol Equation}

-0.5 0 0.5 1.5 2 2.5

[image:40.541.38.500.165.634.2]

Figure 4-2

Various

Solutions

ofthe

Van

der Pol Equation

Figure4-3

4.6 FloquetTheory for Systems

Now let'stakealookatasystemoftheform:

y=

A(t)y

(4-7)

where

A(t)

is acontinuous periodicn

by

n matrixof periodT. Dueto the periodic nature

Nowthere exists a set ofn

linearly

independent

solutions x,,x2,...

,xn whichadsatisfy

(4-7).

Arranging

thesevectors as columns of a matrix denotedas

X(t)

resultsin amatrix of

the form

X(t)=[x1,x2,...,xJ.

X(t)

will nowbe afundamentalsolution matrixof

(4-7),

therefore:

^[X]=[A(t)]X

dt

Since

(4-7)

is true forall_time, it is true that:

^-[X(t+T)]=A(t+T)[X(t+T)]

dt

Since A isperiodic, A(t+T)=A(t). Therefore this equation canbe rewrittenas:

^[X(t+T)]=A(t)[X(t+T)]

dt

So

X(t+T)

mustalsobeafundamentalsolutionmatrix.

From thegeneral_theoryofdifferentialequationsit is true that:

X(t+T)=X(t)C

(4-8)

where C isaconstant nonsingular matrix. Inotherwords,a constant multiple ofa

solution ofalinearsystem is also asolution.

Now,

from linearalgebra, ifC isnonsingular,there is amatrixR=

log(C)

suchthat

eTO=C,whereTisascalar and Ris amatrix.

Wecan assumethat

X(T)

is theprinciplefundamentalsolution,inwhich:

1

X(0)=|

"-.

Since

X(t)

isafundamentalmatrix,det[X(t)]*0 Then from (4-8),

X(T)=era

4.7 Roquet'sTheorem forn-Dimensions

Let

A(t)

bea continuous periodic n

by

n matrix of periodTforwhich

A(t+T)=A(t)

and

X(t)

is a fundamental solution matrix of(4-7). There is a periodic matrix

P(t)

ofperiod T and a constant matrix R suchthatX(t)=P(t)etR.

Corollary:

By

_setting y=[P(t)]u,

(4-7)

istransformed to a constantcoefficient system:

u=Ru

(4-9)

tR-.

The solution of

(4-9)

is ii (t)=e _u0.

However,

itshouldbe notedthattheseresults require

completeknowledgeofthe matrices

P(t)

andR.

Restricting

this discussionto

2-dimensions,

thereare3 genericforms matthe matrixRcan

have. Inequation(4-8), C is a constantmatrix,soC is similartooneof3 generic matrices

denoted asG. That

is,

thereis a change ofbasismatrix

Q

such that:

Q

'CQ

=G

or

C=QGQ

Furthermore, log(C)=Q[Log[G]]Q_1.

Case 1:

IfC has distincteigenvalues

Xi

and

X2

then:

G=

X,

Xj.

then

Log[G]=

logX,

0 1

0

ogX2.

Ifeither of

Xi

and

X2

isnegative, its

log

is notreal. However

L0g[G]l

0 log(X2)2

R

J;l0g

C=Q

log

A.;

T 0 0

logA2

T J so: e,R: iQ =e T 0 0 log/., T . Q~ t e*=Qe log^i T 0 0 log A.2

T JQ"1

e*=Q

log..,

e T 0

logA, Q1

L

o T

Case 2:

IfC hasequal_eigenvalues,

X\=X2=X

(and

taO),

then:

GH

X

1

0 X

then,

r

. i

logGJ10^

X

L

0

log A..

Thus,

R=-logC=Q

T ^

log

A

J_

1

XT

L-i

log

A Q

T

J

T

0

Qt <_* e =e T XT iog^ T . 0 e =Qe logX 1 l~ A o JsSh. T Q" etR=Qi

IL

o T 0

log/v.

AT

!

