EXISTENCE AND

Internat. ^{J. Math.} & Math. Sci.

VOL. 17 NO. 2 (1994) 315-322

315

EXISTENCE AND

^UNIQUENESS

OF

EQUILIBRIUM

STATES

OF

A ROTATING ELASTIC ROD

M.B.M. ELGINDI

Department

ofMathematics University ofWisconsin-EauClaire

Eau Claire,WI54702-4004

(Received

April 7, 1992andin revisedform June25,

1992)

ABSTRACT.

A

flexiblerodisrotatedfromoneend. Theequilibrium equationisafourth order nonlinear two-point boundary value problem which depends on two parameters

,

and a

representing the importance ofcentrifugal effects to flexural rigidity and the angle between the rotation axis and the clamped end, respectively. Previous studies on the existence and uniqueness of solutionof the equilibrium equation assumed ^a 0.

Among

thefindings ofthese studiesis the existenceofacriticalvalue

c

^{beyondwhichthe}uniqueness of the "trivial" solution is lost. The computations of

c

required thesolutionofanonlinear bifurcationproblem. Onthe other hand, this work is concerned with the existence and uniqueness of solution of the equilibrium equation when ^a#^{0 andinparticularin} the computations of^a critical value

’c

^such

that the equilibrium equation has ^a unique solution for each a {} provided <

’c"

^{For small}

a 0thisrequires thesolutionofanonlinearperturbedbifurcationproblem.

KEY

WORDS AND PHRASES. Existenceand uniqueness of equilibrium states, rotating rods, nonlinear eigenvalue problems, fourth order two-point nonlinear boundary value problems, Schauderfixedpoint theorem, perturbedbifurcationproblems, perturbationsolution.

AMSSUBJECT

CLASSIFICATION CODES.

49G99, 73H05,73K15.

1.INTRODUCTION.

Consideran elastic rod ofuniform cross sectionand density whichmakes an anglea,0<a_<

,

with thehorizontal. The rodis being rotated from^oneend that issupported by abushing. The other end is free (Figure

1).

This problem serves as a model for many important engineering problems.

Among

thepreviousstudies arethe papersby OdehandTadjbakshs

[1]

^and

Wang [2], [3].

^Both

[1]

^and

[2]

^{consideredthecasewhena 0.}

In [1]

^{it is shown that} the "trivial" solution becomes unstableas therotationalvelocity exceeds acertaincritical valueand that theproblem possesses nontrivial solutions for such rotationalvelocity. Theproofof existence of the critical rotational velocity required thesolution ofa.nonlinear bifurcation problem and theproofof the existenceof thenontrivial solutions was done using several applied analysis techniques, some of whichwill beusedinthiswork.

[2]

^{is concernedwith}both the analytical(usingthe perturbation

r is

methods)

^{and the} numerical computations of the nontrivial solutions. The case 0<^a_<

treatedanalyticallyandnumericallyin

[3].

ofthe rotational velocity such that for each

.

0the equilibriumequation hasa unique solution for rotational velocities that are less than the critical value. For small 0 this requires the solutionofanonlinearperturbedbifurcationproblem.

Our formulations of the problem go along the same lines as those of

[1]

^and

[2]

^{for the case}

when 0. In fact, the only change ^{is inone}ofthe boundaryconditions. Ve sumethat the gravity effects are negligible and that the rod is thin enough so that it can be treated as an elastica. Figure 2 shows the crdinatesystem.

A

moment balanceon an element oflengthds"

gives (Figure 3)

m m+dm

+

^p

a2y’ds"

ds"cosO,

(1.1)

Figure1" The axially rotating rod

Figure^2: The coordinatesystem

EXISTENCE AND UNIQUENESS OF EQUILIBRIUM STATES OF A ROD 317

y ds

m

dm

ds’

Figure3: Forces andmomentsactingonanelementds’.

where m is the local moment, is the length of the rod, p is the density, f is the rotational velocity,

9"

^{is the distancefrom the rotation} axis, s"is thearclength and0 is the localangleof inclination.