LO 1

J

fQ

Case 3:

IfC hascomplex eigenvalues:

then,

Ai=a+i(_

A2

=a-ip and, thus, so,

G=\

a -(3

P

a. logG= log(a2 +

p2)

t ~^) tan

1/2 /ft\

-tan va;

log(a2+p2)

1/2

R=-log

C=Q

T ^

log(or+

p2)

Qt e**e log^+p2) etR=Qe tan T \a1 log(a T ~4 \1'-tan T log(o. +|_-t_n" T logfaft+p2

t-Q

Theeigenvalues ofthematrix

C=eTO

aredefinedas characteristic multipliers.

Furthermore,

theeigenvalues ofthematrix

R=l/Tlog(C)

areknown as characteristic exponents. Itshouldbenotedthatcharacteristic exponents arethe generalization of eigenvaluesofconstant-coefficient systems.

Based onFloquet's

Theorem,

the _stabilityofasystem withperiodic coefficientsis

governed

by

the characteristic exponents. This follows from thefactthat:

tR

X(t)=P(t)e

where

P(t)

isperiodic and continuous, and

hence,

bounded.

4.8 HiU's Equation

Wecan now move onto the_studyofHill's Equationwhichis:

y+_p(t)y=₀

again withaperiodic coefficientp(t)resulting inthefactthatp(t+T)=p(t). Thisequation

can be changedinto a set oftwocoupledfirstorderequations

by

_{setting xi=y}and _x2=y.

The resultingsystem will appearas:

x,

0 1

-p(t) o

Let X be thefundamentalmatrix _satisfyingX(0)=I. Inotherwords,

X(t)=

'y

i

(0

y2(t)

where_yi(t) and _{y2(t)are solutionsto HiU's Equation} which satisfy: yi(0)=l y2(0)=0 y,(0)=0 y2(0)=l or X(0)=I

In the previous chapterwe sawthat

X(t+T)=X(t)C.

_{Therefore fort=0 weknow that:}

C=X(T)= Y.(T) y2(T)

y.(T) y2(T)

Thecharacteristic multipliersofthismatrix wouldbetherootsoftheequation

det[C-A_I]=det[X(T)-A.I]=0

Written explicidy thisequationis:

X2 - (y,

(T)

+_y2(T))A+_y2

(Dy,

(T)

-y2 (T)y.

(T)

=0

or

A2-(yi(T)

+y2(T))A+

det(C)

=₀

If AandX aren

by

n matrices and X=AX then it is true that

|x|=tr(A)|X|.

For Hill's dt

equationtr(A)=0,where tris the traceof amatrix,or thesum ofthe diagonalelements.

Therefore for

|X|=tr(A)|X|

to _{be true,}

Ixl

mustbe a constant.

However,

forthis tobe dt

thecase theequation

|X(t

+

T)|

=

|X(t)|-|C|

impliesthat

|C|

mustequal 1.

Going

back to thecharacteristicequation above and_substituting

|C|=1

we gettheresult:

X2-(y1(T)

+y2(T))>.+l=0

This isoftheformX2+ZX+

1=0,

which canbe factored into (A-Ai)(A-

A2)=0

whichcould

thenbemultipliedbackoutintotheformA2+-(Ai+

A.2)+AiA2

=0. Thereforewith

Ai

and

X2

as theroots ofthisequation(the characteristic

factors),

_they must_satisfy

A,iA.2=l.

Since thecharacteristic exponents aredefined

by

Ai=eTr' and

A2

=eTr:

then

l=A,iA,2

=eT(r'+A

Using

thecompletedefinitionofthelogarithmforacomplexnumber z,ln z=ln r+i(0+ 2k

Remembering

the condition

AiA2

=1 there arethe

following

results:

(1)

If

A.i5*A,2

thenHtils' Equation hastwo

linearly

independentsolutions, thesecan be expressed as:

yi(t)=erA(t)

y2(t)=er;tf2(t)

whereri=r2 and

f,(t)

has period T.

(2)

If

A,i=A.2

=1 then Hill's Equation has a solution of periodT.

(3)

If

Ai=A2

=-1 thenHill's Equation has a solutionofperiod2T.

Proofofthe firstresult,was shown earlierinthischapterfora one dimensionalsystem.

4.8 The Mathieu EquationForm ofHill's Equation

Aspecial form ofHill's Equation occurswhen p(t)=a+bcos(t). This is knownas theMathieu Equation andhas the

following

form:

0+(a+bcos(t))0=_O

Thistype ofequation wouldbeencountered in a _{parametrically}excited pendulum as shownin

the

following

figure:

4. S

& s=A_sin(Qt)

Theequationof motion canbe determined_{using Lagrange's Equation:}

dt_ydQj

3L

19=

WhereL=T-V, Trepresentskineticenergy, and Vrepresents potential energy. Additionally,

thesystem is assumedtobeconservative (no friction is present).

The_resultingequation of motionis:

0+ g

AQ2

cos(Qx) 3=0 (4-10)

now make thesubstitutionthat

therefore

t=Qx

dt=Qdx

resultsin:

An

dx dt

Q: d^0

du: +

g

Ail2

h+"h cos(t) 3

=₀

d^0

dt2 + T + COs(t)

0

=₀

hQ1

h

nowlet:

a=- g

h_T

and

A

b=

Substitutionyields:

0+(a+bcos(t))0=_O

whichisthe MathieuEquation as stated earlier. This isthe formofHill's Equationwhich we

CHAPTER

5

5.1 AnalysisofMatheiu's Equation

Nowwe turnour attention to theequation of motion forthe_{parametricaUy}excited pendulumderivedinthelastchapter whichhas the

following

form

0+

(a

+bcos(t))0= _{O (5-1)}

This equation can be transformed into two coupled first order equations

by

_assigning

Assign thevariables q, =

0,

q2 =

0,

leading

to thesystem offirstorder equations

qi=q2

q2

=-(a+

bcos(t))q,

Letthe matrix

<&(*) =

0i

(0

<M0

l<mo

<j>2a).

(5-2)

representthe principalfundamentalsolutionof(5-2). This meansthat

"l

KO)-l0

,

Note,

theprincipalfundamentalsolutionswdlbe denotedas<j)iinsteadof_q* which

represents_{any arbitrary}solution.

By

theprevious section, and sincetheperiod ofthe

coefficients of

(5-1)

is

T=2k,

thecharacteristic multipliers aretheeigenvalues

A,;

of:

<D(2j.) =

<>i(27c)

4>2

(2tt)

L<M2ji)

02

(2k).

Furthermore,

thecharacteristic exponentsare

1 r, = InA,.

1

2k

(5-3)

Inequation

(5-3),

thecomplex-valuedlogarithmisassumed.

Consequently,

the

characteristic multiplierssatisfythecharacteristicequation:

X2

By

Abel's

formula,

det[<Kf)]

=

det[<K0)]

traceU)__.l (5"5)

Hence:

det[<K.)]

=

det[<K0)]

= 1

Therefore,

the characteristicequation

(5-4)

_is:

A2

-[<(,

(2k)+<$>2(2k)]X+\

=₀ (5-6)

Unfortunately,

we can_{onlynumerically} solve forthe

fundamental

solutions_<|>.(t) and 4>2(t).

Thesystem:

0

+

(a

+bcos(t))<t>=_O

mustbe integrated_twice,once withinitialconditions:

<J>X(0)=

1,

(()1(0) =0

and once withinitialconditions:

<t>2.0)=

0,<j>2(0)

= l

The characteristic equation

(5-6)

is obtained

by integrating

these initialconditions_upto

thefinal time t=

2k,

_todetermine the traceof

[<_X23t)]

.

By

solvingthe characteristic

equation

(5-4),

forthecharacteristic multipliers

A,

the_stabilityofthesolutions are

determined with respectto the parameters a andb. These parameters, of_course, relate

backto thephysical parameters g/hQr and _{A/h respectively}as shown inChapter4.