For anelasticathelocal momentisrelated to thelocalcurvatureby

rn=EI

s,

dO

where EI is theflexural rigidity of therod. Weintroducethe nondimensional variables

s=,

^{9= u= 9ds, J=} n"

Using

(1.2)

^and

(1.3)

the equation of equilibrium

(1.1)

becomes

(1.2)

(1.3)

d20

_j4

8--

C08O

sinO,

which is asystem oftwo secondordernonlinear differentialequationsinthetwo variables u and O. The boundaryconditionsare

Upon

using thetransformations

o(o)

,,,

du

-

^(o)

^o,

dO(1) u(1) O,

the equilibrium equations andtheboundaryconditions

(1.4), (1.5)

take the form

d2 ,

vos(

+

o,)

d2v

_sin(

₊

_a)

(0) ^dv

&P

-

⁽⁰⁾

-

^{(1) v(1) 0}

(1.6)

(:.7)

where

()

(.s)

J0 J 0

(1.9)

{i’,

^o<<

^,

Gl(S,) (1.10)

<s<

^1,

0<s<L

GI(X,

⁾⁼

(1.11)

-s, {<s<l

Therestofthis paper isorganizedin threesections.

In

thesecond sectionsomepreliminariesare given whichwill be used in the later sections. Thethird section is concernedwith proofs of the existence and uniqueness ofthe solutionsof

(1.7). In

thefourthsectionaperturbation^solutionof

(1.7)

is presentedandcomparedwiththenumerical solution obtained in

[3].

2.PRELIMINARIES.

Considerthelineareigenvalueproblem

s2= d2v

^Av

d-- d2v ^A

(0) ^dv do

-

^(0)-

-

^{(1) v(1)-0}

^(2.1)

It^{iswell known}

[2]

^{that the}eigenvalues

n

^of

^(2.1)

^{arethesquaresof therootsof theequation}

cosxcosh

+

O, corresponding^tothe "normalized"eigenfunctions

(2.2)

(2.3)

Itis alsoknown

[2]

^that

(2.1)

^{canbewritten inthe}integral^form

where

A(),

(2.4)

A()(s)

21111

₀

Gl(S,)G2([,rl)(q)dtld,

0

(2.5)

and

G1,G

₂ ^are the positive symmetric Green’s functions defined, respectively, by

(1.10)

^and

(1.11),

^{andthat Aas an}operatoron

L2(0,1

^satisfies

A

, ^(2.6)

where

"0

^isthe smallesteigenvalueof thelinearproblem

(2.1).

EXISTENCE AND UNIQUENESS OF EQUILIBRIUM STATES OF A ROD 319 3.EXISTENCE AND

UNIQUENESS

OFSOLUTIONS.

In this section we show that the integral equation

(1.8)

^{has at least one solution for each}

r The proofofthe

0<a<gr

^{and each I} and that for <

10

this solution is unique for all0<

-<

_3"

existence of solution is based upon a corollary of the Schauder fixed point theorem which we statehereforcompleteness.

LEMMA

3.1 LetBbe arealBanach space, and let ^K:B---,B becompact. Suppose that thereis apriori bound m>0 such that every solutionofo-tko ^{0, 0< <1, satisfies} o

-<

^{m. Then K}

hasafixedpoint such that _<m.

rand each ,Xconsists of verifying the The proofof existenceofsolutionof

(1.8)

for each 0<^a<

conditions ofLemma 3.1. The compactness of the operator K defined by

(1.9)

is proved in the followinglemma.

a, and each ,X the operator K defined by

(1.9)

^is compact

LEMMA

3.2. For each

0<<7

operatoron

L2(0,1).

r and ,X befixed. Theboundedness of the operator A definedby

(2.5),

PROOF. Let

0<<

Isin(o(s)+[ < and Icos(o(s)+)l _< imply that K as an operator from

/;2(0,1)

^{into 6’([0,1]) is}

bounded in the sense that it carries bounded subsets of

L2(0,1

^into bounded subsets in C([0,1]).