Inthe

following,

let:

S(a,b)

=

trace[0(2Tc)]

=_{<t>,(2TC) + <j>2(27.)}

The characteristicequationbecomes:

hencethe characteristic multipliersare:

S 1

A-u=--Vs^4

Thecharacteristicexponentsare:

r,_, = _ln 1,2

271

S-^4

2 2

5.2Illustrative Examples

Threecaseswillbediscussed in

detail,

withdifferentvaluesof a andb

depending

onthe

values of

S(a,b)

where

S(a,b)

isthe_stabilitysurface.

The stabilityoftheMathieu Equation dependsonS(a,b). The areasinwhichthevalueof

S(a,b)

resultsina stable system can besketched asinthe

following

Strutt diagram. The

hatchedregionsare areasforwhichtheequation willbestable.

This surfaceis plottedoutinthreedimensions_using thecollectionofMATLAB _programs

related to

RUNPTS.

The

following

figure

isone such plotfortheareawhere0<a<2.5and0<b<.25:

Stability

Surface

for

Mathieu

Equation

2.5

b

values

0

0 _{a values}

Note as canbeseen in

Figure

5-1,

the_{stabditysurfacewdlbe}

symmetric aboutthe _ therefore to save_considerable

computing

time,

the surface_{is only}plottedin the

first

quadrant.

a-axis

Anotherplotofthe

stabdity

_{surface appears in the}

next

figure,

this timewith

0<a<2.5

_and

0<b<2.5:

Stability

Surface

for

Mathieu Equation

b

values

0

0

a values

Figure5-3

The largemagnitudesforlargervalues of

'b'

make thedetails_alongthea-axisharder to

The

Runge-Kutta

algorithm usedto

numerically

_{solve forthe fundamental}

solution at each

point on these

stabdity

surface plotsisa built-in

function

on MATLAB accessed through

thecommandode45. Asimplied

by

the name, thealgorithm usesfourth and fifth order

Runge-Kutta formulas.

Accuracy

ofthesolution is determined

by

the variable

*tol*

within

MATLAB. The default value for

'tol',

and the value usedforall_stability surface plots in

this thesis, was

l.e-6,

or

0.000001.

Therefore,

_{thesurface plots}

areaccurate to within plus or minus 0.000002 since S isthe sum ofthe two

fundamental

solutions.

The unstable regions inFigure5-1 can beshownin the

following

plot. Itwas made _using

the MATLAB program calledDOTS. DOTS findsthe value of

S(a,b)

and ifits magnitude is less than 2.0thatpointis marked as_stable, otherwise itis markedasunstable. This

particular regionis_mostly stable, as can beseenintheplot. Alargerregion with more

signdicant unstable regionswidbeshown onthe

following

page.

0.25

0.2

0.15

(0 > 0.1 0.05-xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx

'\&m

xxxx xxxx xxxx xxxx

::Sgg:

XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX xxx

88i

'.$%?

:x xxx XX X

a

X X X X

. stable x unstable

0.5 1.5

2.5

a values

2-^

stable x unstable

xx xxxxxx :xxxxxx ;xxxxxx :xxxxxx .xxxxxx .xxxxxx

'nnnnnn

xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx

nnnnnn

xxxxxx xxxxxx xxxxxx xxxxxx :xxxxxx .xxxxxx

\nnnnnn

'.XXXXXX '.XXXXXX

\nnnnnn

mn

xxxx xxxx xxxx xxxx xxxx xxxx

ms

xxxx xxxx xxxx xxxx XXX xxxx xxxx

mn

xxxx xxxx

m

.xxxx: :xxxx; xxxxxxxx: xxxxxxxx: ^ xxxxxxx: xxxxxxx: xxxxxx: xxxxxx

nnnn?

xxx XX-X :xxxxx :xxxxx ;xxxxx :xxxxx xxxxx .xxxxx .xxxxx

'xxxxx

nmn

xnnnn

xxxx xxxx

xm

xxx xxx

XBB

xxxxx xxxxx

mm

nnm

nnm

xxxxx xxxxx xxxxx xxxxx

num

xxxxx xxxxx xxxxx xxxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx

ms

xxxx xxxx xxxx xxxx xxxxxxxxxx xxxxxxxxx xxxxxxxxx xxxxxxxx

BjgBBBBB

mnnnn

xxxxxxx xxxxxx

mm

nnnnnnnnn

xxxxx xxxxx xxxxx xxxxx XX

m

xxxx xxxx X xxxx xxxx

nm

xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx

mn

xxxx xxxx xxxx xxxx XXX XXX XX XX nx XX

nn

0.5 1.5 a values Figure5-5 2.5

This final plot is starting toresembletheStrutt diagram (Figure 5-2). Although it would

be _{theoretically} possibleto reconstructtheentire plot_usingtheMATLAB programs used

here,

the solutionsare_extremelycalculationintensive andwould requireenormous

Note thatwhenb = _0:

S(a,0)

=

2cos(27w).

Thiscasecorresponds to the equation:

which has fundamentalsolutions

and

0+a0

=_O

<t).(t)=_cosvat

<t>-j(t)=_-7=sinVat Va

resulting intheprincipal fundamentalmatrix:

1

r^y 4, cosVat

-?=_sin

Vat

[<Kt)J= _Va

cos

Vat

Thus,

thecharacteristic multipliers aretheeigenvalues of:

IX27C)]

=

cos(Va2Tc)

-7=sin(Va27c)Va

-Vasin(Va2rc)

cos(Va2Tc)

Thecharacteristicequation is:

withsolution

A?-2cos(2TcVa)A.-. 1=₀

A.=_cos2kVa isin2rcVa

In ordertounderstandthe more generalcase, b*

0,

the abovecan alsoberealized

by

the

following:

We startwith:

where:

R=

ln[<K2T.)]

IK

Next,

we obtain themodal _matrix,

Q,

whichhastheeigenvectorsof

[<(2tc)]

asits

columns.

Thus:

where:

[<K2tc)]

=

Q.G-Q-cos(2T.Va)

-sin(2rcVa)

sin(27cVa)

cos(27rVa)

Theeigenvalues ofGarethecharacteristic multipliers:

X

-exp(i27wa)

Tocompute the characteristic exponents, wenotethat:

ln[<D(2Tc)]

=Qln[G]Q-1

Whichresultsin:

R=^_Qln[G]Q-Thecharacteristic exponents aregivenby:

1

=

(eigenvalues

of

ln[G])

IK

So that:

By

making a

slicing

_{planethrough the}

stabilitysurface whereb=0_{the curve,}

S(a,b)

_which

appears in

Figure

_{5-4canbeseen.}

When_increasing b to .25and then to.5, the

stability

surfaceexpands _{outward. Forsmall}

valuesof

'a'

it_expands

downward,

_{outside ofthe}

stablerange

between

_{-2 andpositive2. These}

additional slices throughthe_stability surfacecan also be seenin

Figure

5-4.

S(a,c)

Setting

a =

0,

_thus

looking

alongthe b-axis, the valuesof

S(0,b)

which appearsin Figure

5-5 below. The stable range is in the band-2 < S < 2whichis markedon_{the graph. As}

can be seen, the system_{is only}stable fora _{veryrestrictivenumber of values ofb}

when a

-0.

S(0,b)

Toemphasize how unstable the system becomes,the value ofLog(

I

S(0,b)

|

) has been

plotted_up tob = _{15. This}appearsin Figure 5-6.

Log(|S(0,b)|)

b

values

RecaU the characteristicequation:

5.3 GeneralCases

A2-S(a,b)A.+

l=0

andtheeigenvalues:

A.U=||VSA4

Depending

on thevalueof

S,

the solutionwiU fall intoone ofthreedifferentcases.