Since the identity operator i:C([O,

1])L2(O,

^{is compact it follows that K is a} compact operator

on

L2(0,1).

The uniqueness of the solutionof

(1.8)

^for <

0

follows from

(2.6)

^{and the}observation that Frechet derivativeof the operatorKisgivenby

K’()

A2 J

⁰

J

⁰

Gl(S’)G2(’q)

^{Cos((l)/()/2a)dtld.}

We gathertheresultsofthis section iathefollowingtheorem.

and each THEOREM 3.3. Equation

(1.8)

^{has at least one solution for each 0}<^a

this solutionisunique for ^,Xless than the smallest eigeavalue^,x₀of Furthermore,for each 0<a_<

the linearproblem

(2.1).

4.PERTURBATIONSOLUTION FOR SMALL _>0.

In

this sectionwe present asymptotic expansionsofthesolution of

(1.7)

^{for small a}>⁰in two cases.

In

the firstce wetake ,X<^,X_0.

In

this casethesolution is unique and, as wewill show below, it becomesinvalid as,x approaches ^,X_0.

In

the secondcase weta&e ^,x>^,X₀ andassumethat ,x-,x

0is small.

In

thiscase,^wegiveanapproximation ofacritical

(a)

>^,X₀such that for ,x >

(a)

the uniqueness of the solution of

(1.7)

^is lost. Finadly,wecompeotiranalytical approximation of

(a)

to its numericalapproximation obta3ned in

[3].

CASEI"

<0.

Forsmall >⁰welinearize

(1.7)

^{about the}"trivial" solution--0,^_=0. Tothisendwewrite

o(s)

aOl(S ⁺

^O(a

2) (4.1)

v()

va() +

⁰⁽

2)

Substituting

(4.1)

^into

(1.7)

^andcollectingtermsthatarelinear inayields

d2o AVl, d2v

d--d- ^{,X(91 +}

^1),

^(4.2)

-1(0)=

_ds

,-

ds

(1)=Vl(1)=0.

For <

0’ ^(4.2)

^{hasauniq solutiongivenDy}

when A=Oand

whenA 0, whereA,B,CandDae the constants determined^by

B

2(o,- o,hv

+

cos

v/

sinh

/

2 cosh

V/

sinh

v/

^s,n

c

^cosh

^v/

I+A-B I+A+B

(4.4)

(4.5)

From

(4.5)

^{we see} that the solutions

(4.4)

^{become unbounded as A-,,}₀ which render the expansions

(4.1)

invalid. Thislattercaseistreated differently below.

CASE2: A>A

0andA-A 0issmall.

Asmentionedabove,^as

AA0,

the expansions

(4.1)

^{arenot valid. For A A}₀^thelinearproblem

(2.1)

^{has the}unique (up to multiplicative constant) solution

(2.3)

with A_n A_0. For^a=0,A=A₀ is a bifurcation point where a new branch of solutions emerges. For a

:

⁰ this bifurcation is perturbed. Our goal hereistoanalyzethisperturbedbifurcation situation.

The present problem now involves two small parameters a and A-A_0. We introduce asmall

"amplitude" parameter and seek solutionsof

(1.7)

in the form

7" e7"

+ e27"2 ⁺ e37"3 ^{+ O(e4),}

v v

+ e2v2 ^{+ e3v}

³

^{+ O(e4), (4.6)}

such that

(oi,

vi), ^{>1, axe}orthogonal to

(7"1’ Vl)"

^{Wealsoexpand and A- A}0inpowers ofe. It turns outthat, forbounded solutions toexist, theseexpansionsmustbe of thefollowingform

a=e

3+0(e4),

A A₀

+ 2A + 3A

₂

+ O(e4). ^(4.7)

Substituting

(4.6)

^and

(4.7)

^into

(1.7)

and collectinglike powersofeyieldstothelinearboundary valueproblem

(2.1)

^{with A A}_0. ^{Thuswe canwritev and}_7’1 as

7"1

A0’ ^(4.8)

Av_O,

where 7"0 ^{and v}0 axe those functions defined by

(2.3)

with A_n A₀ and A is an undetermined constant.