Cases:

Case

1)

IS(a,b)l < 2

Again,

the eigenvalues, or characteristic multipliers are:

S 1

A^-ti-V^S2

=

e"

andthecharacteristic exponents are:

r12 =

(ia)

ia ~ 2k

Therefore,

ln[G]

= 0 -a a 0

andthe_resulting

Q

matrixis:

Using

therelation:

the matrixR is foundtobe:

R=-^-Qln[G]Q1

2tc

R=

0^

a2

4tc2

Andthen

by

manipulation:

e*=e 0 t a2' .tR cos-at 2k

2k . at sin

a 2k

a . at sin 2k 2k cos-at 2k Notethat e*=

a 2k . a

cos t sin t

2tc a 2k

a . a a

sin t cos t

2tc 2jt 2k _

istheprincipal fundamentalsolution ofthesystem:

x= 0 1 L 4k' or jA-=o 4k2

Inthiscasetheperiod ofthesesolutions is

Ta

=

47t2

a

[<Kt)]

=

[P(t)]exp(tR)

a

where

[P{t)]

is periodic of period T= 2k.

Thus,

we_only finda periodicsolutionif is 2k

rational.

In particular, if S=

0,

_a= _and

rl2 =- _andwe

obtain a period-4 solution. That

is,

the

solution has period 8k!

Case

2)

S(a,b)

=2

The roots ofthecharacteristic equation are

Ai =_K2 = ₁

Forthecase ofequaleigenvalues,

[<_>(2jr)]

is similartoeither

G=

"l

or G=

"l

f

1.

1.

Inthe formercase,

R=

0 0

0 0

and

[<_)(.)]

=

[/'(.)]

isperiodic with period2k.

Inthelattercase,

R=

0 1

0 0

and

. . 1

exp(tR)=

Q

IK

0 t

Hence thesystem hasatleastone periodic solution,as

long

astheinitialcondition is a

multiple ofthe sole eigenvectorof

[<K2tx)]

, that

is,

a multiple ofthefirstcolumnofthe

modal matrixQ.

Case

3)

IS(a,b)l >2

Theroots ofthecharacteristic equation

A2-S(a,b)A+l

=0

areboth real. Butsince the product oftherootsis

X\-X2

= ₁

This impliesthatatleastoneofthemhasmagnitude_strictlygreaterthan one.

Here

A,

0

0

A,

and

InG

InA, 0

0

lnA.,

Thus thecharacteristic exponents are

r,_'1,2, = ln

A

2tc

which_{may be}complex-valued,butatleastone ofthemwiU haveapositive real part.

5.4 _{Numerical Examples}

Aseries of sample plots were created_{using MATLAB} todemonstrate_{how stability}

dependson thevalues of a andb andhow thesepatterns correspond to the Strutt diagram

andthe _stabditysurfaceswhichwere lookedat earlier.

These plots areincludedinthe Appendix. Onecan observethat

'a'

and

'b'

forstable and

unstable cases correspondswith the stable and unstable regionsinthe Strutt diagram.

5.5 Effects of

Damping

Having

understood how the parameters of the Mathieu equation affect its stabdity, we now add

damping

to thesystem. The modified equation menbecomes

0+2c0+

(a

+bcos(t))0=O

Herethe

damping

coefficientisrelated to the original system parameters

by

Q. Vh

where is theviscous

damping

ratio.

Instatevariable

form,

wehave

<_i=q2

q2 =-(a+bcost)q, _-2cq2

or

q=

o 1

-(a+

bcost)

-2c

The analysis now proceeds as before.

First,

a principal fundamental matrix is computed.

Aspreviously, this matrix isdenoted

by

<_>(_).

Onesigndicant difference isthat

trace[A(t)]=_-2c

Hence

det[<Kt)]

=

det[<KO)]

expfftr(A)dt

det[<D(2K)]

=

exp(-4Kc)

Therefore,

the characteristic_multipliers, which aretheeigenvalues of<_>

(2tc),

are the roots

of

A2-5(a,_>)A

+exp(-4rtc) =₀

As in the undamped case, the stability of the solutions depends on the magnitude ofthe

characteristic multipliersA,< and

X2.

Inparticular,if

|a,|>i

thenthe solutions are unstable. Thecharacteristic exponents aregiven

by

r,=-LogA,

Thus,

the system isstable,ifand_only

if,

therealpartsofr,are negative.

Returning

to thecharacteristic multiplier equation

X2

-S{a,b)X+ exp(-4Kc)=0

it is_easytoshowthat

|A]

<

1,

provided that

|S(a,b)|<2cosh(27Cc)-e"-2tcc

or

cosi

h(2rcc)

|S(a,b)<2

Notethat thisequationis adirect generalizationoftheundampedcase.

Onceagain,

by

Roquet'sTheorem,thegeneral solutionisgiven

by

Qi

q2

[P(t)]exp[tR]

q_(0)

where

R=

^Log(<_>(2jT))

and

P(t)

isacontinuous and2jr-periodicmatrix.

To see theeffectthatlight

damping

hasonthe shapeofthe_stabilitysurfaces, the

following

stabilitysurface plotshavebeenconstructed_usingtheMATLAB programs relatedto RUNPTS. The cases plottedhere include

damping

constants of

0,

0.1,

0.2,

0.3,

0.4and0.5. Ascan_{easily be}seen, the largerthe

damping

theflatterthe_stabditysurface,

thus thelargerthestable range of 'a'

and 'b'. Thisis intuitive sincea more

heavdy

dampedsystemwouldbeexpectedtobemore stable.

Stability

Surface

for

Mathieu Equation

2.5

b

values

0

0

a values

Stability

Surface for Damped

Mathieu

Equation

(c-.1)

0.15

0.1

0.05

b

values

0.5

0

0

Figure5-10

1.5

a values

Stability

Surface for

Damped Mathieu

Equation

(c-.2)

2.5

0.15

0.1 1.5

0.05

0.5

bvalues 0 0

Figure 5-_{1 1}

Stability

Surface for Damped Mathieu

Equation

(c-.3)

2.5

0.15

0.1

1.5

0.05 _0.5

b

values

0

0

Figure 5-12

Stability

Surfacefor Damped

Mathieu

Equation

(c=.4)

0.2

0.15

0.1

0.05

bvalues

0.5 0 0

Figure5-13

1.5

a values

Stability

Surface for

Damped

mathieu Equation

(c-.5)

0.2

0.15

0.1

0.05

b values

0

0

0.5

1.5

a values

2.5

AnotherMATLABprogramcalled

DOTS

wasutilizedto gainaclearerpicture ofthe

effect oflight

damping

onoursystem.

The

resultsare a two-dimensionalviewofthea-b

planein which each nodeinan 1 1

by

1 1 gridwasanalyzedforstability. Thiscriterion was

thattheabsolutevalue ofthevalueof_{the stabdity surface}atthatpoint mustbe lessthan

2.0. Asexpected, asthe

damping

israised,thesize ofthe stableareasin increased.

Although theresolution on these plotsis

limited,

thetrendisunmistakable.

C--0

ostable x unstable

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9

10

a values

C'-OA

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5

6 7 8 9 10

a values

o stable x _unstable

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a values

The changes in the shapeofthe_stabilitysurface canalso beseen in thecross sectional cuts made through the b=0plane fora fewcases of

damping.

This plotis also

included

to

demonstrate this effect. The maximum limits ofthe

Stability

Surface

at b=0 is

greatly

reduced as

damping

is increased.

This

effect_along the a-axis isrepresentative oftheeffect that

damping

has onthe entire_stability surface.

S(a,0)

with

damping

co 0

-0.5-5 6

a values

8 10

5.6

Summary

The

foUowing

outlines the procedure required to analyze the _stability of linear systems

withperiodic coefficients.

Let q =

A(t)q

be

defined,

where

A(t)

iscontinuous and hasperiodT.

1. Determine theprincipalfundamentalmatrix [<_>(.)]. If

[X(t)]

is any fundamental

matrix, then

[<_>(_)]

=

[X(t)][X(0)]_1

2. Determine theeigensystem

Q1[0>(T)]Q

= G

whereG is incanonical form. Thecharacteristic multipliers,

Ai,

aretheeigenvalues

of

0(231)

and G.

3. Thecharacteristic exponentsaregiven

by

r. _=-LogA;

whicharetheeigenvalues of

R=

^Q[logG]Q"1

1 aswel