EXISTENCE AND UNIQUENESS OF EQUILIBRIUM STATES OF A ROD 321 The systemfor the second orderterms is alsothe sameas

(2.1)

with ,x ,x_0.

In

ordertomake the solutions of this systems unique we require that they areorthogonal ^totheir respective first order quantities. This leads to

This leaves Aand

Xl

undetermined.

The system for thethirdorder quantitiesis

O,

(4.9)

v2=0.

d2o

3

s

²

^{oV3 + AlV 1-1/2 AOOl 2,}

d2v

₃

+ + 4.1o

3(0)

_W^{(0) (1)}

v3(1)

^0.

The solvability condition for

(3,v3)

requires the right hd side of

(4.10)

^to be orthogonM ^to

(0’ v0)"

^Aftersomesimplificationthisleads to the equation

A3-7

^{A A-6=0,}

00)

2

7

(4.11)

XO [ 0(0

⁰

⁾²

^d

6=

2

AoIl ^{0(00 o0)2}

^ds,

Thefirst equation of

(4.7)

implies that e

=

^a1/3 and hencein this "singular" case thesolution (o,v) depends on a through powers of a

1/3.

Equation

(4.11)

reveals the following information aboutthebifurcationpicture. For

A1

^<

’ " ^()2/3

^equation

^(4.11)has

^aunique realsolutionfor

A and for

1

^>

"

^it has three realsolutions. Thisenables ustoapproximate the critical value of

,,

beyondwhichthe uniqueness of the solution of

(1.7)

^{islost, by}

= ^{A0 +} . ^{()2/3 a2/3, (4.12)}

for smalla>0.

Finally, we compare our analytical results obtained from equation

(4.12)

^{to the}

(exact)

numerical approximations given by Figure ^{9 of}

[3].

Evaluating the parameters $,- defined by

(4.11)

^andsubstitutinginto

(4.12)

^weobtainthefollowingapproximation of the critical value of

,

beyondwhichmultiplesolutionsof

(1.7)

^exist

for small a>^0.

approximated by

3.516015

+

3.385993^a

2]3, (4.13)

It follows from

(1.6)

^{that the critical values}

JMin

^{defined in}

[3]

^{can be}

S_Min

" ^I

^=_

1/2, (4.14)

difference

dMm

⁾

->

^{0, isverysmallforsmallvalues of} and increasesas increases.

E 2.0-

0.349 0.524

0.000 0.175 0.698 0.873 1.047 1.222 1.396 1.571

ALPHA

Figure4: Critical values

JMin

^{beyondwhichmultiple}solutions exist. Dashedcurve is exact.

ACKNOWLEDGEMENT.

would like to thank Professor

C.Y. Wang, Department

^of Mathematics, Michigan State University, for suggesting the problem of this paper and for ^his helpfuldiscussions and comments, ProfessorD. Lund,

Department

of Mathematics, University of Wisconsin-Eau Claire, forhishelp withthe graphsof Figure4 and Mrs. Sue Johnson for typing the manuscript ofthispaper.

REFERENCES

1.

ODEH,

F.

& TADJBAKHSH, I., ^A

^nonlinear eigenvalue problem for rotating rods, Arch. Ration. Mech. Anal. 20

(1965),

^81-94.

2.

WANG, C.Y.,

Onthe bifurcation solutions ofan axially rotating rod,

Q.J.

^Mech. Appl. Math.

35

(3) (1982),

^391-402.

3.

WANG, C.Y.,

Nonlinear deformation ofa slender rod rotated at oneend,

ZMAP

41

(1990),

900-912.